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--- abstract: 'Motivated by recent experimental processes, we systemically investigate strongly correlated spin-1 ultracold bosons trapped in a three-dimensional optical lattice in the presence of an external magnetic field. Based on a recently developed bosonic dynamical mean-field theory (BDMFT), we map out complete phase diagrams of the system for both antiferromagnetic and ferromagnetic interactions, where various phases are found as a result of the interplay of spin-dependent interaction and quadratic Zeeman energy. For antiferromagnetic interactions, the system demonstrates competing magnetic orders, including nematic, spin-singlet and ferromagnetic insulating phase, depending on longitudinal magnetization, whereas, for ferromagnetic case, a ferromagnetic-to-nematic-insulating phase transition is observed for small quadratic Zeeman energy, and the insulating phase demonstrates the nematic order for large Zeeman energy. Interestingly, at low magnetic field and finite temperature, we find an abnormal multi-step condensation of the strongly correlated superfluid, i.e. the critical condensing temperature of the $m_F=-1$ component with antiferromagnetic interactions demonstrates an increase with longitudinal magnetization, while, for ferromagnetic case, the Zeeman component $m_F = 0$ demonstrates a local minimum for the critical condensing temperature, in contrast to weakly interacting cases.' author: - Xiaolei Zan - Jing Liu - Jianhua Wu - Yongqiang Li bibliography: - 'apstemplate.bib' title: 'Phase diagrams and multistep condensations of spin-1 bosonic gases in optical lattices' --- Introduction {#introduction .unnumbered} ============ Spin-1 bosons, the simplest nontrivial spinor bosonic system, play an important role in understanding quantum magnetism [@WU2006; @Lewenstein2007; @KAWAGUCHI2012253; @RevModPhys.85.1191; @KRUTITSKY20161; @CAPPONI201650]. Due to the development of optical cooling and trapping techniques, ultracold quantum gases have provided an excellent laboratory for investigating many-body quantum systems with an unprecedented level of precision, and ultracold spinor gases have been realized for both bosons [@Stenger1998Spin; @Barrett2001; @Schmaljohann2004; @Higbie2005; @Zhao2015] and fermions [@Taie2010; @DeSalvo2010; @Lompe940; @Zhang1467; @PEbling2014; @Pagano2014A; @Scazza2014Observation; @Hofrichter2016]. Experimental progresses motivate theoretical studies on spinor bosonic gases, which manifest a range of phenomena absent in scalar Bose-Einstein condensates (BECs) [@Batrouni2009; @Soltanpanahi2010Multi; @Heinze2011; @Alexander2013; @Mahmud2013; @Natu2015; @Zhu2017]. Normally, the spinor system is prepared in a weak magnetic field to avoid blocking spin-exchange processes in the experimental timescale, and in parallel theoretical studies mainly focus on rich phase diagrams of spinor gases at zero magnetic field, where numerical exact methods, including quantum Monte-Carlo (QMC) or density-matrix renormalization group simulations, are mainly developed for low-dimensional cases [@PhysRevLett.95.240404; @PhysRevA.74.053419; @PhysRevA.74.035601; @Batrouni2009; @PhysRevB.88.104509]. Actually, magnetic field lifts the degeneracy of the ground states, and the interplay of spin-dependent and Zeeman interactions enriches strongly-correlated quantum phases [@WU2006; @Lewenstein2007; @KAWAGUCHI2012253; @RevModPhys.85.1191; @KRUTITSKY20161; @CAPPONI201650; @Mohamed2013; @Mobarak2013Tuning; @PhysRevA.90.043609; @PhysRevA.97.023628]. Magnetism in ultracold atomic systems may be ascribed generally to local onsite interactions, and one can load the spinor Bose gas into an optical lattice so as to increase the low-energy density of states and enhance the role of interactions [@WU2006; @Lewenstein2007; @KAWAGUCHI2012253; @RevModPhys.85.1191; @KRUTITSKY20161; @CAPPONI201650]. Actually, one indeed tunes the spin-dependent interactions to the strength of the order of external Zeeman interactions (or microwave dressing field interactions) [@Jiang2016First], and study strongly correlated phenomena of the multiple spin states. Recently, a antiferromagnetic spinor condensate has been experimentally realized in a two-dimensional optical lattice, where, in a sufficiently deep lattice, a phase transition from a longitudinal polar phase to a broken-axisymmetry phase has been observed in steady states of spinor condensates [@Zhao2015]. In a following-up experiment, they demonstrate evidence of first-order superfluid-Mott-insulating phase transitions in a lattice-confined antiferromagnetic spinor Bose-Einstein condensate [@Jiang2016First]. These experiments on lattice systems have deepened the understanding of spinor systems and observed new phenomena which have not been predicted in theory [@Uesugi2003; @Svidzinsky2003; @Mohamed2013]. Thus the influence of both the lattice setups and external magnetic fields should be reconsidered. In contrast, the ferromagnetic spinor condensates have been studied less extensively [@Imambekov; @Tsuchiya; @Kimura; @Yamamoto]. Experimental studies on the ferromagnetic spin-1 bosons have been carried out mainly with $^{87}$Rb [@Luo620], which has a small spin-dependent interaction compared with the spin-independent interaction. Theoretical studies demonstrate that the system exhibits saturated ferromagnetism over the entire zero-temperature phase diagram in the absence of external magnetic fields [@Katsura2013], where properties of the system are similar to those of scalar bosons and the phase transitions are of second-order. However, the system presents rich phases in the presence of an external magnetic field, due to the competition between the quadratic Zeeman effect and the ferromagnetic interactions [@KAWAGUCHI2012253; @RevModPhys.85.1191]. Recently, a theoretical research is carried out for ferromagnetic spin-1 gases under an external magnetic field, which indicates discontinuous first-order phase transitions [@PhysRevA.96.023628]. Despite the rich and interesting results obtained above, there is still a lack of relevant researches about competing spin-ordered Mott phases. Exploring the thermodynamics of interacting many-body systems has been arguably one of the most important achievements of cold-atomic gases. It is an interesting topic examining the connection between magnetic orders and Bose-Einstein condensates of ultracold multispecies atomic gases [@WU2006; @Lewenstein2007; @KAWAGUCHI2012253; @RevModPhys.85.1191; @KRUTITSKY20161; @CAPPONI201650]. For spinor gases, a multi-step condensation has been predicted theoretically [@Isoshima2000Double; @zhang2003; @zhang2004bose; @Kis2006Phases; @Phuc2011effect; @Lang2014Therm] and observed experimentally [@PhysRevLett.102.125301; @PhysRevA.86.061601; @PhysRevA.90.023610]. For small Zeeman field, for example, antiferromagnetic interactions qualitatively change the phase diagram and lead to condensation in $m_F\pm1$ state, which is a phenomenon that cannot occur for an ideal gas [@Frapolli2017Stepwise]. As far as we know, however, there are still lack experiments on multistep condensations of the spinor bosonic gases in an optical lattice, and it is still unclear for condensing sequences of strongly correlated bosonic gases in optical lattices. In this paper, we focus on strongly correlated spinor ultracold gases in a three-dimensional (3D) optical lattice and systemically investigate the system for both negative and positive on-site spin-dependent interactions in the presence of an external Zeeman interaction, based on bosonic dynamical mean-field theory. Our study here is an extensive calculations performed in Ref. [@PhysRevA.93.033622]. For antiferromagnetic interactions, we take $^{23}$Na as examples, and determine the many-body phase diagrams, including nematic phase, ferromagnetic phase, spin-singlet insulator and different types of superfluid phase. As for the ferromagnetic case, we take $^7$Li and $^{87}$Rb as examples, and map out the phase diagrams, including ferromagnetic and nematic insulating phase, and superfluid phase. Moreover, we study the stability of these quantum phases against thermal fluctuations, obtaining finite temperature phase diagrams. Interestingly, a multi-step condensation for the Zeeman components is explored, and an abnormal condensation observed, in contrast to weakly interaction cases. Methods {#methods .unnumbered} ======= The Model {#the-model .unnumbered} --------- We consider spin-1 bosonic gases with spin-dependent interactions loaded in an optical lattice. We assume that an external magnetic field is applied along the quantization axis. In sufficiently deep lattices and the single-mode approximation, the physics of the system is described by the extended Bose-Hubbard model  [@RevModPhys.85.1191]: $$\hat H=-t\sum_{\langle i,j\rangle,\sigma}(a^\dagger_{i,\sigma}a_{j,\sigma}+a^\dagger_{j,\sigma}a_{i,\sigma})+\frac{U_{0}}{2}\sum_{i}\hat n_{i}(\hat n_{i}-1)+\frac{U_2}{2}\sum_{i}(S^{2}_{i}-2\hat n_{i})+pS_{iz}+q\sum_{i,\sigma}\sigma^{2}\hat n_{i,\sigma} -\mu\sum_{i}\hat n_{i} \label{eq:hubbard}$$ where, $a_{i,\sigma}$ ($a^\dagger_{i,\sigma}$) denotes the annihilation (creation) operator of a boson at lattice site $i$, and spin $\sigma\in\{-1,0,1\}$. The particle-number operators are defined by $\hat n_{i,\sigma}:=a^\dagger_{i,\sigma}a_{i,\sigma}$ and $\hat n_{i}:=\sum_{\sigma}\hat n_{i,\sigma}$, the spin operators by $S^{(\alpha)}_{i}:=\sum_{\sigma,\sigma'}a^\dagger_{i,\sigma}S^{(\alpha)}_{\sigma,\sigma'}a_{i,\sigma'}$ for $\alpha=x,y,z$, where $S^{(\alpha)}_{\sigma,\sigma'}$ denotes the elements of spin-1 matrices $S^{(\alpha)}$. $t$ is the hopping amplitude, and $\langle i,j\rangle$ means the sum over nearest neighbors. The second term $U_{0}$ is the on-site interaction between atoms, and the third term $U_{2}$ represents spin-dependent interaction on the same site. The coefficient $p$ denotes linear Zeeman energy (equivalent to magnetization $M_z$), $q$ quadratic Zeeman energy, and $\mu$ the chemical potential. In the following we consider both antiferromagnetic $U_2/U_0>0$ and ferromagnetic interactions $U_2/U_0<0$. For instance, in experiments with $^{23}$Na atoms the spin-dependent interaction $U_2/U_0\simeq 0.037$ [@Zhao2015], with $^7$Li atoms with $U_2/U_0 \simeq -0.7$ [@PhysRevA.68.063602], and with $^{87}$Rb atoms with $U_2/U_0\simeq -0.005$ [@Luo620]. Bosonic Dynamical Mean-Field Theory {#bosonic-dynamical-mean-field-theory .unnumbered} ----------------------------------- To investigate quantum phases of spinor Bose gases loaded into a cubic optical lattice, described by Eq. (\[eq:hubbard\]), we recently establish a bosonic version of dynamical mean-field theory on the generic three-dimensional situation, where details can be found in Ref. [@PhysRevA.93.033622]. Here we only show the basic idea of this method. As in fermionic dynamical mean-field theory, the main idea of our BDMFT approach is to map the quantum lattice problem with many degrees of freedom onto a single site coupled self-consistently to a noninteracting bath [@Georges1996Dynamical; @Vollhart_08; @Hubur_08; @Hu2009Dynamical; @Li2011Tunable; @He2012Quantum; @Li2013Lattice; @Li2012Anisotropic]. The dynamics at the impurity site can thus be thought of as the hybridization of this site with the bath. Therefore, properties of the many-body system can be captured by a single impurity model. In a more formal language, Hamiltonian  (\[eq:hubbard\]) is mapped onto a single-site problem and described by a local effective action: $$\begin{aligned} S_{\mathrm{imp}}=&-&\int^{\beta}_{0}\!d\tau d\tau'\!\sum_{\sigma\sigma'} \!\left( \begin{array}{c} a^\ast_{0,\sigma}(\tau)\quad a_{0,\sigma}(\tau) \end{array} \right) {\bm {\mathcal{G}}}^{-1}_{0,\sigma\sigma'} \!\left( \begin{array}{c} a_{0,\sigma'}(\tau')\\ a^\ast_{0,\sigma'}(\tau')\\ \end{array} \right) +\int^{\beta}_{0}d\tau\{\frac{U_0}{2}n_{0}(\tau)(n_{0}(\tau)-1)+\frac{U_2}{2}({\bf{S}}^{2}_{0}(\tau)-2n_{0}(\tau)) \nonumber\\ &-&\,t\sum_{\langle0,i\rangle,\sigma}(a^\ast_{0,\sigma}(\tau)\phi_{i,\sigma}(\tau)+a_{0,\sigma}(\tau)\phi^\ast_{i,\sigma}(\tau))+\,p S_{iz}(\tau)+ q\sum_{i,\sigma}\sigma^{2} n_{i,\sigma}(\tau)\}, \label{eq:BDMFT}\end{aligned}$$ where $0$ is index of the impurity site, $\tau$ is the imaginary time, $\bm{\mathcal{G}}^{-1}_{0,\sigma\sigma'}(\tau-\tau')$ denotes a local non-interacting propagator interpreted as a local dynamical Weiss Green’s function, $$\bm{\mathcal{G}}^{-1}_{0,\sigma\sigma'}(\tau-\tau')\equiv- \left( \begin{array}{cc} (\partial_{\tau'}-\mu_{\sigma})\delta_{\sigma\sigma'}+t^2\sum\limits_{\langle0i\rangle,\langle0j\rangle}G^{1}_{\sigma\sigma',ij}(\tau,\tau') & t^{2}\sum\limits_{\langle0i\rangle,\langle0j\rangle}G^{2}_{\sigma\sigma',ij}(\tau,\tau')\\ t^{2}\sum\limits_{\langle0i\rangle,\langle0j\rangle}G^{2\ast}_{\sigma\sigma',ij}(\tau',\tau) & (-\partial_{\tau'}-\mu_{\sigma})\delta_{\sigma\sigma'}+t^2\sum\limits_{\langle0i\rangle,\langle0j\rangle}G^{1}_{\sigma\sigma',ij}(\tau',\tau)\\ \end{array} \right) \label{eq:weiss}$$ and the superfluid order parameters is given by $\phi_{i,\sigma}(\tau)\equiv\langle a_{i,\sigma(\tau)}\rangle_{0}$ with $\langle\cdots\rangle_{0}$ indicating that the expectation value is calculated in the cavity system[@Hubur_08; @Li2011Tunable]. We find that the effective action (\[eq:BDMFT\]) is represented by an Anderson impurity Hamiltonian $$\begin{aligned} \label{eq:AndersonHamiltion} \hat H_{A}=& &-t\sum_{\sigma}(\phi^{\ast}_{\sigma}\hat a_{0\sigma}+\mathrm{h.c.})+\frac{U_{0}}{2}\hat n_{0}(\hat n_{0}-1)+\frac{U_{2}}{2}(\bm{\hat{S}}^{2}_{0}-2\hat n_{0})+q\sum_{i,\sigma}\sigma^{2}\hat n_{i,\sigma}-\sum_{\sigma}\mu_{0\sigma}\hat n_{0\sigma}+\sum_{l}\epsilon_{l}\hat b^{\dagger}_{l}\hat b_{l} \\ & &+\sum_{l,\sigma}(V_{\sigma,l}\hat a_{0\sigma}\hat b^{\dagger}_{l}+W_{\sigma,l}\hat a_{0\sigma}\hat b_{l}+\mathrm{h.c.}). \nonumber\end{aligned}$$ We remark three points here. First the chemical potential and interaction term are directly inherited from the Hubbard Hamiltonian. Second, there are two baths coupling to the impurity site: the bath of condensed bosons is represented by the Gutzwiller term with superfluid order parameter $\phi_{\sigma}$ for each component, and the bath of normal bosons is described by a finite number of orbitals with creation operators $\hat b^{\dagger}_{l}$ and energies $\epsilon_{l}$, where these orbitals are coupled to the impurity via normal-hopping amplitudes $V_{\sigma,l}$ and anomalous-hopping amplitudes $W_{\sigma,l}$. Third, we employ the exact diagonalization (ED) solver for the Anderson impurity problem [@Hubur_08]. The maximal number of normal bath orbitals in ED is limited to $n_s = 4$, and the Fock space for the impurity bosons is truncated at a maximum occupation number $n=6$. We systematically verified that the used maximal numbers in the present study are sufficient for ED. By exact diagonalization of Ham. (\[eq:AndersonHamiltion\]), we obtain the local Green’s function $\bm{G}_{\sigma\sigma'}(i\omega_{n})$, and the local self-energy is then obtained from the local Dyson equation $$\bm{\mathcal{G}}^{-1}_{\sigma\sigma'}(i\omega_{n})=(i\omega_{n}\sigma_{z}+\mu_{\sigma})\delta_{\sigma\sigma'}-\bm{\Delta}_{\sigma\sigma'}(i\omega_{n})=\bm{\Sigma}_{\sigma\sigma'}(i\omega_{n})+\bm{G}^{-1}_{\sigma\sigma'}(i\omega_{n}), \label{eq:Weiss Green's function}$$ where we have defined the hybridization functions: $$\begin{aligned} \Delta^{1}_{\sigma\sigma'}(i\omega_{n})\equiv-\sum_{l}(\frac{V_{\sigma,l}V_{\sigma',l}}{\epsilon_{l}-i\omega_{n}}+\frac{W_{\sigma,l}W_{\sigma',l}}{\epsilon_{l}+i\omega_{n}}) &\nonumber\\ \Delta^{2}_{\sigma\sigma'}(i\omega_{n})\equiv-\sum_{l}(\frac{V_{\sigma,l}W_{\sigma',l}}{\epsilon_{l}-i\omega_{n}}+\frac{W_{\sigma,l}V_{\sigma',l}}{\epsilon_{l}+i\omega_{n}}). & \label{eq:hybridization}\end{aligned}$$ We approximate the lattice self-energy $\bm{\Sigma}_{\mathrm{lat},\sigma\sigma'}$ by the impurity self-energy $\bm{\Sigma}_{\sigma\sigma'}$, and the self-consistency loop is then completed by the conditions for lattice Green’s function $$\bm{G}_{\mathrm{lat}}(i\omega_{n})=\int \epsilon \rho(\epsilon)\frac{1}{i\omega_{n}\sigma_{z}+\mu_{\sigma}-\bm{\Sigma}(i\omega_{n})-\epsilon}, \label{eq:lattice Green's function}$$ where $\rho(\epsilon)$ denotes the density of states for three-dimensional cubic lattices at energy $\epsilon$. Note here that this method is exact for infinite dimensions, and is a reasonable approximation for high but finite dimensions, where the reliability of this method has been verified by comparison with quantum Monte-Carlo simulation [@QMC_2007; @Werner_12]. Our approach is a non-perturbative method, including the local quantum fluctuations of the strongly correlated systems exactly, but neglecting non-local spatial correlations. For most of cases, the self-consistent BDMFT loop yields stable solutions, but, for some cases, it produces multiple stable results, especially around the phase boundary for first-order transition. To find the many-body ground states in these cases, we calculate the energy within BDMFT, and for the Bose-Hubbard model of spin-1 bosons, the local energy is given by: $$\begin{aligned} E=\Bigg[ \frac12\sum_i\Big\langle U_0 n_i(n_i-1) + U_2 ({\bf S}^2_i - 2n_i)+2\Big(pS_{iz} + q\sum_\sigma \sigma^2 n_\sigma\Big) \Big\rangle -k_BT\sum_{i,\sigma,n}\int d\epsilon\, \epsilon \rho(\epsilon) G_{i\sigma}(i\omega_n) - t\sum_{\langle ij \rangle \sigma} \phi^\ast_{i\sigma}\phi_{j\sigma} \Bigg] / N.\end{aligned}$$ Here, $N$ is the total number of lattice sites, $i\omega_n=2n\pi/\beta$, and $G_\sigma(i\omega_n)$ denotes the local Green’s function. RESULTS {#results .unnumbered} ======= Antiferromagnetic interactions {#subsec:ferromagnetic interactions .unnumbered} ------------------------------ In this section, we focus on experimentally accessible parameters for $^{23}{\rm Na}$, ${\it i.e.}$ $U_2/U_0=0.037$, in a 3D optical lattice constructed from a single-mode laser at wavelength $\lambda = 1064$ nm with recoil energy $E_R = 2h^2/m \lambda^2$. In this limit of spin-dependent interaction $U_2 \ll U_0$, a first superfluid-insulator transition featured by hysteresis effect and significant heating [@Jiang2016First] is observed experimentally by changing the ratio $U_0/t$. Motivated by the recent experimental progresses, we investigate the spin-1 bosonic gases in a 3D optical lattice in the presence of an external magnetic field. Actually, the interplay of spin-dependent and Zeeman interactions in the system gives rise to an abundant phase diagram. At finite temperature, interestingly, the strongly correlated system may demonstrate an abnormal multi-step condensation for different species, due to quantum and thermal fluctuations. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(Color online) Zero-temperature phase transition for spin-1 ultracold bosons in a 3D cubic lattice with an antiferromagnetic interaction $U_2/U_0=0.037$ for fixed filling $n=2$. []{data-label="Ne_1"}](FF1.eps "fig:"){width="0.9\linewidth"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(Color online) Zero-temperature phase transition for spin-1 ultracold bosons in a 3D cubic lattice with an antiferromagnetic interaction $U_2/U_0=0.037$ for fixed filling $n=2$. []{data-label="transition"}](V20.eps "fig:"){width="0.6\linewidth"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ### Zero temperature {#zero-temperature .unnumbered} #### Phase diagram in deep Mott insulator. First, we study strongly correlated phases in the deep Mott insulator (MI) and resolve its long-range spin order in the presence of an external magnetic field. We summarized our calculations of the zero-temperature case in Fig. \[Ne\_1\], where two phase diagrams are shown in the $p$-$q$ plane with an antiferromagnetic interaction $U_2/U_0\approx0.037\;(\rm ^{23}Na)$ and filling $n=2$ (left) and $3$ (right), respectively. The depth of the optical lattice is chosen to be $V=20\, E_R$ (t/$U_{0} \approx 0.012$), where the system is in a typical Mott insulating regime, with $E_R$ being the recoil energy of atoms. Three distinct phases, namely ferromagnetic phase (FM), nematic insulator (NI) and spin-singlet insulator (SSI), are observed and we employ the values of the condensate order parameter $\phi^1_\alpha\equiv\langle b_\alpha\rangle$, the nematic order parameter $\phi^2_{\alpha\beta} \equiv \langle S_\alpha S_\beta \rangle -\delta_{\alpha\beta}/3 \langle S^2 \rangle$, and the local magnetization $\bf {M}\equiv\langle {\bf S} \rangle$ to characterize them respectively. For filling $n=2$ and $V=20\, E_R$, our BDMFT calculations predict that different parameters will result in different Mott insulating phases with different types of spin order including FM featured by $\phi^1_\alpha =0 $, ${\bf M} \neq 0$, NI by $\phi^1_\alpha =0 $, $\phi^2_{\alpha\beta}>0$ and ${\bf M}=0$ and SSI by $\phi^1_\alpha =0$, $\phi^2_{\alpha\beta}=0$ and $\langle {\bf S}^2\rangle=0$ with $S$ being the local total spin (due to the symmetry of the spin wave-function on each site, $n+S=({\rm even})$ [@Imambekov]). It is worth mention that the formation of singlet pairs will give rise to an even number of atoms per site which characterize SSI phase. Actually, for the small linear Zeeman energy $p$ (equivalent to magnetization $M_z$) and quadratic Zeeman energy $q$, the system is in a regime dominated by spin-dependent interactions, and favors the SSI phase by forming a singlet pair and lowering the spin-dependent energy. For larger quadratic Zeeman energy $q$, the degeneracy of the three magnetic species is lifted, and the system goes through a phase transition from the SSI to the NI phase, $\it i.e.$ the system enters into the NI phase with $n_0 > n_{\pm1}$ for positive $q$, and with $n_{\pm1} > n_0$ for negative $q$, as shown in Fig. \[transition\] with $p=0$. For larger $p$ (magnetization $M_z$), one species of $n_{\pm1}$ is energetically favored and the FM phase develops. We notice that the phase diagrams are symmetric upon linear Zeeman interactions and this symmetry is also manifested in the Hamiltonian (\[eq:hubbard\]). We next move to three identical particles with $n=3$, and the corresponding phase diagram is shown in the right panel of Fig. \[Ne\_1\]. For small magnetic field, the system favors the NI phase with $\langle S^2 \rangle =2$ by lowering spin angular momentum, where the detailed discussion can be found in Ref. [@PhysRevA.93.033622]. For large quadratic Zeeman energy $q$, FM phase with $|{\bf M}| =3$ is energetically favored by populating the $n_1$ species for $p>0$ and by the $n_{-1}$ species for $p<0$, respectively. For intermediate $p$ and negative $q$, the system demonstrates a FM phase with $\langle S^2 \rangle =2$ and $|{\bf M}| =1$. Note here that, along the line $p\approx0$, the system favors the NI phase for both positive and negative quadratic Zeeman energies (see the red lines in the right panel of Fig. \[Ne\_1\]). {width="0.9\linewidth"} #### Hopping dependent phase diagram. Away from the deep Mott-insulating regime, quantum fluctuations should be more and more important with the increase of tunneling amplitudes, and the stability of magnetic phases within the Mott lobes, such as the NI and SSI phase, is needed to be addressed against quantum fluctuations in the presence of the external magnetic field. We remark here that our results correctly recover unconventional spin ordering both in the atomic limit $U_0/t=\infty$ and in the weakly interacting regime. For even filling, our calculations prove that the SF-MI transition is first order for small external magnetic field (equivalent to small magnetism $M_z$ and small quadratic Zeeman interaction $q$), while it is second order for large magnetic field, as shown in Fig. \[order\]. We believe that the spin-dependent interaction $U_2$ will lead to the formation of spin-singlet pairs in the SSI phase which supports a first-order phase transition, while disappearance of the singlet pairs, due to the interplay of spin-dependent and Zeeman interactions, demonstrates a second-order transition. Remarkably, we notice four distinctive types of SF phases in the parameters studied here, characterized by $\phi^{1}_{0}\neq0$, $\phi_{1}^{1}\neq0$ ($\phi^{1}_{-1}\neq0$), $\phi^{1}_{\pm1}\neq0$ or $\phi^{1}_{1,0}\neq0$, which emerge in the phase diagrams for different external linear Zeeman fields $p$ (magnetization $M_z$) and quadratic Zeeman fields $q$. The order parameters as a function of interaction strengths are presented in Fig. \[order\](d),(e),(f) for $U_2/U_0 = 0.037$, $\mu/U_0=1.5$, and different external magnetic fields. For zero magnetization $M_z$ and small quadratic Zeeman interaction, we find that a small region of the SSI phase ($n=2$) is occupied by the NI phase around the tip of the MI-SF transition as a result of the interplay of tunneling and spin-dependent interaction, as shown in the inset of Fig. \[order\](d), where the corresponding MI-SF transition is first order. We remark here that the nematic order $\phi^2_{\alpha\beta}$ demonstrates a small but nonzero value in between SF and SSI phases, as shown in the inset of Fig. \[order\](d), which indicates a possible nematic phase but may be an artifact of the BDMFT method, requiring further studies based on more accurate methods such as quantum Monte-Carlo simulations. While for large quadratic Zeeman interaction, we observe that the spin-order in MI ($n=2$) is the NI type with $\phi^{1}_{\alpha}= 0$ and $\phi^2_{\alpha\beta}\neq0$, instead of SSI, and the corresponding transition is second order, as shown in Fig. \[order\](f). Interestingly, for small but nonzero magnetization $M_z\neq 0$, we observe a FM phase with $\phi^{1}_{\alpha}= 0$ and ${\bf M}\neq 0$ in between SF and SSI, as shown in Fig. \[order\](e). Interestingly, we observe a first-order SSI-FM and a second-order FM-SF transition, with lowering the optical depth $V$. ### Finite temperature {#section:finite .unnumbered} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(Color online) Finite-temperature phase diagram for spin-1 ultracold bosons in a 3D cubic lattice with an antiferromagnetic interaction $U_2/U_0=0.037$ for a fixed temperature $T/U_0=0.01$, and a magnetic field $p=20\, Hz$ and $q=120\, Hz$. Note here that the normal state is shown by the shaded region in between FM.[]{data-label="finite_T"}](T1.eps "fig:"){width="0.6\linewidth"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![(Color online) Abnormal multi-step condensation of spin-1 ultracold bosons in a 3D cubic lattice ($V=21\,{\rm Hz}$) with an antiferromagnetic interaction $U_2/U_0=0.037$ and filling $n=2.5$ for quadratic Zeeman field $q=20\, Hz$ (left), and $-20\, Hz$ (right), respectively. Contrary to weakly interacting gases, we observe a non-smooth change of the critical condensing temperature of the $m_F=-1$ component.[]{data-label="multi_step"}](FF5.eps "fig:"){width="0.9\linewidth"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Stability of the above spin-dependent phases against thermal fluctuations is the first thing to consider in order to obtain a direct observation of them in practical experiments. We investigate this issue for a typical case with $U_2/U_0 = 0.037$ ($^{23}\rm Na$) at a fixed finite temperature $T/U_0=0.01$, and find that four different phases are involved as shown in Fig. \[finite\_T\], including SF ($\phi^1_{0}$ or $\phi^1_{\pm 1}$), FM, and normal state (NS) characterized both by $\phi^1_\alpha = 0$ and large density fluctuations $\Delta^2\equiv \langle n^2\rangle - \langle n\rangle ^2$. Due to thermal fluctuations, we find that the singlet pairs in $n=2$ are broken and the FM phase develops in the parameters studied here. Moreover, BDMFT predicts that the superfluid-Mott-insulating phase transition for even fillings is second order, and the transition between the SF phases ($\phi_{\pm}$ and $\phi_{+1}$) is second order as well. Notice that the temperature in our calculations can be obtained via present cooling schemes, for instance, the spin-gradient cooling [@Weld2009; @PhysRevA.92.041602], and an external harmonic trap can be employed to maintain the coexistence of these phases. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[fig:1\] (Color online) Zero-temperature phase diagrams of spin-1 ultracold bosonic gases in a 3D optical lattice for (a) $(U_{2}/U_{0},\,q/U_{0})=(-0.005,\,0.0085)$ ($^{87}$Rb) and (b) $(U_{2}/U_{0},\,q/U_{0})=(-0.7,\,0.1)$ ($^{7}$Li) respectively by BDMFT (blue circle and green cross), and via Gutzwiller mean-field theory in Ref. [@PhysRevA.96.023628] (red cross). BDMFT predicts four phases in the system, including nematic insulator (NI), ferromagnetic (FM), polar and broken-axisymmetry (BA) superfluid (SF) phase. ](FF6.eps "fig:"){width="0.9\linewidth"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Another crucial question regarding experimental observations is the multi-step condensation of strongly correlated superfluid. For weakly interacting spinor gases, a two-step condensation with $m_F=1$ and $m_F = 0$ has been observed for large quadratic Zeeman energy, while there exists a three-step condensation for small quadratic Zeeman energy [@zhang2003; @Frapolli2017Stepwise], due to the interplay of spin-dependent and Zeeman interactions. However, the condensation of strongly correlated superfluid is still unknown. Fig. \[multi\_step\] shows the finite-temperature phase diagram of the spinor bosonic gases at fixed filling $n=2.5$ and quadratic Zeeman interaction $q=20\, {\rm Hz}$ (left) and $q=-20\, {\rm Hz}$ (right), respectively. An interesting feature is revealed in the neighbouring of the superfluid to Mott-insulating transition with $V=21\, {\rm Hz}$ ($t/U_0\approx0.01$). For example, for positive quadratic Zeeman field and small $M_z$, majority component $m_F = 0$ condenses first at a critical temperature, followed by the $m_F = 1$ component at a lower one, while, for large $M_z$, coexisting the $m_F = \pm 1$ components is energetically favored. This behavior is a result of the interplay of spin-dependent and quadratic Zeeman interactions (favor the $pair$ of the $m_F=\pm1$ components), and longitudinal magnetization (favor the $m_F=1$ component). Our prediction here is consistent with weakly interacting condensate, where a phase transition occurs between the broken-axisymmetry ($\phi^1_{0,1}\neq0$) and the antiferromagnetic phase ($\phi^1_{\pm 1} \neq 0$) for low temperature [@zhang2003], even though here we only observe a two-step condensation for small magnetization. Interestingly, for negative quadratic Zeeman interaction, we observe an abnormal transition from the normal to the superfluid phase via cooling the system, $\it i.e.$ the critical condensing temperature of the $m_F=-1$ component first decreases into zero and then demonstrates an abnormal increase with magnetization $M_z$, which is a remarkable phenomenon never been predicted ever before, indicating unique features of strongly correlated superfluid. In order to offer quantitative guidance for the direct observation of multi-step condensations in realistic experiments, we estimate the critical temperatures. For experiments with $^{23}{\rm Na}$ ($U_{2}/U_{0}\approx0.037$) in a 3D cubic lattice generated by laser beams of wavelength 1064 nm and intensity $V\approx21\, E_{R}$, we find that the system should be cooled down to $T_c\approx2\, {\rm nK}$ ($n=2.5$, $p=\pm20\, {\rm Hz}$). One can employ the excitation spectra or density correlations [@Imambekov; @Natu2015] to reflect these correlated insulating states by occupying, for example, Bragg scattering [@Imambekov], quantum gas microscopy [@Gericke2008High; @Bakr2009A; @Sherson2010Single] or optical birefringence [@Natu2015], as long as the lifetimes of these states can meet the requirement of observation [@Zhao2015]. Recently, a steady state of spinor bosons in optical lattices with a sufficiently long lifetime was reported [@Jiang2016First], as well as the existence of spin-nematic ordering in a spherical trap [@PhysRevA.93.023614]. Ferromagnetic interactions {#subsec:ferromagnetic interactions .unnumbered} --------------------------- In this section, we mainly investigate experimentally accessible systems for $^{87}{\rm Rb}$ with a ferromagnetic interaction $U_2/U_0=-0.005$ and for $^{7}{\rm Li}$ with $U_2/U_0=-0.7$. Actually, the magnetic order is the trivial ferromagnetic order for zero magnetic field. At finite temperature, interestingly, the strongly correlated system may demonstrate a multistep condensation for different species, due to the interplay of quantum and thermal fluctuations. ### Zero-temperature phase diagrams {#zero-temperature-phase-diagrams .unnumbered} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[fig:2\] (Color online) Influence of quadratic Zeeman interactions on zero-temperature phase diagrams of spinor bosonic gases in a 3D optical lattice. We choose the $^{7}$Li atom as examples with $(U_2/U_0=-0.7)$, and the other parameters are set to (a) $q/U_{0}=0$, (b) $q/U_{0}=0.005$, (c) $q/U_{0}=0.03$ and (d) $q/U_{0}=0.1$. Inside the Mott lobes, a competition between nematic insulating and ferromagnetic order is observed, as a result of the interplay of spin-dependent and quadratic Zeeman interactions.](FF7.eps "fig:"){width="0.9\linewidth"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- In this subsection, we discuss zero-temperature phase diagrams of spinor ultracold gases in a 3D optical lattice. For comparison, we firstly study the cases in Ref. [@PhysRevA.96.023628], where a Gutzwiller mean-field calculation is performed for different interactions. Fig. \[fig:1\] shows the obtained phase diagrams for $(U_{2}/U_{0},q/U_{0})=(-0.7,0.1)$ for $^7$Li and $(-0.005,0.0085)$ for $^{87}$Rb, based on BDMFT. Here, we notice that there exists four distinct phases obtained from BDMFT, namely polar superfluid (polar SF), broken-axisymmetry superfluid (BA SF), nematic insulator (NI) and ferromagnetic (FM). In the superfluid phase near to the Mott lobes, only the $\sigma\!=\!0$ component shows superfluidity, which we identify as the polar superfluid phase. The other superfluid phase has three superfluid components, and possesses a non-zero transverse magnetization $M_{tr} := \sqrt{\langle S_{(x)}\rangle^2+\langle S_{(y)}\rangle^2}$; hence we identify it as the broken-axisymmetry superfluid phase. The nematic insulator phase is characterized by $\phi^{1}_{\alpha}=0,\phi^{2}_{\alpha\beta}>0$ and $\bm{M}=0$; and the ferromagnetic phase for $\phi^{1}_{\alpha}=0$ and $\bm{M}\neq0$. We remark here that our method clearly resolves the long-range spin order. As shown in Fig. \[fig:1\](a), there are two different Mott-insulating phases inside the $n=3$ and $4$ Mott lobes, $\it i.e.$ the NI phase for small $zt/U_{0}$ and the FM phase for large $zt/U_{0}$, while, for $n=1$ and $2$, the Mott-insulating phase favors the nematic order, as a result of the interplay of spin-dependent and quadratic Zeeman interactions. ![\[fig:3\] (Color online) Influence of thermal fluctuations on phase diagram of spinor bosonic gases in a 3D optical lattice for $(U_{2}/U_{0},q/U_{0},T/U_{0})=(-0.7,0.005,0.01)$ ($^{7}$Li). The diagram manifests five different phases: polar superfluid (polar SF), broken-axisymmetry superfluid (BA SF), nematic insulator (NI), ferromagnetic phase(FM), and normal state (NS). We observe a second-order NI-SF ($n=1$ and $2$), and a first-order NI-FM-SF ($n=3$ and $4$) phase transition, respectively, as shown by the inset with $\mu/U_0=0.72$. ](t_1_new){width="0.75\linewidth"} Next, we compare our results with those obtained from Gutzwiller mean-field theory in Ref. [@PhysRevA.96.023628]. For example, both methods predict a MI-polar SF and a polar SF-BA SF phase transition. We also find that phase boundaries are well matched with previous results, except that the Mott tip here is slightly bigger than that from the Gutzwiller variational ansatz, since that BDMFT takes quantum fluctuations into account. However, two major differences are also found between the two theories. First, magnetic spin orders of the Mott-insulating phases are resolved via BDMFT, which includes nematic order and ferromagnetic order in the parameters studied here. Second, Gutwziller mean-field theory underestimates the stabilized region of the polar SF phase for $^{87}$Rb atoms, $\it i.e.$ there is an obvious mismatch of the polar SF-BA SF transition boundary. This discrepancy is expected to be due to the lack of quantum fluctuations involved in the Gutzwiller variational ansatz. To figure out the relationship and competition between the long-range spin orders, we focus on a $^{7}$Li gas with spin-dependent interaction $U_2/U_0=-0.7$, and investigate the influence of quadratic Zeeman interactions. It is expected that the NI phase appears in the Mott-insulating lobe and occupies the region of the FM phase with increasing $q$, as a result of the interplay of spin-dependent and quadratic Zeeman interactions, which supports the $m_F=1$ and the $m_F=\pm 1$ components, respectively. The corresponding phase diagrams for different parameters are obtained and summarized in Fig. \[fig:2\]. As examples, we choose $q/U_{0}=0,0.005,0.03$ and $0.1$, respectively. When $q/U_{0}=0$, the Mott lobes are occupied by the FM phase, which is consistent with previous conclusions [@Katsura2013], while there is only broken-axisymmetry superfluid phase outside the Mott lobe. With the increasing of quadratic Zeeman energy $q/U_{0}$, the region of the MI phase gradually widen with the appearance of the NI order from the lower hopping regime, as shown in Fig. \[fig:2\](b), where the polar SF phase emerges at $q/U_{0}=0.005$. Increasing $q$ further, the polar SF phase expands and squeezes the region of the BA SF phase, as shown in Fig. \[fig:2\](b),(c),(d). Inside the Mott lobes, a competition between the NI and the FM order is observed, $\it i.e.$ the region of the NI phase expands larger and the region of the FM phase shrinks with the increasing of $q/U_{0}$. Finally, the FM phase disappears at $(U_{2}/U_{0},q/U_{0})=(-0.7,0.1)$, as shown in Fig. \[fig:2\](d). ### Finite temperature phase diagrams {#finite-temperature-phase-diagrams .unnumbered} Up to now, we have illustrated the zero-temperature case and studied quantum-fluctuation-induced phase transitions. In this subsection, we investigate the influence of thermal fluctuations on phase diagrams of spinor ultracold gases in an optical lattice with ferromagnetic interactions. We take $(U_{2}/U_{0},q/U_{0})=(-0.7,0.005)$ as an example. For a typical parameter with $T/U_{0}=0.01$, the finite-temperature phase diagram is shown in Fig. \[fig:3\]. We observe there are five phases in the system, including NI, FM, BA SF, polar SF, and normal state (NS) characterized both by $\phi^1_\alpha = 0$ and large density fluctuations $\Delta^2\equiv \langle n^2\rangle - \langle n\rangle ^2$. Compared with the zero-temperature case, the nonzero-temperature one possesses some features worth mentioning. While the SF-MI phase boundary expands, the Mott-insulating lobes show major changes. For example, the $n=1$ Mott lobe exhibits the saturated NI phase and the FM phase disappears completely, since thermal fluctuations smooth the population difference between the three species and the FM phase is energetically unfavored. Similarly, for the Mott lobes with $n=2,3,4$, the NI phase expands and squeeze the rest region of the FM phase. We notice some odd behaviors relating to NI-FM phase boundaries for $n=3,4$, while the reasons behind this abnormal behavior remain unknown. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[fig:4\] (Color online) Abnormal multistep condensation for spin-1 ultracold bosons in a 3D cubic lattice with a ferromagnetic interaction $U_2/U_0=-0.7$ ($^{7}$Li) for fixed filling $n=1$ (a),(b) and $n=2$ (c). The other parameters are set to $zt/U_0=0.096$ (a) and $0.084$ (b), and $zt/U_{0}=0.063$ and $q/U_{0}=0.1$ (c). With increasing $M_z$, the system demonstrates interesting features, where a local minimum appears for the critical condensing temperature of the $m_F=0$ component.](FF9.eps "fig:"){width="0.9\linewidth"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- One crucial question regarding experimental observations is the multi-step condensation of strongly correlated superfluid. While the connection between magnetic order and Bose-Einstein condensation of weakly interacting spin-1 bosons with ferromagnetic interactions is debated [@RevModPhys.85.1191], the investigation on multistep condensations of strongly correlated lattice bosons is still missing. Here, we focus on the equilibrium states of the strongly interacting superfluids near the Mott lobes under the constraint of constant longitudinal magnetization (in the absence of dipolar relaxation, longitudinal magnetization is constant, instead of a constant magnetic field). We examine how an applied external magnetic field affects the critical condensing temperature $T_{c}$. In our calculations, we fix the value of the hopping matrix element $zt/U_{0}$ and filling number of per site $n$, and obtain the relationships between $T_{c}$ and the local magnetization $M_{z}$, as shown in Fig. \[fig:4\]. In Fig. \[fig:4\](a),(b), we choose $(U_{2}/U_{0},q/U_{0})=(-0.7,0.03)$, $n=1$, and $zt/U_{0}=0.096$ and $0.084$, respectively. The selected hopping amplitudes have unique properties, where $zt/U_0=0.096$ locates in the BA SF phase at zero magnetization with $\phi^{1}_{0,\pm1}\neq0$, and $zt/U_0=0.084$ sits in the polar SF phase with $\phi_0\neq0$. In Fig \[fig:2\](c), we focus on another case with large filling $n=2$. We find that the system demonstrates interesting features as a function of longitudinal magnetization. With increasing $M_z$, the critical condensing temperature of the $m_F=+1$ component increase monotonously, which is consistent with weakly interacting case [@RevModPhys.85.1191], since this component is energetically favored. However, the critical temperature of the $m_F=-1$ component increases at first and then decreases to zero, while the $m_F=0$ component decreases initially, then increases, and finally decreases to zero, which is inconsistent with weakly interacting case [@RevModPhys.85.1191]. The abnormal changes of condensing temperatures are the results of the interplay of spin-dependent interactions and quadratic Zeeman effect, and longitudinal magnetization, where the population of the $m_F=+1$ component increases with longitudinal magnetization, and quadratic Zeeman interactions favors the “pair” of the $m_F=\pm1$ component. Surprisingly, we observe a local minimum of the critical temperature of the $m_F=0$ component, which is a phenomenon has never been observed ever before. To obtain a better understanding of this behavior, we plot the multistep condensation of the spinor gases for another parameters, as shown in Fig. \[fig:4\](c), with $(U_{2}/U_{0}, q/U_{0})=(-0.7,0.1)$, $n=2$, and $zt/U_{0}=0.063$. We observe a minimum of the critical temperature of the $m_F=0$ component. We remark that the spinor lattice bosons with antiferromagnetic interactions also demonstrate abnormal features for critical condensing temperatures. Discussion {#discussion .unnumbered} ========== In conclusion, we employ spinor bosonic dynamical mean-field theory to carry out extensive calculations of ultracold spinor Bose gases loaded into a cubic optical lattice. Complete phase diagrams of the system with both antiferromagnetic and ferromagnetic interactions are obtained. Various phases, including nematic, ferromagnetic and spin-singlet insulator, polar superfluid, and broken-axisymmetry superfluid, are found. In particular, the competition between above phases are investigated by varying related physical influences such as spin-dependent interactions, quadratic Zeeman energy, and thermal fluctuations. Multistep condensations of the strongly correlated superfluids are explored as a function of longitudinal magnetization and temperature. Interestingly, the critical temperature of the $m_F=-1$ component increases firstly and then decreases to zero with increasing $M$ for antiferromagnetic interactions, while the critical temperature of the $m_F=0$ component demonstrates a local minimum for ferromagnetic cases, which is inconsistent with weakly interacting spinor gases. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== We acknowledge useful discussions with Y.-M. Liu, J.-M. Yuan, Z.-X. Zhao. This work was supported by the National Natural Science Foundation of China under Grants No. 11304386, 11774428, and No. 11104350. Author Contributions Statement {#author-contributions-statement .unnumbered} ============================== J.W. and Y.L. designed the study. X.Z. and J.L. performed calculations. J.H. analyzed the data. X.Z. and Y.L. wrote and reformulated the manuscript. All authors discussed the results and contributed to the manuscript. Additional information {#additional-information .unnumbered} ====================== **Competing interests:** The authors declare no competing interests.

--- abstract: 'This article presents the convergence analysis of a sequence of piecewise constant and piecewise linear functions obtained by the Rothe method to the solution of the first order evolution partial differential inclusion $u''(t)+Au(t)+\iota^*\partial J(\iota u(t))\ni f(t)$, where the multivalued term is given by the Clarke subdifferential of a locally Lipschitz functional. The method provides the proof of existence of solutions alternative to the ones known in literature and together with any method for underlying elliptic problem, can serve as the effective tool to approximate the solution numerically. Presented approach puts into the unified framework known results for multivalued nonmonotone source term and boundary conditions, and generalizes them to the case where the multivalued term is defined on the arbitrary reflexive Banach space as long as appropriate conditions are satisfied. In addition the results on improved convergence as well as the numerical examples are presented.' author: - | Piotr Kalita,\ Faculty of Mathematics and Computer Science,\ Institute of Computer Science,\ Jagiellonian University,\ ul. prof. S. Łojasiewicza 6, 30-348 Kraków, Poland\ `piotr.kalita@uj.edu.pl` title: Convergence of Rothe scheme for hemivariational inequalities of parabolic type --- Introduction ============ Partial differential inclusions with the multivalued term given in the form of Clarke subdifferential are known as hemivariational inequalities (HVIs). HVIs are the natural generalization of the inclusions with monotone multivalued term (which lead to variational inequalities) and were firstly considered by Panagiotopoulos in early 1980s. For the description of the origins of HVIs and underlying mathematical theory we refer the reader to the book [@Naniewicz1995]. This paper deals with the first order evolution inclusion of type $u'(t)+A(u(t))+\iota^*\partial J(\iota u(t)) \ni f(t)$. Such problems are known as parabolic HVIs or boundary parabolic HVIs depending whether an operator $\iota$ is the embedding operator from $H^1(\Omega)$ to $L^2(\Omega)$ or the trace operator from $H^1(\Omega)$ to $H^{\frac{1}{2}}(\partial\Omega)$. The first case corresponds to multivalued and nonmonotone source term in the equation and the second one to multivalued and nonmonotone boundary conditions of Neumann-Robin type. Such inclusions are used to model the diffusive transport through semipermeable membranes where the multivalued term represents the semipermeability relation [@Miettinen1999] and the temperature control problems where the multivalued term represents the feedback control [@Haslinger1999], [@Guanghui2010]. The existence of solutions to problems governed by inclusions of considered type was investigated by many authors. There are several techniques used to obtain the existence results: - Classical Faedo-Galerkin approach combined with the regularization of the multivalued term by means of a standard mollifier; solutions of underlying system of ordinary differential equations are proved to converge (in appropriate sense) to the function which is shown to be the solution of analyzed HVI. This technique was used in context of parabolic HVIs by Miettinen [@Miettinen1996], Miettinen and Panagiotopoulos [@Miettinen1999] and Goeleven et al. [@Goeleven2003]. - The approach based on the notion of upper and lower solutions. The solution is shown to be the limit of solutions of problems governed by the equations obtained by the regularization of the multivalued term together with the truncation by the lower and upper solutions. The distinctive feature of this approach is that the growth conditions on the multivalued term are replaced by the assumption of the existence of lower and upper solutions. The technique was used for parabolic HVIs by Carl [@Carl1996] and developed in [@Carl2002], [@Carl2003], [@Carl2004], [@Carl2008]. - The technique based on showing that the analyzed HVI satisfies the assumptions of the general framework for which the appropriate surjectivity result holds. This approach was used by Liu [@Liu2000] and by Migórski [@Migorski2001] and developed for the boundary case in [@Migorski2004]. - The technique based on adding to the inclusion the regularizing term multiplied by $\epsilon>0$, showing that the solutions to obtained problems satisfy some bounds uniformly in $\epsilon$ and passing to the limit $\epsilon\to 0$. This technique was used for parabolic HVIs by Liu and Zhang [@Liu1998] and Liu [@Liu1999] and developed in [@Liu2003], [@Liu2005]. It should be remarked that above techniques are either nonconstructive (i.e. they are based on surjectivity result) or constructive but not effective (i.e. require a priori knowledge of lower and upper solutions, or require additional or smoothing terms in the problem). In contrast to the existence theory, numerical methods to approximate effectively the solutions to parabolic HVIs were not considered by many authors. In the book of Haslinger, Miettinen and Panagiotopoulos [@Haslinger1999] the convergence of solutions obtained by the finite element approximation of the space variable and finite difference approximation of the time variable is proved. However only the case of the linear operator $A$ and the multivalued source term (and not boundary conditions) is considered (see Remark 4.10 in [@Haslinger1999]). In [@Guanghui2010] the authors proved the convergence of the finite difference scheme (with respect to both time and space variable) for the case of multivalued source term (i.e. $U=H$ in the sequel). Our approach uses the so-called Rothe method (known also as time approximation method) and allows to extend any numerical method that is used to solve the stationary, elliptic inclusions with the multivalued term given as the Clarke subdifferential, to time dependent, parabolic problems. The key idea is the replacement of time derivative with the backward difference scheme and solve the associated elliptic problem in every time step to find the solution in the consecutive points of the time mesh. It is proved that the results obtained by such approach approximate the solution of the original problem. On the other hand, the Rothe metod provides the proof of existence of solutions. In contrast to other approaches this metod, as long as one can solve underlying elliptic problems, does not require any smoothing or other additional regularizing terms in the inclusion. Furthermore the presented approach allows to study the inclusions with multivalued term given on the domain and on the domain boundary within the unified framework in which the multifunction that appears in the problem is defined on an arbitrary reflexive Banach space, which satisfies the appropriate assumption ($H(U)$ in the sequel). This assumption is proved to generalize the case of inclusions with multivalued boundary conditions and the ones with multivalued source term (see Section \[sec:lions\] and examples of problem settings in Section \[sec:settings\]). The Rothe method for parabolic nonlinear PDEs with pseudomonotone operators is described in the monograph of Roubicek [@Roubicek2005], where also the results for the monotone multivalued problems are presented. In the context of parabolic HVIs the variant of the Rothe method was used to show existence of solutions to problems with hysteresis in [@Miettinen1998] and [@Miettinen2003], but there only the case of linear operator $A$ and $f\in L^2(0,T;L^2(\Omega))$ (which excludes nonhomogeneous Neumann conditions) was considered and besides only the case of the multivalued and nonmonotone term source term was analyzed. In Section \[sec:prel\] some basic definitions are recalled. Section \[sec:lions\] presents the generalization of the Lions-Aubin Compactness Lemma that justifies the usage of the assumption $H(U)$ in the sequel. Problem setup and the assumptions are presented in Section \[sec:prblm\]. The auxiliary elliptic problems solved in every time step, which are the key idea of the Rothe method, are formulated and analyzed in Section \[sec:rothe\]. Convergence of piecewise linear and piecewise constant functions constructed basing on the solutions of auxiliary problems as well as the fact that the limit solves the original problem is proved in Section \[sec:limit\]. Some stronger convergence and uniqueness results are established in Section \[sec:uniq\]. Finally in Section \[sec:settings\] it is shown that the cases of multivalued boundary condition and source term are the special cases of presented general framework and a simple numerical example is delivered. Preliminaries {#sec:prel} ============= In this section we recall several key definitions that will be used in the sequel. For a locally Lipschitz functional $j:X\to \mathbb{R}$, where $X$ is a Banach space, generalized directional derivative (in the sense of Clarke) at $x\in X$ in the direction $z\in X$ is defined as $$j^0(x;z)=\limsup_{y\to x, \lambda \to 0^+}\frac{j(y+\lambda z)-j(y)}{\lambda}.$$ Generalized gradient of $j$ (in the sense of Clarke) is the multifunction $\partial j:X\to 2^{X^*}$ defined by $$\partial j(x)=\{\xi\in X^*: j^0(x;y)\geq\langle \xi,y\rangle \ \ \mbox{for all}\ \ y\in X\},$$ where $\langle\cdot,\cdot\rangle$ stands for the duality pairing between $X$ and $X^*$. For the properties and the calculus of the Clarke gradient see [@Clarke1990]. Recall that the multifunction $A:X\to 2^{X^*}$, where $X$ is a real and reflexive Banach space is pseudomonotone if - $A$ has values which are nonempty, weakly compact and convex, - $A$ is usc from every finite dimensional subsepace of $X$ into $X^*$ furnished with weak topology, - if $v_n\to v$ weakly in $X$ and $v_n^*\in A(v_n)$ is such that $\limsup_{n\to\infty}\langle v_n^*,v_n-v\rangle \leq 0$ then for every $y\in X$ there exists $u(y)\in A(v)$ such that $\langle u(y),v-y\rangle\leq \liminf_{n\to\infty}\langle v_n^*,v_n-y\rangle$. Note that sometimes it is useful to check the pseudomonotonicity of an operator via the following sufficient condition (see Proposition 1.3.66 in [@Denkowski2003] or Proposition 3.1 in [@Carl2006]). \[prop:pseudo\] Let X be a real reflexive Banach space, and assume that $A: X\to 2^{X^*}$ satisfies the following conditions - for each $v\in X$ we have that $A(v)$ is a nonempty, closed and convex subset of $X^*$. - $A$ is bounded. - If $v_n\to v$ weakly in $X$ and $v_n^*\to v^*$ weakly in $X^*$ with $v_n^*\in A(v_n)$ and if $\limsup_{n\to\infty}\langle v_n^*, v_n - v\rangle \leq 0$, then $v^*\in A(v)$ and $\langle v_n^*,v_n\rangle \to \langle v^*,v\rangle$. Then the operator $A$ is pseudomonotone. We also recall (see for instance Proposition 1.3.68 [@Denkowski2003]) that the sum of two pseudomonotone multifunctions is pseudomonotone. Generalization of Lions-Aubin Lemma {#sec:lions} =================================== For a Banach space $X$, $1\leq p\leq \infty$ and a finite time interval $I=(0,T)$ we consider the standard spaces $L^p(I;X)$. Furthermore we denote by $BV(I;X)$ the space of functions of bounded total variation on $I$. Let $\pi$ denote any finite partition of $I$ by a family of disjoint subintervals $\{\sigma_i=(a_i,b_i)\}$ such that $\bar{I}=\bigcup_{i=1}^n\bar{\sigma}_i$. Let $\C{F}$ denote the family of all such partitions. Then we define the total variation as $$\|x\|_{BV(I;X)} = \sup_{\pi\in \C{F}}\left\{\sum_{\sigma_i\in\pi}\|x(b_i)-x(a_i)\|_X\right\}.$$ As a generalization of above definition for $1\leq q < \infty$ we can define a seminorm $$\|x\|_{BV^q(I;X)}^q = \sup_{\pi\in \C{F}}\left\{\sum_{\sigma_i\in\pi}\|x(b_i)-x(a_i)\|^q_X\right\}.$$ For Banach spaces $X,Z$ such that $X\subset Z$ we introduce a vector space $$M^{p,q}(I;X,Z)= L^p(I;X)\cap BV^q(I;Z).\nonumber$$ Then $M^{p,q}(I;X,Z)$ is also a Banach space for $1\leq p,q <\infty$ with the norm given by $\|\cdot\|_{L^p(I;X)}+\|\cdot\|_{BV^q(I;Z)}$. Let us recall Theorem 1 of [@Simon1987] (see also Theorem 1 of [@Rossi2003] and Proposition 2.1 of [@Rossi2001]). \[thm:simon\] Let $1\leq p<\infty$ and $X$ be a real Banach space. A subset $\C{G}\subset L^p(0,T;X)$ is relatively compact in a Banach space $L^p(0,T;X)$ provided the following two conditions hold - for every $0<t_1<t_2<T$ the set $$G(t_1,t_2) := \left\{\int_{t_1}^{t_2}u(t)\, dt: u\in \C{G} \right\}$$ is relatively compact in $X$, - $\C{G}$ is strongly integrally equicontinuous i.e. $$\lim_{h\to 0}\sup_{u\in \C{G}}\int_0^{T-h}\|u(t+h)-u(t)\|_{X}^p\ dt = 0.$$ The following proposition is a consequence of Theorem \[thm:simon\]. \[prop:compactembedding\] Let $1\leq p,q<\infty$. Let $X_1\subset X_2\subset X_3$ be real Banach spaces such that $X_1$ is reflexive, the embedding $X_1\subset X_2$ is compact and the embedding $X_2\subset X_3$ is continuous. If a subset $\C{G}\subset M^{p,q}(I; X_1,X_3)$ is bounded, then it is relatively compact in $L^p(I;X_2)$. We apply Theorem \[thm:simon\] with $X=X_2$. Let us fix $0<t_1<t_2<T$ and let $v\in G(t_1,t_2)$. For $u \in \C{G}$ we have $$\begin{aligned} \|v\|_{X_1}=\left\|\int_{t_1}^{t_2}u(t)\, dt\right\|_{X_1}\leq \int_{0}^{T} \|u(t)\|_{X_1}\, dt\leq T^{1-\frac{1}{p}}\|u\|_{L^p(I;X_1)}.\end{aligned}$$ Thus $G(t_1,t_2)$ is bounded in $X_1$ and therefore relatively compact in $X_2$. It suffices to show the strong integral equicontinuity of $\C{G}$. Let $\sup_{u\in \C{G}}\|u\|_{L^p(I;X_1)}^p=M$. We will use the Ehrling Lemma (see for instance [@Roubicek2005], Lemma 7.6). Let us fix $\varepsilon>0$. There exists $C>0$ such that for $v\in X_1$ we have $\|v\|_{X_2}^p\leq \frac{\varepsilon}{2^pM}\|v\|_{X_1}^p+C\|v\|_{X_3}^p$. In particular, fixing $h\in (0,T)$, for $u\in \C{G}$ and almost every $t\in (0,T-h)$ we have $\|u(t+h)-u(t)\|_{X_2}^p\leq \frac{\varepsilon}{2^pM}\|u(t+h)-u(t)\|_{X_1}^p+C\|u(t+h)-u(t)\|_{X_3}^p$. Integrating this inequality we get $$\begin{aligned} &&\int_0^{T-h}\|u(t+h)-u(t)\|_{X_2}^p\ dt\leq \nonumber\\ &&\leq\frac{\varepsilon}{2^pM}\int_0^{T-h}\|u(t+h)-u(t)\|_{X_1}^p\ dt+C\int_0^{T-h}\|u(t+h)-u(t)\|_{X_3}^p\ dt \leq \label{ineq:equicont}\nonumber\\ &&\leq \frac{\varepsilon}{2M}\int_0^{T-h}\|u(t+h)\|_{X_1}^p+\|u(t)\|_{X_1}^p\ dt+C\int_0^{T-h}\|u(t+h)-u(t)\|_{X_3}^p\ dt\leq \nonumber\\ &&\leq \varepsilon + C\int_0^{T-h}\|u(t+h)-u(t)\|_{X_3}^p\ dt.\end{aligned}$$ Now let $\sup_{u\in \C{G}}\|u\|_{BV^q(I;X_3)}^q=S$. If $p \leq q$, then by the Hölder inequality, we have $$\label{ineq:case1}\int_0^{T-h}\|u(t+h)-u(t)\|_{X_3}^p\ dt \leq T^{1-\frac{p}{q}} \left(\int_0^{T-h}\|u(t+h)-u(t)\|_{X_3}^q\ dt\right)^{\frac{p}{q}}.$$ If in turn $q<p$, then $$\label{ineq:case2}\int_0^{T-h}\|u(t+h)-u(t)\|_{X_3}^p\ dt %\leq \int_0^{T-h}\|u(t+h)-u(t)\|_{X_3}^{p-q}\|u(t+h)-u(t)\|_{X_3}^q\ dt\leq \nonumber\\ \leq S^{\frac{p}{q}-1}\int_0^{T-h}\|u(t+h)-u(t)\|_{X_3}^q\ dt.$$ We estimate the last term in (\[ineq:case1\]) and (\[ineq:case2\]) from above (taking, if necessary, $u(t)=u(T)$, if $t>T$) $$\begin{aligned} && \int_0^{T-h}\|u(t+h)-u(t)\|_{X_3}^q\ dt\leq \sum_{i=0}^{\lceil T/h-2\rceil}\int_{ih}^{ih+h}\|u(t+h)-u(t)\|_{X_3}^q\ dt=\nonumber\\ &&=\sum_{i=0}^{\lceil T/h-2\rceil}\int_{0}^{h}\|u(t+ih+h)-u(t+ih)\|_{X_3}^q\ dt=\nonumber\\ &&=\int_{0}^{h}\sum_{i=0}^{\lceil T/h-2\rceil}\|u(t+ih+h)-u(t+ih)\|_{X_3}^q\ dt\leq Sh.\end{aligned}$$ Thus the last term in (\[ineq:equicont\]) tends to $0$ uniformly in $u$ as $h\to 0$ and, since $\varepsilon$ was arbitrary, we get the thesis. **Remark 1.** Note, that Theorem 3.2 in [@Ahmed2003] is a consequence of above theorem. Compare also the Corollary 7.9 in [@Roubicek2005] where the case $p=1$ is excluded and $X_3$ is assumed to have a predual space. Problem formulation and assumptions {#sec:prblm} =================================== Let $V\subset H\subset V^*$ be an evolution triple, where $V$ is a reflexive and separable Banach space and $H$ is a separable Hilbert space with the embeddings being continuous, dense and compact. Embedding between $V$ and $H$ will be denoted by $i$. Furthermore let $U$ be a reflexive Banach space on which the multivalued term will be defined. We use the notation $\C{V}=L^2(0,T;V)$, $\C{H}=L^2(0,T;H)$, $\C{U}=L^2(0,T;U)$ and $\C{W}=\{u\in \C{V}, u' \in \C{V}^*\}$, where the derivative is understood in the sense of distibutions. Duality parings and norms for all the spaces will be denoted by the appropriate subscripts, for the space $V$ no subscript will be used. Scalar product in $H$ will be denoted by $(\cdot,\cdot)$ and norm in $\mathbb{R}^n$ by $|\cdot|$. We consider the operator $A:V\to V^*$ and the functional $J:U\to \mathbb{R}$ such that the following assumptions hold - - $A$ is pseudomonotone, - $A$ satisfies the growth condition $\|A(v)\|_{V^*}\leq a+b\|v\|$ for every $v\in V$ with $a\geq 0, b>0$, - $A$ is coercive $\langle A(v),v\rangle\geq \alpha \|v\|^2-\beta\|v\|^2_{H}$ for every $v\in V$ with $\alpha> 0$ and $\beta\geq 0$, - - $J$ is locally Lipschitz, - $\partial J$ satisfies the growth condition $\|\xi \|_{U^*}\leq c(1+\|u\|_U)$ for every $u\in U$ and $\xi \in \partial J(u)$ with $c>0$. Moreover we assume that - $f\in \C{V}^*$ and $u_0\in H$. We also impose the assumption concerning the space $U$ - There exists the linear, continuous and compact mapping $\iota:V\to U$ such that the associated Nemytskii mapping $\bar{\iota}:M^{2,2}(0,T;V,V^*)\to \C{U}$ defined by $(\bar{\iota} v)(t)=\iota(v(t))$ is also compact. Finally we impose the last assumption - One of the following holds - There exists a linear and continuous mapping $p:H\to U$ such that for $v\in V$ we have $p(i(v))=\iota(v)$. - The constants $\alpha$ and $c$ satisfy the inequality $\alpha > c \|\iota\|_{\C{L}(V;U)}^2$. - For every $u\in U$ we have $J^0(u;-u)\leq d(1+\|u\|_U^\sigma)$ with $d\geq 0$ and $1\leq \sigma <2$. The problem under consideration is as follows $$\begin{aligned} && \mbox{find}\ u\in \C{W}\ \mbox{such that}\ u(0)=u_0 \ \mbox{and for a.e.} \ t\in (0,T)\ \mbox{we have}\nonumber\\ &&u'(t)+Au(t)+\iota^* \partial J (\iota u(t)) \ni f(t).\label{eq:inclusionmain}\end{aligned}$$ The last inclusion is understood in the following sense $$\begin{aligned} &&\mbox{there exists}\ \ \eta\in \C{V}^* \ \mbox{such that}\ u'(t)+Au(t)+\eta(t) = f(t)\ \mbox{for a.e.}\ t\in(0,T)\nonumber\\ &&\mbox{and}\ \ \langle\eta(t),v\rangle\in \langle \partial J(\iota u(t)),\iota v\rangle_{U^*\times U} \ \mbox{for a.e.}\ t\in(0,T)\ \mbox{and}\ v\in V.\end{aligned}$$ **Remark 2.** The formulation (\[eq:inclusionmain\]) puts into a unified framework hemivariational inequalities originating from the initial and boundary value problem with multivalued term defined on the problem domain (in this case we have multivalued source term, and $U=H$, see [@Miettinen1996; @Miettinen1999; @Migorski2001]) and on the part $\Gamma_C$ of domain boundary $\partial\Omega$ (this is the case if we have the multivalued, nonlinear and nonmonotone boundary condition of Neumann-Robin type, $U=L^2(\Gamma_C)$ or $U=L^2(\Gamma_C;\mathbb{R}^n)$, see [@Migorski2004]). A detailed discussion as well as examples of problems which satisfy the assumptions will be given in Section \[sec:settings\]. **Remark 3.** For the sake of simplicity of further argument the assumptions given above are not the most general ones under which the results hold. Possible generalizations include: - The dependance of $A$ and $J$ on time variable. Time dependent operator $A$ for parabolic HVI is considered in [@Miettinen1999] and the case of both $A$ and $J$ depending on time is considered in [@Migorski2004] (see Remark 8.21 in [@Roubicek2005] on the Rothe method for the problem with the operator depending on time). - Instead of pseudomonotonicity one could assume that $A$ is a sum of two operators, one of which is pseudomonotone and the second one is weakly continuous. Such weak continuity allows to take into account the nonlinear terms of lower order which are not of monotone type (see [@Francu1994]). - More general coercivity conditions on $A$ can be assumed. For instance $\langle Av,v\rangle\geq c\|v\|^2-a\|v\|_V-\gamma(t)$ with $c>0$, $a\geq 0$ and $\gamma\in L^1(0,T)$ cf. [@Migorski2001]. - The case when the space $\C{V}$ is defined as $L^p(0,T;V)$ with $2<p<\infty$ can be considered. Then we can assume more general growth conditions on $A$ and $J$. For instance in [@Migorski2001] it is assumed that $\|A(t,v)\|_{V^*}\leq \beta(t)+c_1\|v\|^{p-1}$ and that for $\eta\in \partial j(x,\xi)$ we have $|\eta|\leq c(1+|\xi|^{p-1})$. Note that $J$ is defined typically as the integral functional $J(u)=\int_{\omega}j(x,u(x))\, dx$ and assumptions on the integrand $j$ are given. **Remark 4.** In this paper the abstract setting is considered. For a divergence differential operator of Leray - Lions type on a Sobolev space pseudomonotnicity is implied by the appropriate Leray - Lions type conditions (see, for instance, [@Berkovits1996] where conditions that guarantee pseudomonotonicity on $W^{m,p}(\Omega)$, $1<p<\infty$, $m\geq 1$ are considered). We conclude this section with the Lemma on pseudomonotonicity of Nemytskii operator with respect to the space $M^{2,2}(0,T;V,V^*)$. Note that the proof of this lemma is analogous to the proof of Theorem 2 (b) in [@Berkovits1996] (see also Proposition 1 from [@Papageorgiou1997] and Lemma 8.8 in [@Roubicek2005] for similar results). Lemma 8.8 of [@Roubicek2005] is most similar to Lemma \[lem:nemytskii\], but note that here no a priori bound in $L^\infty(0,T;H)$ is needed and the assumption on the bound of 2-variation which is used here is weaker then the bound on 1-variation as in [@Roubicek2005]. \[lem:nemytskii\] Let $A: V\to V^*$ satisfy $H(A)$ and let $\C{A}:\C{V}\to \C{V}^*$ be a Nemytskii operator for $A$ defined by $(\C{A}u)(t)=A(u(t))$. Then if, for a uniformly bounded sequence $\{u_n\}\subset M^{2,2}(0,T;V,V^*)$ such that $u_n\to u$ weakly in $\C{V}$ we have $\limsup_{n\to\infty} \langle \C{A}u_n,u_n-u\rangle_{\C{V}^*\times \C{V}}\leq 0$, then $\C{A}u_n \to \C{A}u$ weakly in $\C{V}^*$. It is enough to show that the thesis holds for a subsequence. By the generalized Lions Aubin Compactness Lemma (see Proposition \[prop:compactembedding\]) for a subsequence (still denoted by $n$) we have $u_n\to u$ strongly in $\C{H}$. Moreover, for yet another subsequence $u_n(t)\to u(t)$ strongly in $H$ for a.e. $t\in (0,T)$. We denote the set of measure zero on which the convergence does not hold by $N$. Now let us define $\xi_n(t)=\langle Au_n(t),u_n(t)-u(t)\rangle$. We have $$\begin{aligned} \label{eq:boundbelowxi} && \xi_n(t)\geq \alpha\|u_n(t)\|^2-\beta\|u_n(t)\|_H^2-\|u(t)\|(a+b\|u_n(t)\|)\geq\\ &&\geq \frac{\alpha}{2}\|u_n(t)\|^2-\beta \|u_n(t)\|_H^2 - a\|u(t)\| - \frac{b^2}{2\alpha}\|u(t)\|^2.\nonumber\end{aligned}$$ Now let $C=\{t \in [0,T]: \liminf_{n\to\infty}\xi_n(t)< 0\}$. This is the Lebesgue measurable subset of $[0,T]$. Suppose that $m(C)>0$, $m$ being one dimensional Lebesgue measure. For every $t\in C\setminus N$ the sequence $u_n(t)$ has a subsequence (still denoted by $n$) which is bounded in $V$ by (\[eq:boundbelowxi\]) such that $\lim_{n\to\infty}\langle Au_n(t),u_n(t)-u(t) \rangle<0$. Again for a subsequence we have $u_n(t)\to u(t)$ weakly in $V$, where the limit equals $u(t)$ since we can consider only $t\notin N$. By the pseudomonotonicity of $A$ we get $0\leq \liminf_{n\to \infty} \langle Au_n(t), u_n(t)-u(t)\rangle$, which is a contradiction. So $m(C)=0$, which means that $\liminf_{n\to\infty} \xi_n(t)\geq 0$ a.e. on $(0,T)$. From the Fatou Lemma we have $$\begin{aligned} &&\beta\|u\|_H^2 \leq \int_0^T\liminf_{n\to\infty} \xi_n(t)\, dt + \beta\|u\|_H^2 \leq \int_0^T \liminf_{n\to\infty} (\xi_n(t)+\beta\|u_n(t)\|_H^2) \, dt \leq \nonumber \\ && \leq \liminf_{n\to\infty } \int _0^T \xi_n(t)+ \beta\|u_n(t)\|_H^2 \, dt \leq \nonumber\\ && \leq \liminf_{n\to\infty } \int _0^T \xi_n(t)\, dt + \beta \|u\|_H^2\leq \limsup_{n\to\infty } \int _0^T \xi_n(t) \, dt + \beta \|u\|_H^2 \leq \beta \|u\|_H^2.\nonumber\end{aligned}$$ So $\int_0^T\xi_n(t)\, dt\to 0$ as $n\to \infty$. Now note that $|\xi_n(t)|=\xi_n(t)+2\xi_n^-(t)$ and $\xi_n^-(t)\to 0$ for a.e. $t\in (0,T)$. Since, by (\[eq:boundbelowxi\]), for a.e. $t\in (0,T)$ we have $\xi_n(t) + \beta \|u_n(t)\|_H^2 \geq f(t)$ with $f \in L^1(0,T)$, then $ \xi_n^-(t)-\beta\|u_n(t)\|_H^2 \leq f^-(t)$. Invoking Fatou Lemma again we have $\limsup\int_0^T\xi^-(t)\, dt \leq 0$ and furthermore $\int_0^T\xi_n^-(t)\, dt \to 0$ as $n\to \infty$. We deduce that $\xi_n\to 0$ in $L^1(0,T)$ and, for a subsequence (still denoted by the same subscript), $\xi_n(t)\to 0$ for a.e. $t\in (0,T)$. Since, for this subsequence, $u_n(t)\to u(t)$ weakly in $V$, then by pseudomonotonicity of $A$ it follows that $Au_n(t)\to Au(t)$ weakly in $V^*$ and $\langle Au_n(t), u_n(t)\rangle \to \langle Au(t), u(t) \rangle$. For any $v\in \C{V}$ we have $$\begin{aligned} && \langle \C{A}u, u-v\rangle_{\C{V}^*\times \C{V}} = \int_0^T\langle Au(t), u(t)-v(t) \rangle\, dt = \int_0^T\lim_{n\to\infty} \langle Au_n(t), u_n(t)-v(t)\rangle\, dt=\nonumber \\ &&= -\beta\|u\|_H^2 + \int_0^T\lim_{n\to\infty} (\langle Au_n(t), u_n(t)-v(t)\rangle + \beta\|u_n(t)\|_H^2)\, dt .\nonumber\end{aligned}$$ We can apply Fatou Lemma one last time to get $$\begin{aligned} && \langle \C{A}u, u-v\rangle_{\C{V}^*\times \C{V}} \leq \liminf_{n\to\infty}\int_0^T\langle Au_n(t), u_n(t)-v(t)\rangle\, dt =\nonumber\\ &&= \liminf_{n\to\infty} (\langle \C{A}u_n, u_n-u\rangle_{\C{V}^*\times \C{V}}+\langle \C{A}u_n, u-v\rangle_{\C{V}^*\times \C{V}})\leq\nonumber \\ &&\leq\liminf_{n\to\infty} \langle \C{A}u_n, u-v\rangle_{\C{V}^*\times \C{V}}.\end{aligned}$$ Since $v$ is arbitrary we obtain the thesis. The Rothe problem {#sec:rothe} ================= In this section we will work with a sequence of time-steps $\tau_n\to 0$ such that each time step $\tau_n>0$ and the value $T/\tau_n$ is an integer, which we denote by $N_n$. The subscipt $n$ will be omitted in the sequel in order to simplify the notation, so we will write $N, \tau$ instead of $N_n, \tau_n$. We define the piecewise constant approximation of the function $f\in \C{V}^*$. For this purpose we take the sequence of positive numbers $\epsilon(\tau)\to 0$ and the sequence of mollifiers $\rho_\epsilon:\mathbb{R}\to\mathbb{R}$ which belong to $C^\infty(\mathbb{R})$ and are nonnegative, supported on $[-\epsilon,\epsilon]$ and $\int_{\mathbb{R}}\rho_{\epsilon}(x)\, dx = 1$. The function $f$ is regularized according to the formula $$f_\epsilon(t)=\int_0^T \rho_\epsilon\left(t+\epsilon\frac{T-2t}{T}-s\right)f(s)\, ds.$$ Note that $f_\epsilon\in C^1(0,T;V^*)$ (see [@Roubicek2005], Lemma 7.2). The piecewise constant approximation for $f$ is given by $$\bar{f}_\tau(t):=f^k_\tau=f_{\epsilon(\tau)}(k\tau)\ \mbox{for}\ t\in((k-1)\tau,k\tau],\ k\in\{1,\ldots,N\}.$$ Following [@Roubicek2005], Lemma 8.7, we have $\bar{f}_\tau\to f$ in $\C{V}^*$ when $\tau\to 0$. Note (see Remark 8.15 in [@Roubicek2005]) that the smoothing of $f$ is not the only possible approach here. It is also possible to take the Clément zero-order quasi interpolant $f_\tau^k=\frac{1}{\tau}\int_{(k-1)\tau}^{k\tau}f(\theta)\,d\theta$. We approximate the initial condition by elements of $V$. Let $\{u_{0\tau}\}\subset V$ be a sequence such that $u_{0\tau}\to u_0$ strongly in $H$ and $\|u_{0\tau}\|\leq C/\sqrt{\tau}$ for some constant $C>0$. We define the following Rothe problem $$\begin{aligned} && \mbox{find the sequence}\ \{u_\tau^k\}_{k=0}^N \subset V\ \mbox{such that}\ u_\tau^0=u_{0\tau} \ \mbox{and}\nonumber\\ && \left(\frac{u^{k}_\tau-u^{k-1}_\tau}{\tau},v\right)_H+\langle Au_\tau^k,v\rangle+ \langle\partial J (\iota u^k_\tau), \iota v\rangle_{U^*\times U} \ni \langle f^k_\tau, v\rangle \label{eq:rotheproblem}\\ && \mbox{for all}\ v\in V\ \mbox{and}\ k=1,\ldots,N.\nonumber\end{aligned}$$ The above formula is known as the implicit or backward Euler scheme. Existence of solutions to the Rothe problem follows from the following Under assumptions $H(A), H(J), H_0, H(U)$ and $H_{aux}$ there exists $\tau_0>0$ such that the problem (\[eq:rotheproblem\]) has a solution for $\tau\in (0,\tau_0)$. We show that, given $u^{k-1}_\tau \in V$, we can find $u^k_\tau\in V$ such that (\[eq:rotheproblem\]) holds. We need to show that the range of multifunction $V\ni v\to Lv=\frac{i^* iv}{\tau}+Av+\iota^* \partial J(\iota v)\in 2^{V^*}$ constitutes the whole space $V^*$. We will use the surjectivity theorem for pseudomonotone operators (see for instance Theorem 1.3.70 in [@Denkowski2003]). We need to show that $L$ is coercive (in the sense that $\lim_{\|v\|\to\infty}\frac{\inf_{v^*\in Lv}\langle v^*,v\rangle}{\|v\|}=\infty$) and pseudomonotone. **Claim 1.** *$L$ is pseudomonotone.* We verify this condition for all components of $L$ separately. For this purpose we use Proposition \[prop:pseudo\]. The operator $\frac{i^*i}{\tau}$ satisfies the conditions $(i)-(iii)$ trivially. As for $\iota^* \partial J(\iota u)$ the condition $(i)$ follows from the fact that the Clarke subdifferential has nonempty, convex and (for reflexive space) weakly compact values. The condition $(ii)$ follows from the growth assumption on $\partial J$. In order to verify $(iii)$ let us take $v_n\to v$ weakly in $V$ and $\xi_n\to \xi$ weakly in $V^*$ with $\xi_n\in \iota^*\partial J(\iota v_n)$. Obviously $\iota v_n\to \iota v$ strongly in $U$. Define $\eta_n \in \partial J(\iota v_n)$ such that $\xi_n=\iota^*\eta_n$. By the growth condition $H(J)(ii)$ it follows that, for a subsequence still denoted by the same subscript, $\eta_n\to \eta$ weakly in $U^*$. By the closedness of the graph of $\partial J$ in $U\times U^*_{w}$ topology (see [@Clarke1990], Proposition 2.1.5), we get $\eta \in \partial J(\iota v)$. Obviously $\xi=\iota^*\eta$ and $\xi\in \iota^*\partial J(\iota v)$. Moreover $\langle v_n,\xi_n\rangle=\langle \iota v_n, \eta_n\rangle_{U^*\times U} \to \langle \iota v, \eta\rangle_{U^*\times U}=\langle v,\xi\rangle$, where by uniqueness convergence holds for the whole sequence. **Claim 2.** *$L$ is coercive.* Assume that $v^*\in Lv$. We estimate $\langle v^*,v\rangle$ from below. For some $\eta\in \partial J(\iota v)$ we have $$\label{eq:estimatecorec} \langle v^*,v \rangle \geq \frac{1}{\tau}\|v\|_H^2+\alpha \|v\|^2 - \beta \|v\|_H^2+\langle \eta, \iota v\rangle_{U^*\times U}.$$ We proceed for cases $A), B), C)$ separately. For $A)$ and $B)$, by the growth condition $$\langle v^*,v \rangle \geq \left(\frac{1}{\tau}-\beta\right)\|v\|_H^2+\alpha \|v\|^2-c(1+\|\iota v\|_U)\|\iota v\|_U.$$ In the case $A)$ we have $\|\iota v\|_U^2\leq \|p\|_{\C{L}(H,U)}^2\|v\|_H^2$, so $$\langle v^*,v \rangle \geq \left(\frac{1}{\tau}-\beta-c\|p\|_{\C{L}(H,U)}^2\right)\|v\|_H^2+\alpha \|v\|^2-c\|\iota\|_{\C{L}(V;U)}\|v\|.$$ We require $\tau_0 = \frac{1}{\beta+c\|p\|_{\C{L}(H,U)}}$. In the case $B)$ we get $$\langle v^*,v \rangle \geq \left(\frac{1}{\tau}-\beta\right)\|v\|_H^2+(\alpha-c\|\iota\|^2_{\C{L}(V;U)})\|v\|^2-c\|\iota\|_{\C{L}(V;U)}\|v\|.$$ To have coercivity we need to set $\tau_0=\frac{1}{\beta}$. Finally if $C)$ holds, then we get $$\begin{aligned} &&\langle \eta, \iota v\rangle_{U^*\times U}\geq -J^0(\iota v;-\iota v) \geq\nonumber\\ &&\geq - d(1+\|\iota v\|_U^\sigma)\geq -d -\|\iota\|_{\C{L}(V;U)}^\sigma\|v\|^\sigma\geq -d -\frac{\alpha}{2}\|v\|^2-C,\nonumber\end{aligned}$$ where $C>0$ depends on $\alpha, \sigma$ and $\|\iota\|_{\C{L}(V;U)}$. Combining the last estimate with (\[eq:estimatecorec\]) we get $$\langle v^*,v \rangle \geq \left(\frac{1}{\tau}-\beta\right)\|v\|_H^2+\frac{\alpha}{2} \|v\|^2 - d - C.$$ Again setting $\tau_0=\frac{1}{\beta}$ we get the desired property. Next lemma establishes the estimates which are satisfied by the solutions of Rothe problem. \[lem:bounds\] Under assumptions $H(A), H(J), H_0, H(U)$ and $H_{aux}$ there exists $\tau_0>0$ such that for all $\tau\in (0,\tau_0)$ the solutions of Rothe problem (\[eq:rotheproblem\]) satisfy $$\begin{aligned} &&\max_{k=1,\ldots,N}\|u_\tau^k\|_H\leq\mbox{const},\label{eq:bound1}\\ &&\sum_{k=1}^N\|u_\tau^k-u_\tau^{k-1}\|_H^2\leq\mbox{const},\\ &&\tau \sum_{k=1}^N\|u_\tau^k\|^2\leq\mbox{const},\label{eq:bound3}\end{aligned}$$ with the constants independent on $\tau$. We take $v=u^k_\tau$ in (\[eq:rotheproblem\]), which gives for $\varepsilon > 0$ and $k=1,\ldots,N$ $$\begin{aligned} && \frac{1}{\tau}\|u^k_\tau\|^2_H+\alpha\|u^k_\tau\|^2+\langle \xi_\tau^k,\iota u^k_\tau\rangle_{U^*\times U}\leq\nonumber\\ &&\leq \beta\|u^k_\tau\|_H^2+\frac{1}{2\varepsilon}\|f^k_\tau\|^2_{V^*}+\frac{\varepsilon}{2}\|u^k_\tau\|^2+\frac{1}{\tau}(u^{k-1}_\tau,u^k_\tau)\nonumber\end{aligned}$$ with $\xi_\tau^k\in \partial J(\iota u^k_\tau)$. We use the relation $\|a\|^2-(a,b)=\|a\|^2/2 - \|b\|^2/2+\|a-b\|^2/2$ to obtain $$\begin{aligned} \label{eq:estimateonestep} &&\left(\frac{1}{2\tau}-\beta\right)\|u^k_\tau\|^2_H+\frac{1}{2\tau}\|u^{k}_\tau-u^{k-1}_\tau\|_H^2+\left(\alpha-\frac{\varepsilon}{2}\right)\|u^k_\tau\|^2+\\ &&+\langle \xi_\tau^k,\iota u^k_\tau\rangle_{U^*\times U}\leq\frac{1}{2\varepsilon}\|f^k_\tau\|^2_{V^*}+\frac{1}{2\tau}\|u^{k-1}_\tau\|_H^2.\nonumber\end{aligned}$$ Recall that $$\langle \xi^k_\tau, \iota u^k_\tau\rangle_{U^*\times U}\geq \begin{cases}-C_1-c\|p\|^2_{\C{L}(H,U)}\|u^k_\tau\|^2_H-\frac{\alpha}{2}\|u^k_\tau\|^2& \mbox{if $A)$ holds,}\\ (-c\|\iota\|^2_{\C{L}(U,V)}-\delta)\|u^k_\tau\|^2-C_2& \mbox{if $B)$ holds,}\\-C_3-\frac{\alpha}{2}\|u^k_\tau\|^2 & \mbox{if $C)$ holds,}\end{cases}$$ where $C_1>0$ depends on $c,\alpha, \|\iota\|_{\C{L}(V;U)}$, $\delta>0$ is arbitrary, $C_2>0$ depends on $c, \delta, \|\iota\|_{\C{L}(V;U)}$ and $C_3>0$ depends on $d, \alpha, \sigma, \|\iota\|_{\C{L}(V;U)}$. From now on we proceed separately for the cases $A), B)$ and $C)$. In the case $A)$ we take $\varepsilon = \frac{\alpha}{2}$ to get $$\begin{aligned} &&\left(\frac{1}{2\tau}-\beta-c\|p\|^2_{\C{L}(H,U)}\right)\|u^k_\tau\|^2_H+\frac{1}{2\tau}\|u^{k}_\tau-u^{k-1}_\tau\|_H^2+\\ &&+\frac{\alpha}{4}\|u^k_\tau\|^2\leq \frac{1}{\alpha}\|f^k_\tau\|^2_{V^*}+\frac{1}{2\tau}\|u^{k-1}_\tau\|_H^2+C_1.\nonumber\end{aligned}$$ Summing above inequalities for $k=1,\ldots,n$, where $1\leq n\leq N$, we have $$\begin{aligned} &&\|u^n_\tau\|_H^2+\sum_{k=1}^n\|u^k_\tau-u^{k-1}_\tau\|_H^2+\frac{\alpha\tau}{2}\sum_{k=1}^n\|u_\tau^k\|^2\leq\\ &&\leq 2TC_1 + 2 \tau (\beta + c\|p\|^2_{\C{L}(H,U)}) \sum_{k=1}^n\|u^k_\tau\|_H^2+\frac{2\|\bar{f}_\tau\|^2_{\C{V}^*}}{\alpha}+\|u^0_\tau\|_H^2.\nonumber\end{aligned}$$ Now if $\tau < 1 / (4(\beta + c\|p\|^2_{\C{L}(H,U)}))$, by a discrete Gronwall inequality (see e.g. [@Roubicek2005] (1.68)-(1.69)), we have (\[eq:bound1\])-(\[eq:bound3\]). In the case $B)$, for $\delta = \varepsilon = (\alpha-c\|\iota\|^2_{\C{L}(U,V)})/2$ we get $$\begin{aligned} &&\left(\frac{1}{2\tau}-\beta\right)\|u^k_\tau\|^2_H+\frac{1}{2\tau}\|u^{k}_\tau-u^{k-1}_\tau\|_H^2+\frac{\alpha-c\|\iota\|^2_{\C{L}(U,V)}}{4}\|u^k_\tau\|^2\leq\\ &&\leq C_4\|f^k_\tau\|^2_{V^*}+\frac{1}{2\tau}\|u^{k-1}_\tau\|_H^2+C_5,\nonumber\end{aligned}$$ where $C_4 = \frac{1}{\alpha-c\|\iota\|^2_{\C{L}(U,V)}}$ and $C_5>0$ depends on $\alpha, c, \|\iota\|^2_{\C{L}(U,V)}$. In analogy to the previous case we get (\[eq:bound1\])-(\[eq:bound3\]) for $\tau < 1/\beta$. Bounds in the case $C)$ are obtained in an analogous way. Convergence of the Rothe method {#sec:limit} =============================== We define piecewise linear and piecewise constant interpolants $u_\tau \in C([0,T];V)$ and $\bar{u}_\tau\in L^\infty(0,T; V)$ by the formulae $$\begin{aligned} &&u_\tau(t) = \left(\frac{t}{\tau}-k+1\right)u^k_\tau + \left(k-\frac{t}{\tau}\right)u^{k-1}_\tau\ \mbox{for}\ t\in[(k-1)\tau,k\tau],\nonumber\\ &&\bar{u}_\tau(t) = u^k_\tau\ \mbox{for a.e.}\ t\in((k-1)\tau,k\tau].\nonumber\end{aligned}$$ where $k=1,\ldots,T/\tau$. The sequences $\{u_{\tau_n}\}_{n=1}^\infty$ and $\{\bar{u}_{\tau_n}\}_{n=1}^\infty$ are known as the Rothe sequences. Observe, that $u_\tau$ has a distributional derivative $u'_\tau\in L^\infty(0,T;V)$ given by $u'_\tau(t) = \frac{u^k_\tau-u^{k-1}_\tau}{\tau}$ for almost every $t\in ((k-1)\tau,k\tau)$. So, since $u^k_\tau$ solves the Rothe problem, we have for almost every $t\in (0,T)$ $$(u'_\tau(t),v)_H + \langle A \bar{u}_\tau(t), v\rangle + \langle \xi_\tau(t), \iota v\rangle_{U^*\times U} = \langle\bar{f}_\tau(t),v\rangle\ \mbox{for}\ v\in V,$$ with $u_\tau(0)=u_{0\tau}$ and $\xi_\tau(t)=\xi^k_\tau\in\partial J(\iota u^k_\tau)=\partial J(\iota \bar{u}_\tau(t))$ for $t\in ((k-1)\tau,k\tau]$. Defining the Nemytskii operator $\C{A}:\C{V}\to \C{V}^*$ as $(\C{A}v)(t)=A(v(t))$, we have $$(u'_\tau,v)_{\C{H}} + \langle \C{A} \bar{u}_\tau, v\rangle_{\C{V}^*\times\C{V}} + \langle \xi_{\tau}, \bar{\iota} v\rangle_{\C{U}^*\times \C{U}}=\langle\bar{f}_\tau,v\rangle_{\C{V}^*\times\C{V}}\ \mbox{for}\ v\in \C{V}.\label{eq:nemytskii}$$ \[lem:boundsnemytskii\] Under assumptions $H(A), H(J), H_0, H(U)$ and $H_{aux}$ there exists $\tau_0>0$ such that for all $\tau\in (0,\tau_0)$, the piecewise constant and piecewise linear interpolants built on the solutions of the Rothe problem satisfy $$\begin{aligned} &&\|\bar{u}_{\tau}\|_{\C{V}}\leq \mbox{const},\label{eq:bigbound1}\\ &&\|\bar{u}_{\tau}\|_{L^\infty(0,T;H)}\leq \mbox{const},\\ &&\|u_{\tau}\|_{C(0,T;H)}\leq \mbox{const},\label{eq:bigbound4}\\ &&\|u_\tau\|_{\C{V}}\leq \mbox{const},\label{eq:bigbound5}\\ &&\|u'_\tau\|_{\C{V}^*}\leq \mbox{const}\label{eq:bigbound6},\\ &&\|\C{A}\bar{u}_\tau\|_{\C{V}^*}\leq \mbox{const}\label{eq:bigbound7},\\ &&\|\xi_\tau\|_{\C{U}^*}\leq \mbox{const},\label{eq:bigbound8}\\ &&\|\bar{u}_{\tau}\|_{BV^2(0,T;V^*)}\leq \mbox{const}\label{eq:bigboundvar}.\end{aligned}$$ with the constants independent on $\tau$. Estimates (\[eq:bigbound1\])-(\[eq:bigbound4\]) follow directly from Lemma \[lem:bounds\], since $\|\bar{u}_{\tau}\|^2_{\C{V}}=\tau \sum_{i=1}^{N}\|u^k_\tau\|^2$, $\|\bar{u}_{\tau}\|_{L^\infty(0,T;H)}=\max_{k=1,\ldots,N}\|u^k_\tau\|_H$ and $\|u_{\tau}\|_{C(0,T;H)}\leq \max_{k=0,\ldots,N}\|u^k_\tau\|_H$. The simple calculation shows us that $\|u_\tau\|_{\C{V}}^2\leq \tau\sum_{k=0}^N\|u^k_\tau\|_V^2$. This, together with the fact, that $\|u^0_\tau\|\leq C/\sqrt{\tau}$, by Lemma \[lem:bounds\] gives (\[eq:bigbound5\]). To prove (\[eq:bigbound6\]) let us consider the inclusion (\[eq:nemytskii\]). We have $$\begin{aligned} &&\|u'_\tau\|_{\C{V}^*}=\sup_{\|v\|_{\C{V}}\leq 1}\left|(u'_\tau,v)_{\C{H}}\right| = \nonumber\\ &&=\sup_{\|v\|_{\C{V}}\leq 1}\left|\langle\bar{f}_\tau,v\rangle_{\C{V}^*\times\C{V}} - \langle \C{A} \bar{u}_\tau, v\rangle_{\C{V}^*\times\C{V}} - \int_0^T \langle \xi_\tau(t), \iota v(t)\rangle_{U^*\times U}\ dt\right|\leq \nonumber\\ &&\leq \|\bar{f}_\tau\|_{\C{V}^*}+\sqrt{\int_0^T\|A \bar{u}_\tau(t)\|_{V^*}^2\ dt}+\|\iota\|_{\C{L}(V;U)}\sqrt{\int_0^T\|\xi_\tau(t)\|^2_{U^*}\ dt}\leq \nonumber \\&&\leq\|\bar{f}_\tau\|_{\C{V}^*}+\sqrt{2a^2T+2b^2\|\bar{u}_{\tau}\|^2_{\C{V}}}+ \|\iota\|_{\C{L}(V;U)}\sqrt{2c^2T+2c^2\|\iota\|_{\C{L}(V;U)}^2\|\bar{u}_{\tau}\|^2_{\C{V}}}.\label{eq:derivative}\end{aligned}$$ Desired bound is obtained by (\[eq:bigbound1\]). Estimates that appear in (\[eq:derivative\]) prove also (\[eq:bigbound7\]) and (\[eq:bigbound8\]). It remains to prove (\[eq:bigboundvar\]). Let us assume that the seminorm $BV^2(0,T;V^*)$ of piecewise constant function $\bar{u}_\tau$ is realized by some division $0=t_0<t_1<\ldots< t_k=T$. Each $t_j$ is in some interval $((m_j-1)\tau, m_j\tau]$, so $\bar{u}_\tau(t_j) = u^{m_j}_\tau$ with $m_0=0$ and $m_k=N$ and $m_{i+1}>m_i$ for $i=1,\ldots,N-1$. Thus $$\|\bar{u}_{\tau}\|^2_{BV^2(0,T;V^*)}=\sum_{j=1}^{k}\|u^{m_j}_\tau-u^{m_{j-1}}_\tau\|^2_{V^*}.$$ We use the inequality $$\|u^{m_j}_\tau-u^{m_{j-1}}_\tau\|^2_{V^*} \leq (m_j-m_{j-1}) \sum_{i=m_{j-1}+1}^{m_j} \|u^{i}_\tau-u^{i-1}_\tau\|^2_{V^*}.$$ Thus $$\begin{aligned} &&\|\bar{u}_{\tau}\|^2_{BV^2(0,T;V^*)}\leq \sum_{j=1}^{k} \left((m_j-m_{j-1}) \sum_{i=m_{j-1}+1}^{m_j} \|u^{i}_\tau-u^{i-1}_\tau\|^2_{V^*}\right)\leq\nonumber\\ && \leq \left( \sum_{j=1}^{k} (m_j-m_{j-1}-1) \right)\sum_{i=1}^N \|u^{i}_\tau-u^{i-1}_\tau\|^2_{V^*} \leq N \tau\tau \sum_{i=1}^N \left\|\frac{u^{i}_\tau-u^{i-1}_\tau}{\tau}\right\|^2_{V^*} =\nonumber \\ && = T \int_0^T\|u_\tau'(t)\|_{V^*}^2\ dt.\nonumber\end{aligned}$$ The last term is bounded by (\[eq:bigbound6\]), which ends the proof. \[thm:existence\] Under assumptions $H(A), H(J), H_0, H(U)$ and $H_{aux}$ the problem (\[eq:inclusionmain\]) has a solution $u$. Furthermore if $\bar{u}_\tau$ and $u_\tau$ are piecewise constant and piecewise linear interpolants built on the solutions of the Rothe problem, then, for a subsequence, $u_\tau\to u$ weakly in $\C{W}$ and weakly$*$ in $L^\infty(0,T;H)$ and $\bar{u}_\tau\to u$ weakly in $\C{V}$ and weakly$*$ in $L^\infty(0,T;H)$. From the bounds obtained in Lemma \[lem:boundsnemytskii\], possibly for a subsequence, we get $$\begin{aligned} &&\bar{u}_{\tau}\to u\ \mbox{weakly in}\ \C{V}\ \mbox{and weakly$*$ in}\ L^\infty(0,T;H),\label{eq:convuvstar}\\ &&u_{\tau}\to u_1\ \mbox{weakly in}\ \C{V} \ \mbox{and weakly$*$ in}\ L^\infty(0,T;H),\\ &&u'_{\tau}\to u_2\ \mbox{weakly in}\ \C{V}^*,\label{eq:convuprimevstar}\\ &&\C{A}\bar{u}_{\tau}\to \eta \ \mbox{weakly in}\ \C{V}^*,\\ &&\xi_\tau\to \xi \ \mbox{weakly in}\ \C{U}^*.\label{eq:convxiustar}\end{aligned}$$ A standard argument shows that $u_1'=u_2$. To show that $u=u_1$ we observe that $$\|\bar{u}_{\tau}-u_\tau\|_{\C{V}^*}^2=\sum_{k=1}^N\int_{(k-1)\tau}^{k\tau}\left(k\tau-t\right)^2\left\|\frac{u^k_\tau-u^{k-1}_\tau}{\tau}\right\|_{V^*}^2\ dt=\frac{\tau^2}{3}\|u'_\tau\|_{\C{V^*}}^2,$$ which means that $\bar{u}_\tau-u_{\tau} \to 0$ strongly in $\C{V}^*$ as $\tau\to 0$, and, in consequence $u=u_1$. It follows that $u_\tau\to u$ strongly in $L^2(0,T;H)$ and weakly in $C([0,T];H)$. This also implies that $u_{0\tau}=u_\tau(0)\to u(0)$ weakly in $H$, so $u(0)=u_0$. A passage to the limit in (\[eq:nemytskii\]) gives $$u'+\eta+\bar{\iota}^*\xi=f.$$ We observe that, by $H(U)$, we have $\bar{\iota}\bar{u}_{\tau}\to \bar{\iota}u$ strongly in $\C{U}$ and, furthermore, for a subsequence $\iota \bar{u}_{\tau}(t)\to \iota u(t)$ strongly in $U$ for a.e. $t\in (0,T)$. Moreover $\xi_\tau\to\xi$ weakly in $L^1(0,T;U^*)$. Since $\partial J:U\to 2^{U^*}$ has nonempty, closed and convex values and is upper semicontinuous from $U$ furnished with strong topology into $U^*$ furnished with weak topology (see [@Denkowski2003a], Proposition 5.6.10), by the Convergence Theorem of Aubin and Cellina (see [@Aubin1984], Theorem 1, Section 1.4), we deduce that $\xi(t)\in \partial J(\iota u(t))$ for a.e. $t\in (0,T)$. In order to show that $u$ satisfies the inclusion (\[eq:inclusionmain\]), it suffices to prove that $\eta=\C{A}u$. To this end, let us estimate $$\begin{aligned} &&\limsup_{\tau\to 0}\langle\C{A}\bar{u}_\tau,\bar{u}_\tau-u\rangle_{\C{V}^*\times\C{V}}\leq \limsup_{\tau\to 0}\langle\bar{f}_\tau,\bar{u}_\tau-u\rangle_{\C{V}^*\times\C{V}} -\\ &&-\liminf_{\tau\to 0}\langle u_\tau ' ,\bar{u}_\tau-u\rangle_{\C{V}^*\times\C{V}}-\liminf_{\tau\to 0}\langle \xi_\tau, \bar{\iota}(\bar{u}_\tau-u)\rangle_{\C{U}^*\times\C{U}}.\nonumber\end{aligned}$$ Since $\bar{f}_\tau\to f$ strongly in $\C{V}^*$, by (\[eq:convuvstar\]), we get $\lim_{\tau\to 0}\langle\bar{f}_\tau,\bar{u}_\tau-u\rangle_{\C{V}^*\times\C{V}}=0.$ Moreover, since $\bar{\iota}\bar{u}_\tau\to \bar{\iota}u$ strongly in $\C{U}$, by (\[eq:convxiustar\]), we have $\lim_{\tau\to 0}\langle \xi_\tau, \bar{\iota}(\bar{u}_\tau-u)\rangle_{\C{U}^*\times\C{U}}=0$. Now we observe that $$\begin{aligned} &&\langle u_\tau ' ,\bar{u}_\tau-u\rangle_{\C{V}^*\times\C{V}} = \langle u_\tau ' ,\bar{u}_\tau-u_\tau\rangle_{\C{V}^*\times\C{V}} +\nonumber\\ &&+\frac{1}{2}(\|u_\tau(T)-u(T)\|_H^2-\|u_\tau(0)-u(0)\|_H^2) + \langle u', u_\tau - u\rangle_{\C{V}^*\times\C{V}},\nonumber\end{aligned}$$ so, noting that $\langle u_\tau ' ,\bar{u}_\tau-u_\tau\rangle_{\C{V}^*\times\C{V}} \geq 0$, we obtain $$\liminf_{\tau\to 0}\langle u_\tau ' ,\bar{u}_\tau-u\rangle_{\C{V}^*\times\C{V}} \geq 0.$$ Thus we have $$\limsup_{\tau\to 0}\langle\C{A}\bar{u}_\tau,\bar{u}_\tau-u\rangle_{\C{V}^*\times\C{V}}\leq 0.$$ We are in a position to apply Lemma \[lem:nemytskii\] which gives $\eta=\C{A}u$. Thus $u$ solves (\[eq:inclusionmain\]). **Remark 5.** Note that we have also proved that any cluster point of $u_\tau$ and $\bar{u}_\tau$, in the sense (\[eq:convuvstar\])-(\[eq:convuprimevstar\]), solves the problem (\[eq:inclusionmain\]). It is not known, however, whether there are solutions which are not limits of the interpolants built on the solutions of Rothe problem. Uniqueness and strong convergence {#sec:uniq} ================================= In this section we assume the strong monotonicity type relation for $A$ and relaxed monotonicity on $J$. - : assumptions $H(A)$ hold and $A$ satisfies the monotonicity type relation $\langle Au-Av,u-v\rangle\geq m_1 \|u-v\|^2-m_2\|u-v\|_H^2$ for every $u, v\in V$ with $m_1 \geq 0$ and $m_2 > 0$, - : assumptions $H(A)$ hold and the Nemytskii mapping $\C{A}:\C{V}\to\C{V}^*$ is of class $(S_+)$ with respect to the space $M^{2,2}(0,T;V,V^*)$, that is if $u_n\to u$ weakly in $\C{V}$ and $u_n$ is bounded in $M^{2,2}(0,T;V,V^*)$ then $\limsup_{n\to\infty}\langle\C{A} u_n,u_n-u\rangle_{\C{V}^*\times\C{V}}\leq 0$ implies that $u_n\to u$ strongly in $\C{V}$, - : assumptions $H(J)$ hold and $J$ satisfies the relaxed monotonicity condition $\langle \xi - \eta, u - v\rangle_{U\times U^*} \geq - m_3\|u-v\|_U^2$ for every $u, v\in V$ and $\xi\in \partial J(u), \eta\in \partial J(v)$ with $m_3 > 0$, - : either $H_{aux}$ A) holds or $m_1 \geq m_3 \|\iota\|_{\C{L}(V;U)}^2$. **Remark 6.** The assumption $H(A)_1$ for the divergence differential Leray-Lions operator is guaranteed by appropriate Leray-Lions type conditions. For $H(A)_2$ to hold it suffices that the operator $A$ is of class $(S_+)$, by an argument analogous to Theorem 2(c) in [@Berkovits1996]. **Remark 7.** The relaxed monotonicity condition $H(J)_1$ (which is associated with the semiconvexity of the functional $J$) was already used to prove the uniqueness of solutions to the first order evolution parabolic hemivariational inequalities in [@Liu2005] and second order ones in [@Migorski2005]. **Remark 8.** Note that $H(A)_1$ allows the case $m_1=0$, but if the inequality in $H_{const}$ holds, then it must be $m_1>0$. \[thm:unique\] Under assumptions $H(A)_1$, $H(J)_1$, $H_0$, $H(U)$, $H_{aux}$ and $H_{const}$, the solution to the problem (\[eq:inclusionmain\]) is unique. Assume that $u_1, u_2$ are two distinct solutions to the problem (\[eq:inclusionmain\]). We have, for $v\in V$ and a.e. $t\in [0,T]$ $$\langle (u_1-u_2)'(t), v\rangle + \langle Au_1(t) - Au_2(t), v \rangle + \langle\xi(t)-\eta(t),\iota v \rangle_{U\times U^*} = 0,$$ where $\xi(t)\in \partial J(\iota u_1(t))$ and $\eta(t)\in \partial J(\iota u_2(t))$ for a.e. $t\in (0,T)$. Taking $v = u_1(t)-u_2(t)$, we obtain $$\begin{aligned} &&\frac{1}{2}\frac{d}{dt}\|u_1(t)-u_2(t)\|^2_H + \langle Au_1(t) - Au_2(t), u_1(t)- u_2(t)\rangle +\\ &&+\langle\xi(t)-\eta(t),\iota u_1(t)-\iota u_2(t)) \rangle_{U\times U^*} = 0.\nonumber\end{aligned}$$ Application of $H(A)_1$ and $H(J)_1$ gives for a.e. $t\in (0,T)$ $$\begin{aligned} &&\frac{1}{2}\frac{d}{dt}\|u_1(t)-u_2(t)\|^2_H + m_1 \|u_1(t) - u_2(t)\|^2 -\\ &&-m_2\|u_1(t)-u_2(t)\|_H^2-m_3 \|\iota(u_1(t)-u_2(t))\|_U^2\leq 0.\nonumber\end{aligned}$$ By $H_{const}$ we have either $$\frac{1}{2}\frac{d}{dt}\|u_1(t)-u_2(t)\|^2_H \leq m_2\|u_1(t)-u_2(t)\|_H^2,$$ or, in the case of $H_{aux}$ A), $$\frac{1}{2}\frac{d}{dt}\|u_1(t)-u_2(t)\|^2_H \leq (m_2+\|p\|_{\C{L}(H;U)})\|u_1(t)-u_2(t)\|_H^2,$$ which, by the Gronwall lemma, gives the thesis. **Remark 9.** Under assumptions of Theorem \[thm:unique\], the convergences in Theorem \[thm:existence\] hold for the whole sequences $u_\tau$ and $\bar{u}_\tau$. \[thm:converge\] Let assumptions $H(A)_1$, $H(J)$, $H_0$, $H(U)$, $H_{aux}$ hold and the subsequences $u_\tau, \bar{u}_\tau$ converge in the sense (\[eq:convuvstar\])-(\[eq:convuprimevstar\]). Then $u_\tau\to u$ strongly in $C([0,T];H)$. If instead of $H(A)_1$ we assume $H(A)_2$, then $\bar{u}_\tau\to u$ strongly in $\C{V}$. Let $u_\tau$ and $\bar{u}_\tau$ be interpolants built on the solutions of the Rothe problem and let $u$ be the solution to (\[eq:inclusionmain\]) obtained in Theorem [\[thm:existence\]]{}. For $v\in V$ and a.e. $t\in (0,T)$ we get $$\langle u_\tau '(t) - u'(t),v\rangle + \langle A\bar{u}_\tau (t) - Au(t),v\rangle+\langle\xi_\tau(t)-\eta(t),\iota v\rangle_{U^*\times U}=\langle \bar{f}_\tau(t)-f(t),v\rangle,$$ where $\xi_\tau(t) \in \partial J(\iota \bar{u}_\tau(t))$ and $\eta(t) \in \partial J(\iota u(t))$ for a.e. $t\in (0,T)$. Choosing $v=\bar{u}_\tau(t)-u(t)$, we get $$\begin{aligned} &&\langle u_\tau '(t) - u'(t),\bar{u}_\tau (t) -u_\tau (t)\rangle + \frac{1}{2}\frac{d}{dt}\|u_\tau (t)- u(t)\|_H^2 +\nonumber\\ && +\langle A\bar{u}_\tau (t) - Au(t),\bar{u}_\tau (t) - u(t)\rangle\leq \nonumber\\ &&\leq\langle \bar{f}_\tau(t)-f(t),\bar{u}_\tau(t)-u(t)\rangle + \|\iota \bar{u}_\tau(t)-\iota u(t)\|_U\|\xi_\tau(t)-\eta(t)\|_{U^*}.\nonumber\end{aligned}$$ Since $\langle u_\tau '(t), \bar{u}_\tau (t) -u_\tau (t)\rangle = \|\frac{d}{dt}u_\tau '(t)\|_H^2(k\tau-t) \geq 0$ for any $t\in ((k-1)\tau, k\tau)$ for a.e. $t\in (0,T)$ we have $$\begin{aligned} \label{eqn:integration} &&\frac{1}{2}\frac{d}{dt}\|u_\tau (t)- u(t)\|_H^2 + \langle A\bar{u}_\tau (t) - Au(t),\bar{u}_\tau (t) - u(t)\rangle \leq \nonumber\\ &&\leq \|\iota \bar{u}_\tau(t)-\iota u(t)\|_U\|\xi_\tau(t)-\eta(t)\|_{U^*}+\nonumber\\ &&+ \langle \bar{f}_\tau(t)-f(t),\bar{u}_\tau(t)-u(t)\rangle+\langle u'(t),\bar{u}_\tau (t) -u_\tau (t)\rangle.\end{aligned}$$ Using $H(A)_1$ and integrating the last inequality, for $t\in[0,T]$, we get $$\begin{aligned} &&\frac{1}{2}\|u_\tau (t)- u(t)\|_H^2 \leq m_2\int_0^t\|u_\tau (s)- u(s)\|_{H}^2 \ dt + \nonumber\\ &&+\sqrt{2}c\|\bar{\iota} (\bar{u}_\tau-u)\|_{\C{U}}(2T+\|\bar{\iota}\bar{u}_\tau\|_{\C{U}}+\|\bar{\iota} u\|_{\C{U}})+\nonumber\\ &&+ \|\bar{f}_\tau-f\|_{\C{V}^*}(\|\bar{u}_\tau\|_{\C{V}}+\|u\|_{\C{V}})+\langle u',\bar{u}_\tau -u_\tau\rangle_{\C{V}^*\times \C{V}}+\frac{1}{2}\|u_{0 \tau}- u_0\|_H^2.\nonumber\end{aligned}$$ The Gronwall lemma gives the strong convergence of $u_\tau$ to $u$ in $C([0,T];H)$. In order to obtain the strong convergence in $\C{V}$, let us integrate (\[eqn:integration\]) over $(0,T)$. We have $$\begin{aligned} &&\frac{1}{2}\|u_\tau (T)- u(T)\|_H^2 + \langle\C{A}\bar{u}_\tau-\C{A}u,\bar{u}_\tau-u\rangle_{\C{V}^*\times\C{V}}\leq \nonumber\\ &&\leq \sqrt{2}c\|\bar{\iota} (\bar{u}_\tau-u)\|_{\C{U}}(2T+\|\bar{\iota}\bar{u}_\tau\|_{\C{U}}+\|\bar{\iota} u\|_{\C{U}})+\nonumber\\ &&+ \|\bar{f}_\tau-f\|_{\C{V}^*}(\|\bar{u}_\tau\|_{\C{V}}+\|u\|_{\C{V}})+\langle u',\bar{u}_\tau -u_\tau\rangle_{\C{V}^*\times \C{V}}+\frac{1}{2}\|u_{0 \tau}- u_0\|_H^2.\nonumber\end{aligned}$$ Passing to the limit, we get $$\limsup_{\tau\to 0}\langle \C{A}\bar{u}_\tau-\C{A}u,\bar{u}_\tau-u\rangle_{\C{V}^*\times\C{V}} \leq 0.$$ The thesis is implied by $H(A)_2$. **Remark 10.** If, in addition to assumptions of the Theorem \[thm:converge\], also $H(J)_1$ and $H_{const}$ hold, then, by Theorem \[thm:unique\], the whole sequences $u_\tau$ and $\bar{u}_\tau$ converge strongly in $C([0,T];H)$ and $\C{V}$ respectively. Examples {#sec:settings} ======== In this section we provide examples of that problem setup which are particular case of the general problem considered previously. Moreover, we present a simple numerical example. *Problem settings* We assume that $\Omega\subset \mathbb{R}^n$ is an open and bounded domain with smooth boundary. The space $V$ is either $H^1(\Omega;\mathbb{R}^m)$ with $m\in\mathbb{N}$ (possibly, but not necessarily, $m=n$) or its closed subspace (which originates from homogeneous Dirichlet boundary condition on $\Gamma_D\subset \partial \Omega$). Furthermore let $H=L^2(\Omega;\mathbb{R}^m)$. Then the embedding $i:V\to H$ is continuous and compact. We consider two examples. - Multivalued term is defined on $\Omega$. We specify $\Lambda \subset \Omega$ to be an open subset on nonzero measure and fix $d\in \mathbb{N}$. Furthermore we assume that $\C{M}\in L^\infty(\Lambda;\C{L}(\mathbb{R}^m;\mathbb{R}^d))$. Now $U=L^2(\Lambda;\mathbb{R}^d)$. The mapping $\iota$ is defined by $(\iota v)(x)=\C{M}(x)((iv)|_{\Lambda}(x))$. We observe that $\iota:V\to U$ is linear, continuous and compact. By Proposition \[prop:compactembedding\], the embedding $M^{2,2}(0,T;V,V^*)\subset L^2(0,T;H)$ is compact, which implies $H(U)$. Defining $p:H\to U$ by $(pv)(x)=\C{M}(x)(v|_{\Lambda}(x))$, we see that $A)$ of $H_{aux}$ is satisfied. The solution exists under assumptions $H(A), H(J)$ and $H_0$ (Theorem \[thm:existence\]). Additional assumptions $H(A)_1$ and $H(J)_1$ imply uniqueness of solution by Theorem \[thm:unique\] and strong convergence of the Rothe sequence $u_\tau$ in $C([0,T];H)$. Furthermore, if $H(A)_2$ holds, then the sequence $\bar{u}_\tau$ converges strongly in $\C{V}$. As the special case we can consider $\Lambda=\Omega$, $m=n=d$ and $\C{M}(x)\equiv I$ (identity) for all $x\in \Omega$. Then we recover $U=H$, which gives existence results in spirit of [@Migorski2001]. - Multivalued term is defined on the boundary of $\Omega$. We specify $\Gamma_C \subset \partial \Omega$ disjoint with $\Gamma_D$. We take $Z=H^{\delta}(\Omega;\mathbb{R}^m)$ with $\delta\in[\frac{1}{2},1)$. The continuous and compact embedding $V\to Z$ is denoted by $\bar{i}$ and the trace operator is given by $\bar{\gamma}:Z\to L^2(\Gamma_C;\mathbb{R}^m)$. Furthermore let $d\in \mathbb{N}$ and $\C{M}\in L^\infty(\Gamma_C;\C{L}(\mathbb{R}^m;\mathbb{R}^d))$. Now $U=L^2(\Gamma_C;\mathbb{R}^d)$. The mapping $\iota$ is defined by $(\iota v)(x)=\C{M}(x)((\bar{\gamma}\bar{i}v)(x))$. The mapping $\iota:V\to U$ is linear, continuous and compact. The spaces $V\subset Z\subset V^*$ satisfy the assumptions of Proposition \[prop:compactembedding\], so $M^{2,2}(0,T;V,V^*)$ is embedded in $L^2(0,T;Z)$ compactly. Therefore the assumption $H(U)$ is satisfied. Since claim $A)$ of $H_{aux}$ does not hold in this case, in order to obtain the existence of solutions (Theorem \[thm:existence\]) we need to assume $H(A), H(J), H_0$ and either $B)$ or $C)$ of $H_{aux}$. Furthermore, if $H(A)_1$ and $H(J)_1$ hold, then a subsequence of the Rothe sequence $u_\tau$ converges strongly in $C([0,T];H)$ (Theorem \[thm:converge\]). If moreover $H(A)_2$ holds, then we also have the strong convergence of $\bar{u}_\tau$ in $\C{V}$. If furthermore the relation between $m_1$ and $m_3$ given by $H_{const}$ holds, then the solution is unique (Theorem \[thm:unique\]) and the whole Rothe sequences $u_\tau$ and $\bar{u}_\tau$ converge strongly in $C([0,T];H)$ and $\C{V}$ respectively. In the case $m=d=1$ and $\C{M}(x)\equiv I$ we recover the results of [@Migorski2004]. If $m=n>1$ and $\nu$ is the unit outer normal versor on the boundary $\partial \Omega$, then two special cases are $d=1$, $\C{M}(x)(a)=\nu(x)\cdot a$ and $d=m$, $\C{M}(x)(a)=a-(\nu(x)\cdot a)\nu(x)$. We recover the cases of the boundary conditions given in normal and tangent directions, respectively. *Numerical example*. Let us take $\Omega=(0,1)$. The problem under consideration will be $$\begin{aligned} &&u_t(x,t)=u_{xx}(x,t)\ \ \mbox{for}\ \ (x,t)\in \Omega\times(0,T),\\ &&u(0,t)=0\ \ \mbox{for}\ \ t\in(0,T),\\ &&u_x(1,t)\in -\partial j(u(1,t))\ \ \mbox{for}\ \ t\in(0,T),\label{eq:boundex}\\ &&u(x,0)=u_0(x)\ \ \mbox{for}\ \ x\in(0,1).\end{aligned}$$ We set $V=\{v\in H^1(0,1): v(0)=0\}$ and $H=L^2(0,1)$. Taking $t_k=k\Delta t$, $u(x,t_k):=u^k(x)$ and $v\in V$, the Rothe problem has the following form $$\begin{aligned} &&\int_0^1\frac{u^{k+1}(x)-u^k(x)}{\Delta t}v(x)\ dx+\int_0^1u^{k+1}_x(x)v_x(x)\ dx + \partial j(u^{k+1}(1))v(1)\ni 0,\nonumber\\ &&u^0(x)=u_0(x).\end{aligned}$$ The problem in each time step will be solved by the Galerkin scheme. Let $V_n$ be a subspace of $V$ consisting of piecewise linear functions constructed on a uniform mesh $x_0=0, \ldots,x_i=i\Delta x,\ldots, x_n=n\Delta x=1$ in $(0,1)$ such that $\mbox{dim} V_n = n$. Let furthermore $u^k(x)$ be approximated by $\sum_{i=1}^{n}\alpha_i^k v_i$, where $\{v_i\}_{i=1}^{n}$ forms the base of $V_n$ given by the duality condition $v_j(x_i)=\delta_{ij}$. We assume that $\alpha_n^k$ is the value of the solution at the last mesh point ($x=1$). The values $\alpha_i^{k+1}$ satisfy, for $j=1,\ldots, n$ $$\begin{aligned} &&\sum_{i=1}^n\alpha_i^{k+1}\left(\frac{1}{\Delta t}\int_0^1v_i(x)v_j(x)\ dx+\int_0^1 v_i'(x)v_j'(x)\ dx\right)-\nonumber\\ &&-\sum_{i=1}^n\alpha_i^k\frac{1}{\Delta t}\int_0^1v_i(x)v_j(x)\ dx + \partial j(\alpha_n^{k+1}) v_j(1)\ni 0.\nonumber\end{aligned}$$ Calculating the integrals and denoting $d=\frac{\Delta t}{\Delta x^2}$, for $j=2,\ldots,n-1$, we obtain $$\label{eqn:j_1_to_k_m_1} \alpha_{j-1}^{k+1}\left(\frac{1}{6}-d\right)+\alpha_j^{k+1}\left(\frac{2}{3}+2d\right)+ \alpha_{j+1}^{k+1}\left(\frac{1}{6}-d\right)=\alpha_{j-1}^k\frac{1}{6}+\alpha_j^k\frac{2}{3}+ \alpha_{j+1}^k\frac{1}{6},$$ and for $j=n$, we have $$\label{eqn:j_k} \alpha_{n-1}^{k+1}\left(\frac{1}{6}-d\right)+\alpha_n^{k+1}\left(\frac{1}{3}+d\right)+ \frac{\Delta t}{\Delta x}\partial j(\alpha_n^{k+1})\ni\alpha_{n-1}^k\frac{1}{6}+\alpha_n^k\frac{1}{3}.$$ Finally for the left, Dirichlet, boundary point we have $$\label{eqn:j_1} \alpha_1^{k+1}\left(\frac{2}{3}+2d\right)+ \alpha_{2}^{k+1}\left(\frac{1}{6}-d\right)=\alpha_1^k\frac{2}{3}+ \alpha_{2}^k\frac{1}{6}.$$ ![[]{data-label="fig:examples"}](wykres.png){width="\textwidth"} We consider two examples of the locally Lipschitz functionals $j$: $$j_1(r)=\begin{cases} 0\ \mbox{for}\ r\leq 0 \\ \frac{r^2}{2}\ \mbox{for}\ r\in(0,1)\\ \frac{1}{2}\ \mbox{for}\ r\geq 1\end{cases}\ \ j_2(r)=\begin{cases} 0\ \mbox{for}\ r\leq 1 \\ \frac{1-(r-2)^2}{2}\ \mbox{for}\ r\in(1,2)\\ \frac{1}{2}\ \mbox{for}\ r\geq 2.\end{cases}$$ Their subdifferentials in the sense of Clarke are given by $$\label{eqn:subdiff_egzamples} \partial j_1(r)=\begin{cases} 0\ \mbox{for}\ r\leq 0\ \mbox{or}\ r>1 \\ r\ \mbox{for}\ r\in(0,1)\\ [0,1]\ \mbox{for}\ r=1\end{cases}\ \ \partial j_2(r)=\begin{cases} 0\ \mbox{for}\ r<1\ \mbox{or}\ r\geq 2 \\ 2-r\ \mbox{for}\ r\in(1,2)\\ [0,1]\ \mbox{for}\ r=1.\end{cases}$$ ![[]{data-label="fig:j2"}](j_2.png){width="\textwidth"} ![[]{data-label="fig:j1"}](j_1.png){width="\textwidth"} The graphs of $\partial j_1$ and $\partial j_2$ are presented in Figure \[fig:examples\]. Both potentials satisfy $H(J)$. The potential $j_1$ does not satisfy $H(J)_1$ since its subdifferential has a nonmonotone jump. The potential $j_2$ satisfies $H(J)_1$ since its subdifferential has a monotone jump and nonmonotonicity is Lipschitz. In the case of $j_1$ the question of multiplicity of solutions remains an open problem (however the numerical simulation below show that it is more likely that there are multiple solutions) and in the case of $j_2$, a single solution is expected (at least as long as the inequality in $H_{const}$ holds). In both cases we take $u_0(x)\equiv 2$. The following scheme is used to find solutions of (\[eqn:j\_1\_to\_k\_m\_1\])-(\[eqn:j\_1\]). In every time step at most three solutions can be found: - Assume that the element of $\partial j(\alpha_{n}^{k+1})$ for which (\[eqn:j\_k\]) holds is equal to $0$ i.e. we fall on the horizontal line in the graph of $\partial j$. Solve the system of $n$ equations (\[eqn:j\_1\_to\_k\_m\_1\])-(\[eqn:j\_1\]) and verify whether obtained $\alpha_n^{k+1}$ falls in the corresponding interval. - Assume that the element of $\partial j(\alpha_{n}^{k+1})$ for which (\[eqn:j\_k\]) holds is on the oblique line in the graph of $\partial j$. Solve the system of $n$ equations (\[eqn:j\_1\_to\_k\_m\_1\])-(\[eqn:j\_1\]) and verify whether obtained $\alpha_n^{k+1}$ is in the corresponding interval. - Assume that we fall on the vertical line in the graph of $\partial j$, i.e. $\alpha_n^{k+1}=1$. Solve the system of $n-1$ equations (\[eqn:j\_1\_to\_k\_m\_1\]) and (\[eqn:j\_1\]). Then verify if $\partial j(\alpha_n^{k+1})$ calculated from (\[eqn:j\_k\]) falls in the interval $[0,1]$. The simulations were run for $\Delta t = 0.01$ and $\Delta x = 0.01$. For the case of $j_2$ only one numerical solution was obtained (i.e. in every time step only one of above three cases occurred). The result is presented in Figure \[fig:j2\]. For the case of $j_1$ many solutions were obtained (i.e. there were time steps in which more then one of above three cases occurred). Figure \[fig:j1\] shows two solutions with respectively maximum and minimum value of $\alpha_n^{k}$ chosen in each time step in which multiple solutions were found. [99]{} N.U. Ahmed, K.L. Teo, S.H. Hou, Nonlinear impulsive systems on infinite dimensions, Nonlinear Analysis, 54 (2003), 907–925. J.P. Aubin, A. Cellina, Differential Inclusions, Springer, 1984. J. Berkovits, V. Mustonen, Monotonicity methods for nonlinear evolution equations, Nonlinear Analysis, 27 (1996), 1397–1405. S. Carl, Enclosure of solutions for quasilinear dynamic hemivariational inequalities, Nonlinear World, 3 (1996), 281–298. S. Carl, R.P. Gilbert, Extremal solutions of a class of dynamic boundary hemivariational inequalities, Journal of Inequalities and Applications, 7 (2002), 479-502. S. Carl, D. 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--- abstract: 'For two-dimensional complex tori, we characterize the set of all values of positive entropy that arise from automorphisms. For K3 surfaces, we give sufficient conditions for a positive value to be the entropy of some automorphism.' author: - Paul Reschke bibliography: - 'ReschkeP-refs-2012.02.bib' title: Salem Numbers and Automorphisms of Complex Surfaces --- Overview ======== In this note, we examine complex surface automorphisms with positive entropy. We focus on automorphisms of complex tori and K3 surfaces.\ \ For complex tori, we determine precisely which positive values arise as entropies of such automorphisms. Each of these values is the natural logarithm of a Salem number of degree two, four, or six. (See §2 in this note.) We specify necessary and sufficient conditions for a Salem number of degree four or six to give the entropy of some torus automorphism, and we show that every Salem number of degree two gives the entropy of some torus automorphism. Let $S(t)$ be the minimal polynomial for a Salem number $\lambda$. [***)***]{}[ ]{} If $\lambda$ has degree six, then $\log(\lambda)$ is the entropy of some automorphism of a two-dimensional complex torus if and only if $S(1)=-m^2$ for some integer $m$ and $S(-1)=n^2$ for some integer $n$. If $\lambda$ has degree four, then $\log(\lambda)$ is the entropy of some automorphism of a two-dimensional complex torus if and only if one of the following three cases holds: (a) $S(1)=-m^2$ for some integer $m$; (b) $S(-1)=n^2$ for some integer $n$; or (c) $S(1)=-\frac{1}{2}m^2$ and $S(-1)=\frac{1}{2}n^2$ for some integers $m$ and $n$. If $\lambda$ has degree two, then $\log(\lambda)$ is the entropy of some automorphism of a two-dimensional complex torus. These cases constitute all possible positive entropies for an automorphism of a two-dimensional complex torus. (See §3 in this note.) This theorem extends results in [@gmc], [@mc1], and [@mc2] which provide specific examples of Salem numbers whose logarithms are entropies of automorphisms of two-dimensional complex tori. We discuss some additional results specific to automorphisms of projective tori and tori which are products of one-dimensional complex tori. (See §5 in this note.)\ \ We consider an analogous investigation of automorphisms of K3 surfaces. The entropy of any such automorphism must be zero or the natural logarithm of a Salem number of degree at most twenty-two. Kummer surfaces (K3 surfaces arising from quotients of tori) have automorphisms whose entropies are natural logarithms of Salem numbers of degree at most six. Indeed, any Salem number that gives the entropy of a torus automorphism must also give the entropy of the corresponding Kummer surface automorphism. We identify a sufficient condition for a Salem number of degree fourteen to give the entropy of some automorphism of a K3 surface. A result in [@gmc] is that the same condition is sufficient for a Salem number of degree twenty-two to give the entropy of some K3 surface automorphism. Let $S(t)$ be the minimal polynomial for a Salem number $\lambda$ of degree fourteen satisfying $S(-1)S(1)=-1$. Then $\log(\lambda)$ is the entropy of some K3 surface automorphism. (See §6 in this note.) This theorem comes from combining results in [@gmc], [@mc2], and [@ogu]. We note that there are examples of K3 surface automorphisms with positive entropy not accounted for by either this theorem or the Kummer construction. (See [@mc2], and §6 in this note.)\ \ If a compact $\operatorname{K\ddot{a}hler}$ surface admits an automorphism with positive entropy, then (after any exceptional curves with finite order under the automorphism are contracted) the surface can only be a complex torus, a K3 surface, an Enriques surface, or a rational surface. (See [@can], §2.) The set of possible entropies for rational surface automorphisms is described in [@ueh]. Computing Entropies on a Torus ============================== Let $X=\mathbb{C}^2/\Lambda$ be a two-dimensional complex torus; so $\Lambda \subseteq \mathbb{C}^2$ is a four-dimensional lattice acting holomorphically on $\mathbb{C}^2$ by translations. If $F$ is an automorphism of $X$, then $F$ is necessarily the quotient of some invertible linear map $\widetilde{F} \in \operatorname{GL}_2(\mathbb{C})$ with $\widetilde{F}(\Lambda)=\Lambda$. In this sense, $F$ can be expressed as a $2 \times 2$ matrix over $\mathbb{C}$.\ \ Given coordinates $z_1$ and $z_2$ for each factor of $\mathbb{C}^2$, the one-forms $dz_1$, $dz_2$, $d\overline{z}_1$, and $d\overline{z}_2$ on $\mathbb{C}^2$ are all invariant under translations and hence descend to one-forms on $X$. Moreover, these four one-forms generate the cohomology ring for $X$ with coefficients in $\mathbb{C}$. If $F$ is an automorphism of $X$, then $F$ induces an automorphism $F^*$ of $H^*(X;\mathbb{C})$ via the pull-back action. As an element of $\operatorname{GL}_4(\mathbb{C})$, the pull-back action of $F$ on $H^1(X;\mathbb{C})$ is the direct sum of the transpose of $F$ and the conjugate transpose of $F$; that is, $F^*=F^T \oplus \bar{F}^T$. The pull-back action of $F$ gives a convenient means of computing the entropy of $F$ via the spectral radius $\rho(F^*)$. Indeed, the entropy of $F$ is given by $$h(F)=\log(\rho(F^*:H^2(X;\mathbb{C}) \rightarrow H^2(X;\mathbb{C}))).$$ This formula is a special case of a general statement for automorphisms of compact $\operatorname{K\ddot{a}hler}$ manifolds due to Gromov and Yomdin. (See [@can], §2.1.)\ \ If the eigenvalues of $F$ are $\gamma_1$ and $\gamma_2$, then the eigenvalues of $F^*$ restricted to $H^1(X;\mathbb{C})$ are $\gamma_1$, $\gamma_2$, $\overline{\gamma_1}$, and $\overline{\gamma_2}$. Since the pull-back commutes with the wedge product, the eigenvalues of $F^*$ restricted to $H^2(X;\mathbb{C})$ are $|\gamma_1|^2$, $|\gamma_2|^2$, $\gamma_1 \gamma_2$, $\gamma_1 \overline{\gamma_2}$, $\overline{\gamma_1} \gamma_2$, and $\overline{\gamma_1} \overline{\gamma_2}$. Thus the entropy of $F$ is given by $$h(F)=\log(\max\{|\gamma_1|^2,|\gamma_2|^2\}).$$\ Since $F^*$ preserves the lattice $H^1(X;\mathbb{Z})$ inside $H^1(X;\mathbb{C})$, $F^*$ must be an element of $\operatorname{GL}_4(\mathbb{Z})$ and hence must have determinant one or negative one; so $|\gamma_2|=|\gamma_1|^{-1}$ and the eigenvalues of $F^*$ restricted to $H^2(X;\mathbb{C})$ are some $\lambda \geq 1$, $\lambda^{-1}$, $\alpha$, $\overline{\alpha}$, $\beta$, and $\overline{\beta}$ with $|\alpha|=|\beta|=1$. The entropy of $F$ is then $\log(\lambda)$. Since $F^*$ also preserves the lattice $H^2(X;\mathbb{Z})$ inside $H^2(X;\mathbb{C})$, the characteristic polynomial for $F^*$ restricted to $H^2(X;\mathbb{C})$ must be some monic reciprocal polynomial $Q(t) \in \mathbb{Z}[t]$ of degree six. The minimal polynomial for $\lambda$ is then some irreducible factor of $Q(t)$; if $\lambda$ is greater than one, then this irreducible factor is a reciprocal polynomial with exactly two roots off the unit circle, both of which are positive. Thus $\lambda$ is either one or a Salem number of degree two, four, or six. (See also [@mc1], §3.) Constructing Torus Automorphisms ================================ Let $F$ be an automorphism of a two-dimensional complex torus $X$ with eigenvalues $\gamma_1$ and $\gamma_2$. Let $Q(t) \in \mathbb{Z}[t]$ and $P(t) \in \mathbb{Z}[t]$ be the characteristic polynomials of $F^*$ restricted to $H^2(X;\mathbb{C})$ and $H^1(X;\mathbb{C})$, respectively. Then $$Q(t)=(t-|\gamma_1|^2)(t-|\gamma_2|^2)(t-\gamma_1\gamma_2)(t-\gamma_1\overline{\gamma_2})(t-\overline{\gamma_1}\gamma_2)(t-\overline{\gamma_1}\overline{\gamma_2})$$ $$\indent =t^6+at^5+bt^4+ct^3+bt^2+at+1$$ and $$P(t)=(t-\gamma_1)(t-\gamma_2)(t-\overline{\gamma_1})(t-\overline{\gamma_2})$$ $$\indent =t^4+jt^3-at^2+kt+1$$ for some integers $a$, $b$, $c$, $j$, and $k$ with $jk=b+1$ and $j^2+k^2=-c-2a$. So $Q(1)=-(j-k)^2=-m^2$ for some integer $m$ and $Q(-1)=(j+k)^2=n^2$ for some integer $n$; we say that $Q(t)$ has the “square property”. (Compare [@gmc], §1.)\ \ In particular, any Salem number of degree six that gives the entropy for some torus automorphism must have a minimal polynomial with the square property. The following existence theorem provides a means of determining precisely when a Salem number gives the entropy of some torus automorphism. \[[@mc1], Theorem 4.4\] Let $P(t) \in \mathbb{Z}[t]$ be monic of degree four with roots $\gamma_1$, $\gamma_2$, $\overline{\gamma_1}$, and $\overline{\gamma_2}$ such that $|\gamma_2|=|\gamma_1|^{-1}$. Then there is some two-dimensional complex torus $X$ and some automorphism $F$ of $X$ such that $P(t)$ is the characteristic polynomial of $F^*$ restricted to $H^1(X;\mathbb{C})$. Now let $Q(t) \in \mathbb{Z}[t]$ be a monic reciprocal polynomial of degree six with the square property; so $$Q(t)=t^6+at^5+bt^4+ct^3+bt^2+at+1$$ for some integers $a$, $b$, and $c$, and $$Q(1)=2+2a+2b+c=-m^2$$ for some integer $m$, and $$Q(-1)=2-2a+2b-c=n^2$$ for some integer $n$. Then $m$ and $n$ are either both odd or both even, and thus $j=\frac{1}{2}(m+n)$ and $k=\frac{1}{2}(n-m)$ are both integers. Since $jk=b+1$ and $j^2+k^2=-c-2a$, the roots of $Q(t)$ must be the products of distinct roots of the polynomial $$P(t)=t^4+jt^3-at^2+kt+1.$$ If $Q(t)$ has exactly two roots off the unit circle, both positive, then the roots of $P(t)$ occur in conjugate pairs. These calculations yield the following proposition. Let $Q(t) \in \mathbb{Z}[t]$ be monic and reciprocal of degree six with exactly two roots off the unit circle, both positive. Then the following two statements are equivalent: [***)***]{}[ ]{} there is a two-dimensional complex torus $X$ and an automorphism $F$ of $X$ such that $F^*$ restricted to $H^2(X;\mathbb{C})$ has characteristic polynomial $Q(t)$; and $Q(t)$ has the square property. Since the entropy of a torus automorphism must be zero or a Salem number, no polynomial with more than two roots off the unit circle can be the characteristic polynomial for the induced action on second cohomology. If some automorphism gives rise to a characteristic polynomial with no roots off the unit circle, then the automorphism must have entropy zero.\ \ In particular, if $\lambda$ is a Salem number of degree six, then $\log(\lambda)$ is the entropy of some torus automorphism if and only if the minimal polynomial for $\lambda$ has the square property. Similarly, a Salem number of degree two or four gives the entropy of some torus automorphism if and only if its minimal polynomial is a factor of some monic reciprocal polynomial of degree six with the square property and only two roots off the unit circle.\ \ *Proof of Theorem 1.1.* The degree-six case is evident from Proposition 3.2. In the degree-two case, if $S(t)$ is the minimal polynomial for $\lambda$, then $S(t)(t-1)^2(t+1)^2$ is a monic reciprocal polynomial of degree six with only two roots off the unit circle satisfying $S(1)=S(-1)=0$. In the degree-four case, if $S(t)$ is the minimal polynomial for $\lambda$, then $S(t)C(t)$ is a monic reciprocal polynomial of degree six with only two roots off the unit circle if and only if $C(t)$ is one of $t^2-2t+1$, $t^2-t+1$, $t^2+1$, $t^2+t+1$, or $t^2+2t+1$; furthermore, $S(t)C(t)$ has the square property for one of these choices of $C(t)$ if and only if one of the three stated cases holds for $S(t)$. $\Box$\ \ If $F$ is a torus automorphism with positive entropy, then $F$ must have eigenvalues $\gamma_1$ and $\gamma_2$ with $|\gamma_2|=|\gamma_1|^{-1} \neq 1$. So the eigenvalues of $F^*$ restricted to $H^1(X;\mathbb{C})$ cannot have magnitude one and in particular cannot be roots of unity. This result implies that $F$ must be ergodic. (See [@mc1], §4, and [@man], III, Theorems 3.1, 8.5, and 8.6.) Lattice Isometries ================== Given a two-dimensional complex torus $X$, the intersection form makes $H^2(X;\mathbb{Z})$ into an even unimodular lattice of signature (3,3), which we denote $L_X$. If $F$ is an automorphism of $X$, then $F^*$ is an isometry from $L_X$ to itself. The isometry $F^*$ extends to an isometry on $L_X \otimes \mathbb{Q}$ and hence can be expressed in rational canonical form via some change of basis over $\mathbb{Q}$. The invariant factors of $F^*$ correspond to $F^*$-invariant subspaces of $L_X \otimes \mathbb{Q}$; the same is true for any irreducible factors with multiplicity one within an invariant factor of $F^*$. Since any subspace of $L_X \otimes \mathbb{Q}$ must intersect $L_X$ itself, any $F^*$-invariant subspace of $L_X \otimes \mathbb{Q}$ must contain some $F^*$-invariant sublattice of $L_X$. Moreover, since classes in $H^2(X;\mathbb{Z})$ can be represented as isotopy classes of smoothly embedded real surfaces in $X$ (whose intersection numbers are compatible with the intersection form on $L_X$), $F^*$-invariant sublattices of $L_X$ must correspond to $F$-invariant spaces of isotopy classes. (See [@gom], §1.2, and [@mc2], §2.)\ \ The study of torus automorphisms fits as a special case into the study of isometries of even unimodular lattices in general. Results in [@gmc] give a sufficient condition for a Salem number to be the entropy of an isometry of an even unimodular lattice: let $\lambda$ be a Salem number with minimal polynomial $S(t)$ of degree $d$ such that $$d \equiv 2 \mod{4},$$ and let $p$ and $q$ be positive integers satisfying $p+q=d$ and $$p \equiv q \mod{8};$$ then $\log(\lambda)$ is the entropy of some isometry of the even unimodular lattice of signature ($p$,$q$) if $S(-1)S(1)=-1$. (See [@gmc], §1, and [@boy], §2.) Moreover, a necessary condition for $\log(\lambda)$ to be the entropy of such an isometry is that $|S(-1)|$, $|S(1)|$, and $-S(-1)S(1)$ are all squares of integers. Let $L_{3,3}$ be the (unique up to isometry) even unimodular lattice of signature (3,3). In this case, the classification of Salem numbers with respect to entropies of torus automorphisms gives a complete statement: let $S(t)$ be the minimal polynomial for a Salem number of degree six; then $S(t)$ is the characteristic polynomial for some isometry of $L_{3,3}$ if and only if $S(t)$ has the square property. (For a degree-six Salem polynomial $S(t)$, the condition $S(-1)S(1)=-1$ forces $S(-1)=-1$ and $S(1)=1$.) Projective Tori =============== Any two-dimensional complex torus $X$ has as part of its Hodge structure $$H^2(X;\mathbb{C}) \cong H^{2,0} \oplus H^{1,1} \oplus H^{0,2},$$ and the intersection form extends to a Hermitian inner product on $H^2(X;\mathbb{C})$ with signature (2,0) on $H^{2,0} \oplus H^{0,2}$ and signature (1,3) on $H^{1,1}$. If $F$ is an automorphism of $X$, then $F^*$ must preserve the Hodge structure and the Néron-Severi group $$\operatorname{NS}(X) = H^{1,1} \cap H^2(X;\mathbb{Z}).$$ The torus $X$ is projective if and only if $\operatorname{NS}(X)$ contains some element with positive self-intersection. So $F$ can only be an automorphism of a projective torus if $F^*$ preserves a sublattice of $H^2(X;\mathbb{Z})$ of rank at least one and at most four. In particular, the characteristic polynomial of $F^*$ cannot be irreducible if $X$ is projective. Thus no Salem number of degree six can give the entropy of a projective torus automorphism. The projective torus $\mathbb{C}/\mathbb{Z}[\zeta_3] \times \mathbb{C}/\mathbb{Z}[\zeta_3]$, where $\zeta_3$ is a third root of unity, does admit an automorphism whose entropy is the logarithm of a Salem number of degree four (the smallest Salem number of degree four, in fact). (See [@mc2], §1.)\ \ Any complex torus of the form $E \times E$, where $E$ is a one-dimensional complex torus, is necessarily projective. Any linear map $A \in \operatorname{GL}_2(\mathbb{Z})$ with $\det(A)=1$ gives an automorphism of $E \times E$ with eigenvalues $\frac{1}{2}(\operatorname{tr}(A)+\sqrt{\operatorname{tr}(A)^2-4})$ and $\frac{1}{2}(\operatorname{tr}(A)-\sqrt{\operatorname{tr}(A)^2-4})$. (See [@mc1], §4.) The characteristic polynomial for $A^*$ restricted to $H^2(E \times E;\mathbb{C})$ is then $$(t^2-(\operatorname{tr}(A)^2-2)t+1)(t-1)^4,$$ and $A$ has positive entropy if $|\operatorname{tr}(A)|>2$. Similarly, if $\det(A)=-1$, then the characteristic polynomial for $A^*$ restricted to $H^2(E \times E; \mathbb{C})$ is $$(t^2-(\operatorname{tr}(A)^2+2)t+1)(t+1)^4,$$ and $A$ has positive entropy if $|\operatorname{tr}(A)|>0$. These calculations give sufficient conditions for a Salem number of degree two to give the entropy of an automorphism of $E \times E$: let $\lambda$ be a Salem number of degree two and let $t^2-at+1$ (with $a>2$) be its minimal polynomial; then $\log(\lambda)$ is the entropy of some automorphism of $E \times E$ if $a=m^2+2$ for some integer $m$ or $a=m^2-2$ for some integer $m$. These examples do not include the case referred to above of an automorphism of $\mathbb{C}/\mathbb{Z}[\zeta_3] \times \mathbb{C}/\mathbb{Z}[\zeta_3]$ with entropy the logarithm of a degree-four Salem number. The complete picture for entropies of projective torus automorphisms remains to be determined. K3 Surfaces =========== A K3 surface is a simply connected two-dimensional compact complex manifold with a nowhere-vanishing holomorphic (2,0)-form. If $Y$ is a K3 surface, then the intersection form makes $H^2(Y;\mathbb{Z})$ into an even unimodular lattice of signature (3,19). As with two-dimensional complex tori, the induced action on second cohomology determines the entropy of any K3 surface automorphism. Indeed, the entropy of any K3 surface automorphism must be zero or the logarithm of a Salem number of degree at most twenty-two. (If the K3 surface is projective, then the degree of the Salem number cannot be greater than twenty.) (See [@mc1], §3.)\ \ A Kummer surface is a K3 surface constructed from a complex torus: let $X$ be a two-dimensional complex torus and let $i$ be the involution on $X$ sending $x$ to $-x$; then the Kummer surface associated to $X$ is the quotient $X/i$ with the sixteen singularites resolved by blowing up; the blown-up quotient is simply-connected, and it inherits a nowhere-vanishing (2,0)-form from $X$. If $F$ is a torus automorphism, then $F$ descends to an automorphism of the associated Kummer surface with the same entropy. Thus any Salem number that gives the entropy of some torus automorphism also gives the entropy of some K3 surface automorphism. A Kummer surface is projective if and only if it is constructed from a projective torus. If a Salem number gives the entropy of a Kummer surface automorphism that is induced (by the Kummer construction) from a torus automorphism, then the degree of the Salem number cannot be greater than six (and cannot be greater than four if the Kummer surface is projective). (See [@mc1], §4.)\ \ The following existence theorem leads to one method for identifying positive values that can be realized as entropies of K3 surface automorphisms. \[[@mc2], Theorem 6.2\] Let $L_{3,19}$ be the unique even unimodular lattice of signature $(3,19)$. Let $f$ be an isometry of $L_{3,19}$ with spectral radius $\rho(f)>1$ such that $\rho(f)$ is an eigenvalue of $f$. Suppose that there is some $\tau \in (-2,2)$ such that $$E_\tau=\ker(f+f^{-1}-\tau I) \subseteq L_{3,19} \otimes \mathbb{R}$$ has signature $(2,0)$. Then $f$ is realizable by a K3 surface automorphism if $f$ is the identity on $L_{3,19} \cap E_\tau^\perp$. Let $L_{3,11}$ and $L_{3,19}$ be the unique even unimodular lattices of signatures (3,11) and (3,19), respectively. Let $S(t)$ be the minimal polynomial of a Salem number of degree fourteen or twenty-two. As in the degree-six case, results in [@gmc] show that $S(t)$ is the characteristic polynomial of some isometry of $L_{3,11}$ or $L_{3,19}$ (depending on the degree) if $S(-1)S(1)=-1$. In the degree-twenty-two case, this condition guarantees immediately that the Salem number gives the entropy of some K3 surface automorphism; this condition is also essential to the proof of Theorem 1.2. (Compare [@gmc], [@mc2], and [@ogu].)\ \ *Proof of Theorem 1.2.* Let $f$ be the isometry of $L_{3,11}$ with characteristic polynomial $S(t)$, let $R(t)$ be the trace polynomial for $S(t)$ given by $S(t)=t^7R(t+t^{-1})$, and let $E_8(-1)= \textless e_1, \ldots ,e_8 \textgreater$ be the even unimodular lattice of signature (0,8) whose intersection form is given by $$(\langle e_i,e_j \rangle)= \left( \begin{array}{cccccccc} -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & -2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & -2 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 \end{array} \right).$$ Then $f$ extends to an isometry of $L_{3,11} \oplus E_8(-1) \cong L_{3,19}$ which is the identity on the second summand, and there is a unique root $\tau$ of $R(t)$ giving $E_\tau$ with signature (2,0). Since $R(t)$ is irreducible over $\mathbb{Z}$ and $f+f^{-1}$ preserves $E_\tau$, $E_\tau^\perp$, and $L_{3,11}$, it follows that $$(L_{3,11} \oplus E_8(-1)) \cap E_\tau^\perp \subseteq E_8(-1),$$ on which $f$ is the identity. $\Box$\ \ Let $L_{1,9}$ be the unique even unimodular lattice of signature (1,9) and let $S(t)$ be the minimal polynomial of a Salem number of degree ten. The results in [@gmc] show again that $S(t)$ is the characteristic polynomial of some isometry of $L_{1,9}$ if $S(-1)S(1)=-1$. However, in this case, no root of the trace polynomial $R(t)$ given by $S(t)=t^5R(t+t^{-1})$ gives rise to a space $E_\tau$ with signature (2,0). Thus the approach used in the degree-twenty-two and degree-fourteen cases is not immediately applicable. In [@mc2], there is a construction of a specific (non-projective) K3 surface automorphism with entropy the logarithm of a Salem number of degree ten (the smallest Salem number of degree ten, in fact). The construction relies on careful twisting and gluing of $L_{1,9}$. Improvements to this process, in [@mc2] and [@mc3], give constructions of projective K3 surface automorphisms whose entropies include logarithms of Salem numbers of degree six, eight, ten, and eighteen. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Laura DeMarco, Curt McMullen, and the referee for helpful comments.

--- abstract: | Using data drawn from the DEEP2 and DEEP3 Galaxy Redshift Surveys, we investigate the relationship between the environment and the structure of galaxies residing on the red sequence at intermediate redshift. Within the massive $(10 < \log_{10}({\rm M}_{\star} / h^{-2}\ {\rm M}_{\sun}) < 11)$ early-type population at $0.4 < z < 1.2$, we find a significant correlation between local galaxy overdensity (or environment) and galaxy size, such that early-type systems in higher-density regions tend to have larger effective radii (by $\sim \! 0.5\ h^{-1}$ kpc or $25\%$ larger) than their counterparts of equal stellar mass and Sérsic index in lower-density environments. This observed size-density relation is consistent with a model of galaxy formation in which the evolution of early-type systems at $z < 2$ is accelerated in high-density environments such as groups and clusters and in which dry, minor mergers (versus mechanisms such as quasar feedback) play a central role in the structural evolution of the massive, early-type galaxy population. author: - 'Michael C. Cooper Roger L. Griffith, Jeffrey A. Newman, Alison L. Coil, Marc Davis, Aaron A. Dutton, S. M. Faber, Puragra Guhathakurta, David C. Koo, Jennifer M. Lotz, Benjamin J. Weiner, Christopher N. A. Willmer, Renbin Yan' title: 'The DEEP3 Galaxy Redshift Survey: The Impact of Environment on the Size Evolution of Massive Early-type Galaxies at Intermediate Redshift' --- Introduction {#sec_intro} ============ Recent observations of galaxies at intermediate redshift ($z \sim 2$) have identified a significant population of massive ($\sim \! 10^{11}$ M$_{\sun}$), quiescent, early-type systems with old, metal-rich stellar populations and remarkably small sizes relative to their local counterparts [e.g., @daddi05; @labbe05; @papovich06; @kriek06; @trujillo06; @trujillo07; @zirm07; @vandokkum08; @damjanov09; @damjanov11; @toft09; @taylor10]. The stellar densities of these massive, intermediate-redshift galaxies (as measured within one effective radius, $r_{e}$) are typically two orders of magnitude greater than quiescent ellipticals of the same mass at $z \sim 0.1$. Within the central $1$ kpc (physical), however, the densities of early-types at $z \sim 2$ are found to only exceed local measurements by a factor of $2$–$3$ [@bezanson09; @hopkins09b; @vandokkum10]. Altogether, the observations suggest that there is significant evolution in the size of massive ellipticals over the past $10$ Gyr, likely proceeding in an inside-out manner, without the addition of much stellar mass. Several physical processes have been proposed to explain this strong size evolution within the massive, early-type population at $z < 2$. In particular, gas-poor, collisionless (or “dry”) minor mergers are often invoked as a means for puffing up the stellar component of these massive systems [e.g., @naab06; @naab07; @naab09; @khochfar06; @bournaud07; @bk07; @vdw09; @cenarro09; @hopkins09a; @hopkins10b; @trujillo11]. However, a variety of alternative mechanisms have also been proposed, including scenarios in which the observed structural evolution may be driven by secular processes such as adiabatic expansion resulting from stellar mass loss and/or strong AGN-fueled feedback (@fan08 [@fan10]; @damjanov09; @hopkins10a [@hopkins10b]; see also @nipoti09 and @williams10). One way to possibly discriminate between these scenarios (minor mergers versus secular processes) is by quantifying the role of environment in the structural evolution of the massive galaxy population. While secular processes are largely independent of environment and quasars are not preferentially found in overdense regions at $z \sim 1$ [@coil07], mergers are more common in higher-density environments such as galaxy groups [@cavaliere92; @mcintosh08; @wetzl08; @fakhouri09; @lin10; @darg10]. Thus, if mergers are the dominant mechanism by which the sizes of massive early-types evolve at $z < 2$, we should expect to find a variation in the structural properties of galaxies as a function of environment at $z \sim 1$. To test this, we use data drawn from the DEEP2 and DEEP3 Galaxy Redshift Surveys [@davis03; @newman12; @cooper11] to investigate the correlation between the local overdensity of galaxies (which we generally refer to as “environment”) and the sizes of massive galaxies on the red sequence at intermediate redshift. In Section \[sec\_data\], we describe our data set, with results and discussion presented in Sections \[sec\_results\] and \[sec\_disc\], respectively. Throughout, we employ a $\Lambda$CDM cosmology with $w = -1$, $\Omega_m = 0.3$, $\Omega_{\Lambda} = 0.7$, and a Hubble parameter of $H_{0} = 100\ h$ km s$^{-1}$ Mpc$^{-1}$, unless otherwise noted. All magnitudes are on the AB system [@oke83]. Data {#sec_data} ==== To characterize both the environment and the structure of galaxies accurately requires spectroscopic observations [or deep, multi-band photometric observations, @cooper05] as well as high-resolution imaging across a sizable area of sky. Given the limitations of ground-based adaptive-optics observations, the latter is only possible at intermediate redshift via space-based observations (e.g., with [*HST*]{}). Among the fields covered by deep, multi-band imaging with [*HST*]{}, the Extended Groth Strip (EGS) is by far the most complete with regard to spectroscopic coverage at intermediate redshift. The EGS is one of four fields surveyed by the DEEP2 Galaxy Redshift Survey [@davis03; @davis07; @newman12], yielding high-precision ($\sigma_{z} \sim 30$ km s$^{-1}$) secure redshifts for $11,701$ sources at $0.2 < z < 1.4$ over roughly $0.5$ square degrees in the EGS. Building upon the DEEP2 spectroscopic sample, the recently-completed DEEP3 Galaxy Redshift Survey (@cooper11; Cooper et al., in prep) has brought the target sampling rate to $\sim \! 90\%$ at $R_{\rm AB} < 24.1$ over the central $0.25$ square degrees of the EGS — the portion of the field imaged by [*HST*]{}/ACS [see Fig. \[fig\_egs\]; @davis07; @lotz08]. Among the current generation of deep spectroscopic redshift surveys at $z \sim 1$, the combination of the DEEP2 and DEEP3 spectroscopic datasets provides the largest sample of accurate spectroscopic redshifts, the highest-precision velocity information, and the highest sampling density (@newman12; Cooper et al., in prep).[^1] Combined with the relatively wide area imaged with [*HST*]{}/ACS [an area $> \! 2\times$ larger than that surveyed as part of the GOODS program, @giavalisco04], these attributes make the EGS one of the best-suited fields in which to study the relationship between environment and galaxy structure at $z \sim 1$. In this paper, we utilize a parent sample of $11,493$ galaxies drawn from the joint DEEP2/DEEP3 dataset in the EGS with secure redshifts [quality $Q = 3$ or 4 as defined by @davis07; @newman12] in the range $0.4 < z < 1.2$. Rest-frame Colors, Luminosities, and Stellar Masses --------------------------------------------------- For each galaxy in the DEEP2/DEEP3 sample, rest-frame $U-B$ colors and absolute $B$-band magnitudes, $M_{B}$, are calculated from CFHT $BRI$ photometry [@coil04b] using the $K$-correction procedure described in @willmer06. For a subset of the galaxy catalog, stellar masses are calculated by fitting spectral energy distributions (SEDs) to WIRC/Palomar $J$- and $K_{s}$-band photometry in conjunction with the DEEP2 $BRI$ data, according to the prescriptions described by [@bundy05; @bundy06]. However, the near-infrared photometry, collected as part of the Palomar Observatory Wide-field Infrared [POWIR, @conselice08] survey, does not cover the entire DEEP2/DEEP3 survey area, and often faint blue galaxies at the high-$z$ end of the DEEP2 redshift range are not detected in $K_{s}$. Because of these two effects, the stellar masses of @bundy06 have been used to calibrate stellar mass estimates for the full DEEP2 sample that are based on rest-frame $M_{B}$ and $B-V$ values derived from the DEEP2 data in conjunction with the expressions of @bell03, which relate mass-to-light ratio to optical color. We empirically correct these stellar mass estimates to the @bundy06 measurements by accounting for a mild color and redshift dependence [@lin07]; where they overlap, the two stellar masses have an rms difference of approximately $0.3$ dex after this calibration. Local Galaxy Overdensity ------------------------ To characterize the local environment, we compute the projected third-nearest-neighbor surface density $(\Sigma_3)$ about each galaxy in the joint DEEP2/DEEP3 sample, where the surface density depends on the projected distance to the third-nearest neighbor, $D_{p,3}$, as $\Sigma_3 = 3 / (\pi D_{p,3}^2)$. In computing $\Sigma_3$, a velocity window of $\pm 1250$ km s$^{-1}$ is utilized to exclude foreground and background galaxies along the line of sight. Varying the width of this velocity window (e.g., using $\pm 1000$–$2000$ km s$^{-1}$) or tracing environment according to the projected distance to the fifth-nearest neighbor has no significant effect on our results. In the tests of @cooper05, this projected $n^{\rm th}$-nearest-neighbor environment estimator proved to be the most robust indicator of local galaxy density for the DEEP2 survey. To correct for the redshift dependence of the DEEP2 and DEEP3 sampling rates, each surface density value is divided by the median $\Sigma_3$ of galaxies at that redshift within a window of $\Delta z = 0.04$; correcting the measured surface densities in this manner converts the $\Sigma_3$ values into measures of overdensity relative to the median density (given by the notation $1 + \delta_{3}$ here) and effectively accounts for the redshift variations in the selection rate [@cooper05]. Finally, to minimize the effects of edges and holes in the survey geometry, we exclude all galaxies within $1$ $h^{-1}$ comoving Mpc of the DEEP3 survey boundary (see Fig. \[fig\_egs\]), reducing our sample to $7,257$ galaxies in the redshift range $0.4 < z < 1.2$. Sérsic Indices and Sizes ------------------------ To quantify the sizes of the DEEP2 and DEEP3 galaxies, we utilize morphological measurements extracted from the Advanced Camera for Surveys General Catalog (ACS-GC, Griffith et al. 2012, in prep). The ACS-GC analyzed the [*HST*]{}/ACS $V_{\rm F606W}$ and $I_{\rm F814W}$ imaging in the EGS using GALAPAGOS [@haussler11], a semi-automated tool for measuring sizes and spatial profiles via the parametric fitting code GALFIT [@peng02; @peng10]. To determine the galaxy profile shape, each radial profile was fit using a simple Sérsic measurement [@sersic68] of the form $$\Sigma(r) = \Sigma_{e} \: {\rm exp}[-\kappa ((r/r_{e})^{-n} -1)]$$ where $r_{e}$ is the effective radius of the galaxy, $\Sigma_{e}$ is the surface brightness at $r_{e}$, $n$ is the power-law index, and $\kappa$ is coupled to $n$ such that half of the total flux is always within $r_e$. Here, we employ the profile fits to the [*HST*]{}/ACS $I_{\rm F814W}$ imaging for all sources, independent of galaxy redshift. At $z < 1.2$, the $I_{\rm F814W}$ passband samples the rest-frame optical ($\lambda > 3700$Å), which minizes morphological biases associated with observations made in the rest-frame ultraviolet where galaxies typically exhibit more irregular morphologies. The impact (or lack thereof) of any morphological $K$-correction is addressed further in §\[sec\_results\]. Throughout our analysis, all sizes ($r_{e}$) have been converted to physical kpc, according to the DEEP2/DEEP3 spectroscopic redshift and assuming a Hubble parameter of $h = 1$. Finally, note that the multidrizzled [*HST*]{}/ACS images, from which structural properties were measured, have a pixel scale of $0.03^{\prime\prime}$ per pixel and a point-spread function (PSF) of $0.12^{\prime\prime}$ FWHM; from $z=0.4$ to $z = 1.2$, the spatial resolution therefore varies from $\sim \! 0.5$ $h^{-1}$ kpc per PSF FWHM to $\sim \! 0.95$ $h^{-1}$ kpc per PSF FWHM. Sample Selection ---------------- To investigate the relationship between galaxy structure and environment amongst the high-mass portion of the red sequence, we define a subsample of DEEP2/DEEP3 galaxies at $0.4 < z < 1.2$ with stellar mass in the range $10 < \log_{10}({\rm M}_{\star} / h^{-2}\ {\rm M}_{\sun}) < 11$, on the red sequence (i.e., rest-frame color of $U -B > 1$),[^2] and with robust environment and morphology measurements (i.e., away from a survey edge and with $\sigma_{n} < 0.75$).[^3] The median redshift for this subsample of $623$ galaxies is $0.76$ and the median stellar mass is $\log_{10}({\rm M}_{\star} / h^{-2}\ {\rm M}_{\sun}) \sim 10.6$. In Figure \[fig\_cmdz\], we show the redshift distribution for all sources in the EGS with a secure redshift ($Q = -1$, $3$, $4$) in the joint DEEP2/DEEP3 sample alongside the color-magnitude distribution for all galaxies at $0.4 < z < 1.2$, with lines of constant stellar mass overlaid and with lines illustrating the survey magnitude limit at several discrete redshift values. Note that we restrict our primary subsample to a redshift range over which the DEEP2 and DEEP3 selection function is relatively flat. However, over this somewhat broad redshift range the sample is incomplete at the adopted mass limit. For example, at $z=0.9$ the $R_{\rm AB}=24.1$ magnitude limit of DEEP2 includes all galaxies with stellar mass $> \! 10^{10.8}\ {\rm M}_{\star}/h^{-2}\ {\rm M}_{\sun}$ independent of color, but preferentially misses red galaxies at lower masses (see Fig.\[fig\_cmdz\]). This incompleteness in the galaxy population is addressed in more detail in §\[sec\_results\]. Analysis {#sec_results} ======== In order to study the relationship between galaxy properties and environment at fixed stellar mass, as we aim to do here, the galaxy sample under study is often restricted to a narrow range in stellar mass such that correlations between stellar mass and environment are negligible. However, at intermediate redshift, sample sizes are generally limited in number such that using a particularly narrow stellar mass range (e.g., $\sim \! 0.1$–$0.2$ dex in width) significantly reduces the statistical power of the sample. For this reason, broader stellar mass bins (e.g., $\sim \! 0.5$ dex) are commonly employed. However, if the shape of the stellar mass function depends on environment (as suggested by @balogh01 [@kauffmann04; @croton05; @baldry06; @rudnick09; @cooper10b; @bolzonella10]), then the typical stellar mass within a broad mass bin may differ significantly from one density regime to another. Such an effect would clearly impact the ability to study the relationship between galaxy size and environment at fixed stellar mass. For this reason, we instead select those galaxies within the top $15\%$ of the overdensity distribution for all red galaxies at $10 < \log_{10}({\rm M}_{\star} / h^{-2}\ {\rm M}_{\sun}) < 11$ and $0.4 < z < 1.2$ (a subsample of $93$ galaxies), and from the corresponding bottom $50\%$ of the overdensity distribution we randomly draw 1000 subsamples (each composed of $93$ galaxies) so as to match the joint redshift, stellar mass, and Sérsic index distributions of the galaxies in the high-density subsample.[^4] The average environment for the high-density subset is $\log_{10}(1+\delta_3) = 1.31$, with an interquartile (25%–75%) range of $1.15$–$1.41$, while the low-density subsample is biased to considerably less-dense environs with an average overdensity of $\log_{10}(1+\delta_3) = -0.12$ and an interquartile range of $-0.30$–$0.12$. As shown by @cooper06, the high-density subsample is comprised largely of group members, while the low-density population is dominated by “field” galaxies [see also @cooper07; @gerke07]. By matching in redshift, we remove the projection of any possible residual correlation between our environment measurements and redshift (in concert with the known redshift dependence of the survey’s stellar-mass limit) onto the observed size-environment relation. In addition, matching according to redshift alleviates any possible impact from morphological $K$-corrections associated with measuring all structural parameters in the [*HST*]{}/ACS $I_{\rm F814W}$ passband. Finally, recognizing the correlations between environment and parameters such as color, star-formation rate, and morphology (i.e., early- versus late-type) at $z \sim 1$ [e.g., @cooper06; @cooper08; @cooper10b; @capak07; @elbaz07; @vdw07], we also force the Sérsic indices of the low-density subsample to match those of the high-density population, which controls for the contribution of dusty disk galaxies to our red-sequence population. Matching based on rest-frame color, in lieu of Sérsic index, would confuse reddened disk galaxies with red early-type systems. Members of the low-density subsample are drawn randomly from within a three-dimensional window with dimensions of $|\Delta z| < 0.04$, $|\Delta \log_{10}({\rm M}_{\star} / h^{-2}\ {\rm M}_{\sun})| < 0.2$, and $|\Delta n| < 1.5$ of a randomly-selected object in the high-density subsample. Varying the size of this window by factors of a few in each dimension has no significant effect on our results. Given the random nature of the matching, some objects are repeated in the low-density subsamples. However, for each subsample of $93$ galaxies, $> \! 70\%$ of the galaxies are unique; requiring all members of a subsample to be unique would skew the statistics [@efron81]. By matching our high- and low-density subsamples in stellar mass, redshift, as well as Sérsic index, we are able to effectively study the correlation between galaxy size and environment at fixed stellar mass. To test whether our high-density subsample and the random low-density subsamples are consistent with being drawn from the same underlying stellar mass distribution, we apply two non-parametric (i.e., independent of Gaussian assumptions) tests, the two-sided Kolmogorov-Smirnov (KS) test [@press86; @wall03] and the one-sided Wilcoxon-Mann-Whitney (WMW) $U$ test [@mann47]. The result of each test is a $P$-value: the probability that a value of the KS or $U$ statistic equal to the observed value or more extreme would be obtained, if the “null” hypothesis holds that the samples are drawn from the same parent distribution. The WMW $U$ test is computed by ranking all elements of the two data sets together and then comparing the mean (or total) of the ranks from each data set. Because it relies on ranks rather than observed values, it is highly robust to non-Gaussianity. The WMW $U$ test is particularly useful for small data sets (e.g., compared to other related tests such as the chi-square two-sample test, @wall03), as we have when selecting galaxies from a narrow stellar mass range and in extreme environments, due to its insensitivity to outlying data points, its avoidance of binning, and its high efficiency. Note that since this test is one-sided, possible $P_{U}$ values range from $0$ to $0.5$ (versus $P_{\rm KS}$ which ranges from $0$ to $1$); for a $P_{U}$ value below $0.025$ (corresponding closely to $2\sigma$ for a Gaussian), we can reject the null hypothesis (that the two samples have the same distribution) at greater than $95\%$ significance.[^5] In Figure \[fig\_cdists\], we plot the cumulative distribution of stellar masses for the $93$ sources in the high-density subsample alongside that for the 1000 random subsamples (each consisting of $93$ galaxies) matched in redshift but residing in low-density environments. Performing a one-sided WMW $U$ test (and a two-sided KS test) on the size ($r_{e}$) measurements for the low- and high-density populations, we find that the size distribution for the galaxies in high-density environments is skewed to larger sizes, with a probability of $P_{U} < 0.01$ (and $P_{\rm KS} < 0.02$). Meanwhile, the cumulative stellar mass, redshift, and Sérsic index distributions for the low- and high-density subsamples, shown in the inset of Figure \[fig\_cdists\], are well-matched with the WMW $U$ test yielding a $P_{U} > 0.4$. This confirms that our sample-construction procedure has yielded sets of galaxies in low- and high-density environments whose redshift, mass, and Sérsic index distributions match closely. While not directly matched, the rest-frame color distributions for the two samples are also indistinguishable — not a surprising result given that the color-density relation shows no significant variation across the red sequence at a given luminosity [@blanton05; @cooper06]. See Table \[res\_tab1\] for a complete summary of the probability values given by both the WMW $U$ and KS tests. The results of the WMW $U$ test are confirmed by a comparison of the Hodges-Lehmann (H-L) estimator of the mean sizes for the low- and high-density subsamples, which differ by $0.54 \pm 0.22$ $h^{-1}$ kpc. This reinforces the conclusion that there is a non-negligible size-environment relation on the red sequence at $z \sim 0.75$. The Hodges-Lehmann (H-L) estimator of the mean is given by the median value of the mean computed over all pairs of galaxies in the sample [@hodges63]. Like taking the median of a distribution, the H-L estimator of the mean is robust to outliers, but, unlike the median, yields results with scatter (in the Gaussian case) comparable to the arithmetic mean. Thus, by using the H-L estimator of the mean, we gain robustness as in the case of the median, but unlike the median, our measurement errors are increased by only a few percent. In Figure \[fig\_hlmeans\], we show the distribution of the differences between the Hodges-Lehmann estimator of the mean size, stellar mass, redshift, Sérsic index, and color for the high-density subsample relative to that for each of the $1000$ low-density subsamples, where the median difference in the H-L estimate of the mean size is $\Delta r_{e} = 0.559$ (as illustrated by the dashed vertical line) versus $\Delta \log_{10}({\rm M}_{\star} / h^{-2}\ {\rm M}_{\sun}) = -0.002$, $\Delta z = -0.003$, $\Delta n = 0.057$, and $\Delta (U-B) = -0.001$ for stellar mass, redshift, Sérsic index, and color, respectively (see Table \[res\_tab2\]). Within the stellar mass range of $10 < \log_{10}({\rm M}_{\star} / h^{-2}\ {\rm M}_{\sun}) < 11$, we find significant evidence for a correlation between galaxy size and environment at $z \sim 0.75$, such that higher-density regions play host to larger galaxies at a given stellar mass on the red sequence. To test the robustness of our results to the particularities of the sample selection, we repeat the analysis described above for several samples spanning varying redshift and stellar mass regimes. For example, restricting the redshift range over which we select galaxies to $0.7 < z < 1.2$, thereby decreasing the size of the sample, we again find a statistically significant relationship between galaxy structure and environment within our adopted stellar mass bin of $10 < \log_{10}({\rm M}_{\star} / h^{-2}\ {\rm M}_{\sun}) < 11$. For the $64$ galaxies in the high-density regime (again the highest $15\%$ of the overdensity distribution) at $0.7 < z < 1.2$, the cumulative distribution of galaxy size ($r_{e}$) is skewed towards larger effective radii relative to the comparison set of galaxies in low-density environments, yielding $P_{U} < 0.01$ and a median difference in the H-L estimator of the mean size of $\Delta r_{e} \sim 0.63$ $h^{-1}$ kpc. An obvious concern when studying the structure of galaxies on the red sequence is the relative contribution of early-type systems and dusty disk galaxies to the sample. Especially at fainter magnitudes/lower stellar mass, reddened star-forming galaxies comprise a significant portion of the red galaxy population [e.g., @lotz08; @bundy10; @cheng11], and as many studies of environment and clustering at intermediate redshift have shown the star-forming population tends to reside in lower-density regions relative to their passive counterparts [e.g., @cooper06; @cooper07; @capak07; @coil08; @kovac10]. Even with our primary sample selected to be at the massive end of the red sequence, disk galaxies ($n < 2.5$) still account for $\sim \! 25\%$ of the population (see Figure \[fig\_cdists\]). It is unlikely, however, that a difference in the relative contribution of dusty disk galaxies to the high- and low-density populations is driving the observed size-density relation since galaxy size correlates with Sérsic index such that red disk galaxies (systems with $n < 2.5$) tend to have slightly larger (not smaller) measured sizes than red galaxies of slightly larger Sérsic index. Still, to minimize any potential impact from late-type systems, we define two subsamples: one selected according to $n > 2.5$ and a second further constrained to systems with $n > 2.5$ and with axis ratios, $(b/a)$, greater than $0.4$ (see Table \[res\_tab1\]). For these subsamples of galaxies with early-type morphology, we find that the correlation between structure and environment persists, reinforcing the conclusion that early-type systems (and not dusty disk galaxies) are responsible for the observed size-density relation at fixed stellar mass. To illustrate the correlation between size and environment in a more physically intuitive manner, in Figure \[fig\_smrel\], we show the size-stellar mass relations for early-type galaxies in overdense and underdense regions. In the top portion of Fig. \[fig\_smrel\], the high- and low-density samples are simply defined to be the extreme quartiles of the environment distribution for all galaxies with $n > 2.5$, $0.7 < z < 1.2$, and $U-B > 1$. In the middle and bottom panel, however, a more controlled comparison is made, with the high-density sample selected as the top $15\%$ of the environment distribution for all galaxies with $n > 2.5$, $0.4 < z < 1.2$, and $U-B > 1$ and the low-density sample is comprised of $\sim \! 400$ galaxies randomly drawn from the bottom $50\%$ of the environment distribution to match the $z$ distribution (middle) and the joint $n$ and $z$ distribution (bottom) of the high-density sample. In agreement with our previous analysis, we find that within the high-mass segment of the early-type galaxy population the size-stellar mass relation varies with environment, such that galaxies in high-density regions are larger at a given stellar mass. Finally, at low surface brightness levels (i.e., low signal-to-noise per pixel in the [*HST*]{}/ACS imaging), GALFIT tends to underestimate both galaxy size ($r_{e}$) and Sérsic index. While such effects are minimal for objects brighter than the sky background [@haussler07], we define a subsample limited to those sources with surface brightness in the [*HST*]{}/ACS $I_{\rm F814W}$ passband of $\mu < 23.5$ magnitudes per arcsec$^{2}$ ($\mu_{\rm sky} \sim 27.4$), which excludes $58$ of $623$ red galaxies at $0.4 < z < 1.2$ and $10 < \log_{10}({\rm M}_{\star} / h^{-2}\ {\rm M}_{\sun}) < 11$. For our samples with closely matched stellar mass, $z$, and color (i.e., effectively matched apparent magnitude), such a cut on surface brightness imposes an upper limit on physical size, which therefore may impact the observed strength of the size-density relation. However, in spite of this conservative surface brightness limit, we still find a significant relationship between size and local overdensity on the red sequence, with a median difference in the H-L estimator of the mean size of $\Delta r_{e} \sim 0.47$ $h^{-1}$ kpc. In Table \[res\_tab1\] and Table \[res\_tab2\], we list the results from similar analyses of several different galaxy samples. When varying the Sérsic index, stellar mass, surface brightness, axis ratio, and/or redshift regimes probed, we continue to find a significant size-density relation at fixed stellar mass on the red sequence. [c c c c c c c c c c c c]{} $0.4 < z < 1.2$ & & & & & & & & & & &\ $10 < \log_{10}({\rm M}_{\star}) < 11$ & & & & & & & & & & &\ $0.7 < z < 1.2$ & & & & & & & & & & &\ $10 < \log_{10}({\rm M}_{\star}) < 11$ & & & & & & & & & & &\ $0.4 < z < 1.05$ & & & & & & & & & & &\ $10 < \log_{10}({\rm M}_{\star}) < 11$ & & & & & & & & & & &\ $0.4 < z < 1.2$ & & & & & & & & & & &\ $10.5 < \log_{10}({\rm M}_{\star}) < 11$ & & & & & & & & & & &\ $0.7 < z < 1.2$ & & & & & & & & & & &\ $10 < \log_{10}({\rm M}_{\star}) < 11$ & & & & & & & & & & &\ $n > 2.5$ & & & & & & & & & & &\ $0.7 < z < 1.2$ & & & & & & & & & & &\ $10 < \log_{10}({\rm M}_{\star}) < 11$ & & & & & & & & & & &\ $n > 2.5$, $b/a > 0.4$ & & & & & & & & & & &\ $0.4 < z < 1.2$ & & & & & & & & & & &\ $10 < \log_{10}({\rm M}_{\star}) < 11$ & & & & & & & & & & &\ $\mu < 23.5$ & & & & & & & & & & &\ [c cc cc cc cc cc]{} $0.4 < z < 1.2$ & $-0.003$ & $-0.004$ & $0.047$ & $0.001$ & $0.543$ & & & & &\ $10 < \log_{10}({\rm M}_{\star}) < 11$ & $\pm 0.030$ & $\pm 0.013$ &$\pm 0.248$ &$\pm 0.011$ & $ \pm 0.220$ & & & & &\ $0.7 < z < 1.2$ & $-0.022$ & $-0.004$ & $0.023$ & $-0.008$ & $0.688$ & & & & &\ $10 < \log_{10}({\rm M}_{\star}) < 11$ & $\pm 0.026$ & $\pm 0.008$ &$\pm 0.272$ &$\pm 0.012$ & $ \pm 0.251$ & & & & &\ $0.4 < z < 1.05$ & $0.010$ & $-0.005$ & $0.085$ & $-0.005$ & $0.393$ & & & & &\ $10 < \log_{10}({\rm M}_{\star}) < 11$ & $\pm 0.024$ & $\pm 0.011$ &$\pm 0.226$ &$\pm 0.011$ & $ \pm 0.209$ & & & & &\ $0.4 < z < 1.2$ & $-0.008$ & $0.001$ & $0.150$ & $0.010$ & $0.619$ & & & & &\ $10.5 < \log_{10}({\rm M}_{\star}) < 11$ & $\pm 0.017$ & $\pm 0.015$ &$\pm 0.285$ &$\pm 0.012$ & $ \pm 0.274$ & & & & &\ $0.7 < z < 1.2$ & & & & & & & & & &\ $10 < \log_{10}({\rm M}_{\star}) < 11$ & $-0.002$ & $-0.002$ & $0.145$ & $-0.010$ & $0.802$ & & & & &\ $n > 2.5$ & $\pm 0.030$ & $\pm 0.009$ & $\pm 0.277$ & $\pm 0.013$ & $\pm 0.239$ & & & & &\ $0.7 < z < 1.2$ & & & & & & & & & &\ $10 < \log_{10}({\rm M}_{\star}) < 11$ & $-0.003$ & $-0.005$ & $-0.128$ & $-0.006$ & $0.909$ & & & & &\ $n > 2.5$,$b/a > 0.4$ & $\pm 0.030$ & $\pm 0.016$ & $\pm 0.301$ & $\pm 0.013$ & $\pm 0.273$ & & & & &\ $0.7 < z < 1.2$ & & & & & & & & & &\ $10 < \log_{10}({\rm M}_{\star}) < 11$ & $0.000$ & $0.001$ & $0.083$ & $-0.001$ & $0.426$ & & & & &\ $\mu < 23.5$ & $\pm 0.024$ & $\pm 0.011$ & $\pm 0.214$ & $\pm 0.010$ & $\pm 0.224$ & & & & &\ Discussion {#sec_disc} ========== Previous efforts to study the environmental dependence of the size-stellar mass relation at $z < 2$ within the early-type galaxy population have been relatively few in number and have often found no significant trends with local galaxy density. Comparing the morphologies of massive galaxies in the low-redshift ($z=0.165$) Abell 901/902 supercluster to those of comparable field samples selected from the Space Telescope A901/2 Galaxy Evolution Survey [STAGES, @gray09], @maltby10 detect no significant relationship between galaxy structure and environment within the early-type population. Focusing on galaxy groups identified in the Sloan Digital Sky Survey [SDSS, @york00], a less extreme subdivision of the environment distribution, @guo09 also find no significant evidence for a correlation between local environment and the size or Sérsic index within the local early-type population [see also @weinmann09; @nair10]. In contrast, previous studies of brightest cluster galaxies (BCGs) in the local Universe find that BCGs tend to be larger than early-types of comparable stellar mass in less-dense environs [e.g., @bernardi07; @vdl07; @desroches07; @liu08 see also @bk06]. However, this environmental dependence apparent in the BCG population is likely the result of cluster-specific mechanisms that drive the formation of this rare subset of the massive early-type galaxy population and may not be indicative of the massive early-type population as a whole. We note that our DEEP2/DEEP3 sample includes very few (if any) systems that will evolve into BCGs at $z \sim 0$. Beyond the local Universe, @valentinuzzi10 find no significant variation in the size-stellar mass relation for massive early-types in clusters at $z \sim 0.7$ relative to that for comparable systems in the field, using data drawn from the ESO Distant Clusters Survey [EDisCS, @white05]. At yet higher redshift, @rettura10 compare the sizes of massive ellipticals in an X-ray-luminous cluster at $z = 1.237$ to those of correspondingly-massive systems in the field, finding no significant variation in size with environment. Utilizing the same data set, however, @cimatti08 propose (though without quantifying) a possible correlation between size and environment similar in nature to that found within our DEEP2/DEEP3 sample (that is, higher-density regions favoring less-compact galaxies). Nevertheless, using largely the same galaxy samples as these two previous studies, recent work from @raichoor11 argues for the opposite trend such that galaxies in high-density regions are smaller than their field counterparts. The significance of the measured correlation between size and environment, however, is dramatically overstated by @raichoor11, with their results actually consistent with no environment dependence.[^6] The lack of a significant environment dependence reported in these previous studies is likely due to \[*i*\] the smaller sample sizes employed (the @rettura10 and @cimatti08 analyses included a total of $45$ ellipticals across all environments), \[*ii*\] the use of less-precise environment measures (e.g., relying on photometric redshifts such that “field” or low-density samples can be strongly contaminated by group members), and/or \[*iii*\] differences in the redshift range probed. Our results, which show a significant size-density relation at fixed stellar mass within the massive, red galaxy population, suggest that the structural evolution of massive early-type systems occurs preferentially in overdense environments (i.e., groups, given the lack of massive clusters in our sample, @gerke05 [@gerke12]). This environmental dependence is in general agreement with a model of galaxy formation in which minor, “dry” mergers are a critical mechanism in driving structural evolution at $z < 2$. Moreover, our DEEP2/DEEP3 results also suggest that early-types in higher-density regions evolved structurally prior to their counterparts in low-density regions, growing from highly-compact systems at $z \sim 2$–$3$ to more extended systems at $z \sim 0$. The earlier onset of this evolution in overdense environments is in agreement with studies of stellar populations locally [e.g., @cooper10a] as well as studies of the color-density relation at intermediate redshift [e.g., @cooper06; @cooper07; @gerke07], which find that galaxies in high-density environments have typically ceased their star formation earlier than those in less-dense environs. Studies of the Fundamental Plane [FP, @dd87; @dressler87] support this picture of accelerated evolution in high-density environments, with several analyses finding that galaxies in high-density regions tend to reach the FP more quickly than those in low-density regions [@vandokkum01; @gebhardt03; @treu05; @moran05]. A correlation between size and environment at fixed stellar mass within the massive early-type population is arguably in conflict with scenarios in which the observed size evolution at intermediate redshift is driven by quasar feedback, as quasars at $z \sim 1$ are generally not found to reside in overdense environments, especially in relation to the early-type galaxy population. Using cross-correlation techniques and measurements of local environment analogous to those presented herein, @coil07 find that quasars at $z \sim 1$ cluster like blue galaxies, such that they favor regions of average galaxy density ($\log_{10}(1+\delta_3) \sim 0$). While these results suggest that quasars are unlikely to be responsible for the larger sizes of early-types in high-density regions, it should be noted that the clustering measurements of @coil07 are roughly $1$–$2\sigma$ lower than that of similar studies. For example, @croom05 [see also @porciani04 [@grazian04; @myers06]] find that quasars at $0.3 < z < 2.2$ have a bias $\gtrsim \! 2\sigma$ higher than that found by @coil07, while @serber06 find an excess of $\sim \! L^{*}$ galaxies on $25$ kpc to $1$ Mpc ($h=0.7$) projected comoving scales around quasars at $z < 0.4$ [see also @hennawi06; @myers08]. For comparison, the median third-nearest-neighbor distance for our sample of early-type galaxies (over all environments) is $\sim \! 0.6$ $h^{-1}$ comoving Mpc in projection. In addition, @croom05 find that the average dark matter halo mass for quasars at intermediate redshift is roughly consistent with (within a factor of a few of) the minimum halos mass inferred for groups at $z \sim 1$ [@coil06 see also @hopkins07]. Altogether, clustering and environment studies do not support (though also do not clearly exclude) quasars as a viable mechanism for size evolution at $z < 2$; regardless, questions still remain as to how quasar activity would be fueled in a massive early-type system at $z > 1$, as the standard scenario involving the major merger of two gas-rich systems [e.g., @springel05; @dimatteo05; @hopkins06] fails to accurately describe massive early-type systems at $z \gtrsim 1$, which are relatively gas-poor and have stellar populations with luminosity-weighted ages of $> \! 1$–$2$ Gyr [e.g., @daddi05; @longhetti05; @treu05b; @schiavon06; @combes07]. In contrast to quasars, Seyfert galaxies at $z \sim 1$ typically reside in higher-density environs, similar to those of galaxies on the red sequence [@georgakakis07; @georgakakis08; @coil09; @bradshaw11; @digby11 but see also @silverman09]. In addition, systems exhibiting line ratios consistent with Low Ionization Nuclear Emission-line Regions [LINERS, @heckman80] at $z \sim 1$, which tend to reside on the red sequence, are likely to inhabit slightly more overdense regions even relative to galaxies of like color and luminosity [i.e., stellar mass, @yan06; @yan11; @montero-dorta09; @juneau11] — though, recent work suggests that LINERs may not be the product of AGN activity [@yan11b]. While lower-luminosity AGN are found in high-density regions at $z \sim 1$, consistent with being the driving mechanism behind the observed size evolution of early-type galaxies, the lack of variation in the outflow velocities of winds observed in AGN hosts versus star-forming galaxies suggests that low-luminosity AGN may not play a dominant role in galactic feedback [@rupke05; @weiner09; @rubin10; @rubin11; @coil11 but see also @hainline11]. Furthermore, the mass loss needed to produce a factor of $\gtrsim \! 2$ increase in size is of order $30\%$–$50\%$ [@zhao02; @hopkins10b], beyond the expected impact of feedback from lower-luminosity AGN or evolved stars [e.g., @damjanov09]. Summary {#sec_summary} ======= Using data from the DEEP2 and DEEP3 Galaxy Redshift Surveys, we have completed a detailed study of the relationship between galaxy structure and environment on the massive ($10 < \log_{10}({\rm M}_{\star}/h^{-2}\ {\rm M}_{\sun}) < 11$) end of the red sequence at intermediate redshift. Our principal result is that at fixed stellar mass, redshift, Sérsic index, and rest-frame color we find a significant relationship between galaxy size and local galaxy density at $z \sim 0.75$, such that early-type galaxies in high-density regions are more extended than their counterparts in low-density environs. This result is robust to variations in the sample selection procedure, including selection limits based on axis ratio, surface brightness, and Sérsic index. The observed correlation between size and environment is consistent with a scenario in which minor, dry mergers play a critical role in the structural evolution of massive, early-type galaxies at $z < 2$ and in which the evolution of massive ellipticals is accelerated in high-density regions. Future work, for example from the CANDELS [*HST*]{}/WFC3-IR imaging program, will soon enable complementary analyses at yet higher redshift and in more extreme environments such as massive clusters [e.g., @papovich11]. MCC acknowledges support for this work provided by NASA through Hubble Fellowship grant \#HF-51269.01-A, awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. This work was also supported in part by NSF grants AST-0507428, AST-0507483, AST-0071048, AST-0071198, AST-0808133, and AST-0806732 as well as [*Hubble Space Telescope*]{} Archival grant HST-AR-10947.01 and NASA grant HST-G0-10134.13-A. Additional support was provided by NASA through the Spitzer Space Telescope Fellowship Program. MCC acknowledges support from the Southern California Center for Galaxy Evolution, a multi-campus research program funded by the University of California Office of Research. MCC thanks Mike Boylan-Kolchin for helpful discussions in preparing this manuscript and also thanks Greg Wirth and the entire Keck Observatory staff for their help in the acquisition of the DEEP2 and DEEP3 Keck/DEIMOS data. Finally, MCC thanks the anonymous referee for their insightful comments and suggestions that improved this work. 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[^2]: We adopt this simplified color-cut to be more restrictive at the faint end of the red sequence, where dusty star-forming galaxies are more prominent [@lotz08]. However, using a luminosity-dependent color-cut [e.g., Equation 19 of @willmer06] yields no significant changes in our results. [^3]: Limiting the sample to those sources with $\sigma_{n} < 0.75$ excludes very few (only $2$ out of $625$) objects. Removing this selection criterion or making it more restrictive (e.g., $\sigma_{n} < 0.5$) yields no significant changes in our results. [^4]: Selecting the top 10% or 20% of the environment distribution yields similar results. [^5]: Note that the two-sided probability for the WMW $U$ test is given by doubling the one-sided probability. Here, we report only the one-sided probability. [^6]: The average sizes for cluster, group, and field early-types as given in Table 1 of @raichoor11 are all consistent at the $1\sigma$ level.

--- abstract: 'The recently proposed (*Phys. Rev.* **A90** (2014), 062121 and *Phys. Rev.* **A91** (2015), 052110) group theoretical approach to the problem of breaking the Bell inequalities is applied to $S_4$ group. The Bell inequalities based on the choice of three orbits in the representation space corresponding to standard representation of $S_4$ are derived and their breaking is described. The corresponding nonlocal games are analyzed.' --- <span style="font-variant:small-caps;"></span> [^1]\ **\ [^2]\ **\ Introduction ============ The famous Bell inequalities [@bell] provide the necessary conditions for any theory to be a local realistic one. Their importance stems from the observation that they can be violated in quantum theory. As a result the Bell inequalities can be used for test of entanglement and as a basis for protocols in quantum cryptography [@ekert]. Bell inequalities have been studied intensively by numerous authors. Their various forms characterized by the number of parties, measurement settings and outcomes for each measurement have been derived [@clauser]$\div$[@cabello] (for a review, see [@liang] and [@brunner]). Recently, there appeared interesting papers [@ugur], [@ugur1] where the group theoretical method have been proposed as a tool for analyzing the quantum mechanical violation of Bell inequalities. Examples of Bell inequalities based on representations of some finite groups were presented there. Further example has been considered in Ref. [@bolonek]. It is based on $S_4$ symmetry and its standard irreducible representation. The resulting Bell inequality is obtained by selecting two generic orbits determined by the geometry of tetrahedron. In the present paper we provide further examples of Bell inequalities related to the symmetry of tetrahedron. They result from the particular choices of three generic orbits. The paper is organized as follows. In Sec. II we discuss the Bell inequalities from the point of view of the existence of joint probability distribution and describe the group theoretical approach to the problem of their breaking proposed by Gűney and Hillery. This approach is then applied in Sec. III to the symmetric group $S_4$. Three examples of quantum mechanical breakdown of Bell inequalities are presented. They are based on the specific choice of orbits in standard representation of $S_4$. In each of three cases we consider the set of states arising from the choice of three orbits. The results are interpreted in Sec. IV in the framework of game theory. Some technical detailes are relegated to the Appendix. Bell inequalities ================= Quantum mechanical violation of Bell inequalities is closely related to the existence of noncommuting observables. In two elegant papers [@fine], [@fine1] Fine provided a particulary transparent interpretation of Bell inequalities (see also [@halliwell], [@halliwell1]). Assume we have a number of random variables possessing joint probability distribution. Bell inequalities concern the joint probability distributions of some subsets of the initial set of random variables. They result from the assumption that the latter can be obtained as marginals from the original joint probability distribution. What is even more important, the Bell inequalities form also the sufficient conditions for the existence of joint distribution returning other probabilities as marginals. In fact, the latter condition provides a set of linear equations for the joint distribution which possess the whole family of solutions. We are interested in solutions belonging to the interval ${\left<0,1\right>}$. The possibility of selecting such solutions relies on the validity of Bell inequalities. Fine’s theorem explains the origin of quantum machanical violation of Bell inequalities. Due to the uncertainty principle the joint probability can be constructed only for the set of mutually commuting observables. Therefore, no inequality of Bell type could be derived for joint probabilities of commuting observables if they would follow from the assumption that these probabilities emerge as marginals from joint distribution for larger set of in general noncommuting observables. Let us illustrate the above discussion by a simple example. Let $\hat{A}$ be some observable with the spectral decomposition $$\hat{A}=\sum_{i}a_i\hat{\Pi}_i$$ where $\hat{\Pi}_i$ are the projectors on the relevant eigenspaces (we shall assume our space of states in finitedimensional). Consider any state $\hat{\rho}$ and let [@santos] $$C{\left(\zeta\right)}=\text{Tr}{\left(e^{i\zeta\hat{A}}\hat{\rho}\right)}=\sum_i e^{i\zeta a_i}\text{Tr}{\left(\hat{\Pi}_i\hat{\rho}\right)}\equiv\sum_i e^{i\zeta a_i}p_i$$ be the generating function for the moments of $\hat{A}$: $${\left<\hat{A}\right>}=\text{Tr}{\left(\hat{A}^n\hat{\rho}\right)}=\sum_i a_i^n p_i={\left(-i\frac{d}{d\zeta}\right)}^n C{\left(\zeta\right)}\Big |_{\zeta=0}.$$ The probability distribution is obtained by Fourier transform $$p{\left(a\right)}=\frac{1}{2\pi}\int d\zeta e^{-i\zeta a}C(\zeta)=\sum_i p_i\delta{\left(a-a_i\right)}.$$ Assume now we have two observables, $$\hat{A}_1\equiv \sum_i a_{1i}\hat{\Pi}_{1i},\qquad \hat{A}_2\equiv\sum_k a_{2k}\hat{\Pi}_{2k}.$$ The generating function for the moments ${\left<A_1^{n_1}A_2^{n_2}\right>}$ reads $$\begin{split} & C{\left(\zeta_1,\zeta_2\right)}=\text{Tr}{\left(e^{i\zeta_1\hat{A}_1}e^{i\zeta_2\hat{A}_2}\hat{\rho}\right)}=\sum_{i,k}e^{i\zeta_1 a_{1i}}e^{i\zeta_2 a_{2k}}\text{Tr}{\left(\hat{\Pi}_{1i}\hat{\Pi}_{2k}\hat{\rho}\right)}\equiv\\ &\qquad\qquad \equiv \sum_{i,k}e^{i\zeta_1 a_{1i}}e^{i\zeta_2 a_{2k}}p_{ik}. \end{split}$$ We are tempetd to define the joint probability as $$\begin{split} & p{\left(a_1,a_2\right)}\equiv\frac{1}{4\pi^2}\int d\zeta_1 d\zeta_2 e^{-i{\left(\zeta_1a_1+\zeta_2a_2\right)}}C{\left(\zeta_1,\zeta_2\right)}=\\ &\qquad\qquad =\sum_{i,k}p_{ik}\delta{\left(a_1-a_1i\right)}\delta{\left(a_2-a_{2k}\right)}.\label{a1} \end{split}$$ Due to $$\sum_{k}p_{ik}=\sum_k\text{Tr}{\left(\hat{\Pi}_{1i}\hat{\Pi}_{2k}\hat{\rho}\right)}=\text{Tr}{\left(\hat{\Pi}_{1i}{\left(\sum_k\hat{\Pi}_{2k}\right)}\hat{\rho}\right)}=\text{Tr}{\left(\hat{\Pi}_{1i}\hat{\rho}\right)}=p_{1i}$$ single probability densities can be obtained as marginals $$p_1{\left(a_1\right)}=\int da_2 p{\left(a_1,a_2\right)}.$$ To have the genuine probability distribution we must assume $p_{ik}\geq 0$. Then the last expression (\[a1\]) provides a finite positive measure on $\mathbb{R}^2$. Therefore, by Bochner theorem $C{\left(\zeta_1,\zeta_2\right)}$ is positive definite function [@reed]. In particular $$C{\left(\zeta_1,\zeta_2\right)}=\overline{C{\left(-\zeta_1,-\zeta_2\right)}}$$ or $$\text{Tr}{\left(e^{i\zeta_1\hat{A}_1}e^{i\zeta_2\hat{A}_2}\hat{\rho}\right)}=\text{Tr}{\left(e^{i\zeta_2\hat{A}_2}e^{i\zeta_1\hat{A}_1}\hat{\rho}\right)}.\label{a2}$$ Assuming that (\[a2\]) holds for all states $\hat{\rho}$ we find $$e^{i\zeta_1\hat{A}_1}e^{i\zeta_2\hat{A}_2}=e^{i\zeta_2\hat{A}_2}e^{i\zeta_1\hat{A}_1}$$ or ${\left[\hat{A}_1,\hat{A}_2\right]}=0$. We see that the joint probability can be defined only for commuting variables. Taking into account Fine’s results one concludes that the general scheme for deriving the Bell inequalities is quite simple. The relevant combination of probabilities is written in terms of marginals of the joint probability distribution, assumed to exist, arriving at the expression $\sum_{\alpha}c{\left(\alpha\right)}p{\left(\alpha\right)}$, where $c{\left(\alpha\right)}$ are integers equal to the number of times $p{\left(\alpha\right)}$ appears in the sum. Due to $0\leq p{\left(\alpha\right)}\leq 1$, $\sum\limits_{\alpha}p{\left(\alpha\right)}=1$ one obtains $$\min\limits_{\alpha} c{\left(\alpha\right)}\leq\sum_{\alpha}c{\left(\alpha\right)}p{\left(\alpha\right)}\leq \max\limits_{\alpha} c{\left(\alpha\right)}.\label{a4}$$ In order to get the standard form of Bell inequalities one should express $p{\left(\alpha\right)}$ in terms of relevant correlation functions. In order to establish the violation of Bell inequalities in quantum mechanics one has to construct the particular examples. In two papers mentioned above [@ugur], [@ugur1] Gűney and Hillery proposed to use the group theoretical methods. Consider some finite group $G$ and its irreducible representation $D$. The space carrying the representation $D$ becomes the space of states of one party. One selects an orbit ${\left\{D{\left(g\right)}{\left|\varphi\right>}\right\}}_{g\in G}$ in such a way that it decomposes into disjoint sets of orthonormal bases. These bases define the spectral decompositions of observables entering the example. The space of states of the second party carries the second representation in the product $D\otimes D$; the corresponding orbits read ${\left\{D{\left(g\right)}{\left|\psi\right>}\right\}}_{g\in G}$ and defines the observables of second party.\ Let us construct the operator $$X{\left(\varphi,\psi\right)}\equiv\sum_{g\in G}{\left(D{\left(g\right)}{\left|\varphi\right>}\otimes D{\left(g\right)}{\left|\psi\right>}\right)}{\left({\left<\varphi\right|}D^+{\left(g\right)}\otimes{\left<\psi\right|}D^+{\left(g\right)}\right)}.$$ Defining $$\begin{split} & {\left|g,\varphi\right>}\equiv D{\left(g\right)}{\left|\varphi\right>},\qquad {\left|g,\psi\right>}\equiv D{\left(g\right)}{\left|\psi\right>}\\ & {\left|g,\varphi,\psi\right>}\equiv {\left|g,\varphi\right>}\otimes {\left|g,\psi\right>} \end{split}$$ one finds for arbitrary bipartite state ${\left|\chi\right>}$ $${\left<\chi\right|}X{\left|\chi\right>}=\sum_{g\in G}{\left|{\left<g,\varphi,\psi | \chi\right>}\right|}^2.\label{a3}$$ The right hand side of eq. (\[a3\]) represents the sum of probabilities of particular outcomes of measurement performed on observables defined by the orbits ${\left\{{\left|g,\varphi\right>}\right\}}$ and ${\left\{{\left|g,\psi\right>}\right\}}$. Its maximal value corresponds to maximal eigenvector of $X$. In this way we obtain a kind of Cirel’son bound [@cirelson] for the class of states under consideration. On the other hand one easily derives the Bell inequality involving the sum of probabilities on the right hand side of eq. (\[a3\]). To this end one assumes the existence of joint probability distribution for all observables defined by both orbits (note that the ones belonging to one orbit in general do not commute) and uses the inequalities (\[a4\]). It remains to find the maximal eigenvalue of $X$. To this end assume that in the decomposition of $D\otimes D$ into irreducible pieces, $$D\otimes D=\bigoplus\limits_{s}D^{{\left(s\right)}}$$ each $D^{{\left(s\right)}}$ appears only once. Then, by Schur’s lemma, $X{\left(\varphi,\psi\right)}$ is diagonal and reduces to a multiple of unity on each irreducible component. Using the orthogonality relations it is easy to see that the relevant eigenvalues of $X{\left(\varphi,\psi\right)}$ are [@ugur1] $$\frac{{\left|G\right|}}{d_s}\parallel{\left({\left|\varphi\right>}\otimes{\left|\psi\right>}\right)}_s\parallel^2\label{b1}$$ where ${\left|G\right|}$ is the order of $G$, $d_s$ - the dimension of $D^{{\left(s\right)}}$ and ${\left({\left|\varphi\right>}\otimes{\left|\psi\right>}\right)}_s$ is the projection of ${\left|\varphi\right>}\otimes {\left|\psi\right>}$ on the carrier space of $D^{{\left(s\right)}}$. In general, in order to break the Bell inequality it is necessary to consider a number of orbits. To this end one considers the orbits generated by $N$ pairs of vectors ${\left({\left|\varphi_n\right>},{\left|\psi_n\right>}\right)}$ and the corresponding operators $X{\left(\varphi_n,\psi_n\right)}$. They mutually commute so the eigenvalues of $$X=\sum_{n=1}^N X{\left(\varphi_n,\psi_n\right)}\label{b4}$$ are the sums of eigenvalues of all $X{\left(\varphi_n,\psi_n\right)}$. In this way one can maximize the sum of probabilities $$\sum_{n=1}^N\sum_{g\in G}{\left|{\left<g,\varphi_n,\psi_n |\chi\right>}\right|}^2$$ and proceed as above. The $S_4$ group: three orbits ============================= $S_4$ is the group of order 24. It has 6 conjugancy classes. There exist six irreducible representations of $S_4$: trivial representation, the alternating representation, the homomorphic twodimensional one and two threedimansional representations, $D$ and $\widetilde{D}$; $\widetilde{D}$ is obtained from $D$ by multiplication by the alternating representation. All representations can be made orthogonal. Consider the threedimensional representation $D$. It can be described by writing out the matrices representing the transpositions: $$D{\left(12\right)}=\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1 \end{array}\right],\qquad D{\left(13\right)}=\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & -\frac{1}{2} & -\frac{\sqrt{3}}{2}\\ 0 & -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{array}\right]$$ $$D{\left(14\right)}=\left[\begin{array}{ccc} -\frac{1}{3} & -\frac{\sqrt{2}}{3} & -\frac{\sqrt{6}}{3}\\ -\frac{\sqrt{2}}{3} & \frac{5}{6} & -\frac{\sqrt{3}}{6}\\ -\frac{\sqrt{6}}{3} & -\frac{\sqrt{3}}{6}& \frac{1}{2} \end{array}\right],\qquad D{\left(23\right)}=\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & -\frac{1}{2} & \frac{\sqrt{3}}{2}\\ 0 & \frac{\sqrt{3}}{2} & \frac{1}{2} \end{array}\right]$$ $$D{\left(24\right)}=\left[\begin{array}{ccc} -\frac{1}{3} & -\frac{\sqrt{2}}{3} & \frac{\sqrt{6}}{3}\\ -\frac{\sqrt{2}}{3} & \frac{5}{6} & \frac{\sqrt{3}}{6}\\ \frac{\sqrt{6}}{3} & \frac{\sqrt{3}}{6}& \frac{1}{2} \end{array}\right],\qquad D{\left(34\right)}=\left[\begin{array}{ccc} -\frac{1}{3} & \frac{\sqrt{8}}{3} & 0\\ \frac{\sqrt{8}}{3} & \frac{1}{3} & 0\\ 0 & 0 & 1 \end{array}\right].$$ $S_4$ is the symmetry group of regular tetrahedron. One can make the correspondence between the symmetries of tetrahedron and the representation $D$. To this end we find an (degenerate) orbit which forms a tetrahedron. It is easy to check that the vectors: $\vec{a}_1={\left(-\frac{1}{3},-\frac{\sqrt{2}}{3},-\frac{\sqrt{6}}{3}\right)}$, $\vec{a}_2={\left(-\frac{1}{3},-\frac{\sqrt{2}}{3},\frac{\sqrt{6}}{3}\right)}$, $\vec{a}_3={\left(-\frac{1}{3},\frac{\sqrt{8}}{3},0\right)}$ and $\vec{a}_4={\left(1,0,0\right)}$ form the orbit of $S_4$ and are the vertices of regular tetrahedron. Now, let us select the orbits we will use in the construction of our examples of breaking the Bell inequalities. A generic orbit consists of 24 states. According to the discussion in Sec. II we should choose the orbit consisting of eight triples of orthonormal vectors, each providing the spectral decomposition of one of the eight observables. The elements of the orbit are numbered as ${\left|x_\alpha^i\right>}$, $i=1,\ldots,8$, $\alpha=0,1,2$. Then we demand ${\left<x_\alpha^i | x_\beta^i\right>}=\delta_{\alpha\beta}$, $i=1,\ldots ,8$ (no summation over $i$). Consequently, we are dealing with twice the eight observables ($a_i$ stands for $"$Alice$"$ and $b_i$ for $"$Bob$"$) $$a_i=\sum_{\alpha=0}^2\alpha{\left|x_\alpha^i\right>}{\left<x_\alpha^i\right|},\qquad b_i=\sum_{\beta=0}^2\beta{\left|x_\beta^i\right>}{\left<x_\beta^i\right|}.$$ Taking into account that any element of $S_4$ is a product of transpositions and each transposition is represented by a reflection in the symmetry plane orthogonal to the edge connecting the transposed verticles we find the relevant orbit by simple examination of terahedron geometry\ $a_1:\quad {\left(x_0^1,x_1^1,x_2^1\right)}$ $${\left|x_0^1\right>}=\left[\begin{array}{c} \frac{\sqrt{3}}{3}\\ \frac{\sqrt{3}}{3}\\ -\frac{\sqrt{3}}{3} \end{array}\right], \quad {\left|x_1^1\right>}=\left[\begin{array}{c} \frac{\sqrt{3}}{3}\\ \frac{1}{2}{\left(1-\frac{\sqrt{3}}{3}\right)}\\ \frac{1}{2}{\left(1+\frac{\sqrt{3}}{3}\right)}\\ \end{array}\right],\quad {\left|x_2^1\right>}=\left[\begin{array}{c} \frac{\sqrt{3}}{3}\\ -\frac{1}{2}{\left(1+\frac{\sqrt{3}}{3}\right)}\\ -\frac{1}{2}{\left(1-\frac{\sqrt{3}}{3}\right)}\\ \end{array}\right]\label{a5}$$ $a_2:\quad {\left(x_0^2,x_1^2,x_2^2\right)}$ $${\left|x_0^2\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(-3\sqrt{2}-\sqrt{3}-\sqrt{6}\right)}\\ \frac{1}{18}{\left(-3+5\sqrt{3}-2\sqrt{6}\right)}\\ \frac{1}{6}{\left(-1-2\sqrt{2}+\sqrt{3}\right)}\\ \end{array}\right],\, {\left|x_1^2\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(-\sqrt{3}+2\sqrt{6}\right)}\\ \frac{1}{18}{\left(9-\sqrt{3}-2\sqrt{6}\right)}\\ -\frac{1}{6}{\left(1+2\sqrt{2}+\sqrt{3}\right)}\\ \end{array}\right],\, {\left|x_2^2\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(3\sqrt{2}-\sqrt{3}-\sqrt{6}\right)}\\ -\frac{1}{9}{\left(3+2\sqrt{3}+\sqrt{6}\right)}\\ \frac{1}{3}{\left(1-\sqrt{2}\right)}\\ \end{array}\right],$$ $a_3:\quad {\left(x_0^3,x_1^3,x_2^3\right)}$ $${\left|x_0^3\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(3\sqrt{2}-\sqrt{3}-\sqrt{6}\right)}\\ \frac{1}{18}{\left(3+5\sqrt{3}-2\sqrt{6}\right)}\\ \frac{1}{6}{\left(1+2\sqrt{2}+\sqrt{3}\right)}\\ \end{array}\right],\, {\left|x_1^3\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(-\sqrt{3}+2\sqrt{6}\right)}\\ -\frac{1}{18}{\left(9+\sqrt{3}+2\sqrt{6}\right)}\\ \frac{1}{6}{\left(1+2\sqrt{2}-\sqrt{3}\right)}\\ \end{array}\right],\, {\left|x_2^3\right>}=\left[\begin{array}{c} -\frac{1}{9}{\left(3\sqrt{2}+\sqrt{3}+\sqrt{6}\right)}\\ \frac{1}{9}{\left(3-2\sqrt{3}-\sqrt{6}\right)}\\ \frac{1}{3}{\left(-1+\sqrt{2}\right)}\\ \end{array}\right],$$ $a_4:\quad {\left(x_0^4,x_1^4,x_2^4\right)}$ $${\left|x_0^4\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(3\sqrt{2}-\sqrt{3}-\sqrt{6}\right)}\\ \frac{1}{18}{\left(3-\sqrt{3}+4\sqrt{6}\right)}\\ \frac{1}{6}{\left(3+\sqrt{3}\right)}\\ \end{array}\right],\, {\left|x_1^4\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(-\sqrt{3}+2\sqrt{6}\right)}\\ \frac{1}{9}{\left(\sqrt{3}+2\sqrt{6}\right)}\\ -\frac{\sqrt{3}}{3} \end{array}\right],\, {\left|x_2^4\right>}=\left[\begin{array}{c} -\frac{1}{9}{\left(3\sqrt{2}+\sqrt{3}+\sqrt{6}\right)}\\ \frac{1}{18}{\left(-3-\sqrt{3}+4\sqrt{6}\right)}\\ \frac{1}{6}{\left(-3+\sqrt{3}\right)}\\ \end{array}\right],$$ $a_5:\quad {\left(x_0^5,x_1^5,x_2^5\right)}$ $${\left|x_0^5\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(-\sqrt{3}+2\sqrt{6}\right)}\\ \frac{1}{18}{\left(9-\sqrt{3}-2\sqrt{6}\right)}\\ \frac{1}{6}{\left(1+2\sqrt{2}+\sqrt{3}\right)}\\ \end{array}\right],\, {\left|x_1^5\right>}=\left[\begin{array}{c} -\frac{1}{9}{\left(3\sqrt{2}+\sqrt{3}+\sqrt{6}\right)}\\ \frac{1}{18}{\left(-3+5\sqrt{3}-2\sqrt{6}\right)}\\ \frac{1}{6}{\left(1+2\sqrt{2}-\sqrt{3}\right)}\\ \end{array}\right],\, {\left|x_2^5\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(3\sqrt{2}-\sqrt{3}-\sqrt{6}\right)}\\ -\frac{1}{9}{\left(3+2\sqrt{3}+\sqrt{6}\right)}\\ \frac{1}{3}{\left(-1+\sqrt{2}\right)}\\ \end{array}\right],$$ $a_6:\quad {\left(x_0^6,x_1^6,x_2^6\right)}$ $${\left|x_0^6\right>}=\left[\begin{array}{c} -\frac{1}{9}{\left(3\sqrt{2}+\sqrt{3}+\sqrt{6}\right)}\\ \frac{1}{18}{\left(-3-\sqrt{3}+4\sqrt{6}\right)}\\ \frac{1}{6}{\left(3-\sqrt{3}\right)} \end{array}\right],\, {\left|x_1^6\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(3\sqrt{2}-\sqrt{3}-\sqrt{6}\right)}\\ \frac{1}{18}{\left(3-\sqrt{3}+4\sqrt{6}\right)}\\ -\frac{1}{6}{\left(3+\sqrt{3}\right)} \end{array}\right],\, {\left|x_2^6\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(-\sqrt{3}+2\sqrt{6}\right)}\\ \frac{1}{9}{\left(\sqrt{3}+2\sqrt{6}\right)}\\ \frac{\sqrt{3}}{3} \end{array}\right],$$ $a_7:\quad {\left(x_0^7,x_1^7,x_2^7\right)}$ $${\left|x_0^7\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(3\sqrt{2}-\sqrt{3}-\sqrt{6}\right)}\\ \frac{1}{18}{\left(3+5\sqrt{3}-2\sqrt{6}\right)}\\ -\frac{1}{6}{\left(1+2\sqrt{2}+\sqrt{3}\right)} \end{array}\right],\, {\left|x_1^7\right>}=\left[\begin{array}{c} \frac{1}{9}{\left(-\sqrt{3}+2\sqrt{6}\right)}\\ -\frac{1}{18}{\left(9+\sqrt{3}+2\sqrt{6}\right)}\\ \frac{1}{6}{\left(-1-2\sqrt{2}+\sqrt{3}\right)}\\ \end{array}\right],\, {\left|x_2^7\right>}=\left[\begin{array}{c} -\frac{1}{9}{\left(3\sqrt{2}+\sqrt{3}+\sqrt{6}\right)}\\ \frac{1}{9}{\left(3-2\sqrt{3}-\sqrt{6}\right)}\\ \frac{1}{3}{\left(1-\sqrt{2}\right)}\\ \end{array}\right],$$ $a_8:\quad {\left(x_0^8,x_1^8,x_2^8\right)}$ $${\left|x_0^8\right>}=\left[\begin{array}{c} \frac{\sqrt{3}}{3}\\ -\frac{1}{2}{\left(1+\frac{\sqrt{3}}{3}\right)}\\ \frac{1}{2}{\left(1-\frac{\sqrt{3}}{3}\right)} \end{array}\right],\, {\left|x_1^8\right>}=\left[\begin{array}{c} \frac{\sqrt{3}}{3}\\ \frac{1}{2}{\left(1-\frac{\sqrt{3}}{3}\right)}\\ -\frac{1}{2}{\left(1+\frac{\sqrt{3}}{3}\right)} \end{array}\right],\, {\left|x_2^8\right>}=\left[\begin{array}{c} \frac{\sqrt{3}}{3}\\ \frac{\sqrt{3}}{3}\\ \frac{\sqrt{3}}{3} \end{array}\right].\label{a6}$$ For all examples given below the states describing both parties beolng to the same orbit defined by eqs. (\[a5\])$\div$(\[a6\]). However, in each case the orbit of second party ($"$Bob$"$) will be shifted with respect to the one of first party ($"$Alice$"$). In order to compute the eigenvalues of the operator $X$ we should know the matrix which relates the product basis to the one in which the decomposition $$D\otimes D=D\oplus \widetilde{D}\oplus D_2\oplus D_0\label{b3}$$ is explicit. It reads $$C=\left [\begin{array}{ccccccccc} \sqrt{\frac{2}{3}} & 0 & 0 & 0 & -\frac{1}{\sqrt{6}} & 0 & 0 & 0 & -\frac{1}{\sqrt{6}}\\ 0 & -\frac{1}{\sqrt{6}} & 0 & -\frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} & 0 & 0 & 0 & -\frac{1}{\sqrt{3}}\\ 0 & 0 & -\frac{1}{\sqrt{6}} & 0 & 0 & -\frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{6}} & -\frac{1}{\sqrt{3}} & 0\\ 0 & \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & -\frac{1}{\sqrt{2}} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} & 0\\ 0 & \frac{1}{\sqrt{3}} & 0 & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{6}} & 0 & 0 & 0 & -\frac{1}{\sqrt{6}}\\ 0 & 0 & \frac{1}{\sqrt{3}} & 0 & 0 & -\frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{6}} & 0\\ \frac{1}{\sqrt{3}} & 0 & 0 & 0 & \frac{1}{\sqrt{3}} & 0 & 0 & 0 & \frac{1}{\sqrt{3}} \end{array}\right].\label{b2}$$ Eqs. (\[b1\]) and (\[b2\]) allow us to compute all eigenvalues of arbitrary operator $X{\left(\varphi,\psi\right)}$. In all examples given below the largest eigenvalue $\lambda_{max}$ correspond to the scalar component in the decomposition (\[b3\]). All of them involve three orbits (N=3 in eq. (\[b4\])). : - First orbit\ ${\left|\varphi_1\right>}={\left|x_0^1\right>},\quad {\left|\psi_1\right>}={\left|x_1^4\right>}$\ $\lambda_{max}\simeq 7,40$ - Second orbit\ ${\left|\varphi_2\right>}={\left|x_0^1\right>},\quad {\left|\psi_2\right>}={\left|x_0^7\right>}$\ $\lambda_{max}\simeq 4,57$ - Third orbit\ ${\left|\varphi_3\right>}={\left|x_0^1\right>},\quad {\left|\psi_3\right>}={\left|x_1^5\right>}$\ $\lambda_{max}\simeq 4,12$ The maximal eigenvalue of the operator $X$ defined by eq. (\[b4\]):\ $\lambda_{max}(X)\simeq 16,09$.\ : - First orbit\ ${\left|\varphi_1\right>}={\left|x_0^1\right>},\quad {\left|\psi_1\right>}={\left|x_2^3\right>}$\ $\lambda_{max}\simeq 5,21$ - Second orbit\ ${\left|\varphi_2\right>}={\left|x_0^1\right>},\quad {\left|\psi_2\right>}={\left|x_1^6\right>}$\ $\lambda_{max}\simeq 5,30$ - Third orbit\ ${\left|\varphi_3\right>}={\left|x_0^1\right>},\quad {\left|\psi_3\right>}={\left|x_0^1\right>}$\ $\lambda_{max}\simeq 8,00$ The maximal eigenvalue of $X$:\ $\lambda_{max}(X)\simeq 18,51$. : - First orbit\ ${\left|\varphi_1\right>}={\left|x_0^1\right>},\quad {\left|\psi_1\right>}={\left|x_2^5\right>}$\ $\lambda_{max}\simeq 3,35$ - Second orbit\ ${\left|\varphi_2\right>}={\left|x_0^1\right>},\quad {\left|\psi_2\right>}={\left|x_1^4\right>}$\ $\lambda_{max}\simeq 7,40$ - Third orbit\ ${\left|\varphi_3\right>}={\left|x_0^1\right>},\quad {\left|\psi_3\right>}={\left|x_1^8\right>}$\ $\lambda_{max}\simeq 6,63$ The maximal eigenvalue of $X$:\ $\lambda_{max}(X)\simeq 17,38$.\ The corresponding sums of probabilities appearing on the right hand side of eq. (\[a3\]) are written out explicitly in Appendix. Having computed the (maximal) quantum mechanical values of the relevant sums of probabilities one can study the corresponding Bell inequalities. To this end we compute the coefficients $c(\alpha)$ entering the inequalities (\[a4\]). There are 16 observables 8 for $"$Alice$"$ and 8 for $"$Bob$"$. Therefore, the assumed joint probability is defined for $3^{16}$ configurations. We used the computer to check, for three examples above, how many times any given configuration appears in 72 terms in $"$classical$"$ counterpart of the right hand side of eq. (\[a3\]). The result are summarized in Appendix. It follows that the relevant sums of probabilities have the upper bounds 16, 18 and 16 for the examples I, II and III, respectively. This implies that in all three examples the Bell inequalities are broken. Interpretation in terms of game theory ====================================== As it has been described in Refs. [@ugur] and [@ugur1] the Bell inequalities can be discussed in terms of a nonlocal game. To this end we assume there are two players, Alice and Bob and an arbitrator who sends Alice a value $s$ and Bob a value $t$, $s=1,2,\ldots,8$, $t=1,2\ldots,8$; assume that all of 64 possible values of ${\left(s,t\right)}$ are equally likely. After receiving the numbers $s$ and $t$ from an arbitrator both Alice nad Bob transmit back the numbers $a$ and $b$, respectively, where $a=0,1,2$, $b=0,1,2$. They win iff the configuration ${\left(a_s=a,b_t=b\right)}$ appears in the sum of probabilities corresponding to the right hand side of eq. (\[a3\]). Let us consider for definiteness the example I. Using (\[d\]) we get the set of winning values given in Table 1. s, t Alice, Bob ------ ------------ 14 01, 10, 22 15 01, 10, 22 17 00, 12, 21 24 02, 11, 20 25 01, 10, 22 28 02, 11, 20 34 00, 11, 22 37 00, 11, 22 38 02, 10, 21 41 01, 10, 22 42 02, 11, 20 43 00, 11, 22 : Winning configurations for nonlocal game defined by three orbits from example I s, t Alice, Bob ------ ------------ 51 01, 10, 22 52 01, 10, 22 56 02, 10, 21 65 01, 12, 20 67 02, 10, 21 68 00, 11, 22 71 00, 12, 21 73 00, 11, 22 76 01, 12, 20 82 02, 11, 20 83 01, 12, 20 86 00, 11, 22 : Winning configurations for nonlocal game defined by three orbits from example I Following Ref. [@ugur1] we can show that the maximal classical probability of winning the game is determined by Bell inequality. In fact, let $f_A{\left(s\right)}$ and $f_B{\left(t\right)}$ be the strategies of Alice and Bob, respectively; the function $f_{A,B}$ take their values in the set ${\left\{0,1,2\right\}}$. Let $F{\left(a,b;s,t\right)}$ be the characteristic function for the set of winning strategies. Then the winning probability for the given strategies $f_A$, $f_B$ is $$\frac{1}{64}\sum_{a,b=0}^2\sum_{s,t=1}^8 F{\left(a,b;s,t\right)}\delta_{a,f_A{\left(s\right)}}\delta_{b,f_B{\left(t\right)}}.\label{c}$$ Now, the sum entering the left hand side of Bell inequality can be written as $$\sum_{a,b=0}^2\sum_{s,t=1}^8 F{\left(a,b;s,t\right)}p{\left(a_s=a, b_t=b\right)}$$ which is bounded, in example I, by 16 provided $p{\left(a_s=a,b_t=b\right)}$ can be derived from a joint probability distribution. However, defining $$p{\left(a_1,\dots,a_8,b_1,\ldots,b_8\right)}\equiv \prod_{k=1}^8 \delta_{a_k, f_A(k)}\delta_{b_k, f_B(k)}$$ we find that $p{\left(a_s,b_t\right)}$ are derived as marginals from the above joint probability. Therefore, the success probability for any classical strategy ${\left(f_A(s),f_B(t)\right)}$ cannot exceed $\frac{16}{64}=0,25$.\ Note that the optimal strategy saturating this limit always exists. To see this let $\alpha={\left(\underline{a}_1,\ldots,\underline{a}_8,\underline{b}_1,\ldots,\underline{b}_8\right)}$ be one of configurations for which $c(\alpha)$ attains its maximal value. Then the Bell inequality is saturated for the joint distribution probability $p(\alpha)=1$, $p(\alpha')=0$ for $\alpha '\neq\alpha$. Such distribution can be written in form (\[c\]) with $f_A(s)=\underline{a}_s$, $f_B(t)=\underline{b}_t$. In the quantum strategy Alice and Bob share the state corresponding to the maximal eigenvalue of $\sum_{n=1}^3 X{\left(\varphi_n,\psi_n\right)}$. If they receive the numbers $s$, $t$ from an arbitrator, they measure $a_s$ (Alice) and $b_t$ (Bob), respectively, and send the result to the arbitrator. The probability of winning in example I is then $\frac{16,09}{64}\simeq 0,2514$ which exceeds (although only slightly) the classical bound. Other examples can be treated similarly. To make the results slightly more transparent we write out explicitly the sum of probabilities appearing on the right hand side of eq. (\[a3\]). They read: Example I $$\begin{split} & S_1\equiv P{\left(a_1=0,b_4=1\right)}+P{\left(a_1=1,b_5=0\right)}+P{\left(a_1=2,b_7=1\right)} +P{\left(a_2=0,b_4=2\right)}+\\ & \quad +P{\left(a_2=1,b_8=1\right)}+P{\left(a_2=2,b_5=2\right)}+P{\left(a_3=0,b_4=0\right)}+P{\left(a_3=1,b_8=0\right)}+\\ & \quad +P{\left(a_3=2,b_7=2\right)}+P{\left(a_4=0,b_3=0\right)}+P{\left(a_4=1,b_1=0\right)}+P{\left(a_4=2,b_2=0\right)}+\\ & \quad +P{\left(a_5=0,b_1=1\right)}+P{\left(a_5=1,b_6=0\right)}+P{\left(a_5=2,b_2=2\right)}+P{\left(a_6=0,b_5=1\right)}+\\ & \quad +P{\left(a_6=1,b_7=0\right)}+P{\left(a_6=2,b_8=2\right)}+P{\left(a_7=0,b_6=1\right)}+P{\left(a_7=1,b_1=2\right)}+\\ & \quad +P{\left(a_7=2,b_3=2\right)}+P{\left(a_8=0,b_3=1\right)}+P{\left(a_8=1,b_2=1\right)}+P{\left(a_8=2,b_6=2\right)}+\\ & \quad +P{\left(a_1=0,b_7=0\right)}+P{\left(a_1=1,b_4=0\right)}+P{\left(a_1=2,b_5=2\right)} +P{\left(a_2=0,b_5=1\right)}+\\ & \quad +P{\left(a_2=1,b_4=1\right)}+P{\left(a_2=2,b_8=0\right)}+P{\left(a_3=0,b_8=2\right)}+P{\left(a_3=1,b_7=1\right)}+\\ & \quad +P{\left(a_3=2,b_4=2\right)}+P{\left(a_4=0,b_1=1\right)}+P{\left(a_4=1,b_2=1\right)}+P{\left(a_4=2,b_3=2\right)}+\\ & \quad +P{\left(a_5=0,b_6=2\right)}+P{\left(a_5=1,b_2=0\right)}+P{\left(a_5=2,b_1=2\right)}+P{\left(a_6=0,b_7=2\right)}+\\ & \quad +P{\left(a_6=1,b_8=1\right)}+P{\left(a_6=2,b_5=0\right)}+P{\left(a_7=0,b_1=0\right)}+P{\left(a_7=1,b_3=1\right)}+\\ & \quad +P{\left(a_7=2,b_6=0\right)}+P{\left(a_8=0,b_2=2\right)}+P{\left(a_8=1,b_6=1\right)}+P{\left(a_8=2,b_3=0\right)}+\\ & \quad +P{\left(a_1=0,b_5=1\right)}+P{\left(a_1=1,b_7=2\right)}+P{\left(a_1=2,b_4=2\right)} +P{\left(a_2=0,b_8=2\right)}+\\ & \quad +P{\left(a_2=1,b_5=0\right)}+P{\left(a_2=2,b_4=0\right)}+P{\left(a_3=0,b_7=0\right)}+P{\left(a_3=1,b_4=1\right)}+\\ & \quad +P{\left(a_3=2,b_8=1\right)}+P{\left(a_4=0,b_2=2\right)}+P{\left(a_4=1,b_3=1\right)}+P{\left(a_4=2,b_1=2\right)}+\\ & \quad +P{\left(a_5=0,b_2=1\right)}+P{\left(a_5=1,b_1=0\right)}+P{\left(a_5=2,b_6=1\right)}+P{\left(a_6=0,b_8=0\right)}+\\ & \quad +P{\left(a_6=1,b_5=2\right)}+P{\left(a_6=2,b_7=1\right)}+P{\left(a_7=0,b_3=0\right)}+P{\left(a_7=1,b_6=2\right)}+\\ & \quad +P{\left(a_7=2,b_1=1\right)}+P{\left(a_8=0,b_6=0\right)}+P{\left(a_8=1,b_3=2\right)}+P{\left(a_8=2,b_2=0\right)} \end{split}\label{d}$$ Example II: $$\begin{split} & S_2\equiv P{\left(a_1=0,b_3=2\right)}+P{\left(a_1=1,b_2=0\right)}+P{\left(a_1=2,b_6=0\right)} +P{\left(a_2=0,b_1=1\right)}+\\ & \quad +P{\left(a_2=1,b_3=0\right)}+P{\left(a_2=2,b_6=2\right)}+P{\left(a_3=0,b_2=1\right)}+P{\left(a_3=1,b_6=1\right)}+\\ & \quad +P{\left(a_3=2,b_1=0\right)}+P{\left(a_4=0,b_7=1\right)}+P{\left(a_4=1,b_5=2\right)}+P{\left(a_4=2,b_8=0\right)}+\\ & \quad +P{\left(a_5=0,b_7=0\right)}+P{\left(a_5=1,b_8=1\right)}+P{\left(a_5=2,b_4=1\right)}+P{\left(a_6=0,b_1=2\right)}+\\ & \quad +P{\left(a_6=1,b_3=1\right)}+P{\left(a_6=2,b_2=2\right)}+P{\left(a_7=0,b_5=0\right)}+P{\left(a_7=1,b_4=0\right)}+\\ & \quad +P{\left(a_7=2,b_8=2\right)}+P{\left(a_8=0,b_4=2\right)}+P{\left(a_8=1,b_5=1\right)}+P{\left(a_8=2,b_7=2\right)}+\\ & \quad +P{\left(a_1=0,b_6=1\right)}+P{\left(a_1=1,b_3=0\right)}+P{\left(a_1=2,b_2=2\right)} +P{\left(a_2=0,b_6=0\right)}+\\ & \quad +P{\left(a_2=1,b_1=0\right)}+P{\left(a_2=2,b_3=1\right)}+P{\left(a_3=0,b_6=2\right)}+P{\left(a_3=1,b_1=2\right)}+\\ & \quad +P{\left(a_3=2,b_2=0\right)}+P{\left(a_4=0,b_5=0\right)}+P{\left(a_4=1,b_8=1\right)}+P{\left(a_4=2,b_7=2\right)}+\\ & \quad +P{\left(a_5=0,b_8=2\right)}+P{\left(a_5=1,b_4=2\right)}+P{\left(a_5=2,b_7=1\right)}+P{\left(a_6=0,b_3=2\right)}+\\ & \quad +P{\left(a_6=1,b_2=1\right)}+P{\left(a_6=2,b_1=1\right)}+P{\left(a_7=0,b_4=1\right)}+P{\left(a_7=1,b_8=0\right)}+\\ & \quad +P{\left(a_7=2,b_5=1\right)}+P{\left(a_8=0,b_5=2\right)}+P{\left(a_8=1,b_7=0\right)}+P{\left(a_8=2,b_4=0\right)}+\\ & \quad +P{\left(a_1=0,b_1=0\right)}+P{\left(a_1=1,b_1=1\right)}+P{\left(a_1=2,b_1=2\right)} +P{\left(a_2=0,b_2=0\right)}+\\ & \quad +P{\left(a_2=1,b_2=1\right)}+P{\left(a_2=2,b_2=2\right)}+P{\left(a_3=0,b_3=0\right)}+P{\left(a_3=1,b_3=1\right)}+\\ & \quad +P{\left(a_3=2,b_3=2\right)}+P{\left(a_4=0,b_4=0\right)}+P{\left(a_4=1,b_4=1\right)}+P{\left(a_4=2,b_4=2\right)}+\\ & \quad +P{\left(a_5=0,b_5=0\right)}+P{\left(a_5=1,b_5=1\right)}+P{\left(a_5=2,b_5=2\right)}+P{\left(a_6=0,b_6=0\right)}+\\ & \quad +P{\left(a_6=1,b_6=1\right)}+P{\left(a_6=2,b_6=2\right)}+P{\left(a_7=0,b_7=0\right)}+P{\left(a_7=1,b_7=1\right)}+\\ & \quad +P{\left(a_7=2,b_7=2\right)}+P{\left(a_8=0,b_8=0\right)}+P{\left(a_8=1,b_8=1\right)}+P{\left(a_8=2,b_8=2\right)} \end{split}$$ Example III: $$\begin{split} & S_3\equiv P{\left(a_1=0,b_5=2\right)}+P{\left(a_1=1,b_7=0\right)}+P{\left(a_1=2,b_4=0\right)} +P{\left(a_2=0,b_8=0\right)}+\\ & \quad +P{\left(a_2=1,b_5=1\right)}+P{\left(a_2=2,b_4=1\right)}+P{\left(a_3=0,b_7=1\right)}+P{\left(a_3=1,b_4=2\right)}+\\ & \quad +P{\left(a_3=2,b_8=2\right)}+P{\left(a_4=0,b_2=1\right)}+P{\left(a_4=1,b_3=2\right)}+P{\left(a_4=2,b_1=1\right)}+\\ & \quad +P{\left(a_5=0,b_2=0\right)}+P{\left(a_5=1,b_1=2\right)}+P{\left(a_5=2,b_6=2\right)}+P{\left(a_6=0,b_8=1\right)}+\\ & \quad +P{\left(a_6=1,b_5=0\right)}+P{\left(a_6=2,b_7=2\right)}+P{\left(a_7=0,b_3=1\right)}+P{\left(a_7=1,b_6=0\right)}+\\ & \quad +P{\left(a_7=2,b_1=0\right)}+P{\left(a_8=0,b_6=1\right)}+P{\left(a_8=1,b_3=0\right)}+P{\left(a_8=2,b_2=2\right)}+\\ & \quad +P{\left(a_1=0,b_4=1\right)}+P{\left(a_1=1,b_5=0\right)}+P{\left(a_1=2,b_7=1\right)} +P{\left(a_2=0,b_4=2\right)}+\\ & \quad +P{\left(a_2=1,b_8=1\right)}+P{\left(a_2=2,b_5=2\right)}+P{\left(a_3=0,b_4=0\right)}+P{\left(a_3=1,b_8=0\right)}+\\ & \quad +P{\left(a_3=2,b_7=2\right)}+P{\left(a_4=0,b_3=0\right)}+P{\left(a_4=1,b_1=0\right)}+P{\left(a_4=2,b_2=0\right)}+\\ & \quad +P{\left(a_5=0,b_1=1\right)}+P{\left(a_5=1,b_6=0\right)}+P{\left(a_5=2,b_2=2\right)}+P{\left(a_6=0,b_5=1\right)}+\\ & \quad +P{\left(a_6=1,b_7=0\right)}+P{\left(a_6=2,b_8=2\right)}+P{\left(a_7=0,b_6=1\right)}+P{\left(a_7=1,b_1=2\right)}+\\ & \quad +P{\left(a_7=2,b_3=2\right)}+P{\left(a_8=0,b_3=1\right)}+P{\left(a_8=1,b_2=1\right)}+P{\left(a_8=2,b_6=2\right)}+\\ & \quad +P{\left(a_1=0,b_8=1\right)}+P{\left(a_1=1,b_8=2\right)}+P{\left(a_1=2,b_8=0\right)} +P{\left(a_2=0,b_7=2\right)}+\\ & \quad +P{\left(a_2=1,b_7=0\right)}+P{\left(a_2=2,b_7=1\right)}+P{\left(a_3=0,b_5=0\right)}+P{\left(a_3=1,b_5=2\right)}+\\ & \quad +P{\left(a_3=2,b_5=1\right)}+P{\left(a_4=0,b_6=2\right)}+P{\left(a_4=1,b_6=1\right)}+P{\left(a_4=2,b_6=0\right)}+\\ & \quad +P{\left(a_5=0,b_3=0\right)}+P{\left(a_5=1,b_3=2\right)}+P{\left(a_5=2,b_3=1\right)}+P{\left(a_6=0,b_4=2\right)}+\\ & \quad +P{\left(a_6=1,b_4=1\right)}+P{\left(a_6=2,b_4=0\right)}+P{\left(a_7=0,b_2=1\right)}+P{\left(a_7=1,b_2=2\right)}+\\ & \quad +P{\left(a_7=2,b_2=0\right)}+P{\left(a_8=0,b_1=2\right)}+P{\left(a_8=1,b_1=0\right)}+P{\left(a_8=2,b_1=1\right)} \end{split}$$ Therefore, the corresponding Bell inequalities take the form $$S_1\leq 16$$ $$S_2\leq 18$$ $$S_3\leq 16.$$ They were obtained by assuming the existence of joint of random variables $a_1,\ldots,a_8,$ $b_1,\ldots,b_8$ and computing the coefficients $c(\alpha)$ defined in Sec. II. More precisely, for each example we write all probabilities entering the sums $S_1$, $S_2$, $S_3$ as the marginals of joint probability distribution. As a result we obtain the expressions of the form $$S=\sum_\alpha c(\alpha)p(\alpha)$$ where $\alpha$ runs over all $3^{16}$ configurations of the variables $a_1,\ldots,a_8,b_1,\ldots,b_8$. The results of numerical computations are summarized in the Table below. ------------- ----------------------- ------------- ----------------------- ------------- ----------------------- $c(\alpha)$ No. of configurations $c(\alpha)$ No. of configurations $c(\alpha)$ No. of configurations 1 12 960 1 9 720 1 18 360 2 159 408 2 126 576 2 115 596 3 645 408 3 510 480 3 474 696 4 1 729 188 4 1 514 862 4 1 445 778 5 3 479 760 5 3 182 904 5 3 286 224 6 5 424 408 6 5 374584 6 5 510 160 7 6 896 016 7 7 139 664 7 7 178 976 8 7 261 569 8 7 822 791 8 7 670 547 9 6 410 016 9 6 903 648 9 6 795 936 10 4 866 480 10 5 058 216 10 5 012 208 11 3 176 496 11 3 006 000 11 3 087 504 12 1 758 348 12 1 506 186 12 1 567 458 13 808 704 13 613 800 13 638 280 14 311 040 14 208 008 14 196 812 15 90 720 15 55 584 15 41 400 16 15 876 16 11 673 16 4 761 17 0 17 1 656 17 0 18 0 18 144 18 0 19 0 19 0 19 0 20 0 20 0 20 0 ------------- ----------------------- ------------- ----------------------- ------------- ----------------------- Acknowledgement {#acknowledgement .unnumbered} --------------- Katarzyna Bolonek-Lasoń would like to acknowledge Prof. Piotr Kosiński for helpful discussions and suggestions. Research of Katarzyna Bolonek-Lasoń was supported by the NCN Grant no. DEC-2012/05/D/ST2/00754. [99]{} J.S. Bell, *Physica* **1**, 195 (1964) A.K. Ekert, *Phys. Rev. Lett.* **67**, 661 (1997) J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, *Phys. Rev. Lett.* **23**, 880 (1969) J.F. Clauser, M.A. Horne, *Phys. Rev.* **D10**, 526 (1974) D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, A. Zeilinger, *Phys. Rev. Lett.* **85**, 4418 (2000) R.F. Werner, M.M. Wolf, *Phys. Rev.* **A64**, 032112 (2001) D. Collins, N. Gisin, N. Linden, S. Massar, S. Popescu, *Phys. Rev. Lett.* **88**, 040404 (2002) W. Son, J. lee, M.S. Kim, *Phys. Rev. Lett.* **96**, 060406 A. Cabello, S. Severini, A. Winter, *Phys. Rev. Lett.* **112**, 040401 (2014) Y-C. Liang, R. Spekkens, H. Wiseman, *Phys. Rep.* **506**, 1 (2011) N. Brunner, D. Cavalcanti, S. Pironio, V. Scami, S. Wehner, *Rev. Mod. Phys.* **86**, 419 (2014) V. Ugǔr Gűney, M. Hillery, *Phys. Rev.* **A90**, 062121 (2014) V. Ugǔr Gűney, M. Hillery, *Phys. Rev.* **A91**, 052110 (2015) K. Bolonek-Lasoń, Breaking of Bell inequality based on $S_4$ symmetry, arXiv: 1603.07740 (quant-ph) A. Fine, *Phys. Rev. Lett.* **48**, 291 (1982) A. Fine, *Journ. Math. Phys.* **23**, 1306 (1982) J.J. Halliwell, J.M. Yearsley, *Phys. Rev.* **A87**, 022114 (2013) J.J. Halliwell, *Phys. Lett.* **A378**, 2945 (2014) E. Santos, Mathematical and physical meaning of the bell inequalities, arXiv: 1410.4935 M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier analysis, self-adjointness, Academic Press, New York (1975) B.S. Cirel’son, *Lett. Math. Phys.* **4**, 93 (1980) [^1]: kbolonek@uni.lodz.pl [^2]: scibor.sobieski@uni.lodz.pl

--- abstract: 'When exploring large time-varying data sets, visual summaries are a useful tool to identify time intervals of interest for further consideration. A typical approach is to represent the data elements at each time step in a compact one-dimensional form or via a one-dimensional ordering. Such 1D representations can then be placed in temporal order along a time line. There are two main criteria to assess the quality of the resulting visual summary: *spatial quality* – how well does the 1D representation capture the structure of the data at each time step, and *stability* – how coherent are the 1D representations over consecutive time steps or temporal ranges? We focus on techniques that create such visual summaries using 1D orderings for entities moving in 2D. We introduce stable techniques based on well-established dimensionality-reduction techniques: Principle Component Analysis, Sammon mapping, and t-SNE. Our Stable Principal Component method is explicitly parametrized for stability, allowing a trade-off between the two quality criteria. We conduct computational experiments that compare our stable methods to various state-of-the-art approaches using a set of well-established quality metrics that capture the two main criteria. These experiments demonstrate that our stable algorithms outperform existing methods on stability, without sacrificing spatial quality or efficiency.' author: - | \ [ ]{} bibliography: - 'references.bib' --- Introduction ============ Time-varying data is ubiquitous and the size and the variety of the corresponding data sets is ever increasing. For the purpose of this paper, we distinguish three different types of time-varying data, which might appear in isolation or in combination. The first type are the quintessential *time-varying* data which reflect statistical attributes, such as population, or more general attributes such as size and price, in one or often many more dimensions. A second type are *dynamic* data which are characterized by insertions and deletions. The third type are *kinetic* data which have a spatial location (coordinates) that changes continuously over time and is sampled at discrete moments. Kinetic data typically represent moving entities. Any given time-varying data set can be both dynamic and kinetic. {width="\linewidth"} When exploring large time-varying data sets of any type, visual summaries are a useful tool to identify time intervals for further consideration. A typical approach is to represent the data elements at each time step in a compact form, which can then be placed in temporal order along a time line. For example, there are a variety of methods to handle time-varying graphs, such as Parallel Edge Splatting [@DBLP:journals/tvcg/BurchVBDW11] (Fig. \[fig:relworklabeled\]D) and Extended Massive Sequence Views [@van2014dynamic] (Fig. \[fig:relworklabeled\]E), that show the temporal evolution by drawing the graph at each time step in a narrow vertical strip. Similarly, Temporal Treemaps [@DBLP:journals/tvcg/KoppW19] (Fig. \[fig:relworklabeled\]F) encode hierarchies via (essentially) one-dimensional intervals and show the temporal evolution by placing these intervals consecutively along a line. Also Storyline Visualizations [@DBLP:journals/jgaa/DijkFFLMRSW17; @DBLP:journals/tvcg/LiuWWLL13] (Fig. \[fig:relworklabeled\]A) use a compact representation at each time step (essentially a pixel per protagonist); these representations must be coherent between consecutive time steps and as such trace a trajectory for each actor. Arguably the most compact representation for one time step is a one-dimensional ordering of the data objects. Such an ordering directly translates to a grid-based visualization of associated attributes, where a vertical strip of $n$ grid cells encodes $n$ objects. Conceptually, computing a single linear order (linearization) for a given time step is a form of dimensionality reduction: after projecting to 1D, we sort the points by coordinate to obtain the order. Features visualized in the ordering enable an aggregated view on emerging trends, their composition and development over time. While the ordering prohibits the accurate perception of the spatial development of the tracked objects, a general spatial dynamic is encoded by the preservation of neighborhoods. Hence, spatial ordering approaches are best used for the in-situ visualization of large-scale kinetic data, enabling a quick estimation of general trends and developments. In principle, any aspect of the data can be used to create the ordering. For example, both MotionRugs [@BJC+19] (Fig. \[fig:teaser\]) and ParaGlide[@DBLP:journals/tvcg/BergnerSMAS13] (Fig. \[fig:relworklabeled\]B) compute orders from spatial locations for kinetic data, whereas Cui [*et al.*]{} [@DBLP:conf/apvis/CuiWLRMMG14] (Fig. \[fig:relworklabeled\]C) use node degree to order dynamic graph data. For an ordering to meaningfully represent the input data, it needs to satisfy certain properties: (1) the ordering needs to accurately represent the data, as much as is reasonably possible; (2) the ordering needs to allow a user to follow data objects over time. For example, the ordering for dynamic graphs should capture the network structure, under insertions and deletions. For hierarchical data the ordering should capture the implied tree structure, under value changes and insertions and deletions. Finally, for kinetic data, the ordering should capture the spatial proximity under continuous movement of the entities. To facilitate the second requirement, the ordering needs to be temporally coherent and stable over consecutive time steps or temporal ranges. In this paper we focus on methods that create meaningful orderings for kinetic data. [[**Formal problem statement.**]{}]{} Our input is a kinetic set $P = \{ p_1, \ldots, p_n\}$ of $n$ point objects moving in the plane. We sample their positions at $T$ consecutive time steps. That is, each object $p_i$ is a sequence of $T$ locations or points in the plane. We use $p_{i}(t)$ to denote the location of $p_i$ at time $t$, $1 \leq t \leq T$, and, correspondingly, $P(t)$ to denote the complete point set at time $t$. A visual summary $S$ of $P$ is a sequence of linear orders (permutations) of the points in $P$, one per time step. We denote the linear order at time $t$ by $S_t$. We consider $S_t$ to be a bijective function $S_t \colon P \rightarrow \{ 1, \ldots, n \}$. Thus, $S_t(p_i)$ denotes the rank of point object $p_i$ in the linear order at time $t$, and $S_t^{-1}(k)$ denotes the $k$^th^ point object in $S_t$. The quality of a visual summary $S$ is determined by two criteria: Spatial quality. : How well does $S_t$ capture the spatial structure of $P(t)$? We characterize the spatial structure via local neighborhoods: we say that an order has high spatial quality if points that are spatially close in the input are also close in the order. Stability. : How consistent are the linear orders over time? Here we can consider absolute changes between linear orders or changes in local neighborhoods, represented by nearest neighbors in the ordering. Both types of measures can be considered for consecutive time steps or over temporal ranges. Clearly, a visual summary that uses the same order for all time steps is maximally stable. However, the spatial quality of this order will typically be low. Conversely, optimizing spatial quality for each time step in isolation tends to result in unstable summaries which make it more difficult for the user to track objects. Hence, we need algorithms that incorporate the temporal dimension into solving each time step. Ideally, techniques allow an explicit trade-off between spatial quality and stability of visual summaries. [[**Contributions and organization.**]{}]{} We propose three dimensionality-reduction techniques to 1D that take stability into account. Two of our techniques require but simple modifications to existing methods: t-SNE and Sammon mapping. Our third novel algorithm, *Stable Principal Component* \[SPC\], is based on Principal Component Analysis (PCA) and allows an explicit trade-off between spatial quality and stability. The theoretical foundations for SPC lie in the stability analysis for kinetic shape descriptors by Meulemans, Verbeek and Wulms [@meulemans2019stability]. Despite the extensive underlying theory, SPC is simple and straightforward to implement. We describe all three methods in Section \[sec:stabledimreduc\] for kinetic data, but dynamic aspects (insertions and deletions) can be easily incorporated. To order points, there are essentially three approaches via: (1) dimensionality reduction; (2) spatial subdivisions such as partitioning trees and space-filling curves; (3) hierarchical clustering. Generally, these methods are intended to perform well on a single time step and are not designed to be stable. In Section \[sec:algorithms\] we survey a representative set of state-of-the-art ordering methods. To capture spatial quality and stability we rely on a set of well-established quality metrics. In particular, for spatial quality we use the so-called *Key Similarity* measures proposed by Guo and Gahegan [@guo2006spatial] that characterize spatial proximity via $k$ nearest neighbors. For stability we consider two different types of measures: absolute or neighborhood changes in the linear order. For absolute stability we use the number of *Jumps* and *Crossing* as proposed by Buchm[ü]{}ller [*et al.*]{} [@BJC+19]. We model neighborhood changes between orders via changes in the $k$ nearest neighbors, and again use the Key Similarity measures by Guo and Gahegan [@guo2006spatial]. We discuss metrics in detail in Section \[sec:metrics\]. In Section \[sec:experiments\] we report on computational experiments: we compare our stable algorithms with state-of-the-art ordering techniques, measuring the resulting quality using a variety of established metrics. Based on both real-world and synthetic data, these experiments demonstrates that our stable algorithms outperform existing methods on stability, without sacrificing spatial quality or computational efficiency. In Section \[sec:discussion\] we discuss the implications of our results, as well as current limitations and directions for future work. Stable Dimensionality Reduction {#sec:stabledimreduc} =============================== We describe three techniques for computing stable projections into a linear order for moving entities, which are based on well-established dimensionality-reduction techniques: PCA, Sammon mapping, and t-SNE. PCA has a clear indication of quality and representation, which permits interpolation between time steps. We use this fact to develop a stable PCA method which allows an explicit, user-configurable trade-off between spatial quality and stability (see Section \[sec:stablePCA\]). Sammon mapping and t-SNE are both based on gradient descent. In Section \[subsec:dimreduc\] we show how to leverage the existence of local minima to improve stability. Stable Principal Component Analysis {#sec:stablePCA} ----------------------------------- [r]{}[0.2]{} {width="\linewidth"} Our algorithm with an explicit, user-configurable trade-off between spatial quality and stability, uses the first principal component: an output of PCA originally introduced by Pearson [@pearson1901liii]. The first principal component is a vector in the direction along which the point set has most variance. Equivalently, it is the orientation of the line that minimizes the mean squared error to the points. As the most discriminative direction, we thus choose the first principal component vector to sort the points. Meulemans, Verbeek and Wulms [@meulemans2019stability] study the stability of the first principal component from a theoretical point of view. Intuitively, they define stability as “small changes in the input lead to small changes in the output”. The paper analyzes the trade-off between spatial quality and the stability of various ways to summarize the shape of a point set. It shows that the first principal component exhibits unstable behavior, when the point set is not very stretched, and thus does not have a clear direction along which there is most variance. Our approach leverages this result by explicitly enforcing stability, whenever the variance along the first principal component is not high enough with respect to the variance in any other direction. The intuition behind this approach is as follows. If the variance along the first principal component vector is clearly higher than the variance along the second principal component, the direction is very discriminative: the point set is clearly stretched in this direction and sorting the points along this vector tends to lead to high spatial quality. If this is not the case, then the point set is “round” and the spatial quality is roughly equivalent for other directions as well. Our goal is to move between these directions of similar spatial quality in a stable way. Meulemans, Verbeek and Wulms [@meulemans2019stability] introduce a “chasing algorithm”, which follows the optimal direction at fast as possible given a maximum rotation speed. This allows them to bound the solution quality and uses only the data of the “current time step”, which is particularly useful in a streaming setting. Here, however, we want to produce a visual summary and may reasonably assume that all data has already been recorded and is accessible. We can hence “look ahead” from the current time step to identify instabilities, before they actually occur. [r]{}[0.2]{} {width="\linewidth"} [[**\[SPC$_\sigma$\] Stable Principal Component.**]{}]{} To create a stable version of principal component analysis, we use the optimal direction (first principal component) for any $t$ where $P(t)$ is stretched, as well as for the first and last time step. We then interpolate linearly between the orientations for these time steps to define directions also for any time steps that are not clearly stretched. Concretely, the Stable Principal Component algorithm is implemented as follows, using a parameter $\sigma$, $0 \leq \sigma \leq 1$, that determines when a point set is considered stretched. See Algorithm \[alg:stablepc\] for an overview. The algorithm starts by finding the first principal component vector on $P(1)$ and then loops through time steps $t$, from $2$ to $T$ doing the following: We determine the first principal component vector $\textsc{pc}[t]$; as both $\textsc{pc}[t]$ and $-\textsc{pc}[t]$ describe the same orientation, we take $\textsc{pc}[t]$ to have the smallest absolute angle with respect to $\textsc{pc}[t-1]$, the direction of the previous time step. We add the signed angle between these vectors to $\alpha$, which accumulates the angle to interpolate over since the previous stretched (or first) time step. Let $v_1, v_2$ be the eigenvalues for the first and second principal component vectors; note that $v_2 \leq v_1$ by definition. Similar eigenvalues imply similar variances in those orthogonal directions: we determine how stretched $P(t)$ is via the ratio $r = \frac{v_2}{v_1}$. If $r > \sigma$, we consider the point set to be round and skip to the next time step. If $r \leq \sigma$ we consider it to be stretched. We remember the last time step $t'$ where the point set was stretched (or $t' = 1$). When we find a stretched point set at time $t$, we linearly interpolate the direction (first principal component) at time step $t'$ to the direction at time step $t$ for all skipped steps (round point sets), ensuring to turn the direction with an angle $\alpha$ in total (note that $\alpha$ can exceed $360^\circ$). After this loop, we have decided on a direction for each time step; we project the points at each time to the line defined by this direction and use that to infer the linear order $S_t$. The last time step $T$ is always considered “stretched” in the above, such that we end (as well as begin) with the actual first principal component. Point set $P$ over $T$ time steps, and $\sigma \in [0,1]$ Visual summary $S$ for $P$ Set $\textsc{pc}[1]$ to the first principal component vector for $P(1)$ Set $t'$ to $1$ and $\alpha$ to $0$ Calculate first principal component vector $\textsc{pc}[t]$ and eigenvalues $v_1,v_2$ for $P(t)$ Add the signed angle between $\textsc{pc}[t]$ and $\textsc{pc}[t-1]$ to $\alpha$ Set $\textsc{pc}[t_s]$ to $\textsc{pc}[t']$ rotated over $\alpha \cdot \frac{t_s - t'}{t - t'}$ \[algline:interpolate\] Set $t'$ to $t$ and $\alpha$ to $0$ Define $S_t$ by projecting $P(t)$ on line through $\textsc{pc}[t]$ $S$ As Line \[algline:interpolate\] is executed at most once for each time step and computing two principal components and their eigenvalues of two-dimensional data takes linear time, the first loop runs in $O(nT)$ time. To sort the projections, the algorithm thus runs in $O(nT \log n)$ time, on $n$ points moving for $T$ time steps. The explicit trade-off between spatial quality and stability can be configured via parameter $\sigma$. If $\sigma$ is set to a value close to $1$, the focus of the algorithm is on spatial quality, and only when the point set is very round, stability will be enforced; $\sigma=1$ eliminates interpolation and always uses the first principle component in every time step. However, if $\sigma$ is set closer to $0$, the focus will be on stability and even for moderately stretched point sets, linear interpolation can occur, thereby sacrificing spatial quality for stability; $\sigma = 0$ causes one interpolation, from the first principle component at $t = 0$ to the first principle component at $t = T$. Gradient-descent methods {#subsec:dimreduc} ------------------------ The (spatial) quality of dimensionality reduction is a complex optimization function; as such, various local-search heuristics are commonly applied. For example, gradient descent is used by Sammon mapping to preserve distances, and by t-SNE to preserve local neighborhoods. We leverage the nature of gradient descent to increase stability and make it effective for creating visual summaries. There are other dimensionality-reduction techniques, such as MDS [@Kruskal1964] and Isomap [@Tenenbaum2000], but based on their optimization functions we believe that they give similar results. For example, in the Euclidean plane, classical MDS is equivalent to PCA. We first recall Sammon mapping and t-SNE for a single time step $t$, before explaining our adaptation for improved stability. [r]{}[0.2]{} {width="\linewidth"} [[**\[SAM\] Sammon Mapping.**]{}]{} Sammon mapping [@Sammon1969] aims to preserve distances. Let $d_{ij}$ denote the Euclidean distance between points $p_i(t)$ and $p_j(t)$, denote the projected (1D) coordinates by $x_i$, and let $\delta_{ij} = |x_i - x_j|$. Sammon mapping computes coordinates $x_i$, attempting to minimize this cost function: $$C = \frac{1}{\sum_{1 \leq i < j \leq n} d_{ij}} \sum_{1 \leq i < j \leq n} \frac{(d_{ij} - \delta_{ij})^2}{d_{ij}}$$ We start with random initial coordinates and use a steepest (gradient) descent algorithm to minimize cost C. [r]{}[0.2]{} {width="\linewidth"} [[**\[SNE\] t-Distributed Stochastic Neighbor Embedding.**]{}]{} The goal of t-SNE [@tsne2008] is to preserve local neighborhoods in the dimensionality reduction. Like Sammon mapping, it attempts to minimize a particular cost function. Again, let $d_{ij}$ denote the Euclidean distance between points $p_i(t)$ and $p_j(t)$. Similarities between points are captured by a probability distribution: $$\mathcal{P}_{j | i} = \frac{\exp\left(-\frac{d_{ij}^2}{2 \sigma_i^2}\right)}{\sum_{k \neq i} \exp\left(-\frac{d_{ik}^2}{2 \sigma_i^2}\right)}$$ The values $\sigma_i$ are chosen depending on the predefined *perplexity* $\kappa$ (see [@tsne2008] for details); in our experiments we use $\kappa = 40$. We further define $\mathcal{P}_{ij} = \frac{1}{2n}(\mathcal{P}_{j | i} + \mathcal{P}_{i | j})$ and we set $\mathcal{P}_{ij} = 0$ if $i = j$. Denote the projected (1D) coordinates by $x_i$, and define $\delta_{ij}$ as $$\delta_{ij} = \frac{(1 + |x_i - x_j|^2)^{-1}}{\sum_{k \neq l} (1 + |x_k - x_l|^2)^{-1}}$$ The cost function is defined by the Kullback-Leibler divergence as: $$C = \sum_{i \neq j} \mathcal{P}_{ij} \log\frac{\mathcal{P}_{ij}}{\delta_{ij}}$$ Finally, this cost function is minimized in the same way as for Sammon mapping: starting with random initial coordinates and using gradient descent[^1] to minimize cost C. [[**Stability improvements.**]{}]{} Sammon mapping (SAM) and t-SNE (SNE) use random initial coordinates. To improve the stability of both algorithms we initialize them with the solution of the previous time step, resulting in two stable versions, **\[SAMp\]** and **\[SNEp\]**. Recently, Rauber [*et al.*]{} [@DBLP:conf/vissym/RauberFT16] described Dynamic t-SNE: a more explicit way of making t-SNE stable over multiple time steps. Their approach performs a global optimization over all time steps simultaneously, using a separate copy of each point for each time step. They enforce temporal coherence by adding a term to the optimization function depending on the distance between two copies of the same point at consecutive time steps. For two reasons we were not able to include this algorithm in our experiments. First, it is very slow. The paper reports a running time of about 6 minutes per time step. Although a single time step of our data consists of only hundreds of points, we consider thousands of time steps, making their algorithm prohibitively slow for our experiments. Second, the implementation of Dynamic t-SNE rarely gives meaningful output when run on our data[^2], which is much more kinetic than the data experimented on in [@DBLP:conf/vissym/RauberFT16]. We further believe that Dynamic t-SNE would converge slowly on a time-varying data set with many time steps: it would take at least $T$ gradient descent iterations for frames that are $T$ time steps apart to affect each other. Since t-SNE is already known to converge quite slowly, the combination may simply require too many iterations to obtain a reasonable solution. Thus, Dynamic t-SNE exacerbates the usual downsides of t-SNE, namely black-box parameter tuning and slow convergence. Other Techniques {#sec:algorithms} ================ In addition to dimensionality reduction, various other techniques can be found in literature that are used to compute linear orders for a set of points. We compare dimensionality-reduction techniques and our new adaptations not only to each other, but also to several of such existing algorithms. Most of these algorithms are not designed to be stable and typically consider different time steps in isolation. To select suitable algorithms for our comparison, we choose the algorithms that performed best in the experiments by Guo and Gahegan [@guo2006spatial] and Buchm[ü]{}ller [*et al.*]{} [@BJC+19]. We may classify these remaining algorithms based on how they compute a linear order: (1) via spatial subdivisions; (2) via clustering. Finally, we also include a baseline algorithm that is solely focused on stability. Figure \[fig:orderings\] shows an example of the orderings generated by a selection of the algorithms, including the dimensionality-reduction techniques, for one time step of our test data for reference. We give a short overview of the algorithm here, while a full explanation can be found in Appendix \[app:algorithms\]. ![ Orderings using (SPC) and (SNE) dimensionality reduction, space-filling curves (HIL) and clustering (CLC).[]{data-label="fig:orderings"}](orderings2.png){width="1\linewidth"} [[**\[FXD\] Fixed order.**]{}]{} This algorithm outputs the same arbitrary linear order for every time step and hence serves as reference baseline for our experiments. With FXD, each horizontal line always represents the same moving entity. [[**Spatial subdivisions.**]{}]{} Several well-known linearization approaches, which are primarily used for spatial-indexing applications, are based on iterating through some spatial subdivision. These approaches encompass tree data structures and space-filling curves. Many variations exist; see [@DBLP:journals/debu/LuO93] for an overview. Here we focus on four established, representative techniques from this area: \[HIL\] Hilbert curve [@hilbert1891ueber], \[ZOR\] Z-order curve, \[PQR\] Point Quadtree [@DBLP:journals/acta/FinkelB74; @DBLP:books/daglib/0032640] and \[RTR\] R-tree [@DBLP:conf/sigmod/Guttman84]. [[**Clustering.**]{}]{} Another approach is to first compute a hierarchical clustering on the point set, and then order the points in such a way that clusters stay together. These algorithms are defined by how the points are clustered, and how the linear order is computed from the clustering. We use \[CLC\] Complete Linkage Clustering [@gordon1987review] and \[SNN\] Shared Nearest Neighbors [@Jarvis1973] to cluster points and derive an ordering from the cluster hierarchy as follows. The hierarchical clustering is represented by a tree with the individual points stored in the leaves. We aim to order to leaves of such a tree without changing the cluster structure, that is, by only changing the order of the children of any internal node. We follow the algorithm by Bar-Joseph [*et al.*]{} [@Bar-Joseph2003] to efficiently compute the optimal order that minimizes the length of the path formed by visiting the input points in that order. Metrics {#sec:metrics} ======= To evaluate the various methods for computing a visual summary, we need metrics to quantify spatial quality and stability. Spatial Quality --------------- Spatial quality measures the correspondence between $P(t)$ and the linear order $S_t$. We capture this by considering the local neighborhood of a point, as characterized by its nearest neighbors. One way to measure changes in local neighborhoods is using an evaluation of dimensionality reduction via persistent homology as introduced by Rieck and Leitte [@DBLP:journals/cgf/RieckL15]. However, we choose not to use this type of measure. While this approach is more recent than the measure we are using, it does not compare to older results, it is more complex, and most importantly it is an indirect approach. Hence, we use the *Keys Similarity* measures as described by Guo and Gahegan [@guo2006spatial] to directly measure the changes in nearest neighbors. To simplify notation, we omit dependencies on time step $t$, as the metrics consider each time step in isolation. Thus, $P$ denotes a point set in the plane, and $S$ denotes a linear order. Let $n(i,j) \in P$ denote the $j^{th}$ nearest neighbor of $p_i$ in $P$, for each $j$ with $1 \leq j \leq k$ for some constant $k$. We use $r(i,j)$ to denote the neighbor rank in $S$ between $p_i$ and $n(i,j)$. However, the difference in rank $|S(n(i,j)) - S(p_i)|$ is not unique. There are two neighbors at rank difference 1, two at rank difference 2, until we reach one end of a linear order. To avoid arbitrariness, we do not break ties but rather consider each pair with the same rank difference to have the same value for $r(i,j)$. Thus, there are two nodes with $r(i,j) = 1$ (rank difference 1), two nodes with $r(i,j) = 3$ (rank difference 2), etc. Generally, Keys Similarity at time $t$ is then defined as $$KS(P, S) = \frac{\sum_{p_i \in P} \sum^k_{j=1} w(i,j) \cdot r(i,j)}{\sum_{p_i \in P} \sum^k_{j=1} w(i,j)},$$ where $w(i,j)$ denotes the weight or importance of maintaining the $j^{th}$ nearest neighbor of $p_i$ at time $t$ – note that these weights need not be the same at every time step. We use two variants of Keys Similarity, as introduced by Guo and Gahegan [@guo2006spatial]. [[**\[KSra\] Rank-weighted Keys Similarity.**]{}]{} We define $w(i,j) = 1/j$ inversely proportional to the rank, such that maintaining the closest neighbors is considered more important than the more distant neighbors. This gives the following metric, where $H_k$ is the $k^{th}$ harmonic number: $$KSra(P,S) = \frac{\sum_{p_i \in P} \sum^k_{j=1} r(i,j) / j}{\sum_{p_i \in P} \sum^k_{j=1} 1/j} = \frac{\sum_{p_i \in P} \sum^k_{j=1} r(i,j) / j}{n \cdot H_k}$$ [[**\[KSdi\] Distance-weighted Keys Similarity.**]{}]{} We define $w(i,j) = 1/\| p_i - n_t(i,j) \|$ inversely proportional to the Euclidean distance, such that maintaining close neighbors is considered more important than distant neighbors. In contrast to KSra, this variant does not treat neighbors at (nearly) identical distances differently. $$KSdi(P,S) = \frac{\sum_{p_i \in P} \sum^k_{j=1} r(i,j) / \| p_i - n_t(i,j) \|}{\sum_{p_i \in P} \sum^k_{j=1} 1/\| p_i - n_t(i,j) \|}$$ [[**Other facets.**]{}]{} Our metrics focus on combinatorial aspects of the position of the point objects. Spatial structure in general knows many other facets, such as distances and directions between points, as well as density. However, a linear order inherently does not lend itself to represent such concepts. Stability --------- Stability or temporal coherence measures the similarity between consecutive orders in $S$. In our evaluation, we use the following three measures for stability. The first two are based on absolute changes in the order and match the measures used by Buchmüller [*et al.*]{} [@BJC+19] to evaluate MotionRugs. The latter uses neighborhoods, based on the concepts by Guo and Gahegan [@guo2006spatial]. We aim to compare the similarity between two linear orders, $S_t$ and $S_{t+1}$ for each $t$ with $1 \leq t < T$. We could easily use the same metrics to compare nonconsecutive orders, but this provides little insight for such inherently sequential data. To consider the stability over a temporal range $[t,t']$, we use standard summary statistics (e.g., average, minimum, or maximum) over all consecutive pairs. [[**\[JMP\] Jump distance.**]{}]{} We quantify the jump distance for a single point object $p_i$ as the difference between its ranks in the two orders, that is, $|S_t(p_i) - S_{t+1}(p_i)|$. The jump distance between two orders is then the sum over all jump distances for each point object. $$\mathit{JMP}_t(P, S) = \sum_{p_i \in P} |S_t(p_i) - S_{t+1}(p_i)|$$ The value for $\mathit{JMP}_t(P,S)$ lies between $0$ (perfectly stable) and $n(n-1)/2$ (complete inversion of the order). [[**\[CRS\] Crossings.**]{}]{} Whereas JMP penalizes any change in the order, many points moving up together may not constitute much change. Instead we may count the number of inversions or crossings in the order, that is, the pairs $p_i$, $p_j$ for which $S_t(p_i) < S_t(p_j)$ and $S_t(p_i) > S_t(p_j)$. The metric $\mathit{CRS}_t(P,S)$ lies between $0$ (perfectly stable) and $n(n-1)/2$ (complete inversion of the order). Buchmüller et al. [@BJC+19] also use Kendall’s $\tau$ coefficient to evaluate stability. We choose to omit this, as it is equivalent to $1-2\cdot \mathit{CRS}_t(P,S)/(n(n-1)/2)$. That is, Kendall’s $\tau$ is the same as CRS up to normalization to the range $[-1,1]$. [[**\[KSte\] Temporal Keys Similarity.**]{}]{} We may also take the same approach as for spatial similarity and consider the similarity of local neighborhoods in both orders. As distances are not inherently meaningful in the combinatorial order and simply correspond to ranking differences, we use only the rank-weighted version of Keys Similarity. Also for this metric $\mathit{KSte}_t(P,S)$, we do not break ties in either order, but rather give them the same rank. Experimental Evaluation {#sec:experiments} ======================= To demonstrate the effects of taking stability into account when creating visual summaries, we compare the three stable methods to the other ordering algorithms, presented in Sections \[sec:stabledimreduc\] and \[sec:algorithms\], in a quantitative evaluation. For each algorithm, we assess the spatial quality and stability of the computed visual summaries according to the metrics discussed in Section \[sec:metrics\]. The parameter for SPC is explored after settling on the most effective measures for spatial quality and stability. Furthermore, we report on run time measurements. Before discussing the results in detail, we describe our test data. ![Shoal movement over 80 seconds. Several behaviors can be observed: (straight) translation, turning, milling (circling).[]{data-label="fig:movement"}](overview.png){width="1\linewidth"} Data {#sec:data} ---- For comparability, we use the same data as MotionRugs [@BJC+19] along with a synthetic data set generated using Netlogo [@netlogo]. The first data set tracks 151 fish of the Notemigonus crysoleucas species (Golden shiner). Golden shiner fish live in large groups called “shoals”, moving in coordination at almost any given time. The 151 fish were tracked optically while moving through a 2.1m by 1.2m shallow water tank, thus avoiding movement in the third dimension. The tank did not feature any obstacles or hindrances besides the side walls. Different movement patterns can be observed in the data, which allows us to test quality in different situations. Among these patterns are uniform group movements, partial and complete changes of direction, circular movement patterns and changes in group density, speed, and acceleration. 2000 frames of movement were recorded at a rate of 25 frames per second, resulting in 80 seconds of available collective movement data. For each frame, the spatial coordinates of each fish are recorded in a Cartesian coordinate system. During this period, the fish first move straight through the tank (Figure \[fig:movement\], 00:10), then turn, move downwards, turn again, move straight and finally enter a so-called milling formation, moving in a circular shape. In addition to the full data set, we also zoom in on four excerpts (see Figure \[fig:strategycomparison\]). The first three excerpts emphasize typical movements of the entities: translation represents straight group movements with predominantly stable internal group structure, turns involve the group turning with some change in the group structure, and milling show the fish entering and maintaining circular movement patterns. The fourth excerpt does not feature a particular movement pattern but triggers so-called “phantom splits” [@BJC+19] for certain ordering methods, most notably HIL, PQR, or SNEp. The shoal of fish appears to split, but this is purely an artifact of the method and not reflecting the data. The second data set is generated with Netlogo using the Flocking model [@netlogoflocking] from the openly available Models Library within the Netlogo application. Minimal adaptations were made to the model to ensure the boundaries of the canvas do not wrap around, and the trajectories of the moving entities could be extracted easily. In this section we mainly focus on the fish data set, while highlighting results of the Netlogo data set only whenever these are distinctively different from the fish data set, which is during the parameter experiment. The full analysis for the Netlogo data set is given in Appendix \[app:netlogo\]. ![Spatial-quality metrics: mean KSra (left) and KSdi (right) for all algorithms over all frames of the fish data set.[]{data-label="fig:spatialquality-chart"}](spatialquality-chart.pdf){width="\linewidth"} ![Stability metrics: mean JMP, CRS (left axis), and KSte (right axis) for all methods over all frames of the fish data set.[]{data-label="fig:stability-chart"}](stability-chart.pdf){width="\linewidth"} ![MotionRugs for all algorithms, color encodes speed (red = fast, blue = slow). Below each we show KSdi (yellow) and KSte (blue), capped at 37.5 and 6.33, respectively. Four highlighted sections indicate interesting excerpts. []{data-label="fig:strategycomparison"}](stratsoverview.png){width="0.99\linewidth"} Running Time ------------ We implemented and executed all algorithms in Java 11 on a workstation with two Intel Xeon E5-2687W CPUs at 3.10GhZ, 16 Cores, 128GB Ram and an NVidia Quadro M600 GPU, running Windows 10. We measure the running time only for computing the orderings excluding reading input, color mapping and rendering. The running times range from a few milliseconds for the Z-Order curve (ZOR) to just over 8 hours for t-SNE (SNE). General observations include comparably good performance for the subdivision methods (ZOR, HIL, PQR, RTR), with values under one second. Only SPC variants are on par with this speed. Quality Results --------------- ![SPC examples showing how $\sigma$ affects interpolation. Arrows roughly show which frames are interpolated. []{data-label="fig:spcoverview"}](spcoverview.png){width=".99\linewidth"} Figures \[fig:strategycomparison\] and \[fig:spcoverview\] show MotionRugs for all algorithms for the complete fish data set. The MotionRugs are accompanied by a visualization of the mean KSdi and KSte values for each frame, cut off slightly above the mean values of most algorithms. This ensures that the differences between the average behavior of the algorithms becomes visible at a glance. Tables \[sup-tab:experiments\] and \[sup-tab:experiments2\] in Appendix \[app:summarystats\] provide summary statistics over all time steps and for each metric, for both data sets. Below, we first discuss spatial quality and stability statistics separately, along with a discussion of how the metrics are reflected in the full visual summaries and the four excerpts. We follow up with an exploration into the effects of the parameter value on the outcome of SPC and finally consider the trade-off between spatial quality and stability for all methods. [[**Spatial quality.**]{}]{} Figure \[fig:spatialquality-chart\] compares the spatial-quality measures KSra and KSdi, as measured on all algorithms used in our experiments. For both measures lower values indicate higher spatial quality. Overall, we see that the KSra measurements are slightly lower for all algorithms, except SNEp where KSdi has a minimal edge over KSra. As expected FXD achieves the worst spatial quality. Furthermore, SNEp and the algorithms using spatial subdivisions are outperformed by the clustering algorithms, and other dimensionality-reduction techniques. Comparing the spatial quality of SPC to the algorithms that perform best on spatial quality, we see that SPC achieves comparable spatial quality. The choices for parameter $\sigma$ of SPC on the fish data set are 0, 1, and variables $a=0.35, b=0.53, c=0.78$. The choice for the intermediate values $a,b$ and $c$ is different for the two data sets and will be justified in the parameter exploration. Due to the strong correlation of both measures, we focus only on KSdi in the remainder. [[**Stability.**]{}]{} Figure \[fig:stability-chart\] compares the stability measures: JMP, CRS and KSte. While JMP and CRS measure absolute changes between orders, KSte captures changes in local neighborhoods. For each measure lower values indicate higher stability. We see that CRS results in lower values than JMP, which is expected: two entities can jump to different positions in the next frame without crossing, but they cannot cross each other without jumping. We do see some differences between the two data sets, as opposed to the results for spatial quality. For FXD the result is again obvious: all measures are at their minimum. While JMP and CRS are generally low, CLC, SNN and SNE show very high numbers. Those three algorithms also perform worst according to the KSte metric. Another outlier that performs poorly on KSte is RTR, which also performs comparatively poorly on JMP and CRS. Of the remaining algorithms, the spatial subdivisions perform worst on KSte. The SAM, SAMp and SNEp algorithms and the SPC variants show similar and very low mean values of KSte. Again, we observe a strong correlation between the three metrics, and thus consider only KSte in the remainder. [[**Excerpts.**]{}]{} Let us briefly consider the excerpts of motion (translation, split, turn, milling) as highlighted in Figure \[fig:strategycomparison\]. The ordering method clearly defines the resulting visual patterns. Some algorithms, such as CLC and SNE, produce visually rather cluttered results despite a good spatial quality. This clutter is a consequence of instability: the summaries fail to convey patterns over time despite individual frames being objectively good. For some algorithms, for example SAM or SPC with high parameter values, stability is challenged at key moments during the turn and milling patterns. The KSte values are very high for only a few frames. Here we clearly see that even when on average the stability is good, spikes in stability lead to visual artifacts that disrupt patterns that can arise over consecutive frames. Interestingly, while the spatial quality deteriorates for SNEp when it comes to the milling patterns, the other algorithms do not seem to be sensitive to that. Visually, the undesirable phantom split pattern [@BJC+19] can be identified in the subdivision methods (HIL, ZOR, PQR, RTR) and SNEp, while others do not seem to be prone to these kind of visual artifacts or generally produce too fuzzy visual results for such patterns to appear. [[**SPC parameter.**]{}]{} We now investigate the parameter $\sigma$ of SPC and its effect on the results. We run SPC for values for $\sigma$, specifically, for $101$ different values from $0$ and $1$ with increments of $0.01$. As discussed before, we use KSdi to asses the spatial quality of the visual summaries, and specifically we use the mean over all frames. For stability we use the mean as well as the max KSte to quantify stability. As we saw before, mean KSte captures cohesion over time, while max KSte should be low to prevent visual artifacts from disrupting temporal patterns. The results for the fish data set are shown in Figures \[fig:spcoverview\] and \[fig:parameter\]. Note that the highest plotted value of $\sigma$ is $0.95$, while the lowest is $0.29$. Values above and below these extremes are identical to results with $0.95$ and $0.29$ respectively. The values of $\sigma$ that are indicated by labels in the figures are chosen as representatives and used in our other experiments. ![A comparison between the mean and mean (left) as well as max and mean (right) for KSte and for KSdi, for uniformly distributed parameter settings of SPC on the fish data set.[]{data-label="fig:parameter"}](parameter-mean.pdf){width="\linewidth"} ![A comparison between the mean and mean (left) as well as max and mean (right) for KSte and for KSdi, for uniformly distributed parameter settings of SPC on the fish data set.[]{data-label="fig:parameter"}](parameter-max.pdf){width="\linewidth"} Overall, we see an inverse relation between stability and spatial quality. Values of $\sigma$ closer to $1$ result in better spatial quality, while values closer to $0$ sacrifice some spatial quality for more stability. This is to be expected, as SPC$_{1}$ always projects the fish to the first principal component; this will likely lead to the best spatial quality that can be achieved for any parameter value. As $\sigma$ is decreased, SPC increasingly uses interpolated lines for projection instead. This interpolation smooths changes in angle of the line, but the projection reflects spatial relations less accurately as a result. When $\sigma$ drops below $0.30$, the interpolation happens purely between the first and last frame of the data set. Contrary to expectation, this negatively affects both spatial quality and stability: the first principal component rotates both clockwise and counterclockwise at varying speeds, not matching the uniform interpolation over such a long time period; as a result, the interpolated lines do not correspond at all to the first principal components, neither in angles nor in rotation direction. This mismatch in angles leads to poor spatial quality per frame, while the mismatch in rotation direction also decreases stability. ![A comparison between the max for KSte and the mean for KSdi for all algorithms on the fish data set.[]{data-label="fig:spatialqualityVSmaxstability-chart"}](spatialqualityVSstability-chart.pdf){width="\linewidth"} ![A comparison between the max for KSte and the mean for KSdi for all algorithms on the fish data set.[]{data-label="fig:spatialqualityVSmaxstability-chart"}](spatialqualityVSmaxstability-chart.pdf){width="\linewidth"} The Netlogo data set shows similar results albeit more surprising. We again observe an inverse relation between stability and spatial quality. However, for $\sigma$ between $0.45$ and $0.59$ this relation is absent. Increasing $\sigma$ also leads to worse stability and worse spatial quality. During instabilities we observe that at certain frames where SPC projects to an interpolated line, the spatial quality is better than when we project to the first principal component. It is hence not unreasonable that interpolating less (and using first principal component more) when increasing $\sigma$ from $0.46$ to $0.56$ can negatively affect spatial quality. Below $0.40$ both spatial quality and stability change erratically: certain instabilities are no longer interpolated over, creating bigger intervals of consecutive interpolation. [[**Trade-offs.**]{}]{} Our main goal is to investigate the trade-off between spatial quality and stability. Figure \[fig:spatialqualityVSstability-chart\] shows a scatterplot on the means of KDdi and KDte of all algorithms. Since lower values indicate better quality for both, methods in the bottom-left corner perform well on both aspects. In both figures SPC variants are colored in shades of red, Sammon mapping variants in blue and t-SNE variants in green. The most stable variants according to KDte have fully opaque colors, while the unstable variants have a lighter shade. These results clearly show that methods based on spatial subdivisions (ZOR, HIL) and space-filling curves (PQR, RTR), albeit fast to compute, perform poorly on spatial quality and stability. The clustering methods (CLC and SNN) as well as SNE, on the other hand, perform quite well on spatial quality, but exhibit very poor stability. Recall that these methods are also slow to compute. The fixed order (FXD) and SNEp are on the other extreme, having good stability, but very poor spatial quality. Furthermore, the strong influence of initialization for t-SNE stands out. When initialized with random coordinates (SNE), the spatial quality is very good, but the stability is extremely poor. On the other hand, initializing t-SNE with the embedding of the previous time step (SNEp) greatly improves stability, but spatial quality suffers greatly. That leaves SAM, SAMp, and SPC variants, which perform well on both aspects. We note that SAM and SAMp perform very similarly on KDdi (difference of $0.03$), but SAMp performs significantly better in terms of stability. Finally, SPC variants also strike a good balance between spatial quality and stability. All SPC variants have slightly worse spatial quality than SAM variants, but improve stability. However, recall that SPC is significantly faster to compute than the Sammon mapping algorithms SAM and SAMp. The overall composition remains similar, but differences in stability are highlighted. Note that SAM, SAMp and SNEp are deteriorating with respect to other methods; we can also see clear bursts of instability in Figure \[fig:strategycomparison\] for these methods. Figure \[fig:spatialqualityVSmaxstability-chart\] also highlights stability differences between SPC variants. SPC$_1$ always uses the first principal component, which can behave erratically for round point sets, decreasing stability. Our new method SPC overcomes this problem by interpolating over these bursts of instability. Indeed, SPC is largely unaffected for lower parameter values, having the smallest standard deviation overall (see Table \[sup-tab:experiments\] in Appendix \[app:summarystats\]). Overall, stable methods such as SAMp, SNEp, and SPC for parameter values lower than $1$, performs very well in terms of average and worst-case stability, while only marginally sacrificing spatial quality in the case of SAMp and SPC. SPC does so at a fraction of the computational cost necessary for more complex dimensionality reduction techniques. Considering all the above, we conclude that stable methods are the best for computing visual summaries for time-varying (kinetic) data. Discussion & Future Work {#sec:discussion} ======================== The experiments indicate that our stable methods perform best, in particular SPC: it performs better in stability, and performs as well as or better than its competitors in terms of spatial quality and computational efficiency. One the one hand, we leverage interpolation in our adaptation of PCA to SPC, allowing explicit parametrization. SAM and SNE were modified by changing the initial state of the gradient-descent computation; while this generally improves stability, the effect on spatial quality depends on the sensitivity of the underlying measure to local minima. [[**Movement characteristics.**]{}]{} Our data, by nature and design, describes moving entities that primarily form a single, mostly convex cluster. Thus, proximity is the primary concern for determining neighborhoods and hence indicates which entities should be close to each other in the linearization. By definition, the first principal component captures the most discriminating axis – for our single cluster data this is most indicative of neighborhoods, explaining the performance of our method in terms of spatial quality. Methods based on finding clusters suffer in quality (either spatially or temporally) as there are no clear clusters to exploit. With multiple clusters it may be desirable to separate the different clusters in the linear order. The first principal component may do that to some degree, but it is not difficult to imagine a situation where clusters project to the same interval along the principal component vector, interleaving those clusters in the order. By their nature, clustering-based methods will perform better in separating the clusters. But our experiments show that such methods will nonetheless struggle to find a good, stable order within the clusters. Our method and cluster-based methods therefore seem complementary – the one suitable for ordering within a cluster and the other suitable for identifying higher-level structures. It may thus be of interest to develop a new hybrid algorithm by combining cluster detection with dimensionality reduction, as explored also in [@DBLP:journals/tvcg/WenskovitchCRHL18]. This algorithm should identify and maintain clear, well-separated clusters that also persist over time, and order points of individual clusters using our algorithm. In the case of a complexly shaped cluster, we face yet another issue. Clustering detection might not be adequate to find the necessary structure. As above, a single, straight axis for determining the order does not necessarily capture proximity or neighborhood structure well and is hence likely to give unsatisfactory results as well. Perhaps methods from topological persistence can play an important role to identify the necessary structures in such clusters. We leave the development and evaluation of such algorithms for more complex data as future work. Our results show the potential here, for adapting existing methods to explicitly consider stability. [[**Beyond spatial data.**]{}]{} Our stable methods can be used in any situation with time-varying in at least two (numeric) dimensions, to determine the ordering. In a MotionRug, another dimension is then used to color the elements in each order. Such an approach may thus be useful for providing an overview also for abstract data. However, we expect it to be primarily useful when proximity (or more generally, neighborhoods) of items are meaningful in the dimensions used to derive principle components. Investigating precise conditions under which this approach is effective is left to future work. [[**Overview-first.**]{}]{} Visual summaries are primarily an overview-first tool: an analyst is not expected to use only the summary. Visual summaries are intended to give an analyst a rough idea of what happens during the motion of the entities, as a first entry point to find time spans or sets of entities to further investigate. It is therefore important to understand how movement patterns relate to patterns visible in the summary and vice versa. To ensure that *collective* movement of subgroups leads to observable patterns in a MotionRug, we need the attribute used for coloring to be similar for spatially close entities. In our data this is the case for speed and inherent in properties derived from the spatial arrangement. Without a relation between spatial proximity and attribute value, the colors in the MotionRug may jump and it becomes difficult to follow entities or subgroups. Furthermore, we may want to augment a MotionRug or a visual summary in general with information about its spatial and temporal quality. We have augmented our MotionRugs in Figure \[fig:strategycomparison\], using a simple bar chart to show spatial quality and stability per time step. Various other encodings could be considered, e.g. reducing the saturation of the colors or underlining the MotionRug with two lines where the pixel colors indicate the spatial and temporal quality. How to best visually convey the spatial and temporal quality, and how this effects user understanding are left to future work. [[**Acknowledgments.**]{}]{} The authors wish to thank Prof. Dr. Iain Couzin and Dr. Alex Jordan of the Max-Planck-Institute for Ornithology, Radolfzell, Germany, and Dominik Jäckle for helpful insights and discussions, and for providing the fish data set. Other Linearization Techniques {#app:algorithms} ============================== In addition to dimensionality reduction, various other techniques can be found in literature that are used to compute linear orders for a set of points. We compare dimensionality-reduction techniques and our new adaptations not only to each other, but also to several of such existing algorithms. Most of these algorithms are not designed to be stable and typically consider different time steps in isolation. To select suitable algorithms for our comparison, we choose the algorithms that performed best in the experiments by Guo and Gahegan [@guo2006spatial] and Buchm[ü]{}ller [*et al.*]{} [@BJC+19]. We may classify these remaining algorithms based on how they compute a linear order: (1) via spatial subdivisions; (2) via clustering. Finally, we also include a baseline algorithm that is solely focused on stability. Figure \[app:orderings\] shows an example of the orderings generated by a selection of the algorithms, including the dimensionality-reduction techniques, for one time step of our test data for reference. ![ Orderings for one time step generated using linear (SPC) and nonlinear (SNE) dimensionality reduction, space-filling curves (HIL) and clustering (CLC).[]{data-label="app:orderings"}](orderings2.png){width="1\linewidth"} [[**\[FXD\] Fixed Order.**]{}]{} This algorithm outputs the same arbitrary linear order for every time step and hence serves as reference baseline for our experiments. With FXD, each horizontal line always represents the same moving entity. Spatial Subdivisions -------------------- Several well-known linearization approaches, which are primarily used for spatial-indexing applications, are based on the principle of iterating through some spatial subdivision. These approaches encompass tree data structures and space-filling curves. We focus on four established, representative techniques from this area, though many variations exist; see [@DBLP:journals/debu/LuO93] for an overview. [r]{}[0.2]{} {width="\linewidth"} {width="\linewidth"} [[**\[HIL\] Hilbert curve and \[ZOR\] Z-order curve.**]{}]{} The Hilbert curve [@hilbert1891ueber] is a continuous space-filling curve. It can be applied to cover a spatial region in arbitrary precision by repeating the construction pattern recursively. A set of points in space can then be linearized by sampling the curve and noting the order in which the points are encoded on the curve. Another representative of space-filling curves is the Z-order curve, which differs from the Hilbert curve in its geometrical construction pattern resembling a Z shape, where the space is partitioned in four quadrants in the order NW, NE, SW, SE (see figure on the right). Both approaches differ in neighborhood retention and construction complexity, as Lu and Ooi describe [@DBLP:journals/debu/LuO93]. In their comparison, Guo and Gahegan [@guo2006spatial] found that Hilbert curves avoid long jumps better than the Z-Order curve, which in turn outperforms the Hilbert curve in the average of the compared metrics. Since both produce visually different outcomes, we include both strategies in the comparison. [r]{}[0.2]{} {width="\linewidth"} [[**\[PQR\] Point Quadtree.**]{}]{} Quadtrees [@DBLP:journals/acta/FinkelB74] partition space recursively in four parts, until each part contains only a single point. Consequently, sparse areas cause fewer splits than dense areas. Standard quadtrees divide the space in equal parts, while Point Quadtrees split at an input point and thus potentially unevenly in terms of area. To derive the 1D ordering, a depth-first tree-iteration strategy is used; given the neighborhood structure in the tree, this is more suitable than a breadth-first strategy. See [@DBLP:books/daglib/0032640] for details on tree-iteration strategies. The standard quadtree essentially reflects a Z-Order curve linearization if the same quadrant iteration is applied. Hence, we use the point quadtree variant which produces different orderings due to the intermittent partition. [r]{}[0.2]{} {width="\linewidth"} [[**\[RTR\] R-tree.**]{}]{} In R-Trees [@DBLP:conf/sigmod/Guttman84] objects are stored recursively in minimum bounding rectangles (MBR). Each MBR can hold at most a predefined number of objects, thus ensuring a minimum fill. In comparison to quadtrees, more complex balancing is necessary, recomputing the MBRs, when the object limit is reached. Note that MBRs can overlap. Again, a depth-first iteration strategy is used to order points in an R-Tree. Clustering ---------- Another method to compute a linear order from a point set is to first compute a hierarchical clustering on the point set, and then order the points in such a way that clusters stay together. Algorithms of this type are defined by two aspects: (1) how the points are clustered, and (2) how the linear order is computed from the clustering. In the algorithms we consider below, we always use the following method to compute the linear order from the clustering. The hierarchical clustering is represented by a tree with the individual points stored in the leaves. We aim to order to leaves of such a tree without changing the cluster structure: we can change only the order of the children of any internal node. We follow the algorithm by Bar-Joseph [*et al.*]{} [@Bar-Joseph2003] to compute the order that minimizes the length of the path formed by visiting the input points in that order. The algorithm uses dynamic programming to efficiently find the optimal order for every subtree placing two specific points at the first and last position in the order. [r]{}[0.2]{} {width="\linewidth"} [[**\[CLC\] Complete Linkage Clustering.**]{}]{} Initially, every point is considered as a separate cluster to be hierarchically merged in a bottom-up fashion [@gordon1987review]. We do so by repeatedly merging the closest two clusters, until we obtain a single cluster. Distance between clusters is measured as the distance between their farthest points. ![The two quality metrics: mean for KSra (left) and KSdi (right column) for all algorithms over all Netlogo data set frames.[]{data-label="fig:spatialquality-chart2"}](spatialquality-chart2.pdf){width="\linewidth"} [r]{}[0.2]{} {width="\linewidth"} [[**\[SNN\] Shared Nearest Neighbors.**]{}]{} This clustering algorithm [@Jarvis1973] works the same as CLC, but it uses a different metric than Euclidean distance to measure the dis(similarity) between two points. For two points $p$ and $q$, we first count the number of points $x$ that are in the set of $k$ nearest neighbors for both $p$ and $q$. We then define the *shared nearest neighbor* (SNN) distance between $p$ and $q$ as $1/(x+1)$. The SNN clustering is computed using the SNN distance instead of the Euclidean distance. In case of ties in the SNN distance, we use the Euclidean distance to break ties. In our experiments we use $k = 10$. Experimental evaluation Netlogo data set {#app:netlogo} ======================================== We go over the statistics for the Netlogo data set used in our experiments in more detail in this section. Again, we first consider the spatial quality and stability separately, followed by the parameter exploration for SPC. Finally we discuss the trade-off between spatial quality and stability as observed on the Netlogo data set. [[**Spatial quality.**]{}]{} As can be seen in Figure \[fig:spatialquality-chart2\], while the absolute values for the Netlogo data set are higher than for the fish data set, the relative values are very similar. The fixed order gives very bad results on spatial quality, followed by SNEp and RTR. The spatial subdivision techniques all perform similarly, and are slightly better than the previously mentioned techniques. Of the remaining algorithms, the clustering techniques (CLC and SNN) perform slightly worse than all remaining dimensionality reduction techniques (SAM, SAMp, SNE and SPC). For the Netlogo data set we have chosen different parameter values for SPC, specifically $a=0.40, b=0.59$ and $c=0.62$. ![The three stability metrics: mean for JMP, CRS (left), and KSte (right) for all algorithms over all Netlogo data set frames.[]{data-label="fig:stability-chart2"}](stability-chart2.pdf){width="\linewidth"} [[**Stability.**]{}]{} In Figure \[fig:stability-chart2\] we plot the stability statistics for the Netlogo data set. While the chart looks quite different from the stability chart for the fish data set, this is mostly due to the fact that JMP and CRS count the absolute number of changes in the orders, whereas KSte is normalized. Since the Netlogo data set behaves less stable than the fish data set, all metrics show higher values. However, the Netlogo data set also contains more moving entities, which increases the absolute number of changes even further. Comparing the statistics of JMP and CRS, we see very similar performances of all algorithms, with SNE being the least stable, while SAMp is the most stable. On the fish data set SNE was the least stable method overall, and while it still has the highest number of absolute changes on the Netlogo data set, it performs a lot better according to KSte, meaning neighborhoods are perserved relatively well over time. Now we see that RTR performs worst on KSte, followed by the space-filling curves (HIL and ZOR) and the clustering algorithms (CLC and SNN) together with PQR. The SPC variants perform relatively worse than on the fish data set, with parameter values close to but lower than 1 being optimal for stability. Finally SAM, SAMp and SNEp are the most stable according to KSte. ![A comparison between the max for KSte and the mean for KSdi, for uniformly distributed $\sigma$ of SPC$_{\sigma}$ on the Netlogo data set.[]{data-label="fig:parameter-max2"}](parameter-mean2.pdf){width="\linewidth"} ![A comparison between the max for KSte and the mean for KSdi, for uniformly distributed $\sigma$ of SPC$_{\sigma}$ on the Netlogo data set.[]{data-label="fig:parameter-max2"}](parameter-max2.pdf){width="\linewidth"} [[**Parameter experiment.**]{}]{} As already explained in the main text, the results of the parameter experiment are slightly different for the Netlogo data set. In Figures \[fig:parameter-mean2\] and \[fig:parameter-max2\] the results are plotted. For the Netlogo data set, the cut-off values are $0.76$ and $0.32$, so everything above $0.76$ uses exactly the first principal component per frame, and similarly for values below $0.32$ we always interpolate between the first and last frame. The parameter values that are indicated by labels in the figures are the values we used in our other experiments. Note that there are two blue labels, which represent other values of interest that we will use in the analysis below. We will consider the results between the values indicated by black labels in the figures. Starting from the lowest parameter value $0.32$ we see that increasing $\sigma$ has chaotic effects on both spatial quality and stability up to $0.40$ where this fickle behaviour ends. On closer inspection, the values between $0.32$ and $0.40$ constantly pick up more frames where the entities are stretched enough to use the actual first principal component. Since the intervals between which interpolation happens, constantly change, the results do not steadily change, but are quite erratic. Parameter value $0.38$ shows the worst combination, having both bad spatial quality and stability (max and mean). ![A comparison between the max for KSte and the mean for KSdi for all algorithms on the Netlogo data set.[]{data-label="fig:spatialqualityVSmaxstability-chart2"}](spatialqualityVSstability-chart2.pdf){width="\linewidth"} ![A comparison between the max for KSte and the mean for KSdi for all algorithms on the Netlogo data set.[]{data-label="fig:spatialqualityVSmaxstability-chart2"}](spatialqualityVSmaxstability-chart2.pdf){width="\linewidth"} From $0.40$ to $0.45$ we see a steady decrease in stability and increase in spatial quality, as expected when increasing $\sigma$. Further increasing the parameter to $0.59$ has negative effects on both the spatial quality and stability. On closer inspection, this increase in spatial quality can be attributed to properties of the Netlogo data set. During instabilities we can observe that at certain frames where SPC projects to an interpolated line, the spatial quality is better than when we project to the first principal component. While we expect projections to the first principal component to have high spatial quality, it is not always the case, as we see here. In the Netlogo data set this occurs when the cluster of points changes direction and shortly does not form a convex shape. It is therefore not unreasonable that interpolating less (and using first principal component more) when increasing $\sigma$ from $0.46$ to $0.56$ can negatively effect spatial quality. At $0.61$ SPC splits the interpolation over the two consecutive instabilities that were seen as one big instability. This split improves both spatial quality and stability, up to $0.62$ where there are a couple of non-interpolation frames between the instabilities. Increasing the parameter further leads to certain instabilities not being interpolated over any longer, which negatively affects the maximum KSte values observed for those runs of SPC. Summary Statistics {#app:summarystats} ================== The two tables on the next page provide the statistics over all time steps, for each metric and on every algorithm in our experiment. Each table shows the results for one of the data sets, the fish data set and Netlogo data set respectively. The data in the tables is used throughout the paper in a variety of charts and diagrams. In the main text we explained that t-SNE was implemented from scratch, using the simplest form of the algorithm. There are extensions that approximate the gradient during gradient descent to improve run times. These extensions are integrated in most libraries that are currently available. When using the libraries we indeed saw faster run times for both SNE and SNEp, but stability was influenced by the approximations used in the libraries. We therefore chose to implement t-SNE from scratch to show its true capability to produce stable visual summaries. While this lead to slower run times than one would expect from state-of-the-art t-SNE, the run times of the libraries still greatly exceeded the run times observed for the other algorithms in our experiments. [^1]: We tried using the existing implementation at <https://github.com/lejon/T-SNE-Java> to compute the t-SNE mapping. This implementation uses approximations to speed up the computation, which lead to artifacts in our results. We therefore implemented the default version of t-SNE ourselves. See Appendix \[app:summarystats\] for more details. [^2]: The implementation often gives NaN as output. The authors [@Telea2019] have verified that this is a known problem with the implementation.

--- abstract: 'Investigating the influence of quantum information scrambling on quantum correlations in a physical system is an interesting problem. In this article we establish the mathematical connections among the quantifiers known as quantum information scrambling, Uhlmann fidelity, Bures metric and bipartite concurrence. We study these relations via four point out-of-time-order correlation function (OTOC) used for quantum information scrambling. It is investigated that, the quantum information scrambling is a squared root function with Uhlmann fidelity, squared function with Bures metric and liner function with bipartite concurrence. Further we study the dynamics of all the quantifiers and investigate the influence of quantum information scrambling on entanglement in two qubits prepared in Bell states. We also figure out the quantum information scrambling and entanglement balancing point in bipartite system.' author: - | Kapil K. Sharma$^\ast$\ **\ E-mail: $^\ast$iitbkapil@gmail.com\ **\ title: 'Quantum Information Scrambling and Entanglement in Bipartite Quantum States.' --- Introduction ============ Quantum information scrambling (QI scrambling)[@qi1; @qi2; @qi3; @vqi] is known as the spreading of the quantum information over the physical system. The phenomenon of quantum information scrambling may takes place in any physical system because of the chaotic situations. From classical mechanics perspective, the chaos is observed by studying the dynamics of classical trajectories in phase space. If the initial condition is very sensitive than the trajectories diverge in the space and follow the Lyapunov exponent as $e^{\lambda t}$[@lay1; @lay2]. This diversion of trajectories in the phase space with the sensitivity of initial conditions is known as butterfly effect[@bf1]. It is difficult to study butterfly effect in quantum mechanics because of the missing notion of trajectories. The chaos in quantum domain does not have a single definition; there are many ways to look into quantum chaos from quantum mechanics point of view[@qc1]. A popular approach to study the chaos in the physical system is to measure the degree of Irreversibility by using the mismatch between forward-backward evolution of the system. To quantify the chaos by using forward-backward evolution approach; the famous quantifiers are known as Loschdimt Echo and irreversible entropy production[@los1; @los2; @ep1]. These quantifiers also have experimental manifestations in varieties of physical systems. In the connection of forward-backward evolution approach, recently out of time order correlators (OTOC) attracted much attention to measure the quantum information scrambling in thermal density matrices[@ot1; @ot2]. OTOC originally discovered by Larkin and Ovchinnikov, while studying the quasi classical method in theory of superconductivity in 1968[@ot1]. He has been studied the behavior of classical correlator function ie. $C_{c}(t)=\langle [p(t)p(0)]^{\dagger}.[p(t)p(0)]\rangle=e^{2\lambda t}$ in Fermi gas with the Laypunov exponent $\lambda$. The Laypunov exponent quantifies the strength of the chaos which is unbounded for classical physical systems while it has bound for quantum systems with the limit $\lambda\leq 2\pi KT/\hbar$. Recent trends in quantum chaos deal with the quantum mechanical version of $C_{c}(t)$, represented in terms of quantum operators. The quantum version of OTOC has close connection with black hole information problem, holographic theories and quantum chaos in many body physics. It has been shown by Hayden and Preskill by considering a simple model of random unitary evolution that black holes rapidly process the quantum information and exhibit the fastest information scrambling[@qi1]. Temperature is natural recourse of energy for black holes, which is a major factor for information scrambling; hence it is customary to study the quantum information scrambling in thermal density matrices. In continuation of OTOC discussion; here we mention that, the investigation of different versions of OTOC is also an active area. The impact of OTOC can disturb the quantum correlations in a physical system but simultaneously the deep distinction between quantum information scrambling and decoherence is not clear[@qid1]. A lot of work have been carried out on OTOC in different physical systems dealing with varieties of domains like conformal field theories, quantum phase transition, Luttinger liquids, quantum Ising chain, symmetric Kitaev chain, quadratic fermions, hardcore boson model, XX spin chain with random filed [@cf1; @lt1; @lt2; @qch1; @fot1; @xot1]. Often, the OTOC in spin chains have been studied as a function of the distance between two arbitrary spins which are imposed by actions of the local non-commutative operators[@scv1]. Further the laypunov exponent as a function of the velocity ie. $\lambda(v)$, has been studied in classical and semi classical regime and early time behavior of quantum information scrambling has been investigated with Baker-Campbell-Hausdorff (BCH) formula[@bch1; @bch2]. Here we recall, at present the quantum chaos community is bound to study OTOC in thermal density matrices by following the analogy of the involvement of the temperature with black holes. However dynamical studies in varieties of non thermal quantum states are missing. To the best of our knowledge, the present study is totally new in this direction. In the present work, we establish the direct mathematical connection between quantum information scrambling and concurrence in pure bipartite quantum states[@con1; @con2; @con3]. With this mathematical connection, it is easy to study the direct influence of quantum information scrambling on the entanglement. However establishing such direct connections is a hard problem in larger Hilbert spaces. We also establish mathematical connections between quantum information scrambling, Uhlamann fidelity and Bures metric[@uh1; @bm1]. These quantifiers are helpful to study the degree of mismatch between forward and backward evolution and the influence of this mismatch on QI scrambling. Further we study the dynamical behavior of above mentioned quantifiers in two qubits Bell states[@bl1], which prove the strength of of mathematical relations. To start with, the quantum mechanical version of OTOC is given below, $$\langle C(t) \rangle=\langle [W(t),V]^{\dagger}].[W(t),V]\rangle \label{e1}$$ The expression of $\langle C(t) \rangle$ is represented in Heisenberg picture. Here we consider that the operators $W(t)$ and $V$ are Hermitian as well as Unitary. For initial stage at $(t=0)$, the operators $\{W(0),V\}$ commute and no QI scrambling takes place; this is expressed with the following condition, $$[W(0),V]=0.\label{c1}$$ As time advances, the commutativity between $W(t)$ and $V$ may break; which produce QI scrambling. So the condition for the existence of QI scrambling can be considered as, $$[W(t),V]\neq 0.$$ The unitary time evolution of the operator $W(t)$ under certain Hamiltonian govern the degree of commutativity and hence information scrambling. The unitary time evolution of the operator $W(t)$ is give by the series $$\begin{aligned} W(t)=e^{iHt}W(0)e^{-iHt} =W(0)+it[H,W(0)]+\nonumber \\ \frac{t^{2}}{2!}[H,[H,W(0)]]+\frac{it^{3}}{3!}[H,[H,[H,W(0)]]]+......... \label{se1}\end{aligned}$$ At $(t=0)$, the series terminates to $W(0)$ and no scrambling takes place. For the existence of QI scrambling, the following condition should also satisfy, $$[H,W(0)]\neq 0.$$ Here we mention that if $\{H,W(0)\}$ are bounded operators with $||H||\leq \epsilon$ and $||W(0)||\leq \epsilon$, then the series is a convergent series. Linking QI scrambing, Uhlamann Fedility and Bures metric {#a1} ======================================================== In this section we explore the derivation of the OTOC and establish the mathematical connections among the QI scrambling, Uhlmann fidelity and Beures metric for pure quantum states. Here we recall the definition of OTOC given in Eq.\[e1\] as below, $$\langle C(t)\rangle=\langle [W(t),V]^{\dagger}.[W(t),V]\rangle=\langle P.Q\rangle$$ Where, $$\begin{aligned} P=[W(t),V]^{\dagger}=V^{\dagger}.W(t)^{\dagger}-W(t)^{\dagger}.V^{\dagger}\end{aligned}$$ and $$Q=[W(t),V]=W(t).V-V.W(t)$$ In brief, we can obtain the factor $P.Q$ as, $$\begin{aligned} P.Q=V^{\dagger}.W(t)^{\dagger}.W(t).V-V^{\dagger}.W(t)^{\dagger}.V.W(t)\\-W(t)^{\dagger}.V^{\dagger}.W(t).V+W(t)^{\dagger}.V^{\dagger}.V.W(t)\label{eq:7}\end{aligned}$$ Here $W(0)$and $V$ are Hermitian Operators, so these must satisfy the following conditions, $$W(0)^{\dagger}=W(0); \quad W(t)^{\dagger}=W(t) \label{o1}$$ and $$V^{\dagger}=V\label{o2}$$ Applying these conditions to Eq.$\text{\ref{eq:7}}$, we can obtain, $$P.Q=2-V.W(t).V.W(t)-W(t).V.W(t).V\label{eq:12}$$ Taking the average on both the sides we obtain, $$\langle P.Q\rangle=2-\langle\psi|V.W(t).V.W(t)|\psi\rangle-\langle\psi|W(t).V.W(t).V|\psi\rangle$$ $$=2-Tr[V.W(t).V.W(t).\rho]-Tr[W(t).V.W(t).V.\rho.]$$ Assuming $(\rho=I)$ and applying the cyclic property of the trace operation, our simplification leads, $$\langle P.Q\rangle=2.\{1-Tr[W(t).V.W(t).V]\}$$ $$=2.[1-F(t)]\label{ot1}$$ Where, $$F(t)=Tr[W(t).V.W(t).V] \label{ex1}$$ $F(t)$ is called as OTOC with special time ordering of the operators. The expectation value of an operator $O$ in quantum mechanics is defined as, $$<.>=Tr[O.\rho]. \label{ex2}$$ where $O$ is the operator and $(\rho)$ represents the density matrix of the quantum state. Comparing Eq.\[ex1\] with Eq.\[ex2\], we conclude the Eq.\[ex1\] represents the expectation value over the Identity matrix $(I)$. In general this Eq.\[ex1\] can be written over the density matrix $(\rho)$, as below, $$F(t)=Tr[W(t).V.W(t).V.\rho]\label{d1}$$ As per the definition of OTOC; more accurately we can write, $$F(t)=Re[F(t)] \label{r1}$$ Finally the expectation value of $\langle P.Q\rangle$ in Eq.\[ot1\] can be written over a density matrix $(\rho)$ as below, $$\langle P. Q\rangle=2[1-Re.F(t)]\label{is1}$$ Or, $$\langle C(t)\rangle=2[1-Re.F(t)]\label{is1}$$ We emphasize that, the above equations involve $F(t)$ as expressed in Eq.\[r1\] over a density matrix $(\rho)$. To establish the OTOC connection with the Uhlamann fidelity of pure quantum states, we re-look into the Eq.\[d1\] and rewrite the equation by using the cyclic property of trace operation, $$F(t)=\langle\psi|W(t).V.W(t).V|\psi\rangle =\langle y|x\rangle$$ With, $$|x\rangle=W(t).V|\psi\rangle \label{ef1}$$ and $$|y\rangle=V.W(t)|\psi\rangle \label{eb1}$$ Here the state $|x\rangle$ represents forward evolution of the state $|\psi\rangle$ and $\langle y|$ represent the backward evolution of the state $|\psi\rangle$ under the actions of the operators $\{W(t),V\}$. The operators $W(t)$ and $V$ are unitary, so the state $|\psi\rangle$ does not lost its purity during complete evolution. Here the complete evolution we mean, the total evolution involving the forward and backward evolution. The Uhlamann fidelity of two quantum states gives the information about the overlapping of quantum states and measure the similarity or probability of transition between two states. The Uhlmann fidelity between two pure quantum states $\{|x \rangle,|y\rangle\}$ is defined as, $$f=|\langle y|x\rangle|^{2}$$ The above equation gives the degree of mismatch between the forward and backward evolution of the quantum state $|\psi\rangle$. Exploring the above equation we obtain, $$f=\langle y|x\rangle\langle y|x\rangle^{\dagger}=\langle y|x\rangle.\langle x|y\rangle \label{fu1}$$ We plug-in the values from the Eqs.\[ef1\] and \[eb1\] in Eq.\[fu1\] and applying the cyclic property of the trace operation, we obtain the following, $$f=Tr[W(t).V.W(t).V.\rho].Tr[V.W(t).V.W(t).\rho]$$ Assuming $(\rho=I)$, we obtain, $$f=Tr[W(t).V.W(t).V].Tr[V.W(t).V.W(t)]$$ Applying the cyclic property of trace operation we can get, $$\begin{aligned} f=Tr[W(t).V.W(t).V].Tr[W(t).V.W(t).V]\\=\{Tr[W(t).V.W(t).V]\}^{2}\end{aligned}$$ By using the definition in Eq.\[ex2\], in general the above equation can be written over the density matrix $(\rho)$, as below, $$f=\{Tr[W(t).V.W(t).V.\rho]\}^{2} \label{fe1}$$ Which leads, $$\sqrt{f}=Tr[W(t).V.W(t).V.\rho]=Tr[M] \label{ft1}$$ Here $M$ represent the density matrix after the complete evolution mentioned below, $$M=W(t).V.W(t).V.\rho \label{ce1}$$ The dimension of the matrix $M$ depends on the size of the Hilbert space, in which the density matrix $(\rho)$ exists. By utilizing the Eqs.\[d1\] and \[r1\], the Eq.\[ft1\] may be rewritten as, $$\sqrt{f}=Re[F(t)]\label{f1}$$ Putting the value of the above factor in Eq.\[is1\] we obtain the connection between QI scrambling and Uhlamann fidelity as below, $$\langle C(t) \rangle=2[1-\sqrt{f}] \label{es1}$$ The above equation shows that, the QI scrambling is the square root function of Ulhmann fidelity for pure states. Referring the Ref\[x\], we know the relation between Bures metric $(D)$ and Uhlamann fidelity as below, $$D=\sqrt{2(1-\sqrt{f})} \label{bm1}$$ Hence the Eq.\[es1\] can be re written as, $$\langle C(t) \rangle=D^{2} \label{bm2}$$ The above equation establish the connection between QI scrambling and Bures metric. We mention that, Bures metric is another measure of closeness of two quantum states and connected with Uhlmann fidelity. Here we find the QI scrambling is a square function with Bures metric. In the present work we only focus on Uhlamann fidelity as this has lucid properties and widely used in literature.  Important properties of QI scrambling ------------------------------------- Bases on the Eq.\[es1\] we find the following properties of QI scrambling, - Positivity; $\langle C(t) \rangle\geq 0$ - Bounded limits; $0\leq\langle C(t) \rangle\leq 2.$ - Unitary invariant; $U\langle C(t) \rangle U^{\dagger}=\langle C(t) \rangle$ - Exchange symmetry over forward and backward evolution; The Uhlmann fidelity follow the following symmetry, $$f=|\langle y|x\rangle|^{2}=|\langle x|y\rangle^{2}$$ Hence we conclude that under the exchange of forward and backward evolution the QI scrambling $(\langle C(t)\rangle)$ will remain same. Linking QI scrambling and two qubits concurrence ------------------------------------------------ In this section we establish the mathematical connection between QI scrambling and two qubits concurrence. Here we mention that, concurrence is a good measure of entanglement for a bipartite system which also has its experimental manifestations, it is defined as below, $$C(|\psi\rangle)=|\langle\psi|\sigma_{y}\otimes\sigma_{y}|\psi^{\star}\rangle| \label{conc1}$$ Or, $$C(|\psi\rangle)=|Tr(\sigma_{y}\otimes \sigma_{y}).(|\psi^{\star}\rangle\langle\psi|)| \label{c11}$$ Here $(\sigma_{y} \otimes \sigma_{y})$ is the spin flip matrix, which flip both the spins under the action of Pauli Y operators and $|\psi^{\star}\rangle$ is the complex conjugate of the state $|\psi\rangle$. If we consider the case $(|\psi^{\star}\rangle=|\psi\rangle)$; which means the density matrix $(\rho)$ will have only real elements and the condition $(\rho^{\star}=\rho)$ is satisfied. By using this condition, we may write the Eq.\[c11\] as, $$C(\rho)=|Tr[(\sigma_{y}\otimes \sigma_{y}).(\rho)]|$$ Further we may straightforward write the concurrence in the state $(M)$ after complete evolution given in Eq.\[ce1\], as below, $$C(M)=|Tr[(\sigma_{y}\otimes \sigma_{y}).M|$$ Or, $$C(M)=|Tr[(\sigma_{y}\otimes \sigma_{y}).W(t).V.W(t).V.\rho]| \label{cc1}$$ We know the product of two Hermitian matrices $A$ and $B$ is Hermitian matrix if and only if they commute ie.$[A,B]=0$. The factor $W(t).V$ may not be a hermitian matrix in general since the condition $[W(t).V]\neq 0$ is satisfied for QIS. But this factor may convert into a hermitian matrix, it depends on the parameter time $(t)$ involved in the equation. Hence overall the matrix $M$ may not be a Hermitian matrix in general ie. $(M^{\dagger}\neq M)$. For two qubits the structure of the operators $\{W(t).V\}$ can be expanded in composite Hilbert space $(H_{1}\otimes H_{2})$ with the dimension $(2^{2}\times 2^{2})$. We analytically find the structure of the density matrix $M$, which trace is invariant under the action of the operator $(\sigma_{y} \otimes \sigma_{y})$ in the composite Hilbert space $(H_{1}\otimes H_{2})$. This form of the matrix $M$ with its transpose is given as below, $$M=\left( \begin{array}{cccc} a & b & c & -a \\ d & e & e & f \\ g & h & h & i \\ j & k & l & -j \\ \end{array} \right);\quad M^{T}=\left( \begin{array}{cccc} a & d & g & j \\ b & e & h & k \\ c & e & h & l \\ -a & f & i & -j \\ \end{array} \right) \label{xt}$$ with, $$Tr[M]=Tr[M^{T}]$$ Where $(a,b,c,d,e,f,g,h,i,j,k,l)$ are complex numbers. The transpose of the matrix $M$ ie. $M^{T}$ is also trace invariant under the action of the operator $(\sigma_{y}\otimes \sigma_{y})$. Investigation of the matrix $M$ depends on many factors such as the Hamiltonian of the physical system $(H)$, the nature of the scrambling operators $\{W(0),V\}$ and the density matrix $(\rho)$ of the state. Finding out the exact form of the constitutes $\{H,W(0),V.\rho\}$, which produces the structure of $M$ is mathematically difficult problem, even in larger Hilbert spaces. We recall that the trace of $\{M,M^{T}\}$ is invariant under the action of $(\sigma_{y}\otimes \sigma_{y})$, so we can easily write the following relation, $$Tr[M]=|Tr[(\sigma_{y} \otimes \sigma_{y}).M]| \label{te1}$$ Or, $$Tr[M^{T}]=|Tr[(\sigma_{y} \otimes \sigma_{y}).M^{T}]|$$ By substituting the value of $M$ from Eq.\[ce1\], we can write the Eq.\[te1\] as below , $$Tr[W(t).V.W(t).V.\rho]=|Tr[(\sigma_{y} \otimes \sigma_{y}).W(t).V.W(t).V.\rho]|$$  Using the above relation, we can easily obtain the new form of the Eq.\[cc1\] as, $$C(M)=Tr[W(t).V.W(t).V.\rho] \label{con1}$$ Putting the value from Eq.\[ft1\] in Eq.\[con1\], we can establish the connection between concurrence and Uhlmann fidelity as, $$C(M)=\sqrt{f} \label{cf}$$ Obtaining the value of the factor $\sqrt{f}$ from the Eq.\[bm1\] and plug-in the Eq.\[cf\], we can establish the direct connection between concurrence and Bures metric as, $$C(M)=1-\frac{D^{2}}{2}$$ Putting the value of the factor $\sqrt{f}$ from Eq.\[cf\] in Eq.\[es1\], we can obtain the direct connection between QI scrambling and concurrence as, $$\langle C(t)\rangle=2[1-C(M)] \label{cm1}$$ Here we find that the QI scrambling is the liner function with the concurrence during complete evolution. Rewriting the above equation we obtain, $$C(M)=1-\frac{\langle C(t)\rangle}{2}\label{qie}$$ This mathematical relation is helpful to study the direct influence of QI scrambling on two qubits concurrence during complete evolution with the following conditions, - The density matrix $(\rho)$ deals with pure bipartite quantum states having real elements. - The density matrix after the complete evolution has the form given in Eq.\[xt\] In case if the exact structure of $M$ given in Eq.\[xt\] is not obtained, then one is forced to study the concurrence by using the Eq.\[conc1\]. Uhlamann Fidelity and Bures metric, both are measures of the degree of mismatch of quantum state during forward and backward evolution. We recall that, in the present work we only focus on Uhlamann Fidelity. We plot QI scrambling and concurrence vs. Uhlamann fidelity $(f)$ by using the Eqs.\[es1\] and \[cf\] in the left part of Fig.\[f1\]. We find the QI scrambling is monotonically decreasing and entanglement is monotonically increasing function. The decreasing nature of QI scrambling also comes from the series given in Eq.\[se1\], as this series is a convergent series. Both the graphs intersect at $(f=0.44)$, hence QI scrambling and entanglement both have equal amplitude at this point; we call this point as QI scrambling and concurrence balancing point. At $(f=1)$, the QI scrambling becomes zero; which means the operators $\{W(0),V\}$ commute and contributes in no scrambling but concurrence sustains to $C(\rho)=1$. The decreasing nature of QI scrambling decay the entanglement and may not be a favorable candidate in quantum information. This may also be observed in the right figure incorporated in Fig.\[f1\], which is a plot of entanglement vs. QI scrambling by using the Eq.\[qie\]. We find, when QI scrambling is minimum as $(\langle C(t)\rangle=0)$, the entanglement is high but increasing QI scrambling, decreases the entanglement linearly. Dynamics of QI scrambling, Ulhmann fidelity and concurrence in Bells states =========================================================================== In this section we study the dynamics of QI scrambling, Uhlmann fidelity and concurrence by considering two qubits prepared in Bell states, written as below, $$\begin{aligned} |\psi_{\pm}\rangle_{1}=\frac{1}{\sqrt{2}}[|00\rangle\pm|11\rangle]\label{b1} \\ |\psi_{\pm}\rangle_{2}=\frac{1}{\sqrt{2}}[|01\rangle\pm|10\rangle]\label{b2}\end{aligned}$$ The corresponding density matrices of these states are expressed as $(\rho_{\pm})_{1}$ and $(\rho_{\pm})_{2}$. We assume two qubits are prepared in Bell states and carry the Ising interaction with external imposed magnetic filed in z direction. This Ising Hamiltonian is expressed as, $$H=-j_{z}\sum_{i=1}^{2}\sigma^{z}_{1}\sigma^{z}_{2}-b\sum_{i=1}^{2}\sigma_{z}.$$ The operator $W(0)$ evolve under the Hamilton $(H)$ and follow the series given in Eq.\[se1\]. This series can be easily derived by using the famous Baker-Campbell-Hausdorff formula[@bch1; @bch2]. The rate of change this series play an important role in QI scrambling. To proceed the dynamical study of QI scrambling and concurrence, here we consider the scrambling operators as Pauli operators such that, $$\{W(0),V \}=\{\sigma_{i},\sigma_{j}\}$$ With, $$\{\sigma_{i},\sigma_{j}\} \in \{\sigma_{x},\sigma_{y},\sigma_{z}\};\quad (i,j)\in \{x,y,z\}$$ All the Pauli operators satisfy the property of Hermition as well as Unitary matrix given in the Eqs\[o1\],\[o2\] and fulfill’ the need of choosing $\{W(0),V\}$. We consider that any one of the two qubits goes under the action of Pauli operator; here we consider the action of Pauli operators on first qubit. In this direction we can develop the structure of the operators $\{W(0),V\}$ for two qubits in composite Hilbert space $(H_{1}\otimes H_{2})$ as below, $$\{W(0),V\}_{(i,j)}=\{\sigma_{i}\otimes I,\sigma_{j}\otimes I\}\label{w1}$$ Here we mention that the factor $Tr[W(t).V.W(t).V.\rho]$ involved in Eq.\[f1\] is responsible for QI scrambling and concurrence obtained in Eqs.\[es1\] and \[con1\]. This factor is invariant under the cyclic permutations of the operators; if we consider the following option, $$\{W(0),V\}_{(j,i)}=\{\sigma_{j}\otimes I,\sigma_{i}\otimes I\}\label{w2}$$ Then, because of the cyclic permutation property both the Eqs.\[w1\] and \[w2\] will produce the same results of QI scrambling and entanglement. So we are intended to choose the following combinations of the operators $\{W(t),V\}$ for our study as below, $$\begin{aligned} \{\sigma_{x}\otimes I,\sigma_{x}\otimes I \};\quad \label{op1} \{\sigma_{x}\otimes I,\sigma_{y}\otimes I \} \label{op2}\nonumber \\ \{\sigma_{x}\otimes I,\sigma_{z}\otimes I \}\label{op3};\quad \{\sigma_{y}\otimes I,\sigma_{y}\otimes I \}\label{op4} \nonumber \\ \{\sigma_{y}\otimes I,\sigma_{z}\otimes I \};\quad \label{op5} \{\sigma_{z}\otimes I,\sigma_{z}\otimes I \}\nonumber \label{op6} \end{aligned}$$ We study all the Bell states given in Eqs.\[b1\] and \[b2\] under the above mentioned combinations of operators. We find, all the bell states $(\rho_{\pm})_{1}$ and $(\rho_{\pm})_{2}$ under the actions of the combinations $\{\sigma_{x}\otimes I,\sigma_{x}\otimes I \},\{\sigma_{x}\otimes I,\sigma_{y}\otimes I\},\{\sigma_{y}\otimes I,\sigma_{y}\otimes I\}$ produce the same results for all the quantifiers such as QI scrambling, Uhlmann fidelity, Bures metric and concurrence. The functions of these quantifiers are obtained as below, $$\begin{aligned} \text{QI scrambling:}\quad \langle C(t)\rangle= 2 \left(1-\cos[4 t (b+j_{z})]\right)\label{p1} \\ \text{Uhlmann Fidelity:}\quad f=\cos^2[(4 t (b+j_{z}))]\label{p2} \\ \text{Bures Metric:}\quad D=\sqrt{2}\sqrt{1-\cos[4 t (b+j_{z})]}\label{p3} \\ \text{Concurrence:}\quad C(M)=\cos[4 t (b+j_{z})]\label{p4}\end{aligned}$$ On the other hand the actions of the operators $\{\sigma_{x}\otimes I,\sigma_{z}\otimes I\},\{\sigma_{y}\otimes I,\sigma_{z}\otimes I\},\{\sigma_{z}\otimes I,\sigma_{z}\otimes I\}$ do not produce any scrambling for all the states $(\rho_{\pm})_{1}$ and $(\rho_{\pm})_{2}$. Corresponding to these action operators, the values of all the quantifiers {QI scrambling, Uhlmann Fidelity, Bures Metric and Concurrence} are obtained as $\{0,1,0,1\}$. The functions in Eqs.\[p1\] to \[p4\] of all the quantifiers are oscillating functions with the parameters $(b,j_{z},t)$. Here we plot the dynamical behavior of the quantifiers QI scrambling, Uhlmann fidelity and concurrence with the varying parameters $(b,j_{z})$ vs. the parameter time $(t)$ in Fig.\[f2\]. We find the peak value of the QI scrambling vanish the entanglement in the system and makes it zero. The QI scrambling and entanglement balancing points are periodic in Bell states. On the other hand we also do analysis of QI scrambling vs. Uhlmann fidelity. As the Uhlmann fidelity $(f)$ is zero; it means the degree of mismatch between forward and backward evolution of the quantum state $(|\psi\rangle)$ is very high and hence the QI scrambling is also very high, which consequently kills the entanglement in the system and makes it zero. If Ulhmann fidelity $(f)$ approaches to the amplitude as unity, then the forward and backward evolution of the state $(|\psi\rangle)$ is same and QI scrambling is zero in the states, which helps to keep the high amount of the entanglement in Bell states. Conclusion ========== In this article we established the mathematical relations among the quantifiers QI scrambling, Uhlamann fidelity, Bures metric and bipartite concurrence. Most importantly the mathematical relation is developed between QI scrambling and concurrence, which is useful to study the direct influence of QI scrambling on bipartite entanglement in real density matrices. Further we have studied the dynamical behavior of quantum information scrabling, Uhlamann fidelity and entanglement with Ising Hamiltonian in two qubits prepared in Bell states. We have used the scrambling operators as Pauli operators and find out the combinations of scrambling operators under which no scrambling takes place in Bell states. The influence of QI scrambling has been studied on entanglement and it has been found, the increasing QI scrambling decreases the entanglement in the system and may not be a good candidate in quantum information. 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--- abstract: | In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in [@Rob].\ \ [**Keywords**]{}. Finite groups; Sylow subgroups; simple groups\ \ [**Mathematics Subject Classification (2010)**]{}. 20D20; 20D05. address: 'Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj, Iran' author: - 'M. Zarrin' title: An affirmative answer to a conjecture related to the solvability of groups --- **Introduction** ================= we know that, according to Feit-Thompson Theorem, every group with an odd order is solvable. As a consequence of this Theorem, one can say that every finite group $G$ that has a normal Sylow $2$-subgroup, i.e., with $v_2(G) = 1$, is a solvable group, where $v_p(G)$ is the number of Sylow p-subgroups of $G$. Also, if $v_p(G)=1$ for each prime $p$, then $G$ is nilpotent and reciprocally. This result show that the number of Sylow $p$-subgroups for a prime p is restricted arithmetically by the properties of a group $G$. Most recently, the author in [@Rob], proved that if $v_p(G)\leq p^2-p+1$ for each prime p, then $G$ is solvable. Here, first we show that it is not necessary to consider the amount of $v_p(G)$ for each prime number $p$ and also improve the upper bound to $p^2$. In fact, we give a substantial generalization as follows: Every finite group $G$ containing at most $4$ Sylow $2$-subgroups, is a solvable group. Also, the author, raised the following conjecture. Let G be a finite group. If $v_p(G)\leq p^2-p+1$ for each odd prime number $p$, then $G$ is solvable. Finally, we give the positive answer to this conjecture and improve it as follows: Every finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is solvable. **The Proofs** ============== $\mathbf{Proof~~ of ~~Theorem ~~1.1}.$ Suppose, on the contrary, that there exists a non-solvable finite group $G$ of the least possible order with $v_2(G)\leq 4$. In this case $G$ should be a simple group. Otherwise, if there exists a non-trivial proper normal subgroup $M$ of $G$, then as $v_2(M)\leq v_2(G)\leq 4$ and $ v_2(G/M)\leq v_2(G)\leq 4$, both $M$ and $G/M$ are soluble (note that if $N$ or $G/N$ is group with odd order, then by Feit-Thompson Theorem they are solvable). It follows that $G$ is solvable, which is a contradiction. Therefore $G$ is a minimal simple group with $v_2(G)\leq 4$. By Thompson’s classification of minimal simple groups [@Tho], $G$ is isomorphic to one of the following simple groups: $A_5$ the alternating group of degree $5$, $L_2(2^p )$, where $p$ is an odd prime, $L_2(3^p)$, where $p$ is an odd prime, $L_2(p)$, where $5 < p$ is prime and $p \equiv 2 (mod~~5)$, $L_3(3)$, and $^2B_2(q)$ where $q=2^{2m+1}\geq 8$.\ Now we show that in each case we obtain a contradiction. This completes the proof. Clearly $v_2(A_5)=5$ and $v_2(L_3(3))=351$, a contradiction. If $G$ is isomorphic to $L_2(2^p)$, then by Case 2 of the proof of Proposition 2.4 of [@Shi], we get that $n_2(G)=2^p+1\geq 9$, a contradiction. If $G$ is isomorphic to $L_2(3^p)$, then one can again imply, from Proposition 2.4 of [@Shi], that $5 < v_2(G)=3^{2p}-1$ or $(3^{3p}- 3^{p})/24$, a contradiction. If $G$ is isomorphic to $L_2(p)$, where $5 < p$ is prime and $p = 2 (mod~~5)$, then by an argument similar to $L_2(3^p)$ we obtain that $5 < v_2(G)=p^{2}-1$ or $(p^{3}- p)/24$, a contradiction. If $^2B_2(q)$, $q=2^{p}$ and $p$ an odd prime, then by Theorem 3.10 (and its proof) of Chapter XI of [@Hup], we have $|G| = (q-1)(q^{2})(q^{2} +1)$ and $v_2(G) = q^{2} + 1> 65$, a contradiction.\ We note that the bound 4 in Theorem 1.1 is the best possible, as $v_2(A_5)=5$. Now by similar argument we prove Theorem 1.3.\ \ $\mathbf{Proof~~ of ~~Theorem ~~1.3}.$ Suppose, on the contrary, that there exists a non-solvable finite group $G$ of the least possible order with $v_p(G)\leq p^2$ for all its prime odd divisors. In the sequel, by an argument similar to the proof of Theorem 1.1, to prove it is enough to consider the following groups (note that $v_3(A_5)=10$, $v_3(L_3(3))=52$):\ If $G$ is isomorphic to $L_2(q)$ with $q=2^p$, then we consider an odd prime divisor of $|G|$, like $r$. Then it is easy to see that $r$ divides either $q+1$ or $q-1$. Now if $R$ is a Sylow $r$-subgroup of $G$, then $R$ is cyclic such that $N_G(R)=D_{q-1}$ or $D_{q+1}$, where $D_m$ is the dihedral group of order $m$. Therefore, the number of Sylow $r$-subgroups is $q(q + 1)/2$ or $q(q - 1)/2$ and so $n_r(G)> r^2$, a contradiction. If $G$ is isomorphic to $L_2(q)$ with $q=3^p$ and $p$ is an odd prime, then it is easy to see that $$v_3(L_2(q))=v_3(SL(2,q)/Z(SL(2,q)))=v_3(SL(2,q)),$$where $Z(SL(2,q))$ is the center of the group $SL(2,q)$. Assume that $R \in Syl_G(3)$, then $N_G(R)$ is the set of upper triangular matrices with determinant 1. Therefore, the order of the normalizer $N_G(R)$ is $q(q-1)$. Thus $v_3(G)=q(q^2 -1)/q(q-1)=q +1 > 3^2$, a contradiction. If $G$ is isomorphic to $L_2(p)$, where $5 < p$ is prime and $p\equiv 2 (mod~~5)$, then by an argument similar to $L_2(2^p)$ we obtain that $v_r(G)=q(q+1)/2$ or $q(q-1)/2$, where $r\neq 2$, and so $n_r(G)> r^2$, a contradiction. If $G=^2B_2(q)$, where $q=2^{2m+1}\geq 8$, then it is well-known that the Suzuki group $^2B_2(q)$ contains a maximal subgroup like $T$ of order $4(q-r+1)$, where $r=2^{m+1}$ and also $T$ has a normal cyclic subgroup $C$ in which $|C|=q-r+1$. Moreover, $C$ includes a Sylow $5$-subgroup like $P$ (note that as $q^2+1\equiv 0 (mod ~5)$ so $5$ is a prime divisor of $|G|$). From this one can follow that $T \leq N_G(P)$ and so $T=N_G(P)$, as $T$ is maximal. Since $|G| = (q-1)(q^{2})(q^{2} +1)$, the number of conjugates of $P$ in $G$ is $$v_5(G)=|G|/|T|=(q-1)(q^{2})(q^{2} +1)/4(q-r+1)> 25.$$ Thus in each case we obtain a contradiction. This completes the proof.\ Finally, it is well-known that the only nonabelian simple finite groups in which its order is not divisible by 3 are the Suzuki groups. From this one can show, by induction on the order, that: if $H$ is a group such that $v_3(H) = 1$ and has no composition factor isomorphic to $^2B_2(q)$, then $H$ is a solvable group. As a result, one can find out that some of the odd prime numbers (for instance, 3) have stronger influence on the solvability of groups. In fact, most probably for the solvability of finite groups in terms of the number of Sylow $p$ subgroups, we do not need to consider all odd prime numbers. Therefore, it might seem reasonable to pose the following question: What is the smallest positive integer $n$ such that whenever there exist a finite group $G$ satisfying $v_{p_i}(G )\leq p_i^2$ where $p_i$ is odd number and $i\in \{1,\dots, n\}$, which guarantees the solvability of $G$? [99]{} Huppert. B, Endliche Gruppen I, Springer-Verlag, Berlin, 1967. Robati, S. M. A solvability criterion for finite groups related to the number of Sylow subgroups, Comm. Algebra https://doi.org/10.1080/00927872.2020.1782418. Shi, J. (2014). A note on the normalizer of Sylow 2-subgroup of special linear group. Int, J. Group Theory 3(4):33-36. Thompson, J. G. (1968). Nonsolvable finite groups all of whose local subgroups are soluble. Bull. Amer. Math. Soc. 74: 383-437.

--- abstract: | The abundant $\psi'$ events have been collected at the Beijing Electron Positron Collider-II (BEPCII) that could undoubtedly provide us with a great opportunity to study the more attractive charmonium decays. As has been noticed before, in the process of $J/\psi$ decaying to the baryonic final states, $p K^- \overline{\Lambda}$, the evident $\Lambda^*$ and $N^*$ bands have been observed. Similarly, by using the product of $\chi_{cJ}$ from $\psi'$ radiative decay, we may confirm it or find some extra new resonances. $\chi_{c0}$’s data samples will be more than $\chi_{c1,2}$ taking into account the larger branching ratio of $\psi'\to\gamma\chi_{c0}$. Here, we provide explicit partial wave analysis formulae for the very interesting channel $\psi'\to\gamma\chi_{c0}\rightarrow \gamma p K^- \overline{\Lambda}$. author: - 'Xian-Wei Kang$^{1,2}$' - 'Hai-Bo Li$^2$' - 'Gong-Ru Lu$^1$' - 'Bing-Song Zou$^2$' title: | **Partial wave analysis of $\psi'\to\gamma\chi_{c0}\rightarrow \gamma p K^- \overline{\Lambda}$\ being used for searching for baryon resonance** --- Introduction ============ In experiment at BESIII, about $10\times10^9 J/\Psi$ and $3\times10^9 \psi'$ events can be collected per year’s running according to the designed luminosity of BEPCII in Beijing [@besiii] [@bepcii]. These large data samples will provide great opportunities to study the attractive $\chi_{cJ}$ decay and some hyperon interactions. The product of $\chi_{cJ}$ in $\psi'$ radiative decays may provide useful information on two-gluon hadronization dynamics and glueball decays. On the other side, the radiative decays of $\psi'\to\gamma\chi_{cJ}$ are expected to be dominated by electric dipole (E1) transitions, with higher multipoles suppressed by powers of photon energy divided by quark mass [@Karl] and searching for contributions of higher multipoles is promissing. The possibility of anomalous magnetic moments of higher quark being lager than those for light ones may exist [@Geffen]. Thus,$\chi_{cJ}$ decays contain abundant interesting physics. In this paper, the idea is motivated by that there may exist some baryon resonances in the process of $\chi_{cJ}$ decays to baryons. In fact, in our preliminary Monte Carlo study, we found an explicit band structure, most possibly $\Lambda(1520)$, so it necessitate this PWA. In order to get more useful information about the resonance properties such as $J^{PC}$ quantum numbers, mass, width, production and decay rates,[*[etc.]{}*]{}, partial wave analysis (PWA) are necessary. PWA is an effective method for analyzing the experimental data of hadron spectrums. There are two methods of PWA, one is based on the covariant tensor (also named Rarita-Schwinger) formalism [@Schwinger] and the other is based on the original helicity formalism [@Jackson] [@Jacob] and one covariant helicity one developed by Chung [@Chung1] [@Chung2]. For a more basic exposition, the reader may wish to consult the CERN Yellow Report [@yellow; @report]. Ref.[@Filippini] showed the connection between the covariant tensor formalism and helicity one. In this short paper, we will pay more attention to the more popular one: covariant tensor format also append the helicity one for the specific process for $\Lambda(1520)$. The organization of the paper is as follows. In Sec.I, the general formalism is given. In Sec.II, we will present the covariant tensor amplitude for $\psi'\to\gamma\chi_{c0}\rightarrow \gamma P K^- \overline{\Lambda}$. In Sec.III,the corresponding helicity formula are provides. At last, In Sec.IV,there is the conclusion. General formalism {#Sec:formalism} ================= In this part,the general formalism which will be used in the following have been mentioned in Ref.[@psi; @decay; @to; @mesons] [@N*NM; @couplings] [@gamma; @chicJ], including formalism for $\psi$ radiative decay to mesons(denoted by $M$),$M\to N^* N$,$N^*\to N M$,where $N^*$ and $N$ has the half integer spin. As discussed in Ref.[@psi; @decay; @to; @mesons],we denote the $\psi$ polarization four-vector by $\psi_{\mu}(m_1)$ and the polarization vector of the photon by $e_{\nu}(m_2)$.Then the general form for the decay amplitude is $$A=\psi_{\mu}(m_1)e^*_{\nu}(m_2)A^{\mu\nu}=\psi_{\mu}(m_1)e^*_{\nu}(m_2)\underset{i}{\sum}\Lambda_i U^{\mu\nu}_i$$ here, $U^{\mu}_{i}$ is the $i-$th partial wave amplitude with coupling strength determined by a complex parameter $\Lambda_i$. Because of massless properties,there are two additional conditions,$(1)$ the usual orthogonality condition $e_{\nu}q^{\nu}=0$,where $q$ is the photon momentum;$(2)$ gauge invariance condition (assuming the Coulomb gauge in $\psi$ rest system) $e_{\nu}p^{\nu}_{\psi}=0$,where $P_{\psi}$ is the momentum of vector meson $\psi$.Then we have $$\begin{aligned} \underset{m}{\sum}e^*_{\mu}(m)e_{\nu}(m)&=&-g_{\mu\nu}+\frac{q_{\mu}K_{\nu}+K_{\mu}q_{\nu}}{q\cdot K}-\frac{K \cdot K}{(q \cdot K)^2}q_{\mu}q_{\nu}\nonumber\\ &\equiv& -g_{\mu\nu}^{(\perp\perp)}\end{aligned}$$ with $K=p_{\psi}-q $ and $ e_{\nu}K^{\nu}=0.$.To compute the differential cross section,we need an expression for $|A|^2$ ,the square modulus of the decay amplitude,which gives the decay probability of a certain configuration should be independent of any particular frame.Thus the radiative cross section is : $$\begin{aligned} \frac{d\sigma}{d\Phi_n}&=&\frac{1}{2}\sum^2_{m_1=1}\sum^2_{m_2=1}\psi_{\mu}(m_1)e^*_{\nu}(m_2)A^{\mu\nu}\psi^*_{\mu'}(m_1)e_{\nu'}(m_2)A^{*\mu'\nu'} \nonumber\\ &=&-\frac{1}{2}\sum^2_{m_1=2}\psi_{\mu}(m_1)\psi_{\mu'}(m_1)g^{(\perp \perp)}_{\nu\nu'}A^{\mu\nu}A^{*\mu'\nu'}\nonumber\\ &=&-\frac{1}{2}\sum^2_{\mu=1}A^{\mu\nu}g^{(\perp\perp)}_{\nu\nu'}A^{*\mu\nu'}\nonumber\\ &=&-\frac{1}{2}\sum_{i,j}\Lambda_i\Lambda_j^*\sum^2_{\mu=1}U^{\mu\nu}_ig^{(\perp\perp)}_{\nu\nu'}U_j^{*\mu\nu'}\equiv\sum_{i,j}P_{ij}\cdot F_{ij}\end{aligned}$$ with definition $$\begin{aligned} P_{ij}&=&P_{ji}^*=\Lambda_i\Lambda_j^*, \\F_{ij}&=&F_{ji}^* =-\frac{1}{2}\sum^2_{\mu=1}=-\frac{1}{2}U^{\mu\nu}_i g^{(\perp\perp)}_{\nu\nu'}U_j^{*\mu\nu'}.\end{aligned}$$ note the relation $$\sum^2_{m=1}\psi_{\mu}(m)\psi_{\mu'}^*(m)=\delta_{\mu\mu'}(\delta_{\mu1}+\delta_{\mu2}).$$ The partial wave amplitude $U$ in the covariant Rarita-Schwinger tensor formalism [@Schwinger] can be constructed by using pure orbital angular momentum covariant tensor $\widetilde{t}^{(L)}_{\mu_1\mu_2\cdots\mu_L}$ and covariant spin wave functions $\phi_{\mu_1\mu_2\cdots\mu_S}$ together with the metric tensor $g^{\mu\nu}$, the totally antisymmetric Levi-Civita tensor $\epsilon_{\mu\nu\lambda\sigma}$ and the four momenta of participating particles.For a process $a\to bc$ ,if there exists a relative orbital angular momentum $L_{bc}$ between the particle $b$ and $c$ ,then the pure orbital angular momentum $L_{bc}$ state can be represented by the covariant tensor wave function $\widetilde{t}^{(L)}_{\mu_1\mu_2\cdots\mu_L}$ which is built of the relative momentum.Here,we list the amplitude for pure $S-,P-,D-,$ and $F-$ wave orbital angular momentum: $$\begin{aligned} \widetilde{t}^{(0)}&=&1,\\ \widetilde{t}^{(1)}_{\mu}&=&\widetilde{g}_{\mu\nu}(p_a)r^{\nu}B_1(Q_{abc})\equiv \widetilde{r}^{\mu}B_1(Q_{abc}),\\ \widetilde{t}^{(2)}_{\mu\nu}&=&[\widetilde{r}^{\mu}\widetilde{r}^{\nu}-\frac{1}{3}(\widetilde{r}\cdot\widetilde{r})\widetilde{g}_{\mu\nu}(p_a))]B_2(Q_{abc}),\end{aligned}$$ $$\begin{aligned} \widetilde{t}^{(3)}_{\mu\nu\lambda}&=[\widetilde{r}_{\mu}\widetilde{r}_{\nu}\widetilde{r}_{\lambda}-\frac{1}{5}(\widetilde{r}\cdot\widetilde{r})(\widetilde{g}_{\mu\nu}(p_a)\widetilde{r}_{\lambda}\nonumber \\ &\quad+\widetilde{g}_{\nu\lambda}(p_a)\widetilde{r}_{\mu}+\widetilde{g}_{\lambda\mu}(p_a)\widetilde{r}_{\nu})]B_3(Q_{abc}),\end{aligned}$$ where $r=p_b-p_c$ is the relative momentum of the two decay products in the parent particle rest frame,$\widetilde{r}\cdot\widetilde{r}=\vec{r}\cdot\vec{r}$ where $\vec{r}$ is the magnitude of three-vector , with $$\widetilde{g}_{\mu\nu}(p_a)=-g_{\mu\nu}+\frac{p_{a\mu}p_{a\nu}}{p_a^2}$$ which is the vector boson polarization sum relation, and $$Q^2_{abc}=\frac{(s_a+s_b-s_c)^2}{4s_a}-s_b$$ where $s_a=E_a^2-p_a^2$ and $B_l(Q_{abc})$ is the Blatt-Weisskopf barrier factor [@Hippel],explicitly, $$\begin{aligned} B_1(Q_{abc})=&\sqrt{\frac{2}{Q_{abc}^2}+Q_0^2}\\ B_2(Q_{abc})=&\sqrt{\frac{13}{Q_{abc}^4}+3Q_{abc}^2Q_0^2+9Q_0^4},\\ B_3(Q_{abc})=&\sqrt{\frac{277}{Q_{abc}^6}+6Q_{abc}^4Q_0^2+45Q_{abc}^2Q_0^4+225Q_0^6}\end{aligned}$$ Here $Q_0$ is a hadron scale parameter $Q_0=0.197321/R GeV/c$,in which $R$ is the radius of the centrifugal barrier in fm. If $a$ is an intermediate resonance decaying into $bc$,one needs to introduce into the amplitude a Breit-Wigner propergator [@primer] $$f^{(a)}_{(bc)}=\frac{1}{m_a^2-s_{bc}-im_a\Gamma_a}$$ In this equation,$s_{bc}=(p_b+p_c)^2$ is the invariant mass-squared of $b$ and $c$; $m_a,\Gamma_a$ are the resonance mass and width. Additionally,some expressions depend also on the total momentum of the $ij$ pair,$p_{(ij)}=p_i+p_j$.When one wants to combine two angular momenta $j_b$ and $j_c$ into a total angular momentum $j_a$,if $j_a+j_b+j_c$ is an odd number,then a combination $\epsilon_{\mu\nu\lambda\sigma}p^{\mu}_{a}$ with $p_a$ the momentum of the parent particle is needed,otherwise it is not needed. For a given hadronic decay process $A\rightarrow BC$ (B,C are fermions),in the $L-S$ scheme on hadronic level,the initial state is described by its $4-$ momentum $P_{\mu}$ and its spin state $S_A$,the final state is described by the relative orbital angular momentum state of $BC$ system and their spin state $(S_B,S_C)$.The spin states $(S_A,S_B,S_C)$ can be well represented by the relativistic Rarita-Schwinger spin wave functions for particles of arbitrary spin.As is well known that,spin-$\frac{1}{2}$ wavefunction is the standard Dirac spinor $U(p,s)$ and $V(p,s)$ ;spin-$1$ wavefunction is the standard spin-$1$ polarization four-vector $\epsilon^{\mu}(p,s)$ for particle with momentum $p$ and spin projection $s$.(1)For the case of A as a meson,B as $N^*$ with spin $n+\frac{1}{2}$ and C as $\overline{N}$ with spin $\frac{1}{2}$ ,the total spin of BC $(S_{BC})$ can be either $n$ or $n+1$. The two $S_{BC}$ states can be represented as [@N*NM; @couplings] $$\begin{aligned} \psi^{(n)}_{\mu_1\mu_2\cdots\mu_n}=&\quad \bar{u}_{\mu_1\mu2\cdots\mu_n}(p_B,s_B)\gamma_5 v(p_C,s_C),\\ \Psi^{(n+1)}_{\mu_1\mu_2\cdots\mu_{n+1}}=&\quad \bar{u}_{\mu_1\mu2\cdots\mu_n}(\gamma_{\mu_{n+1}}-\frac{r_{\mu_{n+1}}}{m_A+m_B+m_C}v(p_c,s_C)) \nonumber\\&+(\mu_1\leftrightarrow\mu_{n+1})+\cdots+(\mu_{n}\leftrightarrow\mu_{n+1})\end{aligned}$$ (2)For the case of A as $N^*$ with spin $n+\frac{1}{2}$,B as $N$ and C as a meson,one needs to couple $-S_A$ and $S_B$ first to get $S_{AB}=-S_A+S_B$ states,which are $$\begin{aligned} \phi^{(n)}_{\mu_1\mu_2\cdots\mu_n}=&\bar{u}(p_b,s_B)u_{\mu_1\mu_2\cdots\mu_n}(p_A,s_A),\\ \Phi^{n+1}_{\mu_1\mu_2\cdots\mu_n}=&\bar{u}(p_b,s_B)\gamma_5\widetilde{\gamma}_{\mu_1\mu_2\cdots\mu_n}(p_A,s_A)\nonumber\\ &+(\mu_1\leftrightarrow\mu_{n+1})+\cdots+(\mu_{n}\leftrightarrow\mu_{n+1})\end{aligned}$$ Up to now,we have introduced all knowledges for constructing the covariant tensor amplitude.In the concrete case,the P parity conservation may be applied,which expression is $$\label{parity} \eta_A=\eta_B\eta_C(-1)^{L}$$ where $\eta_A$,$\eta_B$ and $\eta_C$ are the intrinsic parities of particles A, B, and C, respectively.From this relation,L can be even or odd for one case,which guarantee a pure L final state,which is the soul of covariant $L-S$ coupling scheme. analysis for $\psi'\rightarrow \gamma \chi_{c0}\rightarrow \gamma P K^- \overline{\Lambda}$ =========================================================================================== From now on,we denote $P$,$K^-$,$\overline{\Lambda}$ by number $1,2,3$.Firstly,for $\psi'\to \gamma \chi_{c0}$,from the helicity formalism,it is easy to show that there is only one independent amplitude for $\psi'$ radiative deca y to a spin $0$ meson.Hense,the amplitude is $$U^{\mu\nu}_{\gamma \chi_{c0}}=g^{\mu\nu}f^{(\chi_{c0})}.$$ For sequential $\chi_{c0}$ decay,there may be the following modes: $\chi_{c0}\to\Lambda_{x}\overline{\Lambda},\Lambda_x\rightarrow P K^-$,where $\Lambda_{x}$ can be\ $\Lambda(1520)\frac{3}{2}^-,\Lambda(1600)\frac{1}{2}^+, \Lambda(1670)\frac{1}{2}^-,\Lambda(1690)\frac{3}{2}^-,\Lambda(1800)\frac{1}{2}^-,\\ \Lambda(1810)\frac{1}{2}^+,\Lambda(1820)\frac{5}{2}^+,\Lambda(1830)\frac{5}{2}^-,\Lambda(1890)\frac{3}{2}^+, \Lambda(2100)\frac{7}{2}^-,\\ \Lambda(2110)\frac{5}{2}^+;$\ $\chi_{c0}\rightarrow\overline{N}\overline{\Lambda},\overline{N}\rightarrow \overline{\Lambda}K^-$,where $\overline{N}$ is the anti-partner of hyperon $N$,with that $N$ can be\ $N(1650)\frac{1}{2}^-$,$N(1675)\frac{5}{2}^-$,$N(1700)\frac{3}{2}^-$,$N(1710)\frac{1}{2}^+$ or $N(1720)\frac{3}{2}^+$.\ Another possibility that $P \overline{\Lambda}$ may be generated from an unknown intermediate resonance $K_x$ is also taken into account.Amplitudes for up to $K_x's$ spin-4 are given. For $\Lambda_x$ being $\Lambda(1520)\frac{3}{2}^-$,the total spin of $\Lambda(1520)$ and $\overline{\Lambda}\frac{1}{2}^-$ can be $1$ or $2$,corresponding to the $P$ wave and $D$ wave respectively,because of the special property of spin-$0$ of $\chi_{c0}$.The parity relation makes $P$ wave impossible.Considering that this channel is recognized as a meson decaying to two fermions,now one can write the covariant amplitude as $$\Phi_{(\mu\nu)}^{(2)}\widetilde{t}^{(2)\mu\nu}.$$ And then considering $\Lambda(1520)\rightarrow P K^-$,the total spin of particle $1$ and $2$ can only be $\frac{1}{2}$,corresponding to the $P$ wave and $D$ wave,after the parity formula being applied,$L$ must be $2$.As it belongs to a fermion decay to a fermion and a meson,the covariant amplitude can be expressed as $\Phi^{(2)_{\mu\nu}}\widetilde{t}^{(2)\mu\nu}$. Here,we list all the amplitudes for the whole decay chain $\psi'\to\gamma\chi_{c0},\chi_{c0}\rightarrow\Lambda_x\overline{\Lambda},\Lambda_x\rightarrow P K^-$ up to spin-$\frac{7}{2}$ for $\Lambda_x$ according to the above principles. $$\begin{aligned} \Lambda_x(\frac{1}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(1)}_{\lambda}\widetilde{t}^{(1)\lambda}\Phi^{(1)}_{\sigma}\widetilde{t}^{(1)\sigma} \\ \Lambda_x(\frac{1}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{\chi_{c0}}_{(123)}\psi^{(0)}\phi^{(0)} \\\Lambda_x(\frac{3}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(1)}_{\lambda}\widetilde{t}^{(1)\lambda}\phi^{(1)}_{\sigma}\widetilde{t}^{(1)\sigma} \\\Lambda_x(\frac{3}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(2)}_{\lambda\delta}\widetilde{t}^{(2)\lambda\delta}\Phi^{(2)}_{\rho\sigma}\widetilde{t}^{(2)\rho\sigma} \\\Lambda_x(\frac{5}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(3)}_{\lambda\delta\beta}\widetilde{t}^{(3)\lambda\delta\beta}\Phi^{(3)}_{\rho\sigma\eta}\widetilde{t}^{(3)\rho\sigma\eta} \\\Lambda_x(\frac{5}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(2)}_{\lambda\delta}\widetilde{t}^{(2)\lambda\delta}\phi^{(2)}_{\rho\sigma}\widetilde{t}^{(2)\rho\sigma} \\\Lambda_x(\frac{7}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(3)}_{\lambda\delta\beta}\widetilde{t}^{(3)\lambda\delta\beta}\phi^{(3)}_{\rho\sigma\eta}\widetilde{t}^{(3)\rho\sigma\eta} \\\Lambda_x(\frac{7}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(4)}_{\lambda\delta\beta\xi}\widetilde{t}^{(4)\lambda\delta\beta\xi}\Phi^{(4)}_{\rho\sigma\eta\zeta}\widetilde{t}^{(4)\rho\sigma\eta\zeta}\end{aligned}$$note that $\widetilde{t}^{(0)}=1$.For channel $\chi_{c0}\rightarrow \overline{N_x}P,\overline{N_x}\rightarrow K^-\overline{\Lambda}$,we can imitate the amplitude up to spin-$\frac{7}{2}$ for $N_x$ without any difficulty,even through the highest spin for $N_x$ decaying into $K^-\Lambda$ can only be $\frac{5}{2}$ presently [@pdg2008]. $$\begin{aligned} \overline{N_x}(\frac{1}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(0)}\phi^{(0)} \\\overline{N_x}(\frac{1}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(1)}_{\lambda}\widetilde{t}^{(1)\lambda}\Phi^{(1)}_{\sigma}\widetilde{t}^{(1)\sigma} \\\overline{N_x}(\frac{3}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(2)}_{\lambda\delta}\widetilde{t}^{(2)\lambda\delta}\Phi^{(2)}_{\rho\sigma}\widetilde{t}^{(2)\rho\sigma} \\\overline{N_x}(\frac{3}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(1)}_{\lambda}\widetilde{t}^{(1)\lambda}\phi^{(1)}_{\sigma}\widetilde{t}^{(1)\sigma} \\\overline{N_x}(\frac{5}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(2)}_{\lambda\delta}\widetilde{t}^{(2)\lambda\delta}\phi^{(2)}_{\rho\sigma}\widetilde{t}^{(2)\rho\sigma} \\\overline{N_x}(\frac{5}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(3)}_{\lambda\delta\beta}\widetilde{t}^{(3)\lambda\delta\beta}\Phi^{(3)}_{\rho\sigma\eta}\widetilde{t}^{(3)\rho\sigma\eta} \\\overline{N_x}(\frac{7}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(4)}_{\lambda\delta\beta\xi}\widetilde{t}^{(4)\lambda\delta\beta\xi}\Phi^{(4)}_{\rho\sigma\eta\zeta}\widetilde{t}^{(4)\rho\sigma\eta\zeta} \\\overline{N_x}(\frac{7}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(3)}_{\lambda\delta\beta}\widetilde{t}^{(3)\lambda\delta\beta}\phi^{(3)}_{\rho\sigma\eta}\widetilde{t}^{(3)\rho\sigma\eta}\end{aligned}$$ For channel $\chi_{c0}\rightarrow K^+_x K^-,K^+_x\rightarrow P \overline{\Lambda}$,the amplitudes are also given.$J^{P}=0^+,1^-,2^+,3^-,4^+ \cdots $ are forbidden by the parity relation[@Parity].The partial wave amplitude is denoted by $U^{\mu\nu}_{(LS)}$, $L,S$ means the orbital angular momentum number and spin angular momentum number between $P$ and $\overline{\Lambda}$. $$\begin{aligned} K_x^+(0^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(0)} \\K_x^+(1^+)\qquad U^{\mu\nu}_{(10)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(0)}\widetilde{T}^{(1)\sigma}\phi_{\sigma}\epsilon_{\lambda}\widetilde{t}^{(1)\lambda} \\U^{\mu\nu}_{(11)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\epsilon_{\sigma}\widetilde{t}^{(1)}_{\eta}\Psi^{(1)}_{\zeta}\widetilde{T}^{(1)\sigma}\phi_{\sigma} \\K_x^+(2^-)\qquad U^{\mu\nu}_{(20)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\widetilde{T}^{(2)\eta\zeta}\phi_{\eta\zeta}\widetilde{t}^{(2)\rho\sigma}\psi^{(0)}\phi_{\rho\sigma}\nonumber\\ &=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}P^{(2)}_{\rho\sigma\eta\zeta}\psi^{(0)}\widetilde{T}^{(2)\eta\zeta}\widetilde{t}^{(2)\rho\sigma} \\U^{\mu\nu}_{(21)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\widetilde{T}^{(2)\beta\lambda}\phi_{\beta\lambda}\widetilde{t}^{(2)\iota}_{\sigma}\phi_{\iota\eta}\Psi^{(1)}_{\zeta} \nonumber\\&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}P^{(2)}_{\beta\lambda\iota\eta}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\widetilde{T}^{(2)\beta\lambda}\widetilde{t}^{(2)\iota}_{\sigma}\Psi^{(1)}_{\zeta} \\K_x^+(3^+)\qquad U^{\mu\nu}_{(30)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\widetilde{T}^{(3)\lambda\delta\beta}\phi_{\lambda\delta\beta}\psi^{(0)}\widetilde{t}^{(3)\rho\sigma\eta}\phi_{\rho\sigma\eta}\nonumber \\&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}P^{(3)}_{\lambda\delta\beta\rho\sigma\eta}\widetilde{T}^{(3)\lambda\delta\beta}\psi^{(0)}\widetilde{t}^{(3)\rho\sigma\eta} \\U^{\mu\nu}_{(31)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\widetilde{T}^{(3)\lambda\delta\beta}\phi_{\lambda\delta\beta}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\widetilde{t}^{(3)\kappa\xi}_{\sigma}\Psi^{(1)}_{\eta}\cdot\nonumber \\ &\quad\phi^{(3)}_{\zeta\kappa\xi}\nonumber\\ &=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}P^{(3)}_{\lambda\delta\beta\zeta\kappa\xi}\widetilde{T}^{(3)\lambda\delta\beta}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\widetilde{t}^{(3)\kappa\xi}_{\sigma}\Psi^{(1)}_{\eta}\end{aligned}$$ In the above formulas,$\phi$ implies the spin wave functions for $K_x$,for example,$\phi_{\sigma}$ corresponding to the spin-1,$\phi_{\rho\sigma}$ corresponding to spin-2.There is a general wave fuction for a particle of spin $J$ [@Chung1],which is a rank-$J$ tensor $$\begin{aligned} \phi^{\alpha_1\alpha_2\cdots\alpha_J}(m)&=\underset{m_1m_2\cdots}{\sum}\langle 1m_11m_2|2n_1\rangle\langle 2n_11m_3|3n_2\rangle\nonumber\\ &\quad\cdots\langle J-1n_{J-2}1m_J|Jm\rangle\phi^{\alpha_1}(m_1)\nonumber\\ &\quad\phi^{\alpha_2}(m_2)\cdots\phi^{\alpha_J}(m_J)\end{aligned}$$ where $\phi^{\alpha}(m)$ is the familiar polarization four-vector of spin-1 particle, $$\phi^{\alpha}(1,-1)=\mp\frac{1}{\sqrt{2}}(0;1,\pm i,0), \phi^{\alpha}(0)=(0;0,0,1).$$ note the following useful relationship: $$\phi(-m)=(-)^m\phi^*(m).$$ It is best to illustrate these formulas with some examples.For $J=1$,one finds that it reduces to identities for $\phi(1)$ and $\phi(0)$.For $J=2$,one has $$\begin{gathered} \qquad\qquad\phi^{\alpha\beta}(+2)=\phi^{\alpha}(1)\phi^{\beta}(1) \\ \phi^{\alpha\beta}(+1)=\frac{1}{\sqrt{2}}[\phi^{\alpha}(1)\phi^{\beta}(0)+\phi^{\alpha}(0)\phi^{\beta}(1)]\\ \phi^{\alpha\beta}(0)=\frac{1}{\sqrt{6}}[\phi^{\alpha}(1)\phi^{\beta}(-1)+\phi^{\alpha}(-1)\phi^{\beta}(1)+\sqrt{\frac{2}{3}}\phi^{\alpha}(0)\phi^{\beta}(0)]\end{gathered}$$ $\phi\cdot\phi$ is in reality its spin projection operator $P^{(S)}$. $$\begin{aligned} P^{(2)}_{\rho\sigma\eta\zeta}(p_{K_x})&=\underset{m}{\sum}\phi_{\rho\sigma}(p_{K_x},m)\phi^*_{\eta\zeta}(p_{K_x,m})\nonumber\\&=\frac{1}{2}(\widetilde{g}_{\rho\eta} \widetilde{g}_{\sigma\zeta}+\widetilde{g}_{\rho\zeta}\widetilde{g}_{\sigma\eta})-\frac{1}{3}\widetilde{g}_{\rho\sigma}\widetilde{g}_{\eta\zeta} \\\quad P^{(3)}_{\lambda\delta\beta\zeta\kappa\xi}(p_{K_x})&=\underset{m}{\sum}\phi_{\lambda\delta\beta}(p_{K_x},m)\phi^*{\zeta\kappa\xi}(p_{K_x},m)\nonumber\\&=\frac{1}{6}(\widetilde{g}_{\lambda\zeta}\widetilde{g}_{\delta\kappa}\widetilde{g}_{\beta\xi} +\widetilde{g}_{\lambda\zeta}\widetilde{g}_{\delta\xi}\widetilde{g}_{\beta\kappa}+\widetilde{g}_{\lambda\kappa}\widetilde{g}_{\delta\zeta}\widetilde{g}_{\beta\xi} \nonumber\\&\quad\quad+\widetilde{g}_{\lambda\zeta}\widetilde{g}_{\delta\xi}\widetilde{g}_ {\beta\kappa}+\widetilde{g}_{\delta\xi}\widetilde{g}_{\beta\zeta}\widetilde{g}_{\lambda\xi}+\widetilde{g}_{\delta\kappa}\widetilde{g}_{\lambda\xi}\widetilde{g}_{\beta\zeta}) \nonumber\\&\quad-\frac{1}{15}(\widetilde{g}_{\lambda\delta}\widetilde{g}_{\zeta\kappa}\widetilde{g}_{\beta\xi}+\widetilde{g}_{\lambda\delta}\widetilde{g}_{\kappa\xi}\widetilde{g}_{\beta\zeta}+\widetilde{g}_{\lambda\delta}\widetilde{g}_{\zeta\xi}\widetilde{g}_{\beta\kappa} \nonumber\\&\quad\quad+\widetilde{g}_{\lambda\beta}\widetilde{g}_{\zeta\xi}\widetilde{g}_{\delta\kappa}+\widetilde{g}_{\lambda\beta}\widetilde{g}_{\zeta\kappa}\widetilde{g}_{\delta\xi} +\widetilde{g}_{\lambda\beta}\widetilde{g}_{\kappa\xi}\widetilde{g}_{\delta\zeta} \nonumber\\&\quad\quad+\widetilde{g}_{\delta\beta}\widetilde{g}_{\kappa\xi}\widetilde{g}_{\lambda\kappa}+\widetilde{g}_{\delta\beta}\widetilde{g}_{\zeta\kappa}\widetilde{g}_{\lambda\xi}+\widetilde{g}_{\delta\beta}\widetilde{g}_{\kappa\xi}\widetilde{g}_{\lambda\kappa})\end{aligned}$$ So far,we have given the covariant tensor amplitude formula for the process $\psi'\to\gamma\chi_{c0}\rightarrow \gamma P K^- \overline{\Lambda}$. helicity formula ================ For completeness,we also give the helicity format in comparison with tensor formula.Helicity formalism has an explicit advantage, the angular dependence can be easily seen.In this section ,we will give the amplitude for $\chi_{c0}\to\Lambda(1520)\overline{\Lambda},\Lambda(1520)\to P K^-,\overline{\Lambda}\to\overline{P}\pi^+$.$\Lambda(1520)$ the most possible resononse,has been mentioned above. Firstly,we want to introduce the general helicity formula expression.Consider a state with spin(parity) $=J(\eta_J)$ decaying into two states with $S(\eta_s)$ and $\sigma(\eta_\sigma)$.The decay amplitudes are given,in the rest frame of $J$ [@helicity; @guide] [@Jacob], $$\mathcal{M}^{J\to s\sigma}_{\lambda\nu}=\sqrt{\frac{2J+1}{4\pi}}D^{J*}_{M\delta}(\phi,\theta,0)H^J_{\lambda\nu},$$ where $\lambda$ and $\nu$ are the helicities of the two final state particles $s$ and $\sigma$ with $\delta=\lambda-\nu$.The symbol $M$ stands for the $z$ component of the spin $J$ in a coordinate system fixed by production process.The helicities $\lambda$ and $\nu$ are rotational invariants by definition.The direction of the break-up momentum of the decaying particle $s$ is given by the angles $\theta$ and $\phi$ in the $J$ rest frame.Let $\hat{x},\hat{y}$ and $\hat{z}$ be the coordinate system fixed in the $J$ rest frame.It is important to recognize,for applications to sequential decays,the exact nature of the body-fixed (helicity) coordinate system implied by the arguments of the $D$ function given above.Let $\hat{x}_h,\hat{y}_h$ and $\hat{z}_h$ be the helicity coordinate system fixed by the $s$ decay.Then by definition $\hat{z}_h$ describes the direction of $s$ in the $J$ rest frame (also termed the helicity axis) and the $y$ axis is given by $\hat{y}_h=\hat{z}\times\hat{z}_h$ and $\hat{x}_h=\hat{y}_h\times\hat{z}_h$.Parity conservation in the decay leads to the relationship $$H^{J}_{\lambda\nu}=\eta_J\eta_s\eta_\sigma(-)^{J-s-\sigma}H^{J}_{-\lambda-\nu}$$ Let us consider a full process $A\to B+C$ where $B$ and $C$ are also unstable particles decaying to $B_1+B_2$ and $C_1+C_2$ respectively.The decay amplitude is simply [@PhD; @thesis] $$\begin{aligned} \mathcal{M}(\lambda_{B_1},\lambda_{B_2},\lambda_{C_1},\lambda_{C_2})=\underset{\lambda_B,\lambda_C}{\sum}& \mathcal{M}^{A\to B+C}_{\lambda_B,\lambda_C}\cdot \mathcal{M}^{B\to B_1+B_2}_{\lambda_{B_1},\lambda_{B_2}}\cdot\nonumber \\&\mathcal{M}^{C\to C_1+C_2}_{\lambda_{C_1},\lambda_{C_2}}\end{aligned}$$ with $$\begin{aligned} \mathcal{M}^{A\to B+C}_{\lambda_B,\lambda_C}&=\sqrt{\frac{2J_A+1}{4\pi}}D^{J_A*}_{M_A,\lambda_B-\lambda_C}(\phi_A,\theta_A,0)H^A_{\lambda_B,\lambda_C}, \\ \mathcal{M}^{B\to B_1+B_2}_{\lambda_{B_1},\lambda_{B_2}}&=\sqrt{\frac{2J_B+1}{4\pi}}D^{J_B*}_{\lambda_B,\lambda_{B_1}-\lambda_{B_2}}(\phi_B,\theta_B,0)H^B_{\lambda_{B_1},\lambda_{B_2}}, \\ \mathcal{M}^{C\to C_1+C_2}_{\lambda_{C_1},\lambda_{C_2}}&=\sqrt{\frac{2J_C+1}{4\pi}}D^{J_C*}_{-\lambda_C,\lambda_{C_1}-\lambda_{C_2}}(\phi_C,\theta_C,0)H^C_{\lambda_{C_1},\lambda_{C_2}}, \label{sub3}\end{aligned}$$ Please note in the first subscript of $D^{J_C*}$ is $-\lambda_C$ and NOT $\Lambda_C$ although it also gives the correct result,because the quantization axis is along the direction of the momentum of particle $B$ so that the spin-quantization projection $M_C$ in the particle $C$ rest frame verifies $M_C=-\lambda_C$. The unpolarized angular distribution is then given by averaging over initial spins and by summing over final spins: $$\begin{aligned} \frac{d^3\Gamma}{\mathcal{N}d\Omega_A d\omega_B d\Omega_C}=\frac{1}{2S_A+1}&\underset{\lambda_{B_1},\lambda_{C_1},\lambda_{B_2},\lambda_{C_2}}{\sum}\nonumber \\&|\mathcal{M}(\lambda_{B_1},\lambda_{B_2},\lambda_{C_1},\lambda_{C_2})|^2.\end{aligned}$$ where $\mathcal{N}$ is the normalization factor. Using the parity conservation formula,one has $H^{\overline{\Lambda}}_{\frac{1}{2}0}=H^{\overline{\Lambda}}_{-\frac{1}{2}0}$ and $H^{\Lambda(1520)}_{\frac{1}{2}0}=-H^{\Lambda(1520)}_{-\frac{1}{2}0}$.By applying the above amplitude expression,after a lengthy,tedious but trivial evaluating,one can get the helicity amplitude, $$\begin{aligned} \label{helicity distribution} \frac{d^3\Gamma}{\mathcal{N}d\Omega_A d\omega_B d\Omega_C}=&[\frac{3}{2}\cos^2\theta_{\Lambda(1520)}-\frac{3}{2}\cos\theta_{\Lambda(1520)}\nonumber \\&+\frac{9}{2}\cos^2\theta_{\Lambda(1520)}\sin\theta_{\Lambda(1520)}\cos\phi_{\Lambda(1520)}\nonumber \\&+\frac{\sqrt{3}}{2}\cos^2\theta_{\Lambda(1520)}\cos2\phi_{\Lambda(1520)}\nonumber \\&-\frac{\sqrt{3}}{4}\cos\theta_{\Lambda(1520)}\cos2\phi_{\Lambda(1520)}\nonumber \\&-\frac{3\sqrt{3}}{4}\cos2\phi_{\Lambda(1520)}+1]|H^{\overline{\Lambda}}_{\frac{1}{2}0}|^2H^{\Lambda(1520)}_{\frac{1}{2}0}|^2\end{aligned}$$ where the subscript $\Lambda(1520)$ denotes that the angle defined in the rest frame of $\Lambda(1520)$. After integrating $\phi_{\Lambda(1520)}$’s from $[0,2\pi]$,the Eq. becomes $$\frac{d^3\Gamma}{\mathcal{N'}d\Omega_A d\omega_B d\Omega_C}=\frac{3}{2}\cos^2\theta_{\Lambda(1520)}-\frac{3}{2}\cos\theta_{\Lambda(1520)}+1$$ where $\mathcal{N'}=\mathcal{N}|H^{\overline{\Lambda}}_{\frac{1}{2}0}|^2H^{\Lambda(1520)}_{\frac{1}{2}0}|^2$ is redefined normalization factor. conclusion ========== In this short note,firstly,the relevant general tensor formalism hac been introduced,and then give the covariant tensor amplitudes.At last,for completeness and some experimental reasons,the helicity amplitude expression has also been provided and a figure attached. acknowledgments =============== The author acknowledges greatly helpful discussions with B. S. Zou.This work is supported in part by the National Natural Science Foundation of China under contracts Nos. 10521003,10821063, the 100 Talents program of CAS, and the Knowledge Innovation Project of CAS under contract Nos. U-612 and U-530 (IHEP). 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--- bibliography: - 'gapbib.bib' --- [**Learning theories reveal loss of pancreatic electrical connectivity in diabetes as an adaptive response** ]{}\ Pranay Goel$^{1,\ast}$, Anita Mehta$^{2}$\ **[1]{} Mathematics and Biology, Indian Insitute of Science Education and Research Pune, Pune, Maharashtra, India\ **[2]{} Department of Physics, S. N. Bose National Centre for Basic Sciences, Kolkata, West Bengal, India\ $\ast$ E-mail: pgoel@iiserpune.ac.in**** Abstract {#abstract .unnumbered} ======== Cells of almost all solid tissues are connected with gap junctions which permit the direct transfer of ions and small molecules, integral to regulating coordinated function in the tissue. The pancreatic islets of Langerhans are responsible for secreting the hormone insulin in response to glucose stimulation. Gap junctions are the only electrical contacts between the beta-cells in the tissue of these excitable islets. It is generally believed that they are responsible for synchrony of the membrane voltage oscillations among beta-cells, and thereby pulsatility of insulin secretion. Most attempts to understand connectivity in islets are often interpreted, bottom-up, in terms of measurements of gap junctional conductance. This does not, however explain systematic changes, such as a diminished junctional conductance in type 2 diabetes. We attempt to address this deficit via the model presented here, which is a learning theory of gap junctional adaptation derived with analogy to neural systems. Here, gap junctions are modelled as bonds in a beta-cell network, that are altered according to homeostatic rules of plasticity. Our analysis reveals that it is nearly impossible to view gap junctions as homogeneous across a tissue. A modified view that accommodates heterogeneity of junction strengths in the islet can explain why, for example, a loss of gap junction conductance in diabetes is necessary for an increase in plasma insulin levels following hyperglycemia. Introduction {#introduction .unnumbered} ============ Gap junctions are clusters of intercellular channels between cells formed by the membrane proteins connexins (Cx), that mediate rapid intercellular communication via direct electric contact and diffusion of metabolites [@goodpaul]. In excitable cells such as neurons, cardiac myocytes and smooth muscles, gap junctions provide efficient low-resistance pathways through which membrane voltage changes can be shared across the tissue. Besides excitable cells, gap junctions are found between cells in almost every solid tissue [@goodpaul]. Gap junctions are thus central to multicellular life [@nicholson], with numerous diseases linked to connexin disorders [@willecke], including type 2 diabetes mellitus [@elke; @ravier; @medachatroom]. The islets of Langerhans in the pancreas are clusters of largely alpha-, beta- and delta-cells that respectively control secretion of the hormones glucagon, insulin and somatostatin central to energy regulation. Gap junctions form direct connections between beta-cells [@orci73; @boscomeda11; @cabrera] in islets, and are important for normal glucose-stimulated insulin secretion (GSIS) [@medachatroom; @macd; @head12]. Gap junctions are generally believed to be important for coordinating the beta-cell electrical oscillations known as bursting, which in turn, can then support pulsatile insulin secretion [@chansharing; @macd; @medachatroom]; this view is supported by theoretical studies [@rinzel; @shermansmolen; @bss] as well. The conductance strength of gap junctions evolves by the insertion or deletion of connexin proteins (Fig. \[gj\]) into junctional plaques, and by altering the single-channel conductance and probability of channel opening [@goodpaul]. Whether these molecular changes constitute a systematic adaptive response of the endocrine tissue to its metabolic environment remains to be investigated, in particular from a theoretical point of view. As with many other excitable cells, the information content of bioelectric signals [@levin] in islets is yet unclear. The mechanisms underlying bursting are well understood [@glycophan; @goel]; however, how those temporal properties regulate energy homeostasis is not. While slow (5 – 15 minute period) bursts are generally thought to drive secretion at stimulatory concentrations of glucose, faster (periods less than 5 minutes) oscillations are also found, typically at sub-stimulatory (basal) glucose levels; the average calcium signal, however, is comparable in either case (such as in simulations from [@glycophan], not shown). The hypothesis that a synchronous bursting of beta-cells [@medarojas; @chansharing; @macd] is essential to GSIS is guided by the observation of pulsatile insulin secretion from islets [@bergsten] and in vivo [@bergdiab]. Gap junctions can certainly mediate synchrony in principle, as shown in both simulations [@rinzel; @chansharing] and experiments [@calabrese; @ravier]. Whether this is their role in vivo is debatable. In general, in vitro studies do not address this question completely, because they are typically carried out with glucose perifusion. Since glucose is microscopically delivered to beta-cells via a rich blood vessel supply in the islet in vivo, oscillator entrainment by junction coupling may be far less important than expected from experiments on isolated islets, especially if the beta-cells are not too heterogeneous in frequency [@nunemaker]. In fact, Rocheleau et al. [@piston] have performed experiments using a microfluidic chip taking care to see that glucose stimulates an islet only partially; they find partially propagated waves but not synchrony. Their result shows that gap junctions are limited in their ability to support uniform synchronization across the entire islet in the presence of a glucose gradient in the islet. It is possible that even with glucose micro-delivery as in vivo, synchronization may be a more local phenomenon than has been previously appreciated. Stozer et al. [@stozer] have recently demonstrated that in islet slices only local synchronization is seen across groups of beta-cells. Another theory, different from one that anticipates gap junctions serve to synchronize an islet uniformly, thus appears to be necessary to explain some of the phenomena associated with insulin secretion, and it is this that we attempt in the rest of this paper. A paradigm that is gaining increasing recognition is that bioelectric and (epi-)genetic signaling are related as a cyclical dynamical system [@levin]: membrane voltage activity induces changes in mRNA expression and transcriptional regulation, which in turn leads to altered membrane channel proteins. Here we develop a theory to study an adaptive response of gap junctions to islet firing activity. Bioelectric cues are encoded as bursting, these determine junctional conductance states, and junctions respond in turn by translation modifications that alter firing rates. In this way, electric and genetic components “learn" from each other, iteratively. While learning is integral to neural systems and functionally beneficial at the level of a single individual, many studies have focused on the collective effects of \[simple forms of\] individual learning and decision-making, e.g. in populations of interacting individuals, or agents. Such distributed systems, exemplifying social or ecological group behavior, also share similarities with interacting systems of statistical physics, in the nature of the local “rules" followed by the individual units as well as in the emergent behavior at the macro level. Game-theoretic approaches [@game1; @game2; @game3] are sometimes brought to bear on such issues, their underlying idea being that the behavior of an individual (its “strategy") is to a large extent determined by what the other individuals are doing. The strategic choices of an individual are thus guided by those of the others, through considerations of the relative “payoffs" (returns) obtainable in interactive games. In this context, a stochastic model of strategic decision-making was introduced in [@amjml99], which captures the essence of the above-stated notion, i.e. selection from among a set of [*competing*]{} strategies based on a comparison of the [*expected*]{} payoffs from them. Depending upon which of the available strategic alternatives (that are being wielded by the other agents) is found to have the most favorable “outcome" in the local vicinity, every individual appropriately revises its strategic choice. Competition between prevalent strategies and adaptive changes at the individual level characterize the sociologically motivated model of [@amjml99]. Given that these two features of competition and adaptation also generally occur across the framework of activity-induced synaptic plasticity, a translation of the notions in [@amjml99] to the latter context was attempted in  [@gmmam] and  [@plosone]. A model was delineated in ref. [@gmmam] along these lines, with the types or weights of a plastic synapse taking the place of strategies. In the next subsections, we will extend these concepts to formulate a theory of ‘competing’ gap junctions in a network. Voltage gating of junctional conductance and homeostatic adaptation {#voltage-gating-of-junctional-conductance-and-homeostatic-adaptation .unnumbered} ------------------------------------------------------------------- Gap junctions are known to adapt on at least two timescales: trans-junctional currents are gated on a fast timescale of the order of a few milliseconds to seconds in response to a trans-junctional voltage difference ($\Delta V$) [@goodpaul]. Voltage gated currents of Cx36 channels (the connexin isoform relevant to islets [@ravier]), expressed in Xenopus oocytes and transfected human HeLa cells, were recorded in [@teubner] (Fig. \[vg\]). Haefliger et al. [@haefliger] have shown hyperglycemia decreases Cx expression in adult rats. Paulauskas et al. [@bakauskas] have recently described a 16-state stochastic model of gap junctional currents that are voltage gated by altering, amongst other things, unitary single channel conductance and the probability of opening [@goodpaul; @bakauskas4]. On much slower timescales of hours to days, gap junctions are regulated by the events that alter the insertion and deletion of channels in the junctional plaque, connexin proteins synthesis, trafficking to the membrane and degradation. We propose to study adaptation in gap junction strength on slow timescales; this is the natural setting for a mean field theory of gap junction modification, that is, over suitably long periods that averages over cellular firing rates can be treated as adiabatic. Interestingly, the voltage-gated gap junction appears to conform to a homeostatic principle with respect to transjunctional current, $I_{gap}=g_{gap}\;\Delta V$: when $\Delta V$ is small, such as during synchronous bursting for example, gap junctional conductance is large, while a large $\Delta V$, as in anti-synchrony, is compensated with a small $g_{gap}$. That is, firing patterns $\Delta V$ result in changes in $g_{gap}$ that stabilize $I_{gap}$. We extrapolate from this argument to construct a [*homeostatic learning rule*]{} for (slow) modification of gap junctions, as described below. Model and Results {#model-and-results .unnumbered} ================= Model – A learning theory of gap junctional adaptation {#model-a-learning-theory-of-gap-junctional-adaptation .unnumbered} ------------------------------------------------------ Our starting point is a model of competitive learning introduced in [@amjml99] and applied, in [@gmmam] and [@plosone] to look at the optimisation of learning via a model of competing synapses. Proceeding by analogy, we consider a network consisting of $\beta$-cells connected by gap junctions, where the latter are treated as mutual neighbors if they are connected by a $\beta$-cell. In a one-dimensional formulation, each gap junction will thus be associated with two gap junctional neighbors. For simplicity the $\beta$-cells can be represented by binary threshold units, and the two states of the binary gap junction, which are inter-convertible by definition, are assumed to have different weights, which we label as ‘strong’ and ‘weak’ types. A weak gap junction is characterized, for example, by fewer connexin proteins in the junctional plaque. When the middle gap junction is under consideration for a state update, the $\beta$-cells A and B (Fig. \[bonds\]) share this middle gap junction in common; thus, in comparing how often the two $\beta$-cells are found activated, one can factor out the influence of the common gap junction, when considering averages, and effectively treat the time-averaged activation frequency of either $\beta$-cell as being determined only by the single, [*other*]{} gap junction that the $\beta$-cell is connected to. This essentially implies that the state of $\beta$-cell A, say, can be considered quite reasonably as an “outcome" to be associated with gap junction $n - 1$, and similarly with $\beta$-cell B and gap junction $n+1$; thus, $\beta$-cells can be thought of as taking on the identities of the respective gap junctions. There are few general principles that can organize an argument to discuss plastic behavior in excitable cells; Hebb’s postulate is one such. In common colloquialism this learning rule is stated as “cells that fire together, wire together"; in other words, temporal association between pairs of firing neurons is successively encoded in synaptic coupling between those neurons. A Hebbian philosophy asserts that the direction of adaptation is such as to reinforce coordinated activity between cells. One can now set forth some rules governing the above weight changes, which may have a Hebbian or anti-Hebbian flavor as the situation demands, and depend on the outcomes of the surrounding $\beta$-cells. Hebbian rules in the case of synaptic plasticity favour synchrony, so that e.g. a synapse is strengthened if its surrounding neurons fire or do not fire together; the opposite is the case with anti-Hebbian rules. In the present context, we use this concept analogously: for Hebbian rules, synchronous activity causes a strengthening of conductance while anti-synchronous activity causes a weakening of conductance.Thus, loosely speaking, two gap junctions adjacent to any given gap junction “compete" to decide its type, and this continues to happen repeatedly across the entire network. Let us now consider the update dynamics of a single [*effective*]{} gap junction, that in some sense represents the average state of the whole network. To begin with, in such a picture, the outcomes are assumed to be uncorrelated at different locations, and treated as independent random variables, with the probability for activation being obtainable from the time-averaged activation frequency of the $\beta$-cell. Consistent with the situation described in the previous paragraph, that the effect of the common gap junction can be left out on average in comparing the outcomes of its connected $\beta$-cells, we associate, with each $\beta$-cell, a probability for activation at any instant that is [*only*]{} a function of the other neighboring gap junction, being equal to $p_+$ ($p_-$) for a strong (weak) type gap junction. We now consider a mean-field version of the model. The idea behind the mean-field approximation is that we look at the average behavior in an infinite system. This, at one stroke, deals with two problems: first, there are no fluctuations associated with system size, and second, the approximation that we have made in ignoring the “self-coupling" of the gap junction is better realized. In the mean-field representation, every gap junction is assigned a probability (uniform over the lattice) to be either strong ($f_+$) or weak ($f_-$), so that spatial variation is ignored, as are fluctuations and correlations. This single effective degree of freedom allows for a description of the system in terms of its fixed point dynamics. The rate of change of the probability $f_+$, say, (which in the limit of large system size is equivalent to the fraction of strong units) with time, is computed by taking into account only the nearest-neighbor gap junctional interactions, via specific rules. To design a transition rule for gap junctions that is consistent with a Hebbian theory, and at the same time tunes gap junctional plasticity to voltage activity in the network, we mimic the homeostatic adaptation implicit in (fast) voltage-gating of conductance (Fig. \[vg\]): to reinforce synchronous activity conductance, changes must be directed towards a maximal state of conductance, while anti-synchronous activity is best served by a weakening of conductance. The [*homeostatic learning rule*]{} is summarised as follows: if $\beta$-cells (Fig. \[bonds\]) fire simultaneously $\Delta V_{AB}$ is zero and gap junction, $g$, strengthens to one, while if one $\beta$-cell fires but not the other, $\Delta V_{AB}$ is one and junction strength weakens to zero. We write equations for the probability $f_+ (t+1)$ that the intermediate gap junction (Figure \[bonds\]) is in the strong state, say, at time $t+1$ in terms of the same probability at time $t$, $f_+ (t)$, the (complementary) probability that it was in the weak state at time $t$, $f_- (t)$ and Prob($\Delta F$), the probability of a change in strength of a given magnitude: $$f_+ (t+1) = f_+ (t)\times \text{Prob}(\Delta F = 1) + f_- (t)\times \text{Prob}(\Delta F = 1)$$ The first term on the right hand side represents the probability that the strong state at time $t$ stays strong at time $t+1$; since the gap junctions are binary, a strong junction cannot get any stronger. Since $f_+ (t)+f_- (t) = 1$, this reduces to the equation $f_S (t+1) = \text{Prob}(\Delta F = 1)$, independent of the initial state of the gap junction. We now write down all possible scenarios for $\text{Prob}(\Delta F = 1)$: in words, these correspond to the sum of the following probabilities: (Prob that both $g_L$ and $g_R$ are in the strong state)$\times$(Prob that A and B both fire, AND both don’t fire) + (Prob that both $g_L$ and $g_R$ are in the weak state)$\times$(Prob that A and B both fire, AND both don’t fire) + (Prob that $g_L$ and $g_R$ are in disparate states)$\times$(Prob. that A and B both fire, AND both don’t fire). For example: if $g_L$ and $g_R$ (see Fig. \[bonds\]) are both strong – with probability $f_+^2$ – the firing pattern that leads to a strong middle junction, $g$, according to the homeostatic learning rule is when $\Delta V_{AB}=0$, i.e. either when both A and B fire simultaneously (probability, $p_+^2$), or both do not fire (probability, $(1-p_+)^2$). All such combinations are enumerated in Table \[combotable\], this leads to an equation for the evolution of $g$: $$\begin{aligned} f_+(t+1) &= &f_+^2(t)\; (p_+^2 + (1-p_+)^2) \nonumber \\ && +\; f^2_-(t) \;(p_-^2 +(1-p_-)^2) \label{eq:de}\\ && +\; 2f_+(t)f_-(t)\;(p_+p_- + (1-p_+)(1-p_-) \nonumber\end{aligned}$$ This evolution equation thus embodies that if $\beta$-cells (Fig. \[bonds\]) fire simultaneously, $\Delta V_{AB}$ is zero and gap junctions strengthen, while if $\Delta V_{AB}$ is one, junction strength weakens. Results – the steady state distribution of gap junctions {#results-the-steady-state-distribution-of-gap-junctions .unnumbered} -------------------------------------------------------- The steady-state distribution of weak and strong junctions is obtained as the fixed point solution of Eq. (\[eq:de\]): $$f_+^* = \frac{ -4p_-(p_+ - p_-) + 2(p_+ - p_-) + 1- \sqrt{-4(p_+^2-p_-^2)+4(p_+-p_-)+1} }{4(p_+ - p_-)^2}.$$ $f_+^*$ is stable in the entire $0<(p_-,p_+)<1$ domain. Perturbations from $f_+^*$ relax at a rate $\lambda =1-\sqrt{1 - 4p_+^2 + 4p_+ - 4p_- + 4p_-^2}$. The physically reasonable condition on the firing probabilities is $p_- < p_+$. The minimum $f_+^*$ is $0.5$ which occurs for $p_- + p_+=1$ (Fig. \[fig:pd2a\]). That is, the theory predicts that in vivo at least half of the gap junctions in an islet will be of the strong type. It is possible for strong junctions to dominate the islet completely, $f_+^* \approx 1$, but this is seen to be an extreme scenario and requires either: $p_+$ is very low and $p_-$ as well, or $p_+$ is very high and $p_-$ is greater than about half. For the large part of the $(p_-,p_+)$ parameter space $f_+^*$ is predominantly between 0.5 and 0.7. For low firing probabilities, such as for example $(p_-,p_+)=(0.1,0.15)$ the beta-cells A and B (Fig. \[bonds\]) seldom fire and $\Delta V_{AB}$ is invariably close to zero; $g$ therefore adapts towards the strong state. Likewise, when $p_-$ and $p_+$ are both high, such as for example at $(0.85,0.9)$ beta-cells A and B fire with a high rate and $\Delta V_{AB}$ is again close to zero and $g$ adapts towards the strong state. When the probabilities $p_-$ and $p_+$ are considerably different, however, for example when $(p_-,p_+)=(0.1,0.9)$ four possibilities arise: either A and B are both associated with weak (strong) junctions and $g$ adapts towards 1; or one of A or B is associated with a weak (strong) junction, but since one beta-cell then fires with a probability much larger than the other, $g$ adapts towards 0. Thus $f_+^*$ is close to half in this case ($g$ equally likely to be 0 or 1), as is the firing rate (Fig. \[frate\]). We see thus that similar behaviour for the two gap junctions induces strengthening, while dissimilar behaviour induces weakening, in line with the Hebbian viewpoint adopted above. Discussion {#discussion .unnumbered} ========== One major interest in developing a theory of gap junction adaption is to understand the changes in junctional conductance that take place in type 2 diabetes. It has been suspected from animal studies that loss of Cx36 is phenotypically similar to a prediabetic condition characterized by glucose intolerance, diminished insulin oscillations and first and second phases of insulin secretion, and a loss of beta-cell mass [@bavamian07; @hamelinmeda09; @medachatroom; @poto12]. Head et al. [@head12] have recently confirmed this [*in vivo*]{} via the observation that Cx36 conductance loss induces postprandial glucose intolerance in mice. These observations suggest that a loss of electrical connectivity in islets may underlie type 2 diabetes by disrupting insulin oscillations and reducing first-phase insulin secretion [@head12; @medachatroom]. Benninger et al. [@benningerbasal] have found yet another effect that could be relevant to diabetes, that a loss of gap junctions in islets leads to increased basal (i.e. when minimally stimulated by glucose) insulin release. If this were to hold in vivo it could explain hyperinsulinaemia as a result of gap junction loss as well, when steady state levels of circulating plasma insulin in diabetics continue to be high even in fasting conditions. A word about dimensionalities – while we recognise that the geometries of real synaptic networks are complex and that they are embedded in three dimensions, our choice of working in one dimension is based as much on simplicity as on the absence of a reason to choose a more complex geometry. Working on a three-dimensional lattice would only increase the complexity of our algebra, while not really getting closer to the real geometry of synaptic networks, which are, as the name suggests, probably embedded on abstract graphs. However, the fact that we have worked in mean field (ignoring correlations and going to the limit of infinite systems) in a one-dimensional embedding makes our results less reliant on the embedding geometry than they otherwise might have been. We mean by this that while specific quantitative estimates might well be affected by the inclusion of more neighbours in higher dimensionalities, the qualitative outlines of our calculations will remain very similar. Our choice of mean field dynamics both in this case (as well as in the original learning model of [@gmmam])was very purposeful: in both cases, the exact geometries/connectivities of islets/synapses are imprecisely known, and infinitely variable. Under these conditions mean field theory is the tool most widely resorted to by modellers, since it is able to predict general features based on minimalistic assumptions. The game-theoretic formalism presented here provides a high-level explanation why a loss of junctional conductance would be necessary in diabetes. In the healthy individual insulin secretion occurs relatively sparingly, for a few hours at regularly spaced intervals following glucose ingestion (breakfast, lunch and dinner). The low firing rates in a healthy individual are accompanied by a high proportion of strong gap junctions (that is, near the region marked by A, Fig. \[fig:pd2a\], where $f_+^*$ is close to 1). Diabetes is associated with overnutrition among various other factors [@nathan], and invariably involves combating an increased glucose load [@ada12; @dcare12]. Several authors that proposed that a substantial loss of Cx36 could ocur in type 2 diabetes (reviewed for example in [@hamelinmeda09]). Much of the evidence that connexins expression or signaling are altered in models of type 2 diabetes comes from rodents; however, because Cx36 is present in human islets, this gives rise to the speculation (see e.g. [@head12]) that a loss of Cx36 gap junction conductance may occur in type 2 diabetes. Thus, based on glucose intolerance measured in the conscious mouse Head et al. [@head12], as well as others [@ravier; @benzhang; @speier; @medachatroom], have estimated that a loss of nearly 50% in junctional conductance could occur in diabetes. In Fig. \[fig:pd2a\] the locus of a 50% connectivity loss is the line $p_+ + p_-=1$, where the fraction of strong gap junctions is halved ($f_+^*=0.5$) but the firing rates are higher (Fig. \[frate\]). That is, the islet stressed by an increase glycemic stimulation is forced to respond with an increase in its firing and insulin secretion rate, which it does by degrading strong gap junctions to weaker ones. In this way, the islet is able to accommodate a stimulus stronger than that for which its physiology had evolved. A change in $f_+^*$ is accomplished largely through altering the probabilities of junction-induced firing, $p_+$ and $p_-$. As mentioned in the introduction, the classical view of diabetes is that it results from gap junction dysfunction. Instead, the game-theoretic theory we have presented relates a conductance decrease to an *adaptive response of an islet that sacrifices strong gap junctions in order to maintain insulin control over hyperglycemia*. At the heart of our game-theoretic theory is its use of stochasticity in gap junction synchronisation. Classically, strong gap junctions entrain beta-cells to fire, the entire assembly is assumed to be fairly homogeneous in gap junction strength, and the resultant synchronous bursting is seen to be essential to GSIS. Our theory on the other hand, introduces the possibility that beta-cells coupled even to strong gap junctions may not fire, and likewise, weak gap junctions may induce simultaneous firing. Further, synchronous bursting, as well as the simultaneous *absence* of bursting, induces stronger junctions, while antisynchrony weakens them. The result is that gap junctional strengths are constantly updated as a result of the synchronous or asynchronous bursting of beta-cells. In other words, the core idea of our paper is that disparate firing patterns lead to changes in gap junctional strength – which provides a hitherto unexplored scenario for synchrony. This then naturally leads to a situation where heterogeneity prevails in the distribution of gap junctional strengths in the islet. The heterogeneity of gap junctions in turn determines more complex patterns of activity in the network, beyond the simple categories of (anti-)synchronous bursting. In principle it is possible to explain observations of junctional strengths such as in [@benningerbasal] individually, without recourse to a general theory of gap junction function. Typically, a lot of the focus is on studying the heterogeneity of beta-cells in an islet. Indeed, Benninger et al. verify that different thresholds exist for calcium excitations among the beta-cells of a (Cx36 null) islet, and conclude therefore that beta-cells with high thresholds create oscillator death [@od] through gap junctions to decrease basal secretion. The other question to ask, however, is: can heterogeneous gap junctions within an islet shape the emergent properties of bursting? Once the heterogeneity of the gap junctions themselves is recognized as crucial, that leads, ipso facto, to an alternate view, one in which changes in junctional conductance are seen as solutions to an optimization problem. The essential ingredients of a theory of gap junction adaptation include keeping track of the propensities with which strong and weak junctions influence firing rates in beta-cells, and transition rules that determine how gap junctions will respond to local firing patterns. We have concentrated on learning rules that embody homeostatic principles, which are a central feature of the energy maintenance pathways of the body. However our general formalism is certainly applicable to other forms of adaptation rules that may be uncovered in future experiments. We have constructed a theory that offers an alternative explanation to the classical view that gap junctions primarily function to synchronize beta-cells in an islet so the entire islet behaves like a syncytium and a uniform period emerges. When gap junction adaptation is considered, partial synchronization can occur even in networks fully coupled with (strong) gap junctions. This learning framework predicts in a natural fashion that a full synchrony across the islet is very unlikely, that synchronization is a local phenomenon and happens across a few groups of cells. Thus the view that emerges instead is that the islet is sensitive to a glucose demand in secreting insulin and uses gap junctions as a tuning parameter in this adaptation. Paradoxically, an increase in secretion efficiency can come not by strengthening junctions, but down-regulating them instead. Thus, a lowered conductance need not necessarily be interpreted as “failing” gap junctions. On the contrary, they are judiciously adapting to the increased glucose load to cope with an increased demand for insulin secretion. At the moment there does not seem to be direct experimental evidence that a reduction of gap junctions occurs in human type 2 diabetes. Additionally, although it is very attractive from a theoretical viewpoint, it is not proven that gap junctions are altered in response to altered islet firing activity in diabetes. Our model is a complementary line of evidence, albeit theoretical, in these directions. Further, the model makes another related prediction, that gap junction expression and coupling strength are very likely to occur as heterogeneous across the islet, in both health as well as diabetes. If the naturally heterogeneous nature of gap junctions is acknowledged, this could be critical in designing appropriate clinical interventions, since connexins are potential targets for diabetes therapy. Indeed, we hope that our work will be helpful to researchers seeking to clarify the adaptive dynamics of gap junctions in diabetes. Figure Legends {#figure-legends .unnumbered} ============== ![[**Gap junctions between cells permit intercellular communication.**]{} Figure credit: Mariana Ruiz LadyofHats, http://en.wikipedia.org/wiki/File:Gap\_cell\_junction\_en.svg. []{data-label="gj"}](gj.eps){width="60.00000%"} ![[**Voltage gating of Cx36 gap junctions, adapted from [@teubner].**]{} Steady-state junctional currents from HeLa-Cx36 cell pairs indicate conductance, $G_j$, varies with transjunctional potential difference, $\Delta V_j$ . If two neighboring coupled cells fire nearly together, or do not simultaneously fire, trans-junctional conductance is high, but when one fires and the other does not conductance is low. This compensatory behavior inspires our [*homeostatic*]{} learning rule, see text. []{data-label="vg"}](vgating.eps "fig:"){width="50.00000%"}\ ![[**The bonds formalism of an islet.**]{} $\beta$-cells, A and B, are dominated by gap junctions $g_L$ and $g_R$ respectively. Each junction ($g_L$ and $g_R$) can be in either strong (with probability $f_+$) or weak state (with probability $f_-$). A weak (strong) junction is likely to fire with a probability $p_-$ ($p_+$). The central gap junction $g$ is altered in response to the average potential difference of cells A and B, $\Delta V_{AB}$, across it, according to a specified learning rule, such as the homeostatic rule of Fig. \[vg\] that is considered here. For example, if cell A (red) here is assumed to fire in response to a strong $g_L$ (this occurs with probability $p_+$) while cell B is silent (the probability with which it could have been active is $p_-$) in response to a weak $g_R$, then the bond, g, will be weakened since $\Delta V_{AB}=1$.[]{data-label="bonds"}](bonds_new.eps "fig:"){width="70.00000%"}\ ![[**The $f_+^*$ contour plot in the $p_-$ – $p_+$ plane.**]{} The physically relevant ($p_- < p_+$) region is the triangle ABC above the line $p_+ = p_-$. $f_+^*=0$ along BD, $p_+ +p_- =1$. The region near A where $f_+^*$ is close to $1$ represents healthy individuals while diabetics are assumed to lie along BD where $f_+^*=0.5$.[]{data-label="fig:pd2a"}](phasediag2d.eps){width="70.00000%"} ![[**Evolution of gap junctions with network activity.**]{} Beta-cells were initialized as firing (1) or not (0), and gap junctions as weak (0) or strong (1) with equal probability. 5000 beta-cell–gap junction pairs (Fig. \[bonds\]) were iterated according to the learning rules described in the text. The legend indicates the ($p_-,p_+$) values for a computation. The top panel shows the evolution of the fraction of strong gap junctions, $f_+^*$, in the network. The bottom panel shows the corresponding fraction of beta-cells that are active. Note that $f_+^*$ as well as firing rate in the simulation are both 0.5 along $p_-+p_+=1$ as expected from the theory, Fig. \[fig:pd2a\]. A transition from health with low firing and high proportion of strong gap junctions (black curves) to diabetes takes place with degrading the gap junctions to increase firing rates (red curves).[]{data-label="frate"}](firingrate.eps "fig:"){width="70.00000%"}\ Tables {#tables .unnumbered} ====== $g_L$ $g_R$ $P$ --------------- --------------- --------------------------------------------- Strong Strong $f_+^2 \{p_+^2 + (1-p_+)^2\}$ Strong (Weak) Weak (Strong) $2f_+f_-\{p_+p_- + (1-p_+)(1-p_-)\}$ Weak Weak $f^2_- \{p_-^2 +(1-p_-)^2\}$ \[combotable\] : **[The probability of a gap junction adapting to a strong, high conductance state is determined by the current state of the bonds $g_L$ and $g_R$ (Fig. \[bonds\]).]{}**

--- abstract: 'We develop a model for stochastic acceleration of electrons in solar flares. As in several previous models, the electrons are accelerated by turbulent fast magnetosonic waves (“fast waves”) via transit-time-damping (TTD) interactions. (In TTD interactions, fast waves act like moving magnetic mirrors that push the electrons parallel or anti-parallel to the magnetic field). We also include the effects of Coulomb collisions and the waves’ parallel electric fields. Unlike previous models, our model is two-dimensional in both momentum space and wavenumber space and takes into account the anisotropy of the wave power spectrum $F_k$ and electron distribution function $f_{\rm e}$. We use weak turbulence theory and quasilinear theory to obtain a set of equations that describes the coupled evolution of $F_k$ and $f_{\rm e}$. We solve these equations numerically and find that the electron distribution function develops a power-law-like non-thermal tail within a restricted range of energies $E\in (E_{\rm nt}, E_{\rm max})$. We obtain approximate analytic expressions for $E_{\rm nt}$ and $E_{\rm max}$, which describe how these minimum and maximum energies depend upon parameters such as the electron number density and the rate at which fast-wave energy is injected into the acceleration region at large scales. We contrast our results with previous studies that assume that $F_k$ and $f_{\rm e}$ are isotropic, and we compare one of our numerical calculations with the time-dependent hard-x-ray spectrum observed during the June 27, 1980 flare. In our numerical calculations, the electron energy spectra are softer (steeper) than in models with isotropic $F_k$ and $f_{\rm e}$ and closer to the values inferred from observations of solar flares.' author: - 'Peera Pongkitiwanichakul & Benjamin D. G. Chandran' bibliography: - 'articles.bib' nocite: '[@Miller96]' title: 'Stochastic Acceleration of Electrons by Fast Magnetosonic Waves in Solar Flares: the Effects of Anisotropy in Velocity and Wavenumber Space' --- Introduction {#sec:intro} ============ Solar flares involve a rapid increase in the number of photons emitted at energies exceeding $\sim 10$ keV. The photon spectra at these energies are typically non-thermal [@Lin81; @Lin03; @Grigis04; @Liu09; @Krucker10; @Krucker11; @Caspi10; @Ishikawa11], indicating the presence of non-thermal electrons [@Brown71; @Miller97]. One of the proposed mechanisms for generating these energetic electrons is stochastic particle acceleration [@Eichler79; @Miller96; @Miller97; @Petrosian06; @Benz08]. In stochastic-particle-acceleration (SPA) models, energy is initially released from the coronal magnetic field by magnetic reconnection [@Carmichael64; @Hirayama74; @Kopp76; @Tsuneta92; @Tsuneta96; @Priest2000]. A portion of the released energy is in the form of plasma outflows. Downward-directed outflows collide with closed magnetic loops lower in the corona, generating electromagnetic fluctuations. These fluctuations interact with electrons stochastically, accelerating some of the electrons to high energies. For the purposes of studying fluctuations with lengthscales much smaller than the flare acceleration region, the acceleration site can be modeled as a homogeneous, magnetized plasma with a uniform background magnetic field $\bm{B}_0$. Electromagnetic fluctuations with magnetic fluctuations $\delta b\ll B_0$ can then be approximated as waves in a homogeneous plasma. At wavelengths exceeding the ion inertial length $v_{\rm A}/\Omega_{\rm p}$, these waves can be approximated as magnetohydrodynamic (MHD) waves, i.e., Alfvén waves, fast magnetosonic waves (fast waves), slow magnetosonic waves, and entropy waves. (The quantity $v_{\rm A} = B_0/\sqrt{4\pi \rho}$ is the Alfvén speed, $\rho$ is the mass density, and $\Omega_{\rm p}$ is the proton cyclotron frequency.) Of these wave types, fast waves are thought to be the most effective at accelerating electrons [@Miller96; @Schlickeiser98; @chandran03; @Selkowitz04; @Yan04; @Yan08]. Fast waves are compressive and modify the magnitude of the magnetic field as they propagate. These waves act like moving magnetic mirrors, exerting forces on the electrons, which enables waves and electrons to exchange energy. Such interactions are called transit-time-damping (TTD) interactions, or simply TTD. In order for TTD to cause a secular increase in an electron’s energy, the electron and the wave it interacts with must satisfy the resonance condition, $$\omega_{kr}-k_\parallel v_\parallel = 0, \label{eq:Landau_resonance}$$ where $\omega_{kr}$ is the real part of the wave frequency, $k_\parallel$ is the component of the wavevector $\bm{k}$ parallel to $\bm{B}_0$, and $v_\parallel$ is the component of the electron’s velocity parallel to $\bm{B}_0$. The dispersion relation of fast waves in low-$\beta$ plasmas such as those found in solar flares (where $\beta$ is the ratio of plasma pressure to magnetic pressure) is $ \omega_{kr} = k v_{\rm A}$, and so the resonance condition reduces to $$v_\parallel = v_{\rm A}/\cos\theta, \label{eq:resonance2}$$ where $\theta$ is the angle between $\bm{k}$ and $\bm{B}_0$. In the non-relativistic limit, TTD increases only the parallel kinetic energy $m_{\rm e} v^2_\parallel/2$ of the electrons, where $m_{\rm e}$ is the electron mass. The same is true of Landau-damping (LD) interactions, which are mediated by the waves’ parallel electric fields. On the other hand, Coulomb collisions and possibly others processes (e.g., pitch-angle scattering by whistler waves) can convert parallel kinetic energy into perpendicular kinetic energy, which is an important process in SPA models, as we discuss further in Section \[sec:fe\]. Although fast waves are initially excited at large wavelengths by the interaction between reconnection outflows and magnetic loops, the energy of these fast waves cascades turbulent wave-wave interactions. Fast-wave turbulence is similar to acoustic turbulence, which transfers wave energy from small k to large k along radial lines in k-space [@Zakharov70; @Cho02; @Chandran05]. This turbulent cascade is important, because it is the largest-$k$ fast waves in such turbulent systems that lead to the strongest TTD interactions (Miller et al. 1996; see also Equation (\[eq:dzz\]) below). Turbulence also introduces disorder or randomness into the wave field, causing wave-particle interactions to become stochastic. In this work, we extend previous SPA models to allow for anisotropy in both the fast-wave power spectrum and the electron velocity distribution. To the best of our knowledge, this is the first time that both types of anisotropy have been accounted for within a single SPA model. In addition to TTD interactions, we account for LD interactions and Coulomb collisions. Our treatment of wave-particle and wave-wave interactions is based on quasilinear theory and weak turbulence theory. We describe our model in detail in Section \[sec:model\]. In Section \[sec:method\], we describe the numerical method that we use to solve the equations of our model. We compare numerical results from our model to results from [@Miller96] in Section \[sec:miller\]. In Section \[sec:spectrum\] we discuss the evolution of the wave power spectrum in our model. In Section \[sec:fe\] we derive analytic expressions describing the anisotropy and maximum energy of the non-thermal tail of the electron distribution function, which we compare with new numerical results. In Section \[sec:appflare\] we compare one of our numerical calculations with X-ray observations from the June 27, 1980 flare. We discuss and summarize our principal findings in Section \[sec:con\]. Model {#sec:model} ===== We model the electron acceleration region as a box located $\sim 20,000$ km above the chromosphere [@Aschwanden07], filled with a homogeneous proton-electron plasma pervaded by a uniform magnetic field $$\bm{B}_0 = B_0 \bm{\hat{z}}, \label{eq:B0z}$$ where $(x, y, z)$ are Cartesian coordinates. We define the fast-wave power spectrum in the acceleration region $F(\bm{k})$, abbreviated $F_k$, to be twice the energy per unit mass per unit volume in $\bm{k}$-space, where $\bm{k}$ is the wavevector. The total fast-wave fluctuation energy per unit mass is given by $$U_{\rm t} = \frac{1}{2} \, \int F_k\, d^3 k. \label{eq:Ut}$$ For simplicity, we assume reflectional symmetry, $F(- \bm{k}) = F(\bm{k})$. We take $F_k$ to evolve in time according to the equation $$\frac{\partial F_k}{\partial t}= S_k+\left(\frac{\partial F_k}{\partial t}\right)_{\rm turb}+\left(\frac{\partial F_k}{\partial t}\right)_{\rm res}- k^8 \sin^2 (\theta) \nu F_k . \label{eq:dFdt}$$ The term $$S_k = \left\{ \begin{array}{cc}\displaystyle \frac{4 \dot{E}_0 }{3 \pi^{3/2} k^3_0} \left(\frac{k}{k_0}\right)^2 \exp\left(-\frac{k^2}{k^2_0}\right) \sigma(\theta) & \mbox{ if $0 < t\leq t_{\rm inj}$} \vspace{0.2cm} \\ 0 & \mbox{ if $t > t_{\rm inj}$} \end{array} \right. \label{eq:S}$$ is a source term representing fast-wave injection from reconnection outflows, and $k_0$ is the wavenumber at which $S_k$ peaks. The term $\sigma(\theta)$ determines the $\theta$-dependence of $S_k$, where $\theta$ is the angle between $\bm{k}$ and $\bm{B}_0$. We normalize $\sigma(\theta)$ so that $0.5 \int_0^\pi \sigma(\theta) \sin \theta\, d\theta = 1$. The quantity $\dot{E}_0$ is then the total wave energy injection rate per unit mass. We set $$\sigma(\theta) = \left\{ \begin{array}{cl}\displaystyle \frac{3}{2} \sin^2\theta & \mbox{\,\,\,\, for model solutions A1, A2, A3, and A4} \vspace{0.2cm} \\ 1 & \mbox{\,\,\,\, for model solutions B, C and D} \end{array} \right. \label{eq:Sigma}$$ where the labels A1, A2, A3, A4, B, C, and D refer to numerical calculations that we will discuss further in Sections \[sec:miller\] through \[sec:appflare\]. The parameter values for these solutions are listed in Table \[table:set\]. We have considered different values of $\sigma$ because the best value for the modeling turbulence in flares is not known. We take the wave injection to last for a time $t_{\rm inj}$, where $t_{\rm inj}$ is an adjustable parameter. [ccccccccc]{} &&&&&&&\ \ Model solution & $B_0 $ & $n_{\rm e}$ & $v_{\rm A}$ & $T_{e\,\, \rm initial}$ & $\beta_{\rm e, initial}$ & $\dot{E}_0$ & $\tau_{\rm cas}$ & $t_{\rm inj}$\ & (G) & $(\mbox{cm}^{-3})$ & $(\mbox{cm} \mbox{ s}^{-1})$ & $(\mbox{K})$ & & $(v^2_{\rm A} \Omega_{\rm p})$ & $(\Omega_{\rm p}^{-1})$ & $(\Omega_{\rm p}^{-1})$\ \ \ A1 & 500 & $10^{10}$ & $1.1 \times 10^9$ & $3\times 10^6$ &$4.2\times 10^{-4}$ & $2\times10^{-10}$ & $9.8\times10^5 $ & $3\times10^6 $\ A2, A3, A4 & 500 & $10^{10}$ & $1.1 \times 10^9$ & $3\times 10^6$ & $4.2\times 10^{-4}$& $1.8\times 10^{-9}$ & $(3.25-3.3) \times10^5 $ & $3\times10^6$\ B & 500 & $10^{10}$ & $1.1 \times 10^9$ & $10^6$ & $1.4\times 10^{-4}$& $5\times10^{-10}$ & $7.8\times10^5 $ & $\infty$\ C & 250 & $3\times 10^9$ & $1.0 \times 10^9$ & $3\times 10^6$ & $5.0\times 10^{-4}$& $1.5\times10^{-10}$ & $1.3\times10^6 $ & $\infty$\ D & 150 & $10^9$ & $1.0 \times 10^9$ & $3\times 10^6$ &$4.6\times 10^{-4}$& $1.25\times10^{-11}$ & $4.4\times10^6 $ & $3\times 10^8$\ \ The term $(\partial F_k/\partial t)_{\rm turb}$ in Equation (\[eq:dFdt\]) is the so-called “collision integral" in the wave kinetic equation for weakly turbulent fast waves in low-$\beta$ plasmas derived by [@Chandran05; @chandran08], where $\beta = 8\pi p/B_0^2$ and $p$ is the plasma pressure. In particular, we set $(\partial F_k/\partial t)_{\rm turb}$ equal to the right-hand side of Equation (\[eq:turb\]) of [@Chandran05], with the Alfvén-wave power spectrum $A_k$ in that equation set equal to zero: $$\left(\frac{\partial F_k}{\partial t} \right)_{\rm\!\! turb} = \frac{ 9 \pi \sin^2\theta}{8 v_{\rm A} } \int \! d^3\!p\,\,d^3\!q\, \Big[\delta (k-p-q)kq F_p\big( F_q - F_k\big)$$ $$\hspace{0.1cm} + \hspace{0.1cm} \delta (k+p-q)k \big(k F_p F_q + pF_q F_k - qF_p F_k\big)\Big]\, \delta(\bm{k} - \bm{p} - \bm{q}) . \label{eq:turb}$$ We have neglected Alfvén waves for simplicity, but we expect that their inclusion would not change our conclusions about electron acceleration by fast waves. This is because superthermal, super-Alfvénic electrons interact with fast waves with $\theta > 45^\circ$, which interact only weakly with Alfvén waves [@Chandran05; @chandran08]. In weak fast-wave turbulence, waves with collinear wavevectors $\bm{k}$, $\bm{p}$, and $\bm{q}$ that satisfy the wavenumber resonance condition $\bm{k} = \bm{p} + \bm{q}$ and frequency matching condition $k = p + q$ interact to produce a weak form of wave steepening, which transfers wave energy from small $k$ to large $k$ along radial lines in $\bm{k}$-space. As $\sin \theta$ decreases, fast waves become less compressive, the fast-wave cascade weakens, and the energy cascade time increases. This anisotropy is represented mathematically by the coefficient of $\sin^2 \theta$ in Equation (\[eq:turb\]). When $\sigma(\theta) \propto \sin^2\theta$, the weakening of $S_k$ at small $\theta$ combined with the weakening of the cascade rate at small $\theta$ causes $F_k$ to become isotropic [@Chandran05]. We have chosen $\sigma(\theta) \propto \sin^2\theta$ in numerical calculations A1 through A4 in order to compare our model with a previous SPA model based on an isotropic $F_k$ [@Miller96]. The quantity $$\tau_{\rm cas} = \frac{U_{\rm t}}{\dot{E}_0} \label{eq:deftaucas}$$ is the approximate energy cascade timescale at the forcing wavenumber $k_0$, near which most of the fast-wave energy is concentrated. Because the energy cascade timescale is a decreasing function of $k$, $\tau_{\rm cas}$ is also approximately the time required for fast-wave energy to cascade from $k= k_0$ to $k= \Omega_{\rm p}/v_{\rm A}$. We list the values of $\tau_{\rm cas}$ in our numerical calculations in Table \[table:set\]. For these values, we evaluate $U_{\rm t}$ after the total fast-wave energy has reached an approximate steady state. The second-to-last term in Equation (\[eq:dFdt\]) is a damping term representing resonant interactions between electrons and waves with $k < k_{\rm max}$. We set $$\left(\frac{\partial F_k}{\partial t}\right)_{\rm res} = \left\{\begin{array}{ll} 2 \gamma^{(e)}_k F_k & \mbox{ if $k < k_{\rm max}$} \\ 0 & \mbox{ if $k\geq k_{\rm max}$} \end{array} \right., \label{eq:damp2}$$ where $$k_{\rm max} = \frac{\Omega_{\rm p}}{3 v_{\rm A}} \label{eq:defkappa}$$ is roughly the maximum wavenumber at which the waves can be approximated as fast waves. At $k \gtrsim k_{\rm max}$, the fast-wave branch of the dispersion relation transitions to the whistler branch. We have set $(\partial F_k/\partial t)_{\rm res} = 0$ at $k> k_{\rm max}$ in order to exclude the contribution of whistler waves to electron heating and acceleration. Although potentially important, the role of whistler waves is beyond the scope of this paper. The quantity $\gamma^{(e)}_k$ is the imaginary part of the wave frequency, which we determine using quasilinear theory, as described below. The last term on the right-hand side of Equation (\[eq:dFdt\]) is a hyperviscous dissipation term, which we include in order to model all dissipation mechanisms operating at $k > k_{\rm max}$. Although we do not account for the way that electrons are affected by waves at $k> k_{\rm max}$ in our model, the power that is dissipated by hyperviscosity corresponds to power that would, in a real plasma, be available for electron heating and/or acceleration via whistler-electron interactions. We take the electron distribution function $f_{\rm e}$ to evolve according to the equation $$\frac{\partial f_{\rm e}}{\partial t} = \left(\frac{\partial f_{\rm e}}{\partial t}\right)_{\rm res}+\left(\frac{\partial f_{\rm e}}{\partial t}\right)_{\rm coll}. \label{eq:electron}$$ The first term in Equation (\[eq:electron\]) is the rate of change of $f_{\rm e}$ resulting from resonant interactions with fast waves, and is the counterpart to the term $(\partial F_k/\partial t)_{\rm res}$ in Equation (\[eq:dFdt\]). The last term in Equation (\[eq:electron\]) is the rate of change of $f_{\rm e}$ due to Coulomb collisions (see Equation (\[eq:coll\]) below). We model resonant wave-particle interactions using quasilinear theory. In this theory, the Vlasov equation is averaged over many wave periods and wavelengths. It is assumed that the fluctuations in the electric and magnetic fields are from small-amplitude waves, and that the imaginary parts of the wave frequencies are much smaller than the real parts. The averaged particle distribution function of species $s$, denoted $f_s$, then evolves according to the equation [@Kennel66; @Stix92] $$\left(\frac{\partial f_s}{\partial t}\right)_{\rm res} =\lim_{L \to \infty} \sum_{n=-\infty}^\infty \pi q_s^2 \left(\frac{2\pi}{L}\right)^3 \int \frac{d^3 k}{p_\perp} G p_\perp \nonumber \delta (\omega_{kr}-k_\parallel v_\parallel-n\Omega_s)| \psi_{n,k}^{(s)} | ^2 G f_s, \label{eq:QLT0}$$ where $$\Omega_{s} = \frac{q_s B_0}{m_s \gamma c} \label{eq:defOmegas}$$ is the signed, relativistic cyclotron frequency of species $s$, $q_s$ and $m_s$ are the charge and mass of a particle of species $s$, $\gamma= (1 - v^2/c^2)^{-1/2}$ is the Lorentz factor, $c$ is the speed of light, $p_\parallel$ ($p_\perp$) is the component of the particle momentum ${\bm p}$ parallel (perpendicular) to ${\bm B}_0$, $k_\parallel$ ($k_\perp)$ is the component of ${\bm k}$ parallel (perpendicular) to ${\bm B}_0$, $$G=\left(1-\frac{k_\parallel v_\parallel}{\omega_{kr}}\right)\frac{\partial}{\partial p_\perp} +\left ( \frac{k_\parallel v_\perp}{\omega_{kr}}\right)\frac{\partial}{\partial p_\parallel}, \label{:G}$$ $$\psi_{n,k}^{(s)}=\frac{1}{\sqrt{2}}[E^+_k e^{i \phi}J_{n+1}(z)+E^-_k e^{-i \phi} J_{n-1}(z)]+\frac{p_\parallel}{p_\perp}E_{kz} J_n(z), \label{:psi}$$ $z=k_\perp v_\perp/\Omega_s$, $J_n$ is the Bessel function of order $n$, $E^{\pm}_k=(E_{kx} \mp i E_{ky})/\sqrt{2} $, ${\bm E}_k$ (${\bm B}_k$) is the Fourier transform of the electric (magnetic) field, and $\phi$ is the azimuthal angle in $\bm{k}$-space. The quantity $L$ is the length scale of the window function that multiplies functions of position before we take a Fourier transform. Our Fourier-transform convention, described further in Appendix \[appen:reldamping\], differs from that of [@Stix92] by factors of $2\pi$, which accounts for why the right-hand side of Equation (\[eq:QLT0\]) is a factor of $(2\pi)^3$ larger than the right-hand side of Equation (17-41) of [@Stix92]. The species subscript $s$ is “p” for protons or “e” for electrons. The delta function in Equation (\[eq:QLT0\]) implies that strong interactions occur only when waves and particles satisfy the resonance condition $$\omega_{kr}-k_{\parallel} v_\parallel=n\Omega_s. \label{eq:res}$$ TTD and Landau damping arise when the resonance condition with $n=0$ is satisfied. To evaluate $|\psi_{n,k}^{(s)}|^2$ and $\omega_{kr}$, we treat the fast waves as if they were propagating in a plasma with the (non-relativistic) bi-Maxwellian distribution function $$f_{\rm BM} = \frac{n_{\rm e}}{\pi^{3/2} m_{\rm e}^3 v_{\perp \rm T}^2 v_{\parallel \rm T}} \exp\left(-\frac{v_\perp^2}{v_{\perp \rm T}^2} - \frac{v_\parallel^2}{v_{\parallel \rm T}^2}\right), \label{eq:fM}$$ where $n_{\rm e}$ is the electron density, $$v_{\perp \rm T} = \sqrt{\frac{2k_{\rm B} T_{\perp \rm e}}{m_{\rm e}}} \qquad v_{\parallel \rm T} = \sqrt{\frac{2k_{\rm B} T_{\parallel \rm e}}{m_{\rm e}}} \label{eq:defvperpT}$$ are the perpendicular and parallel electron thermal speeds, and $T_{\perp \rm e}$ and $T_{\parallel \rm e}$ are the perpendicular and parallel electron temperatures. The factor of $m_{\rm e}^3$ is included in the denominator of Equation (\[eq:fM\]) because we have defined $f_{\rm e}$ to be the number of particles per unit volume in physical space per unit volume in momentum space (i.e., $\int d^3 p f_{\rm e} = n_{\rm e}$). After setting $f_{\rm e} = f_{\rm BM}$, we expand the hot-plasma dispersion relation in the limit that $|\omega_{kr}| \ll \Omega_{\rm p}$, $k_\perp v_\perp \ll |\Omega_{\rm e}|$ , and $\omega_{kr}/k_\parallel v_{\parallel \rm T} \sim {\cal O}(1)$. The details of this procedure are given in Section 4 of Chapter 11 of [@Stix92]. Fast waves in this limit satisfy $$\omega_{kr} = k v_{\rm A}. \label{eq:disp}$$ For the case in which $\bm{k}$ is in the $x-z$ plane, $$\frac{i E_{kz}}{E_{ky}} = -\frac{k_\perp k_\parallel v^2_{\perp \rm T}}{2 \omega_{kr} \Omega_{\rm e}}, \label{eq:polarization}$$ $|E_{kx}| \ll |E_{ky}|$, and $$\psi_{0,k}^{(e)} = -\, \frac{ik_\perp v_\perp}{2\Omega_{\rm e}} \left( 1 - \frac{v_{\perp \rm T}^2}{v_\perp^2} \right) E_{ky}. \label{eq:psi0}$$ We note that Equation (\[eq:polarization\]) differs from Equation (31) of Chapter 17 of [@Stix92], because the latter equation only applies when $v_{\perp \rm T} = v_{\parallel \rm T}$. Restricting Equation (\[eq:QLT0\]) to “Landau-resonant” interactions (i.e., $n=0$), we rewrite Equation (\[eq:QLT0\]) in the form $$\left(\frac{\partial f_{\rm e}}{\partial t}\right)_{\rm res} = \frac{\partial}{\partial p_\parallel} \left(D_{\rm res} \frac{\partial f_{\rm e}}{\partial p_\parallel}\right), \label{eq:ttd_diffusion}$$ where $$D_{\rm res} = \frac{\pi^2 m_{\rm e}^2 \gamma^2 }{4} \frac{(v_\perp^2 - v_{\perp \rm T}^2)^2}{|v_\parallel^3|} \left(1 - \frac{v_{\rm A}^2}{v_\parallel^2}\right) \int_{-1}^1 d(\cos\theta)\; \delta\Big(\!\cos\theta - \frac{v_{\rm A}}{v_\parallel}\Big) \int_0^{\rm k_{\rm max}} dk \,k F_k . \label{eq:dzz}$$ Ordinarily, the upper limit on the $k$ integration would be $+ \infty$, as in Equation (\[eq:QLT0\]). However, in Equation (\[eq:damp2\]) we have restricted the $k$ integration to $k < k_{\rm max} = \Omega_{\rm p}/3v_{\rm A}$, in order to exclude wave-particle interactions involving whistler waves. We therefore must do the same in Equation (\[eq:dzz\]) in order to maintain energy conservation. To express $D_{\rm res}$ in Equation (\[eq:dzz\]) in terms of $F_k$ instead of $|E_{ky}|^2$, we have made use of Equations (\[eq:FW\]) and (\[eq:WE\]) below and our assumption of spherical symmetry about the $z$ axis, which allows us to evaluate the $\phi$ integral in Equation (\[eq:QLT0\]) by taking $\bm{k}$ to be in the $x-z$ plane and then replacing $\int_0^{2\pi} d\phi(\dots)$ with $2\pi \times (\dots)$. Equation (\[eq:dzz\]) differs from the momentum diffusion coefficient $D_p$ given in Equation (2.2a) of [@Miller96] in two ways. First, $D_{\rm res}$ in Equation (\[eq:dzz\]) is the coefficient for diffusion in $p_\parallel$, whereas $D_p$ in Equation (2.2a) of [@Miller96] is the coefficient for diffusion in $p$ when rapid pitch-angle scattering isotropizes $f_{\rm e}$. Second, Equation (\[eq:dzz\]) accounts for LD interactions mediated by the parallel component of the electric field, $E_{kz}$. The parallel electric field is responsible for the terms proportional to $v_{\perp \rm T}^2$ in Equations (\[eq:psi0\]) and (\[eq:dzz\]). The minus signs preceding these terms reflect the fact that the electric force on electrons is $180^\circ$ out of phase with the $\mu\nabla B$ force on the electrons [@Stix92]. For fast waves in Maxwellian plasmas, the effects of the parallel electric field are quite important. As noted by [@Stix92], the parallel electric field in a Maxwellian plasma reduces the fast-wave damping rate by a factor of 2 relative to the case in which $E_{kz}$ is neglected (i.e., the case in which the fast waves are damped only by TTD). On the other hand, for electrons with $v_\perp \gg v_{\perp \rm T}$, TTD interactions are much stronger than LD interactions, and the parallel electric field leads to only a small reduction in $D_{\rm res}$. Returning to Equation (\[eq:damp2\]), when the imaginary part of the wave frequency $\gamma_k$ is much less than the real part, $\gamma_k$ can be determined using quasilinear theory [@kennel67]. In Appendix \[appen:reldamping\], we show that the general form of $\gamma_k$, allowing for relativistic particles and cyclotron $(n\neq0)$ interactions, is given by $\gamma_{\bm{k}} = \sum_s \gamma^{(s)}$, where $$\gamma_{\bm{k}}^{\rm (s)}= \sum_{n=-\infty}^\infty \frac{\pi^2 q^2_s}{2} \int^{\infty}_0 d p_\perp\int^{\infty}_{-\infty} d p_\parallel \frac{p^2_\perp c^2}{\sqrt{p^2c^2+m^2_sc^4}} \frac{|\psi_{n,\bm{k}}^{(s)} | ^2}{W_{\bm{k}}}\delta (\omega_{kr}-k_{\parallel} v_\parallel-n\Omega_s) G f_s, \label{eq:reldamp}$$ $$W_{\bm{k}}=\frac{1}{16 \pi}\left [ \textbf{B}_k^* \cdot \textbf{B}_k +\textbf{E}_k^*\cdot\frac{\partial (\omega \underline{\underline{\epsilon}}_{\,h})}{\partial \omega}\cdot \textbf{E}_k \right ] \label{eq:defW}$$ is one half the wave energy per unit $k$-space volume divided by $(2\pi)^3$ (see Equation (\[eq:epsw\])), and $\underline{\underline{\epsilon}}_{\,h}$ is the hermitian part of the dielectric tensor $\underline{\underline{\epsilon}}$. Since $F_k$ is twice the fast-wave energy per unit mass per unit volume in $k$ space (see Equation (\[eq:Ut\])), $$W_{\bm{k}} = \left(\frac{L}{2\pi}\right)^3 \frac{\rho F_k}{4}. \label{eq:FW}$$ To evaluate the right-hand side of Equation (\[eq:defW\]), we again follow the development in Chapter 11 of [@Stix92] and expand $\underline{ \underline{ \epsilon}}$ in the that $|\omega_{kr}| \ll \Omega_{\rm p}$, $k_\perp v_\perp \ll |\Omega_{\rm e}|$ , and $\omega_{kr}/k_\parallel v_{\parallel \rm T} \sim {\cal O}(1)$. For fast waves in this limit with $\bm{k}$ in the $x-z$ plane, $$W_{\bm{k}} = \frac{c^2}{8 \pi v^2_{\rm A}} |E_{ky}|^2. \label{eq:WE}$$ Given our assumption of cylindrical symmetry about the $z$ axis, we can evaluate $\gamma_k$ at any $\bm{k}$ by first rotating $\bm{k}$ about the $z$ axis until it lies in the $x-z$ plane, and then making use of Equations (\[eq:psi0\]) and (\[eq:WE\]). In the non-relativistic limit, Equation (\[eq:reldamp\]) reduces to the value of $\gamma_k^{\rm (s)}$ derived by [@kennel67]. If we set $n=0$ and consider only interactions involving electrons, then Equation (\[eq:reldamp\]) gives the value of $\gamma^{(e)}_k$ in Equation (\[eq:damp2\]). As a check on our results, we note that for $n=0$ interactions with non-relativistic, Maxwellian electrons, Equation (\[eq:reldamp\]) yields $$\gamma^{(e)}_k= -\,\frac{\pi^{1/2}}{4}\;\frac{k^2_{\perp}v_{\rm A}}{|k_{\parallel}|} \sqrt{\frac{m_{\rm e} \beta_{\rm e}}{m_{\rm p}}} \;\exp\left(- \frac{m_{\rm e}}{\beta_{\rm e} m_{\rm p}\cos^2\theta}\right), \label{eq:ken67}$$ where $$\beta_{\rm e} = \frac{8\pi n_{\rm e} k_B T_{\rm e}}{B^2_0}. \label{eq:betae}$$ This expression is equivalent to the fast-wave damping rate for Maxwellian plasmas derived by [@ginz60] (see also [@Petrosian06]). To determine the value of the collision term $(\partial f_{\rm e}/\partial t)_{\rm col}$ in Equation (\[eq:electron\]), we make the following approximations. First, we neglect electron-proton collisions. We also work in the non-relativistic limit, setting $$\bm{v} = \frac{\bm{p}}{m_{\rm e}}, \label{eq:vp}$$ which is a reasonable simplification because we focus on electron energies $\lesssim 100 \mbox{ keV}$. The Coulomb collision operator for electron-electron collisions can be written in the form [@rosenbluth57] $$\left (\frac{\partial f_{\rm e}}{\partial t} \right )_{\rm coll}=- C \nabla_v \cdot \bm{J}, \label{eq:coll}$$ where $$C = \frac{{\bf2}\pi\Lambda e^4}{m^2_{\rm e}}, \label{eq:defC}$$ $$\Lambda = 24 - \ln\left[\left(\frac{n_{\rm e}}{1 \mbox{ cm}^{-3}}\right)^{1/2} \left( \frac{ k_{\rm B} T_{\rm e}}{1 \mbox{ eV}}\right)^{-1}\right] \label{eq:Coullog}$$ is the Coulomb logarithm, $$\bm{J} = \frac{2}{m_{\rm e}}f_{\rm e} \nabla_v K_1 -\frac{1}{m_{\rm e}}\nabla_v \nabla_v K_2 \cdot \nabla_v f_{\rm e}, \label{eq:jcoll}$$ $$K_1= \int \frac{f_{\rm e}}{U} d^3 p, \label{eq:kcoll1}$$ $$K_2= \int U f_{\rm e} d^3 p, \label{eq:kcoll2}$$ and $U=|\vec{v}-\vec{v}'|$. To evaluate Equation (\[eq:coll\]) numerically would require a number of operations per time step $\propto N^2_v$, where $N_v$ is the number of velocity grid points in the numerical calculation. In order to reduce the number of operations required, we replace $f_{\rm e}$ in Equations (\[eq:kcoll1\]) and (\[eq:kcoll2\]) with a Maxwellian distribution $f_{\rm M}$ of temperature $T_{\rm e}$. In numerical calculations A1 and A2, we keep $T_{\rm e}$ fixed at the initial electron temperature. (As we will discuss further in Section 4, this is to compare our model to the model of of [@Miller96].) In numerical calculations A3, A4, B, C and D, we pick $T_{\rm e}$ so that $f_{\rm M}$ and $f_{\rm e}$ have the same total energy. This allows $T_{\rm e}$ to increase during a flare, as seen in hard X-ray observations (see, e.g., Figure 3 of Lin et al. 1981.) In Appendix \[sec:b\], we estimate the error introduced by our approximations of $K_1$ and $K_2$ in numerical calculations A4, B, C, and D. We find that the maximum error is $\lesssim 6\%$ for $K_1$ and $\lesssim 18\%$ for $K_2$. Using these approximated values of $K_1$ and $K_2$, we can rewrite Equation (\[eq:coll\]) as $$\left (\frac{\partial f_{\rm e}}{\partial t} \right )_{\rm Coll}=4\pi\Lambda e^4 n_{\rm e}m_{\rm e} \nabla_p \cdot \left [\frac{\nu_s}{2} \frac{\bm{p}}{p^3} f_{\rm e}+\frac{1}{2}\frac{\nu_\parallel}{p^3} \bm{p}\bm{p}\cdot \nabla_p f_{\rm e} +\frac{1}{4}\frac{\nu_\perp}{p^3} (p^2\mathbb{I} -\bm{p}\bm{p})\cdot\nabla_p f_{\rm e} \right], \label{eq:coll2}$$ where $\mathbb{I}$ is the unit matrix, $$\nu_s = 2 \chi(x_\beta), \label{eq:gs}$$ $$\nu_\parallel = \frac{\chi(x_\beta)}{x_\beta}, \label{eq:gpar}$$ $$\nu_\perp = 2\left[ \left ( 1-\frac{1}{2 x_\beta}\right )\chi(x_\beta)+\chi'(x_\beta) \right ], \label{eq:gperp}$$ $$\chi(x) = \frac{2}{\sqrt{\pi}} \int^{x}_0 t^{1/2} e^{-t} dt , \label{eq:psix}$$ and $$x_\beta = \frac{p^2}{2 m_{\rm e} k_B T_{\rm e}}. \label{eq:xbeta}$$ Numerical Method {#sec:method} ================ In order to solve for the time evolution of $F_k$ and $f_{\rm e}$, we integrate Equations (\[eq:dFdt\]) and (\[eq:electron\]) numerically. We use an explicit method to integrate Equation (\[eq:dFdt\]) — the numerical algorithm employed by [@Chandran05] with a trivial extension to account for the damping term $(\partial F_k/\partial t)_{\rm res} = -2\gamma_{\rm k} F_{\rm k}$. If we were to use an explicit method to integrate Equation (\[eq:electron\]), we would need to make the time step $\Delta t$ exceedingly small in order to maintain numerical stability. We therefore integrate Equation (\[eq:electron\]) using the implicit biconjugate gradient-stabilized method [@Vorst03]. We evaluate $v_{\perp \rm T}$ in Equation (\[eq:dzz\]) by setting $n_{\rm e} v_{\perp \rm T}^2 = \int d^3 p f_{\rm e} v_\perp^2$. To simplify the numerical algorithm, we treat the following quantities as constant within a single time step: the damping rate $\gamma_k$ used to calculate $(\partial F_k/\partial t)_{\rm res}$ in Equation (\[eq:dFdt\]), the momentum diffusion coefficient $D_{\rm res}$ used to calculate $(\partial f_{\rm e}/\partial t)_{\rm res}$ in Equation (\[eq:ttd\_diffusion\]), and the electron temperature $T_{\rm e}$ in Equation (\[eq:xbeta\]). After each time step, we update the values of $T_{\rm e}$ in the collision operator for numerical calculations A3, A4, B, C, and D, but we keep $T_{\rm e}$ fixed in model solutions A1 and A2, as discussed further in Section \[sec:miller\]. After each time step, we also update the values of $\gamma_k$ and $D_{\rm res}$. To calculate $\gamma_k^{(e)}$ numerically, we use the procedure described in Appendix \[appen:reldamping\] following Equation (\[eq:defI\]). With this approach, our numerical treatment of wave-particle interactions conserves energy to machine accuracy. In wavenumber space, we use a logarithmic wavenumber grid in both $k_\perp$ and $k_\parallel$ (the components of $\bm{k}$ perpendicular and parallel to $\bm{B}_0$), with $k_{\perp i} = (0.2 k_0 ) 2^{i/4}$ for $i = 0, 1, 2, \dots, N-1$, $k_{\parallel 0} = 0$, $k_{\parallel j} = (0.2 k_0) 2^{(j-1)/4}$ for $j= 1, 2, 3, \dots, N-1$, and $N = 62$. In all of our calculations, we choose  the hyperviscosity coefficient $\nu$ so that dissipation is negligible at $k\le k_{\rm max} = \Omega_{\rm p}/3v_{\rm A}$ but strong enough at $k> k_{\rm max}$ to truncate the cascade. In momentum space, we use a pseudo-logarithmic grid in $p_\perp$ and $p_\parallel$. In $p_\perp$, cell centers are given by $$p^g_{\perp i}=\frac{p_0 [e^{\alpha (2i-1)}-1]}{e^{\alpha}-1} \label{:rhoc}$$ and cell boundaries are given by $$p^l_{\perp i}=\frac{p_0 [e^{\alpha (2i-2)}-1]}{e^{\alpha}-1}, \label{:rhob}$$ where $p_0 = 2.02\times10^{-2} m_{\rm e} v_{\rm A}$ and $\alpha = 1.83\times 10^{-2}$ for $i=1,2,...,N_p$. We choose this grid because it extends to $p^l_\perp = 0$ and has the property that $\Delta p_{\perp i+1} = e^{2\alpha} \Delta p_{\perp i}$, where $\Delta p_{\perp i} = p^l_{\perp i+1}-p^l_{\perp i}$ is the “bin width" in $p_\perp$. The $p_\parallel$ grid is identical to the $p_\perp$ grid. Before discretizing Equation (\[eq:electron\]), we write this equation in the form $$\frac{\partial f}{\partial t} = -\nabla \cdot \bm{J}_{\rm tot}, \label{eq:genfe}$$ where $\bm{J}_{\rm tot}$ is the total electron flux in momentum space. We then obtain a set of discrete equations by integrating Equation (\[eq:genfe\]) over each grid cell in momentum space and applying Gauss’s theorem, so that $\partial f/\partial t$ within each cell is given by the electron fluxes through the faces of the cell. Except at the edges of the simulated portion of momentum space, the flux through each cell face appears twice in the calculation: as an increase in the number of electrons in one cell and an equal and opposite decrease in the number of electrons in an adjacent cell. Summing over all cells, we conserve the total particle number, except for a tiny flow of particles out of the numerical domain at large momenta. Comparison with Miller et al (1996) {#sec:miller} =================================== In this section, we compare our model with one of the numerical solutions from [@Miller96], hereafter “MLM96.” In particular, we compare our results with MLM96’s “Case 4,” which is based on Kraichnan’s (1965) phenomenology of MHD turbulence. Since MLM96 only considered TTD, we set $E_{kz}$ to zero in numerical calculations A1, A2, and A3 in order to compare with their results. This has the effect of eliminating the $v_{\perp \rm T}^2$ term in Equation (\[eq:dzz\]). (We retain the parallel electric field and the $v_{\perp \rm T}^2$ term in Equation (\[eq:dzz\]) in model solutions A4, B, C, and D) The acceleration region in MLM96’s model is homogeneous and has dimension $L_{\rm f} =10^9 \mbox{ cm}$, volume $10^{27} \mbox{ cm}^3$, electron density $n_{\rm e} =10^{10} \mbox{ cm}^{-3}$, and a uniform background magnetic field of strength 500 G. The electrons are initially Maxwellian with a temperature of $3 \times 10^6$ K. As time progresses, the electrons in MLM96’s model undergo Coulomb collisions with a background electron population that remains at $T_{\rm e} = 3\times 10^6$ K, even though the simulated electrons are heated and accelerated. For these parameters, $\beta_{\rm e}$ (defined in Equation (\[eq:betae\])) is $ 4.16\times 10^{-4}$, the electron thermal speed $v_{T_{\rm e}} = \sqrt{k_B T_{\rm e}/m_{\rm e}}$ is initially $0.62\,v_{\rm A}$, and electrons with energy equal to $20$ keV move at speed 7.8 $v_{\rm A}$. Fast waves are not present at the beginning of MLM96’s numerical calculations, but are instead injected at the wave number $k_0 = 1.4\times10^{-3}\Omega_{\rm p} /v_{\rm A}$ from $t=0$ to $ t= t_{\rm inj} = 3\times 10^6 \Omega^{-1}_p$ at the rate $\dot{E}_0 = 2 \times10^{-10} v^2_{\rm A} \Omega_{\rm p}$. As a first comparison between our 2D model and MLM96’s isotropic model, we carry out numerical calculation A1 in Table \[table:set\], which has the same parameters as MLM96’s Case 4 and the same treatment of collisions (fixed $T_{\rm e}$ in Equation (\[eq:xbeta\])). Our choice of $\sigma(\theta)$ in Equation (\[eq:S\]) for this calculation results in a steady-state inertial-range fast-wave power spectrum that is independent of $\theta$, as discussed following Equation (\[eq:turb\]). We find that in numerical calculation A1 the maximum number of electrons with energies $>20$ keV, denoted $N_{20, \rm max}$, is $2.5\times10^4$, and the maximum rate at which electrons are accelerated to energies $>20$ keV, denoted $R_{20, \rm max}$, is $9.2\times10^4$ s$^{-1}$. These values are, respectively, $\sim 1600$ and $\sim 1500$ times smaller than the corresponding values in MLM96’s case 4. Only 20% of the total energy injected into waves in our numerical calculation is transferred to electrons, while the remainder is dissipated by hyperviscosity at large $k$. As mentioned in Section 2, the energy dissipated by hyperviscosity in our model serves as a proxy for the amount of energy that cascades to whistler-scale wavelengths $\gtrsim \Omega_{\rm p}/v_{\rm A}$. In a real plasma, this energy would also presumably be transferred to electrons, but electron heating and acceleration by whistlers is beyond the scope of our model. In MLM96’s case 4, almost all of the wave energy is transferred to electrons. One of the reasons that electron acceleration is less efficient in our model is that in weak turbulence theory the fast-wave energy cascade is more rapid than in the simple phenomenological model employed by MLM96. For example, if $k^2 F_k = c_1 k^{-3/2}$, where $c_1$ is a constant, Equation (\[eq:dFdt\]) leads to a cascade rate that is $\simeq 9$ times larger than the cascade rate assumed by MLM96 (see Appendix \[appen:wave\]) — hence, $c_1$ would be smaller in our model in order to achieve the same value of $\dot{E}_0$. A second reason that electron acceleration is less efficient in our model is the anisotropy of $f_{\rm e}$. Transit-time damping increases only the parallel kinetic energy $m_{\rm e}v_\parallel^2/2$ of the superthermal electrons, and thus leads to anisotropic electron distributions in which $v_\perp^2 < v_\parallel^2$ for most of the electrons. For non-thermal electrons with $|v_\parallel| \gg v_{\rm A}$, $D_{\rm ttd} \propto \gamma^2 v_\perp^4/|v_\parallel|^3$ (see Equation (\[eq:d\_ttd\]) below), and thus transit-time damping is less effective in our model than in models in which $f_{\rm e}$ is isotropic. In order to isolate the effects of $f_{\rm e}$ anisotropy on electron acceleration, we carry out a second numerical calculation (A2 in Table \[table:set\]) in which $\dot{E}_0$ is increased by a factor of 9 so that the wave amplitudes in our model are roughly the same as in MLM96’s Case 4. We note that increasing $\dot{E}_0$ reduces the wave cascade time and causes TTD to start earlier in our larger-$\dot{E}_0$ calculation than in MLM96’s Case 4. With this larger value of $\dot{E}_0$, the value of $N_{20, \rm max}$ becomes $2.1\times 10^6$ and the value of $R_{20, \rm max}$ is $7.2\times 10^6$ s$^{-1}$. These values are both $\sim 20$ times smaller than the corresponding values in MLM96’s Case 4. We conclude that $f_{\rm e}$ anisotropy reduces the efficiency of electron acceleration by fast magnetosonic waves by a factor of $\sim 20$ for fixed wave amplitudes. We note, however, that in model solution A2, only 10% of the total energy injected into waves is transferred to electrons. The remaining energy cascades to wavenumbers $\gtrsim \Omega_{\rm p}/v_{\rm A}$, at which it would, in a real plasma, contribute to further electron heating and acceleration, but via mechanisms not included in our model. In model solutions A1 and A2, $T_{\rm e}$ is fixed in our approximate collision operator (Equation (\[eq:xbeta\])). However, as mentioned previously, $T_{\rm e}$ can increase during a flare. To investigate the effect of this increase, we carry out numerical calculation A3, which is identical to numerical calculation A2 except that $T_{\rm e}$ is now allowed to evolve so that $(3/2) n_{\rm e} k_B T_{\rm e}$ is the total energy density of the instantaneous electron distribution. In model solution A3, the value of $N_{20 \rm max}$ is $3.1\times 10^7$ and $R_{20 \rm max}$ is $2.0\times 10^8$ s$^{-1}$. These values are roughly 15 and 30 time larger than in solution A2. The reason that increasing $T_{\rm e}$ in the collision operator enhances the electron acceleration rate is that the simulated electrons lose less energy through collisions because they are colliding with hotter target electrons. The time evolution of $N_{20 \rm max}$ and $R_{20 \rm max}$ in solution A3 are shown in Figure \[fig:m20\]. About 35% of the total energy injected into waves is transferred to electrons. ![The number $N$ of electrons with energies $E$ exceeding 20 keV (solid line) and the acceleration rate $dN/dt$ in model solution A3. At $t= t_{\rm inj}$, wave injection ceases. Subsequently, the waves decay, and $N$ decreases because of Coulomb collisions. \[fig:m20\]](m20.eps){width="12.cm"} In Figure \[fig:fe\] we plot the electron energy spectrum $$N(E) = \frac{2\pi}{c^2} \,p \,\sqrt{p^2 c^2+m^2_{\rm e} c^4}\, \int^1_{-1} d\mu\, f_{\rm e} (p, \mu), \label{eq:NE}$$ in numerical calculation A3, where $\mu = p_\parallel/p$ and $E =\sqrt{p^2 c^2+m^2_{\rm e} c^4}-m_{\rm e} c^2$. As this figure shows, a power-law-like structure develops over a narrow range of energies. At the end of the wave-injection period (i.e., at $t= t_{\rm inj} = 3\times 10^6 \mbox{ eV}$), this approximate power law extends from  $\sim 7 \mbox{ keV}$ to $\sim 25 \mbox{ keV}$, and $N(E)$ is roughly proportional to $E^{-3.3}$ in this range, shown in Figure \[fig:fe\]. A similar power-law-like feature appears in Case 4 of MLM96 (their Figure 11). However, their approximate power law is much flatter than ours ($\sim E^{-\eta}$ with $\eta$ as small as 1.2) and extends to larger energies ($> 100 \mbox{ keV})$. ![The electron energy spectrum $N(E)$ at three different times in model solution A3. \[fig:fe\]](fe_kev.eps){width="12.cm"} For reference, we carry out a fourth numerical calculation, A4, that is identical to A3, except that $E_{kz}$ is included. In this calculation, about 16% of the total energy injected into waves is transferred to electrons, which is about half as much as in solution A3. The values of $N_{20 \rm max}$ and $R_{20 \rm max}$ are $3.7\times 10^{6}$ and $2.0\times 10^7$ s$^{-1}$, respectively. These values are $\sim$ 8 times and 10 times smaller than those in solution A3. These reductions occur for the same reasons that the inclusion of $E_{kz}$ reduces the linear damping rate of fast waves in Maxwellian plasmas by a factor of 2 relative to the case in which $E_{kz}$ is neglected [@Stix92]: the parallel electric force on electrons is $180^\circ$ degrees out of phase with the magnetic-mirror force, as discussed in Section \[sec:model\]. Evolution of the Wave Power Spectrum $F_k$ {#sec:spectrum} ========================================== In this section, we describe the characteristic way that $F_k$ evolves in our numerical calculations, using solutions A1 and A2 as examples. In Figure \[fig:w0\], we plot the energy-weighted average wavenumber $$\langle k \rangle \equiv \frac{\int d^3k \; k F_k}{\int d^3 k\; F_k} \label{eq:avk}$$ for solution A2 (dashed line) and for a modified version of solution A2 in which transit-time damping is turned off (dash-dot-dash line). In this modified version of numerical calculation A2, the value of $\langle k \rangle$ is somewhat larger than in the original solution A2, consistent with the fact that TTD preferentially removes fast-wave energy at large $k$. ![The total wave energy $U_t$ in model solution A2 (solid line) and solution A1 (dotted line). The dashed line is $\langle k \rangle$ in solution A2, and the dash-dot-dash line is $\langle k \rangle$ in a modified version of solution A2 in which transit-time damping is turned off. \[fig:w0\]](w0.eps){width="12.cm"} In Figure \[fig:w0\] we also plot the total fast-wave fluctuation energy $U_t$ in numerical calculations A1 and A2. In Figure \[fig:fk\] we plot the angle-integrated, $k^2$-compensated power spectrum $$E_k = 2\pi \int_0^{\pi} d\theta \, \sin(\theta) k^2 F_k \label{eq:defWk}$$ in numerical calculation A3 at three different times. At early times, $U_t$ grows, but this growth saturates while wave energy is still being injected. The reason for this saturation is that $F_k$ approaches a state in which energy injection at small $k$ is balanced by energy dissipation at large $k$. At early times, $\langle k \rangle$ also grows, as $F_k$ evolves towards a broad power-law-like spectrum. As can be seen in Figure \[fig:w0\], $U_t$ reaches its maximum value at an earlier time in solution A2 than in solution A1. This is because the larger values of $\dot{E}_0$ and $F_k$ in solution A2 reduce the energy cascade timescale at the forcing wavenumber $k_0$. ![The angle-integrated, $k^2$-compensated fast-wave power spectrum $E_k$ at three different times in model solution A3. \[fig:fk\]](fk.eps){width="12.cm"} As mentioned in Section \[sec:miller\], less than half of the energy that is injected into waves in all previously described numerical calculations is transferred to the electrons, and more than half cascades to $k> k_{\rm max}$ where it is dissipated by hyperviscosity. We note that much of the wave energy that cascades to $k> k_{\rm max}$ in our numerical calculations is in highly oblique waves with comparatively large values of $\sin \theta$. There are two reasons for this. As discussed in Section \[sec:model\], the energy cascade time in fast-wave turbulence decreases as $\sin \theta$ increases. In addition, because of the TTD resonance condition, waves with $\sin \theta \sim 1$ interact with only a small number of high-speed electrons, and thus experience comparatively little damping. The Anisotropic Electron Distribution Function {#sec:fe} ============================================== In this section, we focus on how resonant wave-particle interactions and Coulomb collisions affect the anisotropic electron distribution function. We begin with an example, solution B of Table \[table:set\], in which $t_{\rm inj} = \infty$, so that wave-injection is never shut off. Figure \[fig:fec2\] shows $f_{\rm e}$ at three different times in this numerical calculation. In the middle and right panels of this figure, and at a fixed $p$, $f_{\rm e}$ peaks at a pitch angle corresponding approximately to the black line. (We discuss the precise way in which this black line is determined later in this section.) The electron distribution becomes increasingly anisotropic at higher energies, in the sense that the value of $p_\parallel/p_\perp$ along the black line increases as $p_\parallel$ increases. As we will argue in this section, the anisotropic structure of $f_{\rm e}$ reflects a balance between resonant interactions, which accelerates electrons to larger $|p_\parallel|$, and collisions, which isotropize the distribution. For reference, we plot the curve $p=p_{\rm T}$ (white quarter circles) in Figure \[fig:fec2\], where $$p_{\rm T} = \sqrt{2 k_B m_{\rm e} T_{\rm e} } \label{eq:defpt}$$ is the thermal momentum. ![Grey-scale plot of the distribution function $f_{\rm e}$ in model solution B at $t = 0$, $2.5\times10^7 \Omega^{-1}_p$, and $4.0\times10^7 \Omega^{-1}_p$. The solid lines are plots Equation (\[eq:blackcurve\]), which represents the condition that the TTD timescale $\tau_{\rm ttd}$ equals the collisional timescale $\tau_{\perp \rm col}$. \[fig:fec2\]](fec2_log.eps){width="17.cm"} To describe the interplay between wave-particle interactions and collisions analytically, we begin by obtaining an approximate analytic expression for the momentum diffusion coefficient $D_{\rm res}$ in Equation (\[eq:dzz\]). Although some fast-wave energy at $k< k_{\rm max} = \Omega_{\rm p}/3v_{\rm A}$ is transferred to electrons via wave-particle interactions, we make the approximation that most of the fast-wave energy injected at small wavenumbers cascades to $k> k_{\rm max}$, as in the numerical calculations described in Section \[sec:miller\]. We then model $F_k$ at $k< k_{\rm max}$ using weak-turbulence theory, neglecting losses of fast-wave energy due to wave-particle interactions. If fast-wave energy (per unit mass) is injected into the turbulence isotropically at small $k$ at rate $\dot{E}_0$ (i.e., $\sigma = 1$ in (\[eq:Sigma\])), then at $k_0 \ll k < k_{\rm max}$ $$F_k = \left( \frac{4v_{\rm A} \dot{E}_0}{9 \pi^3 c_2}\right) \frac{k^{-7/2}}{\sin\theta}, \label{eq:Fkapprox}$$ where $ c_2 = \int_0^\infty dx\,\ln(1+x)[x(1+x)]^{-5/2}[(1+x)^{9/2} - x^{9/2} - 1] \simeq 26.2$ [@Chandran05]. We discuss Equation (\[eq:Fkapprox\]) further in Appendix \[appen:wave\]. We restrict our discussion to superthermal electrons, setting $$p_{\perp} \gg p_{\perp \rm T}, \label{eq:vperplim}$$ which implies that TTD interactions dominate over LD interactions, as discussed following Equation (\[eq:dzz\]). Upon substituting Equations (\[eq:Fkapprox\]) and (\[eq:vperplim\]) into Equation (\[eq:dzz\]), we find that $$D_{\rm res} \simeq D_{\rm ttd} \label{eq:d_ttd0}$$ where $$D_{\rm ttd} = \left(\frac{\pi}{9c_2}\right)^{1/2} \frac{m_{\rm e}^2 \gamma^2 v_\perp^4}{v_\parallel^4}\left[ k_{\rm max} v_{\rm A} \dot{E}_0 (v_\parallel^2 - v_{\rm A}^2)\right]^{1/2} \label{eq:d_ttd}$$ is the parallel-momentum diffusion coefficient arising from TTD interactions. We henceforth restrict our analysis to non-relativistic or trans-relativistic electrons, setting $$\gamma \simeq 1. \label{eq:gamma1}$$ The characteristic timescale on which TTD changes an electron’s parallel momentum by a factor of order unity is $$\tau_{\rm ttd} \sim \frac{p_\parallel^2}{D_{\rm ttd}}. \label{eq:deftauttd}$$ We define the perpendicular (parallel) collisional timescale $\tau_{\perp \rm col}$ ($\tau_{\parallel \rm col}$) to be the characteristic time required for Coulomb collisions to change $p_\perp$ ($p_\parallel$) by a factor of order unity. At $p\gg p_{\rm T}$ and below the black line in Figure \[fig:fec2\], $p_\perp \ll p_\parallel$. In this region, $p_\perp$ can change by a factor of order unity when an electron’s pitch angle changes by much less than one radian, which causes $\tau_{\perp \rm col}$ to be $ \ll \tau_{\parallel \rm col}$. We can show from Equation (\[eq:coll2\]) that when $p\gg p_{\rm T}$ and $|p_\parallel| \gg p_\perp$, the momentum diffusion coefficient for diffusion in $p_\perp$ is approximately $$D_{\perp \rm col} \simeq \frac{\nu_0 (m_{\rm e} v_{\rm A})^3}{ 2|p_\parallel|}, \label{eq:perp_coll}$$ where $$\nu_0 = \frac{4 \pi \Lambda e^4 n_{\rm e}}{m_{\rm e}^2 v_{\rm A}^3} \label{eq:defnu0}$$ is the characteristic collision frequency for electrons with momentum $m_{\rm e}v_{\rm A}$. The perpendicular collision timescale is then $$\tau_{\perp \rm col} \sim \frac{ p_\perp^2}{D_{\perp \rm col}}. \label{eq:tauperpcol}$$ The relative importance of TTD and collisions can be determined by comparing $\tau_{\rm ttd}$ and $\tau_{\perp \rm col}$. Since $D_{\rm ttd} \propto v^4_\perp$ for electrons with $\gamma \simeq 1$, and since Coulomb collisions become weaker as $v_\perp$ increases, $\tau_{\rm ttd} \ll \tau_{\perp \rm col}$ at sufficiently large $v_\perp$. When $\tau_{\rm ttd} \ll \tau_{\perp \rm col}$, electrons diffuse primarily in $p_\parallel$ rather than $p_\perp$, which explains why the contours of constant $f_{\rm e}$ are horizontal at large $p_\perp$ in Figure \[fig:fec2\]. On the other hand, at very small $p_\perp$, $\tau_{\perp \rm col} \ll \tau_{\rm ttd}$ and electrons diffuse in $\ln (p_\perp/p_{\rm T})$ much more rapidly than they diffuse in $\ln(p_\parallel/p_{\rm T})$. This explains why the contours of constant $f_{\rm e}$ are nearly vertical at small $p_\perp$ in Figure \[fig:fec2\]. The transition between the TTD-dominated regime at large $p_\perp$ and the collision-dominated regime at small $p_\perp$ occurs when $$\tau_{\rm ttd} \sim \tau_{\perp \rm col}. \label{eq:eqtimes}$$ If we set $\tau_{\rm ttd} = \tau_{\perp \rm col}$, take $|p_\parallel|$ to be $\gg m_{\rm e} v_{\rm A}$, and replace the $\sim$ signs in Equations (\[eq:deftauttd\]) and (\[eq:tauperpcol\]) with equals signs, we obtain $$\frac{p_\perp}{m_{\rm e}v_{\rm A}} = 1.3 c_3 \left( \frac{v_{\rm A} \nu_0^2}{k_{\rm max}\dot{E}_0}\right)^{1/12} \left(\frac{p_\parallel}{m_{\rm e} v_{\rm A}}\right)^{2/3}, \label{eq:blackcurve}$$ where $c_3$ is a dimensionless constant, which we have inserted to account for the uncertainties in replacing the $\sim$ signs with $=$ signs. The black lines in Figure \[fig:fec2\] are plots of Equation (\[eq:blackcurve\]) with $$c_3 = 0.89. \label{eq:c3}$$ As mentioned previously, at a fixed $p>p_{\rm T}$, $f_{\rm e}$ reaches its maximum value close to the black lines in Figure \[fig:fec2\]. To a reasonable approximation, we can thus take the majority of the electrons at any fixed non-thermal energy $E$ to satisfy Equation (\[eq:blackcurve\]) to within a factor of order unity. In this approximation, we can view all properties of the non-thermal electrons as functions of the single variable $p_\parallel$. For example, $p_\perp = p_\perp(p_\parallel)$, $\tau_{\rm ttd} = \tau_{\rm ttd}(p_\parallel)$, etc. The way that electrons diffuse out to larger energies along the black lines in Figure \[fig:fec2\] is through a combination of two processes. TTD causes electrons to diffuse in $p_\parallel$ at a fixed $p_\perp$, and Coulomb collisions scatter electrons to larger values of $p_\perp$. If we focus on one of the horizontal lines of constant $f_{\rm e}$ above the black lines in Figure \[fig:fec2\], the timescale $\tau_{\rm ttd}$ increases as $p_\parallel$ increases. The time it takes an electron to reach a point on one of the black lines in Figure \[fig:fec2\] with parallel velocity $p_\parallel$ is thus $\sim \tau_{\rm ttd}(p_\parallel)$, or equivalently $\tau_{\perp \rm col}(p_\parallel)$. This timescale is the acceleration timescale, denoted $\tau_{\rm acc}$: $$\tau_{\rm acc}(p_\parallel) = \tau_{\perp \rm col}(p_\parallel). \label{eq:deftauacc}$$ With the use of Equations (\[eq:perp\_coll\]), (\[eq:tauperpcol\]), and (\[eq:blackcurve\]), we find that $$\tau_{\rm acc} = 3.4 c_3^2 \left(\frac{v_{\rm A}}{\nu_0^4 \,k_{\rm max} \dot{E}_0 }\right)^{1/6} \left(\frac{p_\parallel}{m_{\rm e} v_{\rm A}}\right)^{7/3}. \label{eq:tau_acc}$$ The largest $|p_\parallel|$ to which an electron can be accelerated, denoted $p_{\parallel \rm max}$, is approximately given by the condition $$\tau_{\rm acc}(p_{\parallel \rm max}) = \Delta t, \label{eq:tauaccDt}$$ where $$\Delta t = t - \tau_{\rm cas} \label{eq:valDelt}$$ is the duration of the acceleration process. The values of the energy cascade timescale $\tau_{\rm cas}$ (defined in Equation (\[eq:deftaucas\])) in our numerical calculations are listed in Table \[table:set\]. (We note that at $0<t< t_{\rm cas}$, $F_k$ is still growing, and Equation (\[eq:d\_ttd\]), which is the basis of our analysis, does not apply.) Equation (\[eq:tauaccDt\]) leads to a maximum parallel momentum of $$p_{\parallel \rm max} = 0.59 c_3^{-6/7} \nu_0^{2/7} \left(\frac{k_{\rm max} \dot{E}_0}{v_{\rm A}}\right)^{1/14} \left(\Delta t\right)^{3/7} m_{\rm e}v_{\rm A}. \label{eq:pparmax}$$ In the $\gamma \simeq 1$ limit that we have been focusing on, the maximum energy $E_{\rm max}$ that electrons can be accelerated to via TTD is then $$E_{\rm max} \simeq \frac{[p_\perp(p_{\parallel \rm max})]^2 + p_{\parallel\rm max}^2}{2m_{\rm e}} \label{eq:Ent}$$ We note that Equations (\[eq:pparmax\]) and (\[eq:Ent\]) are valid only when $t_{\rm cas} < t < t_{\rm inj}$. At larger values of $t$, after wave injection ceases, the fast-wave energy decays away, TTD interactions cease, and the electrons undergo a purely collisional evolution, which is described further in Section \[sec:appflare\]. Referring to Figure \[fig:fe\], the energy $E_{\rm max}$ is the high-energy cutoff of the non-thermal tail in the electron energy distribution. We now discuss, with the aid of Figure \[fig:diagram\], the physics that determines the minimum energy of this non-thermal tail, which we denote $E_{\rm nt}$, again restricting our discussion to $t< t_{\rm inj}$. The vertical dashed line Figure \[fig:diagram\] represents the minimum parallel momentum $p_\parallel = m_{\rm e} v_{\rm A}$ at which electrons can satisfy the TTD resonance condition, Equation (\[eq:resonance2\]). The solid line in this figure is a plot of the solution of Equation (\[eq:blackcurve\]) for some arbitrary choice of parameters. Above this line, and to the right of the dashed line, $\tau_{\rm ttd} < \tau_{\perp \rm col}$ and TTD interactions are dominant. That is, electrons diffuse primarily in $p_\parallel$ rather than in $p_\perp$, as illustrated schematically with the horizontal double-headed arrow. The $p_\perp$ coordinate at the intersection of the solid and dashed lines is denoted $p_{\perp \rm min}$ and is the minimum value of $p_\perp$ for which TTD can dominate over collisions. Assuming that $E<E_{\rm max}$, electrons with $p_\perp > p_{\perp \rm min}$ diffuse rapidly in $p_\parallel$ within the interval $p_\parallel \in [m_{\rm e}v_{\rm A}, p_\parallel(p_\perp)]$, where the function $p_\parallel(p_\perp)$ is obtained by inverting Equation (\[eq:blackcurve\]). In the non-relativistic limit, the energies at the endpoints of this $p_\parallel$ interval are $$E_1(p_\perp) = \frac{1}{2 m_{\rm e}} \left ( p^2_{\perp}+m^2_{\rm e} v^2_{\rm A} \right ) \label{eq:E1}$$ and $$E_2(p_\perp) = \frac{1}{2 m_{\rm e}} \left\{ p^2_{\perp}+[p_{\parallel}(p_\perp)]^2 \right\}. \label{eq:E2}$$ These endpoints are labeled $E_1$ and $E_2$ in Figure \[fig:diagram\]. We define the ratio $$R(p_\perp) = \frac{N_{\rm M}(E_1)}{N_{\rm M}(E_2)} , \label{eq:nr}$$ where $N_{\rm M}(E)$ is the Maxwellian energy spectrum, obtained by replacing $f_{\rm e}$ in Equation (\[eq:NE\]) with $f_{\rm M}$, the Maxwellian distribution that has the same total energy as the instantaneous value of $f_{\rm e}$. When $p_\perp$ is just slightly larger than $p_{\perp \rm min}$, $E_1$ and $E_2$ are not too dissimilar, $R(p_\perp)$ is not very large, and the diffusion of electrons from $p_\parallel = m_{\rm e}v_{\rm A}$ to $p_\parallel = p_\parallel(p_\perp)$ causes only a minor enhancement of the energy spectrum at $E=E_2$ relative to a Maxwellian energy spectrum. Such a minor enhancement is unable to produce a noticeable non-thermal tail in $N(E)$. However, as $p_\perp$ increases, $R(p_\perp)$ grows, and eventually the diffusion of electrons from $p_\parallel = m_{\rm e}v_{\rm A}$ to $p_\parallel = p_\parallel(p_\perp)$ produces a major enhancement in the value of $N(E)$ at $E=E_2$, leading to the presence of a substantial non-thermal tail in the distribution. In our numerical calculations, we find that the non-thermal tail begins at an energy $\sim E_2(p_{\perp \rm nt})$, where $p_{\perp \rm nt}$ is the solution of the equation $$R(p_{\perp \rm nt}) = 100. \label{eq:R100}$$ That is, $$E_{\rm nt} = E_2(p_{\perp \rm nt}). \label{eq:EntE2}$$ ![The solid line is the solution to Equation (\[eq:blackcurve\]) for some arbitrary choice of parameters. This line gives the location in the $(p_\parallel, p_\perp)$ plane at which $\tau_{\rm ttd} = \tau_{\perp \rm col}$. The vertical dashed line $p_\parallel = m_{\rm e} v_{\rm A}$ shows the minimum $p_\parallel$ for which electrons can undergo resonant TTD interactions. The horizontal dotted line represents $p_\parallel$-diffusion due to TTD, which dominates over collisions above the solid line and to the right of the dashed line. In order for TTD to be dominant, $p_\perp$ must exceed $p_{\perp \rm min}$, which is the $p_\perp$ coordinate of the intersection between the solid and dashed lines. The energies $E_1(p_\perp)$ and $E_2(p_\perp)$ are evaluated, respectively, along the dashed and solid lines. \[fig:diagram\]](Peera_fe.eps){width="7.cm"} Qualitatively, there are two main factors that control the value of $E_{\rm nt}$. The first is the amplitude of the fast-wave turbulence. As $\dot{E}_0$ and $F_k$ decrease, $p_{\perp \rm min}$ increases, since electrons need larger values of $p_\perp$ for TTD to dominate over collisions. This causes $E_{\rm nt}$ to increase as a consequence. On the other hand, if $\dot{E}_0$ and $F_k$ are sufficiently large, $E_{\rm nt}$ can be reduced to energies just moderately above the thermal energy. The second factor that influences $E_{\rm nt}$ is the electron temperature. As electrons are heated, the effects of TTD on $f_{\rm e}$ become pronounced only at higher and higher electron energies, causing $E_{\rm nt}$ to increase. For example, if at a fixed $p_\perp$, the difference in the energy between the dashed line and solid line in Figure \[fig:diagram\] is less than $k_{\rm B} T_{\rm e}$, then the diffusion of electrons from $p_\parallel= m_{\rm e} v_{\rm A}$ to $p_\parallel = p_\parallel(p_\perp)$ at that value of $p_\perp$ will have only a minor effect on $N(E_2(p_\perp))$. We note that the location of the black solid lines in Figures \[fig:fec2\] and \[fig:diagram\] do not depend upon $T_{\rm e}$, since $T_{\rm e}$ does not enter into Equation (\[eq:blackcurve\]). In Figures \[fig:as\] and \[fig:as2\], we compare our expressions for $E_{\rm max}$ and $E_{\rm nt}$ in Equations (\[eq:Ent\]) and (\[eq:EntE2\]) with the electron energy spectrum in model solution B at three different times. We show the same comparison for two snapshots of solutions C and D in Figure \[fig:ec2\]. For the most part, our expressions for $E_{\rm max}$ and $E_{\rm nt}$ in Equations (\[eq:Ent\]) and (\[eq:EntE2\]) approximately bound the non-thermal tail in the electron distribution in our numerical calculations. The least successful fit occurs in solution C, for which Equation (\[eq:Ent\]) underestimates $E_{\rm max}$ by a factor of $\sim 2$. A discrepancy of this magnitude, however, is not entirely surprising, given the approximations we have made in deriving Equation (\[eq:Ent\]). ![The electron energy spectrum $N(E)$ in model solution B at $t=0$, $2.5\times10^7\Omega^{-1}_p$, $3.5\times10^7\Omega^{-1}_p$, and $4.5\times10^7\Omega^{-1}_p$. The dotted, dashed, and solid vertical lines indicate the values of $E_{\rm max}$ from Equation (\[eq:Ent\]) at $t=2.5\times10^7\Omega^{-1}_p$, $t=3.5\times10^7\Omega^{-1}_p$, and $t=4.5\times10^7\Omega^{-1}_p$, respectively. \[fig:as\]](as_kev.eps){width="12.cm"} ![The dotted-line, dashed-line, and solid-line curves are plots of $N(E)$ in model solution B at $t = 2.5\times 10^7 \Omega^{-1}_p$, $t = 3.5\times 10^7 \Omega^{-1}_p$, and $t = 4.5\times 10^7 \Omega^{-1}_p$, respectively. The dash-dot-dash curves are Maxwellian energy spectra $N_{\rm M}(E)$ that have the same total energy as $N(E)$ at these same three times. The vertical dotted, dashed, and solid lines show the values of $E_{\rm nt}$ at the times $t = 2.5\times 10^7 \Omega^{-1}_p$, $t = 3.5\times 10^7 \Omega^{-1}_p$, and $t = 4.5\times 10^7 \Omega^{-1}_p$, respectively. \[fig:as2\]](as2_kev.eps){width="12.cm"} ![Solid-line curves are $N(E)$ in model solution C at $t = 2.8\times 10^7 \Omega^{-1}_p$ (left panel) and solution D at $t = 2\times10^8 \Omega^{-1}_p$ (right panel). The thin dashed lines are plots of Maxwellian distributions with the same total energy as the (non-Maxwellian) electron spectra. The vertical dotted (thick-dashed) lines show the locations of $E_{\rm nt}$ ($E_{\rm max}$) in these numerical calculations at these same times. At the moments, the spectral indices are $3.4$ and $2.9$ from the solutions C and D, respectively. The straight solid lines are power-law fits to $N(E)$ in the energy interval $E_{\rm nt} < E < E_{\rm max}$.) \[fig:ec2\]](ec2_kev.eps){width="17.cm"} The Minimum $\beta_{\rm e}$ Required for Efficient TTD {#eq:betamin} ------------------------------------------------------ The fraction of the electron population that is significantly affected by TTD depends strongly on $\beta_{\rm e}$. If $\beta_{\rm e} \ll m_{\rm e}/m_{\rm p}$, then the electron thermal speed is much less than $v_{\rm A}$, and the number of electrons with $|p_\parallel| \gg m_{\rm e} v_{\rm A}$ is exponentially small. Since $|p_\parallel|$ must exceed $m_{\rm e} v_{\rm A}$ in order for electrons to satisfy the TTD resonance condition, TTD interactions with fast waves are exceedingly weak if $\beta _{\rm e} \ll m_{\rm e}/m_{\rm p}$. If $\beta_{\rm e}$ is initially small compared to $m_{\rm e}/m_{\rm p}$ in a flare, there may be a transient early stage in a flare in which some process heats the electrons until $\beta_{\rm e} \sim m_{\rm e}/m_{\rm p}$. During this initial heating stage, TTD is ineffective at accelerating electrons to non-thermal energies since only a minuscule fraction of the electrons have $p_\parallel > m_{\rm e} v_{\rm A}$. However, after this heating stage, a significant fraction of electrons satisfy $p_\parallel > m_{\rm e} v_{\rm A}$, and TTD acceleration to higher energies becomes much more efficient. Power-Law Fits to the Non-thermal Tail {#subsec:index} -------------------------------------- TTD results in a non-thermal tail in the electron energy spectrum that resembles a power law within the energy range $E_{\rm min} < E < E_{\rm max}$. We fit the energy spectra in our numerical calculations within this energy range with a power-law of the form $N(E) \propto E^{-\eta}$ and show these fits in Figures \[fig:fe\], \[fig:ec2\], and \[fig:lin\]. The resulting values of $\eta$ range from $2.9$ to $3.4$. As mentioned in Section \[sec:miller\], our electron energy spectra are steeper than in the isotropic-$f_{\rm e}$ model of [@Miller96], in which the non-thermal tail in $N(E)$ can scale like $E^{-\eta}$ with eta as small as 1.2. Time Evolution of the Electron Energy Spectrum {#sec:appflare} ============================================== In Figure \[fig:lin\] we plot the electron energy spectrum $N(E)$ at different times in model solution D. Between $t=0$ and $t\sim 42 \mbox{ s}$, the electron distribution develops a non-thermal, power-law-like tail extending to $\sim 80 \mbox{ keV}$. As time progresses, this power-law tail shifts to larger energies, and the temperature of the thermal particles increases, so that the thermal distribution shifts into the energy window shown in the figure. After wave injection ceases at $t=t_{\rm inj} =$ 3 min 23 s, the heating of the thermal distribution ends, and the non-thermal particles are gradually pulled back into the thermal distribution by Coulomb collisions. However, the collision frequency is $\propto p^{-3}$ at these non-thermal energies, and thus the low-energy end of the non-thermal tail is affected by collisions earlier than the high-energy end is affected. As a result, during the collisional evolution at $t> t_{\rm inj}$, the non-thermal tail drops to lower amplitudes but becomes flatter, as can be seen in the middle and right panels of Figure \[fig:lin\]. The evolution of $N(E)$ shown in Figure \[fig:lin\] is qualitatively similar to the evolution of the hard x-ray spectrum observed in the June 27, 1980 flare, which is plotted in Figure 3 of [@Lin81]. In both our model and the observations: (1) the power-law part of the spectrum is confined to a fairly narrow energy range, with $N(E)$ steepening at $E\sim 100 \mbox{ keV}$; (2) the thermal distribution and non-thermal tail shift to higher energies as time progresses during the early stages of the flare; and (3) during the late stages of the flare, the non-thermal tail becomes flatter, but drops in amplitude. Although the electron spectrum in solution D qualitatively resembles the photon spectrum in the June 27, 1980 flare, our model is not yet sufficiently sophisticated to produce a synthetic hard x-ray spectrum $I(E)$ for a detailed comparison to the observations. In order for us to map $N(E)$ in our model, which is the electron energy spectrum in the coronal acceleration region, into an x-ray spectrum $I(E)$, we would need to calculate the flux of electrons per unit energy $F(E)$ into the chromosphere, and we would need to account for the way that the escape of particles from the corona modifies $N(E)$. ![The time evolution of the electron energy spectrum $N(E)$ in model solution D. The dotted-line curve in the left panel is the initial Maxwellian spectrum at $t=0$ s. The curves plotted with $+$ signs in all three panels are plots of $N(E)$ at $t=(42 + 27.75 j) \mbox{ s}$ with $j=0, 1, 2, ... 12$. These curves are ordered in time, from top to bottom in each panel, with $j \in (0,4)$ in the left panel, $j\in( 4,8)$ in the middle panel, and $j\in (9,12)$ in the right panel. The scale on the vertical axis applies to the topmost $+$-sign curve in each panel as well as the dotted line in the left panel, with each succeeding plot of $N(E)$ in each panel offset downward by a factor of $10^{-2}$. \[fig:lin\]](lin.eps){width="13.cm"} Discussion and Conclusion {#sec:con} ========================= In this paper, we develop a stochastic-particle-acceleration (SPA) model in which electrons are energized by weakly turbulent fast magnetosonic waves via a combination of transit-time-damping (TTD) interactions, Landau-damping (LD) interactions, and pitch-angle scattering from Coulomb collisions. We use quasilinear theory and weak turbulence theory to describe the time evolution of the electron distribution function $f_{\rm e}$ and the fast-wave power spectrum $F_k$. We solve the equations of this model numerically and find that TTD leads to power-law-like non-thermal tails in the electron energy spectrum $N(E)$ extending from a minimum energy $E_{\rm nt}$ to a maximum energy $E_{\rm max}$. We derive approximate analytic expressions for $E_{\rm nt}$ and $E_{\rm max}$ and find that these expressions agree with our numerical solutions reasonably well. For a fast wave, the parallel electric field exerts a force on electrons that is $180^\circ$ out of phase with the magnetic-mirror force, and thus the inclusion of the parallel electric field (LD interactions) in our model reduces the rate of electron acceleration (see, e.g., the discussion of numerical calculation A4 at the end of Section \[sec:miller\]). The main new feature of our model that distinguishes it from previous studies is our inclusion of anisotropy in both momentum space and wavenumber space. We assume cylindrical symmetry about the magnetic field direction in both velocity space and wavenumber space, but allow $f_{\rm e}$ to depend upon both $p_\perp$ and $p_\parallel$ and $F_k$ to depend on both $k_\perp$ and $k_\parallel$. Another new feature of our work in the context of SPA models is our use of weak turbulence theory to describe the fast-wave energy cascade, which enables us to avoid introducing an adjustable free parameter into the energy cascade rate and to account for the weakening of the energy cascade as $\sin \theta$ decreases, where $\theta$ is the angle between the wavevector $\bm{k}$ and the background magnetic field $\bm{B}_0$. To investigate how much these new features affect our results, we compare one of our numerical solutions with a numerical example (“Case 4”) published by [@Miller96] (MLM96), which is based on their isotropic SPA model. We find that there are two main differences between our model and theirs. The first concerns the energy cascade rate. They modeled the fast-wave energy cascade by solving a nonlinear diffusion equation for $F_k$, in which the diffusion coefficient contained an adjustable free parameter. If we set the injection rate to produce $k^2 F_k = A k^{-3/2}$ in both models, where $A$ is some constant, then the energy cascade rate in our model is roughly 9 times faster than in their model. Conversely, if we set the energy cascade rates to be equal in the two models, then $F_k$ is smaller in our model than in theirs by a factor of $\simeq 3$, which weakens electron acceleration by fast waves in our model relative to theirs. The second main difference between the two models is that the anisotropy of $f_{\rm e}$ reduces the efficiency of electron acceleration via TTD. This is because TTD accelerates electrons to larger values of $|p_\parallel|$, but not to larger values of $p_\perp$, which causes most of the electrons in our model to satisfy $|p_\parallel|> p_\perp$. The TTD momentum diffusion coefficient $D_{\rm ttd}$ for energetic electrons (with $|v_\parallel| \gg v_{\rm A}$), however, is $\propto \gamma^2 v_\perp^4/|v_\parallel|^3$, and the electrons in our anisotropic model thus have smaller values $D_{\rm ttd}$ than in MLM96’s isotropic model. Because of these differences, the total number of electrons accelerated to energies $> 20 \mbox{ keV}$ is smaller in our model than in MLM96’s, the power-law-like non-thermal tails in the electron energy spectrum are steeper in our model, and these tails are limited to lower maximum energies in our model. Beyond the comparison with MLM96, our principal results are the following: 1. In the presence of TTD and Coulomb collisions, the electron distribution function at non-thermal energies approaches a specific characteristic form, which is shown in Figure \[fig:fec2\]. At a fixed $p$, $f_{\rm e}$ peaks at a pitch angle that corresponds to the black line in Figure \[fig:fec2\]. This line is a plot of Equation (\[eq:blackcurve\]) and corresponds to the locations in the $p_\perp$-$p_\parallel$ plane at which the TTD timescale $\tau_{\rm ttd}$ equals the collisional timescale $\tau_{\perp \rm col}$. Above this black line (at large $p_\perp$), TTD dominates over collisions, and rapid $p_\parallel$-diffusion of electrons causes $f_{\rm e}$ to become almost independent of $p_\parallel$. Below this curve, collisions dominate over TTD, and $f_{\rm e}$ depends more strongly on $\ln p_\parallel$ than on $\ln p_\perp$. 2. As can be seen in our expression for $E_{\rm max}$ in Equation (\[eq:Ent\]), the maximum energy of the non-thermal tail increases with increasing electron density $n_{\rm e}$. This is because collisions help electrons to reach higher energies by converting some of the parallel kinetic energy ($m_{\rm e} v_\parallel^2/2$) gained via TTD interactions into perpendicular kinetic energy ($m_{\rm e} v_\perp^2/2$), which increases the rate of TTD acceleration (since $D_{\rm ttd} \propto \gamma^2 v_\perp^4/|v_\parallel|^3$ for energetic electrons). Another way of thinking about this is that an electron can only reach a point $(p_\parallel , p_\perp)$ on the black line in Figure \[fig:fec2\] after the elapsed time $\Delta t$ has grown to a value of order the collisional timescale $\tau_{\perp \rm col}$ at that point (which equals the TTD timescale $\tau_{\rm ttd}$ at that point). Consistent with this reasoning, $E_{\rm max}$ increases with $t$ up until the wave injection ceases and the waves decay away. 3. One of the ways that the magnetic field strength $B_0$ and the initial electron temperature affect electron acceleration via TTD is through their influence on the value of $\beta_{\rm e}$. If the initial value of $\beta_{\rm e}$ is $\ll m_{\rm e}/m_{\rm p}$, then only an exponentially small fraction of the electrons have large enough values of $|v_\parallel|$ that they can satisfy the TTD resonance condition Equation (\[eq:resonance2\]), and TTD acceleration is exceedingly weak. 4. The time evolution of $N(E)$ in our model solution D qualitatively resembles the time evolution of the hard x-ray spectrum $I(E)$ in the June 27, 1980 flare reported by [@Lin81]. However, our model is not yet sophisticated enough to produce synthetic x-ray spectra, because we have neglected the escape of electrons from the acceleration region, which alters $N(E)$ and is needed to determine $I(E)$. There are several processes that we have not included in our model. As just mentioned, we have not accounted for the escape of electrons from the flare acceleration region or the flow of low-energy electrons into the acceleration region from the chromosphere. We have also neglected the escape of fast waves from the acceleration region (see [@Peera12a] for a detailed discussion of wave escape) and resonance broadening in wave-particle interactions [@shalchi04a; @shalchi04b; @yan08b; @lynn12; @lynn13; @lynn14]. In the numerical calculations we have carried out so far, a significant amount of the power injected into fast waves at small $k$ cascades to $k>\Omega_{\rm p}/v_{\rm A}$, where it presumably initiates a cascade of whistler waves. However, our model neglects the effects of whistler waves and other waves at $k\gtrsim\Omega_{\rm p}/v_{\rm A}$ on the electrons. A related point is that we have neglected non-collisional forms of pitch-angle scattering. One of the effects that waves at $k>\Omega_{\rm p}/v_{\rm A}$ could have is to enhance the electron pitch-angle scattering rate. By converting perpendicular electron kinetic energy into parallel kinetic energy, such enhanced pitch-angle scattering would increase the efficiency of TTD electron acceleration in flares. A useful direction for future research would be to incorporate some or all of these processes into the type of anisotropic SPA model that we have developed. Another valuable direction for future research would be to determine the amplitude of fast-wave turbulence in solar flares using large-scale direct numerical simulations. Because the turbulence amplitude plays a critical role in SPA models, a determination of this amplitude would lead to much more rigorous tests of SPA models than have previously been possible. This work benefited from valuable discussions with our colleagues in a NASA Living-With-a-Star Focused-Science-Topic team working on “Flare Particle Acceleration Near the Sun and Contribution to Large SEP Events.” This work was supported in part by NASA grants NNX07AP65G, NNX11AJ37G, and NNX12AB27G, DOE grant DE-FG02-07-ER46372, and NSF grant AGS-1258998. Analytic Expression for the Wave Damping Rate Allowing for Relativistic Particles {#appen:reldamping} ================================================================================= We follow the standard approach in quasilinear theory of treating the plasma as infinite and homogeneous. To obtain Fourier transforms of the fluctuating quantities, we define the “windowed” Fourier transform $$\tilde{g}(\bm{k}) = \frac{1}{(2\pi)^3} \int d^3 x\,\, g(\bm{x}) H(\bm{x}) \exp{(-i \bm{k}\cdot \bm{x})}, \label{eq:fourier1}$$ where $$H(\bm{x}) = \left\{ \begin{array}{cc}\displaystyle 1 & \mbox{ if $|x|<\displaystyle \frac{L}{2}$, $|y|<\displaystyle \frac{L}{2}$, and $|z|<\displaystyle \frac{L}{2}$ } \vspace{0.2cm} \\ 0 & \mbox{otherwise} \end{array} \right. . \label{eq:H}$$ As mentioned in Section \[sec:model\], our Fourier-transform convention is the same as that of [@Stix92], except that we have an extra factor of $(2\pi)^{-3/2}$ on the right-hand side of Equation (\[eq:fourier1\]). Accounting for this difference, we can use Eq. (67) of Chapter 4 of [@Stix92] to write the wave energy density $\epsilon_w$ in the form $$\varepsilon_w = \lim_{L\rightarrow \infty} \left(\frac{2\pi}{L}\right)^3 \int d^3k \,2W_k, \label{eq:epsw}$$ where $W_k$ is defined in Equation (\[eq:defW\]). Wave-particle interactions cause the particle kinetic energy density of species $s$ to change at the rate $$\dot{K}_s = \int d^3 p\,\, [(p^2 c^2+m^2_s c^4)^{1/2}-m_s c^2] \left(\frac{\partial f_s}{\partial t}\right)_{\rm res}, \label{:heat_rate}$$ where $(\partial f_s/\partial t)_{\rm res}$ is given in Equation (\[eq:QLT0\]). The second term in brackets in Equation (\[:heat\_rate\]), $m_s c^2$, can be dropped, because $\int d^3p (\partial f_s/\partial t)_{\rm res} = 0$. Equation (\[eq:epsw\]) implies that wave-particle interactions cause the wave energy density to change at the rate $$\dot{\varepsilon}_w = \left(\frac{2\pi}{L}\right)^3 \int d^3 k \, 4\gamma_k W_{\bm{k}},$$ where $\gamma_k$ is the imaginary part of the wave frequency. Because the sum of the wave and particle-kinetic-energy densities is conserved, $$\sum_s \dot{K}_s + \dot{\varepsilon}_w = 0. \label{:e_conserved}$$ Upon substituting Equation (\[eq:QLT0\]) into the right-hand side of Equation (\[:heat\_rate\]), integrating by parts, and using the identity $$\frac{\partial v_\perp}{\partial p_\parallel} = \frac{\partial v_\parallel }{\partial p_\perp}, \label{eq:ID1}$$ we can rewrite Equation (\[:e\_conserved\]) in the form $$\int d^3 k W_k I_k = 0, \label{eq:econ2}$$ where $$I_k = \gamma_k - \sum_s \sum_{n=-\infty}^\infty \frac{\pi^2 q_s^2}{2} \int_0^\infty dp_\perp \int_{-\infty}^\infty dp_\parallel \frac{p_\perp^2 c^2}{\sqrt{p^2 c^2 + m_s^2 c^4}} \frac{|\psi_{n,k}^{(s)}|^2}{W_k} \delta(\omega_{kr} - k_\parallel v_\parallel - n \Omega_s) G f_s. \label{eq:defI}$$ Equation (\[eq:econ2\]) must be satisfied for any function $W_k$, and hence $I_k$ must vanish at all $k$. The condition that $I_k = 0$ at each $k$ reflects the fact that the change in the energy of the waves within any small volume $V$ of wavenumber space is equal and opposite to the change in the particle kinetic energy that results from wave-particle interactions involving waves with $\bm{k} \in V$. We make use of this fact when we evaluate $\gamma_k$ in our numerical calculations. Specifically, when we evaluate $(\partial f_{\rm e}/\partial t)_{\rm res}$, we keep track of the change in particle kinetic energy that results from interactions with waves within each grid cell in wavenumber space. We use the term $\Delta K_{i}$ to denote the change in particle kinetic energy resulting from waves in the $i^{\rm th}$ grid cell. We then evaluate $\gamma_k$ within the $i^{\rm th}$ cell by setting $ (2\pi/L)^3 4\gamma_k W_k (\Delta k)_i^3 \Delta t$ within that cell equal to $-\Delta K_i$, where $(\Delta k)_i^3$ is the volume of the grid cell in $k$ space, and $\Delta t$ is the time step. By using this procedure, we ensure that the changes in $f_{\rm e}$ and $F_k$ that result from wave-particle interactions conserve energy to machine accuracy. From the equation $I_k=0$, we obtain $$\gamma_k = \sum_s \sum_{n=-\infty}^\infty \frac{\pi^2 q_s^2}{2} \int_0^\infty dp_\perp \int_{-\infty}^\infty dp_\parallel \frac{p_\perp^2 c^2}{\sqrt{p^2 c^2 + m_s^2 c^4}} \frac{|\psi_{n,k}^{(s)}|^2}{W_k} \delta(\omega_{kr} - k_\parallel v_\parallel - n \Omega_s) G f_s. \label{eq:rel_damp}$$ In the non-relativistic limit, Equation (\[eq:rel\_damp\]) can be written in the form $$\gamma_{\bm{k}}=\sum_s \sum_{n=-\infty}^\infty \frac{\pi \omega^2_{ps}}{8 n_{\rm e}} \left |\frac{1}{k_{\|}}\right| \int^{\infty}_0 d v_\perp v^2_\perp \int^{\infty}_{-\infty} d v_\parallel \frac{|\psi_{n,\bm{k}}^{(s)} | ^2}{W_{\bm{k}}}\delta\left (v_\| -\frac{\omega_{kr}-n\Omega}{k_\|}\right) G_v f_s^{(v)}, \label{eq:rel_damp3}$$ where $\omega_{ps} = \sqrt{4\pi n_s q_s^2/m_s}$ is the plasma frequency of species $s$, $f_s^{(v)} = f_s/m_{\rm e}^3$ is (in the non-relativistic limit) the usual velocity-space distribution function, and $$G_v=\left(1-\frac{k_\parallel v_\parallel}{\omega_{kr}}\right)\frac{\partial}{\partial v_\perp} +\left ( \frac{k_\parallel v_\perp}{\omega_{kr}}\right)\frac{\partial}{\partial v_\parallel}. \label{:Gv}$$ Equation (\[eq:rel\_damp3\]) is exactly the result derived by [@kennel67]. Equation (\[eq:rel\_damp\]) can thus be viewed as a generalization of Kennel & Wong’s (1967) result that allows for relativistic particles. Estimating the Error in our Approximate Collision Operator {#sec:b} ========================================================== The Coulomb collision operator involves the quantities $$K_1(\bm{p}) = \int d^3 p' f_{\rm e} (\bm{p}') u^{-1},$$ and $$K_2(\bm{p}) = \int d^3 p' f_{\rm e} (\bm{p}') u,$$ where $u = |\bm{p}'-\bm{p}|$. To evaluate these integrals would require a very large number of operations per time step. In order to increase the speed of the calculations, we replace $K_1(\bm{p})$ and $K_2(\bm{p})$, respectively, with $$H_1(\bm{p}) = \int d^3 p' f_{\rm M} (\bm{p}') u^{-1}$$ and $$H_2(\bm{p}) =\int d^3 p' f_{\rm M} (\bm{p}') u,$$ where $f_{\rm M}$ is the Maxwellian distribution that has the same total particle kinetic energy as $f_{\rm e}$. In this case, $H_1(\bm{p})$ and $H_2(\bm{p})$ can be pre-calculated and depend only on $p_\perp$, $p_\parallel$, $n_{\rm e}$, and $T_{\rm e}$. To estimate the error introduced by this approximation we compare $H_1(\bm{p})$ to $K_1(\bm{p})$ and $H_2(\bm{p})$ to $K_2(\bm{p})$ in numerical calculations A3, B, C, and D. The maximum values of $|H_1-K_1|/K_1$ and $|H_2-K_2|/K_2$ increase as $f_{\rm e}$ deviates from a Maxwellian shape. They increase and reach their maximum value approximately at $t=t_{\rm inj}$. In model solutions A3 and D, which have finite injection times, the maximum values of $|H_1-K_1|/K_1$ are 0.06 and 0.004, and the maximum values of $|H_2-K_2|/K_2$ are 0.18 and 0.02, respectively. In numerical calculations B and C, the maximum value of $|H_1-K_1|/K_1$ is 0.06 and the maximum value of $|H_2-K_2|/K_2$ is 0.16 within the time period we consider. The plots of $|H_1-K_1|/K_1$ and $|H_2-K_2|/K_2$ from numerical calculation B at $t = 6.9\times 10^7 \Omega^{-1}_p$ are shown in Figure \[fig:em\]. The reason these errors are not much larger is that most of the electrons in the numerical calculations remain at low energies, where $f_{\rm e}$ is approximately Maxwellian. ![Difference between $H_1$ and $K_1$ (top panel) and $H_2$ and $K_2$ (bottom panel) in model solution B at $t = 6.9\times 10^7 \Omega^{-1}_p$. \[fig:em\]](em.eps){width="12.cm"} Fast-Wave Turbulence {#appen:wave} ==================== In many situations involving turbulence, fluctuations or waves are excited at some large scale $\sim 1/k_0$ and then cascade to smaller scales. In our model, this large-scale excitation is represented by the term $S_k$ in Equation (\[eq:dFdt\]), where $$\dot{E}_0 = \frac{1}{2}\int d^3 k S_k,$$ is the total energy injection rate. Since $S_k \propto e^{-k^2/k^2_0}$ in our model, wave injection is limited to small wavenumbers $\lesssim k_0$. If dissipation is only effective at wavenumbers exceeding some dissipation wavenumber $k_d$, then wavenumbers $\gg k_0$ and $\ll k_d$ are said to be in the inertial range of the turbulence. In steady state, the energy cascade rate in the inertial range must equal $\dot{E}_0$. From Equation (\[eq:turb\]), when waves reach a steady state in which $F_k = A_1 g_{\theta} k^{-7/2}$, the energy cascade rate per solid angle per unit mass density is given by [@Chandran05], $$\epsilon = \frac{9 \pi^2 c_2 A_1^2 g^2_\theta \sin^2\theta}{16 v_{\rm A}}, \label{eq:cas_angle_mass}$$ where $c_2 \simeq 26.2$ is defined following Equation (\[eq:d\_ttd\]). The quantity $g_\theta$ is a function of $\theta$ that depends on the angular distribution of the input power at $k\lesssim k_0$. Equating the total energy injection rate with the total cascade power, we obtain $$\dot{E}_0 = 2\pi \int_0^\pi \epsilon \sin \theta d\theta. \label{eq:valEdot}$$ If waves are injected isotropically (as in numerical calculation A4, B, C, and D, in which $\epsilon$ is independent of $\theta$), then $g_\theta = 1/\sin\theta$ [@Chandran05], and $$\dot{E} = \frac{9 \pi^3 c_2 A_1^2}{4 v_{\rm A}} \label{eq:r1} \hspace{1cm} \mbox{(isotropic $\epsilon$)}.$$ On the other hand, in our numerical calculations A1, A2, and A3, we take $S_k \propto \sin\theta$, which leads to $g_\theta = 1$ [@Chandran05]. Equation (\[eq:valEdot\]) then yields $$\dot{E} = \frac{3 \pi^3 c_2 A^2_1}{2 v_{\rm A}} \label{eq:r2} \hspace{1cm} \mbox{(isotropic $F_k$)}.$$ We can compare [**Equation (\[eq:r2\])**]{} to the cascade rate implied by the equation 3.3 in MLM96, $$\dot{E} = \frac{14 \pi^2 A^2_1}{v_{\rm A}}. \label{eq:rmiller} \hspace{1cm} \mbox{(MLM96)}$$ The cascade rate in our model in Equation (\[eq:r2\]) is larger than MLM96’s by a factor of $3\pi c_2/28 = 8.8$ for a fixed isotropic $F_k$. Therefore, if $\dot{E}$ is the same in our model and MLM96’s, then $F_k$ be will smaller in our model by a factor of $\simeq 3$.

--- abstract: 'The transmission or reception of packets passing between computers can be represented in terms of time-stamped events and the resulting activity understood in terms of point-processes. Interestingly, in the disparate domain of neuroscience, models for describing dependent point-processes are well developed. In particular, spectral methods which decompose second-order dependency across different frequencies allow for a rich characterisation of point-processes. In this paper, we investigate using the spectral coherence statistic to characterise computer network activity, and determine if, and how, device messaging may be dependent. We demonstrate on real data, that for many devices there appears to be very little dependency between device messaging channels. However, when significant coherence is detected it appears highly structured, a result which suggests coherence may prove useful for discriminating between types of activity at the network level.' author: - 'Alex J. Gibberd$^*$, Jordan Noble, Edward A.K. Cohen[^1]' bibliography: - 'cyber.bib' - 'data.bib' title: Characterising Dependency in Computer Networks using Spectral Coherence --- Introduction ============ Understanding how devices on computer networks communicate is a challenging task. While it is possible to gather vast quantities of data from such networks, for instance via packet monitoring, it is difficult to store, let alone process. As a result, protocols such as NetFlow which sample and summarise packet level data are now very popular [@Duffield2004]. Even still, regular monitoring protocols can produce hundreds of gigabytes of summary statistics on a network per day which need to be converted into actionable insights for network administrators. Network defenders should be at a theoretical advantage over attackers, in that they can attempt to model and understand the day-to-day activity of their network. From such models they can then define what anomalous, and/or malicious events may look like. Additionally, in order to enhance detection performance, one may desire to use prior knowledge of what benign network activity should look like in order to define anomalies. For example, and relevant to the approach developed here, one may expect that communication between pairs of devices are not correlated such that their activity should be broadly independent when monitored at a network level. However, if traffic is dependent across pairs then this may indicate potentially malicious behaviour such as lateral movement or tunnelling [@Neil2013]. There are a great variety of measures and methods that can be used to analyse dependency between streams, for instance through measures such as covariance [@Jin2007], correlation [@Neil2013], partial correlation [@Gibberd2017], or higher-order measures such as cross-cumulants [@Brillinger]. A traditional approach to network traffic modelling is to assume it is generated according to a Poisson point process [@Duffield2004; @Murdoch2007]. While such models may be generalised to a multivariate setting [@Baurele2005], they do not allow for us to encode auto-correlation structure within a point-process. As a result, we may be able to describe processes which are dependent on each other across data-streams, but they do not allow for dependency within a data-stream itself. When considering computer network traffic, it is not hard to imagine that events from a device will be somehow dependent on previous events from that same device. Spectral approaches, based on either Fourier [@Brillinger1981] or time-scale wavelet analysis [@Riedi1999; @Scherrer2007] of processes provide a valuable tool in this situation as they allow for both a rich description of auto-correlation and cross-channel dependency [@Cohen2014]. In this paper, we propose to utilise a measure known as spectral coherence [@Carter1987] to characterise dependency between network communication channels. We are not aware of any previous application of such a measure to network traffic analysis, although the method has received great attention in neuroscience for modelling neuron dependency [@Jarvis2001]. Dataset and Preprocessing ========================= Consider network connected devices $A,B,C$ and their associated users, for example, these may be personal computers, DNS servers, authentication servers, or even printers. Typically, we would expect these devices to go about their work as fairly independent actors, i.e. they may browse websites, download material etc., but not in any particularly coordinated manner. Device communication is typically performed through packet transmission. However, given network monitoring limitations, the events that we analyse need not necessarily be packets themselves. More likely, they are aggregates or summaries of communication, for example NetFlow sessions. In our case, we analyse a subset of NetFlow session data from the Los Alamos National Laboratory (LANL) multi-source cyber-security events data [@kent-2015-cyberdata1; @akent-2015-enterprise-data]. More specifically, we create a subset of events (NetFlow session start times) relating to the top $N_{\mathrm{triple}}=500$ busiest edge-pairs in the network over a single day’s (Thursday) worth of data. We assess dependency in a pair-wise manner such that data-streams correspond to directed edge-communication between devices $A\rightarrow B$ and $B\rightarrow C$ for $i=1,\ldots,N_{\mathrm{triple}}$ device triples $(A,B,C)_{i}$. For each triple, the activity for each edge corresponds to the same time frame. The protocol monitored is the same for all edge pairs. To avoid confusion, we exclude all flows from the triple $(C,B,A)$ when the triple $(A,B,C)$ is included. One should note that our selection criteria for data-set construction does not explicitly specify devices which have a particular function on the network. However, if we look at the graph of communication edges in Figure \[fig:Graph\_of\_edge\_pairs\] it appears that many of our triples have repetitive edges, there are only $95$ unique devices in our data-set and $96$ unique edges. Looking at the topology of the network it appears that most of the devices are communicating through the device C5721, while we do not have labelled data relating to the function of devices, it would appear that this node acts as some form of server. We note that in our recordings it is possible to observe two events which have the same start time. This means that the events cannot reasonably be treated as being observed in continuous time, and indeed the timestamps provided with our data are only accurate to the second. As such, the raw events are aggregated into bins of width $\Delta=1$s. The binned bivariate process will be denoted $\{\boldsymbol{X}[k] = [X_{AB}[k],X_{BC}[k]]^T; k\in\mathbb{Z} \}$, for which we observe a portion $\boldsymbol{X}[1], \boldsymbol{X}[2],...,\boldsymbol{X}[K]$. As a pre-processing step, we subtract the empirical mean of the data-streams, relabeling $\boldsymbol{X}[k]:=\boldsymbol{X}[k]-\bar{\boldsymbol{X}}$ so that they can be well approximated as zero-mean processes. ![Graph of messaging channels under analysis. Size of text and node are respectively proportional to the out-going and in-going degree. The weight of the edges (and colour) is proportional to the rate measured on that edge.\[fig:Graph\_of\_edge\_pairs\]](images/graph_rates){width="\columnwidth"} [$N_{\mathrm{triple}}$]{} [500]{} [Average length $T$]{} [17.2 (hours)]{} --------------------------- --------- --------------------------------- ------------------ [Number of nodes]{} [95]{} [Average rate $\lambda_{AB}$]{} [0.101 (1/s)]{} [Unique edges]{} [96]{} [Average rate $\lambda_{BC}$]{} [0.032 (1/s)]{} Spectral Coherence as a Measure of Association ============================================== In this section, we will define the spectral coherence as a property relating to the cross-spectrum of a bivariate process. The discussion here will be developed based on the understanding that we are with observations relating to a discrete-time process. However, it is also possible to perform such analysis at the individual event level, for examples, see the work of [@Cohen2014; @Jarvis2001; @Brillinger1972]. To simplify the notation, let us use $X_{1}[k]\equiv X_{AB}[k]$ and $X_{2}[k]\equiv X_{BC}[k]$. Furthermore, we will assume that $\{X_{1}[k],X_{2}[k]\}$ represent a jointly second-order stationary process, i.e. the covariance $\Sigma_{ij}[\tau]\equiv\mathrm{Cov}(X_{i}[k+\tau],X_{j}[k])$ only depends on $\tau$ for $i,j=1,2$. Provided $\sum_{\tau}|\Sigma_{ij}[\tau]|<\infty$, then for all frequencies $|\omega|\le \pi/\Delta$, the function $S_{ij}(\omega)=\Delta\sum_{\tau=-\infty}^{\infty}\Sigma_{ij}[\tau]e^{-i\omega\tau\Delta}$, $i,j=1,2$, is termed the spectum of $\{X_i[k]\}$ when $i=j$, and the *cross-spectrum* between $\{X_{1}[k]\}$ and $\{X_{2}[k]\}$ when $i\neq j$. These spectra can be conveniently represented with the spectral matrix $\boldsymbol{S}(\omega) = (S_{ij}(\omega))$. The argument we present in this paper, is that the cross-spectrum provides a rich framework within which we may characterise computer network messaging processes. In particular, since we are interested in dependency between data-streams, we will concern ourselves with the squared coherencey, or ordinary coherence, defined as the real-valued quantity $$R(\omega)=\frac{|S_{12}(\omega)|^{2}}{S_{11}(\omega)S_{12}(\omega)}\;.\label{eq:Coherence_def}$$ The coherence provides a useful statistic for assessing dependency between point-processes; not only does it permit a decomposition over frequencies allowing one to highlight periodicities associated with dependence, but is also invariant to scaling of the marginal auto-covariance as the measure is normalised by the on-diagonal spectra. Estimating the Spectra of Point-Processes ----------------------------------------- Since the true values of the spectra, cross-spectra and coherence are unknown to us, we are required to estimate them from data. The approach that we utilise here is based on the work of Thomson [@Thomson1982]. Specifically, we will construct our estimators from the *tapered discrete-time Fourier transform (tDFT)* defined as $$\hat{F}_{j;l}(\omega)\equiv\Delta^{1/2}\sum_{k=1}^{K}h_{l}[k]X_{j}[k]e^{-i\omega k\Delta}\quad j=1,2,\label{eq:raw_DFT}$$ for frequencies $-\pi/\Delta<\omega<\pi/\Delta$ where $\{h_{l}[k];k=1,...,N\}$ for $l=1,\ldots,L$ are a set of taper sequences. The tapers in the above construction are important, in that they enable us to selectively transform data-points prior to taking the Fourier transform. If we temporarily assume that $h_{l}[k]=1$ for all $l,k$, then taking the conjugate outer-product leads to the periodogram $\hat{I}_{ij;l}(\omega)\equiv \hat{F}_{i;l}(\omega)\hat{F}_{j;l}^{*}(\omega)$. Unfortunately, while the periodogram is an asymptotically unbiased estimator of the spectrum $E[\hat{I}_{ij;l}(\omega)]\rightarrow S_{ij}(\omega)$ as $T\rightarrow\infty$, it is not consistent, in that $\mathrm{Var}[\hat{I}_{ij;l}(\omega)]\not\rightarrow0$. Principally, this is due to us attempting to estimate the spectra at an infinite number of frequencies $\omega\in\mathbb{R}$ with only a finite portion of data [@Brillinger1972; @Jarvis2001]. There are several approaches which can be used to sculpt asymptotically consistent estimators of the spectra [@Nuttall1982; @Thomson1982; @Walden2000]. A general strategy [@Walden2000], is to adapt the direct spectral estimate (where $h_{l}[k]=1$) such that the tapers take different shapes, for example they may be supported in disjoint regions [@Brillinger1981], or constitute a set of overlapping windows [@Nuttall1982]. From the Fourier transform of the tapered data, we may then construct vectors $\hat{\boldsymbol{F}}_{l}(\omega)=[\hat{F}_{1;l},\hat{F}_{2;l}]^T$ and compute what is known as a *multi-taper spectral estimator* by averaging: $$\hat{\boldsymbol{S}}(\omega)=\frac{1}{L}\sum_{l=1}^{L}\hat{\boldsymbol{F}}_{l}(\omega)\hat{\boldsymbol{F}}_{l}^{H}(\omega)\quad i,j=1,2,$$ where $^H$ denotes the complex conjugate transpose. From the multi-taper spectral estimate we may then obtain an estimate for the coherence $\hat R_{12}(\omega)$ via (\[eq:Coherence\_def\]) replacing the true spectra $S_{11}(\omega)$, $S_{22}(\omega)$ and $S_{12}(\omega)$ with the estimates $\hat S_{11}(\omega)$, $\hat S_{22}(\omega)$ and $\hat S_{12}(\omega)$, respectively. Taper Specification ------------------- If we consider choosing orthogonal taper sequences whereby $\sum_{k}h_{l}[k]h_{l'}[k]=0$ for $l\ne l'$ then the resultant Fourier transforms will be asymptotically independent [@Brillinger1981]. Averaging over these independent sequences can then reduce the variation in the estimate, the reduction will be related to the number of tapers we average over. It has been demonstrated, c.f. [@Brillinger1981], that in the case of asymptotically orthogonal tapers, the sampling distribution of the spectral matrix $\hat{\boldsymbol{S}}(\omega) = (\hat S_{ij}(\omega))$ is given by a 2D complex Wishart distribution $\hat{\boldsymbol{S}}(\omega)\sim W_{2}^C(L,\boldsymbol{S}(\omega))$ with $L$ degrees of freedom and scale matrix $\boldsymbol{S}(\omega)$. In our application, we utilise a form of taper first demonstrated for spectral estimation by Thomson [@Thomson1982]. Often referred to as the Slepian tapers, these sequences have the beneficial properties that they are mutually orthogonal while maximising energy concentration in a small frequency interval $[-\omega_{W},\omega_{W}]$. If two frequencies are separated by more than this bandwidth, then the bias due to tapering is in some sense minimised. However, as the number of tapers $L$ increases, the width of the side-lobe associated with the Fourier transform of $h_{L}[k]$ necessarily increases. As such, there is a classic bias-variance trade-off, increasing $L$ reduces the variance, but increases bias. The appropriate number of tapers to use is highly dependent on application and something we will shortly revisit in the context of the network traffic dataset. ![Example of estimates for the spectral density. Top: estimation with $L=5$ tapers. Bottom: estimation with $L=40$ tapers. The red and blue lines respectively illustrate the on-diagonal spectral density $\hat{S}_{AB,AB}(\omega)$ and $\hat{S}_{BC,BC}(\omega)$. The grey line indicates the resultant coherence $\hat{R}(\omega)$. \[fig:Example\_spectra\]](images/spectra_ep0_L5_40){width="1\columnwidth"} Dependency in Network Traffic ============================= Applying coherence estimation to edge pairs results in a set of estimates $\{\hat{R}^{(n)}(\omega_{1},\ldots,\omega_{N_{f}})\}$ for $n=1,\ldots,N_{\mathrm{triple}}$. As may be expected there is significant variation of the spectra across the set of edge-pairs. In this analysis, we consider fixing the window of frequencies such that $\omega_{q}=2\pi f_{\max}(q/N_{f})$ for $q=1,\ldots,N_{f}=500$ and $f_{\max}=0.05Hz$. As the length of the edge-pair recordings differ, one may desire to increase $L$ as a function of length $T$. Potentially, this would lead to increased confidence in our spectral estimate as the Wishart degrees of freedom are increased. However, such an adaptive tapering scheme where $L$ depends on $T$ creates challenges when comparing across coherence estimates as it may be hard to disentangle differences due to the tapering treatment from underlying differences in the process spectra. As such, in these experiments we decide to fix the number of tapers at a moderate level $L=40$ for all edge-pairs. The difference between tapering with $L=5$ and $L=40$ is demonstrated in Figure \[fig:Example\_spectra\]. Note, that while the cross-spectra for different edge-pairs may be of a different scale, the coherence (plotted in grey) provides an intuitive measure on $[0,1]$ allowing comparison across many data-stream pairs. As an aside, the individual spectra appear non-Poisson, exhibiting shapes that are characteristic of self-exciting behaviour [@Hawkes1971]. Acknowledging that there will be some error in our spectral estimates, it is desirable to assign some measure of confidence to estimates. A useful corollary of the Wishart asymptotic result for multi-taper estimates is that the coherence is distributed (asymptotically) according to the Goodman distribution [@Carter1987; @Goodman1963]. Based on this distribution, there are a variety of tests that one may perform to assess the *significance* of a coherence estimate. For example, one may test the null hypothesis that states $R(\omega_{q})=0$ for each frequency $\omega_{q}$, $q=1,\ldots,N_{f}$. Rather than test explicitly against a null of zero coherence, in this work we construct two sided confidence intervals in a similar manner to Wang et al. [@ShouYan2004]. Examples of such intervals for $\alpha=0.05$ are reported in Fig. \[fig:significant\_coherence\]. Alongside these intervals denoted $[\hat{a}_{\alpha/2},\hat{a}_{1-\alpha/2}]$, we declare that the coherence at a frequency is significant if the interval excludes zero. For this particular triple, we note what appears to be significant coherent beaconing across the devices at multiples of approx. 0.0017Hz (a periodicity of 10 minutes). ![The coherence as plotted in the bottom of Fig. \[fig:Example\_spectra\] with two-sided 95% confidence intervals. Frequencies where the confidence interval excludes zero are highlighted in red. \[fig:significant\_coherence\]](images/significant_coherence_ep0){width="1\columnwidth"} To assess variation in coherence estimates across the $N_{\mathrm{triple}}=500$ edge-pairs under study we attempt to cluster the resultant coherence estimates. Prior to performing clustering, we threshold coherence estimates according to the confidence intervals such that $\hat{R}^{*}(\omega_{q})=0$ if $0\in[\hat{a}_{\alpha/2},\hat{a}_{1-\alpha/2}]$ and $\hat{R}^{*}(\omega_{q})=\hat{R}(\omega_{q})$ otherwise. The resultant coherence estimates are then modelled as a *Gaussian mixture model (GMM)*, such that $[\hat{R}^*(\omega_{1}),\ldots,\hat{R}^*(\omega_{N_f})]\sim\mathrm{GMM}(\{\boldsymbol{\mu}_{c},\boldsymbol{\Sigma}_{c}\}_{c=1}^{C})$ where $\boldsymbol{\mu}_{c}\in\mathbb{R}^{N_{f}}$ represent cluster means and $\boldsymbol{\Sigma}_{c}$ the cluster covariances. While the Gaussian assumption of the above model contrasts with the Goodman asymptotic distribution for the coherence, the approximation may still hold relevance. For instance, Enochson and Goodman [@Enochson1965] demonstrated that a Gaussian approximation may be effective when calculating confidence intervals. In this example we use the MATLAB implementation of the expectation-maximisation with covariance regularisation set at $\lambda=0.001$. Due to the many local-minima that may be obtained when fitting a GMM, we perform one thousand replications with random initialisation and report the clustering with the largest likelihood. Figure \[fig:K-GMM\] presents the resulting mean profiles of $C=4$ clusters alongside the standard-deviation obtained from the estimated covariance matrices. ![GMM clustering of the significance thresholded coherence estimates with $C=4$. Shaded regions indicate points within one standard-deviation from the cluster mean. Bottom: graph of edges relating to each cluster $c=1,2,3,4$ from left to right.\[fig:K-GMM\]](images/gmm_1000_k4){width="1\columnwidth"} Discussion ========== The clustering results are insightful in that the emergence of clusters $c=3,4$ partially confirm our initial hypothesis that many device pairs exhibit little dependency. Out of the initial 500 edge-pairs 90 are placed into clusters with non-negliable coherence, for reference, the example demonstrated in Fig. \[fig:Example\_spectra\] is placed into cluster $c=1$. These active coherent clusters exhibit pronounced structure across multiple frequencies, once again providing evidence that modelling auto-covariance within network traffic is important. Of some note, is the clear peak at $f=0.018Hz$ corresponding to periodicity of around $57$s . Without a more intimate knowledge of device functionality on the network we can only hypothesise the cause of such a feature, but it is possible this is due to beaconing activities. Interestingly, Heard et al. [@Heard2014] demonstrate strong periodicity at this same frequency for devices connecting with dropbox.com. While most beaconing activities are benign, scanning techniques used by intruders may also create similar strongly periodic activity and it is thus of interest to administrators to detect such patterns. As a direction for future work, it is possible the methods developed here to detect association between event driven data-streams in a defensive context, may also be used as a form of correlation attack, for example to break anonymisation protocols when traffic is transmitted via mixing devices [@Murdoch2007]. It may also be of interest to relax the stationarity assumptions of this work, for instance within a wavelet framework. Indeed an algorithm that could derive the coherence in a streaming manner would be an important step towards building a practical anomaly detector. [^1]: Funded by EPSRC grant EP/P011535/1

--- abstract: | Our previous analyses of radio Doppler and ranging data from distant spacecraft in the solar system indicated that an apparent anomalous acceleration is acting on Pioneer 10 and 11, with a magnitude $a_P\sim 8\times 10^{-8}$ cm/s$^2$, directed towards the Sun. Much effort has been expended looking for possible systematic origins of the residuals, but none has been found. A detailed investigation of effects both external to and internal to the spacecraft, as well as those due to modeling and computational techniques, is provided. We also discuss the methods, theoretical models, and experimental techniques used to detect and study small forces acting on interplanetary spacecraft. These include the methods of radio Doppler data collection, data editing, and data reduction. There is now further data for the Pioneer 10 orbit determination. The extended Pioneer 10 data set spans 3 January 1987 to 22 July 1998. \[For Pioneer 11 the shorter span goes from 5 January 1987 to the time of loss of coherent data on 1 October 1990.\] With these data sets and more detailed studies of all the systematics, we now give a result, of $a_P = (8.74 \pm 1.33) \times 10^{-8} ~~{\rm cm/s}^2$. (Annual/diurnal variations on top of $a_P$, that leave $a_P$ unchanged, are also reported and discussed.) author: - | [John D. Anderson]{},[^1]$^a$ [Philip A. Laing]{},[^2]$^b$ [Eunice L. Lau]{},[^3]$^a$\ [Anthony S. Liu]{},[^4]$^c$ [Michael Martin Nieto]{},[^5]$^d$ and [Slava G. Turyshev]{}[^6]$^a$\      date: 11 April 2002 title: Study of the anomalous acceleration of Pioneer 10 and 11 --- \[intro\]INTRODUCTION ===================== Some thirty years ago, on 2 March 1972, Pioneer 10 was launched on an Atlas/Centaur rocket from Cape Canaveral. Pioneer 10 was Earth’s first space probe to an outer planet. Surviving intense radiation, it successfully encountered Jupiter on 4 December 1973 [@science]-[@pioweb]. In trail-blazing the exploration of the outer solar system, Pioneer 10 paved the way for, among others, Pioneer 11 (launched on 5 April 1973), the Voyagers, Galileo, Ulysses, and the upcoming Cassini encounter with Saturn. After Jupiter and (for Pioneer 11) Saturn encounters, the two spacecraft followed hyperbolic orbits near the plane of the ecliptic to opposite sides of the solar system. Pioneer 10 was also the first mission to enter the edge of interstellar space. That major event occurred in June 1983, when Pioneer 10 became the first spacecraft to “leave the solar system” as it passed beyond the orbit of the farthest known planet. The scientific data collected by Pioneer 10/11 has yielded unique information about the outer region of the solar system. This is due in part to the spin-stabilization of the Pioneer spacecraft. At launch they were spinning at approximately 4.28 and 7.8 revolutions per minute (rpm), respectively, with the spin axes running through the centers of the dish antennae. Their spin-stabilizations and great distances from the Earth imply a minimum number of Earth-attitude reorientation maneuvers are required. This permits precise acceleration estimations, to the level of $10^{-8}$ cm/s$^2$ (single measurement accuracy averaged over 5 days). Contrariwise, a Voyager-type three-axis stabilized spacecraft is not well suited for a precise celestial mechanics experiment as its numerous attitude-control maneuvers can overwhelm the signal of a small external acceleration. In summary, Pioneer spacecraft represent an ideal system to perform precision celestial mechanics experiments. It is relatively easy to model the spacecraft’s behavior and, therefore, to study small forces affecting its motion in the dynamical environment of the solar system. Indeed, one of the main objectives of the Pioneer extended missions (post Jupiter/Saturn encounters) [@extended] was to perform accurate celestial mechanics experiments. For instance, an attempt was made to detect the presence of small bodies in the solar system, primarily in the Kuiper belt. It was hoped that a small perturbation of the spacecraft’s trajectory would reveal the presence of these objects [@jdakuiper]-[@pulsar]. Furthermore, due to extremely precise navigation and a high quality tracking data, the Pioneer 10 scientific program also included a search for low frequency gravitational waves [@anderson85; @anderson93]. Beginning in 1980, when at a distance of 20 astronomical units (AU) from the Sun the solar-radiation-pressure acceleration on Pioneer 10 [*away*]{} from the Sun had decreased to $< 5 \times 10^{-8}$ cm/s$^2$, we found that the largest systematic error in the acceleration residuals was a constant bias, $a_P$, directed [*toward*]{} the Sun. Such anomalous data have been continuously received ever since. Jet Propulsion Laboratory (JPL) and The Aerospace Corporation produced independent orbit determination analyses of the Pioneer data extending up to July 1998. We ultimately concluded [@anderson; @moriond], that there is an unmodeled acceleration, $a_P$, towards the Sun of $\sim 8 \times 10^{-8}$ cm/s$^2$ for both Pioneer 10 and Pioneer 11. The purpose of this paper is to present a detailed explanation of the analysis of the apparent anomalous, weak, long-range acceleration of the Pioneer spacecraft that we detected in the outer regions of the solar system. We attempt to survey all sensible forces and to estimate their contributions to the anomalous acceleration. We will discuss the effects of these small non-gravitational forces (both generated on-board and external to the vehicle) on the motion of the distant spacecraft together with the methods used to collect and process the radio Doppler navigational data. We begin with descriptions of the spacecraft and other systems and the strategies for obtaining and analyzing information from them. In Section \[pioneer\] we describe the Pioneer (and other) spacecraft. We provide the reader with important technical information on the spacecraft, much of which is not easily accessible. In Section \[Exp\_tech\] we describe how raw data is obtained and analyzed and in Section \[navigate\] we discuss the basic elements of a theoretical foundation for spacecraft navigation in the solar system. The next major part of this manuscript is a description and analysis of the results of this investigation. We first describe how the anomalous acceleration was originally identified from the data of all the spacecraft in Section \[results\] [@anderson; @moriond]. We then give our recent results in Section \[recent\_results\]. In the following three sections we discuss possible experimental systematic origins for the signal. These include systematics generated by physical phenomena from sources external to (Section \[ext-systema\]) and internal to (Section \[int-systema\]) the spacecraft. This is followed by Section \[Int\_accuracy\], where the accuracy of the solution for $a_P$ is discussed. In the process we go over possible numerical/calculational errors/systematics. Sections \[ext-systema\]-\[Int\_accuracy\] are then summarized in the total error budget of Section \[budget\]. We end our presentation by first considering possible unexpected physical origins for the anomaly (Section \[newphys\]). In our conclusion, Section \[disc\], we summarize our results and suggest venues for further study of the discovered anomaly. \[pioneer\]THE PIONEER AND OTHER SPACECRAFT =========================================== In this section we describe in some detail the Pioneer 10 and 11 spacecraft and their missions. We concentrate on those spacecraft systems that play important roles in maintaining the continued function of the vehicles and in determining their dynamical behavior in the solar system. Specifically we present an overview of propulsion and attitude control systems, as well as thermal and communication systems. Since our analysis addresses certain results from the Galileo and Ulysses missions, we also give short descriptions of these missions in the final subsection. General description of the Pioneer spacecraft {#sec:pio_description} --------------------------------------------- Although some of the more precise details are often difficult to uncover, the general parameters of the Pioneer spacecraft are known and well documented [@science]-[@pioweb]. The two spacecraft are identical in design [@design]. At launch each had a “weight” (mass) of 259 kg. The “dry weight” of the total module was 223 kg as there were 36 kg of hydrazine propellant [@mass; @gasuse]. The spacecraft were designed to fit within the three meter diameter shroud of an added third stage to the Atlas/Centaur launch vehicle. Each spacecraft is 2.9 m long from its base to its cone-shaped medium-gain antenna. The high gain antenna (HGA) is made of aluminum honeycomb sandwich material. It is 2.74 m in diameter and 46 cm deep in the shape of a parabolic dish. (See Figures \[fig:pio\_design\] and \[fig:trusters\].) -10pt The main equipment compartment is 36 cm deep. The hexagonal flat top and bottom have 71 cm long sides. The equipment compartment provides a thermally controlled environment for scientific instruments. Two three-rod trusses, 120 degrees apart, project from two sides of the equipment compartment. At their ends, each holds two SNAP-19 (Space Nuclear Auxiliary Power, model 19) RTGs (Radioisotope Thermoelectric Generators) built by Teledyne Isotopes for the Atomic Energy Commission. These RTGs are situated about 3 m from the center of the spacecraft and generate its electric power. \[We will go into more detail on the RTGs in Section \[int-systema\].\] A third single-rod boom, 120 degrees from the other two, positions a magnetometer about 6.6 m from the spacecraft’s center. All three booms were extended after launch. With the mass of the magnetometer being 5 kg and the mass of each of the four RTGs being 13.6 kg, this configuration defines the main moment of inertia along the $z$-spin-axis. It is about ${\cal I}_{\tt z} \approx 588.3$ kg m$^2$ [@vanallen]. \[Observe that this all left only about 164 kg for the main bus and superstructure, including the antenna.\] Figures \[fig:pio\_design\] and \[fig:trusters\] show the arrangement within the spacecraft equipment compartment. The majority of the spacecraft electrical assemblies are located in the central hexagonal portion of the compartment, surrounding a 16.5-inch-diameter spherical hydrazine tank. Most of the scientific instruments’ electronic units and internally-mounted sensors are in an instrument bay (“squashed” hexagon) mounted on one side of the central hexagon. The equipment compartment is in an aluminum honeycomb structure. This provides support and meteoroid protection. It is covered with insulation which, together with louvers under the platform, provides passive thermal control. \[An exception is from off-on control by thermal power dissipation of some subsystems. (See Sec. \[int-systema\]).\] Propulsion and attitude control systems {#sec:prop} --------------------------------------- Three pairs of these rocket thrusters near the rim of the HGA provide a threefold function of spin-axis precession, mid-course trajectory correction, and spin control. Each of the three thruster pairs develops its repulsive jet force from a catalytic decomposition of liquid hydrazine in a small rocket thrust chamber attached to the oppositely-directed nozzle. The resulted hot gas is then expended through six individually controlled thruster nozzles to effect spacecraft maneuvers. The spacecraft is attitude-stabilized by spinning about an axis which is parallel to the axis of the HGA. The nominal spin rate for Pioneer 10 is 4.8 rpm. Pioneer 11 spins at approximately 7.8 rpm because a spin-controlling thruster malfunctioned during the spin-down shortly after launch. \[Because of the danger that the thruster’s valve would not be able to close again, this particular thruster has not been used since.\] During the mission an Earth-pointing attitude is required to illuminate the Earth with the narrow-beam HGA. Periodic attitude adjustments are required throughout the mission to compensate for the variation in the heliocentric longitude of the Earth-spacecraft line. \[In addition, correction of launch vehicle injection errors were required to provide the desired Jupiter encounter trajectory and Saturn (for Pioneer 11) encounter trajectory.\] These velocity vector adjustments involved reorienting the spacecraft to direct the thrust in the desired direction. There were no anomalies in the engineering telemetry from the propulsion system, for either spacecraft, during any mission phase from launch to termination of the Pioneer mission in March 1997. From the viewpoint of mission operations at the NASA/Ames control center, the propulsion system performed as expected, with no catastrophic or long-term pressure drops in the propulsion tank. Except for the above-mentioned Pioneer 11 spin-thruster incident, there was no malfunction of the propulsion nozzles, which were only opened every few months by ground command. The fact that pressure was maintained in the tank has been used to infer that no impacts by Kuiper belt objects occurred, and a limit has been placed on the size and density distribution of such objects [@jdakuiper], another useful scientific result. For attitude control, a star sensor (referenced to Canopus) and two sunlight sensors provided reference for orientation and roll maneuvers. The star sensor on Pioneer 10 became inoperative at Jupiter encounter, so the sun sensors were used after that. For Pioneer 10, spin calibration was done by the DSN until 17 July 1990. From 1990 to 1993 determinations were made by analysts using data from the Imaging Photo Polarimeter (IPP). After the 6 July 1993 maneuver, there was not enough power left to support the IPP. But approximately every six months analysts still could get a rough determination using information obtained from conscan maneuvers [@conscan] on an uplink signal. When using conscan, the high gain feed is off-set. Thruster firings are used to spiral in to the correct pointing of the spacecraft antenna to give the maximum signal strength. To run this procedure (conscan and attitude) it is now necessary to turn off the traveling-wave-tube (TWT) amplifier. So far, the power and tube life-cycle have worked and the Jet Propulsion Laboratory’s (JPL) Deep Space Network (DSN) has been able to reacquire the signal. It takes about 15 minutes or so to do a maneuver. \[The magnetometer boom incorporates a hinged, viscous, damping mechanism at its attachment point, for passive nutation control.\] In the extended mission phase, after Jupiter and Saturn encounters, the thrusters have been used for precession maneuvers only. Two pairs of thrusters at opposite sides of the spacecraft have nozzles directed along the spin axis, fore and aft (See Figure \[fig:trusters\].) In precession mode, the thrusters are fired by opening one nozzle in each pair. One fires to the front and the other fires to the rear of the spacecraft [@rearfront], in brief thrust pulses. Each thrust pulse precesses the spin axis a few tenths of a degree until the desired attitude is reached. The two nozzles of the third thruster pair, no longer in use, are aligned tangentially to the antenna rim. One points in the direction opposite to its (rotating) velocity vector and the other with it. These were used for spin control. Thermal system and on-board power {#sec:onboard} --------------------------------- Early on the spacecraft instrument compartment is thermally controlled between $\approx$ $0$ F and 90 F. This is done with the aid of thermo-responsive louvers located at the bottom of the equipment compartment. These louvers are adjusted by bi-metallic springs. They are completely closed below $\sim40$ F and completely open above $\sim 85$ F. This allows controlled heat to escape in the equipment compartment. Equipment is kept within an operational range of temperatures by multi-layered blankets of insulating aluminum plastic. Heat is provided by electric heaters, the heat from the instruments themselves, and by twelve one-watt radioisotope heaters powered directly by non-fissionable plutonium ($^{238}_{~94}$Pu$ \rightarrow ^{234}_{~92}$U$+{}^4_2$He). $^{238}$Pu, with a half life time of 87.74 years, also provides the thermal source for the thermoelectric devices in the RTGs. Before launch, each spacecraft’s four RTGs delivered a total of approximately 160 W of electrical power [@tele; @Rconf]. Each of the four space-proven SNAP-19 RTGs converts 5 to 6 percent of the heat released from plutonium dioxide fuel to electric power. RTG power is greatest at 4.2 Volts; an inverter boosts this to 28 Volts for distribution. RTG life is degraded at low currents; therefore, voltage is regulated by shunt dissipation of excess power. The power subsystem controls and regulates the RTG power output with shunts, supports the spacecraft load, and performs battery load-sharing. The silver cadmium battery consists of eight cells of 5 ampere-hours capacity each. It supplies pulse loads in excess of RTG capability and may be used for sharing peak loads. The battery voltage is often discharged and charged. This can be seen by telemetry of the battery discharge current and charge current At launch each RTG supplied about 40 W to the input of the $\sim 4.2$ V Inverter Assemblies. (The output for other uses includes the DC bus at 28 V and the AC bus at 61 V) Even though electrical power degrades with time (see Section \[subsec:mainbus\]), at $-41$ F the essential platform temperature as of the year 2000 is still between the acceptable limits of $-63$ F to 180 F. The RF power output from the traveling-wave-tube amplifier is still operating normally. The equipment compartment is insulated from extreme heat influx with aluminized mylar and kapton blankets. Adequate warmth is provided by dissipation of 70 to 120 watts of electrical power by electronic units within the compartment; louvers regulating the release of this heat below the mounting platform maintain temperatures in the vicinity of the spacecraft equipment and scientific instruments within operating limits. External component temperatures are controlled, where necessary, by appropriate coating and, in some cases, by radioisotope or electrical heaters. The energy production from the radioactive decay obeys an exponential law. Hence, 29 years after launch, the radiation from Pioneer 10’s RTGs was about 80 percent of its original intensity. However the electrical power delivered to the equipment compartment has decayed at a faster rate than the $^{238}$Pu decays radioactively. Specifically, the electrical power first decayed very quickly and then slowed to a still fast linear decay [@lasher]. By 1987 the degradation rate was about $-2.6$ W/yr for Pioneer 10 and even greater for the sister spacecraft. This fast depletion rate of electrical power from the RTGs is caused by normal deterioration of the thermocouple junctions in the thermoelectric devices. The spacecraft needs 100 W to power all systems, including 26 W for the science instruments. Previously, when the available electrical power was greater than 100 W, the excess power was either thermally radiated into space by a shunt-resistor radiator or it was used to charge a battery in the equipment compartment. At present only about 65 W of power is available to Pioneer 10 [@theorypower]. Therefore, all the instruments are no longer able to operate simultaneously. But the power subsystem continues to provide sufficient power to support the current spacecraft load: transmitter, receiver, command and data handling, and the Geiger Tube Telescope (GTT) science instrument. As pointed out in Sec. \[subs:pioneer\], the science package and transmitter are turned off in extended cruise mode to provide enough power to fire the attitude control thrusters. Communication system -------------------- The Pioneer 10/11 communication systems use S-band ($\lambda\simeq 13$ cm) Doppler frequencies [@sband]. The communication uplink from Earth is at approximately 2.11 GHz. The two spacecraft transmit continuously at a power of eight watts. They beam their signals, of approximate frequency 2.29 GHz, to Earth by means of the parabolic 2.74 m high-gain antenna. Phase coherency with the ground transmitters, referenced to H-maser frequency standards, is maintained by means of an S-band transponder with the 240/221 frequency turnaround ratio (as indicated by the values of the above mentioned frequencies). The communications subsystem provides for: i) up-link and down-link communications; ii) Doppler coherence of the down-link carrier signal; and iii) generation of the conscan [@conscan] signal for closed loop precession of the spacecraft spin axis towards Earth. S-band carrier frequencies, compatible with DSN, are used in conjunction with a telemetry modulation of the down-link signal. The high-gain antenna is used to maximize the telemetry data rate at extreme ranges. The coupled medium-gain/omni-directional antenna with fore and aft elements respectively, provided broad-angle communications at intermediate and short ranges. For DSN acquisition, these three antennae radiate a non-coherent RF signal, and for Doppler tracking, there is a phase coherent mode with a frequency translation ratio of 240/221. Two frequency-addressable phase-lock receivers are connected to the two antenna systems through a ground-commanded transfer switch and two diplexers, providing access to the spacecraft via either signal path. The receivers and antennae are interchangeable through the transfer switch by ground command or automatically, if needed. There is a redundancy in the communication systems, with two receivers and two transmitters coupled to two traveling-wave-tube amplifiers. Only one of the two redundant systems has been used for the extended missions, however. At launch, communication with the spacecraft was at a data rate 256 bps for Pioneer 10 (1024 bps for Pioneer 11). Data rate degradation has been $-1.27$ mbps/day for Pioneer 10 ($-8.78$ mbps/day for Pioneer 11). The DSN still continues to provide good data with the received signal strength of about $-178$ dBm (only a few dB from the receiver threshold). The data signal to noise ratio is still mainly under 0.5 dB. The data deletion rate is often between 0 and 50 percent, at times more. However, during the test of 11 March 2000, the average deletion rate was about 8 percent. So, quality data are still available. Status of the extended mission {#subs:pioneer} ------------------------------ The Pioneer 10 mission officially ended on 31 March 1997 when it was at a distance of 67 AU from the Sun. (See Figure \[fig:pioneer\_path\].) At a now nearly constant velocity relative to the Sun of $\sim$12.2 km/s, Pioneer 10 will continue its motion into interstellar space, heading generally for the red star Aldebaran, which forms the eye of Taurus (The Bull) Constellation. Aldebaran is about 68 light years away and it would be expected to take Pioneer 10 over 2 million years to reach its neighborhood. A switch failure in the Pioneer 11 radio system on 1 October 1990 disabled the generation of coherent Doppler signals. So, after that date, when the spacecraft was $\sim 30$ AU away from the Sun, no useful data have been generated for our scientific investigation. Furthermore, by September 1995, its power source was nearly exhausted. Pioneer 11 could no longer make any scientific observations, and routine mission operations were terminated. The last communication from Pioneer 11 was received in November 1995, when the spacecraft was at distance of $\sim 40$ AU from the Sun. (The relative Earth motion carried it out of view of the spacecraft antenna.) The spacecraft is headed toward the constellation of Aquila (The Eagle), northwest of the constellation of Sagittarius, with a velocity relative to the Sun of $\sim$11.6 km/s Pioneer 11 should pass close to the nearest star in the constellation Aquila in about 4 million years [@pioweb]. (Pioneer 10 and 11 orbital parameters are given in the Appendix.) However, after mission termination the Pioneer 10 radio system was still operating in the coherent mode when commanded to do so from the Pioneer Mission Operations center at the NASA Ames Research Center (ARC). As a result, after 31 March 1997, JPL’s DSN was still able to deliver high-quality coherent data to us on a regular schedule from distances beyond 67 AU. Recently, support of the Pioneer spacecraft has been on a non-interference basis to other NASA projects. It was used for the purpose of training Lunar Prospector controllers in DSN coordination of tracking activities. Under this training program, ARC has been able to maintain contact with Pioneer 10. This has required careful attention to the DSN’s ground system, including the installation of advanced instrumentation, such as low-noise digital receivers. This extended the lifetime of Pioneer 10 to the present. \[Note that the DSN’s early estimates, based on instrumentation in place in 1976, predicted that radio contact would be lost about 1980.\] At the present time it is mainly the drift of the spacecraft relative to the solar velocity that necessitates maneuvers to continue keeping Pioneer 10 pointed towards the Earth. The latest successful precession maneuver to point the spacecraft to Earth was accomplished on 11 February 2000, when Pioneer 10 was at a distance from the Sun of 75 AU. \[The distance from the Earth was $\sim 76$ AU with a corresponding round-trip light time of about 21 hour.\] The signal level increased 0.5-0.75 dBm [@dBm] as a result of the maneuver. This was the seventh successful maneuver that has been done in the blind since 26 January 1997. At that time it had been determined that the electrical power to the spacecraft had degraded to the point where the spacecraft transmitter had to be turned off to have enough power to perform the maneuver. After 90 minutes in the blind the transmitter was turned back on again. So, despite the continued weakening of Pioneer 10’s signal, radio Doppler measurements were still available. The next attempt at a maneuver, on 8 July 2000, turned out in the end to be successful. Signal was tracked on 9 July 2001. Contact was reestablished on the 30th anniversary of launch, 2 March 2002. The Galileo and Ulysses missions and spacecraft {#othercraft} ----------------------------------------------- ### The Galileo mission {#galileocraft} The Galileo mission to explore the Jovian system [@johnson] was launched 18 October 1989 aboard the Space Shuttle Discovery. Due to insufficient launch power to reach its final destination at 5.2 AU, a trajectory was chosen with planetary flybys to gain gravity assists. The spacecraft flew by Venus on 10 February 1990 and twice by the Earth, on 8 December 1990 and on 8 December 1992. The current Galileo Millennium Mission continues to study Jupiter and its moons, and coordinated observations with the Cassini flyby in December 2000. The dynamical properties of the Galileo spacecraft are very well known. At launch the orbiter had a mass of 2,223 kg. This included 925 kg of usable propellant, meaning over 40% of the orbiter’s mass at launch was for propellant! The science payload was 118 kg and the probe’s total mass was 339 kg. Of this latter, the probe descent module was 121 kg, including a 30 kg science payload. The tensor of inertia of the spacecraft had the following components at launch: $J_{\tt xx}= 4454.7, J_{\tt yy}= 4061.2, J_{\tt zz}= 5967.6, J_{\tt xy}= -52.9, J_{\tt xz}= 3.21, J_{\tt yz}= -15.94$ in units of kg m$^2$. Based on the area of the sun-shade plus the booms and the RTGs we obtained a maximal cross-sectional area of 19.5 m$^2$. Each of the two of the Galileo’s RTGs at launch delivered of 285 W of electric power to the subsystems. Unlike previous planetary spacecraft, Galileo featured an innovative “dual spin” design: part of the orbiter would rotate constantly at about three rpm and part of the spacecraft would remain fixed in (solar system) inertial space. This means that the orbiter could easily accommodate magnetospheric experiments (which need to made while the spacecraft is sweeping) while also providing stability and a fixed orientation for cameras and other sensors. The spin rate could be increased to 10 revolutions per minute for additional stability during major propulsive maneuvers. Apparently there was a mechanical problem between the spinning and non-spinning sections. Because of this, the project decided to often use an all-spinning mode, of about 3.15 rpm. This was especially true close to the Jupiter Orbit Insertion (JOI), when the entire spacecraft was spinning (with a slower rate, of course). Galileo’s original design called for a deployable high-gain antenna (HGA) to unfurl. It would provide approximately 34 dB of gain at X-band (10 GHz) for a 134 kbps downlink of science and priority engineering data. However, the X-band HGA failed to unfurl on 11 April 1991. When it again did not deploy following the Earth fly-by in 1992, the spacecraft was reconfigured to utilize the S-band, 8 dB, omni-directional low-gain antenna (LGA) for downlink. The S-band frequencies are 2.113 GHz - up and 2.295 GHz - down, a conversion factor of 240/221 at the Doppler frequency transponder. This configuration yielded much lower data rates than originally scheduled, 8-16 bps through JOI [@LGA]. Enhancements at the DSN and reprogramming the flight computers on Galileo increased telemetry bit rate to 8-160 bps, starting in the spring of 1996. Currently, two types of Galileo navigation data are available, namely Doppler and range measurements. As mentioned before, an instantaneous comparison between the ranging signal that goes up with the ranging signal that comes down would yield an “instantaneous” two-way range delay. Unfortunately, an instantaneous comparison was not possible in this case. The reason is that the signal-to-noise ratio on the incoming ranging signal is small and a long integration time (typically minutes) must be used (for correlation purposes). During such long integration times, the range to the spacecraft is constantly changing. It is therefore necessary to “electronically freeze” the range delay long enough to permit an integration to be performed. The result represents the range at the moment of freezing [@anderson75; @Kinman92]. ### The Ulysses mission {#ulyssescraft} Ulysses was launched on 6 October 1990, also from the Space Shuttle Discovery, as a cooperative project of NASA and the European Space Agency (ESA). JPL manages the US portion of the mission for NASA’s Office of Space Science. Ulysses’ objective was to characterize the heliosphere as a function of solar latitude [@genU]. To reach high solar latitudes, its voyage took it to Jupiter on 8 February 1992. As a result, its orbit plane was rotated about 80 degrees out of the ecliptic plane. Ulysses explored the heliosphere over the Sun’s south pole between June and November, 1994, reaching maximum Southern latitude of 80.2 degrees on 13 September 1994. It continued in its orbit out of the plane of the ecliptic, passing perihelion in March 1995 and over the north solar pole between June and September 1995. It returned again to the Sun’s south polar region in late 2000. The total mass at launch was the sum of two parts: a dry mass of 333.5 kg plus a propellant mass of 33.5 kg. The tensor of inertia is given by its principal components $J_{\tt xx} =371.62, J_{\tt yy} = 205.51, J_{\tt zz} = 534.98$ in units kg m$^2$. The maximal cross section is estimated to be 10.056 m$^2$. This estimation is based on the radius of the antenna 1.65 m (8.556 m$^2$) plus the areas of the RTGs and part of the science compartment (yielding an additional $\approx$ 1.5 m$^2$). The spacecraft was spin-stabilized at 4.996 rpm. The electrical power is generated by modern RTGs, which are located much closer to the main bus than are those of the Pioneers. The power generated at launch was 285 W. Communications with the spacecraft are performed at X-band (for downlink at 20 W with a conversion factor of 880/221) and S-band (both for uplink 2111.607 MHz and downlink 2293.148 MHz, at 5 W with a conversion factor of 240/221). Currently both Doppler and range data are available for both frequency bands. While the main communication link is S-up/X-down, the S-down link was used only for radio-science purposes. Because of Ulysses’ closeness to the Sun and also because of its construction, any hope to model Ulysses for small forces might appear to be doomed by solar radiation pressure and internal heat radiation from the RTGs. However, because the Doppler signal direction is towards the Earth while the radiation pressure varies with distance and has a direction parallel the Sun-Ulysses line, in principle these effects could be separated. And again, there was range data. This all would make it easier to model non-gravitational acceleration components normal to the line of sight, which usually are poorly and not significantly determined. The Ulysses spacecraft spins at $\sim 5$ rpm around its antenna axis (4.996 rpm initially). The angle of the spin axis with respect to the spacecraft-Sun line varies from near zero at Jupiter to near 50 degrees at perihelion. Any on-board forces that could perturb the spacecraft trajectory are restricted to a direction along the spin axis. \[The other two components are canceled out by the spin.\] As the spacecraft and the Earth travel around the Sun, the direction from the spacecraft to the Earth changes continuously. Regular changes of the attitude of the spacecraft are performed throughout the mission to keep the Earth within the narrow beam of about one degree full width of the spacecraft–fixed parabolic antenna. \[Exp\_tech\]DATA ACQUISITION AND PREPARATION ============================================= Discussions of radio-science experiments with spacecraft in the solar system requires at least a general knowledge of the sophisticated experimental techniques used at the DSN complex. Since its beginning in 1958 the DSN complex has undergone a number of major upgrades and additions. This was necessitated by the needs of particular space missions. \[The last such upgrade was conducted for the Cassini mission when the DSN capabilities were extended to cover the Ka radio frequency bandwidth. For more information on DSN methods, techniques, and present capabilities, see [@dsn].\] For the purposes of the present analysis one will need a general knowledge of the methods and techniques implemented in the radio-science subsystem of the DSN complex. This section reviews the techniques that are used to obtain the radio tracking data from which, after analysis, results are generated. Here we will briefly discuss the DSN hardware that plays a pivotal role for our study of the anomalous acceleration. Data acquisition ---------------- The Deep Space Network (DSN) is the network of ground stations that are employed to track interplanetary spacecraft [@dsn; @dsn82]. There are three ground DSN complexes, at Goldstone, California, at Robledo de Chavela, outside Madrid, Spain, and at Tidbinbilla, outside Canberra, Australia. There are many antennae, both existing and decommissioned, that have been used by the DSN for spacecraft navigation. For our four spacecraft (Pioneer 10, 11, Galileo, and Ulysses), depending on the time period involved, the following Deep Space Station (DSS) antennae were among those used: (DSS 12, 14, 24) at the California antenna complex; (DSS 42, 43, 45, 46) at the Australia complex; and (DSS 54, 61, 62, 63) at the Spain complex. Specifically, the Pioneers used (DSS 12, 14, 42, 43, 62, 63), Galileo used (DSS 12, 14, 42, 43, 63), and Ulysses used (DSS 12, 14, 24, 42, 43, 46, 54, 61, 63). The DSN tracking system is a phase coherent system. By this we mean that an “exact” ratio exists between the transmission and reception frequencies; i.e., 240/221 for S-band or 880/221 for X-band [@sband]. (This is in distinction to the usual concept of coherent radiation used in atomic and astrophysics.) Frequency is an average frequency, defined as the number of cycles per unit time. Thus, accumulated phase is the integral of frequency. High measurement precision is attained by maintaining the frequency accuracy to 1 part per $10^{12}$ or better (This is in agreement with the expected Allan deviation for the S-band signals.) [ **The DSN Frequency and Timing System (FTS):  **]{} The DSN’s FTS is the source for the high accuracy just mentioned (see Figure \[fig:dsn\_block\]). At its center is an hydrogen maser that produces a precise and stable reference frequency [@barnes; @vessot74]. These devices have Allan deviations [@SFJ98] of approximately $3\times 10^{-15}$ to $1\times 10^{-15}$ for integration times of $10^2$ to $10^3$ seconds, respectively. -10pt These masers are good enough so that the quality of Doppler-measurement data is limited by thermal or plasma noise, and not by the inherent instability of the frequency references. Due to the extreme accuracy of the hydrogen masers, one can very precisely characterize the spacecraft’s dynamical variables using Doppler and range techniques. The FTS generates a 5 MHz and 10 MHz reference frequency which is sent through the local area network to the Digitally Controlled Oscillator (DCO). [**The Digitally Controlled Oscillator (DCO) and Exciter: **]{} Using the highly stable output from the FTS, the DCO, through digitally controlled frequency multipliers, generates the Track Synthesizer Frequency (TSF) of $\sim 22$ MHz. This is then sent to the Exciter Assembly. The Exciter Assembly multiplies the TSF by 96 to produce the S-band carrier signal at $\sim 2.2$ GHz. The signal power is amplified by Traveling Wave Tubes (TWT) for transmission. If ranging data are required, the Exciter Assembly adds the ranging modulation to the carrier. \[The DSN tracking system has undergone many upgrades during the 29 years of tracking Pioneer 10. During this period internal frequencies have changed.\] This S-band frequency is sent to the antenna where it is amplified and transmitted to the spacecraft. The onboard receiver tracks the up-link carrier using a phase lock loop. To ensure that the reception signal does not interfere with the transmission, the spacecraft (e.g., Pioneer) has a turnaround transponder with a ratio of 240/221. The spacecraft transmitter’s local oscillator is phase locked to the up-link carrier. It multiplies the received frequency by the above ratio and then re-transmits the signal to Earth. [**Receiver and Doppler Extractor: **]{} When the two-way [@way] signal reaches the ground, the receiver locks on to the signal and tunes the Voltage Control Oscillator (VCO) to null out the phase error. The signal is sent to the Doppler Extractor. At the Doppler Extractor the current transmitter signal from the Exciter is multiplied by 240/221 (or 880/241 for X-band)) and a bias, of 1 MHz for S-band or 5 MHz for X-band [@sband], is added to the Doppler. The Doppler data is no longer modulated at S-band but has been reduced as a consequence of the bias to an intermediate frequency of 1 or 5 MHz Since the light travel time to and from Pioneer 10 is long (more than 20 hours), the transmitted frequency and the current transmitted frequency can be different. The difference in frequencies are recorded separately and are accounted for in the orbit determination programs we discuss in Section \[results\]. [**Metric Data Assembly (MDA): **]{} The MDA consists of computers and Doppler counters where continuous count Doppler data are generated. The intermediate frequency (IF) of 1 or 5 MHz with a Doppler modulation is sent to the Metric Data Assembly (MDA). From the FTS a 10 pulse per second signal is also sent to the MDA for timing. At the MDA, the IF and the resulting Doppler pulses are counted at a rate of 10 pulses per second. At each tenth of a second, the number of Doppler pulses are counted. A second counter begins at the instant the first counter stops. The result is continuously-counted Doppler data. (The Doppler data is a biased Doppler of 1 MHz, the bias later being removed by the analyst to obtain the true Doppler counts.) The Range data (if present) together with the Doppler data is sent separately to the Ranging Demodulation Assembly. The accompanying Doppler data is used to rate aid (i.e., to “freeze” the range signal) for demodulation and cross correlation. [**Data Communication: **]{} The total set of tracking data is sent by local area network to the communication center. From there it is transmitted to the Goddard Communication Facility via commercial phone lines or by government leased lines. It then goes to JPL’s Ground Communication Facility where it is received and recorded by the Data Records Subsystem. Radio Doppler and range techniques {#Dopp_tech} ---------------------------------- Various radio tracking strategies are available for determining the trajectory parameters of interplanetary spacecraft. However, radio tracking Doppler and range techniques are the most commonly used methods for navigational purposes. The position and velocities of the DSN tracking stations must be known to high accuracy. The transformation from a Earth fixed coordinate system to the International Earth Rotation Service (IERS) Celestial System is a complex series of rotations that includes precession, nutation, variations in the Earth’s rotation ([UT1-UTC]{}) and polar motion. Calculations of the motion of a spacecraft are made on the basis of the range time-delay and/or the Doppler shift in the signals. This type of data was used to determine the positions, the velocities, and the magnitudes of the orientation maneuvers for the Pioneer, Galileo, and Ulysses spacecraft considered in this study. Theoretical modeling of the group delays and phase delay rates are done with the orbit determination software we describe in the next section. [**Data types:**]{} Our data describes the observations that are the basis of the results of this paper. We receive our data from DSN in closed-loop mode, i.e., data that has been tracked with phase lock loop hardware. (Open loop data is tape recorded but not tracked by phase lock loop hardware.) The closed-loop data constitutes our Archival Tracking Data File (ATDF), which we copy [@datatapes] to the National Space Science Data Center (NSSDC) on magnetic tape. The ATDF files are stored on hard disk in the RMDC (Radio Metric Data Conditioning group) of JPL’s Navigation and Mission Design Section. We access these files and run standard software to produce an Orbit Data File for input into the orbit determination programs which we use. (See Section \[results\].) The data types are two-way and three-way [@way] Doppler and two-way range. (Doppler and range are defined in the following two subsections.) Due to unknown clock offsets between the stations, three-way range is generally not taken or used. The Pioneer spacecraft only have two- and three-way S-band [@sband] Doppler. Galileo also has S-band range data near the Earth. Ulysses has two- and three-way S-band up-link and X-band [@sband] down-link Doppler and range as well as S-band up-link and S-band down-link, although we have only processed the Ulysses S-band up-link and X-band down-link Doppler and range. ### Doppler experimental techniques and strategy {#sec:doppler} In Doppler experiments a radio signal transmitted from the Earth to the spacecraft is coherently transponded and sent back to the Earth. Its frequency change is measured with great precision, using the hydrogen masers at the DSN stations. The observable is the DSN frequency shift [@drift] $$\Delta \nu(t)={\nu_0}\,\frac{1}{c}\frac{d \ell}{dt}, \label{eq:doppler}$$ where $\ell$ is the overall optical distance (including diffraction effects) traversed by a photon in both directions. \[In the Pioneer Doppler experiments, the stability of the fractional drift at the S-band is on the order of $\Delta \nu/\nu_0\simeq10^{-12}$, for integration times on the order of $10^3$ s.\] Doppler measurements provide the “range rate” of the spacecraft and therefore are affected by all the dynamical phenomena in the volume between the Earth and the spacecraft. Expanding upon what was discussed in Section \[data-acquisition\], the received signal and the transmitter frequency (both are at S-band) as well as a 10 pulse per second timing reference from the FTS are fed to the Metric Data Assembly (MDA). There the Doppler phase (difference between transmitted and received phases plus an added bias) is counted. That is, digital counters at the MDA record the zero crossings of the difference (i.e., Doppler, or alternatively the beat frequency of the received frequency and the exciter frequency). After counting, the bias is removed so that the true phase is produced. The system produces “continuous count Doppler” and it uses two counters. Every tenth of a second, a Doppler phase count is recorded from one of the counters. The other counter continues the counts. The recording alternates between the two counters to maintain a continuous unbroken count. The Doppler counts are at 1 MHz for S-band or 5 MHz for X-band. The wavelength of each S-band cycle is about 13 cm. Dividers or “time resolvers” further subdivide the cycle into 256 parts, so that fractional cycles are measured with a resolution of 0.5 mm. This accuracy can only be maintained if the Doppler is continuously counted (no breaks in the count) and coherent frequency standards are kept throughout the pass. It should be noted that no error is accumulated in the phase count as long as lock is not lost. The only errors are the stability of the hydrogen maser and the resolution of the “resolver.” Consequently, the JPL Doppler records are not frequency measurements. Rather, they are digitally counted measurements of the Doppler phase difference between the transmitted and received S-band frequencies, divided by the count time. Therefore, the Doppler observables, we will refer to, have units of cycles per second or Hz. Since total count phase observables are Doppler observables multiplied by the count interval T$_c$, they have units of cycles. The Doppler integration time refers to the total counting of the elapsed periods of the wave with the reference frequency of the hydrogen maser. The usual Doppler integrating times for the Pioneer Doppler signals refers to the data sampled over intervals of 10 s, 60 s, 600 s, or 1980 s. ### Range measurements A range measurement is made by phase modulating a signal onto the up-link carrier and having it echoed by the transponder. The transponder demodulates this ranging signal, filters it, and then re-modulates it back onto the down-link carrier. At the ground station, this returned ranging signal is demodulated and filtered. An instantaneous comparison between the outbound ranging signal and the returning ranging signal that comes down would yield the two-way delay. Cross correlating the returned phase modulated signal with a ground duplicate yields the time delay. (See [@anderson75] and references therein.) As the range code is repeated over and over, an ambiguity can exist. The orbit determination programs are then used to infer (some times with great difficulty) the number of range codes that exist between a particular transmitted code and its own corresponding received code. Thus, the ranging data are independent of the Doppler data, which represents a frequency shift of the radio carrier wave without modulation. For example, solar plasma introduces a group delay in the ranging data but a phase advance in the Doppler data. Ranging data can also be used to distinguish an actual range change from a fictitious range change seen in Doppler data that is caused by a frequency error [@falsedop]. The Doppler frequency integrated over time (the accumulated phase) should equal the range change except for the difference introduced by charged particles ### Inferring position information from Doppler It is also possible to infer the position in the sky of a spacecraft from the Doppler data. This is accomplished by examining the diurnal variation imparted to the Doppler shift by the Earth’s rotation. As the ground station rotates underneath a spacecraft, the Doppler shift is modulated by a sinusoid. The sinusoid’s amplitude depends on the declination angle of the spacecraft and its phase depends upon the right ascension. These angles can therefore be estimated from a record of the Doppler shift that is (at least) of several days duration. This allows for a determination of the distance to the spacecraft through the dynamics of spacecraft motion using standard orbit theory contained in the orbit determination programs. Data preparation {#Data_edit} ---------------- In an ideal system, all scheduled observations would be used in determining parameters of physical interest. However, there are inevitable problems that occur in data collection and processing that corrupt the data. So, at various stages of the signal processing one must remove or “edit” corrupted data. Thus, the need arises for objective editing criteria. Procedures have been developed which attempt to excise corrupted data on the basis of objective criteria. There is always a temptation to eliminate data that is not well explained by existing models, to thereby “improve” the agreement between theory and experiment. Such an approach may, of course, eliminate the very data that would indicate deficiencies in the [*a priori*]{} model. This would preclude the discovery of improved models. In the processing stage that fits the Doppler samples, checks are made to ensure that there are no integer cycle slips in the data stream that would corrupt the phase. This is done by considering the difference of the phase observations taken at a high rate (10 times a second) to produce Doppler. Cycle slips often are dependent on tracking loop bandwidths, the signal to noise ratios, and predictions of frequencies. Blunders due to out-of-lock can be determined by looking at the original tracking data. In particular, cycle slips due to loss-of-lock stand out as a 1 Hz blunder point for each cycle slipped. If a blunder point is observed, the count is stopped and a Doppler point is generated by summing the preceding points. Otherwise the count is continued until a specified maximum duration is reached. Cases where this procedure detected the need for cycle corrections were flagged in the database and often individually examined by an analyst. Sometimes the data was corrected, but nominally the blunder point was just eliminated. This ensures that the data is consistent over a pass. However, it does not guarantee that the pass is good, because other errors can affect the whole pass and remain undetected until the orbit determination is done. To produce an input data file for an orbit determination program, JPL has a software package known as the Radio Metric Data Selection, Translation, Revision, Intercalation, Processing and Performance Evaluation Reporting (RMD-STRIPPER) Program. As we discussed in Section \[sec:doppler\], this input file has data that can be integrated over intervals with different durations: 10 s, 60 s, 600 s and 1980 s. This input Orbit Determination File (ODFILE) obtained from the RMDC group is the initial data set with which both the JPL and The Aerospace Corporation groups started their analyses. Therefore, the initial data file already contained some common data editing that the RMDC group had implemented through program flags, etc. The data set we started with had already been compressed to 60 s. So, perhaps there were some blunders that had already been removed using the initial STRIPPER program. The orbit analyst manually edits the remaining corrupted data points. Editing is done either by plotting the data residuals and deleting them from the fit or plotting weighted data residuals. That is, the residuals are divided by the standard deviation assigned to each data point and plotted. This gives the analyst a realistic view of the data noise during those times when the data was obtained while looking through the solar plasma. Applying an “$N$-$\sigma$” ($\sigma$ is the standard deviation) test, where $N$ is the choice of the analyst (usually 4-10) the analyst can delete those points that lie outside the $N$-$\sigma$ rejection criterion without being biased in his selection. The $N$-$\sigma$ test, implemented in CHASMP, is very useful for data taken near solar conjunction since the solar plasma adds considerable noise to the data. This criterion later was changed to a similar criteria that rejects all data with residuals in the fit extending for more than $\pm 0.025$ Hz from the mean. Contrariwise, the JPL analysis edits only very corrupted data; e.g., a blunder due to a phase lock loss, data with bad spin calibration, etc. Essentially the Aerospace procedure eliminates data in the tails of the Gaussian probability frequency distribution whereas the JPL procedure accepts this data. If needed or desired, the orbit analyst can choose to perform an additional data compression of the original navigation data. The JPL analysis does not apply any additional data compression and uses all the original data from the ODFILE as opposed to Aerospace’s approach. Aerospace makes an additional compression of data within CHASMP. It uses the longest available data integration times which can be composed from either summing up adjacent data intervals or by using data spans with duration $\ge 600$ s. (Effectively Aerospace prefers 600 and 1980 second data intervals and applies a low-pass filter.) The total count of corrupted data points is about 10% of the total raw data points. The analysts’ judgments play an important role here and is one of the main reasons that JPL and Aerospace have slightly different results. (See Sections \[results\]and \[recent\_results\].) In Section \[results\]we will show a typical plot (Figure \[fig:aerospace\] below) with outliers present in the data. Many more outliers are off the plot. One would expect that the two different strategies of data compression used by the two teams would result in significantly different numbers of total data points used in the two independent analyses. The influence of this fact on the solution estimation accuracy will be addressed in Section \[recent\_results\] below. Data weighting {#dataweight} -------------- Considerable effort has gone into accurately estimating measurement errors in the observations. These errors provide the data weights necessary to accurately estimate the parameter adjustments and their associated uncertainties. To the extent that measurement errors are accurately modeled, the parameters extracted from the data will be unbiased and will have accurate sigmas assigned to them. Both JPL and Aerospace assign a standard uncertainty of 1 mm/s over a 60 second count time for the S–band Pioneer Doppler data. (Originally the JPL team was weighting the data by 2 mm/s uncertainty.) A change in the DSN antenna elevation angle also directly affects the Doppler observables due to tropospheric refraction. Therefore, to correct for the influence of the Earth’s troposphere the data can also be deweighted for low elevation angles. The phenomenological range correction is given as $$\sigma= \sigma_{\tt nominal} \left(1+\frac{18}{(1+\theta_E)^2}\right), \label{eq:sig_aer0}$$ where $\sigma_{\tt nominal}$ is the basic standard deviation (in Hz) and $\theta_E$ is the elevation angle in degrees [@cane]. Each leg is computed separately and summed. For Doppler the same procedure is used. First, Eq. (\[eq:sig\_aer0\]) is multiplied by $\sqrt{60 \,{\rm s}/T_c}$, where $T_c$ is the count time. Then a numerical time differentiation of Eq. (\[eq:sig\_aer0\]) is performed. That is, Eq. (\[eq:sig\_aer0\]) is differenced and divided by the count time, $T_c$. (For more details on this standard technique see Refs. [@Moyer71]-[@MuhlemanAnderson81].) There is also the problem of data weighting for data influenced by the solar corona. This will be discussed in Section \[corona+wt\]. Spin calibration of the data {#spincalibrate} ---------------------------- The radio signals used by DSN to communicate with spacecraft are circularly polarized. When these signals are reflected from spinning spacecraft antennae a Doppler bias is introduced that is a function of the spacecraft spin rate. Each revolution of the spacecraft adds one cycle of phase to the up-link and the down-link. The up-link cycle is multiplied by the turn around ratio 240/221 so that the bias equals (1+240/221) cycles per revolution of the spacecraft. High-rate spin data is available for Pioneer 10 only up to July 17, 1990, when the DSN ceased doing spin calibrations. (See Section \[sec:prop\].) After this date, in order to reconstruct the spin behavior for the entire data span and thereby account for the spin bias in the Doppler signal, both analyses modeled the spin by performing interpolations between the data points. The JPL interpolation was non-linear with a high-order polynomial fit of the data. (The polynomial was from second up to sixth order, depending on the data quality.) The CHASMP interpolation was linear between the spin data points. After a maneuver in mid-1993, there was not enough power left to support the IPP. But analysts still could get a rough determination approximately every six months using information obtained from the conscan maneuvers. No spin determinations were made after 1995. However, the archived conscan data could still yield spin data at every maneuver time if such work was approved. Further, as the phase center of the main antenna is slightly offset from the spin axis, a very small (but detectable) $\sin$e-wave signal appears in the high-rate Doppler data. In principle, this could be used to determine the spin rate for passes taken after 1993, but it has not been attempted. Also, the failure of one of the spin-down thrusters prevented precise spin calibration of the Pioneer 11 data. Because the spin rate of the Pioneers was changing over the data span, the calibrations also provide an indication of gas leaks that affect the acceleration of the spacecraft. A careful look at the records shows how this can be a problem. This will be discussed in Sections \[spinhistory\] and \[sec:gleaks\]. \[navigate\]BASIC THEORY OF SPACECRAFT NAVIGATION ================================================= Accuracy of modern radio tracking techniques has provided the means necessary to explore the gravitational environment in the solar system up to a limit never before possible [@massprog]. The major role in this quest belongs to relativistic celestial mechanics experiments with planets (e.g., passive radar ranging) and interplanetary spacecraft (both Doppler and range experiments). Celestial mechanics experiments with spacecraft have been carried out by JPL since the early 1960’s [@anderson74; @dsn86]. The motivation was to improve both the ephemerides of solar system bodies and also the knowledge of the solar system’s dynamical environment. This has become possible due to major improvements in the accuracy of spacecraft navigation, which is still a critical element for a number of space missions. The main objective of spacecraft navigation is to determine the present position and velocity of a spacecraft and to predict its future trajectory. This is usually done by measuring changes in the spacecraft’s position and then, using those measurements, correcting (fitting and adjusting) the predicted spacecraft trajectory. In this section we will discuss the theoretical foundation that is used for the analysis of tracking data from interplanetary spacecraft. We describe the basic physical models used to determine a trajectory, given the data. Relativistic equations of motion -------------------------------- The spacecraft ephemeris, generated by a numerical integration program, is a file of spacecraft positions and velocities as functions of ephemeris (or coordinate) time ([ET]{}). The integrator requires the input of various parameters. These include adopted constants ($c$, $G$, planetary mass ratios, etc.) and parameters that are estimated from fits to observational data (e.g., corrections to planetary orbital elements). The ephemeris programs use equations for point-mass relativistic gravitational accelerations. They are derived from the variation of a time-dependent, Lagrangian-action integral that is referenced to a non-rotating, solar-system, barycentric, coordinate frame. In addition to modeling point-mass interactions, the ephemeris programs contain equations of motion that model terrestrial and lunar figure effects, Earth tides, and lunar physical librations [@Newhall83]-[@Standish95a]. The programs treat the Sun, the Moon, and the nine planets as point masses in the isotropic, parameterized post-Newtonian, N-body metric with Newtonian gravitational perturbations from large, main-belt asteroids. Responding to the increasing demand of the navigational accuracy, the gravitational field in the solar system is modeled to include a number of relativistic effects that are predicted by the different metric theories of gravity. Thus, within the accuracy of modern experimental techniques, the parameterized post-Newtonian (PPN) approximation of modern theories of gravity provides a useful starting point not only for testing these predictions, but also for describing the motion of self-gravitating bodies and test particles. As discussed in detail in [@Will93], the accuracy of the PPN limit (which is slow motion and weak field) is adequate for all foreseeable solar system tests of general relativity and a number of other metric theories of gravity. (For the most general formulation of the PPN formalism, see the works of Will and Nordtvedt [@Will93; @WillNordtvedt72].) For each body $i$ (a planet or spacecraft anywhere in the solar system), the point-mass acceleration is written as [@Moyer71; @Moyer00; @Newhall83; @estabrook69; @Moyer81] $$\begin{aligned} \nonumber \ddot{\bf r}_i&=&\sum_{j\not=i}\frac{\mu_j({\bf r}_j-{\bf r}_i)} {r^3_{ij}}\Bigg(1-\frac{2(\beta+\gamma)}{c^2}\sum_{k\not=i} \frac{\mu_k}{r_{ik}}-\frac{2\beta-1}{c^2}\sum_{k\not=j} \frac{\mu_k}{r_{jk}}-\frac{3}{2c^2}\Big[\frac{({\bf r}_j-{\bf r}_i)\dot{\bf r}_j}{r_{ij}}\Big]^2+\frac{1}{2c^2} ({\bf r}_j-{\bf r}_i)\ddot{\bf r}_j-\frac{2(1+\gamma)}{c^2} \dot{\bf r}_i\dot{\bf r}_j+\\ &+&\gamma\left(\frac{v_i}{c}\right)^2+ (1+\gamma)\left(\frac{v_j}{c}\right)^2\Bigg)+ \frac{1}{c^2}\sum_{j\not=i}\frac{\mu_j}{r^3_{ij}} \Big([{\bf r}_i-{\bf r}_j)]\cdot[(2+2\gamma)\dot{\bf r}_i-(1+2\gamma) \dot{\bf r}_j]\Big)(\dot{\bf r}_i-\dot{\bf r}_j)+ \frac{3+4\gamma}{2c^2}\sum_{j\not=i} \frac{\mu_j\ddot{\bf r}_j}{r_{ij}} \label{eq:rdotdot}\end{aligned}$$ where $\mu_i$ is the “gravitational constant” of body [*i*]{}. It actually is its mass times the Newtonian constant: $\mu_i={\it G}m_i$. Also, ${\bf r}_i(t)$ is the barycentric position of body $i$, $r_{ij}=|{\bf r}_j-{\bf r}_i|$ and $v_i=|\dot{\bf r}_i|$. For planetary motion, each of these equations depends on the others. So they must be iterated in each step of the integration of the equations of motion. The barycentric acceleration of each body $j$ due to Newtonian effects of the remaining bodies and the asteroids is denoted by $\ddot{\bf r}_j$. In Eq. (\[eq:rdotdot\]), $\beta$ and $\gamma$ are the PPN parameters [@Will93; @WillNordtvedt72]. General relativity corresponds to $\beta = \gamma = 1$, which we choose for our study. The Brans-Dicke theory is the most famous among the alternative theories of gravity. It contains, besides the metric tensor, a scalar field $\varphi$ and an arbitrary coupling constant $\omega$, related to the two PPN parameters as $\gamma= \frac{1+\omega}{2+\omega}, ~\beta=1$. Equation (\[eq:rdotdot\]) allows the consideration of any problem in celestial mechanics within the PPN framework. Light time solution and time scales {#sec:time_scales} ----------------------------------- In addition to planetary equations of motion Eq. (\[eq:rdotdot\]), one needs to solve the relativistic light propagation equation in order to get the solution for the total light time travel. In the solar system, barycentric, space-time frame of reference this equation is given by: $$\begin{aligned} \nonumber t_2-t_1&=&\frac{r_{21}}{c}+\frac{(1+\gamma)\mu_\odot}{c^3} \ln\bigg[\frac{r_1^\odot+r_2^\odot+r_{12}^\odot} {r_1^\odot+r_2^\odot-r_{12}^\odot}\bigg]+\\ &+& \sum_{i} \frac{(1+\gamma)\mu_i}{c^3} \ln\bigg[\frac{r_1^i+r_2^i+r_{12}^i} {r_1^i+r_2^i-r_{12}^i}\bigg], \label{eq:lt}\end{aligned}$$ where $\mu_\odot$ is the gravitational constant of the Sun and $\mu_i$ is the gravitational constant of a planet, an outer planetary system, or the Moon. $r_1^\odot, r_2^\odot and r_{12}^\odot$ are the heliocentric distances to the point of RF signal emission on Earth, to the point of signal reflection at the spacecraft, and the relative distance between these two points. Correspondingly, $r_1^i, r_2^i,$ and $r_{12}^i$ are similar distances relative to a particular $i$-th body in the solar system. In the spacecraft light time solution, $t_1$ refers to the transmission time at a tracking station on Earth, and $t_2$ refers to the reflection time at the spacecraft or, for one-way [@way] data, the transmission time at the spacecraft. The reception time at the tracking station on Earth or at an Earth satellite is denoted by $t_3$. Hence, Eq. (\[eq:lt\]) is the up-leg light time equation. The corresponding down-leg light time equation is obtained by replacing subscripts as follows: $1\rightarrow 2 $ and $2\rightarrow 3 $. (See the details in [@Moyer00].) The spacecraft equations of motion relative to the solar system barycenter are essentially the same as given by Eq. (\[eq:rdotdot\]). The gravitational constants of the Sun, planets and the planetary systems are the values associated with the solar system barycentric frame of reference, which are obtained from the planetary ephemeris [@Moyer81]. We treat a distant spacecraft as a point-mass particle. The spacecraft acceleration is integrated numerically to produce the spacecraft ephemeris. The ephemeris is interpolated at the ephemeris time ([ET]{}) value of the interpolation epoch. This is the time coordinate $t$ in Eqs. (\[eq:rdotdot\]) and (\[eq:lt\]), i.e., $t\equiv\,{\tt ET}$. As such, ephemeris time means coordinate time in the chosen celestial reference frame. It is an independent variable for the motion of celestial bodies, spacecraft, and light rays. The scale of [ET]{} depends upon which reference frame is selected and one may use a number of time scales depending on the practical applications. It is convenient to express [ET]{} in terms of International Atomic Time ([TAI]{}). [TAI]{} is based upon the second in the International System of Units ([SI]{}). This second is defined to be the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom [@exp_cat]. The differential equation relating ephemeris time ([ET]{}) in the solar system barycentric reference frame to [TAI]{} at a tracking station on Earth or on Earth satellite can be obtained directly from the Newtonian approximation to the N-body metric [@Moyer81]. This expression has the form $$\begin{aligned} \frac{d \,{\tt TAI}}{d\, \tt ET}= 1-\frac{1}{c^2}\Big(U-\langle U\rangle + \frac{1}{2}v^2-\frac{1}{2}\langle v^2 \rangle\Big) +{\cal O}(\frac{1}{c^{4}}), \label{eq:tai_et}\end{aligned}$$ where $U$ is the solar system gravitational potential evaluated at the tracking station and $v$ is the solar system barycentric velocity of the tracking station. The brackets $\langle ~\rangle$ on the right side of Eq. (\[eq:tai\_et\]) denote long-time average of the quantity contained within them. This averaging amounts to integrating out periodic variations in the gravitational potential, $U$, and the barycentric velocity, $v^2$, at the location of a tracking station. The desired time scale transformation is then obtained by using the planetary ephemeris to calculate the terms in Eq. (\[eq:tai\_et\]). The vector expression for the ephemeris/coordinate time ([ET]{}) in the solar system barycentric frame of reference minus the [TAI]{} obtained from an atomic clock at a tracking station on Earth has the form [@Moyer81] $$\begin{aligned} {\tt ET-TAI} &=& 32.184~{\rm s}+ \frac{2}{c^2}(\dot{\bf r}^\odot_{\tt B}\cdot {\bf r}^\odot_{\tt B})+ \frac{1}{c^2}(\dot{\bf r}^{\tt SSB}_{\tt B} \cdot {\bf r}^{\tt B}_{\tt E})+ \nonumber\\ &+&\frac{1}{c^2}(\dot{\bf r}^{\tt SSB}_{\tt E}\cdot {\bf r}^{\tt E}_A)+\frac{\mu_J}{c^2(\mu_\odot+\mu_J)} (\dot{\bf r}^\odot_J\cdot {\bf r}^\odot_J)+\nonumber\\ &+& \frac{\mu_{Sa}}{c^2(\mu_\odot+\mu_{Sa})} (\dot{\bf r}^\odot_{Sa}\!\cdot{\bf r}^\odot_{Sa})+ \frac{1}{c^2}(\dot{\bf r}^{\tt SSB}_\odot\!\!\cdot {\bf r}^\odot_{\tt B}), \label{eq:time}\hskip 14pt\end{aligned}$$ where ${\bf r}^j_i$ and $\dot{\bf r}^j_i$ position and velocity vectors of point $i$ relative to point $j$ (they are functions of [ET]{}); superscript or subscript ${\tt SSB}$ denotes solar system barycenter; $\odot$ stands for the Sun; ${\tt B}$ for the Earth-Moon barycenter; $ E, J, Sa $ denote the Earth, Jupiter, and Saturn correspondingly, and $A$ is for the location of the atomic clock on Earth which reads [TAI]{}. This approximated analytic result contains the clock synchronization term which depends upon the location of the atomic clock and five location-independent periodic terms. There are several alternate expressions that have up to several hundred additional periodic terms which provide greater accuracies than Eq. (\[eq:time\]). The use of these extended expressions provide transformations of [ET]{} – [TAI]{} to accuracies of 1 ns [@Moyer00]. For the purposes of our study the Station Time ([ST]{}) is especially significant. This time is the atomic time [TAI]{} at a DSN tracking station on Earth, [ST]{}=[TAI]{}$_{\tt station}$. This atomic time scale departs by a small amount from the “reference time scale.” The reference time scale for a DSN tracking station on Earth is the Coordinated Universal Time ([UTC]{}). This last is standard time for 0$^\circ$ longitude. (For more details see [@Moyer00; @exp_cat].) All the vectors in Eq. (\[eq:time\]) except the geocentric position vector of the tracking station on Earth can be interpolated from the planetary ephemeris or computed from these quantities. Universal Time ([UT]{}) is the measure of time which is the basis for all civil time keeping. It is an observed time scale. The specific version used in JPL’s Orbit Determination Program (ODP) is [UT1]{}. This is used to calculate mean sidereal time, which is the Greenwich hour angle of the mean equinox of date measured in the true equator of date. Observed [UT1]{} contains 41 short-term terms with periods between 5 and 35 days. They are caused by long-period solid Earth tides. When the sum of these terms, [$\Delta$UT1]{}, is subtracted from [UT1]{} the result is called [UT1R]{}, where [R]{} means regularized. Time in any scale is represented as seconds past 1 January 2000, 12$^{\tt h}$, in that time scale. This epoch is J2000.0, which is the start of the Julian year 2000. The Julian Date for this epoch is JD 245,1545.0. Our analyses used the standard space-fixed J2000 coordinate system, which is provided by the International Celestial Reference Frame (ICRF). This is a quasi-inertial reference frame defined from the radio positions of 212 extragalactic sources distributed over the entire sky [@Ma98]. The variability of the earth-rotation vector relative to the body of the planet or in inertial space is caused by the gravitational torque exerted by the Moon, Sun and planets, displacements of matter in different parts of the planet and other excitation mechanisms. The observed oscillations can be interpreted in terms of mantle elasticity, earth flattening, structure and properties of the core-mantle boundary, rheology of the core, underground water, oceanic variability, and atmospheric variability on time scales of weather or climate. Several space geodesy techniques contribute to the continuous monitoring of the Earth’s rotation by the International Earth Rotation Service (IERS). Measurements of the Earth’s rotation presented in the form of time developments of the so-called Earth Orientation Parameters ([EOP]{}). Universal time ([UT1]{}), polar motion, and the celestial motion of the pole (precession/nutation) are determined by Very Long-Baseline Interferometry (VLBI). Satellite geodesy techniques, such as satellite laser ranging (SLR) and using the Global Positioning System (GPS), determine polar motion and rapid variations of universal time. The satellite geodesy programs used in the IERS allow determination of the time variation of the Earth’s gravity field. This variation reflects the evolutions of the Earth’s shape and of the distribution of mass in the planet. The programs have also detected changes in the location of the center of mass of the Earth relative to the crust. It is possible to investigate other global phenomena such as the mass redistributions of the atmosphere, oceans, and solid Earth. Using the above experimental techniques, Universal time and polar motion are available daily with an accuracy of about 50 picoseconds (ps). They are determined from VLBI astrometric observations with an accuracy of 0.5 milliarcseconds (mas). Celestial pole motion is available every five to seven days at the same level of accuracy. These estimations of accuracy include both short term and long term noise. Sub-daily variations in Universal time and polar motion are also measured on a campaign basis. In summary, this dynamical model accounts for a number of post-Newtonian perturbations in the motions of the planets, the Moon, and spacecraft. Light propagation is correct to order $c^{-2}$. The equations of motion of extended celestial bodies are valid to order $c^{-4}$. Indeed, this dynamical model has been good enough to perform tests of general relativity [@anderson75; @Will93; @WillNordtvedt72]. Standard modeling of small, non-gravitational forces {#sec:syst0} ---------------------------------------------------- In addition to the mutual gravitational interactions of the various bodies in the solar system and the gravitational forces acting on a spacecraft as a result of presence of those bodies, it is also important to consider a number of non-gravitational forces which are important for the motion of a spacecraft. (Books and lengthy reports have been written about practically all of them. Consult Ref. [@milani; @longuski] for a general introduction.) The Jet Propulsion Laboratory’s ODP accounts for many sources of non-gravitational accelerations. Among them, the most relevant to this study, are: i) solar radiation pressure, ii) solar wind pressure, iii) attitude-control maneuvers together with a model for unintentional spacecraft mass expulsion due to gas leakage of the propulsion system. We can also account for possible influence of the interplanetary media and DSN antennae contributions to the spacecraft radio tracking data and consider the torques produced by above mentioned forces. The Aerospace CHASMP code uses a model for gas leaks that can be adjusted to include the effects of the recoil force due to emitted radio power and anisotropic thermal radiation of the spacecraft. In principle, one could set up complicated engineering models to predict at least some of the effects. However, their residual uncertainties might be unacceptable for the experiment, in spite of the significant effort required. In fact, a constant acceleration produces a linear frequency drift that can be accounted for in the data analysis by a single unknown parameter. The figure against which we compare the effects of non-gravitational accelerations on the Pioneers’ trajectories is the expected error in the acceleration error estimations. This is on the order of $$\sigma_0 \sim 2\times 10^{-8} ~~{\rm cm/s^2}, \label{eq:req}$$ where $\sigma_0$ is a single determination accuracy related to acceleration measurements averaged over number of days. This would contribute to our result as $\sigma_N\sim\sigma_0/\sqrt{N}$. Thus, if no systematics are involved then $\sigma_N$ will just tend to zero as time progresses. Therefore, the important thing is to know that these effects (systematics) are not too large, thereby overwhelming any possibly important signal (such as our anomalous acceleration). This will be demonstrated in Sections \[ext-systema\] and \[int-systema\]. Solar corona model and weighting {#corona+wt} -------------------------------- The electron density and density gradient in the solar atmosphere influence the propagation of radio waves through the medium. So, both range and Doppler observations at S-band are affected by the electron density in the interplanetary medium and outer solar corona. Since, throughout the experiment, the closest approach to the center of the Sun of a radio ray path was greater than 3.5 $R_\odot$, the medium may be regarded as collisionless. The [*one way*]{} time delay associated with a plane wave passing through the solar corona is obtained [@MuhlemanAnderson81; @anderson74; @MuhlemanEspositoAnderson77] by integrating the group velocity of propagation along the ray’s path, $\ell$: $$\begin{aligned} \nonumber \Delta t &=& \pm \frac{1}{2c\,n_{\tt crit}(\nu)} \int_\oplus^{SC}d\ell~n_e(t, {\bf r}), \\ n_{\tt crit}(\nu) &=& 1.240\times 10^4~ \Big(\frac{\nu}{1~{\rm MHz}}\Big)^2~~{\rm cm}^{-3}, \label{eq:sol_plasma} \end{aligned}$$ where $n_e(t, {\bf r})$ is the free electron density in the solar plasma, $c$ is the speed of light, and $n_{\tt crit}(\nu)$ is the critical plasma density for the radio carrier frequency $\nu$. The plus sign is applied for ranging data and the minus sign for Doppler data [@slavanote]. Therefore, in order to calibrate the plasma contribution, one should know the electron density along the path. One usually decomposes the electron density, $n_e$, into a static, steady-state part, $\overline{n}_e(\bf{r})$, plus a fluctuation $\delta n_e(t, {\bf r})$, i.e., $n_e(t, {\bf r})= \overline{n}_e({\bf r})+ \delta n_e(t, {\bf r})$. The physical properties of the second term are hard to quantify. But luckily, its effect on Doppler observables and, therefore, on our results is small. (We will address this issue in Sec. \[solarwind\].) On the contrary, the steady-state corona behavior is reasonably well known and several plasma models can be found in the literature [@MuhlemanEspositoAnderson77]-[@bird]. Consequently, while studying the effect of a systematic error from propagation of the S-band carrier wave through the solar plasma, both analyses adopted the following model for the electron density profile [@MuhlemanAnderson81]: $$\begin{aligned} n_e(t, {\bf r})= A\Big(\frac{R_\odot}{r}\Big)^2+ B\Big(\frac{R_\odot}{r}\Big)^{2.7} e^{-\left[\frac{\phi}{\phi_0}\right]^2}+ C\Big(\frac{R_\odot}{r}\Big)^6. \label{corona_model_content}\end{aligned}$$ $r$ is the heliocentric distance to the immediate ray trajectory and $\phi$ is the helio-latitude normalized by the reference latitude of $\phi_0=10^\circ$. The parameters $r$ and $\phi$ are determined from the trajectory coordinates on the tracking link being modeled. The parameters $A, B, C$ are parameters chosen to describe the solar electron density. (They are commonly given in two sets of units, meters or cm$^{-3}$ [@scunits].) They can be treated as stochastic parameters, to be determined from the fit. But in both analyses we ultimately chose to use the values determined from the recent solar corona studies done for the Cassini mission. These newly obtained values are: $A= 6.0\times 10^3, B= 2.0\times 10^4, C= 0.6\times 10^6$, all in meters [@Ekelund]. \[This is what we will refer to as the “Cassini corona model.”\] Substitution of Eq. (\[corona\_model\_content\]) into Eq. (\[eq:sol\_plasma\]) results in the following steady-state solar corona contribution to the range model that we used in our analysis: $$\begin{aligned} \nonumber \Delta_{\tt SC}{\rm range}&=& \pm \Big(\frac{\nu_0}{\nu}\Big)^2\bigg[ A\Big(\frac{R_\odot}{\rho}\Big)F+ \\ &+& B\Big(\frac{R_\odot}{\rho}\Big)^{1.7} e^{-\left[\frac{\phi}{\phi_0}\right]^2}+ C\Big(\frac{R_\odot}{\rho}\Big)^{5}\bigg]. \hskip 10pt \label{corona_model}\end{aligned}$$ $\nu_0$ and $\nu$ are a reference frequency and the actual frequency of radio-wave \[for Pioneer 10 analysis $\nu_0=2295$ MHz\], $\rho$ is the impact parameter with respect to the Sun and $F$ is a light-time correction factor. For distant spacecraft this function is given as follows: $$\begin{aligned} F&=&F(\rho, r_T, r_E)=\\\nonumber &=&\frac{1}{\pi}\Bigg\{ {\sf ArcTan}\Big[\frac{\sqrt{r_T^2-\rho^2}}{\rho}\Big]+ {\sf ArcTan}\Big[\frac{\sqrt{r_E^2-\rho^2}}{\rho}\Big]\Bigg\}, \label{eq:weight_doppler*}\end{aligned}$$ where $r_T$ and $r_E$ are the heliocentric radial distances to the target and to the Earth, respectively. Note that the sign of the solar corona range correction is negative for Doppler and positive for range. The Doppler correction is obtained from Eq. (\[corona\_model\]) by simple time differentiation. Both analyses use the same physical model, Eq. (\[corona\_model\]), for the steady-state solar corona effect on the radio-wave propagation through the solar plasma. Although the actual implementation of the model in the two codes is different, this turns out not to be significant. (See Section \[Ext\_accuracy\].) CHASMP can also consider the effect of temporal variation in the solar corona by using the recorded history of solar activity. The change in solar activity is linked to the variation of the total number of sun spots per year as observed at a particular wavelength of the solar radiation, $\lambda$=10.7 cm. The actual data corresponding to this variation is given in Ref. [@F10-7]. CHASMP averages this data over 81 days and normalizes the value of the flux by 150. Then it is used as a time-varying scaling factor in Eq. (\[corona\_model\]). The result is referred to as the “F10.7 model.” Next we come to corona data weighting. JPL’s ODP does not apply corona weighting. On the other hand, Aerospace’s CHASMP can apply corona weighting if desired. Aerospace uses a standard weight augmented by a weight function that accounts for noise introduced by solar plasma and low elevation. The weight values are adjusted so that i) the post-fit weighted sum of the squares is close to unity and ii) approximately uniform noise in the residuals is observed throughout the fit span. Thus, the corresponding solar-corona weight function is: $$\sigma_{\tt r}= \frac{k}{2} \Big(\frac{\nu_0}{\nu}\Big)^2 \Big(\frac{R_\odot}{\rho}\Big)^{\frac{3}{2}}, \label{weightrange}$$ where, for range data, $k$ is an input constant nominally equal to 0.005 light seconds, $\nu_0$ and $\nu$ are a reference frequency and the actual frequency, $\rho$ is the trajectory’s impact parameter with respect to the Sun in km, and $R_\odot$ is the solar radius in km [@Muhleman]. The solar-corona weight function for Doppler is essentially the same, but obtained by numerical time differentiation of Eq. (\[weightrange\]). Modeling of maneuvers {#model-maneuvers} --------------------- There were 28 Pioneer 10 maneuvers during our data interval from 3 January 1987 to 22 July 1998. Imperfect coupling of the hydrazine thrusters used for the spin orientation maneuvers produced integrated velocity changes of a few millimeters per second. The times and durations of each maneuver were provided by NASA/Ames. JPL used this data as input to ODP. The Aerospace team used a slightly different approach. In addition to the original data, CHASMP used the spin-rate data file to help determine the times and duration of maneuvers. The CHASMP determination mainly agreed with the data used by JPL. \[There were minor variations in some of the times, one maneuver was split in two, and one extraneous maneuver was added to Interval II to account for data not analyzed (see below).\] Because the effect on the spacecraft acceleration could not be determined well enough from the engineering telemetry, JPL included a single unknown parameter in the fitting model for each maneuver. In JPL’s ODP analysis, the maneuvers were modeled by instantaneous velocity increments at the beginning time of each maneuver (instantaneous burn model). \[Analyses of individual maneuver fits show the residuals to be small.\] In the CHASMP analysis, a constant acceleration acting over the duration of the maneuver was included as a parameter in the fitting model (finite burn model). Analyses of individual maneuver fits show the residuals are small. Because of the Pioneer spin, these accelerations are important only along the Earth-spacecraft line, with the other two components averaging out over about 50 revolutions of the spacecraft over a typical maneuver duration of 10 minutes. By the time Pioneer 11 reached Saturn, the pattern of the thruster firings was understood. Each maneuver caused a change in spacecraft spin and a velocity increment in the spacecraft trajectory, immediately followed by two to three days of gas leakage, large enough to be observable in the Doppler data [@null81]. Typically the Doppler data is time averaged over 10 to 33 minutes, which significantly reduces the high-frequency Doppler noise. The residuals represent our fit. They are converted from units of Hz to Doppler velocity by the formula [@drift] $$[\Delta v]_{\tt DSN} = \frac{c}{2} \frac{[\Delta \nu]_{\tt DSN}}{\nu_0}, \label{hztodoppler}$$ where $\nu_0$ is the downlink carrier frequency, $\sim 2.29$ GHz, $\Delta \nu$ is the Doppler residual in Hz from the fit, and $c$ is the speed of light. As an illustration, consider the fit to one of the Pioneer 10 maneuvers, \# 17, on 22 December 1993, given in Figure \[fig:man17\]. This was particularly well covered by low-noise -10pt -10pt Doppler data near solar opposition. Before the start of the maneuver, there is a systematic trend in the residuals which is represented by a cubic polynomial in time. The standard error in the residuals is 0.095 mm/s. After the maneuver, there is a relatively small velocity discontinuity of $-0.90 \pm 0.07$ mm/s. The discontinuity arises because the model fits the entire data interval. In fact, the residuals increase after the maneuver. By 11 January 1994, 19 days after the maneuver, the residuals are scattered about their pre-maneuver mean of $-0.15$ mm/s. For purposes of characterizing the gas leak immediately after the maneuver, we fit the post-maneuver residuals by a two-parameter exponential curve, $$\Delta v = -v_0 \exp\Big[-\frac{t}{\tau}\,\Big] - 0.15 ~~~{\rm mm/s}.$$ The best fit yields $v_0 = 0.808$ mm/s and the time constant $\tau$ is 13.3 days, a reasonable result. The time derivative of the exponential curve yields a residual acceleration immediately after the maneuver of 7.03 $\times$ 10$^{-8}$ cm/s$^{2}$. This is close to the magnitude of the anomalous acceleration inferred from the Doppler data, but in the *opposite* direction. However the gas leak rapidly decays and becomes negligible after 20 days or so. Orbit determination procedure {#sec:OD} ----------------------------- Our orbit determination procedure first determines the spacecraft’s initial position and velocity in a data interval. For each data interval, we then estimate the magnitudes of the orientation maneuvers, if any. The analyses are modeled to include the effects of planetary perturbations, radiation pressure, the interplanetary media, general relativity, and bias and drift in the Doppler and range (if available). Planetary coordinates and solar system masses are obtained using JPL’s Export Planetary Ephemeris DE405, where DE stands for the Development Ephemeris. \[Earlier in the study, DE200 was used. See Section \[subsec:accel\].\] We include models of precession, nutation, sidereal rotation, polar motion, tidal effects, and tectonic plates drift. Model values of the tidal deceleration, nonuniformity of rotation, polar motion, Love numbers, and Chandler wobble are obtained observationally, by means of Lunar and Satellite Laser Ranging (LLR and SLR) techniques and VLBI. Previously they were combined into a common publication by either the International Earth Rotation Service (IERS) or by the United States Naval Observatory (USNO). Currently this information is provided by the ICRF. JPL’s Earth Orientation Parameters (EOP) is a major source contributor to the ICRF. The implementation of the J2000.0 reference coordinate system in CHASMP involves only rotation from the Earth-fixed to the J2000.0 reference frame and the use of JPL’s   DE200   planetary ephemeris [@Laing91]. The rotation from   J2000.0   to Earth-fixed is computed from a series of rotations which include precession, nutation, the Greenwich hour angle, and pole wander. Each of these general categories is also a multiple rotation and is treated separately by most software. Each separate rotation matrix is chain multiplied to produce the final rotation matrix. CHASMP, however, does not separate precession and nutation. Rather, it combines them into a single matrix operation. This is achieved by using a different set of angles to describe precession than is used in the ODP. (See a description of the standard set of angles in [@Lieske76].) These angles separate luni-solar precession from planetary precession. Luni-solar precession, being the linear term of the nutation series for the nutation in longitude, is combined with the nutation in longitude from the DE200 ephemeris tape [@Standish82]. Both JPL’s ODP and The Aerospace Corporation’s CHASMP use the JPL/Earth Orientation Parameters (EOP) values. This could be a source of common error. However the comparisons between EOP and IERS show an insignificant difference. Also, only secular terms, such as precession, can contribute errors to the anomalous acceleration. Errors in short period terms are not correlated with the anomalous acceleration. Parameter estimation strategies {#sec:PE} ------------------------------- During the last few decades, the algorithms of orbital analysis have been extended to incorporate Kalman-filter estimation procedure that is based on the concept of “process noise” (i.e., random, non-systematic forces, or random-walk effects). This was motivated by the need to respond to the significant improvement in observational accuracy and, therefore, to the increasing sensitivity to numerous small perturbing factors of a stochastic nature that are responsible for observational noise. This approach is well justified when one needs to make accurate predictions of the spacecraft’s future behavior using only the spacecraft’s past hardware and electronics state history as well as the dynamic environment conditions in the distant craft’s vicinity. Modern navigational software often uses Kalman filter estimation since it more easily allows determination of the temporal noise history than does the weighted least-squares estimation. To take advantage of this while obtaining JPL’s original results [@anderson; @moriond] discussed in Section \[results\], JPL used batch-sequential methods with variable batch sizes and process noise characteristics. That is, a batch-sequential filtering and smoothing algorithm with process noise was used with ODP. In this approach any small anomalous forces may be treated as stochastic parameters affecting the spacecraft trajectory. As such, these parameters are also responsible for the stochastic noise in the observational data. To better characterize these noise sources, one splits the data interval into a number of constant or variable size batches and makes assumptions on possible statistical properties of these noise factors. One then estimates the mean values of the unknown parameters within the batch and also their second statistical moments. Using batches has the advantage of dealing with a smaller number of experimental data segments. We experimented with a number of different constant batch sizes; namely, 0, 5, 30, and 200 day batch sizes. (Later we also used 1 and 10 day batch sizes.) In each batch one estimates the same number of desired parameters. So, one expects that the smaller the batch size the larger the resulting statistical errors. This is because a smaller number of data points is used to estimate the same number of parameters. Using the entire data interval as a single batch while changing the process noise [*a priori*]{} values is expected in principle (see below) to yield a result identical to the least-squares estimation. In the single batch case, it would produce only one solution for the anomalous acceleration. There is another important parameter that was taken into account in the statistical data analysis reported here. This is the expected correlation time for the underlying stochastic processes (as well as the process noise) that may be responsible for the anomalous acceleration. For example, using a zero correlation time is useful in searches for an $a_P$ that is generated by a random process. One therefore expects that an $a_P$ estimated from one batch is statistically independent (uncorrelated) from those estimated from other batches. Also, the use of finite correlation times indicates one is considering an $a_P$ that may show a temporal variation within the data interval. We experimented with a number of possible correlation times and will discuss the corresponding assumptions when needed. In each batch one estimates solutions for the set of desired parameters at a specified epoch within the batch. One usually chooses to report solutions corresponding to the beginning, middle, or end of the batch. General coordinate and time transformations (discussed in Section \[sec:time\_scales\]) are then used to report the solution in the epoch chosen for the entire data interval. One may also adjust the solutions among adjacent batches by accounting for possible correlations. This process produces a smoothed solution for the set of solved-for parameters. More details on this so called “batch–sequential algorithm with smoothing filter” are available in Refs. [@Moyer71]-[@Gelb]. Even without process noise, the inversion algorithms of the Kalman formulation and the weighted least-squares method seem radically different. But as shown in [@sherman], if one uses a single batch for all the data and if one uses certain assumptions about, for instance, the process noise and the smoothing algorithms, then the two methods are mathematically identical. When introducing process noise, an additional process noise matrix is also added into the solution algorithm. The elements of this matrix are chosen by the user as prescribed by standard statistical techniques used for navigational data processing. For the recent results reported in Section \[recent\_results\], JPL used both the batch-sequential and the weighted least-squares estimation approaches. JPL originally implemented only the batch-sequential method, which yielded the detection (at a level smaller than could be detected with any other spacecraft) of an annual oscillatory term smaller in size than the anomalous acceleration [@moriond]. (This term is discussed in Section \[annualterm\].) The recent studies included weighted least-squares estimation to see if this annual term was a calculational anomaly. The Aerospace Corporation uses only the weighted least-squares approach with its CHASMP software. A $\chi^2$ test is used as an indicator of the quality of the fit. In this case, the anomalous acceleration is treated as a constant parameter over the entire data interval. To solve for $a_P$ one estimates the statistical weights for the data points and then uses these in a general weighted least-squares fashion. Note that the weighted least-squares method can obtain a result similar to that from a batch-sequential approach (with smoothing filter, zero correlation time and without process noise) by cutting the data interval into smaller pieces and then looking at the temporal variation among the individual solutions. As one will see in the following, in the end, both programs yielded very similar results. The differences between them can be mainly attributed to (other) systematics. This gives us confidence that both programs and their implemented estimation algorithms are correct to the accuracy of this investigation. \[results\]ORIGINAL DETECTION OF THE ANOMALOUS ACCELERATION =========================================================== Early JPL studies of the anomalous Pioneer Doppler residuals {#subsec:accel} ------------------------------------------------------------ As mentioned in the introduction, by 1980 Pioneer 10 was at 20 AU, so the solar radiation pressure acceleration had decreased to $< 5\times 10^{-8}$ cm/s$^2$. Therefore, a search for unmodeled accelerations (at first with the further out Pioneer 10) could begin at this level. With the acceptance of a proposal of two of us (JDA and ELL) to participate in the Heliospheric Mission on Pioneer 10 and 11, such a search began in earnest [@jpl]. The JPL analysis of unmodeled accelerations used the JPL’s Orbit Determination Program (ODP) [@Moyer71]-[@Moyer00]. Over the years the data continually indicated that the largest systematic error in the acceleration residuals is a constant bias of $a_P \sim (8\pm 3) \times 10^{-8}$ cm/s$^2$, directed [*toward*]{} the Sun (to within the beam-width of the Pioneers’ antennae [@sunearth]). As stated previously, the analyses were modeled to include the effects of planetary perturbations, radiation pressure, the interplanetary media, general relativity, together with bias and drift in the Doppler signal. Planetary coordinates and the solar system masses were taken from JPL’s Export Planetary Ephemeris DE405, referenced to ICRF. The analyses used the standard space-fixed J2000 coordinate system with its associated JPL planetary ephemeris DE405 (or earlier, DE200). The time-varying Earth orientation in J2000 coordinates is defined by a 1998 version of JPL’s EOP file, which accounts for the inertial precession and nutation of the Earth’s spin axis, the geophysical motion of the Earth’s pole with respect to its spin axis, and the Earth’s time varying spin rate. The three-dimensional locations of the tracking stations in the Earth’s body-fixed coordinate system (geocentric radius, latitude, longitude) were taken from a set recommended by ICRF for JPL’s DE405. Consider ${\nu}_{\tt obs}$, the frequency of the re-transmitted signal observed by a DSN antennae, and $\nu_{\tt model}$, the predicted frequency of that signal. The observed, two-way anomalous effect can be expressed to first order in $v/c$ as [@drift] $$\begin{aligned} \nonumber \left[\nu_{\tt obs}(t)- \nu_{\tt model}(t)\right]_{\tt DSN} = - \nu_{0}\frac{2a_P~t}{c}, \\ \nu_{\tt model} = \nu_{0}\left[1 - \frac{2v_{\tt model}(t)}{c}\right]. \label{eq:delta_nu}\end{aligned}$$ Here, $\nu_{0}$ is the reference frequency, the factor $2$ is because we use two- and three-way data [@way]. $v_{\tt model}$ is the modeled velocity of the spacecraft due to the gravitational and other large forces discussed in Section \[navigate\]. (This velocity is outwards and hence produces a red shift.) We have already included the sign showing that $a_P$ is inward. (Therefore, $a_P$ produces a slight blue shift on top of the larger red shift.) By DSN convention [@drift], the first of Eqs. (\[eq:delta\_nu\]) is $[\Delta \nu_{\tt obs} - \Delta \nu_{\tt model}]_{\tt usual} = - [\Delta \nu_{\tt obs} - \Delta \nu_{\tt model}]_{\tt DSN}$. Over the years the anomaly remained in the data of both Pioneer 10 and Pioneer 11 [@bled]. (See Figure \[fig:forces\].) -20pt In order to model any unknown forces acting on Pioneer 10, the JPL group introduced a stochastic acceleration, exponentially correlated in time, with a time constant that can be varied. This stochastic variable is sampled in ten-day batches of data. We found that a correlation time of one year produces good results. We did, however, experiment with other time constants as well, including a zero correlation time (white noise). The result of applying this technique to 6.5 years of Pioneer 10 and 11 data is shown in Figure \[fig:correlation\]. The plotted points represent our determination of the stochastic variable at ten-day sample intervals. We plot the stochastic variable as a function of heliocentric distance, not time, because that is more fundamental in searches for trans-Neptunian sources of gravitation. -10pt -20pt As possible “perturbative forces” to explain this bias, we considered gravity from the Kuiper belt, gravity from the galaxy, spacecraft “gas leaks,” errors in the planetary ephemeris, and errors in the accepted values of the Earth’s orientation, precession, and nutation. We found that none of these mechanisms could explain the apparent acceleration, and some were three orders of magnitude or more too small. \[We also ruled out a number of specific mechanisms involving heat radiation or “gas leaks,” even though we feel these are candidates for the cause of the anomaly. We will return to this in Sections \[ext-systema\] and \[int-systema\].\] We concluded [@anderson], from the JPL-ODP analysis, that there is an unmodeled acceleration, $a_P$, towards the Sun of $(8.09\pm0.20)\times 10^{-8}$ cm/s$^2$ for Pioneer 10 and of $(8.56\pm 0.15) \times 10^{-8}$ cm/s$^2$ for Pioneer 11. The error was determined by use of a five-day batch sequential filter with radial acceleration as a stochastic parameter subject to white Gaussian noise ($\sim$ 500 independent five-day samples of radial acceleration) [@tap]. No magnitude variation of $a_P$ with distance was found, within a sensitivity of $\sigma_0=2\times10^{-8}$ cm/s$^2$ over a range of 40 to 60 AU. All our errors are taken from the covariance matrices associated with the least–squares data analysis. The assumed data errors are larger than the standard error on the post–fit residuals. \[For example, the Pioneer S–band Doppler error was set at 1 mm/s at a Doppler integration time of 60 s, as opposed to a characteristic $\chi^2$ value of 0.3 mm/s.\] Consequently, the quoted errors are realistic, not formal, and represent our attempt to include systematics and a reddening of the noise spectrum by solar plasma. Any spectral peaks in the post-fit Pioneer Doppler residuals were not significant at a 90% confidence level [@anderson]. First Aerospace study of the apparent Pioneer acceleration {#subsec:aero} ---------------------------------------------------------- With no explanation of this data in hand, our attention focused on the possibility that there was some error in JPL’s ODP. To investigate this, an analysis of the raw data was performed using an independent program, The Aerospace Corporation’s Compact High Accuracy Satellite Motion Program (CHASMP) [@chasmp] – one of the standard Aerospace orbit analysis programs. CHASMP’s orbit determination module is a development of a program called POEAS (Planetary Orbiter Error Analysis Study program) that was developed at JPL in the early 1970’s independently of JPL’s ODP. As far as we know, not a single line of code is common to the two programs [@poeas]. Although, by necessity, both ODP and CHASMP use the same physical principles, planetary ephemeris, and timing and polar motion inputs, the algorithms are otherwise quite different. If there were an error in either program, they would not agree. Aerospace analyzed a Pioneer 10 data arc that was initialized on 1 January 1987 at 16 hr (the data itself started on 3 January) and ended at 14 December 1994, 0 hr. The raw data set was averaged to 7560 data points of which 6534 points were used. This CHASMP analysis of Pioneer 10 data also showed an unmodeled acceleration in a direction along the radial toward the Sun [@aero]. The value is $(8.65 \pm 0.03) \times 10^{-8}$ cm/s$^{2}$, agreeing with JPL’s result. The smaller error here is because the CHASMP analysis used a batch least-squares fit over the whole orbit [@tap; @chasmp], not looking for a variation of the magnitude of $a_P$ with distance. Without using the apparent acceleration, CHASMP shows a steady frequency drift [@drift] of about $-6 \times 10^{-9}$ Hz/s, or 1.5 Hz over 8 years (one-way only). (See Figure \[fig:aerospace\].) This equates to a clock acceleration, $-a_t$, of $-2.8\times 10^{-18}$ s/s$^{2}$. The identity with the apparent Pioneer acceleration is $$a_t \equiv a_P/c. \label{asubt}$$ The drift in the Doppler residuals (observed minus computed data) is seen in Figure \[fig:pio10best\_fit\]. -10pt -10pt The drift is clear, definite, and cannot be removed without either the added acceleration, $a_P$, or the inclusion in the data itself of a frequency drift, i.e., a “clock acceleration” $a_t$. If there were a systematic drift in the atomic clocks of the DSN or in the time-reference standard signals, this would appear like a non-uniformity of time; i.e., all clocks would be changing with a constant acceleration. We now have been able to rule out this possibility. (See Section \[sec:timemodel\].) Continuing our search for an explanation, we considered the possibilities: i) that the Pioneer 10/11 spacecraft had internal systematic properties, undiscovered because they are of identical design, and ii) that the acceleration was due to some not-understood viscous drag force (proportional to the approximately constant velocity of the Pioneers). Both these possibilities could be investigated by studying spin-stabilized spacecraft whose spin axes are not directed towards the Sun, and whose orbital velocity vectors are far from being radially directed. Two candidates were Galileo in its Earth-Jupiter mission phase and Ulysses in Jupiter-perihelion cruise out of the plane of the ecliptic. As well as Doppler, these spacecraft also yielded a considerable quantity of range data. By having range data one can tell if a spacecraft is accumulating a range effect due to a spacecraft acceleration or if the orbit determination process is fooled by a Doppler frequency rate bias. Galileo measurement analysis {#galileo} ---------------------------- We considered the dynamical behavior of Galileo’s trajectory during its cruise flight from second Earth encounter (on 8 December 1992) to arrival at Jupiter. \[This period ends just before the Galileo probe release on 13 July 1995. The probe reached Jupiter on 7 December 1995.\] During this time the spacecraft traversed a distance of about 5 AU with an approximately constant velocity of 7.19(4) km/s. A quick JPL look at limited Galileo data (241 days from 8 January 1994 to 6 September 1994) demonstrated that it was impossible to separate solar radiation effects from an anomalous constant acceleration. The Sun was simply too close and the radiation cross-section too large. The nominal value obtained was $\sim 8 \times 10^{-8}$ cm/s$^2$. The Aerospace’s analysis of the Galileo data covered the same arc as JPL and a second arc from 2 December 1992 to 24 March 1993. The analysis of Doppler data from the first arc resulted in a determination for $a_P$ of $\sim (8 \pm 3) \times 10^{-8}$ cm/s$^2$, a value similar to that from Pioneer 10. But the correlation with solar pressure was so high (0.99) that it is impossible to decide whether solar pressure is a contributing factor [@aT]. The second data arc was 113 days long, starting six days prior to the second Earth encounter. This solution was also too highly correlated with solar pressure, and the data analysis was complicated by many mid-course maneuvers in the orbit. The uncertainties in the maneuvers were so great, a standard null result could not be ruled out. -10pt -10pt However, there was an additional result from the data of this second arc. This arc was chosen for study because of the availability of ranging data. It had 11596 Doppler points of which 10111 were used and 5643 range points of which 4863 used. The two-way range change and time integrated Doppler are consistent (see Figure \[fig:galileo\_range\]) to $\sim 4$ m over a time interval of one day. For comparison, note that for a time of $t=1$ day, $(a_Pt^2/2)\sim 3$ m. For the apparent acceleration to be the result of hardware problems at the tracking stations, one would need a linear frequency drift at all the DSN stations, a drift that is not observed. Ulysses measurement analysis {#ulysses} ---------------------------- ### JPL’s analysis An analysis of the radiation pressure on Ulysses, in its out-of-the-ecliptic journey from 5.4 AU near Jupiter in February 1992 to the perihelion at 1.3 AU in February 1995, found a varying profile with distance [@uly]. The orbit solution requires a periodic updating of the solar radiation pressure. The radio Doppler and ranging data can be fit to the noise level with a time-varying solar constant in the fitting model [@mcelrath]. We obtained values for the time-varying solar constant determined by Ulysses navigational data during this south polar pass [@uly]. The inferred solar constant is about 40 percent larger at perihelion (1.3 AU) than at Jupiter (5.2 AU), a physical impossibility! We sought an alternative explanation. Using physical parameters of the Ulysses spacecraft, we first converted the time-varying values of the solar constant to a positive (i.e., outward) radial spacecraft acceleration, $a_r$, as a function of heliocentric radius. Then we fit the values of $a_r$ with the following model: $$a_r = \frac{\mathcal{K}f_\odot A}{c M}\frac{\cos\theta(r)}{r^2} - a_{P(U)}, \label{Armodel_corr}$$ where $r$ is the heliocentric distance in AU, $M$ is the total mass of the spacecraft, $f_\odot=1367 ~{\rm W/m}^{2}$(AU)$^2$ is the (effective-temperature Stefan-Boltzmann) “solar radiation constant” at 1 AU, $A$ is the cross-sectional area of the spacecraft and $\theta(r)$ is the angle between the direction to the Sun at distance $r$ and orientation of the antennae. \[For the period analyzed $\theta(r)$ was almost a constant. Therefore its average value was used which corresponded to $\langle{\cos\theta(r)}\rangle\approx 0.82$.\] Optical parameters defining the reflectivity and emissivity of the spacecraft’s surface were taken to yield $\mathcal{K}\approx 1.8$. (See Section \[solarP\] for a discussion on solar radiation pressure.) Finally, the parameter $a_{P(U)}$ was determined by linear least squares. The best–fit value was obtained $$a_{P(U)} = (12 \pm 3)\times 10^{-8}~~{\rm cm/s}^2, \label{SandA}$$ where both random and systematic errors are included. So, by interpreting this time variation as a true $r^{-2}$ solar pressure plus a constant radial acceleration, we found that Ulysses was subjected to an unmodeled acceleration towards the Sun of (12 $\pm$ 3) $\times 10^{-8}$ cm/s$^{2}$. Note, however, that the determined constant $a_{P(U)}$ is highly correlated with solar radiation pressure (0.888). This shows that the constant acceleration and the solar-radiation acceleration are not independently determined, even over a heliocentric distance variation from 5.4 to 1.3 AU. ### Aerospace’s analysis {#sec:AUlysses} The next step was to perform a detailed calculation of the Ulysses orbit from near Jupiter encounter to Sun perihelion, using CHASMP to evaluate Doppler and ranging data. The data from 30 March 1992 to 11 August 1994 was processed. It consisted of 50213 Doppler points of which 46514 were used and 9851 range points of which 8465 were used. Such a calculation would in principle allow a more precise and believable differentiation between an anomalous constant acceleration towards the Sun and systematics. Solar radiation pressure and radiant heat systematics are both larger on Ulysses than on the Pioneers. However, this calculation turned out to be a much more difficult than imagined. Because of a failed nutation damper, an inordinate number of spacecraft maneuvers were required (257). Even so, the analysis was completed. But even though the Doppler and range residuals were consistent as for Galileo, the results were disheartening. For an unexpected reason, any fit is not significant. The anomaly is dominated by (what appear to be) gas leaks [@ulygas]. That is, after each maneuver the measured anomaly changes. The measured anomalies randomly change sign and magnitude. The values go up to about an order of magnitude larger than $a_P$. So, although the Ulysses data was useful for range/Doppler checks to test models (see Section \[sec:timemodel\]), like Galileo it could not provide a good number to compare to $a_P$. \[recent\_results\]RECENT RESULTS ================================= Recent changes to our strategies and orbit determination programs, leading to new results, are threefold. First, we have added a longer data arc for Pioneer 10, extending the data studied up to July 1998. The entire data set used (3 Jan. 1987 to 22 July 1998) covers a heliocentric distance interval from 40 AU to 70.5 AU [@AU]. \[Pioneer 11 was much closer in (22.42 to 31.7 AU) than Pioneer 10 during its data interval (5 January 1987 to 1 October 1990).\] For later use in discussing systematics, we here note that in the ODP calculations, masses used for the Pioneers were $M_{Pio~10}=251.883$ kg and $M_{Pio~11}=239.73$ kg. CHASMP used 251.883 kg for both [@gasuse]. As the majority of our results are from Pioneer 10, we will make $M_0 = 251.883$ kg to be our nominal working mass. Second, and as we discuss in the next subsection, we have studied the spin histories of the craft. In particular, the Pioneer 10 history exhibited a very large anomaly in the period 1990.5 to 1992.5. This led us to take a closer look at any possible variation of $a_P$ among the three time intervals: The JPL analysis defined the intervals as I (3 Jan. 1987 to 17 July 1990); II (17 July 1990 to 12 July 1992) bounded by 49.5 to 54.8 AU; and III (12 July 1992 to 22 July 1998). (CHASMP used slightly different intervals [@I/II]) The total updated data set now consists of 20,055 data points for Pioneer 10. (10,616 data points were used for Pioneer 11.) This helped us to better understand the systematic due to gas leaks, which is taken up in Section \[sec:gleaks\]. Third, in looking at the detailed measurements of $a_P$ as a function of time using ODP, we found an anomalous oscillatory annual term, smaller in size than the anomalous acceleration [@moriond]. As mentioned in Section \[sec:PE\], and as will be discussed in detail in Section \[annualterm\], we wanted to make sure this annual term was not an artifact of our computational method. For the latest results, JPL used both the batch-sequential and the least-squares methods. All our recent results obtained with both the JPL and The Aerospace Corporation software have given us a better understanding of systematic error sources. At the same time they have increased our confidence in the determination of the anomalous acceleration. We present a description and summary of the new results in the rest of this section. Analysis of the Pioneer spin history {#spinhistory} ------------------------------------ Both Pioneers 10 and 11 were spinning down during the respective data intervals that determined their $a_P$ values. Because any changes in spacecraft spin must be associated with spacecraft torques (which for lack of a plausible external mechanism we assume are internally generated), there is also a possibility of a related internally generated translational force along the spin axis. Therefore, it is important to understand the effects of the spin anomalies on the anomalous acceleration. In Figures \[fig:pioneer\_spin\] and \[fig:pio11spin\] we show the spin histories of the two craft during the periods of analysis. -5pt Consider Pioneer 10 in detail. In time Interval I there is a slow spin down at an average rate (slope) of $\sim(-0.0181\pm 0.0001)$ rpm/yr. Indeed, a closer look at the curve (either by eye or from an expanded graph) shows that the spin down is actually slowing with time (the curve is flattening). This last feature will be discussed in Sections \[subsec:katz\] and \[subsec:mainbus\]. Every time thrusters are used, there tends to be a short-term leakage of gas until the valves set (perhaps a few days later). But there can also be long-term leakages due to some mechanism which does not quickly correct itself. The major Pioneer 10 spin anomaly that marks the boundary of Intervals I and II, is a case in point. During this interval there was a major factor of $\sim 4.5$ increase in the average spin-rate change to $\sim(-0.0861\pm0.0009)$ rpm/yr. One also notices kinks during the interval. Few values of the Pioneer 10 spin rate were obtained after mid-1993, so the long-term spin-rate change is not well-determined in Interval III. But from what was measured, there was first a short-term transition region of about a year where the spin-rate change was $\sim-0.0160$ rpm/yr. Then things settled down to a spin-rate change of about $\sim(-0.0073 \pm 0.0015)$ rpm/yr, which is small and less than that of interval I. The effects of the maneuvers on the values of $a_P$ will allow an estimation of the gas leak systematic in Section \[sec:gleaks\]. Note, however, that in the time periods studied, only orientation maneuvers were made, not trajectory maneuvers. Shortly after Pioneer 11 was launched on 5 April 1973, the spin period was 4.845 s. A spin precession maneuver on 18 May 1973 reduced the period to 4.78 s and afterwards, because of a series of precession maneuvers, the period lengthened until it reached 5.045 s at encounter with Jupiter in December 1974. The period was fairly constant until 18 December 1976, when a mid-course maneuver placed the spacecraft on a Saturn-encounter trajectory. Before the maneuver the period was 5.455 s, while after the maneuver it was 7.658 s. At Saturn encounter in December 1979 the period was 7.644 s, little changed over the three-year post maneuver cruise phase. At the start of our data interval on 5 January 1987, the period was 7.321 s, while at the end of the data interval in October 1990 it was 7.238 s. Although the linear fit to the Pioneer 11 spin rate shown in Figure \[fig:pio11spin\] is similar to that for Pioneer 10 in Interval I, $\sim(-0.0234\pm 0.0003)$ rpm/yr, the causes appear to be very different. (Remember, although identical in design, Pioneers 10 and 11 were not identical in quality [@design].) Unlike Pioneer 10, the spin period for Pioneer 11 was primarily affected at the time of spin precession maneuvers. One sees that at maneuvers the spin period decreases very quickly, while in between maneuvers the spin rate actually tends to [*increase*]{} at a rate of $\sim(+0.0073 \pm 0.0003)$ rpm/yr (perhaps due to a gas leak in the opposite direction). All the above observations aid us in the interpretation of systematics in the following three sections. Recent results using JPL software {#jplresults} --------------------------------- The latest results from JPL are based on an upgrade, *Sigma*, to JPL’s ODP software [@sigma]. *Sigma*, developed for NASA’s Cassini Mission to Saturn, eliminates structural restrictions on memory and architecture that were imposed 30 years ago when JPL space navigation depended solely on a Univac 1108 mainframe computer. Five ODP programs and their interconnecting files have been replaced by the single program *Sigma* to support filtering, smoothing, and mapping functions. Program/Estimation method Pio 10 (I) Pio 10 (II) Pio 10 (III) Pio 11 -------------------------------- ------------------ ---------------- --------------- --------------- *Sigma*, [WLS]{}, $$&$$ $$&$$ no solar corona model $8.02\pm0.01$ $ 8.65\pm0.01$ $7.83\pm0.01$ $8.46\pm0.04$ *Sigma*, [WLS]{}, $$&$$ $$&$$ with solar corona model $8.00\pm0.01$ $8.66\pm0.01$ $7.84\pm0.01$ $8.44\pm0.04$ *Sigma*, [BSF]{}, 1-day batch, with solar corona model $7.82\pm0.29$ $8.16\pm0.40$ $7.59\pm0.22$ $8.49\pm0.33$ CHASMP, [WLS]{}, no solar corona model $8.25\pm0.02$ $8.86\pm0.02$ $7.85\pm0.01$ $8.71\pm0.03$ CHASMP, [WLS]{}, with solar corona model $8.22 \pm 0.02 $ $8.89\pm0.02$ $7.92\pm0.01$ $8.69\pm0.03$ CHASMP, [WLS]{}, with corona, weighting, and F10.7 $8.25\pm0.03$ $8.90\pm0.03$ $7.91\pm0.01$ $8.91\pm0.04$ We used *Sigma* to reduce the Pioneer 10 (in three time intervals) and 11 Doppler of the unmodeled acceleration, $a_P$, along the spacecraft spin axis. As mentioned, the Pioneer 10 data interval was extended to cover the total time interval 3 January 1987 to 22 July 1998. Of the total data set of 20,055 Pioneer 10 Doppler points, JPL used $\sim$19,403, depending on the initial conditions and editing for a particular run. Of the available 10,616 (mainly shorter time-averaged) Pioneer 11 data points, 10,252 were used (4919 two-way and 5333 three-way). We wanted to produce independent (i.e., uncorrelated) solutions for $a_P$ in the three Pioneer 10 segments of data. The word independent solution in our approach means only the fact that data from any of the three segments must not have any information (in any form) passed onto it from the other two intervals while estimating the anomaly. We moved the epoch from the beginning of one data interval to the next by numerically integrating the equations of motion and not iterating on the data to obtain a better initial conditions for this consequent segment. Note that this numerical iteration provided us only with an *a priori* estimate for the initial conditions for the data interval in question. Other parameters included in the fitting model were the six spacecraft heliocentric position and velocity coordinates at the 1987 epoch of 1 January 1987, 01:00:00 [ET]{}, and 84 (i.e., $28\times 3$) instantaneous velocity increments along the three spacecraft axes for 28 spacecraft attitude (or spin orientation) maneuvers. If these orientation maneuvers had been performed at exactly six month intervals, there would have been 23 maneuvers over our 11.5 year data interval. But in fact, five more maneuvers were performed than expected over this 11.5 year interval giving a total of 28 maneuvers in all. As noted previously, in fitting the Pioneer 10 data over 11.5 years we used the standard space-fixed J2000 coordinate system with planetary ephemeris DE405, referenced to ICRF. The three-dimensional locations of the tracking stations in the Earth’s body-fixed coordinate system (geocentric radius, latitude, longitude) were taken from a set recommended by ICRF for JPL’s DE405. The time-varying Earth orientation in J2000 coordinates was defined by a 1998 version of JPL’s EOP file. This accounted for the geophysical motion of the Earth’s pole with respect to its spin axis and the Earth’s time varying spin rate. JPL used both the weighted least-squares ([WLS]{}) and the batch-sequential filter ([BSF]{}) algorithms for the final calculations. In the first three rows of Table \[resulttable\] are shown the ODP results for i) [WLS]{} with no corona, ii) [WLS]{} with the Cassini corona model, and iii) [BSF]{} with the Cassini corona model. Observe that the [WLS]{} acceleration values for Pioneer 10 in Intervals I, II, and III are larger or smaller, respectively, just as the spin-rate changes in these intervals are larger or smaller, respectively. This indicates that the small deviations may be due to a correlation with the large gas leak/spin anomaly. We will argue this quantitatively in Section \[sec:gleaks\]. For now we just note that we therefore expect the number from Interval III, $a_P= 7.83 \times 10^{-8}$cm/s$^2$, to be close to our basic (least perturbed) JPL result for Pioneer 10. We also note that the statistical errors and the effect of the solar corona are both small for [WLS]{}, and will be handled in our error budget. In Figure \[ODPall\] we show ODP/*Sigma* [WLS]{} Doppler residuals for the entire Pioneer 10 data set. The residuals were obtained by first solving for $a_P$ with no corona in each of the three Now look at the batch-sequential results in row 3 of Table \[resulttable\]. First, note that the statistical Intervals independently and then subtracting these solutions (given in Table \[resulttable\]) from the fits within the corresponding data intervals. -10pt One can easily see the very close agreement with the CHASMP residuals of Figure \[fig:pio10best\_fit\], which go up to 14 December 1994. The Pioneer 11 number is significantly higher. A deviation is not totally unexpected since the data was relatively noisy, was from much closer in to the Sun, and was taken during a period of high solar activity. We also do not have the same handle on spin-rate change effects as we did for Pioneer 10. We must simply take the number for what it is, and give the basic JPL result for Pioneer 11 as $a_P= 8.46 \times 10^{-8}$ cm/s$^2$. Now look at the batch-sequential results in row 3 of Table \[resulttable\]. First, note that the statistical errors are an order of magnitude larger than for [WLS]{}. This is not surprising since: i) the process noise significantly affects the precision, ii) [BSF]{} smoothes the data and the data from the various intervals is more correlated than in [WLS]{}. The effects of all this are that all four numbers change so as to make them all closer to each other, but yet all the numbers vary by less than $2 \sigma$ from their [WLS]{} counterparts. Finally, there is the annual term. It remains in the data (for both Pioneers 10 and 11). A representation of it can be seen in a 1-day batch-sequential averaged over all 11.5 years. It yielded a result $a_P= (7.77 \pm 0.16) \times 10^{-8}$ cm/s$^2$, consistent with the other numbers/errors, but with an added annual oscillation. In the following subsection we will compare JPL results showing the annual term with the counterpart Aerospace results. We will argue in Section \[annualterm\] that this annual term is due to the inability to model the angles of the Pioneers’ orbits accurately enough. \[Note that this annual term is not to be confused with a small oscillation seen in Figure \[fig:aerospace\] that can be caused by mispointing towards the spacecraft by the fit programs.\] Recent results using The Aerospace Corporation software {#aerospaceresults} ------------------------------------------------------- As part of an ongoing upgrade to CHASMP’s accuracy, Aerospace has used Pioneer 10 and 11 as a test bed to confirm the revision’s improvement. In accordance with the JPL results of Section \[jplresults\], we used the new version of CHASMP to concentrate on the Pioneer 10 and 11 data. The physical models are basically the same ones that JPL used, but the techniques and methods used are largely different. (See Section \[Ext\_accuracy\].) The new results from the Aerospace Corporation’s software are based on first improving the Planetary Ephemeris and Earth orientation and spacecraft spin models required by the program. That is: i) the spin data file has been included with full detail; ii) a newer JPL Earth Orientation Parameters file was used; iii) all IERS tidal terms were included; iv) plate tectonics were included; v) DE405 was used; vi) no [*a priori*]{} information on the solved for parameters was included in the fit; vii) Pioneer 11 was considered, viii) the Pioneer 10 data set used was extended to 14 Feb. 1998. Then the Doppler data was refitted. Beginning with this last point: CHASMP uses the same original data file, but it performs an additional data compression. This compression combines the longest contiguous data composed of adjacent data intervals or data spans with duration $\ge 600$ s (effectively it prefers 600 and 1980 second data intervals). It ignores short-time data points. Also, Aerospace uses an N-$\sigma$/fixed boundary rejection criteria that rejects all data in the fit with a residual greater than $\pm 0.025$ Hz. These rejection criteria resulted in the loss of about 10 % of the original data for both Pioneers 10 and 11. In particular, the last five months of Pioneer 10 data, which was all of data-lengths less than 600 s, was ignored. Once these data compression/cuts were made, CHASMP used 10,499 of its 11,610 data points for Pioneer 10 and 4,380 of its 5,137 data points for Pioneer 11. Because of the spin-anomaly in the Pioneer 10 data, the data arc was also divided into three time intervals (although the I/II boundary was taken as 31 August 1990 [@I/II]). In what was especially useful, the Aerospace analysis uses direct propagation of the trajectory data and solves for the parameter of interest only for the data within a particular data interval. That means the three interval results were truly independent. Pioneer 11 was fit as a single arc. Three types of runs are listed, with: i) no corona; ii) with Cassini corona model of Sections \[corona+wt\] and \[sec:corona\]; and iii) with the Cassini corona model, but added are corona data weighting (Section \[corona+wt\]) and the time-variation called “F10.7” [@F10-7]. (The number 10.7 labels the wavelength of solar radiation, $\lambda$=10.7 cm, that, in our analysis, is averaged over 81 days.) The results are given in rows 4-6 of Table \[resulttable\]. The no corona results (row 4) are in good agreement with the *Sigma* results of the first row. This is especially true for the extended-time Interval III values for Pioneer 10, which interval had clean data. However there is more disagreement with the values for Pioneer 10 in Intervals I and II and for Pioneer 11. These three data sets all were noisy and underwent more data-editing. Therefore, it is significant that the deviations between *Sigma* and CHASMP in these arcs are all similar, but small, between $0.20$ to $0.25$ of our units. As before, the effect of the solar corona is small, even with the various model variations. But most important, the numbers from *Sigma* and CHASMP for Pioneer 10 Interval III are in excellent agreement. Further, CHASMP also found the annual term. (Recall that CHASMP can also look for a temporal variation by calculating short time averages.) Results on the time variation in $a_P$ can be seen in Figure \[fig:rec\_res\_comb\]. Although there could possibly be $a_P$ variations of $\pm 2\times10^{-8}$ cm/s$^2$ on a 200-day time scale, a comparison of the variations with the error limits shown in Figure \[fig:rec\_res\_comb\] indicate that our measurements of these variations are not statistically significant. The 5-day averages of $a_P$ from ODP (using the batch-sequential method) are not reliable at solar conjunction in the middle (June) of each year, and hence should be ignored there. The CHASMP 200-day averages suppress the solar conjunction bias inherent in the ODP 5-day averages, and they reliably indicate a constant value of $a_P$. Most encouraging, these results clearly indicate that the obtained solution is consistent, stable, and its mean value does not strongly depend on the estimation procedure used. The presence of the small annual term on top of the obtained solution is apparent. Our solution, before systematics, for the anomalous acceleration {#final_sol} ---------------------------------------------------------------- From Table \[resulttable\] we can intuitively draw a number of conclusions:\ A) The effect of the corona is small. This systematic will be analyzed in Section \[solarwind\].\ B) The numerical error is small. This systematic will be analyzed in Section \[leastsquares\].\ C) The differences between the *Sigma* and CHASMP Pioneer 10 results for Interval I and Interval II, respectively, we attribute to two main causes: especially i) the different data rejection techniques of the two analyses but also ii) the different maneuver simulations. Both of these effects were especially significant in Interval II, where the data arc was small and a large amount of noisy data was present. Also, to account for the discontinuity in the spin data that occurred on 28 January 1992 (see Figure \[fig:pioneer\_spin\]), Aerospace introduced a fictitious maneuver for this interval. Even so, the deviation in the two values of $a_P$ was relatively small, namely $0.23$ and $0.21$, respectively, $\times 10^{-8}$ cm/s$^2$.\ D) The changes in $a_P$ in the different Intervals, correlated with the changes in spin-rate change, are likely (at least partially) due to gas leakage. This will be discussed in Section \[sec:gleaks\]. But independent of the origin, this last correlation between shifts in $a_P$ and changes in spin rate actually allows us to calculate the best “experimental” base number for Pioneer 10. To do this, assume that the spin-rate change is directly contributing to an anomalous acceleration offset. Mathematically, this is saying that in any interval $i=\mathrm{I,II,III}$, for which the spin-rate change is an approximate constant, one has $$a_{P}(\ddot{\theta}) = a_{P(0)} - \kappa~\ddot{\theta}, \label{gasleakeq}$$ where $\kappa$ is a constant with units of length and $ a_{P(0)} \equiv a_P(\ddot{\theta}=0)$ is the Pioneer acceleration without any spin-rate change. One now can fit the data to Eq. (\[gasleakeq\]) to obtain solutions for $\kappa$ and $a_{P(0)}$. The three intervals $i=\mathrm{I,II,III}$ provide three data combinations $\{a_{P(i)}(\ddot{\theta}), \ddot{\theta}_i\}$. We take our base number, with which to reference systematics, to be the weighted average of the *Sigma* and CHASMP results for $a_{P(0)}$ when no corona model was used. Start first with the *Sigma* Pioneer 10 solutions in row one of Table \[resulttable\] and the Pioneer 10 spin-down rates given in Section \[jplresults\] and Figure \[fig:pioneer\_spin\]: $a_{P(i)}^{\tt Sigma}=(8.02\pm 0.01,~ 8.65\pm 0.01,~7.83\pm 0.01)$ in units of $10^{-8}$ cm/s$^2$ and $\ddot{\theta}_{i}=-(0.0181\pm 0.0001,~0.0861\pm 0.0009,~0.0073\pm 0.0015)$ in units of rpm/yr, where $$\begin{aligned} 1~\mathrm{rpm/year}&=& 5.281\times10^{-10}~\mathrm{rev/s}^2 \nonumber\\ &=& 3.318\times10^{-9}~\mathrm{radians/s}^2. \end{aligned}$$ With these data we use the maximum likelihood and minimum variance approach to find the optimally weighted least-squares solution for $a_{P(0)}$: $$\begin{aligned} a^{\tt Sigma}_{P(0)} &=& (7.82\pm 0.01) \times 10^{-8}~\mathrm{cm/s}^2, \label{eq:sol_sigma_a} %\\ \kappa^{\tt Sigma} &=& (29.2 \pm 0.7)~ \mathrm{cm}. %\label{eq:sol_sigma_k} \end{aligned}$$ with solution for the parameter $\kappa$ obtained as $\kappa^{\tt Sigma} = (29.2 \pm 0.7)~ \mathrm{cm}$. Similarly, for CHASMP one takes the values for $a_P$ from row four of Table \[resulttable\]: $a^{\tt CHASMP}_{P(i)}=(8.25\pm0.02,~ 8.86\pm0.02,~7.85\pm0.01)$ and uses them with the same $\ddot{\theta}_{i}$ as above. The solution for $a_{P(0)}$ in this case is $$\begin{aligned} a^{\tt CHASMP}_{P(0)} & = &(7.89\pm 0.02) \times 10^{-8}~\mathrm{cm/s}^2 \label{eq:sol_chasmp}, %\\\kappa^{\tt CHASMP} &= &(32.1 \pm 1.0)~ \mathrm{cm}.\end{aligned}$$ together with $\kappa^{\tt CHASMP} = (34.7 \pm 1.1)~ \mathrm{cm}$. The solutions for *Sigma* and CHASMP are similar, 7.82 and 7.89 in our units. We take the weighted average of these two to yield our base line “experimental” number for $a_P$: $$\begin{aligned} a_{P({\tt exper)}}^{\tt Pio10} &=& (7.84\pm 0.01)~\times~10^{-8}~\mathrm{cm/s}^2. \label{pio10lastresult}\end{aligned}$$ \[The weighted average constant $\kappa$ is $\kappa_0 =(30.7\pm 0.6)$ cm.\] For Pioneer 11, we only have the one 3$\frac{3}{4}$ year data arc. The weighted average of the two programs’ no corona results is $(8.62\pm 0.02) \times 10^{-8}$ cm/s$^2$. We observed in Section \[spinhistory\] that between maneuvers (which are accounted for - see Section \[model-maneuvers\]) there is actually a spin rate [*increase*]{} of $\sim(+0.0073 \pm 0.0003)$ rpm/yr. If one uses this spin-up rate and the Pioneer 10 value for $\kappa_0=30.7$ cm given above, one obtains a spin-rate change corrected value for $a_P$. We take this as the experimental value for Pioneer 11: $$a_{P({\tt exper)}}^{\tt Pio 11}= (8.55\pm 0.02) \times 10^{-8} ~{\rm cm/s}^2. \label{pio11lastresult}$$ \[ext-systema\]SOURCES OF SYSTEMATIC ERROR EXTERNAL TO THE SPACECRAFT ===================================================================== We are concerned with possible systematic acceleration errors that could account for the unexplained anomalous acceleration directed toward the Sun. There exist detailed publications describing analytic recipes developed to account for non-gravitational accelerations acting on spacecraft. (For a summary see Milani et al. [@milani].) With regard to the specific Pioneer spacecraft, possible sources of systematic acceleration have been discussed before for Pioneer 10 and 11 at Jupiter [@null76] and Pioneer 11 at Saturn [@null81]. External forces can produce three vector components of spacecraft acceleration, unlike forces generated on board the spacecraft, where the two non-radial components (i.e., those that are effectively perpendicular to the spacecraft spin) are canceled out by spacecraft rotation. However, non-radial spacecraft accelerations are difficult to observe by the Doppler technique, which measures spacecraft velocity along the Earth-spacecraft line of sight. But with several years of Doppler data, it is in principle possible to detect systematic non-radial acceleration components [@sunearth]. With our present analysis [@sunearth] we find that the Doppler data yields only one significant component of unmodeled acceleration, and that any acceleration components perpendicular to the spin axis are small. This is because in the fitting we tried including three unmodeled acceleration constants along the three spacecraft axes (spin axis and two orthogonal axes perpendicular to the spin axis). The components perpendicular to the spin axis had values consistent with zero to a 1-$\sigma$ accuracy of 2 $\times$ 10$^{-8}$ cm/s$^{2}$ and the radial component was equal to the reported anomalous acceleration. Further, the radial acceleration was not correlated with the other two unmodeled acceleration components. Although one could in principle set up complicated engineering models to predict all or each of the systematics, often the uncertainty of the models is too large to make them useful, despite the significant effort required. A different approach is to accept our ignorance about a non-gravitational acceleration and assess to what extent these can be assumed a constant bias over the time scale of all or part of the mission. (In fact, a constant acceleration produces a linear frequency drift that can be accounted for in the data analysis by a single unknown parameter.) In fact, we will use both approaches. In most orbit determination programs some effects, like the solar radiation pressure, are included in the set of routinely estimated parameters. Nevertheless we want to demonstrate their influence on Pioneer’s navigation from the general physics standpoint. This is not only to validate our results, but also to be a model as to how to study the influence of the other physical phenomena that are not yet included in the standard navigational packages for future more demanding missions. Such missions will involve either spacecraft that will be distant or spacecraft at shorter distances where high-precision spacecraft navigation will be required. In this section we will discuss possible systematics (including forces) generated external to the spacecraft which might significantly affect our results. These start with true forces due to (1) solar-radiation pressure and (2) solar wind pressure. We go on to discuss (3) the effect of the solar corona and its mismodeling, (4) electro-magnetic Lorentz forces, (5) the influence of the Kuiper belt, (6) the phase stability of the reference atomic clocks, and (7) the mechanical and phase stability of the DSN antennae, together with influence of the station locations and troposphere and ionosphere contributions. Direct solar radiation pressure and mass {#solarP} ---------------------------------------- There is an exchange of momentum when solar photons impact the spacecraft and are either absorbed or reflected. Models for this solar pressure effect were developed before either Pioneer 10 or 11 were launched [@rad] and have been refined since then. The models take into account various parts of the spacecraft exposed to solar radiation, primarily the high-gain antenna. It computes an acceleration directed away from the Sun as a function of spacecraft orientation and solar distance. The models for the acceleration due to solar radiation can be formulated as $$a_{\tt s.p.}(r)=\frac{\mathcal{K} f_\odot A }{c~M} \frac{ \cos\theta(r)}{r^2}. \label{eq:srp}$$ $f_\odot=1367 ~{\rm W/m}^{2}$(AU)$^2$ is the (effective-temperature Stefan-Boltzmann) “solar radiation constant” at 1 AU from the Sun and $A$ is the effective size of the craft as seen by the Sun [@solar_irr]. (For Pioneer the area was taken to be the antenna dish of radius 1.73 m.) $\theta$ is the angle between the axis of the antenna and the direction of the Sun, $c$ is the speed of light, $M$ is the mass of the spacecraft (taken to be 251.883 for Pioneer 10), and $r$ is the distance from the Sun to the spacecraft in AU. $\mathcal{K}$ [@Lambda] is the [*effective*]{} [@effect] absorption/reflection coefficient. For Pioneer 10 the simplest approximately correct model yields $\mathcal{K}_{0}=1.71$ [@effect]. Eq. (\[eq:srp\]) provides a good model for analysis of the effect of solar radiation pressure on the motion of distant spacecraft and is accounted for by most of the programs used for orbit determination. However, in reality the absorptivities, emissivities, and effective areas of spacecraft parts parameters which, although modeled by design, are determined by calibration early in the mission [@sunparam]. One determines the magnitude of the solar-pressure acceleration at various orientations using Doppler data. (The solar pressure effect can be distinguished from gravity’s $1/r^2$ law because $\cos\theta$ varies [@massprog].) The complicated set of program input parameters that yield the parameters in Eq. (\[eq:srp\]) are then set for later use [@sunparam]. Such a determination of the parameters for Pioneer 10 was done, soon after launch and later. When applied to the solar radiation acceleration in the region of Jupiter, this yields (from a 5 % uncertainty in $a_{\tt s.p.}$ [@null76]) $$\begin{aligned} a_{\tt s.p.}(r={\tt 5.2 AU})&=& (70.0 \pm 3.5) \times 10^{-8}~{\rm cm/s}^2, \nonumber\\ \mathcal{K}_{\tt 5.2} &=& 1.77. \label{aspS}\end{aligned}$$ The second of Eqs. (\[aspS\]) comes from putting the first into Eq. (\[eq:srp\]). Note, specifically, that in a fit a too high input mass will be compensated for by a higher effective $\mathcal{K}$. Because of the $1/r^2$ law, by the time the craft reached 10 AU the solar radiation acceleration was $18.9\times 10^{-8}$ cm/s$^2$ going down to 0.39 of those units by 70 AU. Since this systematic falls off as $r^{-2}$, it can bias the Doppler determination of a constant acceleration at some level, even though most of the systematic is correctly modeled by the program itself. By taking the average of the $r^{-2}$ acceleration curves over the Pioneer distance intervals, we estimate that the systematic error from solar-radiation pressure in units of 10$^{-8}$ ${\rm cm/s}^2$ is 0.001 for Pioneer 10 over an interval from 40 to 70 AU, and 0.006 for Pioneer 11 over an interval from 22 to 32 AU. However, this small uncertainty is not our main problem. In actuality, since the parameters were fit the mass has decreased with the consumption of propellant. Effectively, the $1/r^2$ systematic has changed its normalization with time. If not corrected for, the difference between the original $1/r^2$ and the corrected $1/r^2$ will be interpreted as a bias in $a_P$. Unfortunately, exact information on gas usage is unavailable [@gasuse]. Therefore, in dealing with the effect of the temporal mass variation during the entire data span (i.e. nominal input mass vs. actual mass history [@mass; @gasuse]) we have to address two effects on the solutions for the anomalous acceleration $a_P$. They are i) the effect of mass variation from gas consumption and ii) the effect of an incorrect input mass [@mass; @gasuse]. To resolve the issue of mass variation uncertainty we performed a sensitivity analysis of our solutions to different spacecraft input masses. We simply re-did the no-corona, WLS runs of Table \[resulttable\] with a range of different masses. The initial wet weight of the package was 259 kg with about 36 kg of consumable propellant. For Pioneer 10, the input mass in the program fit was 251.883 kg, roughly corresponding to the mass after spin-down. By our data period, roughly half the fuel (18 kg) was gone so we take 241 kg as our nominal Pioneer 10 mass. Thus, the effect of going from 251.883 kg to 241 kg we take to be our bias correction for Pioneer 10. We take the uncertainty to be given by one half the effect of going from plus to minus 9 kg (plus or minus a quarter tank) from the nominal mass of 241 kg. For the three intervals of Pioneer 10 data, using ODP/*Sigma* yields the following changes in the accelerations: $$\begin{aligned} \delta a^{\tt mass }_P &=& [(0.040 \pm 0.035),~(0.029 \pm 0.025), ~~~~~~~~~~~~~~~~\nonumber \\ &~& ~~~~~~~~~~~~~~~ (0.020 \pm 0.017)]~\times 10^{-8}~ \mathrm{cm/s}^2. \nonumber\end{aligned}$$ As expected,these results make $a_P$ larger. For our systematic bias we take the weighted average of $\delta a^{\tt mass }_P$ for the three intervals of Pioneer 10. The end result is $$a_{\tt s.p.}= (0.03~ \pm~ 0.01) \times 10^{-8}~ \mathrm{cm/s}^2.$$ For Pioneer 11 we did the same except our bias point was 3/4 of the fuel gone (232 kg). Therefore the bias results by going from the input mass of 239.73 to 232 kg. The uncertainty is again defined by $\pm$ 9 kg. The result for Pioneer 11 is more sensitive to mass changes, and we find using ODP/*Sigma* $$a_{\tt s.p.}= (0.09~ \pm~ 0.21) \times 10^{-8}~ \mathrm{cm/s}^2.$$ The bias number is three times larger than the similar number for Pioneer 10, and the uncertainty much larger. We return to this difference in Section \[twospace\]. The previous analysis also allowed us to perform consistency checks on the effective values of $\mathcal{K}$ which the programs were using. By taking $[r_{\mathrm{min}}r_{\mathrm{max}}]^{-1}= [\int(dr/r^2)/\int dr]$ for the inverse distance squared of a data set, varying the masses, and determining the shifts in $a_P$ we could determine the values of $\mathcal{K}$ implied, We found: $\mathcal{K}_{\tt Pio-10(I)}^{\tt ODP} \approx 1.72$; $\mathcal{K}_{\tt Pio-11}^{\tt ODP} \approx 1.82$; $\mathcal{K}_{\tt Pio-10(I)}^{\tt CHASMP} \approx 1.74$; and $\hat{\mathcal{K}}_{\tt Pio-11)}^{\tt CHASMP} \approx 1.84$. \[The hat over the last $\mathcal{K}$ indicates it was multiplied by (237.73/251.883) because CHASMP uses 259.883 kg instead of 239.73 kg for the input mass.\] All these values of $\mathcal{K}$ are in the region expected and are clustered around the value $\mathcal{K}_{\tt 5.2}$ in Eq. (\[aspS\]). Finally, if you take the average values of $\mathcal{K}$ for Pioneers 10 and 11 (1.73, 1.83), multiply these numbers by the input masses (251.883, 239.73) kg, and divide them by our nominal masses (241, 232) kg, you obtain (1.87, 1.89), indicating our choice of nominal masses was well motivated. The solar wind {#solarwind} -------------- The acceleration caused by the solar wind has the same form as Eq. (\[eq:srp\]), with $f_\odot$ replaced by $m_pv^3n$, where $n \approx5$ cm$^{-3}$ is the proton density at 1 AU and $v\approx400$ km/s is the speed of the wind. Thus, $$\begin{aligned} \sigma_{\tt s.w.}(r)&=&\mathcal{K}_{\tt s.w.}\frac{m_pv^3\,n\, A\cos\theta}{cM \,r^2}\nonumber\\ &\approx& 1.24\times10^{-13} \left(\frac{20 ~\rm AU}{r}\right)^2~{\rm cm/s}^2. \label{eq:sw}\end{aligned}$$ Because the density can change by as much as 100%, the exact acceleration is unpredictable. But there are measurements [@solar_irr] showing that it is about 10$^{-5}$ times smaller than the direct solar radiation pressure. Even if we make the very conservative assumption that the solar wind contributes only 100 times less force than the solar radiation, its smaller contribution is completely negligible. The effects of the solar corona and models of it {#sec:corona} ------------------------------------------------ As we saw in the previous Section \[solarwind\], the effect of the solar wind pressure is negligible for distant spacecraft motion in the solar system. However, the solar corona effect on propagation of radio waves between the Earth and the spacecraft needs to be analyzed in more detail. Initially, to study the sensitivity of $a_P$ to the solar corona model, we were also solving for the solar corona parameters $A$, $B$, and $C$ of Eq. (\[corona\_model\_content\]) in addition to $a_P$. However, we realized that the Pioneer Doppler data is not precise enough to produce credible results for these physical parameters. In particular, we found that solutions could yield a value of $a_P$ which was changed by of order 10 % even though it gave unphysical values of the parameters (especially $B$, which previously had been poorly defined even by the Ulysses mission [@bird]). \[By “unphysical” we mean electron densities that were either negative or positive with values that are vastly different from what would be expected.\] Therefore, as noted in Section \[corona+wt\], we decided to use the newly obtained values for $A$, $B$, and $C$ from the Cassini mission and use them as inputs for our analyses: $A= 6.0\times 10^3, B= 2.0\times 10^4, C= 0.6\times 10^6$, all in meters [@Ekelund]. This is the “Cassini corona model.” The effect of the solar corona is expected to be small for Doppler and large for range. Indeed it is small for *Sigma*. For ODP/*Sigma*, the time-averaged effect of the corona was small, of order $$\sigma_{\tt corona} = \pm 0.02~ \times~ 10^{-8}~ \mathrm{cm/s}^2,$$ as might be expected. We take this number to be the error due to the corona. What about the results from CHASMP. Both analyses use the same physical model for the effect of the steady-state solar corona on radio-wave propagation through the solar plasma (that is given by Eq. (\[corona\_model\])). However, there is a slight difference in the actual implementation of the model in the two codes. ODP calculates the corona effect only when the Sun-spacecraft separation angle as seen from the Earth (or Sun-Earth-spacecraft angle) is less then $\pi/2$. It sets the corona contribution to zero in all other cases. Earlier CHASMP used the same model and got a small corona effect. Presently CHASMP calculates an approximate corona contribution for all the trajectory. Specific attention is given to the region when the spacecraft is at opposition from the Sun and the Sun-Earth-spacecraft angle $\sim \pi$. There CHASMP’s implementation truncates the code approximation to the scaling factor $F$ in Eq. (\[corona\_model\]). This is specifically done to remove the fictitious divergence in the region where “impact parameter” is small, $\rho \rightarrow 0$. However, both this and also the more complicated corona models (with data-weighting and/or “F10.7” time variation) used by CHASMP produce small deviations from the no-corona results. Our decision was to incorporate these small deviations between the two results due to corona modeling into our overall error budget as a separate item: $$\sigma_{\tt corona\_model} = \pm 0.02~\times~ 10^{-8} ~~{\rm cm/s}^2.$$ This number could be discussed in Section \[Int\_accuracy\], on computational systematics. Indeed, that is where it will be listed in our error budget. Electro-magnetic Lorentz forces ------------------------------- The possibility that the spacecraft could hold a charge, and be deflected in its trajectory by Lorentz forces, was a concern for the magnetic field strengths at Jupiter and Saturn. However, the magnetic field strength in the outer solar system is on the order of $<1~\gamma~(\gamma=10^{-5}$ Gauss). This is about a factor of $10^5$ times smaller than the magnetic field strengths measured by the Pioneers at their nearest approaches to Jupiter: 0.185 Gauss for Pioneer 10 and 1.135 Gauss for the closer in Pioneer 11 [@edsmith]. Also, there is an upper limit to the charge that a spacecraft can hold. For the Pioneers that limit produced an upper bound on the Lorentz acceleration at closest approach to Jupiter of $20 \times 10^{-8}$ cm/s$^{2}$ [@null76]. With the interplanetary field being so much lower than at Jupiter, we conclude that the electro-magnetic force on the Pioneer spacecraft in the outer solar system is at worst on the order of $10^{-12}$ cm/s$^{2}$, completely negligible [@lorentz]. Similarly, the magnetic torques acting on the spacecraft were about a factor of $10^{-5}$ times smaller than those acting on Earth satellites, where they are a concern. Therefore, for the Pioneers any observed changes in spacecraft spin cannot be caused by magnetic torques. The Kuiper belt’s gravity {#sec:kuiper} ------------------------- From the study of the resonance effect of Neptune upon Pluto, two primary mass concentration resonances of 3:2 and 2:1 were discovered [@malhotra], corresponding to 39.4 AU and 47.8 AU, respectively. Previously, Boss and Peale had derived a model for a non-uniform density distribution in the form of an infinitesimally thin disc extending from 30 AU to 100 AU in the ecliptic plane [@liupeale]. We combined the results of Refs. [@malhotra] and [@liupeale] to determine if the matter in the Kuiper belt could be the source of the anomalous acceleration of Pioneer 10 [@liudust]. We specifically studied three distributions, namely: i) a uniform distribution, ii) a 2:1 resonance distribution with a peak at 47.8 AU, and iii) a 3:2 resonance distribution with a peak at 39.4 AU. Figure \[fig:pioneer\_kb\] exhibits the resulting acceleration felt by Pioneer 10, from 30 to 65 AU which encompassed our data set at the time. We assumed a total mass of one Earth mass, which is significantly larger than standard estimates. Even so, the accelerations are only on the order of $10^{-9}$ cm/s$^2$, which is two orders of magnitude smaller than the observed effect. (See Figure \[fig:pioneer\_kb\].) Further, the accelerations are not constant across the data range. Rather, they show an increasing effect as Pioneer 10 approaches the belt and a decreasing effect as Pioneer 10 recedes from the belt, even with a uniform density model. For these two reasons, we excluded the dust belt as a source for the Pioneer effect. More recent infrared observations have ruled out more than 0.3 Earth mass of Kuiper Belt dust in the trans-Neptunian region [@backman; @teplitzinfra]. Therefore, we can now place a limit of $\pm 3 \times 10^{-10}$ cm/s$^2$ for the contribution of the Kuiper belt. Finally, we note that searches for gravitational encounters of Pioneer with large Kuiper-belt objects have so far not been successful [@gio]. Phase and frequency stability of clocks {#sec:clocs} --------------------------------------- After traversing the mechanical components of the antenna, the radio signal enters the DSN antenna feed and passes through a series of amplifiers, filters, and cables. Averaged over many experiments, the net effect of this on the calculated dynamical parameters of a spacecraft should be very small. We expect instrumental calibration instabilities to contribute $0.2\times10^{-8} $ cm/s$^2$ to the anomalous acceleration on a 60 s time interval. Thus, in order for the atomic clocks [@vessot_clocks] to have caused the Pioneer effect, all the atomic clocks used for signal referencing clocks would have had to have drifted in the same manner as the local DSN clocks. In Section \[results\] we observed that without using the apparent anomalous acceleration, the CHASMP residuals show a steady frequency drift [@drift] of about $-6 \times 10^{-9}$ Hz/s, or 1.5 Hz over 8 years (one-way only). This equates to a clock acceleration, $-a_t$, of $-2.8\times 10^{-18}$ s/s$^{2}$. (See Eq. (\[asubt\]) and Figure \[fig:aerospace\].) To verify that it is actually not the clocks that are drifting, we analyzed the calibration of the frequency standards used in the DSN complex. The calibration system itself is referenced to Hydrogen maser atomic clocks. Instabilities in these clocks are another source of instrumental error which needs to be addressed. The local reference is synchronized to the frequency standards generated either at the National Institute of Standards and Technology (NIST), located in Boulder, Colorado or at the U. S. Naval Observatory (USNO), Washington, DC. These standards are presently distributed to local stations by the Global Positioning System (GPS) satellites. \[During the pre-GPS era, the station clocks used signals from WWV to set the Cesium or Hydrogen masers. WWV, the radio station which broadcasts time and frequency services, is located in Fort Collins, CO.\] While on a track, the station is “free-running,” i.e., the frequency and timing data are generated locally at the station. The Allan variances are about $10^{-13}$ for Cesium and $10^{-15}$ for Hydrogen masers. Therefore, over the data-pass time interval, the data accuracy is on the order of one part in 1000 GHz or better. Long-term frequency stability tests are conducted with the exciter/transmitter subsystems and the DSN’s radio-science open-loop subsystem. An uplink signal generated by the exciter is translated at the antenna by a test translator to a downlink frequency. (See Section \[Exp\_tech\].) The downlink signal is then passed through the RF-IF downconverter present at the antenna and into the radio science receiver chain [@dsn]. This technique allows the processes to be synchronized in the DSN complex based on the frequency standards whose Allan variances are of the order of $\sigma_y \sim 10^{-14}-10^{-15}$ for integration time in the range from 10 s to 10$^3$ s. For the S-band frequencies of the Pioneers, the corresponding Allan variances are 1.3 $\times$ 10$^{-12}$ and 1.0 $\times$ 10$^{-12}$, respectively, for a 10$^3$ s Doppler integration time. Phase-stability testing characterizes stability over very short integration times; that is, spurious signals whose frequencies are very close to the carrier (frequency). The phase noise region is defined to be frequencies within 100 kHz of the carrier. Both amplitude and phase variations appear as phase noise. Phase noise is quoted in dB relative to the carrier, in a 1 Hz band at a specified deviation from the carrier; for example, dBc-Hz at 10 Hz. Thus, for the frequency 1 Hz, the noise level is at $-51$ dBc and 10 Hz corresponds to $-60$ dBc. This was not significant for our study. Finally, the influence of the clock stability on the detected acceleration, $a_P$, may be estimated based on the reported Allan variances for the clocks, $\sigma_y$. Thus, the standard ‘single measurement’ error on acceleration as derived by the time derivative of the Doppler frequency data is $(c \sigma_y)/\tau$, where the Allan variance, $\sigma_y$, is calculated for 1000 s Doppler integration time, and $\tau$ is the signal averaging time. This formula provides a good rule of thumb when the Doppler power spectral density function obeys a $1/f$ flicker-noise law, which is approximately the case when plasma noise dominates the Doppler error budget. Assume a worst case scenario, where only one clock was used for the whole 11 years study. (In reality each DSN station has its own atomic clock.) To estimate the influence of that one clock on the reported accuracy of the detected anomaly $a_P$, combine $\sigma_y={\Delta\nu}/{\nu_0}$, the fractional Doppler frequency shift from the reference frequency of $\nu_0=\sim 2.29$ GHz, with the estimate for the Allan variance, $\sigma_y =1.3 \times 10^{-12}$. This yields a number that characterizes the upper limit for a frequency uncertainty introduced in a single measurement by the instabilities in the atomic clock: $\sigma_\nu=\nu_0\sigma_y=2.98\times10^{-3}$ Hz for a 10$^3$ Doppler integration time. In order to derive an estimate for the total effect, recall that the Doppler observation technique is essentially a continuous count of the total number of complete frequency circles during observational time. Within a year one can have as many as $N\approx3.156\times10^3$ independent single measurements of the clock with duration $10^3$ seconds. This yields an upper limit for the contribution of atomic clock instability on the frequency drift of $\sigma_{\tt clock} = {\sigma_\nu}/{\sqrt{N}} \approx 5.3\times 10^{-5}$ Hz/year. But in Section \[subsec:aero\] we noted that the observed $a_P$ corresponds to a frequency drift of about 0.2 Hz/year, so the error in $a_P$ is about $0.0003 \times 10^{-8}$ cm/s$^2$. Since all data is not integrated over 1,000 seconds and is data is not available for all time, we increase the numerical factor to $0.001$, which is still negligible to us. \[But further, this upper limit for the error becomes even smaller if one accounts for the number of DSN stations and corresponding atomic clocks that were used for the study.\] Therefore, we conclude that the clocks are not a contributing factor to the anomalous acceleration at a meaningfully level. We will return to this issue in Section \[sec:timemodel\] where we will discuss a number of phenomenological time models that were used to fit the data. DSN antennae complex {#sec:dsn_complex} -------------------- The mechanical structures which support the reflecting surfaces of the antenna are not perfectly stable. Among the numerous effects influencing the DSN antennae performance, we are only interested in those whose behavior might contribute to the estimated solutions for $a_P$. The largest systematic instability over a long period is due to gravity loads and the aging of the structure. As discussed in [@SoversJacobs96], antenna deformations due to gravity loads should be absorbed almost entirely into biases of the estimated station locations and clock offsets. Therefore, they will have little effect on the derived solutions for the purposes of spacecraft navigation. One can also consider ocean loading, wind loading, thermal expansion, and aging of the structure. We found none of these can produce the constant drift in the Doppler frequency on a time scale comparable to the Pioneer data. Also, routine tests are performed by DSN personnel on a regular basis to access all the effects that may contribute to the overall performance of the DSN complex. The information is available and it shows all parameters are in the required ranges. Detailed assessments of all these effect on the astrometric VLBI solutions were published in [@SFJ98; @SoversJacobs96]. The results for the astrometric errors introduced by the above factors may be directly translated to the error budget for the Pioneers, scaled by the number of years. It yields a negligible contribution. Our analyses also estimated errors introduced by a number of station-specific parameters. These include the error due to imperfect knowledge in a DSN station location, errors due to troposphere and ionosphere models for different stations, and errors due to the Faraday rotation effects in the Earth’s atmosphere. Our analysis indicates that at most these effects would produce a distance- and/or time-dependent drifts that would be easily noticeable in the radio Doppler data. What is more important is that none of the effects would be able to produce a constant drift in the Doppler residuals of Pioneers over such a long time scale. The updated version of the ODP, [*Sigma*]{}, routinely accounts for these error factors. Thus, we run covariance analysis for the whole set of these parameters using both [*Sigma*]{} and CHASMP. Based on these studies we conclude that mechanical and phase stability of the DSN antennae together with geographical locations of the antennae, geophysical and atmospheric conditions on the antennae site have negligible effects on our solutions for $a_P$. At most their contributions are at the level of $\sigma_{\tt DSN}\leq10^{-5}a_P$. \[int-systema\]SOURCES OF SYSTEMATIC ERROR INTERNAL TO THE SPACECRAFT ===================================================================== In this section we will discuss the forces that may be generated by spacecraft systems. The mechanisms we consider that may contribute to the found constant acceleration, $a_P$, and that may be caused by the on-board mechanisms include: (1) the radio beam reaction force, (2) RTG heat reflecting off the spacecraft, (3) differential emissivity of the RTGs, (4) non-isotropic radiative cooling of the spacecraft, (5) expelled Helium produced within the RTG, (6) thruster gas leakage, and (7) the difference in experimental results from the two spacecraft. Radio beam reaction force {#radioantbeam} ------------------------- The Pioneer navigation does not require that the spacecraft constantly beam its radio signal, but instead it does so only when it is requested to do so from the ground control. Nevertheless, the recoil force due to the emitted radio-power must also be analyzed. The Pioneers have a total nominal emitted radio power of eight Watts. It is parameterized as $$P_{\tt rp}~=\int_0^{\theta_{\tt max}} d\theta~ \sin\theta~ {\cal P}(\theta),$$ ${\cal P}(\theta)$ being the antenna power distribution. The radiated power has been kept constant in time, independent of the coverage from ground stations. That is, the radio transmitter is always on, even when not received by a ground station. The recoil from this emitted radiation produces an acceleration bias, $b_{\tt rp}$, on the spacecraft away from the Earth of $$b_{\tt rp}= \frac{\beta \,P_{\tt rp}}{Mc}. \label{eq:rp}$$ $M$ is taken to be the Pioneer mass when half the fuel is gone [@mass]. $\beta$ is the fractional component of the radiation momentum that is going in a direction opposite to $a_P$: $$\beta =\frac{1}{P_{\tt rp}} {\int_0^{\theta_{\tt max}} d\theta~ \sin\theta~\cos\theta~ {\cal P}(\theta)}. \label{radiopower}$$ Ref [@piodoc] describes the HGA and shows its downlink antenna pattern in Fig. 3.6-13. (Thermal antenna expansion mismodeling is thought to be negligible.) The gain is given as $(33.3 \pm 0.4)$ dB at zero (peak) degrees. The intensity is down by a factor of two ($-3$ dB) at 1.8 degrees. It is down a factor of 10 ($-10$ dB) at 2.7 degrees and down by a factor of 100 ($-20$ dB) at 3.75 degrees. \[The first diffraction minimum is at a little over four degrees.\] Therefore, the pattern is a very good conical beam. Further, since $\cos [3.75^\circ] = 0.9978$, we can take $\beta = (0.99 \pm 0.01)$, yielding $b_{\tt rp}=1.10$. Finally, taking the error for the nominal 8 Watts power to be given by the 0.4 dB antenna error ($0.10$) and the error due to the uncertainty in our nominal mass ($0.04$), we arrive at the result $$a_{\tt rp} =b_{\tt rp} \pm \sigma_{\tt rp} =(1.10 \pm 0.11)\times 10^{-8} ~{\rm cm/s}^2.$$ RTG heat reflecting off the spacecraft {#subsec:katz} -------------------------------------- It has been argued that the anomalous acceleration seen in the Pioneer spacecraft is due to anisotropic heat reflection off of the back of the spacecraft high-gain antennae, the heat coming from the RTGs [@katz]. Before launch, the four RTGs had a total thermal fuel inventory of 2580 W (now $\sim$ 2070 W). They produced a total electrical power of 160 W (now $\sim$ 65 W). Presently $\sim 2000$ W of RTG heat must be dissipated. Only $\sim63$ W of directed power could explain the anomaly. Therefore, in principle there is enough power to explain the anomaly this way. However, there are two reasons that preclude such a mechanism, namely: i\) [The spacecraft geometry:]{} The RTGs are located at the end of booms, and rotate about the spacecraft in a plane that contains the approximate base of the antenna. From the closest axial center point of the RTGs, the antenna is seen nearly “edge on” (the longitudinal angular width is 24.5$^o$). The total solid angle subtended is $\sim$ 1-2% of $4\pi$ steradians [@s]. Even though a more detailed calculation yields a value of 1.5% [@ss], even taking the higher bound of 2% means this proposal could provide at most $\sim 40$ W. But there is more [@heatreflect]. ii\) [The RTGs’ radiation pattern:]{} The above estimate was based on the assumption that the RTGs are spherical black bodies. But they are not. The main bodies of the RTGs are cylinders and they are grouped in two packages of two. Each package has the two cylinders end to end extending away from the antenna. Every RTG has six fins separated by equal angles of 60 degrees that go radially out from the cylinder. Presumably this results in a symmetrical radiation of thermal power into space. Thus, the fins are “edge on” to the antenna (the fins point perpendicular to the cylinder axes). The largest opening angle of the fins is seen only by the narrow-angle parts of the antenna’s outer edges. Ignoring these edge effects, only $\sim$2.5% of the surface area of the RTGs is facing the antenna. This is a factor 10 less than that from integrating the directional intensity from a hemisphere: $[(\int^{\tt h.sph.}d\Omega\cos\theta)/(4\pi)]=1/4$. So, one has only 4 W of directed power. This suggests a systematic bias of $\sim0.55 \times 10^{-8}$ cm/s$^2$. Even adding an uncertainty of the same size yields a systematic for heat reflection of $$a_{\tt h.r.}= (-0.55 \pm 0.55) \times 10^{-8}~\mathrm{ cm/s}^2.$$ But there are reasons to consider this an upper bound. The Pioneer SNAP 19 RTGs have larger fins than the earlier test models and the packages were insulated so that the end caps have lower temperatures. This results in lower radiation from the end caps than from the cylinder/fins [@tele; @Rconf]. As a result, even though this is not exact, we can argue that the vast majority of the heat radiated by the RTGs is symmetrically directed to space unobscured by the antenna. Further, for this mechanism to work one still has to assume that the energy hitting the antenna is completely reradiated in the direction of the spin axis [@heatreflect]. Finally, if this mechanism were the cause, ultimately an unambiguous decrease in the size of $a_P$ should be seen because the RTGs’ radioactively produced radiant heat is decreasing. As noted previously, the heat produced is now about 80% of the original magnitude. In fact, one would similarly expect a decrease of about $0.75\times 10^{-8}$ cm/s$^2$ in $a_P$ over the 11.5 year Pioneer 10 data interval if this mechanism were the origin of $a_P$. So, even though a complete thermal/physical model of the spacecraft might be able to ascertain if there are any other unsuspected heat systematics, we conclude that this particular mechanism does not provide enough power to explain the Pioneer anomaly [@uskatz]. In addition to the observed constancy of the anomalous acceleration, any explanation involving thermal radiation must also discuss the absence of a disturbance to the spin of the spacecraft. There may be a small correlation of the spin angular acceleration with the anomalous linear acceleration. However, as described in Section \[recent\_results\], the linear acceleration is much more constant than the spin. This suggests that most of the linear acceleration is not caused by whatever disturbs the spin, thermal or not. However, a careful look at the Interval I results of Figure \[fig:pioneer\_spin\] shows that the nearly steady, background spin-rate change of about $6 \times 10^{-5}$ rpm/day is slowly decreasing. In principle this could be caused by heat. The spin-rate change produced by the torque of radiant power directed against the rotation with a lever arm $d$ is $$\ddot{\theta} = \frac {P~d}{c~{\cal I}_{\tt z}}, \label{tdd}$$ where ${\cal I}_{\tt z}$ is the moment of inertia, 588.3 kg m$^2$ [@vanallen]. We take a base unit of $\ddot{\theta}_0$ for a power of one Watt and a lever arm of one meter. This is $$\begin{aligned} \ddot{\theta}_0 &=& 5.63 \times 10^{-12}~\mathrm{rad/s^2} =4.65 \times 10^{-6}~\mathrm{ rpm/day} = \nonumber\\ &=& 1.71 \times 10^{-3}~\mathrm{ rpm/yr}. \end{aligned}$$ So, about 13 Watt-meters of directed power could cause the base spin-rate change. It turns out that such sources could, in principle, be available. There are $3\times 3 = 9$ radioisotope heater units (RHUs) with one Watt power to heat the Thruster Cluster Assembly (TCA). (See pages 3.4-4 and 3.8-1–3.8-17 of Ref. [@piodoc].) The units are on the edge of the antenna of radius 1.37 m, in the housings of the TCAs which are approximately 180$^\circ$ apart from each other. At one position there are six RHUs and at the other position there are three. An additional RHU is near the sun sensor which is located near the second assembly. The final RHU is located at the magnetometer, 6.6 meters out from the center of the spacecraft. The placement gives an “ideal” rotational asymmetry of two Watts. But note, the real asymmetry should be less, since these RHUs do not radiate only in one direction. Even one Watt unidirected at the magnetometer, is not enough to cause the baseline spin rate decrease. Further, since the base line is decreasing faster than what would come from the change cause by radioactive decay decrease, one cannot look for this effect or some complicated RTG source as the entire origin of the baseline change. One would suspect a very small gas leak or a combination of this and heat from the powered bus. (See Section \[subsec:mainbus\].) Indeed, the factor $1/c$ in Eq. (\[tdd\]) is a manifestation of the energy-momentum conservation power needed to produce $\ddot{\theta}$ by heat vs. massive particles. But in any event, this baseline spin-rate change is not significantly correlated with the anomalous acceleration, so we do not have to pursue it further. Differential emissivity of the RTGs {#differemit} ----------------------------------- Another suggestion related to the RTGs is the following [@slusher]: during the early parts of the missions, there might have been a differential change of the radiant emissivity of the solar-pointing sides of the RTGs with respect to the deep-space facing sides. Note that, especially closer in the Sun, the inner sides were subjected to the solar wind. Contrariwise, the outer sides were sweeping through the solar-system dust cloud. Therefore, it can be argued that these two processes could have caused the effect. However, other information seems to make it difficult for this explanation to work. The six fins of each RTG, designed to “provide the bulk of the heat rejection capacity,” were fabricated of HM21A-T8 magnesium alloy plate [@tele]. The metal, after being specially prepared, was coated with two to three mils of zirconia in a sodium silicate binder to provide a high emissivity $(\sim 0.9)$ and low absorptivity $(\sim 0.2)$. Depending on how symmetrically fore-and-aft they radiated, the relative fore-and-aft emissivity of the alloy would have had to have changed by $\sim10$% to account for $a_P$ (see below). Given our knowledge of the solar wind and the interplanetary dust (see Section \[sec:know\]), we find that this amount of a radiant change would be difficult to explain, even if it were of the right sign. (In fact, even the brace bars holding the RTGs were built such that radiation is roughly fore/aft symmetric,) We also have “visual” evidence from the Voyager spacecraft. As mentioned, the Voyagers are not spin-stabilized. They have imaging video cameras attached [@camera]. The cameras are mounted on a scan platform that is pointed under both celestial and inertial attitude control modes [@plate]. The cameras [*do not*]{} have lens covers [@hansen]. During the outward cruise calibrations, the cameras were sometimes pointed towards an imaging target plate mounted at the rear of the spacecraft. But most often they were pointed all over the sky at specific star fields in support of ultraviolet spectrometer observations. Meanwhile, the spacecraft antennae were pointed towards Earth. Therefore, at an angle, the lenses were sometimes hit by the solar wind and sometimes by the interplanetary dust. Even so, there was no noticeable deterioration of the images received, even when Voyager 2 reached Neptune [@neptune]. We infer, therefore, that this mechanism can not explain the Pioneer effect. It turned out that the greatest radiation damage occurred during the flybys. The peak Pioneer 10 radiation flux near Jupiter was about 10,000 times that of Earth for electrons (1,000 times for protons). Pioneer 11 experienced an even higher radiation flux and also went by Saturn [@piopr2]. (We return to this in Section \[twospace\].) Therefore, if radiation damage was a problem, one should have seen an approximately uniform change in emissivity during flyby. Since the total heat flux, $\cal{F}$, from the RTGs was a constant over a flyby, there would have been a change in the RTG surface temperature manifested by the radiation formula ${\cal{F}} \propto \epsilon_1T_1^4 = \epsilon_2T_2^4$, the $\epsilon_i$ being the emissivities of the fin material. There are several temperature sensors mounted at RTG fin bases. They measured average temperatures of approximately 330 F, roughly 440 K. Therefore, a 10% change in the [*total average*]{} emissivity would have produced a temperature change of $\sim$12.2 K $=$ 22 F. Such a change would have been noticed. (Measurements would be compared from, say, 30 days before and after flyby to eliminate the flyby power/thermal distortions.) Since (see below) a 10% [*differential*]{} fore/aft emissivity could cause the Pioneer effect, the lack of observation of a 10% [*total average*]{} emissivity change limits the size of the differential emissivity systematic. To obtain a reasonable estimate of the uncertainty, consider if one side (fore or aft) of the RTGs had its emissivity changed by 1% with respect to the other side. In a simple cylindrical model of the RTGs, with 2000 W power (here we presume only radial emission with no loss out the sides), the ratio of power emitted by the two sides would be 0.99 = 995/1005, or a differential emission between the half cylinders of 10 W. Therefore, the fore/aft asymmetry towards the normal would be $[10~{\mathrm{W}}] \times \int_0^\pi [\sin \phi]d\phi/\pi \approx 6.37$ W. If one does a more sophisticated fin model, with 4 of the 12 fins facing the normal (two flat and two at 30$^\circ$), one gets a number of 6.12 W. We take this to yield our uncertainty, $$\sigma_{\tt d.e.} = 0.85 \times 10^{-8} ~ {\mathrm{cm/s}}^2.$$ Note that $10~ \sigma_{\tt d.e.}$ almost equals our final $a_P$. This is the origin of our previous statement that $\sim 10$% differential emissivity (in the correct direction) would be needed to explain $a_P$. Finally, we want to comment on the significance of radioactive decay for this mechanism. Even acknowledging the Interval jumps due to gas leaks (see below), we reported a one-day batch-sequential value (before systematics) for $a_P$, averaged over the entire 11.5 year interval, of $a_P = (7.77 \pm 0.16)\times 10^{-8}$ cm/s$^2$. From radioactive decay, the value of $a_P$ should have decreased by $0.75$ of these units over 11.5 years. This is 5 times the above variance, which is very large with batch sequential. Even more stringently, this bound is good for [*all*]{} radioactive heat sources. So, what if one were to argue that emissivity changes occurring before 1987 were the cause of the Pioneer effect? There still should have been a decrease in $a_P$ with time since then, which has not been observed. We will return to these points in Section \[twospace\]. Non-isotropic radiative cooling of the spacecraft {#subsec:mainbus} ------------------------------------------------- It has also been suggested that the anomalous acceleration seen in the Pioneer 10/11 spacecraft can be, “explained, at least in part, by non-isotropic radiative cooling of the spacecraft [@murphy].” So, the question is, does “at least in part” mean this effect comes near to explaining the anomaly? We argue it does not [@usmurphy]. Consider radiation of the main-bus electrical systems power from the spacecraft rear. For the Pioneers, the aft has a louver system, and “the louver system acts to control the heat rejection of the radiating platform. A bimetallic spring, thermally coupled radiatively to the platform, provides the motive force for altering the angle of each blade. In a closed position (below 40 F) the heat rejection of the platform is minimized by virtue of the blockage of the blades while open fin louvers provide the platform with a nearly unobstructed view of space [@piodoc].” If these louvers were open (above $\sim$ 88 F) and all the diminishing electrical-power heat was radiated only out of the louvers, this mechanism could produce a significant effect. However, by nine AU the actuator spring temperature had already reached $\sim$40 F [@piodoc]. This means the louver doors were closed (i.e., the louver angle was zero) from where we obtained our data. Thus, from that time on of the radiation properties, the contribution of the thermal radiation to the Pioneer anomalous acceleration should be small. Although one might speculate that a louver stuck, there are 30 louvers on each craft. They clearly worked as designed, or else the temperature of the crafts’ interiors would have fallen to disastrous levels. As shown in Figure \[epower\], in 1984 Pioneer 10 was at about 33 AU and the power was about 105 W. (Always reduce the total power numbers by 8 W to account for the radio beam power.) In (1987, 1992, 1996) the spacecraft was at $\sim$(41, 55, 65) AU and the power was $\sim$(95, 82, 73) W. The louvers were inactive, and no decrease in $a_P$ was seen. In fact, during the entire 11.5 year period from 1987 to 1998 the electrical power decreased from around 95 W to around 68 W, a change of 27 W. Since we already have noted that about $\sim 65$ W is needed to cause our effect, such a large decrease in the “source” of the acceleration would have been seen. But as shown in Section \[recent\_results\], it was not. Even the small differences in the three intervals are most likely to be from gas leaks (as will be demonstrated in Section \[sec:gleaks\]). Later a double modification of this idea was given. It was first suggested that “most, if not all, of the unmodeled acceleration” of Pioneer 10 and 11 is due to an essentially constant supply of heat coming from the central compartment, directed out the front of the craft through the closed louvers [@scheffer]-(a). However, when one studies the electrical power history in both parts (instruments and experimental) of the central compartment, there is no constancy of heat. (See the details in [@usscheffer].) Indeed during our data period the heat from this compartment decreased from about 73 W to about 57 Watts, or a factor of 1.26. This is inconsistent with the constancy of our result. Further, if one looks at the earlier, very roughly analyzed [@earlydata] data in Figure \[fig:correlation\] one sees nothing close to the internal power change of 93 to 57 W (a factor of 1.6) [@usscheffer]. To address this inconsistency a second modification [@scheffer]-b,c was made. It was arbitrarily argued that there was an incorrect determination of the reflection/absorption coefficients by a large factor. But these coefficients are known to 5%. If they were as poorly determined as speculated, the mission would have failed early on. (Further discussion is in [@usscheffer].) We conclude that neither the original proposal [@murphy] nor the modification [@scheffer] can explain the anomalous Pioneer acceleration [@usmurphy; @usscheffer]. A bound on the constancy of $a_P$ comes from first noting the 11.5 year 1-day batch-sequential result, sensitive to time variation: $a_P = (7.77 \pm 0.16)\times 10^{-8}$ cm/s$^2$. Also given the constancy of the earlier imprecise date, it is conservative to take three times this error to be our systematic uncertainty for radiative cooling of the craft, $\sigma_{\tt r.c.}= \pm 0.48 \times 10^{-8}$ cm/s$^2$. Although doubtful, one can also speculate that some mechanism like this might be involved with the baseline spin-rate change discussed in Section \[subsec:katz\]. In 1986-7, Pioneer 10 power was about 97 W, decreasing at about 2.5-3.0 W/yr. If you take a lever arm of 0.71 meters (the hexagonal bus size), this is more than enough to provide the 13 W-meters necessary to produce the baseline spin-rate change of Figure \[fig:pioneer\_spin\]. Further for the first three years the decrease about matches the bus power loss rate. Then after the complex changes associated with the end of 1989 to 1990, there is a decrease in the base rate with a continued similar slope. Perhaps the “baseline" rate is indeed from the heat of the bus being vented to the side. But the much larger gas leaks would be on top of the baseline. Expelled Helium produced within the RTGs {#subsec:helium} ---------------------------------------- Another possible on-board systematic is from the expulsion of the He being created in the RTGs from the $\alpha$-decay of $^{238}$Pu. To make this mechanism work, one would need that the He leakage from the RTGs be preferentially directed away from the Sun, with a velocity large enough to cause the acceleration. The SNAP-19 Pioneer RTGs were designed in a such a way that the He pressure has not been totally contained within the Pioneer heat source over the life of RTGs [@tele]. Instead, the Pioneer heat source contains a pressure relief device which allows the generated He to vent out of the heat source and into the thermoelectric converter. (The strength member and the capsule clad contain small holes to permit He to escape into the thermoelectric converter.) The thermoelectric converter housing-to-power output receptacle interface is sealed with a viton O-ring. The O-ring allows the helium gas within the converter to be released by permeation to the space environment throughout the mission life of the Pioneer RTGs. Information on the fuel pucks [@puck] shows that they each have heights of 0.212 inches with diameters of 2.145 inches. With 18 in each RTG and four RTGs per mission, this gives a total volume of fuel of about 904 cm$^3$. The fuel is PMC Pu conglomerate. The amount of $^{238}$Pu in this fuel is about 5.8 kg. With a half life of 87.74 years, that means the rate of He production (from Pu decay) is about 0.77 gm/year, assuming it all leaves the cermet. Taking on operational temperature on the RTG surface of 320 F = 433 K, implies a $3kT/2$ helium velocity of 1.22 km/s. (The possible energy loss coming out of the viton is neglected for helium.) Using this in the rocket equation, $$a(t) = -v(t) \frac{d}{dt} \Big[\ln M(t)\Big]$$ with our nominal Pioneer mass with half the fuel gone [*and the assumption*]{} that the gas is all unidirected, yields a maximal bound on the possible acceleration of $1.16 \times 10^{-8}$ cm/s$^2$. So, we can rule out helium permeating through the O-rings as the cause of $a_P$ although it is a systematic to be dealt with. Of course, the gas is not totally unidirected. As one can see by looking at Figures \[fig:trusters\] and III-2 of [@tele]: the connectors with the O-rings are on the RTG cylinder surfaces, on the ends of the cylinders where the fins are notched. They are equidistant (30 degrees) from two of the fins. The placement is exactly at the “rear” direction of the RTG cylinders, i.e., at the position closest to the Sun/Earth. The axis through the O-rings is parallel to the spin-axis. The O-rings, sandwiched by the receptacle and connector plates, “see” the outside world through an angle of about 90$^\circ$ in latitude [@hefromrtg]. (Overhead of the O-rings is towards the Sun.) In longitude the O-rings see the direction of the bus and space through about 90$^\circ$, and “see” the fins through most of the rest of the longitudinal angle. If one assumes a single elastic reflection, one can estimate the fraction of the bias away from the Sun. (Indeed, multiple and back reflections will produce an even greater bias. Therefore, we feel this approximation is justified.) This estimate is $(3/4) \sin30^\circ$ times the average of the heat momentum component parallel to the shortest distance to the RTG fin. Using this, we find the bias would be $0.31 \times 10^{-8}$ cm/s$^2$. This bias effectively increases the value of our solution for $a_P$, which we hesitate to accept given all the true complications of the real system. Therefore we take the systematic expulsion to be $a_{\tt He} = (0.15 \pm 0.16) \times 10^{-8}$ cm/s$^2$. Propulsive mass expulsion due to gas leakage {#sec:gleaks} -------------------------------------------- The effect of propulsive mass expulsion due to gas leakage has to be assessed. Although this effect is largely unpredictable, many spacecraft have experienced gas leaks producing accelerations on the order of $10^{-7} ~{\rm cm/s^2}$. \[The reader will recall the even higher figure for Ulysses found in Section \[sec:AUlysses\].\] As noted previously, gas leaks generally behave differently after each maneuver. The leakage often decreases with time and becomes negligibly small. Gas leaks can originate from Pioneer’s propulsion system, which is used for mid-course trajectory maneuvers, for spinning-up or -down the spacecraft, and for orientation of the spinning spacecraft. The Pioneers are equipped with three pairs of hydrazine thrusters which are mounted on the circumference of the Earth-pointing high gain antenna. Each pair of thrusters forms a Thruster Cluster Assembly (TCA) with two nozzles aligned in opposition to each other. For attitude control, two pairs of thrusters can be fired forward or aft and are used to precess the spinning antenna (See Section \[sec:prop\].) The other pair of thrusters is aligned parallel to the rim of the antenna with nozzles oriented in co- and contra-rotation directions for spin/despin maneuvers. During both observing intervals for the two Pioneers, there were no trajectory or spin/despin maneuvers. So, in this analysis we are mainly concerned with precession (i.e., orientation or attitude control) maneuvers only. (See Section \[sec:prop\].) Since the valve seals in the thrusters can never be perfect, one can ask if the leakages through the hydrazine thrusters could be the cause of the anomalous acceleration, $a_P$. However, when we investigate the total computational accuracy of our solution in Section \[Ext\_accuracy\], we will show that the currently implemented models of propulsion maneuvers may be responsible for an uncertainty in $a_P$ only at the level of $\pm0.01\times 10^{-8}$ cm/s$^2$. Therefore, the maneuvers themselves are the main contributors neither to the total error budget nor to the gas leak uncertainty, as we now detail The serious uncertainty comes from the possibility of undetected gas leaks. We will address this issue in some detail. First consider the possible action of gas leaks originating from the spin/despin TCA. Each nozzle from this pair of thrusters is subject to a certain amount of gas leakage. But only a differential leakage from the two nozzles would produce an observable effect causing the spacecraft to either spin-down or spin-up [@leaks]. So, to obtain a gas leak uncertainty (and we emphasize “uncertainty” vs. “error” because we have no other evidence) let us ask how large a differential force is needed to cause the spin-down or spin-up effects observed? Using the moment of inertia about the spin axis, ${\cal I}_{\tt z}=\sim 588.3$ kg$\cdot$m${^2}$ [@vanallen], and the antenna radius, ${\cal{R}}=1.37$ m, as the lever arm, one can calculate that the differential force needed to torque the spin-rate change, $\ddot{\theta}_i$, in Intervals $i=$I,II,III is $$\begin{aligned} F_{\ddot{\theta}_i} &=& \frac{{\cal I}_{\tt z}{\ddot{\theta}_i}}{{\cal{R}}} =\big(2.57, ~12.24, ~1.03\big) \times 10^{-3}~~{\rm dynes}.\hskip 20pt \label{FthetaI} \end{aligned}$$ It is possible that a similar mechanism of undetected gas leakage could be responsible for the net differential force acting in the direction along the line of sight. In other words, what if there were some undetected gas leakage from the thrusters oriented along the spin axis of the spacecraft that is causing $a_P$? How large would this have to be? A force of ($M= 241$ kg) $$F_{a_P}= M \, a_P =21.11\times10^{-3}~{\rm dynes}$$ would be needed to produce our final unbiased value of $a_P$. (See Section \[budget\]. That is, one would need even more force than is needed to produce the anomalously high rotational gas leak of Interval II. Furthermore, the differential leakage to produce this $a_P$ would have had to have been constant over many years and in the same direction for both spacecraft, without being detected as a spin-rate change. That is possible, but certainly not demonstrated. Furthermore if the gas leaks hypothesis were true, one would expect to see a dramatic difference in $a_P$ during the three Intervals of Pioneer 10 data. Instead an almost 500 % spin-down rate change between Intervals I and II resulted only in a less than 8% change in $a_P$. Given the small amount of information, we propose to [*conservatively*]{} take as our gas leak uncertainties the acceleration values that would be produced by differential forces equal to $$\begin{aligned} F_{a_P(i)\tt g.l.}&\simeq & \pm \sqrt{2}F_{\ddot{\theta}_i} = \\ &=& \big(\pm 3.64,~\pm 17.31,~\pm 1.46\big) \times 10^{-3}~~{\rm dynes}.\nonumber \label{eq:diff}\end{aligned}$$ The argument for this is that, in the root sum of squares sense, one is accounting for the differential leakages from the two pairs of thrusters with their nozzles oriented along the line of sight direction. This directly translates into the acceleration errors introduced by the leakage during the three intervals of Pioneer 10 data, $$\begin{aligned} \sigma(a_{P(i) \tt g.l.})&=& \pm F_{a_P(i)\tt g.l.}/M =\\ &=&\big(\pm 1.51,~\pm 7.18,~\pm 0.61\big)\times10^{-8}~{\rm cm/s}^2. \nonumber\end{aligned}$$ Assuming that these errors are uncorrelated and are normally distributed around zero mean, we find the gas leak uncertainty for the entire Pioneer 10 data span to be $$\sigma_{\tt g.l.} = \pm 0.56 \times 10^{-8}~~{\rm cm/s}^2. \label{gluncertC}$$ This is one of our largest uncertainties. The data set from Pioneer 11 is over a much smaller time span, taken when Pioneer 11 was much closer to the Sun (off the plane of the ecliptic), and during a maximum of solar activity. For Pioneer 11 the main effects of gas leaks occurred at the maneuvers, when there were impulsive lowerings of the spin-down rate. These dominated the over-all spin rate change of $\ddot{\theta}_{11}= -0.0234$ rpm/yr. (See Figure \[fig:pio11spin\].) But in between maneuvers the spin rate was actually [*increasing*]{}. One can argue that this explains the higher value for $a_{P(11)}$ in Table \[resulttable\] as compared to $a_{P(10)}$. Unfortunately, one has no [*a priori*]{} way of predicting the effect here. We do not know that the same specific gas leak mechanism applied here as did in the case of Pioneer 10 and there is no well-defined interval set as there is for Pioneer 10. Therefore, although we feel this “spin up” may be part of the explanation of the higher value of $a_P$ for Pioneer 11, we leave the different numbers as a separate systematic for the next subsection. At this point, we must conclude that the gas leak mechanism for explaining the anomalous acceleration seems very unlikely, because it is hard to understand why it would affect Pioneer 10 and 11 at the same level (given that both spacecraft had different quality of propulsion systems, see Section \[sec:prop\]). One also expects a gas leak would obey the rules of a Poisson distribution. That clearly is not true. Instead, our analyses of different data sets indicate that $a_P$ behaves as a constant bias rather than as a random variable. (This is clearly seen in the time history of $a_P$ obtained with batch-sequential estimation.) Variation between determinations from the two spacecraft {#twospace} -------------------------------------------------------- Finally there is the important point that we have two “experimental” results from the two spacecraft, given in Eqs. (\[pio10lastresult\]) and (\[pio11lastresult\]): 7.84 and 8.55, respectively, in units of $10^{-8}$ cm/s$^2$. If the Pioneer effect is real, and not a systematic, these numbers should be approximately equal. The first number, 7.84, is for Pioneer 10. In Section \[final\_sol\] we obtained this number by correlating the values of $a_P$ in the three data Intervals with the different spin-down rates in these Intervals. The weighted correlation between a shift in $a_P$ and the spin-down rate is $\kappa_0 =(30.7\pm 0.6)$ cm. (We argued in the previous Section \[sec:gleaks\] that this correlation is the manifestation of the rotational gas leak systematic.) Therefore, this number represents the entire 11.5 year data arc of Pioneer 10. Similarly, Pioneer 11’s number, 8.55, represents a 3$\frac{3}{4}$ year data arc. Even though the Pioneer 11 number may be less reliable since the craft was so much closer to the Sun, we calculate the time-weighted average of the experimental results from the two craft: $[(11.5)(7.84) + (3.75)(8.55)]/(15.25) = 8.01$ in units of $10^{-8}$ cm/s$^2$. This implies a bias of $b_{\tt 2\_craft}=+0.17\times10^{-8}$ cm/s$^2$ with respect to the Pioneer 10 experimental result $a_{P({\tt exper})}$. We also take this number to be our two spacecraft uncertainty. This means $$\begin{aligned} a_{\tt 2-craft}&=&b_{\tt 2-craft}\pm \sigma_{\tt 2\_craft} = \nonumber\\ &=& (0.17 \pm 0.17)~\times~10^{-8}~\mathrm{cm/s}^2.\end{aligned}$$ The difference between the two craft could be due to different gas leakage. But it also could be due to heat emitted from the RTGs. In particular, the two sets of RTGs have had different histories and so might have different emissivities. Pioneer 11 spent more time in the inner solar system (absorbing radiation). Pioneer 10 has swept out more dust in deep space. Further, Pioneer 11 experienced about twice as much Jupiter/Saturn radiation as Pioneer 10. Further, note that $[a_{P({\tt exper)}}^{\tt Pio11} - a_{P({\tt exper)}}^{\tt Pio10}]$ and the uncertainty from differential emissivity of the RTGs, $\sigma_{\tt d.e.}$, are of the same size: 0.71 and 0.85 $\times10^{-8}$ cm/s$^2$. It could therefore be argued that Pioneer 11’s offset from Pioneer 10 comes from Pioneer 11 having obtained twice as large a differential emissivity bias as Pioneer 10. Then our final value of $a_P$, given in Section \[budget\], would be reduced by about $0.7$ of our units since $\sigma_{\tt d.e.}$ would have become mainly a negative bias, $b_{\tt d.e.}$. This would make the final number closer to $8 \times10^{-8}$ cm/s$^2$. Because this model and our final number are consistent, we present this observation only for completeness and as a possible reason for the different results of the two spacecraft. \[Int\_accuracy\]COMPUTATIONAL SYSTEMATICS ========================================== Given the very large number of observations for the same spacecraft, the error contribution from observational noise is very small and not a meaningful measure of uncertainty. It is therefore necessary to consider several other effects in order to assign realistic errors. Our first consideration is the statistical and numerical stability of of the calculations. We then go on to the cumulative influence of all modeling errors and editing decisions. Finally we discuss the reasons for and significance of the annual term. Besides the factors mentioned above, we will discuss in this section errors that may be attributed to the specific hardware used to run the orbit determination computer codes, together with computational algorithms and statistical methods used to derive the solution. Numerical stability of least-squares estimation {#leastsquares} ----------------------------------------------- Having presented estimated solutions along with their formal statistics, we should now attempt to characterize the true accuracy of these results. Of course, the significance of the results must be assessed on the basis of the expected measurement errors. These expected errors are used to weight a least-squares adjustment to parameters which describe the theoretical model. \[Examination of experimental systematics from sources both external to and also internal to the spacecraft was covered in Sections \[ext-systema\]-\[int-systema\].\] First we look at the numerical stability of the least squares estimation algorithm and the derived solution. The leading computational error source turns out to be subtraction of similar numbers. Due to the nature of floating point arithmetic, two numbers with high order digits the same are subtracted one from the other results in the low order digits being lost. This situation occurs with time tags on the data. Time tags are referenced to some epoch, such as say 1 January 1 1950 which is used by CHASMP. As more than one billion seconds have passed since 1950, time tags on the Doppler data have a start and end time that have five or six common leading digits. Doppler signal is computed by a differenced range formulation (see Section \[Dopp\_tech\]). This noise in the time tags causes noise in the computed Doppler at the 0.0006 Hz level for both Pioneers. This noise can be reduced by shifting the reference epoch closer to the data or increasing the word length of the computation, however, it is not a significant error source for this analysis. In order to guard against possible computer compiler and/or hardware errors we ran orbit determination programs on different computer platforms. JPL’s ODP resides on an HP workstation. The Aerospace Corporation ran the analysis on three different computer architectures: (i) Aerospace’s DEC 64-bit RISC architecture workstation (Alphastation 500/266), (ii) Aerospace’s DEC 32-bit CISC architecture workstation (VAX 4000/60), and (iii) Pentium Pro PC. Comparisons of computations performed for CHASMP in the three machine show consistency to 15 digits which is just sufficient to represent the data. While this comparison does not eliminate the possibility of systematic errors that are common to both systems, it does test the numerical stability of the analysis on three very different computer architectures. The results of the individual programs were given in Sections \[results\]and \[recent\_results\]. In a test we took the JPL results for a batch-sequential [*Sigma*]{} run with 50-day averages of the anomalous acceleration of Pioneer 10, $a_P$. The data interval was from January 1987 to July 1998. We compared this to an Aerospace determination using CHASMP, where the was split into 200 day intervals, over a shorter data interval ending in 1994. As seen in Figure \[fig:rec\_res\_comb\], the results basically agree. Given the excellent agreement in these implementations of the modeling software, we conclude that differences in analyst choices (parameterization of clocks, data editing, modeling options, etc.) give rise to coordinate discrepancies only at the level of $0.3$ cm. This number corresponds to an uncertainty in estimating the anomalous acceleration on the order of $8\times 10^{-12}$ cm/s$^2$. But there is a slightly larger error to contend with. In principle the STRIPPER can give output to 16 significant figures. From the beginning the output was-rounded off to 15 and later to 14 significant figures. When Block 5 came on near the beginning of 1995, the output was rounded off to 13 significant figures. Since the Doppler residuals are 1.12 mm/s this last truncation means an error of order 0.01 mm/s. If we divide this number by 2 for an average round off, this translates to $\pm 0.04\times10^{-8}$ cm/s$^2$. The roundoff occurred in approximately all the data we added for this paper. This is the cleanest 1/3 of the Pioneer 10 data. Considering this we take the uncertainty to be $$\sigma_{\tt num} \pm 0.02 \times 10^{-8} ~~{\rm cm/s}^2. \label{eq:num_st}$$ It needs to be stressed that such tests examine only the accuracy of implementing a given set of model codes, without consideration of the inherent accuracy of the models themselves. Numerous external tests, which we have been discussing in the previous three sections, are possible for assessing the accuracy of the solutions. Comparisons between the two software packages enabled us to evaluate the implementations of the theoretical models within a particular software. Likewise, the results of independent radio tracking observations obtained for the different spacecraft and analysis programs have enabled us to compare our results to infer realistic error levels from differences in data sets and analysis methods. Our analysis of the Galileo and Ulysses missions (reported in Sections \[galileo\] and \[ulysses\]) was done partially for this purpose. Accuracy of consistency/model tests {#Ext_accuracy} ----------------------------------- #### Consistency of solutions: A code that models the motion of solar system bodies and spacecraft includes numerous lengthy calculations. Therefore, the software used to obtain solutions from the Doppler data is, of necessity, very complex. To guard against potential errors in the implementation of these models, we used two software packages; JPL’s ODP/*Sigma* modeling software [@Moyer71; @Moyer81] and The Aerospace Corporation’s POEAS/CHASMP software package [@chasmp; @poeas]. The differences between the JPL and Aerospace orbit determination program results are now examined. As discussed in Section \[sec:OD\], in estimating parameters the CHASMP code uses a standard variation of parameters method whereas ODP uses the Cowell method to integrate the equations of motion and the variational equations. In other words, CHASMP integrates six first-order differential equation, using the Adams-Moulton predictor-corrector method in the orbital elements. Contrariwise, ODP integrates three second-order differential equations for the accelerations using the Gauss-Jackson method. (For more details on these methods see Ref. [@herrick].) As seen in our results of Sections \[results\]and \[recent\_results\], agreement was good; especially considering that each program uses independent methods, models, and constants. Internal consistency tests indicate that a solution is consistent at the level of one part in $10^{15}$. This implies an acceleration error on the order of no more then one part in $10^{4}$ in $a_P$. #### Earth orientation parameters: In order to check for possible problems with Earth orientation, CHASMP was modified to accept Earth orientation information from three different sources. (1) JPL’s STOIC program that outputs [UT1R-UTC]{}, (2) JPL’s Earth Orientation Parameter files ([UT1-UTC]{}), and (3) The International Earth Rotation Service’s Earth Orientation Parameter file ([UT1-UTC]{}). We found that all three sources gave virtually identical results and changed the value of $a_P$ only in the 4th digit [@Folkner]. #### Planetary ephemeris: Another possible source of problems is the planetary ephemeris. To explore this a fit was first done with CHASMP that used DE200. The solution of that fit was then used in a fit where DE405 was substituted for DE200. The result produced a small annual signature before the fit. After the fit, the maneuver solutions changed a small amount (less then 10%) but the value of the anomalous acceleration remained the same to seven digits. The post-fit residuals to DE405 were virtually unchanged from those using DE200. This showed that the anomalous acceleration was unaffected by changes in the planetary ephemeris. This is pertinent to note for the following subsection. To reemphasize the above, a small “annual term” can be introduced by changing the planetary ephemerides. This annual term can then be totally taken up by changing the maneuver estimations. Therefore, in principle, any possible mismodeling in the planetary ephemeris could be at least partially masked by the maneuver estimations. #### Differences in the codes’ model implementations: The impact of an analyst’s choices is difficult to address, largely because of the time and expense required to process a large data set using complex models. This is especially important when it comes to data editing. It should be understood that small differences are to be expected as models differ in levels of detail and accuracy. The analysts’ methods, experience, and judgment differ. The independence of the analysis of JPL and Aerospace has been consistently and strictly maintained in order to provide confidence on the validity of the analyses. Acknowledging such difficulties, we still feel that using the very limited tests given above is preferable to an implicit assumption that all analysts’ choices were optimally made. Another source for differences in the results presented in Table \[resulttable\] is the two codes’ modeling of spacecraft re-orientation maneuvers. ODP uses a model that solves for the resulted change in the Doppler observable $\Delta v$ (instantaneous burn model). This is a more convenient model for Doppler velocity measurements. CHASMP models the change in acceleration, solves for $\Delta a$ (finite burn model), and only then produces a solution for $\Delta v$. Historically, this was done in order to incorporate range observations (for Galileo and Ulysses) into the analysis. Our best handle on this is the no-corona results, especially given that the two critical Pioneer 10 Interval III results differed by very little, $0.02 \times 10^{-8}$ cm/s$^2$. This data is least affected by maneuver modeling, data editing, corona modeling, and spin calibration. Contrariwise, for the other data, the differences were larger. The Pioneer Interval I and II results and the Pioneer 11 results differed, respectively, by (0.21, 0.23, 0.25) in units of $10^{-8}$ cm/s$^2$. In these intervals models of maneuvers and data editing were crucial. Assuming that these errors are uncorrelated, we compute their combined effect on anomalous acceleration $a_P$ as $$\sigma_{\tt consist/model} = \pm 0.13 \times 10^{-8}~\textrm{cm/s}^2.$$ #### Mismodeling of maneuvers: A small contribution to the error comes from a possible mismodeling of the propulsion maneuvers. In Section \[model-maneuvers\] we found that for a typical maneuver the standard error in the residuals is $\sigma_0\sim0.095$ mm/s. Then we would expect that in the period between two maneuvers, which on average is $\tau=$ 11.5/28 year, the effect of the mismodeling would produce a contribution to the acceleration solution with a magnitude on the order of $\delta a_{\tt man}= {\sigma_0}/{\tau} = 0.07 \times 10^{-8}$ cm/s$^2$. Now let us assume that the errors in the Pioneer Doppler residuals are normally distributed around zero mean with the standard deviation of $\delta a_{\tt man}$ that constitute a single measurement accuracy. Then, since there are $N=28$ maneuvers in the data set, the total error due to maneuver mismodeling is $$\sigma_{\tt man} = \frac{\delta a_{\tt man}}{\sqrt{N}} = 0.01 \times 10^{-8} ~~{\rm cm/s}^2. \label{gluncertA}$$ #### Mismodeling of the solar corona: Finally, recall that our number for mismodeling of the solar corona, $ \pm 0.02 \times 10^{-8}$ cm/s$^2$, was already explained in Section \[sec:corona\]. Apparent annual/diurnal periodicities in the solution {#annualterm} ----------------------------------------------------- In Ref. [@moriond] we reported, in addition to the constant anomalous acceleration term, a possible annual sinusoid. If approximated by a simple sine wave, the amplitude of this oscillatory term is about $1.6 \times 10^{-8}$ cm/s$^2$. The integral of a sine wave in the acceleration, $a_P$, with angular velocity $\omega$ and amplitude $A_0$ yields the following first-order Doppler amplitude in two-way fractional frequency: $$\frac{\Delta \nu}{\nu} = \frac{2A_0}{c~ \omega}. \label{lasttwo}$$ The resulting Doppler amplitude for the annual angular velocity $\sim 2 \times 10^{-7}$ rad/s is $\Delta \nu/\nu$ = 5.3 $\times$ 10$^{-12}$. At the Pioneer downlink S-band carrier frequency of $\sim 2.29$ GHz, the corresponding Doppler amplitude is 0.012 Hz (i.e. 0.795 mm/s). This term was first seen in ODP using the [BSF]{} method. As we discussed in Section \[sec:PE\], treating $a_P$ as a stochastic parameter in JPL’s batch–sequential analysis allows one to search for a possible temporal variation in this parameter. Moreover, when many short interval times were used with least-squares CHASMP, the effect was also observed. (See Figure \[fig:rec\_res\_comb\] in Section \[recent\_results\].) The residuals obtained from both programs are of the same magnitude. In particular, the Doppler residuals are distributed about zero Doppler velocity with a systematic variation $\sim$ 3.0 mm/s on a time scale of $\sim$ 3 months. More precisely, the least-squares estimation residuals from both ODP/*Sigma* and CHASMP are distributed well within a half-width taken to be 0.012 Hz. (See, for example, Figure \[fig:pio10best\_fit\].) Even the general structures of the two sets of residuals are similar. The fact that both programs independently were able to produce similar post-fit residuals gives us confidence in the solutions. With this confidence, we next looked in greater detail at the acceleration residuals from solutions for $a_P$. Consider Figure \[annualresiduals\], which shows the $a_P$ residuals from a value for $a_P$ of $(7.77\pm 0.16) \times 10^{-8}$ cm/s$^2$. The data was processed using ODP/*Sigma* with a batch-sequential filter and smoothing algorithm. The solution for $a_P$ was obtained using 1-day batch sizes. Also shown are the maneuver times. At early times the annual term is largest. During Interval II, the interval of the large spin-rate change anomaly, coherent oscillation is lost. During Interval III the oscillation is smaller and begins to die out. In attempts to understand the nature of this annual term, we first examined a number of possible sources, including effects introduced by imprecise modeling of maneuvers, the solar corona, and the Earth’s troposphere. We also looked at the influence of the data editing strategies that were used. We concluded that these effects could not account for the annual term. Then, given that the effect is particularly large in the out-of-the-ecliptic voyage of Pioneer 11 [@moriond], we focused on the possibility that inaccuracies in solar system modeling are the cause of the annual term in the Pioneer solutions. In particular, we looked at the modeling of the Earth orbital orientation and the accuracy of the planetary ephemeris. #### Earth’s orientation: We specifically modeled the Earth orbital elements $\Delta p$ and $\Delta q$ as stochastic parameters. ($\Delta p$ and $\Delta q$ are two of the Set III elements defined by Brouwer and Clemence [@bc].) *Sigma* was applied to the entire Pioneer 10 data set with $a_P$, $\Delta p$, and $\Delta q$ determined as stochastic parameters sampled at an interval of five days and exponentially correlated with a correlation time of 200 days. Each interval was fit independently, but with information on the spacecraft state (position and velocity) carried forward from one interval to the next. Various correlation times, 0-day, 30-day, 200-day, and 400-day, were investigated. The [*a priori*]{} error and process noise on $\Delta p$ and $\Delta q$ were set equal to 0, 5, and 10 $\mu$rad in separate runs, but only the 10 $\mu$rad case removed the annual term. This value is at least three orders of magnitude too large a deviation when compared to the present accuracy of the Earth orbital elements. It is most unlikely that such a deviation is causing the annual term. Furthermore, changing to the latest set of EOP has very little effect on the residuals. \[We also looked at variations of the other four Set III orbital elements, essentially defining the Earth’s orbital shape, size, and longitudinal phase angle. They had little or no effect on the annual term.\] #### Solar system modeling: We concentrated on Interval III, where the spin anomaly is at a minimum and where $a_P$ is presumably best determined. Further, this data was partially taken after the DSN’s Block 5 hardware implementation from September 1994 to August 1995. As a result of this implementation the data is less noisy than before. Over Interval III the annual term is roughly in the form of a sine wave. (In fact, the modeling error is not strictly a sine wave. But it is close enough to a sine wave for purposes of our error analysis.) The peaks of the sinusoid are centered on conjunction, where the Doppler noise is at a maximum. Looking at a CHASMP set of residuals for Interval III, we found a 4-parameter, nonlinear, weighted, least-squares fit to an annual sine wave with the parameters amplitude $v_{\tt a.t.}=(0.1053\pm 0.0107)$ mm/s, phase $(-5.3^\circ \pm 7.2^\circ$), angular velocity $\omega_{\tt a.t}=(0.0177 \pm 0.0001$) rad/day, and bias ($0.0720 \pm 0.0082$) mm/s. The weights eliminate data taken inside of solar quadrature, and also account for different Doppler integration times $T_c$ according to $\sigma = (0.765 {\rm ~mm/s})\,[(60$ s$)/T_c]^{1/2}$. This rule yields post-fit weighted RMS residuals of 0.1 mm/s. The amplitude, $v_{\tt a.t.}$, and angular velocity, $\omega_{\tt a.t.}$, of the annual term results in a small acceleration amplitude of $a_{\tt a.t.}=v_{\tt a.t.}\omega_{\tt a.t.} = (0.215 \pm 0.022) \times 10^{-8}$ cm/s$^2$. We will argue below that the cause is most likely due to errors in the navigation programs’ determinations of the direction of the spacecraft’s orbital inclination to the ecliptic. A similar troubling modeling error exists on a much shorter time scale that is most likely an error in the spacecraft’s orbital inclination to the Earth’s equator. We looked at CHASMP acceleration residuals over a limited data interval, from 23 November 1996 to 23 December 1996, centered on opposition where the data is least affected by solar plasma. As seen in Figure \[opp96\], there is a significant diurnal term in the Doppler residuals, with period approximately equal to the Earth’s sidereal rotation period ($23^{\rm h}56^{\rm m}04^{\rm s}$.0989 mean solar time). After the removal of this diurnal term, the RMS Doppler residuals are reduced to amplitude 0.054 mm/s for $T_c = 660$ s ($\sigma_\nu/\nu = 2.9 \times 10^{-13}$ at $T_c = 1000$ s). The amplitude of the diurnal oscillation in the fundamental Doppler observable, $v_{\tt d.t.}$, is comparable to that in the annual oscillation, $v_{\tt a.t.}$, but the angular velocity, $\omega_{\tt d.t.}$, is much larger than $\omega_{\tt a.t.}$. This means the magnitude of the apparent angular acceleration, $a_{\tt d.t.}=v_{\tt d.t.}\omega_{\tt d.t.} = (100.1 \pm 7.9) \times 10^{-8}$ cm/s$^2$, is large compared to $a_P$. Because of the short integration times, $T_c=660$ s, and long observing intervals, $T\sim 1$ yr, the high frequency, diurnal, oscillation signal averages out to less than $0.03\times 10^{-8}$ cm/s$^2$ over a year. This intuitively helps to explain why the apparently noisy acceleration residuals still yield a precise value of $a_P$. Further, all the residuals from CHASMP and ODP/*Sigma* are essentially the same. Since ODP and CHASMP both use the same Earth ephemeris and the same Earth orientation models, this is not surprising. This is another check that neither program introduces serious modeling errors of its own making. Due to the long distances from the Sun, the spin-stabilized attitude control, the long continuous Doppler data history, and the fact that the spacecraft communication systems utilize coherent radio-tracking, the Pioneers allow for a very sensitive and precise positioning on the sky. For some cases, the Pioneer 10 coherent Doppler data provides accuracy which is even better than that achieved with VLBI observing natural sources. In summary, the Pioneers are simply much more sensitive detectors of a number of solar system modeling errors than other spacecraft. The annual and diurnal terms are very likely different manifestations of the same modeling problem. The magnitude of the Pioneer 10 post-fit weighted RMS residuals of $\approx 0.1$ mm/s, implies that the spacecraft angular position on the sky is known to $\le 1.0$ milliarcseconds (mas). (Pioneer 11, with $\approx 0.18$ mm/s, yields the result $\approx 1.75$ mas.) At their great distances, the trajectories of the Pioneers are not gravitationally affected by the Earth. (The round-trip light time is now $\sim 24 $ hours for Pioneer 10.) This suggests that the sources of the annual and diurnal terms are both Earth related. Such a modeling problem arises when there are errors in any of the parameters of the spacecraft orientation with respect to the chosen reference frame. Because of these errors, the system of equations that describes the spacecraft’s motion in this reference frame is under-determined and its solution requires non-linear estimation techniques. In addition, the whole estimation process is subject to Kalman filtering and smoothing methods. Therefore, if there are modeling errors in the Earth’s ephemeris, the orientation of the Earth’s spin axis (precession and nutation), or in the station coordinates (polar motion and length of day variations), the least-squares process (which determines best-fit values of the three direction cosines) will leave small diurnal and annual components in the Doppler residuals, like those seen in Figures \[annualresiduals\]-\[opp96\]. Orbit determination programs are particularly sensitive to an error in a poorly observed direction [@melbourne]. If not corrected for, such an error could in principle significantly affect the overall navigational accuracy. In the case of the Pioneer spacecraft, navigation was performed using only Doppler tracking, or line-of-sight observations. The other directions, perpendicular to the line-of-sight or in the plane of the sky, are poorly constrained by the data available. At present, it is infeasible to precisely parameterize the systematic errors with a physical model. That would have allowed one to reduce the errors to a level below those from the best available ephemeris and Earth orientation models. A local empirical parameterization is possible, but not a parameterization over many months. We conclude that for both Pioneer 10 and 11, there are small periodic errors in solar system modeling that are largely masked by maneuvers and by the overall plasma noise. But because these sinusoids are essentially uncorrelated with the constant $a_P$, they do not present important sources of systematic error. The characteristic signature of $a_P$ is a linear drift in the Doppler, not annual/diurnal signatures [@myles]. #### Annual/diurnal mismodeling uncertainty: We now estimate the annual term contribution to the error budget for $a_P$. First observe that the standard errors for radial velocity, $v_r$, and acceleration, $a_r$, are essentially what one would expect for a linear regression. The caveat is that they are scaled by the root sum of squares (RSS) of the Doppler error and unmodeled sinusoidal errors, rather than just the Doppler error. Further, because the error is systematic, it is unrealistic to assume that the errors for $v_r$ and $a_r$ can be reduced by a factor 1/$\sqrt{N}$, where $N$ is the number of data points. Instead, averaging their correlation matrix over the data interval, $T$, results in the estimated systematic error of $$\begin{aligned} \sigma_{a_r}^2 = \frac{12}{T^2}~\sigma_{v_r}^2 = \frac{12}{T^2}~\Big(\sigma_{T}^2 + \sigma_{v_{\tt a.t.}}^2+\sigma_{v_{\tt d.t.}}^2\Big). \label{syserror}\end{aligned}$$ $\sigma_{T}=0.1$ mm/s is the Doppler error averaged over $T$ (not the standard error on a single Doppler measurement). $\sigma_{v_{\tt a.t.}}$ and $\sigma_{v_{\tt d.t.}}$ are equal to the amplitudes of corresponding unmodeled annual and diurnal sine waves divided by $\sqrt{2}$. The resulting RSS error in radial velocity determination is about $\sigma_{v_r}= (\sigma_{T}^2 + \sigma_{v_{\tt a.t.}}^2+ \sigma_{v_{\tt d.t.}}^2)^{1/2}=0.15$ mm/s for both Pioneer 10 and 11. Our four interval values of $a_P$ were determined over time intervals of longer than a year. At the same time, to detect an annual signature in the residuals, one needs at least half of the Earth’s orbit complete. Therefore, with $T = 1/2$ yr, Eq. (\[syserror\]) results in an acceleration error of $$\sigma_{{\tt a/d}} = \frac{0.50~~{\rm mm/s}}{T} = 0.32~\times 10^{-8}~{\mathrm{cm/s}}^2. \label{aderror}$$ We use this number for the systematic error from the annual/diurnal term. \[budget\]ERROR BUDGET AND FINAL RESULT ======================================= It is important to realize that our experimental observable is a Doppler frequency shift, i.e., $\Delta \nu (t)$. \[See Figure \[fig:aerospace\] and Eq. (\[eq:delta\_nu\]).\] In actual fact it is a cycle count. We *interpret* this as an apparent acceleration experienced by the spacecraft. However, it is possible that the Pioneer effect is not due to a real acceleration. (See Section \[newphys\].) Therefore, the question arises “In what units should we report our errors?” The best choice is not clear at this point. For reasons of clarity we chose units of acceleration. ------------ ------------------------------------------------------- ----------------------- ----------------------- Item Description of error budget constituents Bias      Uncertainty $10^{-8} ~\rm cm/s^2$ $10^{-8} ~\rm cm/s^2$ 1 [Systematics generated external to the spacecraft:]{} a\) Solar radiation pressure and mass $+0.03$ $\pm 0.01$ b\) Solar wind $ \pm < 10^{-5}$ c\) Solar corona $ \pm 0.02$ d\) Electro-magnetic Lorentz forces $\pm < 10^{-4}$ e\) Influence of the Kuiper belt’s gravity $\pm 0.03$ f\) Influence of the Earth orientation $\pm 0.001$ g\) Mechanical and phase stability of DSN antennae $\pm < 0.001$ h\) Phase stability and clocks $\pm <0.001$ i\) DSN station location $\pm < 10^{-5}$ j\) Troposphere and ionosphere $\pm < 0.001$ \[10pt\] 2 [On-board generated systematics:]{} a\) Radio beam reaction force $+1.10$ $\pm 0.11$ b\) RTG heat reflected off the craft $-0.55$ $\pm 0.55$ c\) Differential emissivity of the RTGs $\pm 0.85$ d\) Non-isotropic radiative cooling of the spacecraft $\pm 0.48$ e\) Expelled Helium produced within the RTGs $+0.15$ $\pm 0.16$ f\) Gas leakage $\pm 0.56$ g\) Variation between spacecraft determinations $+0.17$ $\pm 0.17$ \[10pt\] 3 [Computational systematics:]{} a\) Numerical stability of least-squares estimation $\pm0.02$ b\) Accuracy of consistency/model tests $\pm0.13$ c\) Mismodeling of maneuvers $\pm 0.01$ d\) Mismodeling of the solar corona $\pm 0.02$ e\) Annual/diurnal terms $\pm 0.32$ \[10pt\] Estimate of total bias/error $+0.90$ $\pm 1.33$ ------------ ------------------------------------------------------- ----------------------- ----------------------- The tests documented in the preceding sections have considered various potential sources of systematic error. The results of these tests are summarized in Table \[error\_budget\], which serves as a systematic “error budget.” This budget is useful both for evaluating the accuracy of our solution for $a_P$ and also for guiding possible future efforts with other spacecraft. In our case it actually is hard to totally distinguish “experimental” error from “systematic error.” (What should a drift in the atomic clocks be called?) Further, there is the intractable mathematical problem of how to handle combined experimental and systematic errors. In the end we have decided to treat them all in a least squares [*uncorrelated*]{} manner. The results of our analyses are summarized in Table \[error\_budget\]. There are two columns of results. The first gives a bias, $b_P$, and the second gives an uncertainty, $\pm \sigma_P$. The constituents of the error budget are listed separately in three different categories: 1) systematics generated external to the spacecraft; 2) on-board generated systematics, and 3) computational systematics. Our final result then will become some average $$a_P = a_{P({\tt exper)}}~ + b_P ~\pm \sigma_P,$$ where, from Eq. (\[pio10lastresult\]), $a_{P({\tt exper)}} = (7.84\pm 0.01) \times 10^{-8}$ cm/s$^2$. The least significant factors of our error budget are in the first group of effects, those external to the spacecraft. From the table one sees that some are near the limit of contributing. But in totality, they are insignificant. As was expected, the on-board generated systematics are the largest contributors to our total error budget. All the important constituents are listed in the second group of effects in Table \[error\_budget\]. Among these effects, the radio beam reaction force produces the largest bias to our result, $1.10\times 10^{-8}$ cm/s$^2$. It makes the Pioneer effect larger. The largest bias/uncertainty is from RTG heat reflecting off the spacecraft. We argued for an effect as large as $(-0.55 \pm 0.55) \times 10^{-8}$ cm/s$^2$. Large uncertainties also come from differential emissivity of the RTGs, radiative cooling, and gas leaks, $\pm 0.85$, $\pm 0.48$, and $\pm 0.56$, respectively, $\times 10^{-8}$ cm/s$^2$. The computational systematics are listed in the third group of Table \[error\_budget\]. Therefore, our final value for $a_P$ is $$\begin{aligned} a_P &=& (8.74 \pm 1.33) \times 10^{-8}~{\rm cm/s}^2 \nonumber\\ &\sim& (8.7 \pm 1.3) \times 10^{-8}~{\rm cm/s}^2.\end{aligned}$$ The effect is clearly significant and remains to be explained. \[newphys\]POSSIBLE PHYSICAL ORIGINS OF THE SIGNAL ================================================== A new manifestation of known physics? {#sec:know} ------------------------------------- With the anomaly still not accounted for, possible effects from applications of known physics have been advanced. In particular, Crawford [@crawford] suggested a novel new effect: a gravitational frequency shift of the radio signals that is proportional to the distance to the spacecraft and the density of dust in the intermediate medium. In particular, he has argued that the gravitational interaction of the S-band radio signals with the interplanetary dust may be responsible for producing an anomalous acceleration similar to that seen by the Pioneer spacecraft. The effect of this interaction is a frequency shift that is proportional to the distance and the square root of the density of the medium in which it travels. Similarly, Didon, Perchoux, and Courtens [@courtens] proposed that the effect comes from resistance of the spacecraft antennae as they transverse the interplanetary dust. This is related to more general ideas that an asteroid or comet belt, with its associated dust, might cause the effect by gravitational interactions (see Section \[sec:kuiper\]) or resistance to dust particles. However, these ideas have problems with known properties of the interplanetary medium that were outlined in Section \[sec:kuiper\]. In particular, infrared observations rule out more than 0.3 Earth mass from Kuiper Belt dust in the trans-Neptunian region [@backman; @teplitzinfra]. Ulysses and Galileo measurements in the inner solar system find very few dust grains in the $10^{-18}-10^{-12}$ kg range [@dust]. The density varies greatly, up and down, within the belt (which precludes a constant force) and, in any event, the density is not large enough to produce a gravitational acceleration on the order of $a_P$ [@malhotra]-[@liudust]. One can also speculate that there is some unknown interaction of the radio signals with the solar wind. An experimental answer could be given with two different transmission frequencies. Although the main communication link on the Ulysses mission is S-up/X-down mode, a small fraction of the data is S-up/S-down. We had hoped to utilize this option in further analysis. However, using them in our attempt to study a possible frequency dependent nature of the anomaly, did not provide any useful results. This was in part due to the fact that X-band data (about 1.5 % of the whole data available) were taken only in the close proximity to the Sun, thus prohibiting the study of a possible frequency dependence of the anomalous acceleration. Dark matter or modified gravity? {#sec:DMgr} -------------------------------- It is interesting to speculate on the unlikely possibility that the origin of the anomalous signal is new physics [@photon]. This is true even though the probability is that some “standard physics” or some as-yet-unknown systematic will be found to explain this “acceleration.” The first paradigm is obvious. “Is it dark matter or a modification of gravity?” Unfortunately, neither easily works. If the cause is dark matter, it is hard to understand. A spherically-symmetric distribution of matter which goes as $\rho \sim r^{-1}$ produces a constant acceleration [*inside*]{} the distribution. To produce our anomalous acceleration even only out to 50 AU would require the total dark matter to be greater than $3 \times 10^{-4} M_\odot$. But this is in conflict with the accuracy of the ephemeris, which allows only of order a few times $10^{-6} M_\odot$ of dark matter even within the orbit of Uranus [@ephem]. (A 3-cloud neutrino model also did not solve the problem [@jgscold].) Contrariwise, the most commonly studied possible modification of gravity (at various scales) is an added Yukawa force [@physrep]. Then the gravitational potential is $$V(r) = -\frac{GMm}{(1+\alpha)r}\left[1 +\alpha e^{-r/\lambda}\right], \label{V}$$ where $\alpha$ is the new coupling strength relative to Newtonian gravity, and $\lambda$ is the new force’s range. Since the radial force is $F_r = -d_r V(r) =ma$, the power series for the acceleration yields an inverse-square term, no inverse-$r$ term, then a constant term. Identifying this last term as the Pioneer acceleration yields $$a_P = -\frac{\alpha {a_1}}{2(1+\alpha)} \frac{r_1^2}{\lambda^2}, \label{solution}$$ where $a_1$ is the Newtonian acceleration at distance $r_1 =1$ AU. (Out to 65 AU there is no observational evidence of an $r$ term in the acceleration.) Eq. (\[solution\]) is the solution curve; for example, $\alpha = -1 \times 10^{-3}$ for $\lambda = 200$ AU. It is also of interest to consider some of the recent proposals to modify gravity, as alternatives to dark matter [@milgrom]-[@mil]. Consider Milgrom’s proposed modification of gravity [@mil], where the gravitational acceleration of a massive body is $a \propto 1/r^2$ for some constant $a_0 \ll a$ and $a \propto 1/r$ for $a_0 \gg a$. Depending on the value of $H$, the Hubble constant, $a_0 \approx a_P$! Indeed, as a number of people have noted, $$a_H = cH \rightarrow 8 \times 10^{-8}~ {\rm cm/s}^2, \label{hubble}$$ if $H = 82$ km/s/Mpc. Of course, there are (fundamental and deep) theoretical problems if one has a new force of the phenomenological types of those above. Even so, the deep space data piques our curiosity. In fact, Capozziello et al. [@Capozzielloetal] note the Pioneer anomaly in their discussion of astrophysical structures as manifestations of Yukawa coupling scales. This ties into the above discussion. However, any universal gravitational explanation for the Pioneer effect comes up against a hard experimental wall. The anomalous acceleration is too large to have gone undetected in planetary orbits, particularly for Earth and Mars. NASA’s Viking mission provided radio-ranging measurements to an accuracy of about 12 m [@reasenberg; @mg6]. If a planet experiences a small, anomalous, radial acceleration, $a_A$, its orbital radius $r$ is perturbed by $$\Delta r =-\frac{{\it l}^6 a_A}{(GM_\odot)^4} \rightarrow - \frac{r~ a_A}{a_N} , \label{deltar}$$ where [*l*]{} is the orbital angular momentum per unit mass and $a_N$ is the Newtonian acceleration at $r$. (The right value in Eq. (\[deltar\]) holds in the circular orbit limit.) For Earth and Mars, $\Delta r$ is about $-21$ km and $-76$ km. However, the Viking data determines the difference between the Mars and Earth orbital radii to about a 100 m accuracy, and their sum to an accuracy of about 150 m. The Pioneer effect is not seen. Further, a perturbation in $r$ produces a perturbation to the orbital angular velocity of $$\Delta \omega = \frac{2{\it l}a_A}{GM_\odot} \rightarrow \frac{2 \dot{\theta}~ a_A}{a_N}.$$ The determination of the synodic angular velocity $(\omega_E - \omega_M)$ is accurate to 7 parts in 10$^{11}$, or to about 5 ms accuracy in synodic period. The only parameter that could possibly mask the spacecraft-determined $a_R$ is $(GM_\odot)$. But a large error here would cause inconsistencies with the overall planetary ephemeris [@ephem; @Standish92]. \[Also, there would be a problem with the advance of the perihelion of Icarus [@sanmil].\] We conclude that the Viking ranging data limit any unmodeled radial acceleration acting on Earth and Mars to no more than $0.1 \times 10^{-8}$ cm/s$^2$. Consequently, if the anomalous radial acceleration acting on spinning spacecraft is gravitational in origin, it is [*not*]{} universal. That is, it must affect bodies in the 1000 kg range more than bodies of planetary size by a factor of 100 or more. This would be a strange violation of the Principle of Equivalence [@pe]. (Similarly, the $\Delta \omega$ results rule out the universality of the $a_t$ time-acceleration model. In the age of the universe, $T$, one would have $a_t T^2/2 \sim 0.7~T$.) A new dark matter model was recently proposed by Munyaneza and Viollier [@MunyanezaViollier] to explain the Pioneer anomaly. The dark matter is assumed to be gravitationally clustered around the Sun in the form of a spherical halo of a degenerate gas of heavy neutrinos. However, although the resulting mass distribution is consistent with constraints on the mass excess within the orbits of the outer planets previously mentioned, it turns out that the model fails to produce a viable mechanism for the detected anomalous acceleration. New suggestions stimulated by the Pioneer effect {#sec:neww} ------------------------------------------------ Due to the fact that the size of the anomalous acceleration is of order $cH$, where $H$ is the Hubble constant (see Eq. (\[hubble\])), the Pioneer results have stimulated a number of new physics suggestions. For example, Rosales and Sánchez-Gomez [@rosales] propose that $a_P$ is due to a local curvature in light geodesics in the expanding spacetime universe. They argue that the Pioneer effect represents a new cosmological Foucault experiment, since the solar system coordinates are not true inertial coordinates with respect to the expansion of the universe. Therefore, the Pioneers are mimicking the role that the rotating Earth plays in Foucault’s experiment. Therefore, in this picture the effect is not a “true physical effect” and a coordinate transformation to the co-moving cosmological coordinate frame would entirely remove the Pioneer effect. From a similar viewpoint, Guruprasad [@guru] finds accommodation for the constant term while trying to explain the annual term as a tidal effect on the physical structure of the spacecraft itself. In particular, he suggests that the deformations of the physical structure of the spacecraft (due to external factors such as the effective solar and galactic tidal forces) combined with the spin of the spacecraft are directly responsible for the detected annual anomaly. Moreover, he proposes a hypothesis of the planetary Hubble’s flow and suggests that Pioneer’s anomaly does not contradict the existing planetary data, but supports his new theory of relativistically elastic space-time. [Ø]{}stvang [@ostvang] further exploits the fact that the gravitational field of the solar system is not static with respect to the cosmic expansion. He does note, however, that in order to be acceptable, any non-standard explanation of the effect should follow from a general theoretical framework. Even so, [Ø]{}stvang still presents quite a radical model. This model advocates the use of an expanded PPN-framework that includes a direct effect on local scales due to the cosmic space-time expansion. Belayev [@belayev] considers a Kaluza-Klein model in 5 dimensions with a time-varying scale factor for the compactified fifth dimension. His comprehensive analysis led to the conclusion that a variation of the physical constants on a cosmic time scale is responsible for the appearance of the anomalous acceleration observed in the Pioneer 10/11 tracking data. Modanese [@modan] considers the effect of a scale-dependent cosmological term in the gravitational action. It turns out that, even in the case of a static spherically-symmetric source, the external solution of his modified gravitational field equations contains a non-Schwartzschild-like component that depends on the size of the test particles. He argues that this additional term may be relevant to the observed anomaly. Mansouri, Nasseri and Khorrami [@MansouriNasseriKhorrami] argue that there is an effective time variation in the Newtonian gravitational constant that in turn may be related to the anomaly. In particular, they consider the time evolution of $G$ in a model universe with variable space dimensions. When analyzed in the low energy limit, this theory produces a result that may be relevant to the long-range acceleration discussed here. A similar analysis was performed by Sidharth [@Sidharth], who also discussed cosmological models with a time-varying Newtonian gravitational constant. Inavov [@ivanov] suggests that the Pioneer anomaly is possibly the manifestation of a superstrong interaction of photons with single gravitons that form a dynamical background in the solar system. Every gravitating body would experience a deceleration effect from such a background with a magnitude proportional to Hubble’s constant. Such a deceleration would produce an observable effect on a solar system scale. All these ideas produce predictions that are close to Eq. (\[hubble\]), but they certainly must be judged against discussions in the following two subsections. In a different framework, Foot and Volkas [@foot], suggest the anomaly can be explained if there is mirror matter of mirror dust in the solar system. this could produce a drag force and not violate solar-system mass constraints. Several scalar-field ideas have also appeared. Mbelek and Lachièze-Rey [@rey] have a model based on a long-range scalar field, which also predicts an oscillatory decline in $a_P$ beyond about 100 AU. This model does explain the fact that $a_P$ stays approximately constant for a long period (recall that Pioneer 10 is now past 70 AU). From a similar standpoint Calchi Novati et al. [@novati] discuss a weak-limit, scalar-tensor extension to the standard gravitational model. However, before any of these proposals can be seriously considered they must explain the precise timing data for millisecond binary pulsars, i.e., the gravitational radiation indirectly observed in PSR 1913+16 by Hulse and Taylor [@millipulsar]. Furthermore, there should be evidence of a distance-dependent scalar field if it is uniformly coupled to ordinary matter. Consoli and Siringo [@consoli_siringo] and Consoli [@consoli] consider the Newtonian regime of gravity to be the long wavelength excitation of a scalar condensate from electroweak symmetry breaking. They speculate that the self-interactions of the condensate could be the origins of both Milgrom’s inertia modification [@milgrom; @mil] and also of the Pioneer effect. Capozziello and Lambiase [@CapozzielloLambiase] argue that flavor oscillations of neutrinos in the Brans-Dicke theory of gravity may produce a quantum mechanical phase shift of neutrinos. Such a shift would produce observable effects on astrophysical/cosmological length and time scales. In particular, it results in a variation of the Newtonian gravitational constant and, in the low energy limit, might be relevant to our study. Motivated by the work of Mannheim [@mann; @mann2], Wood and Moreau [@moreau] investigated the theory of conformal gravity with dynamical mass generation. They argue that the Higgs scalar is a feature of the theory that cannot be ignored. In particular, within this framework they find one can reproduce the standard gravitational dynamics and tests within the solar system, and yet the Higgs fields may leave room for the Pioneer effect on small bodies. In summary, as highly speculative as all these ideas are, it can be seen that at the least the Pioneer anomaly is influencing the phenomenological discussion of modern gravitational physics and quantum cosmology [@BertolamiNunes]. Phenomenological time models {#sec:timemodel} ---------------------------- Having noted the relationships $a_P = c~ a_t$ of Eq. (\[asubt\]) and that of Eq. (\[hubble\]), we were motivated to try to think of any (purely phenomenological) “time” distortions that might fortuitously fit the CHASMP Pioneer results shown in Figure \[fig:aerospace\]. In other words, are Eqs. (\[hubble\]) and/or (\[asubt\]) indicating something? Is there any evidence that some kind of “time acceleration” is being seen? The Galileo and Ulysses spacecraft radio tracking data was especially useful. We examined numerous “time” models searching for any (possibly radical) solution. It was thought that these models would contribute to the definition of the different time scales constructed on the basis of Eq. (\[eq:time\]) and discussed in the Section \[sec:time\_scales\]. The nomenclature of the standard time scales [@Moyer81]-[@exp_cat] was phenomenologically extended in our hope to find a desirable quality of the trajectory solution for the Pioneers. In particular we considered: i\) [Drifting Clocks.]{} This model adds a constant acceleration term to the Station Time ([ST]{}) clocks, i.e., in the [ST-UTC]{} (Universal Time Coordinates) time transformation. The model may be given as follows: $$\Delta{\tt ST}={\tt ST}_{\tt received}-{\tt ST}_{\tt sent} ~~\rightarrow ~~\Delta{\tt ST}+\frac{1}{2}a_{\tt clocks} \cdot\Delta{\tt ST}^2$$ where ${\tt ST}_{\tt received}$ and ${\tt ST}_{\tt sent}$ are the atomic proper times of sending and receiving the signal by a DSN antenna. The model fit Doppler well for Pioneer 10, Galileo, and Ulysses but failed to model range data for Galileo and Ulysses. ii\) [Quadratic Time Augmentation.]{} This model adds a quadratic-in-time augmentation to the [TAI-ET]{} (International Atomic Time – Ephemeris Time) time transformation, as follows $${\tt ET} ~~\rightarrow ~~ {\tt ET}+\frac{1}{2}a_{\tt ET}\cdot{\tt ET}^2.$$ The model fits Doppler fairly well but range very badly. iii\) [Frequency Drift.]{} This model adds a constant frequency drift to the reference S-band carrier frequency: $$\nu_{\tt S-band}(t)=\nu_{0} \Big(1+\frac{a_{\tt fr.drift}\cdot{\tt TAI}}{c} \Big).$$ The model also fits Doppler well but again fits range poorly. iv\) [Speed of Gravity]{}. This model adds a “light time” delay to the actions of the Sun and planets upon the spacecraft: $$v_{\tt grav}=c \Big(1+\frac{a_{\tt sp.grav} \cdot|\vec{r}_{\tt body}-\vec{r}_{\tt Pioneer}|}{c^2} \Big).$$ The model fits Pioneer 10 and Ulysses well. But the Earth flyby of Galileo fit was terrible, with Doppler residuals as high as 20 Hz. All these models were rejected due either to poor fits or to inconsistent solutions among spacecraft. Quadratic in time model ----------------------- There was one model of the above type that was especially fascinating. This model adds a quadratic in time term to the light time as seen by the DSN station. Take any labeled time ${\tt T}_a$ to be $${\tt T}_a = t_a - t_0 \rightarrow t_a - t_0 + \frac{1}{2}a_t\left(t_a^2 - t_0^2\right).$$ Then the light time is $$\begin{aligned} \Delta{\tt TAI}&=&{\tt TAI}_{\tt received}-{\tt TAI}_{\tt sent} ~~\rightarrow \nonumber\\ &&\hskip -30pt \rightarrow~~ \Delta{\tt TAI}+\frac{1}{2}a_{\tt quad}\cdot \Big( {\tt TAI}_{\tt received}^2-{\tt TAI}_{\tt sent}^2\Big). \label{eq:aqt}\end{aligned}$$ It mimics a line of sight acceleration of the spacecraft, and could be thought of as an [*expanding space*]{} model. Note that $a_{\tt quad}$ affects only the data. This is in contrast to the $a_t$ of Eq. (\[asubt\]) that affects both the data and the trajectory. This model fit both Doppler and range very well. Pioneers 10 and 11, and Galileo have similar solutions although Galileo solution is highly correlated with solar pressure; however, the range coefficient of the quadratic is negative for the Pioneers and Galileo while positive for Ulysses. Therefore we originally rejected the model because of the opposite signs of the coefficients. But when we later appreciated that the Ulysses anomalous acceleration is dominated by gas leaks (see Section \[sec:AUlysses\]), which makes the different-sign coefficient of Ulysses meaningless, we reconsidered it. The fact that the Pioneer 10 and 11, Galileo, and Ulysses are spinning spacecraft whose spin axis are periodically adjusted so as to point towards Earth turns out to make the quadratic in time model and the constant spacecraft acceleration model highly correlated and therefore very difficult to separate. The quadratic in time model produces residuals only slightly ($\sim20\%$) larger than the constant spacecraft acceleration model. However, when estimated together with no [*a priori*]{} input [ i.e.]{}, based only the tracking data, even though the correlation between the two models is 0.97, the value $a_{\tt quad}$ determined for the quadratic in time model is zero while the value for the constant acceleration model $a_P$ remains the same as before. The orbit determination process clearly prefers the constant acceleration model, $a_P$, over that the quadratic in time model, $a_{\tt quad}$ of Eq. (\[eq:aqt\]). This implies that a real acceleration is being observed and not a pseudo acceleration. We have not rejected this model as it may be too simple in that the motions of the spacecraft and the Earth may need to be included to produce a true expanding space model. Even so, the numerical relationship between the Hubble constant and $a_P$, which many people have observed (cf. Section \[sec:neww\]), remains an interesting conjecture. \[disc\]CONCLUSIONS =================== In this paper we have discussed the equipment, theoretical models, and data analysis techniques involved in obtaining the anomalous Pioneer acceleration $a_P$. We have also reviewed the possible systematic errors that could explain this effect. These included computational errors as well as experimental systematics, from systems both external to and internal to the spacecraft. Thus, based on further data for the Pioneer 10 orbit determination (the extended data spans 3 January 1987 to 22 July 1998) and more detailed studies of all the systematics, we can now give a total error budget for our analysis and a latest result of $a_P = (8.74 \pm 1.33) \times 10^{-8}$ cm/s$^2$. This investigation was possible because modern radio tracking techniques have provided us with the means to investigate gravitational interactions to an accuracy never before possible. With these techniques, relativistic solar-system celestial mechanical experiments using the planets and interplanetary spacecraft provide critical new information. Our investigation has emphasized that effects that previously thought to be insignificant, such as rejected thermal radiation or mass expulsion, are now within (or near) one order of magnitude of possible mission requirements. This has unexpectedly emphasized the need to carefully understand all systematics to this level. In projects proposed for the near future, such as a Doppler measurement of the solar gravitational deflection using the Cassini spacecraft [@GAB] and the Space Interferometry Mission [@sim], navigation requirements are more stringent than those for current spacecraft. Therefore, all the effects we have discussed will have to be well-modeled in order to obtain sufficiently good trajectory solutions. That is, a better understanding of the nature of these extra small forces will be needed to achieve the stringent navigation requirements for these missions. Currently, we find no mechanism or theory that explains the anomalous acceleration. What we can say with some confidence is that the anomalous acceleration is a line of sight constant acceleration of the spacecraft toward the Sun [@sunearth]. Even though fits to the Pioneers appear to match the noise level of the data, in reality the fit levels are as much as 50 times above the fundamental noise limit of the data. Until more is known, we must admit that the most likely cause of this effect is an unknown systematic. (We ourselves are divided as to whether “gas leaks” or “heat” is this “most likely cause.”) The arguments for “gas leaks” are: i) All spacecraft experience a gas leakage at some level. ii) There is enough gas available to cause the effect. iii) Gas leaks require no new physics. However, iv) it is unlikely that the two Pioneer spacecraft would have gas leaks at similar rates, over the entire data interval, especially when the valves have been used for so many maneuvers. \[Recall also that one of the Pioneer 11 thrusters became inoperative soon after launch. (See Section \[sec:prop\].)\] v) Most importantly, it would require that these gas leaks be precisely pointed towards the front [@rearfront] of the spacecraft so as not to cause a large spin-rate changes. But vi) it could still be true anyway. The main arguments for “heat” are: i) There is so much heat available that a small amount of the total could cause the effect. ii) In deep space the spacecraft will be in approximate thermal equilibrium. The heat should then be emitted at an approximately constant rate, deviating from a constant only because of the slow exponential decay of the Plutonium heat source. It is hard to resist the notion that this heat somehow must be the origin of the effect. However, iii) there is no solid explanation in hand as to how a specific heat mechanism could work. Further, iv) the decrease in the heat supply over time should have been seen by now. Further experiment and analysis is obviously needed to resolve this problem. On the Pioneer 10 experimental front, there now exists data up to July 2000. Further, there exists archived high-rate data from 1978 to the beginning of our data arc in Jan 1987 that was not used in this analysis. Because this early data originated when the Pioneers were much closer in to the Sun, greater effort would be needed to perform the data analyses and to model the systematics. As Pioneer 10 continues to recede into interstellar space, its signal is becoming dimmer. Even now, the return signal is hard to detect with the largest DSN antenna. However, with appropriate instrumentation, the 305-meter antenna of the Arecibo Observatory in Puerto Rico will be able to detect Pioneer’s signal for a longer time. If contact with Pioneer 10 can be maintained with conscan maneuvers, such further extended data would be very useful, since the spacecraft is now so far from the Sun. Other spacecraft can also be used in the study of $a_P$. The radio Doppler and range data from the Cassini mission could offer a potential contribution. This mission was launched on 15 October 1997. The potential data arc will be the cruise phase from after the Jupiter flyby (30 December 2000) to the vicinity of Saturn (just before the Huygens probe release) in July 2004. Even though the Cassini spacecraft is in three-axis-stabilization mode, using on-board active thrusters, it was built with very sophisticated radio-tracking capabilities, with X-band being the main navigation frequency. (There will also be S- and K-band links.) Further, during much of the cruise phase, reaction wheels will be used for stabilization instead of thrusters. Their use will aid relativity experiments at solar conjunction and gravitational wave experiments at solar opposition. (Observe, however, that the relatively large systematic from the close in Cassini RTGs will have to be accounted for.) Therefore, Cassini could yield important orbit data, independent of the Pioneer hyperbolic-orbit data. A similar opportunity may exist, out of the plane of the ecliptic, from the proposed Solar Probe mission. Under consideration is a low-mass module to be ejected during solar flyby. On a longer time scale, the reconsidered Pluto/Kuiper mission (with arrival at Pluto around 2029) could eventually provide high-quality data from very deep space. All these missions might help test our current models of precision navigation and also provide a new test for the anomalous $a_P$. In particular, we anticipate that, given our analysis of the Pioneers, in the future precision orbital analysis may concentrate more on systematics. That is, data/systematic modeling analysis may be assigned more importance relative to the astronomical modeling techniques people have concentrated on for the past 40 years [@tuck]-[@bartlett]. Finally, we observe that if no convincing explanation is to be obtained, the possibility remains that the effect is real. It could even be related to cosmological quantities, as ha\]s been intimated. \[See Eq. (\[asubt\]) and Sec. \[newphys\], especially the text around Eq. (\[hubble\]).\] This possibility necessitates a cautionary note on phenomenology: At this point in time, with the limited results available, there is a phenomenological equivalence between the $a_P$ and $a_t$ points of view. But somehow, the choice one makes affects one’s outlook and direction of attack. If one has to consider new physics one should be open to both points of view. In the unlikely event that there is new physics, one does not want to miss it because one had the wrong mind set. Firstly we must acknowledge the many people who have helped us with suggestions, comments, and constructive criticisms. Invaluable information on the history and status of Pioneer 10 came from Ed Batka, Robert Jackson, Larry Kellogg, Larry Lasher, David Lozier, and Robert Ryan. E. Myles Standish critically reviewed the manuscript and provided a number of important insights, especially on time scales, solar system dynamics and planetary data analysis. We also thank John E. Ekelund, Jordan Ellis, William Folkner, Gene L. Goltz, William E. Kirhofer, Kyong J. Lee, Margaret Medina, Miguel Medina, Neil Mottinger, George W. Null, William L. Sjogren, S. Kuen Wong, and Tung-Han You of JPL for their aid in obtaining and understanding DSN Tracking Data. Ralph McConahy provided us with very useful information on the history and current state of the DSN complex at Goldstone. R. Rathbun and A. Parker of TRW provided information on the mass of the Pioneers. S. T. Christenbury of Teledyne-Brown, to whom we are very grateful, supplied us with critical information on the RTGs. Information was also supplied by G. Reinhart of LANL, on the RTG fuel pucks, and by C. J. Hansen of JPL, on the operating characteristics of the Voyager image cameras. We thank Christopher S. Jacobs of JPL for encouragement and stimulating discussions on present VLBI capabilities. Further guidance and information were provided by John W. Dyer, Alfred S. Goldhaber, Jack G. Hills, Timothy P. McElrath, Irwin I. Shapiro, Edward J. Smith, and Richard J. Terrile. Edward L. Wright sent useful observations on the RTG emissivity analysis. We also thank Henry S. Fliegel, Gary B. Green, and Paul Massatt of The Aerospace Corporation for suggestions and critical reviews of the present manuscript. This work was supported by the Pioneer Project, NASA/Ames Research Center, and was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. P.A.L. and A.S.L. were supported by a grant from NASA through the Ultraviolet, Visible, and Gravitational Astrophysics Program. M.M.N. acknowledges support by the U.S. DOE. Finally, the collaboration especially acknowledges the contributions of our friend and colleague, Tony Liu, who passed away while the manuscript was nearing completion. APPENDIX ======== In Table \[poeas00479\] we give the hyperbolic orbital parameters for Pioneer 10 and Pioneer 11 at epoch 1 January 1987, 01:00:00 [UTC]{}. The semi-major axis is $a$, $e$ is the eccentricity, $I$ is the inclination, $\Omega$ is the longitude of the ascending node, $\omega$ is the argument of the perihelion, $M_0$ is the mean anomaly, $f_0$ is the true anomaly at epoch, and $r_0$ is the heliocentric radius at the epoch. The direction cosines of the spacecraft position for the axes used are $(\alpha, \,\beta, \,\gamma)$. These direction cosines and angles are referred to the mean equator and equinox of J2000. The ecliptic longitude $\ell_0$ and latitude $b_0$ are also listed for an obliquity of 23$^\circ$26$^{'}$21.$^{''}$4119. The numbers in parentheses denote realistic standard errors in the last digits. Parameter Pioneer 10 Pioneer 11 -------------------------- --------------------- ---------------------- $a$ \[km\] $-1033394633(4)$ $-1218489295(133)$ \[1pt\] $e$ $1.733593601(88)$ $2.147933251(282)$ \[1pt\] $I$ \[Deg\] $26.2488696(24)$ $9.4685573(140)$ \[1pt\] $\Omega$ \[Deg\] $-3.3757430(256)$ $35.5703012(799)$ \[1pt\] $\omega$ \[Deg\] $-38.1163776(231)$ $-221.2840619(773)$ \[1pt\] $M_0$ \[Deg\] $259.2519477(12)$ $109.8717438(231)$ \[1pt\] $f_0$ \[Deg\] $112.1548376(3)$ $81.5877236(50)$ \[1pt\] $r_0$ \[km\] $5985144906(22)$ $3350363070(598)$ \[1pt\] $\alpha$ $0.3252905546(4)$ $-0.2491819783(41)$ \[1pt\] $\beta$ $0.8446147582(66)$ $-0.9625930916(22)$ \[1pt\] $\gamma$ $0.4252199023(133)$ $-0.1064090300(223)$ \[1pt\] $\ell_0$ \[Deg\] $70.98784378(2)$ $-105.06917250(31)$ \[1pt\] $b_0$ \[Deg\] $3.10485024(85)$ $16.57492890(127)$ \[1pt\] : Orbital parameters for Pioneer 10 and Pioneer 11 at epoch 1 January 1987, 01:00:00 [UTC]{}. \[poeas00479\] [99]{} See the special issue of Science [**183**]{}, No. 4122, 25 January 1974; specifically, J. D. Anderson, G. W. Null, and S. K. Wong, Science [**183**]{}, 322 (1974). R. O. Fimmel, W. Swindell, and E. Burgess, [*Pioneer Odyssey: Encounter with a Giant*]{}, NASA document No. SP-349 (NASA, Washington D.C., 1974). R. O. Fimmel, J. Van Allen, and E. Burgess, [*Pioneer: First to Jupiter, Saturn, and beyond*]{}, NASA report NASA–SP-446 (NASA, Washington D.C., 1980). , Pioneer Project NASA/ARC document No. PC-202 (NASA, Washington, D.C., 1971). , Revised, NASA/ARC document No. PC-1001 (NASA, Washington, D.C., 1994). For web summaries of Pioneer, go to: [http://quest.arc.nasa.gov/pioneer10]{},\ [http://spaceprojects.arc.nasa.gov/\ Space\_Projects/ pioneer/PNhome.html]{} J. D. Anderson, E. L. Lau, K. Scherer, D. C. Rosenbaum, and V. L. Teplitz, [Icarus]{} [**131**]{}, 167 (1998). J. D. Anderson, E. L. Lau, T. P. Krisher, D. A. Dicus, D. C. Rosenbaum, and V. L. Teplitz, [Astrophys. J.]{} [**448**]{}, 885 (1995). K. Scherer, H. Fichtner, J. D. Anderson, and E. L. Lau, [ Science]{} [**278**]{}, 1919 (1997). J. D. Anderson and B. Mashoon, [Astrophys. J.]{} [**290**]{}, 445 (1985). J. D. Anderson, J. W. Armstrong, and E. L.Lau, [Astrophys. J.]{} [**408**]{}, 287 (1993). J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, Phys. Rev. Lett. [**81**]{}, 2858 (1998). Eprint gr-qc/9808081. S. G. Turyshev, J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, and M. M. Nieto, in: [*Gravitational Waves and Experimental Gravity, Proceedings of the [XVIIIth]{} Moriond Workshop of the Rencontres de Moriond*]{}, ed. by J. Dumarchez and J. Tran Thanh Van (World Pub., Hanoi, 2000), pp. 481-486. Eprint gr-qc/9903024. There were four Pioneers built of this particular design. After testing, the best components were placed in Pioneer 10. (This is probably why Pioneer 10 has lasted so long.) The next best were placed in Pioneer 11. The third best were placed in the “proof test model.” Until recently, the structure and many components of this model were included in an exhibit at the National Air and Space Museum. The other model eventually was dismantled. We thank Robert Ryan of JPL for telling us this. Figures given for the mass of the entire Pioneer package range from under 250 kg to over 315 kg. However, we eventually found that the total (“wet”) weight at launch was 259 kg (571 lbs), including 36 kg of hydrazine (79.4 lbs). Credit and thanks for these numbers are due to Randall Rathbun, Allen Parker, and Bruce A. Giles of TRW, who checked and rechecked for us including going to their launch logs. Consistent total mass with lower fuel (27 kg) numbers were given by Larry Kellogg of NASA/Ames. (We also thank V. J. Slabinski of USNO who first asked us about the mass.) Information about the gas usage is by this time difficult to find or lost. During the Extended Mission the collaboration was most concerned with power to the craft. The folklore is that most of Pioneer 11’s propellant was used up going to Saturn and used very little for Pioneer 10. In particular, a Pioneer 10 nominal input mass of 251.883 kg and a Pioneer 11 mass of 239.73 kg were used by the JPL program and the Aerospace program used 251.883 for both. The 251 number approximates the mass lost during spin down, and the 239 number models the greater fuel usage. These numbers were not changed in the programs. For reference, we will use 241 kg, the mass with half the fuel used, as our number with which to calibrate systematics. We take this number from Ref. [@piodoc], where the design, boom-deployed moment of inertia is given as 433.9 slug (ft)$^2$ (= 588.3 kg m$^2$). This should be a little low since we know a small amount of mass was added later in the development. A much later order-of-magnitude number 770 kg m$^2$ was obtained with a too large mass [@mass; @gasuse]. See J. A. Van Allen, [*Episodic Rate of Change in Spin Rate of Pioneer 10,*]{} Pioneer Project Memoranda, 20 March 1991 and 5 April 1991. Both numbers are dominated by the RTGs and magnetometer at the ends of long booms. Conscan stands for conical scan. The receiving antenna is moved in circles of angular size corresponding to one half of the beam-width of the incoming signal. This procedure, possibly iterated, allows the correct pointing direction of the antenna to be found. When coupled with a maneuver, it can also be used to find the correct pointing direction for the spacecraft antenna. The precession maneuvers can be open-loop, for orientation towards or away from Earth-pointing, or closed-loop, for homing on the uplink radio-frequency transmission from the Earth. When a Pioneer antenna points toward the Earth, this defines the “rear” direction on the spacecraft. The equipment compartment placed on the other side of of the antenna defines the “front” direction on the spacecraft. (See Figure \[fig:trusters\].) , Teledyne report IESD 2873-172, June, 1973, tech. report No. DOE/ET/13512-T1; DE85017964, gov. doc. \# E 1.9, and S. T. Christenbury, private communications. F. A. Russo, in: [*Proceedings of the 3rd RTG Working Group Meeting*]{} (Atomic Energy Commission, Washington, DC, 1972), ed. by P. A. O’Rieordan, papers \# 15 and 16. L. Lasher, Pioneer Project Manager, recently reminded us (March 2000) that not long after launch, the electrical power had decreased to about 155 W, and degraded from there. \[Plots of the available power with time are available.\] This is a “theoretical value,” which does not account for inverter losses, line losses, and such. It is interesting to note that at mission acceptance, the total “theoretical” power was 175 Watts. We take the S-band to be defined by the frequencies 1.55-5.20 GHz. We take the X-band to be defined by the frequencies 5.20-10.90 GHz. It turns out there is no consistent international definition of these bands. The definitions vary from field to field, with geography, and over time. The above definitions are those used by radio engineers and are consistent with the DSN usage. (Some detailed band definitions can be found at [http://www.eecs.wsu.edu/$\sim$hudson/ Teaching/ee432/spectrum.htm]{}.) \[We especially thank Ralph McConahy of DSN Goldstone on this point.\] dBm is used by radio engineers as a measure of received power. It stands for decibels in milliwatts. For a description of the Galileo mission see T. V. Johnson, C. M. Yeats, and R. Young, [Space Sci. Rev.]{} [**60**]{}, 3–21 (1992). For a description of the trajectory design see L. A. D’Amario, L. E. Bright, and A. A. Wolf, [Space Sci. Rev.]{} [**60**]{}, 22–78 (1992). The LGA was originally supposed to “trickle” down low-rate engineering data. It was also to be utilized in case a fault resulted in the spacecraft “safing” and shifting to a Sun-pointed attitude, resulting in loss of signal from the HGA. \[“Safing” refers to a spacecraft entering the so called “safe mode.” This happens in case of an emergency when systems are shut down.\] J. D. Anderson, P. B. Esposito, W. Martin, C. L. Thornton, and D. O. Muhleman, [Astrophys. J.]{} [**200**]{}, 221 (1975). P. W. Kinman, IEEE Transactions on Microwave Theory and Techniques [**40**]{}, 1199 (1992). For descriptions of the Ulysses mission see E. J. Smith and R. G. Marsden, Sci. Am. [**278**]{}, No. 1, 74 (1998); B. M. Bonnet, Alexander von Humboldt Magazin, No. 72, 27 (1998). A technical description, with a history and photographs, of the Deep Space Network can be found at [http://deepspace.jpl.nasa.gov/dsn/]{}. The document describing the radio science system is at [http://deepspace.jpl.nasa.gov/dsndocs/810-5/ 810-5.html]{}. N. A. Renzetti, J. F. Jordan, A. L. Berman, J. A. Wackley, T. P. Yunck, [*The Deep Space Network – An Instrument for Radio Navigation of Deep Space Probes*]{}, Jet Propulsion Laboratory Technical Report 82-102 (1982). J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, Jr., W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, [IEEE Transactions on Instrumentation and Measurement]{} [**20**]{}, 105 (1971). R. F. C. Vessot, in: [*Experimental Gravitation*]{}, ed. B. Bertotti, ( Academic Press, New York and London, 1974), p.111. O. J. Sovers, J. L. Fanselow, and C. S. Jacobs, [Rev. Mod. Phys.]{}, [**70**]{}, 1393 (1998). One-way data refers to a transmission and reception, only. Two-way data is a transmission and reception, followed by a retransmission and reception at the original transmission site. This would be, for example, a transmission from a radio antenna on Earth to a spacecraft and then a retransmission back from the spacecraft to the same antenna. Three-way refers to the same as two-way, except the final receiving antenna is different from the original transmitting antenna. Much, but not all, of the data we used has been archived. Since the Extended Pioneer Mission is complete, the resources have not been available to properly convert the entire data set to easily accessible format. The JPL and DSN convention for Doppler frequency shift is $(\Delta \nu)_{\tt DSN} = \nu_0 - \nu$, where $\nu$ is the measured frequency and $\nu_0$ is the reference frequency. It is positive for a spacecraft receding from the tracking station (red shift), and negative for a spacecraft approaching the station (blue shift), just the opposite of the usual convention, $(\Delta \nu)_{\tt usual} = \nu - \nu_0$. In consequence, the velocity shift, $\Delta v = v - v_0$, has the same sign as $(\Delta \nu)_{\tt DSN}$ but the opposite sign to $(\Delta \nu)_{\tt usual}$. Unless otherwise stated, we will use the DSN frequency shift convention in this paper. We thank Matthew Edwards for asking us about this. As we will come to in Section \[sec:timemodel\], this property allowed us to test and reject several phenomenological models of the anomalous acceleration that fit Doppler data well but failed to fit the range data. D. L. Cain, JPL Technical Report (1966). T. D. Moyer, [*Mathematical Formulation of the Double Precision Orbit Determination Program (DPODP)*]{}, Jet Propulsion Laboratory Technical Report 32-1527 (1971). T. D. Moyer, [*Formulation for Observed and Computed Values of Deep Space Network (DSN) Data Types for Navigation*]{}, JPL Publication 00-7 (October 2000). A. Gelb, ed. [*Applied Optimal Estimation*]{} (M.I.T. Press, Cambridge, MA, 1974). D. O. Muhleman and J. D. Anderson, [Astrophys. J.]{} [**247**]{}, 1093 (1981). Once in deep space, all major forces on the spacecraft are gravitational. The Principle of Equivalence holds that the inertial mass ($m_I$) and the gravitational mass ($m_G$) are equal. This means the mass of the craft should cancel out in the dynamical gravitational equations. As a result, the people who designed early deep-space programs were not as worried as we are about having the correct mass. When non-gravitational forces were modeled, an incorrect mass could be accounted for by modifying other constants. For example, in the solar radiation pressure force the effective sizes of the antenna and the albedo could take care of mass inaccuracies. J. D. Anderson, in: [*Experimental Gravitation*]{}, ed. B. Bertotti (New York and London, Academic Press, 1974), p.163. J. D. Anderson, G. S. Levy, and N. A. Renzetti, “Application of the Deep Space Network (DSN) to the testing of general relativity,” in [*Relativity in Celestial Mechanics and Astrometry*]{}, eds. J. Kovalevsky and V. A. Brumberg. (Kluwer Academic, Dordrecht, Boston, 1986), p. 329. X X Newhall, E. M. Standish, and J. G. Williams, Astron. Astrophys. [**125**]{}, 150 (1983). E. M. Standish, Jr., X X Newhall, J. G. Williams, and D. K. Yeomans, “Orbital ephemeris of the Sun, Moon, and Planets,” in: Ref. [@exp_cat], p. 279. Also see E. M. Standish, Jr. and R. W. Hellings, Icarus [**80**]{}, 326 (1989). E. M. Standish, Jr., X X Newhall, J. G. Williams, and W. M. Folkner, [*JPL Planetary and Lunar Ephemeris, DE403/LE403*]{}, Jet Propulsion Laboratory Internal IOM No. 314.10-127 (1995). C. M. Will, [*Theory and Experiment in Gravitational Physics*]{}, (Rev. Ed.) (Cambridge University Press, Cambridge, 1993). C. M. Will and K. Nordtvedt, Jr, [Astrophys. J.]{} [**177**]{}, 757 (1972). F. E. Estabrook, [Astrophys. J.]{} [**158**]{}, 81 (1969). T. D. Moyer, Parts. 1 and 2, Celest. Mech. [**23**]{}, 33, 57 (1981). P. K. Seidelmann, ed., [*Explanatory Supplement to the Astronomical Almanac*]{} (University Science Books, Mill Valley, CA, 1992). C. Ma, E. F. Arias, T. M. Eubanks, A. L. Fey, A.-M. Gontier, C. S. Jacobs, O. J. Sovers, B. A. Archinal, and P. Charlot, Astron. J. [**116**]{}, 516 (1998). A. Milani, A. M. Nobili, and P. Farinella, [*Non-Gravitational Perturbations and Satellite Geodesy*]{}, (Adam Hilger, Bristol, 1987). See, especially, p. 125. J. M. Longuski, R. E. Todd, and W. W. König, J. Guidance, Control, and Dynamics, [**15**]{}, 545 (1992). D.O. Muhleman, P.B. Esposito, and J. D. Anderson, Astrophys. J. [**211**]{}, 943 (1977). The propagation speed for the Doppler signal is the phase velocity, which is greater than $c$. Hence, the negative sign in Eq. (\[eq:sol\_plasma\]) applies. The ranging signal propagates at the group velocity, which is less than $c$. Hence, there the positive sign applies. B.-G. Anderssen and S. G. Turyshev, JPL Internal IOM 1998-0625, and references therein. M. K. Bird, H. Volland, M. Pätzold, P. Edenhofer , S. W. Asmar and J. P. Brenkle, [Astrophys. J.]{} [**426**]{}, 373 (1994). The units conversion factor for $A,B,C$ from m to cm$^{-3}$ is $2 N_c(S)/R_\odot = 0.01877$, where $N_c(S) =1.240 \times 10^4 \nu^2$ is the S band (in MHz) critical plasma density, and $R_\odot$ is the radius of the Sun. These values of parameters $A, B, C$ were kindly provided to us by John E. Ekelund of JPL. They represent the best solution for the solar corona parameters obtained during his simulations of the solar conjunction experiments that will be performed with the Cassini mission spacecraft in 2001 and 2002. This model is explained and described at\ [http://science.msfc.nasa.gov/ssl/pad/solar/ predict.htm]{} These come from the adjustment in the system of data weights (inverse of the variance on each measurement) for Mariner 6/7 range measurements. Private communication by Inter-office Memorandum from D. O. Muhleman of Caltech to P. B. Esposito of JPL, dated 7 July 1971. G. W. Null, E. L. Lau, E. D. Biller, and J. D. Anderson, [Astron. J.]{} [**86**]{}, 456 (1981). P. A. Laing, “Implementation of J2000.0 reference frame in CHASMP,” The Aerospace Corporation’s Internal Memorandum \# 91(6703)-1. January 28, 1991. J. H. Lieske, [ Astron. Astrophys.]{} [**73**]{}, 282 (1979). Also, see [*FK5/J2000.0 for DSM Applications,*]{} Applied Technology Associates, 6 June 1985. E. M. Standish, [Astron. Astrophys.]{} [**114**]{}, 297 (1982) J. Sherman and W. Morrison. Ann. Math. Stat. [**21**]{}, 124 (1949) J. D. Anderson, Quarterly Report to NASA/Ames Research Center, [*Celestial Mechanics Experiment, Pioneer 10/11,*]{} 22 July 1992. Also see the later quarterly report for the period 1 Oct. 1992 to 31 Dec. 1992, dated 17 Dec. 1992, Letter of Agreement ARC/PP017. This last, specifically, contains the present Figure \[fig:correlation\]. We only measure Earth-spacecraft Doppler frequency and, as we will discuss in Sec. \[radioantbeam\], the down link antenna yields a conical beam of width 3.6 degrees at half-maximum power. Therefore, between Pioneer 10’s past and present (May 2001) distances of 20 to 78 AU, the Earth-spacecraft line and Sun-spacecraft line are so close that one can not resolve whether the force direction is towards the Sun or if the force direction is towards the Earth. If we could have used a longer arc fit that started earlier and hence closer, we might have able to separate the Sun direction from the Earth direction. A preliminary discussion of these results appeared in M. M. Nieto, T. Goldman, J. D. Anderson, E. L. Lau, and J. Pérez-Mercader, in: [*Proc. Third Biennial Conference on Low-Energy Antiproton Physics, LEAP’94*]{}, ed. by G. Kernel, P. Krizan, and M. Mikuz (World Scientific, Singapore, 1995), p. 606. Eprint hep-ph/9412234. Since both the gravitational and radiation pressure forces become so large close to the Sun, the anomalous contribution close to the Sun in Figures \[fig:forces\] and \[fig:correlation\] is meant to represent only what anomaly can be gleaned from the data, not a measurement. B. D. Tapley, in [*Recent Advances in Dynamical Astronomy*]{}, eds. B. D. Tapley and V. Szebehely (Reidel, Boston, 1973), p.396. P. A. Laing, [*Thirty Years of CHASMP*]{}, Aerospace report (in preparation). P. A. Laing, [*Software Specification Document, Radio Science Subsystem, Planetary Orbiter Error Analysis Study Program (POEAS)*]{}, Jet Propulsion Laboratory Technical Report DUK-5127-OP-D, 19 February 1981. POEAS was originally developed to support the Mariner Mars program. P. A. Laing and A. S. Liu. NASA Interim Technical Report, Grant NAGW-4968, 4 October 1996. Galileo is less sensitive to either an $a_P$- or an $a_t= a_P/c$-model effect than the Pioneers. Pioneers have a smaller solar pressure and a longer light travel time. Sensitivity to a clock acceleration is proportional to the light travel time squared. T. McElrath, private communication. T. P. McElrath, S. W. Thurman, and K. E. Criddle, in [*Astrodynamics 1993*]{}, edited by A. K. Misra, V. J. Modi,R. Holdaway, and P. M. Bainum (Univelt, San Diego CA, 1994), Ad. Astodynamical Sci. [**85**]{}, Part II, p. 1635, paper No. AAS 93-687. The gas leaks found in the Pioneers are about an order of magnitude too small to explain $a_P$. Even so, we feel that some systematic or combination of systematics (such as heat or gas leaks) will likely explain the anomaly. However, such an explanation has yet to be demonstrated. We will discuss his point further in Sections \[recent\_results\] and \[int-systema\]. More information on the “Heliocentric Trajectories for Selected Spacecraft, Planets, and Comets,” can be found at [http://nssdc.gsfc.nasa.gov/space/ helios/heli.html]{}. ODP/*Sigma* took the Interval I/II boundary as 22 July 1990, the date of a maneuver. CHASMP took this boundary date as 31 August 1990, when a clear anomaly in the spin data was seen. We have checked, and these choices produce less than one percent differences in the results. J. A. Estefan, L. R. Stavert, F. M. Stienon, A. H. Taylor, T. M. Wang, and P. J. Wolff, [*Sigma User’s Guide. Navigation Filtering/Mapping Program*]{}, JPL document 699-FSOUG/NAV-601 (Revised: 14 Dec. 1998). G. W. Null, [Astron. J.]{} [**81**]{}, 1153 (1976). R. M. Georgevic, [*Mathematical model of the solar radiation force and torques acting on the components of a spacecraft,*]{} JPL Technical Memorandum 33-494 (1971). Data is available at [http://www.ngdc.noaa.gov/stp/ SOLAR/IRRADIANCE/irrad.html]{} For an ideal flat surface facing the Sun, $\mathcal{K} = (\alpha + 2\epsilon) = (1 +2\mu + 2 \nu)$. $\alpha$ and $\epsilon$ are, respectively, the absorption and reflection coefficients of the spacecraft’s surface. ODP uses the second formulation in terms of reflectivity coefficients, ODP’s input $\mu$ and $\nu$ for Pioneer 10, are obtained from design information and early fits to the data. (See the following paragraph.) These numbers by themselves yield $\mathcal{K_0}= 1.71$. When a first (negative) correction is made for the antenna’s parabolic surface, $\mathcal{K}\rightarrow 1.66$. There are complicating effects that modify the ideal antenna. The craft actually has multiple different-shaped surfaces (such as the RTGs), that are composed of different materials oriented at different angles to the spin axis, and which degrade with time. But far from the Sun, and given $M$ and $A$, the sum of all such corrections can be subsumed, for our purposes, in an [*effective*]{} $\mathcal{K}$. It is still expected to be of order 1.7. Eq. (\[eq:srp\]) is combined with information on the spacecraft surface geometry and it’s local orientation to determine the magnitude of its solar radiation acceleration as it faces the Sun. As with other non-gravitational forces, an incorrect mass in modeling the solar radiation pressure force can be accounted for by modifying other constants such as the effective sizes of the antenna and the albedo. E. J. Smith, L. Davis, Jr., D. E. Jones, D. S. Colburn, P. J. Coleman, Jr., P. Dyal, and C. P. Sonnett, Science [**183**]{}, 306 (1974); [*ibid*]{}. [**188**]{}, 451 (1975). This result was obtained from a limit for positive charge on the spacecraft [@null76]. No measurement dealt with negative charge, but such a charge would have to be proportionally larger to have a significant effect. R. Malhotra, Astron. J. [**110**]{}, 420 (1995); [*ibid.*]{} [**111**]{}, 504 (1996). A. P. Boss and S. J. Peale, Icarus [**27**]{}, 119 (1976). A. S. Liu, J. D. Anderson, and E. Lau, Proc. AGU (Fall Meeting, San Francisco, 16-18 December 1996), paper \# SH22B-05. G. E. Backman, A. Dasgupta, and R. E. Stencel, [Astrophys. J.]{} [**450**]{}, L35 (1995). Also see S. A. Stern, Astron. Astrophys. [**310**]{}, 999 (1996). V. L. Teplitz, S. A. Stern, J. D. Anderson, D. Rosenbaum, R. J. Scalise, and P. Wentzler, [Astrophys. J.]{} [**516**]{}, 425 (1999). J. D. Anderson, G. Giampieri, P. A. Laing, and E. L. Lau, work in progress. R. F. C. Vessot, “Space experiments with high stability clocks,” in proceedings of the “Workshop on the Scientific Applications of Clocks in Space,” (November 7-8, 1996. Pasadena, CA). Edited by L. Maleki. JPL Publication 97-15 (JPL, Pasadena, CA, 1997), p. 67. O. J. Sovers and C. S. Jacobs, [*Observational Model and Parameter Partials for the JPL VLBI Parameter Estimation Software “MODEST” - 1996*]{}, Jet Propulsion Laboratory Technical Report 83-39, Rev. 6 (1996). J. I. Katz, Phys. Rev. Lett. [**83**]{}, 1892 (1999). There is an intuitive way to understand this. Set up a coordinate system at the closest axial point of an RTG pair. Have the antenna be in the (+z,-x) direction, and the RTG pair in the positive x direction. Then from the RTGs the antenna is in 1/4 of a sphere (positive z and negative x). The ‘antenna occupies about 1/3 of 180 degrees in azimuthal angle. Its form is the base part of the parabola. Thus, it resembles a “flat” triangle of the same width, producing another factor of $\sim (1/2-2/3)$ compared to the angular size of a rectangle. It occupies of order (1/4-1/3) of the latitudinal-angle phase space angle of 90$^o$. This yields a total reduction factor of $\sim(1/96-2/108)$, or about 1 to 2 %. The value of 1.5% is obtained by doing an explicit calculation of the solid angle subtended by the antenna from the middle of the RTG modules using the Pioneer’s physical dimensions. V. J. Slabinski of USNO independently obtained a figure of 1.6%. Our high estimate of 40 W is not compromised by imprecise geometry. If the RTGs were completely in the plane of the top of the dish, then the maximum factor multiplying the 40 W directed power would be $\kappa_z = 1$. This would presume all the energy was reflected and/or absorbed and re-emitted towards the rear of the craft. (If the RTGs were underneath the antenna, then the total factor could ideally go as high as “2”, from adding the RTG heat going out the opposite direction.) The real situation is that the average sine of the latitudinal angle up to the antenna from the RTGs is about 0.3. This means that the heat gong out the opposite direction might cause an effective factor $\kappa_z$ to go as high as 1.3. However, the real reflection off of the antenna is not straight backwards. It is closer to 45$^o$. The absorbed and re-emitted radiation is also at an angle to the rotation axis, although smaller. (This does not even consider reflected/reemitted heat that does not go directly backwards but rather bounces off of the central compartment.) So, the original estimate of $\kappa=1$ is a good bound. J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, Phys. Rev. Lett. [**83**]{}, 1893 (1999). We acknowledge R. E. Slusher of Bell Labs for raising this possibility. B. A. Smith, G. A. Briggs, G. E. Danielson, A. F. Cook, II, M. E. Davies, G. E. Hunt, H. Masursky, L. A. Soderblom, T. C. Owen, C. Sagan, and V. E. Suomi, Space Sci. Rev. [**21**]{}, 103 (1977). C. E. Kohlhase and P. A. Penzo, Space Sci. Rev. [**21**]{}, 77 (1977). We are grateful to C. J. Hansen of JPL, who kindly provided us with operational information on the Voyager video cameras. B. A. Smith, L. A. Soderblom, D. Banfield, C. Barnet, T. Basilevsky, R. F. Beebe, K. Bollinger, J. M. Boyce, A. Brahic, G. A. Briggs, R. H. Brown, C. Chyba, S. A. Collins, T. Colvin, A. F. Cook, II, D. Crisp, S. K. 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Laing, E. L. Lau, M. M. Nieto, and S. G. Turyshev, eprint gr-qc/0107022. These results were not treated for systematics, used different time-evolving estimation procedures, were done by three separate JPL navigation specialists, separated and smoothed by one of us [@jpl], and definitely not analyzed with the care of our recent run (1987.0 to 1998.5). In particular, the first two Pioneer 11 points, included in the early memos [@jpl], were after Pioneer 11 encountered Jupiter and then was going back across the central solar system to encounter Saturn. T. K. Keenan, R. A. Kent, and R. N. R. Milford, [*Data Sheets for PMC Radioisotopic Fuel*]{}, Los Alamos Report LA-4976-MS (1972), available from NTIS. We thank Gary Reinhart for finding this data for us. Diagrams showing the receptacle and the bayonet coupling connector were made by the Deutsch Company of Banning, CA. (The O-ring was originally planned to be silicon.) Diagrams of the receptacle as mounted on the RTGs were made by Teledyne Isotopes (now Teledyne Brown Engineering). Once again we gratefully acknowledge Ted Christenbury for obtaining these documents for us. In principle, many things could be the origin of some spin down: structural deformations due to adjustments or aging, thermal radiation, leakage of the helium from the RTGs, etc. But in the case of Pioneer spacecraft none of these provide an explanation for the spin history exhibited by the Pioneer 10, especially the large unexpected changes among the Intervals I, II, and III. S. Herrick, [*Astodynamics*]{} (Van Nostrand Reinhold Co., London, New York, 1971-72). Vols. 1-2, We thank William Folkner of JPL for his assistance in producing several test files and invaluable advice. D. Brouwer and G. M. Clemence, [*Methods of Celestial Mechanics*]{} (Academic Press, New York, 1961). W.G. Melbourne, Scientific American [**234**]{}, No. 6, 58 (1976). We thank E. 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Also see eprint gr-qc/9901042. R. D. Reasenberg, I. I. Shapiro, P. E. MacNeil, R. B. Goldstein, J. C. Breidenthal, J. P. Brenkle, D. L. Cain, T. M. Kaufman, T. A. Komarek, and A. I. Zygielbaum, [Astrophys. J.]{} [**234**]{}, L219 (1979). J. D. Anderson, J. K. Campbell, R. F. Jurgens, E. L. Lau, X X Newhall, M. A. Slade III, and E. M. Standish, Jr., in: [*Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity*]{}, Part A, ed. H. Sato and T. Nakamura, (World Scientific, Singapore, 1992), p. 353. R. H. Sanders, private communication to M. Milgrom (1998). The Principle of Equivalence figure of merit is $a_P/a_N$. This is worse than for laboratory experiments (comparing small objects) or for the Nordtvedt Effect (large objects of planetary size) [@Will93]. It again emphasizes that the Earth and Mars do not change positions due to $a_P$. F. Munyaneza and R. D. Viollier, eprint astro-ph/9910566. J. L. Rosales and J. L. Sánchez-Gomez, eprint gr-qc/9810085. V. 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But later analyses showed that the experiments had been so precise that small inhomogeneities in the field surveys had introduced anomalies in the results at this newly precise level [@bartlett]. But the very important positive outcome was that geophysicists realized the point had been reached where more precise studies of systematics were necessary. F. D. Stacey, G. J. Tuck, G. J. Moore, S. C. Holding, B. D. Goodwin, and R. Zhou, Rev. Mod. Phys. [**59**]{}, 157 (1987); D. H. Eckhardt, C. Jekeli, A. R. Lazarewicz, A. J. Romaides, and R. W. Sands, Phys. Rev. Lett. [**60**]{}, 2567 (1988). Measurements were more often taken at easily accessible sites, such as roads, rather than at more inaccessible cites at different heights, such as mountain sides or marshes. See D. F. Bartlett and W. L. Tew, Phys. Rev. D [**40**]{}, 673 (1989); [*. ibid.*]{}, J. Geophys. Res. \[Solid Earth Planet\] [**95**]{}, 17363 (1990); C. Jekeli, D. H. Eckhardt, and A. J. Romaides, Phys. Rev. Lett. [**64**]{}, 1204 (1990). For a review, see Section 4 of Ref. [@physrep]. [^1]: Electronic address: john.d.anderson@jpl.nasa.gov [^2]: Electronic address: Philip.A.Laing@aero.org [^3]: Electronic address: Eunice.L.Lau@jpl.nasa.gov [^4]: Deceased (13 November 2000). [^5]: Electronic address: mmn@lanl.gov [^6]: turyshev@jpl.nasa.gov

--- author: - | K. Ziegler\ Institut für Physik, Universität Augsburg, Germany\ e-mail: ziegler@physik.uni-augsburg.de title: 'Spin-1/2 fermions: crossover from weak to strong attractive interaction' --- Abstract: The formation and dissociation of bosonic molecules in an optical lattice, formed by spin-1/2 fermionic atoms, is considered in the presence of an attractive nearest-neighbor interaction. A mean-field approximation reveals three different phases at zero temperature: an empty, a condensate and a Mott-insulating phase. The density of fermionic atoms and the density of bosonic molecules indicate a characteristic behavior with respect to the interaction strength that distinguishes between a dilute and a dense regime. In particular, the attractive interaction favors the formation of molecules in the dilute regime and the dissociation of atoms in the dense regime. Introduction ============ Feshbach resonances provide a powerful tool to vary the interaction strength of atomic gases [@kleppner04]. Physical effects, like the formation of molecules, Bose-Einstein condensation or Cooper pairing in fermionic systems depend strongly on the strength of the interparticle interaction [@strecker03; @salomon03; @zwierlein03; @grimm03; @jin03]. A model for spin-1/2 fermions with attractive nearest-neighbor interaction is studied to describe the formation and dissociation of bosonic molecules in a grand-canonical gas of fermionic atoms. The model depends on three tunable parameters, the fermionic tunneling rate $\bt$, the fugacity $\zeta$ of the grand-canonical gas, and the molecular tunneling rate $J$. $J$ also controls the attractive nearest-neighbor interaction of the model. A difference in comparison with previous works [@holland01; @chiofalo02; @ohashi02; @holland04] is that we study the $T=0$ properties of a Fermi gas in an optical lattice. A consequence of the latter is that a Mott-insulating state can be identified, whereas $T=0$ avoids thermal fluctuations of the system. In this case a mean-field approximation should give reasonable results, provided that quantum fluctuation are not too strong. A qualitative picture of the $T=0$ phase diagram can be given by an estimate of the energies. There is a single-particle potential due to the chemical potential of the fermionic atoms. It is assumed that the chemical potential of the molecules is the same as that of the constituting fermionic atoms, i.e. the effective chemical potential of a molecule is twice as that of a single fermionic atom. If this potential is sufficiently negative the optical lattice is empty, since particles cannot overcome the potential barrier. The kinetic energy, however, represented by the tunneling rates $\bt$ and $J$, competes with the chemical potential and favors the creation of a Fermi gas and/or a molecular Bose gas. The interaction plays a crucial role in this regime: In addition to the formation of (local) bosonic molecules, the attractive interaction between the fermionic atoms may lead to the formation of Cooper pairs, which can be considered as “non-local molecules”. For sufficiently large chemical potential the optical lattice will be filled with two atoms per lattice site, which is a Mott-insulating state. This Mott state differs from the Mott insulator of a half-filled Fermi lattice gas with [*repulsive*]{} interaction (Hubbard model [@fulde]), where the interaction creates a ground state with one fermion per lattice site. In other words, the Mott insulator in the system with attractive interaction is a special kind of insulator that is created by the Pauli exclusion, and not by a finite repulsive interaction, in contrast to the Hubbard model at half filling. A mean-field theory is applied to discuss a Mott insulator and a condensed phase. Depending on the tunable parameters of the model, the density of dissociated atoms and the density of molecules in the condensed phase are investigated. It will be shown that the density of dissociated atoms increases with the total number of particles, reaches a maximum and decreases, whereas the density of molecules is a monotoneously increasing function. Model: Hamiltonian and Functional Integral ========================================== A gas of spin-1/2 fermions in an optical lattice is considered. Using fermionic creation (annihilation) operators $c^\dagger_{r\sigma}$ ($c_{r\sigma}$) for spin $\sigma$ and at site $r$ (a minimum in the optical lattice), the Hamiltonian of the Fermi gas is H=-\_[&lt;r,r’&gt;]{}\_[=,]{}c\^\_[r]{}c\_[r’]{} -J\_[&lt;r,r’&gt;]{}c\^\_[ru]{}c\_[r’u]{}c\^\_[r]{}c\_[r’]{} -\_[r]{}\_[=,]{}c\^\_[r]{}c\_[r]{}. \[hamilton\] $<r,r'>$ refers to nearest-neighbor sites $r$ and $r'$. A chemical potential term for molecules $$-\nu\sum_r c^\dagger_{r\u}c_{r\u}c^\dagger_{r\d}c_{r\d}$$ has been neglected because it leads only to a shift of $J$ in terms of the subsequent mean-field calculation. Similar Hamiltonians were considered in a number of papers [@holland01; @chiofalo02; @ohashi02]. The first term describes tunneling of individual fermions in the optical lattice with rate $\bt$, the second term tunneling of local fermion pairs (i.e. bosonic molecules) between nearest-neighbor sites with rate $J$. It should be noticed that the latter is also responsible for an attractive interaction between fermions with different spin ($J>0$). However, there is no local (diagonal) interaction, all the fermionic interaction is carried by the nearest-neighbor term. The bosonic molecules experience a local repulsive (hard-core) interaction, since their constituent atoms are fermions and obey the Pauli exclusion. For the limiting case $\bt=0$ only bosonic molecules can appear, for $J=0$ only non-interacting fermionic atoms. The chemical potential $\mu$ controls the number of particles in a grand-canonical ensemble. The latter is given by the partition function $$Z=Tr e^{-\beta H}.$$ Space-time correlations of fermions are described by the Green’s function $$G_{r,t,\sigma ;r',0,\sigma'}= {1\over Z} Tr\Big[ e^{-(\beta-t) H} c_{r,\sigma}e^{-tH}c^\dagger_{r',\sigma'}\Big].$$ The partition function can also be written in terms of a Grassmann integral [@negele; @ziegler02] as $$Z=\int e^{-S}{\cal D}[\psi],$$ where the action $S$ for spin-1/2 fermions with attractive interaction, related to the Hamiltonian in Eq. (\[hamilton\]), reads $$S=\sum_x(\psi_{x,1}{\bar\psi}_{x,1} +\psi_{x,2}{\bar\psi}_{x,2}) -\sum_{r,r',t}(\zeta\delta_{r,r'}+\bt w_{r,r'}) (\psi_{r,t,1}{\bar\psi}_{r',t+1,1} +\psi_{r,t,2}{\bar\psi}_{r',t+1,2})$$ -J\_[r,r’,t]{}\_[r,t,1]{}[|]{}\_[r’,t+1,1]{}\_[r,t,2]{} [|]{}\_[r’,t+1,2]{} \[action0\] with space-time coordinates $x=(r,t)$. $w_{r,r'}$ is $1/2d$ for nearest-neighbor sites on a $d$-dimensional cubic optical lattice and zero otherwise, and $\zeta=e^\mu$ is the fugacity. The Green’s function then reads $$G_{x,\sigma ;x',\sigma'}= \langle \psi_{x,\sigma}{\bar\psi}_{x',\sigma'}\rangle \equiv\int \psi_{x,\sigma}{\bar\psi}_{x',\sigma'} e^{-S}{\cal D}[\psi ]/Z.$$ After renaming ${\bar\psi}$ by a time shift $t\to t-1$ \_[r,t,j]{}\_[r,t-1,j]{}   (j=1,2) \[shift\] the action reads $$S'=\sum_x(\psi_{x,1}\partial_t^T{\bar\psi}_{x,1} +\psi_{x,2}\partial_t^T{\bar\psi}_{x,2}) -\sum_{r,r',t}(\zeta\delta_{r,r'}+\bt w_{r,r'}) (\psi_{r,t,1}{\bar\psi}_{r',t,1} +\psi_{r,t,2}{\bar\psi}_{r',t,2})$$ $$-J\sum_{r,r',t}\psi_{r,t,1}{\bar\psi}_{r',t,1}\psi_{r,t,2} {\bar\psi}_{r',t,2}.$$ The quartic interaction term is now diagonal with respect to time. It can be decoupled by two complex Hubbard-Stratonovich fields $\phi_x$, $\chi_x$ [@ziegler02]. The linear combination $i\phi+\chi$ couples to fermions like (i\_[x]{}+\_[x]{})\_[x,1]{}\_[x,2]{}+h.c.. \[eff\] This coupling is similar to the atom-molecule coupling discussed in Refs. [@holland04; @petrov03]. There is a difference, however, by the fact that we started from the fermionic model and derived an effective electron-boson model, whereas the other authors started from an electron-boson model and derived the effective fermion model. In both cases an effective coupling constants can be evaluated from the original model, an effective fermion-fermion coupling in Ref. [@holland04] and an effective fermion-boson coupling in our model. Although this is an interesting problem it will not be pursued in this paper. Instead, the fermions are integrated out in $Z$, since they appear only in a quadratic form in the action. This step provides an effective model only for bosons: the integration gives a fermion determinant and the resulting fuctional $Z$ depends only on the complex fields $\phi$ and $\chi$ with the effective action S\_[eff]{}=(,v\^[-1]{})+[12J]{}(,) -[12]{}(A) \[seff\] with the antisymmetric space-time matrix $$A=\pmatrix{ 0 & i\phi+\chi & \zeta+\bt w-\partial_t^T & 0 \cr -i\phi-\chi & 0 & 0 & \zeta+\bt w - \partial_t^T \cr -\zeta-\bt w +\partial_t & 0 & 0 & i{\bar\phi}+{\bar\chi} \cr 0 & -\zeta-\bt w +\partial_t & -i{\bar\phi}-{\bar\chi} & 0 \cr }$$ with $v^{-1}=(w+2{\bf 1})^{-1}/J$. $\phi$ corresponds to the conventional BCS field in the case of a local interaction. The additional field $\chi$ is necessary for the nearest-neighbor interaction to ensure that the Hubbard-Stratonovich decoupling is well-defined [@ziegler02]. It will be discussed subsequently that the mean-field approximation yields a linear relationship between these two fields. Atomic and Molecular Densities ------------------------------ The densities of atoms and molecules can be measured as expectation values of the Grassmann fields. For this purpose the original fields are used, i.e., the fields before the time shift in Eq. (\[shift\]), to write $$\langle ... \rangle =\int ... e^{-S}{\cal D}[\psi ]/Z.$$ The action $S$ of Eq. (\[action0\]) can be expanded in $Z$ around $$S_0=\sum_x(\psi_{x,1}{\bar\psi}_{x,1}+\psi_{x,2}{\bar\psi}_{x,2})$$ to obtain $$Z=\sum_{l\ge0}{1\over l!}\int e^{-S_0}(S_0-S)^l{\cal D}[\psi ].$$ This expansion can be viewed as an expansion in terms of world lines in a space-time lattice. There are two types of world lines, individual fermion lines with spin $\sigma$ and molecular world lines, as shown in Fig. 1. For a given point in space and time $x$ the contribution from the polynomial $(S_0-S)^l$ is either a factor 1 (i.e. no contribution from $(S_0-S)^l$ at this point), a factor $\psi_{x,\sigma}{\bar\psi}_{x,\sigma}$, or a factor $\psi_{x,1}{\bar\psi}_{x,1}\psi_{x,2}{\bar\psi}_{x,2}$. To measure the probability of the appearence of these factors in $Z$, an expectation value with respect to the Grassmann field can be introduced. In particular, the density of dissociated atoms reads $$n_{f,x}=-\langle\psi_{x,1}{\bar\psi}_{x,1} +\psi_{x,2}{\bar\psi}_{x,2}+2\psi_{x,1}{\bar\psi}_{x,1} \psi_{x,2}{\bar\psi}_{x,2}\rangle$$ and the density of molecules $$n_{m,x}=\langle 1+\psi_{x,1}{\bar\psi}_{x,1} +\psi_{x,2}{\bar\psi}_{x,2}+\psi_{x,1}{\bar\psi}_{x,1} \psi_{x,2}{\bar\psi}_{x,2}\rangle .$$ The truncated expectation value $$C_{12}=\langle\psi_{x,1}{\bar\psi}_{x,1} \psi_{x,2}{\bar\psi}_{x,2}\rangle- \langle\psi_{x,1}{\bar\psi}_{x,1}\rangle\langle \psi_{x,2}{\bar\psi}_{x,2}\rangle$$ gives n\_[f,x]{}=-1+\_[x,1]{}[|]{}\_[x,1]{}\_[x,2]{}[|]{}\_[x,2]{}-1+\_[x,2]{}[|]{}\_[x,2]{}\_[x,1]{}[|]{}\_[x,1]{}-2C\_[12]{} \[denf\] and n\_[m,x]{}=1+\_[x,1]{}[|]{}\_[x,1]{}1+\_[x,2]{}[|]{}\_[x,2]{}+ C\_[12]{}. \[denm\] It will be seen below that $C_{12}$ vanishes in mean-field approximation. In the Mott-insulating phase a fully occupied lattice is expected, i.e., two fermionic atoms (= one molecule) per site, as the only commensurate state, since there is no repulsive interaction which could maintain a commensurate state. Any other groundstate of the system is a condensate, except for the empty lattice. The remaining question is how the condensate varies with the total number of atoms and molecules in the system. It is obvious that the total density of particles increases monotoneously with the fugacity. However, it is less clear how the formation of molecules or the dissociation of atoms is affected by an increasing fugacity. Moreover, a naive picture suggests that the density of molecules increases monotoneously with an increasing attractive coupling of the fermionic atoms $J$. It will be shown in terms of a mean-field approximation that this is not the case. The Saddle-point Equation ========================= The saddle-point condition for the effective action in Eq. (\[seff\]) is S\_[eff]{}=0. \[spe0\] This gives a nonlinear difference equation. A simple ansatz for its solution is given by uniform fields $\phi$ and $\chi$ (mean-field approximation). Then Eq. (\[spe0\]) reads \[spe\] with the integral G=\_[-1]{}\^1[(x)]{} dx \[int1\] and $s=\zeta+\bt x$. $\rho(x)$ is the density of states for the nearest-neighbor tunneling term $w$. There is a trivial solution $\phi=\chi=0$ and a nontrivial solution G=1/J,   =-2i/3. \[bcs\] This is the same mean-field equation as in the BCS theory, if $J$ is considered as the coupling constant of the fermions. The solution in Eq. (\[bcs\]) implies a non-negative value for $$(-i\chi+\phi)(-i{\bar\chi}+{\bar\phi})=\phi{\bar\phi}/9\equiv |\phi|^2/9.$$ Then $G$ in Eq. (\[int1\]) reads G=\_[-1]{}\^1[(x)]{} dx \[spe2\] which descreases monotoneously with $|\phi|^2$. If $(s^2-1)^2$ becomes 0 inside the interval of integration $-1\le x\le 1$ the integral diverges for $|\phi|\to0$. Therefore, there is a non-zero solution of $\phi$ for any value of $J>0$. On the other hand, if $(s^2-1)^2$ does not become 0 inside the interval of integration, there is a non-zero solution only for sufficiently large values of $J$ (cf. Fig. 2). The densities can be evaluated within the saddle-point approximation. A straightforward calculation gives for the expectation values, used in $n_{f,x}$ and $n_{m,x}$, the expression \_[x,j]{}[|]{}\_[x,j]{} =-[12]{}+[12]{}\_[-1]{}\^1 [||\^2/9+s\^2-1]{} (x)dx \[exp1\] and $$\langle\psi_{x,j}{\bar\psi}_{x,j'}\rangle =0\ \ \ (j\ne j').$$ Results ======= Using the mean-field approximation of the previous section the following quantities are evaluated: the order parameter $|\phi|^2$ of the condensate, the $T=0$ phase diagram, and $n_f(|\phi|^2,\zeta)$, $n_m(|\phi|^2,\zeta)$. $\phi=0$ -------- In the case of a vanishing order parameter (i.e. $\phi=0$) only independent fermions are described by the mean-field approach. Then the diagonal element in Eq. (\[exp1\]) is $$\langle\psi_{x,j}{\bar\psi}_{x,j}\rangle =-\int_{-1}^1\Theta(1-(\zeta+tx)^2)\rho(x)dx.$$ If $s^2<1$ for the entire interval of integration, i.e., $$-1+\bt <\zeta<1-\bt$$ both densities vanish: $n_f=n_m=0$. On the other hand, if $s^2>1$ for the entire interval of integration, i.e., $$\zeta>1+\bt$$ one obtains $n_f=0$, $n_m=1$. Thus all lattice sites are occupied by bosonic molecules. There is also an intermediate regime for $1-\bt<\zeta<1+\bt$ where $n_f, n_m\ne0$. Details of this behavior are shown in Figs. 2 and 3. $\phi\ne 0$ ----------- For the beginning the special case $\bt=0$ (molecules cannot dissociate) is considered. Then the integrand in Eq. (\[exp1\]) is constant with respect to the integration variable $x$ and yields $$\langle\psi_{x,j}{\bar\psi}_{x,j}\rangle =-1/2 +{1\over2} {|\phi|^2/9+\zeta^2-1\over\sqrt{ |\phi|^4/81+2(\zeta^2+1)|\phi|^2/9 +(\zeta^2-1)^2 }}.$$ The saddle-point solution $$|\phi|^2=9(\sqrt{4\zeta^2+J^2}-1-\zeta^2)$$ exists for $$\cases{ \sqrt{1-J}<\zeta<\sqrt{1+J} & if $0<J\le1$ \cr 0<\zeta<\sqrt{1+J} & if $J >1$ \cr }.$$ The corresponding phase diagram is shown in Fig. 4. This result yields $$\langle\psi_{x,j}{\bar\psi}_{x,j}\rangle=-1/2-1/J+\sqrt{\zeta^2/J^2+1/4}.$$ In the general case with dissociated atoms (i.e. for $\bt>0$), the density of states in $G$ can be approximated by a constant as $\rho(x)=1/2$. Moreover, for small $J$ the integral in Eq. (\[spe2\]) can be easily performed, giving the saddle-point solution of the order parameter $$|\phi|^2\sim 4\bt^2 e^{-4\bt /J}\ \ \ \ (1-\bt <\zeta< 1+\bt).$$ The model parameters $\zeta$, $J$, and $\bt$ control the densities. However, in a realistic situation the order parameter $\phi$ and the densities $n_f,n_m$ will be measured. Therefore, the densities for a given value of $|\phi|$ are plotted as functions of the fugacity $\zeta$, where increasing of $|\phi|$ means an intereasing interaction parameter $J$. The result is shown in Figs. 2 and 3: (1) The density of dissociated atoms $n_f$ has a maximum. This maximum is at $\zeta=1$ for $\phi=0$ and moves to lower values of $\zeta$ with increasing order parameter values. The maximum value of $n_f$ itself is $0.5$ but decreases for $|\phi|>1$. (2) The density of molecules $n_m$ always increases with $\zeta$. But as a function of $|\phi|$ it increases (decreases) below (above) $\zeta\approx1$. This result indicates that strong interaction favors the formation of molecules in a dilute system and favors the dissociation of atoms in a dense system. The latter can be understood as a special kind of frustration effect because there many ways for a pair of fermionic atoms to form a molecule. (3) The total density $n_f+n_m$ increases always monotoneously with $\zeta$. Conclusions =========== An attractively interacting Fermi gas in an optical lattice was treated in mean-field approximation. At zero temperature three different phases were found: an empty phase, a Mott-insulating phase and a condensed phase. The latter is characterized by a non-vanishing order parameter. The fermions appear in this phase as a mixture of local pairs (molecules) and extended pairs (dissociated atoms). The densities of these two types of fermionic pairs have a characteristic behavior for the crossover from weak to strong attractive interaction. This is indicated by (1) a maximum of the density of dissociated fermionic atoms and (2) support for the formation of molecules (dissociation of atoms) in the dilute (dense) system by the attractive interaction. 0.5cm The author is grateful to G. Shlyapnikov for useful discussions. This research was supported in part by the National Science Foundation under Grant No. PHY99-07949 and by the Deutsche Forschungsgemeinschaft through SFB 484. [99]{} Kleppner, D., 2004, Physics Today August, 12. Strecker, K.E. et al., 2003, Phys. Rev. Lett. [**91**]{}, 080406. Cubizolles, J. et al., 2003, Phys. Rev. Lett. [**91**]{}, 240401. Zwierlein, M.W. et al., 2003, Phys. Rev. Lett. [**91**]{}, 250401. Jochim, S. et al., 2003, Phys. Rev. Lett. [**91**]{}, 240401. Regal, C.A., Greiner, M., and Jin, D.S., 2003, Phys. Rev. Lett. [**92**]{}, 083201. Dickerscheid, D.B.M. et al., 2004, Feshbach resonances in an optical lattice, cond-mat/0409416. Holland, M. et al., 2001, Phys. Rev. Lett. [**87**]{}, 120406. Chiofalo, M.L. et al., 2002, Phys. Rev. Lett. [**88**]{}, 090402. Ohashi, Y. and Griffin, A., 2002, Phys. Rev. Lett. [**89**]{}, 130402. Holland, M.J., Menotti, C., and Viverit, L., 2004, cond-mat/0404234. Petrov, D.S., Salomon, C., and Shlyapnikov, G.V., 2003, cond-mat/0309010. Negele, J.W. and Orland, H., 1988, [*Quantum Many - Particle Systems*]{} (New York: Addison - Wesley). Fulde, P., 1993, [*Electron Correlations in Molecules and Solids*]{} (Berlin: Springer - Verlag). Ziegler, K., 2002, Journ. Low. Temp. Phys. [**126**]{}, 1431.  \   

--- abstract: 'Synthetic spectra generated with the parameterized supernova synthetic–spectrum code SYNOW are compared to photospheric–phase spectra of Type Ib supernovae (SNe Ib). Although the synthetic spectra are based on many simplifying approximations, including spherical symmetry, they account well for the observed spectra. Our sample of SNe Ib obeys a tight relation between the velocity at the photosphere, as determined from the Fe II features, and the time relative to that of maximum light. From this we infer that the masses and the kinetic energies of the events in this sample were similar. After maximum light the minimum velocity at which the He I features form usually is higher than the velocity at the photosphere, but the minimum velocity of the ejected helium is at least as low as 7000 . Previously unpublished spectra of SN 2000H reveal the presence of hydrogen absorption features, and we conclude that hydrogen lines also were present in SNe 1999di and 1954A. Hydrogen appears to be present in SNe Ib in general, although in most events it becomes too weak to identify soon after maximum light. The hydrogen–line optical depths that we use to fit the spectra of SNe 2000H, 1999di, and 1954A are not high, so only a mild reduction in the hydrogen optical depths would be required to make these events look like typical SNe Ib. Similarly, the He I line optical depths are not very high, so a moderate reduction would make SNe Ib look like SNe Ic.' author: - '[David Branch]{}, [S. Benetti]{}, [Dan Kasen]{}, [E. Baron]{}, [David J. Jeffery]{}, [Kazuhito Hatano]{}, [R. A. Stathakis]{}, [Alexei V. Filippenko]{}, [Thomas Matheson]{}, [A. Pastorello]{}, [G. Altavilla]{}, [E. Cappellaro]{}, [L. Rizzi]{}, [M. Turatto]{}, [Weidong Li]{}, [Douglas C. Leonard]{}, and [Joseph C. Shields]{}' title: Direct Analysis of Spectra of Type Ib Supernovae --- INTRODUCTION ============ Supernovae of Type II are those that have obvious hydrogen lines in their optical spectra. Type IIb supernovae have obvious hydrogen lines around the time when they reach their maximum brightness (hereafter just “maximum light”) but later the hydrogen lines become weak or even disappear. Type Ib supernovae do not have obvious hydrogen lines but they do develop conspicuous He I lines after maximum light. Neither hydrogen nor He I lines are conspicuous in the spectra of Type Ic supernovae. Most or all events of these four types — II, IIb, Ib, and Ic — are thought to result from core collapse in massive stars. (Type Ia supernovae, whose spectra lack hydrogen and have a strong absorption feature produced by Si II $\lambda\lambda6347,6371$, are thought to have a fundamentally different origin, as thermonuclear disruptions of accreting or merging white dwarfs.) For a recent review of supernova spectral classification, including its historical development and with illustrations of spectra of each type, see Filippenko (1997). In this paper we are concerned with optical photospheric–phase spectra of Type Ib supernovae (SNe Ib). Until recently, good photospheric–phase spectra had been published for only two SNe Ib: SN 1983N (Richtler & Sadler 1983; Harkness et al. 1987) and SN 1984L (Harkness et al. 1987). The situation has improved substantially now that Matheson et al. (2001) have published spectra of the SNe Ib that were observed at the Lick Observatory during the 1990s. These newly available spectra, together with some additional previously unpublished spectra that are presented in this paper, motivated us to carry out a comparative study of SN Ib spectra. (Matheson et al. also present spectra of SNe IIb and Ic, the study of which we defer to separate papers.) Our method is to compare the observed SN Ib spectra with synthetic spectra that we generate with the fast, parameterized, supernova spectrum–synthesis code, SYNOW. We refer to this approach, in which the goal is to extract some constraints on the ejected matter from the observations in an empirical spirit, as “direct” analysis — to distinguish it from the process of making very detailed non–local–thermodynamic–equilibrium (non–LTE) calculations of synthetic spectra based on supernova hydrodynamical models (e.g., Baron et al. 1999). Issues that we can explore by means of our direct analysis include (1) line identifications; (2) the extent to which synthetic spectra calculated on the basis of simple assumptions can or cannot account for observed SN Ib spectra; (3) the degree to which the rather homogeneous appearance of SN Ib spectra, pointed out by Matheson et al. (2001), reflects a genuine physical homogeneity; (4) the velocities at which the He I lines form, compared to the velocity at the photosphere as determined by the Fe II lines; (5) whether hydrogen lines are present and, when they are, the velocities at which they form. Previous work on the interpretation of photospheric–phase spectra of SNe Ib is briefly summarized in §2. The observed spectra that were selected for this project are discussed in §3 and those that have not been published previously are displayed. The synthetic spectrum calculations are described in §4, and comparisons with observed spectra are presented in §5. The results are summarized and discussed in §6. PREVIOUS WORK ============= The classic early paper on the interpretation of the photospheric–phase spectra of SNe Ib was that of Harkness et al. (1987), who calculated local thermondynamic equilibrium (LTE) synthetic spectra for parameterized model supernovae having power–law density structures and homogeneous chemical compositions, and compared them to observed spectra of SNe 1984L and 1983N. Some observed features that appear in spectra of all supernova types were readily attributed to Ca II and Fe II lines. Most of the remaining conspicuous features were convincingly attributed to lines of He I, even though large ad hoc overpopulations of the highly excited lower levels of the He I lines had to be invoked in order to account for their presence. The explanation for the overpopulations was later shown to be nonthermal excitation and ionization caused by the decay products of radioactive $^{56}$Ni and $^{56}$Co (Lucy 1991; Swartz et al. 1993). A feature of special interest in the spectra of SNe 1984L and 1983N was an absorption near 6300 Å that could not be attributed to Fe II, Ca II, or He I lines. \[Harkness et al. (1987) referred to this as “the 6300 Å absorption” and so will we, although its wavelength can be as short as 6200 Å at early times.\] Harkness et al. tentatively attributed this absorption to C II $\lambda6580$, forming in outer high–velocity ($\ge$14,000 ) layers of the ejected matter. Later, on the basis of LTE synthetic spectra calculated for radially stratified chemical compositions, Wheeler et al. (1994) suggested the absorption to be H$\alpha$, forming at $\ge$13,000 . The issue of whether hydrogen is present in SNe Ib is very important because of its implications for the nature and appearance of the progenitor stars, but it has been a difficult issue to resolve because C II 6580, being less than 800  to the red of H$\alpha$, is usually a plausible alternative identification. Woosley & Eastman (1997) presented a comparison of a non–LTE synthetic spectrum based on a particular explosion model with a photospheric–phase spectrum of SN 1984L. Overall, the synthetic spectrum accounted rather well for the major features in the observed spectrum. The explosion model did not contain any hydrogen, and in the synthetic spectrum the absorption nearest to the 6300 Å feature was produced by Si II, but as we will see below this cannot be the actual identification of the 6300 Å absorption. This is the only comparison of an non–LTE synthetic spectrum with an observed SN Ib photospheric–phase spectrum to be published so far. No other SN Ib was well observed until SN 1999dn. Deng et al. (2000) used the same SYNOW code that we use in this paper to make a detailed study of line identifications in three spectra that were obtained at the Beijing Astronomical Observatory at times of about 10 days before, at, and 14 days after maximum light. In addition to Fe II, Ca II, and He I lines, Deng et al. explored the possible role of lines of other ions (C I, O I, C II, \[O II\], Na I, Mg II, Si II, Ca I, and Ni II) in shaping the spectra of SN 1999dn. They attributed the 6300 Å absorption in the latest of their three spectra of SN 1999dn to C II $\lambda6580$, forming at $\ge$10,000 , but they suggested that in the earlier two spectra the observed feature was more likely to be H$\alpha$, forming at higher velocity. DATA ==== The 11 SNe Ib that were selected for this study are listed in Table 1. An asterisk preceding the recession velocity, $cz$, indicates that it is the value given by Matheson et al. (2001) for an H II region near the site of the supernova; otherwise the listed value is that of the parent galaxy, from the Asiago Supernova Catalog (Barbon et al. 1999; updates are available at [*http://merlino.pd.astro.it/supern/*]{}). All observed spectra displayed in this paper are corrected for redshift using the values of $cz$ listed in Table 1. An asterisk preceding the date of maximum light in the $V$ band, $t_{max}$, indicates that only the date of discovery is listed, because the date of maximum light is unknown. Six of these events — SNe 1991ar, 1997dc, 1998T, 1998dt, 1999di, and 1999dn — were selected from Matheson et al. (2001) because they don’t have obvious hydrogen lines while they do have conspicuous He I lines. \[SN 1991D, which also may be a Type Ib but with fairly weak He I lines, is discussed in a separate paper (S. Benetti et al., in preparation).\] SNe 1998dt, 1999di, and 1999dn are especially useful for our study because on the basis of photometry obtained at the Lick Observatory, Matheson et al. were able to estimate the dates of maximum light in the $R$ band, which we will assume to peak at the same time as the $V$ band \[as was the case for the Type IIb SN 1996cb (Qiu et al. 1999)\]. The three spectra of SN 1999dn that appeared in Deng et al. (2000) also are included in this study. The spectra of SN 1983N are from Richter & Sadler (1983) and Harkness et al. (1987), and the adopted date of maximum light, 1983 July 17, (in the [*IUE*]{} FES band, which is roughly like the $V$ band) is from an unpublished manuscript that was circulated by N. Panagia et al. in 1984. The spectra of SN 1984L are from Harkness et al. (1987). Tsvetkov (1987) estimated that SN 1984L reached maximum light in the $B$ band on 1984 August $20 \pm 4$ days. We assume that the $V$ band peaked two days later (as was the case for SN 1996cb) and adopt August 22 as the date of maximum light in the $V$ band. SN 1954A is a special case because only photographic spectra, obtained by N. U. Mayall at the Lick Observatory and by R. Minkowski at the Mount Wilson and Palomar Observatories, are available. Microphotometer tracings of the spectra of SN 1954A and many other supernovae observed at the Lick Observatory and the Mount Wilson and Palomar Observatories between 1937 and 1971 have been digitized and displayed by Casebeer et al. (2000) and Blaylock et al. (2000). In the Asiago Catalog the date of maximum light of SN 1954A in the $B$ band is estimated as 1954 April 19, so we will adopt April 21 for the $V$ band. One previously unpublished spectrum of SN 1996N is included in this study. The spectrum, obtained at the Anglo–Australian Telescope on 1996 March 23 (Germany et al. 2000), 11 days after discovery, is shown in Figure 1. The date of maximum light of SN 1996N is unknown. The spectrum appears to be that of a typical SN Ib not long after maximum light. Six previously unpublished spectra[^1] of SN 2000H (Pastorello et al. 2000; Benetti et al. 2000) also are included. These are shown in Figure 2. The spectra of SN 2000H resemble those of a typical SN Ib except for an unusually deep 6300 Å absorption in the first four spectra, as well as a weak, narrow absorption near 4650 Å in at least the second and third spectra. Benetti et al. attributed these absorptions to H$\alpha$ and H$\beta$. (The H$\beta$ feature will be seen more clearly in subsequent figures.) This identification of hydrogen lines might raise the question of whether SN 2000H should be regarded as a Type IIb, but we do not favor such a classification because even at the earliest observed times the presence of hydrogen lines was not obvious, as evidenced by initial classifications of SN 2000H as a peculiar Type Ia (Garnavich et al. 2000) and a Type Ic (Pastorello et al. 2000). From unpublished ESO photometry of SN 2000H we estimate that the date of maximum light in the $B$ band was 2000 February $9\pm2$ days, so we adopt February 11 as the date of maximum light in the $V$ band. CALCULATIONS ============ Calculations have been carried out with the fast, parameterized supernova spectrum–synthesis code, SYNOW. Recent applications to Type Ic supernovae and brief descriptions of SYNOW can be found in Millard et al. (1999) and Branch (2001), and technical details of the code are in Fisher (2000). An extensive discussion and illustration of the elements of supernova line formation appears in Jeffery & Branch (1990). The basic assumptions of SYNOW are spherical symmetry; velocity proportional to radius; a sharp photosphere; and line formation by resonant scattering, treated in the Sobolev approximation. Various fitting parameters are available. The parameter $T_{bb}$ is the temperature of the blackbody continuum from the photosphere. The values used in this paper range from 8500 to 3600 K and typically are $\sim$6500 K around the time of maximum light and $\sim$5000 K beginning roughly two weeks after maximum. We do not attach much physical significance to these values because (for one thing) the observed spectra have not been corrected for interstellar extinction. For each ion whose lines are introduced, the optical depth at the photosphere of a “reference line” is a fitting parameter, and the optical depths of the other lines of the ion are calculated assuming Boltzmann excitation at excitation temperature $T_{exc}$. In this paper, to reduce the number of free parameters, we simply fix $T_{exc}$ at a nominal SN Ib value of 7000 K. The relevant lines of a given ion don’t have widely differing excitation potentials so their relative optical depths don’t depend strongly on $T_{exc}$, within the range of temperatures that are relevant here. The line optical depths are taken to vary with ejection velocity as $v^{-n}$. Again for simplicity, in this paper we always use $n = 8$, except for one illustration of the effects of using $n = 5$ instead. In the analysis of the SN 1999dn spectra by Deng et al. (2000), the line optical depths were taken to vary as $e^{-v/v_e}$, with $v_e = 1000$ . Since the exponential distribution has an effective power–law index of $n = v/v_e$, the distribution used by Deng et al. falls off more steeply than $n = 8$ for $v > 8000$  and less steeply for $v < 8000$ . This leads to some differences in the values of $v_{phot}$ and the reference–line optical depths used by Deng et al. and by us to match the same observed spectra. The maximum velocity of the line–forming region is set high enough so that effectively there is no outer boundary. The default minimum velocity of the line–forming region is the velocity at the photosphere; when an ion is assigned a higher minimum velocity, that ion is said to be detached from the photosphere. Reasons that SYNOW spectra cannot be expected to provide exact fits to observed spectra are numerous and obvious: the calculations are based on many simplifiying assumptions, including spherical symmetry, and the oscillator strengths (Kurucz 1993) are good but not perfect. In this paper we are not concerned with proposing a line identification for every weak observed feature. We are more interested in establishing the identities of the major features and then concentrating on a comparative analysis — to investigate the degree to which the SNe Ib of our sample are physically similar, and to look for differences. COMPARISONS =========== The Fiducial SN Ib Spectrum: SN1999dn, 17 Days After Maximum ------------------------------------------------------------ We begin the comparisons of observed and synthetic spectra by concentrating on a “fiducial” SN Ib spectrum — a spectrum of a typical SN Ib that has good signal–to–noise ratio and broad wavelength coverage, and in which most of the major spectral features are well developed. The best available spectrum for this purpose is the Matheson et al. (2001) spectrum of SN 1999dn obtained on 1999 September 17, 17 days after maximum light. In Figure 3, this spectrum is compared with a synthetic spectrum that has =6000  and =4800 K, and contains lines of Fe II, He I, O I, Ca II, Ti II, and Sc II. Almost all of the features in the observed spectrum can be attributed to these ions. The discrepancies will be discussed as we look at the contribution of each ion to the synthetic spectrum. As always when fitting observed spectra with SYNOW spectra, we are more concerned with discrepancies in the wavelengths of absorption features than with discrepancies in flux; the latter are inevitable given the simplicity of our spectrum calculations. Figure 4 is like Figure 3 but with nothing but the Fe II lines in the synthetic spectrum. The optical depth of the reference line, 5018, is 7. The Fe II lines are mainly responsible for the spectral features from about 4300 to 5300 Å, and they have additional effects at shorter wavelengths. At this post–maximum time they may also be responsible for the observed absorptions near 6100 and 6300 Å. (These two features are not strong enough in this particular synthetic spectrum, but a higher value of $T_{exc}$ would increase their strengths relative to the reference line.) For this reason we don’t use H$\alpha$ or C II 6580 to account for the weak 6300 Åabsorption in this observed spectrum. Around maximum light, however, Fe II lines are not strong enough to account for the 6300 Åabsorption that is observed at that time. In this paper we always determine the value of $v_{phot}$ on the basis of the Fe II features in the region from about 4300 to 5300 Å. Figure 5 shows a comparison of two Fe II synthetic spectra that have = 5000 and 10,000 . This figure shows that the spectral signature of Fe II in this wavelength range is quite sensitive to . Our fitting uncertainty in $v_{phot}$ is about 1000 . The top panel of Figure 6 is like Figure 3 but with only the He I lines in the synthetic spectrum. The He I lines are detached at 8000  (recall that =6000 ), where the optical depth of the reference line, 5876, is 10. It is likely that two optical He I lines, 6678 and 7065, are almost entirely responsible for their corresponding observed features. Two other lines, 5876 and 4472, are mainly responsible for their corresponding observed features but they may be blended with the Na I D lines (5890, 5896) and Mg II 4481, respectively. In this spectrum He I 7281 accounts very nicely for an observed feature, but in some other spectra the fit is not so good. The remaining optical He I lines are weaker and in the synthetic spectrum of Figure 3 they are overwhelmed by lines of other ions. In Figures 3 and 6 the blue edge of the synthetic absorption produced by He I 10830 is not blue enough to account for the sharp drop in the observed spectrum near 9000 Å, but we show how to remedy this below. In Figure 3 and the top panel of Figure 6 the He I lines are detached at 8000  in order to fit the wavelengths of the corresponding observed absorption features. The detachment causes the flat tops of the synthetic He I emission components (which are superimposed on a sloping continuum). The rounded emission peak that is observed near 5900 Å could easily be achieved in the synthetic spectrum by including undetached Na I D lines. To illustrate the necessity of detaching the He I lines, the bottom panel of Figure 6 shows how they appear when they are undetached, i.e., when they are allowed to form down to the photospheric velocity of 6000 . These synthetic absorptions obviously are insufficently blueshifted. The top panel of Figure 7 is like Figure 3 but with only the O I lines. The optical depth of the reference line, 7773, is 1. The 7773 line accounts for at least most of an observed feature; in some of the other observed spectra this feature may be partly produced by Mg II 7890. The O I 9264 line may be responsible for a weak observed feature, while the 8446 feature usually is overwhelmed by the Ca II infrared triplet in SNe Ib. Whenever we use O I lines in the synthetic spectra of this paper, the optical depth of the reference line is near 1. The bottom panel of Figure 7 is like Figure 3 but with only the Ca II lines. The optical depth of the reference line, 3933, is 300. Only the H&K lines (3933, 3968) and the infrared triplet (8542, 8662, 8498) produce observable features, both of which are very strong. The notch in the synthetic spectrum near 8400 Ånicely matches an observed feature. In this synthetic spectrum the Ca II lines are detached to 7000  to match the infrared triplet, but in most of our synthetic spectra the Ca II lines are undetached. The top panel of Figure 8 is like Figure 3 but with only the Ti II lines. The optical depth of the reference line, 4550, is 1. The Ti II lines are used to help match the broad observed absorption trough between 4100 and 4500 Å. We consider the presence of Ti II lines in the observed spectrum to be probable but not definite. \[We do consider them to be definite in peculiar subluminous SNe Ia such as SN 1991bg (Filippenko et al. 1992) and SN 1999by (Garnavich et al. 2001), and in the Type Ic SN 1994I (Millard et al. 1999)\]. Whenever we use Ti II lines in this paper, the optical depth is near 1. None of our other conclusions would be affected by omitting the Ti II lines. The bottom panel of Figure 8 is like Figure 3 but with only the Sc II lines. The optical depth of the reference line, 4247, is 0.5. The Sc II lines are used mainly to get a peak in the synthetic spectrum near 5500 Å. The price to be paid is an overly strong synthetic absorption produced by 4247. Sc II lines are plausibly present in SNe Ib because they are expected (in LTE) to appear at low temperatures (Hatano et al. 1999) and they appear in SNe II. Harkness et al. (1987) suggested that in SN 1984L the observed emission peak near 5470 Å was caused by the early emergence of blueshifted \[O I\] 5577 nebular–phase emission, but Swartz et al. (1993) found this to be unlikely for the Type Ic SN 1987M. For a discussion of the possibility of an early emergence of blueshifted \[O I\] 5577 emission in the Type IIb SN 1996cb, see Qiu et al. (1999) and Deng, Qiu, & Hu (2001). In our view the 5500 Å emission in the fiducial spectrum of SN 1999dn probably, but not definitely, is produced by Sc II lines. Whenever we use them, the optical depth of the reference line is near 1. None of our other conclusions would be affected by omitting the Sc II lines. Figure 9 is like Figure 3 except that the power–law index $n$ has been reduced from 8 to 5 (and the optical depths of the reference lines have been correspondingly reduced to keep the synthetic features from becoming too strong). Now the blue wings of the synthetic absorptions produced by the Ca II infrared triplet, Ca II H&K, and He I 10830 fit better than in Figure 3, but the synthetic absorptions produced by 5876 and 6678 extend too far to the blue. This reflects the fact that in real supernovae, contrary to our assumption, the line optical depths do not all follow the same power law (or any power law). A slower radial decline of the optical depth of 10830 line, compared to the optical He I lines, is expected on the basis of the nonthermal–excitation calculations of Lucy (1991; his Figure 3) and Swartz et al. (1993; their Figure 11). Note that if the blue edge of the 10830 absorption really extends to 9000 Å, as it does in the synthetic spectrum of Figure 9, then the line is forming all the way out to 50,000 . Matheson et al. (2001) demonstrated that the absorption produced by 6678 (the lower level of which is 1s2p $^1$P$^0$) becomes weaker with time relative to the absorptions produced by 5876 and 7065 (both 1s2p $^3$P$^0$); this also is to be expected on the basis of the results of Lucy and of Swartz et al., because the singlet resonance transitions to the ground state become less opaque as the ejecta density decreases through expansion. As mentioned above, Deng et al. (2000) identified C II lines in their September 14 spectrum of SN 1999dn, obtained only three days before our fiducial spectrum of September 17. The identification of C II 6580 for the 6300 Å absorption was supported by attributing a weak absorption near 4580 Å to C II 4738, 4745. The reasons that we don’t introduce C II lines for the fiducial spectrum (apart from the fact that the 4580 Å absorption doesn’t appear distinctly in the fiducial spectrum) are that (1) as mentioned above, Fe II lines could be responsible for the 6300 Å absorption, and (2) the absorption near 4580 Å in the Beijing spectrum of September 14 might be produced by He I 4731 (see the top panel of Figure 6) and/or lines of Sc II (see the bottom panel of Figure 8). SNe 2000H, 1999di, and 1954A: Hydrogen in SNe Ib ------------------------------------------------ Now we turn to SN 2000H, an event that has conspicuous He I lines but that according to Benetti et al. (2000) also has hydrogen lines. Figure 10 compares the $+5$ day spectrum of SN 2000H with a synthetic spectrum that has =8000  and =6500 K, and contains hydrogen lines in addition to the ions used above for the fiducial spectrum of SN 1999dn. The He I lines are detached at 9000 , where the optical depth of the reference line is 2. The hydrogen lines are detached at 13,000  where the optical depth of the reference line, H$\alpha$, is 2.5. With this detachment velocity, H$\alpha$ accounts for at least most of the 6300 Å absorption and H$\beta$ accounts for the unusual notch in the emission peak near 4650 Å. A closer view of the H$\beta$ region is provided in Figure 11. The presence of an absorption that is attributable to H$\beta$ provides strong support for the presence of hydrogen in SN 2000H. Matheson et al. (2001) noted the presence of an unusually deep 6300 Å absorption in SN 1999di, and mentioned that it could be Si II 6355, C II 6580, or H$\alpha$. Si II can now be rejected because its absorption would be blueshifted by only 2600 , which is too much lower than than our value of =6000 . Figure 12 compares our earliest spectrum of SN 1999di, obtained 21 days after maximum, with the $+19$ day spectrum of SN 2000H. The similarity of these two spectra is remarkable (apart from differences at wavelengths longer than 9000 Å where both spectra are noisy). The narrow H$\beta$ absorption of SN 2000H also can be seen in SN 1999di. Figure 13 compares the $+21$ day spectrum of SN 1999di with a synthetic spectrum that has =7000  and =4500 K and contains the same ions as Figure 10 for SN 2000H. Hydrogen is detached at 12,000 . Figure 14 shows a closer view of the H$\beta$ region. Could the 6300 Å absorption in SNe 2000H and 1999di be produced by C II 6580 rather than H$\alpha$? We think not. It would be surprising to see a deep C II 6580 absorption in SNe Ib, when even in SNe Ic this line never forms a deep absorption and, if present at all, is hard to identify. It also would be surprising that the C II feature would be so detached in SNe Ib. The top panel of Figure 15 is like Figure 13 except that only C II lines, detached at 13,000 , are used. Although 6580 can account for the 6300 Å absorption as well as H$\alpha$ does, and 7236, 7231 are not a problem because, being more detached than He I, they fall near or within the strong feature produced by He I 7065, the absorption produced by C II 4738, 4745 is much too strong (at least in LTE at 7000 K). Could the 6300 Å absorption in SNe 2000H and 1999di be produced by Ne I $\lambda$6402 rather than H$\alpha$? No, it cannot. The bottom panel of Figure 15 shows that undetached Ne I fails in two ways: (1) the absorption produced by $\lambda$6402 is too far to the blue, and (2) even though the synthetic absorption produced by $\lambda$6402 has not been made strong enough, other unwanted features already are present. In view of these difficulties with C II and Ne I, and the apparent presence of H$\beta$ in the observed spectra, we consider the identification of hydrogen lines in SNe 2000H and 1999di to be definite. SN 1954A appears to have been spectroscopically akin to SNe 2000H and 1999di. McLaughlin (1963) and Branch (1972) identified He I lines in photographic spectra of SN 1954A obtained at the Lick Observatory and the Mount Wilson and Palomar Observatories, respectively. Consequently SN 1954A usually has been regarded to have been a Type Ib supernova, but on occasion doubts has been expressed about the classification because of the low quality of the photographic spectra compared to modern observations. Figure 16 compares microphotometer tracings of the two earliest spectra of SN 1954A (Blaylock et al. 2000), obtained on blue– and red–sensitive emulsions 46 days after maximum light, with the $+19$ day spectrum of SN 2000H. Only this one spectrum of SN 1954 was obtained on a red–sensitive emulsion. The SN 1954A spectra are not actually relative flux but merely a measure of the transmission through the photographic plate. Parts of the spectra were overexposed so the features are distorted; in particular, emission peaks tend to be suppressed. The shapes of these spectra also are strongly influenced by the wavelength dependences of the emulsion sensitivities; e.g., neither emulsion is sensitive around 5200 Åand the sensitivity falls off steeply between 6500 and 7000 Å. Nevertheless, some of the absorptions in SN 1954A can be located and they correspond well with the absorptions in SN 2000H, including the two attributed to H$\alpha$ and H$\beta$. Branch (1972) considered the possibility of H$\alpha$ and H$\beta$ absorptions in SN 1954A, blueshifted by 10,800  in the observer’s frame; the value of $cz$ for the parent galaxy is now known to be 291 , so the hydrogen lines are blueshifted by about 11,000  in the supernova frame. Branch also considered the possibility of Ne I lines in SN 1954A. This is perhaps still not ruled out, but given the apparent resemblance of SN 1954A to SNe 2000H, we prefer the hydrogen identification. In terms of apparent magnitude, SN 1954A was the fourth brightest supernova of the twentieth century, surpassed only by the Type II SN 1987A and the Type Ia SNe 1972E and 1937C (Barbon et al. 1999). SN 1954A occurred in the star–bursting dwarf galaxy NGC 4214, at a distance of only about 4 Mpc (Leitherer et al. 1996), more than 10 times closer than SNe 2000H and 1999di, so it is more amenable to studies of the environment in which it exploded. We note that Van Dyk, Hamuy, & Filippenko (1996) found SN 1954A to be unusual among SNe Ib in that it was not near any visible H II region, and that a deep VLA search for radio emission at the site of SN 1954A carried out by J. Cowan and D. Branch in May, 1986, resulted in a three–sigma upper limit to the flux density at 20 cm of 0.068 mJy (Eck 1998), which corresponds to a monochromatic luminosity of less than one twentieth of Cas A. Other Selected Comparisons -------------------------- Figure 17 compares a spectrum of SN 1984L obtained 9 days after maximum with a synthetic spectrum that has =8000  and = 6500 K, and includes hydrogen lines detached at 15,000 . The Fe II lines fit very well, and the fit to the other features is satisfactory. Given the presence of H$\alpha$ in SNe 2000H, 1999di, and 1954A, we assume that the 6300 Å absorption is produced by H$\alpha$. However, because the H$\alpha$ optical depth at the detachment velocity is only 0.6, there is no support for this identification from H$\beta$ because it is too weak to see. (The oscillator strength of H$\beta$ is about one fifth that of H$\alpha$.) Figure 18 compares the $+9$ day spectrum of SN 1984L with the $+5$ day spectrum of SN 2000H. The spectra are similar except that in SN 1984L the 6300 Å absorption is weaker and the 4560 Å absorption is not visible. This shows that while our assumption that the 6300 Å absorption in SN 1984L is produced by H$\alpha$ is reasonable, it is not proven. It is conceivable that C II 6580 or (more plausibly because it wouldn’t need to be detached) Ne I $\lambda$6402 could be responsible for the 6300 Åabsorption in SN 1984L and other typical SNe Ib, and be overwhelmed by H$\alpha$ only in events such as SNe 2000H, 1999di, and 1954A. We proceed on the assumption that at early times the 6300 Å absorption is produced by H$\alpha$ in all the SNe Ib of our sample. As an example of a comparison at a earlier time when  is higher, Figure 19 compares the earliest spectrum for which good wavelength coverage is available, the Beijing spectrum of SN 1999dn 10 days before maximum, with a synthetic spectrum that has =14,000  and = 6500 K. Now hydrogen lines are detached at 18,000 , where the H$\alpha$ optical depth is 1.3. The fit is good, except near 6600 Å. Figure 20 compares a spectrum of SN 1998dt obtained 32 days after maximum with a synthetic spectrum that has = 9000  and = 5000 K. This comparison is shown because, as will be seen below, although the spectrum has a typical SN Ib appearance the inferred value of  is unusually high for a SN Ib this long after maximum. The fit to the Fe II features is unusually poor, but this value of does give the best fit in the 4300 to 5300 Å region, and a significantly lower value would give a noticeably worse fit. (The narrow H$\alpha$ emission from an H II region in the parent galaxy is very close to 6563 Å, which shows that the high required value of  is not due to the observed spectrum having been inadvertently overcorrected for parent galaxy redshift.) Now we briefly consider the events of our sample for which the times of maximum light are unknown. The spectrum of SN 1996N indicates that it was discovered not long after maximum. Because we use hydrogen lines in the synthetic spectrum for SN 1996N, the comparison with the observed spectrum is shown in Figure 21. Helium lines are undetached and the optical depth of the reference line is 5; hydrogen lines are detached at 17,000  where the optical depth of H$\alpha$ is 0.5. Overall, the fit is good. SNe 1991ar and 1997dc do not appear to be unusual provided that they were discovered well after maximum light. As discussed by Matheson et al. (2001), the spectra of SN 1998T are seriously contaminated by light from the parent galaxy; taking this into consideration, SN 1998T does not appear to be unusual provided that it was discovered about a week after maximum. As far as we can tell, SNe 1991ar, 1997dc, and 1998T were typical SNe Ib, but they were not observed early enough to check on the presence of H$\alpha$. RESULTS AND DISCUSSION ====================== The most important of the fitting parameters that have been used for the synthetic spectra are collected in Table 2. The spectra are listed in order of time with respect to maximum light so only the supernovae for which we have an estimate of the time of maximum light appear in the table. (The date 0703, for example, refers to July 3.) For the six SNe Ib for which we have estimates of the time of maximum light,  is plotted against time in Figure 22. The tightness of the relationship is striking. SN 1998dt at 32 days after maximum seems to stand out; otherwise, the scatter about the mean curve is about what should be expected from our nominal errors of 1000  in  and a few days in the dates of maximum light. For the simple case of constant opacity and a $v^{-n}$ density distribution, the velocity at the photosphere would decrease with time as $v_{phot} \propto t^{-2/(n-1)}$. The line in Figure 22, the best power–law fit to the data (excluding SN 1998dt at 32 days), corresponds to $n=3.6$. Considering that the opacity is not really constant, that the actual density distribution does not really follow a single power law over a wide velocity range, and that the best power–law index for fitting the spectra is not well constrained, not much significance should be attched to the difference between $n=3.6$ and $n=8$. Our adopted values of  can be used to make rough estimates of the mass and kinetic energy above the photosphere. For spherical symmetry and a $v^{-n}$ density distribution, the mass (in $M_\odot$) and the kinetic energy (in 10$^{51}$ ergs) above the electron–scattering optical depth $\tau_{es}$ are (Millard et al. 1999) $$M=1.2 \times 10^{-4}\ v_4^2\ t_d^2\ \mu_e\ {{n-1}\over{n-3}}\ \tau_{es}, \eqno (1)$$ $$E=1.2 \times 10^{-4}\ v_4^4\ t_d^2\ \mu_e\ {{n-1}\over{n-5}}\ \tau_{es}, \eqno (2)$$ where $v_4$ is  in units of 10,000 , $t_d$ is the time since explosion in days, $\mu_e$ is the mean molecular weight per free electron, and the integration is carried out to arbitrarily high velocity. If we assume that maximum light occurs 20 days after explosion, that $n=8$, that $\mu_e = 8$ (e.g., half–ionized helium or singly ionized oxygen), and that  is at $\tau_{es}=1$, then at maximum light =10,000  (Figure 22) gives $M=0.5~M_\odot$ and $E=0.9 \times 10^{51}$ ergs. At 20 days after maximum, using =7000  and keeping the other parameters the same gives $M=1.1~M_\odot$ and $E=0.9 \times 10^{51}$ ergs. In reality, of course, the kinetic energy above the 7000  photosphere must be greater than that above the 10,000  photosphere. If we use $n=4$ instead of $n=8$ between 7000 and 10,000  then we obtain a total of $M=1.5~M_\odot$ and $E=1.4 \times 10^{51}$ ergs above the 7000  photosphere. Figure 22 provides some constraints on models of SNe Ib. The hydrodynamics must account for the velocity at the photosphere as a function of time, and the ensemble of SN Ib progenitors must be consistent with the tightness of the relationship. The small scatter suggests that the masses and the kinetic energies of the SNe Ib of our sample are similar, and it does not leave much room for the influence on  of departures from spherical symmetry. Figure 23 shows the minimum velocities of the He I lines (squares when undetached and diamonds when detached). There appears to be a standard pattern (again with the possible exception of SN 1998dt at 32 days). Before and near the time of maximum the He I lines tend to be undetached. After maximum the lines tend to be detached, but the detachment velocities tend to decrease with time, from about 10,000 to 7000 . This means that the fraction of helium in this velocity range that is in the lower levels of the optical He I lines is increasing with time faster than $t^2$. (The matter density is decreasing as $t^{-3}$ by expansion but the Sobolev optical depth also is proportional to $t$ because it is inversely proportional to the velocity gradient.) The increasing fraction of helium in the excited levels may be understandable in terms of the decreasing column depth between the nickel core and the helium layers, and the decreasing detachment velocity may mean that the fractional helium abundance is lower at lower velocities. In any case, some helium is present at least down to 7000 . Our estimate above for the total mass above the 7000  photosphere was 1.1 to 1.5 M$_\odot$, which is a rough upper limit on the mass of helium above 7000 . There could be more helium below 7000 . Figure 23 provides more constraints on models of SNe Ib. The radial profile of the helium abundance, together with that of the $^{56}$Ni that is responsible for exciting it, should account for the He I velocities and optical depths (Table 2). Figure 23 also shows the minimum velocities of the hydrogen lines (circles), which always are detached. In the three events for which we are convinced of the hydrogen identifications — SNe 2000H, 1999dn, and 1954A — the minimum hydrogen velocity is between 11,000 and 13,000 . In the events in which we assume H$\alpha$ to be present at early times — SNe 1983N, 1984L, 1999dn, and 1996N (the latter is not shown in Figure 23 because the date of maximum light is unknown) — the detachment velocities tend to decrease with time but they are consistent with similar minimum velocities of the hydrogen. Thus the available evidence is consistent with the proposition that SNe Ib in general have hydrogen down to $11,000 - 13,000$ . \[For comparison, in the Type IIb SN 1993J the characteristic velocity of the ejected hydrogen was about 9000  (Patat et al. 1995; Utrobin 1996; Houck & Fransson 1996), and in the Type IIb SN 1996cb it was about 10,000  (Deng et al. 2001).\] A challenge for those who study the complicated evolution of massive stars in binary systems (e.g., Podsiadlowski, Joss, & Hsu 1992; Nomoto, Iwamoto, & Suzuki 1995; Wellstein, Langer, & Braun 2001) is to understand why many or perhaps even all stellar explosions that develop strong He I lines should eject at least a small amount of hydrogen. The hydrogen mass that is required to give an H$\alpha$ optical depth of unity depends on the fraction of hydrogen that is in the Balmer level. In LTE, with an electron density of $10^9$ cm$^{-3}$, the Balmer fraction peaks around $10^{-9}$ near 6000 K. In this case we estimate that a hydrogen mass on the order of $10^{-2}~M_\odot$ would be required. Non–LTE calculations that take nonthermal excitation into account are needed for a more reliable estimate of the hydrogen mass. The optical depths of the helium lines, and especially the hydrogen lines, are not very high, even though the corresponding absorption features are distinct and fairly deep. This reflects a simple geometric aspect of supernova line formation: as explained in Jeffery & Branch (1990), absorption features formed by detached lines are deeper than those formed by undetached lines. This point is illustrated in Figure 24, which shows that when hydrogen is detached from the photosphere by a factor of two and H$\alpha$ has $\tau=2$, its absorption feature is deeper than that of an undetached H$\alpha$ that has $\tau=10$. The H$\alpha$ optical depths in our synthetic spectra for SNe 2000H, 1999di, and 1954A are not high, so if they were only mildly lower these events would look like typical SNe Ib. Similarly, moderately lower He I line optical depths would transform a SN Ib into a SN Ic. Figure 24 also illustrates how undetached hydrogen lines are more “obvious” than detached lines. First, undetached lines have conspicuous narrow, rounded emission peaks while detached lines have inconspicuous broad, flat peaks. Note also that although the H$\alpha$ absorption is deeper in the detached spectrum, the H$\beta$ absorption is deeper in the undetached spectrum. This is because in the undetached spectrum the optical depth at the photosphere of H$\beta$ is about 2 while in the detached spectrum the optical depth at the detachment velocity is only about 0.4. An optical depth as low as 0.4 can produce only a shallow absorption, even when the line is detached. For these reasons, supernovae that have undetached hydrogen lines have obvious hydrogen lines and are classified as Type II. Supernovae that have detached hydrogen lines are classified as Type Ib because the presence of hydrogen is not immediately obvious, even when the H$\alpha$ absorption is as deep as it is in SNe 2000H, 1999di, and 1954A. SNe IIb are those that have undetached hydrogen lines when they are first observed. In some cases, whether an event is classified as Ib or IIb may depend on how early the first spectrum is obtained. The implication of the previous paragraphs is that the spectroscopic differences between SNe IIb, the SNe Ib that have deep H$\alpha$ absorptions, and typical SNe Ib may be caused mainly by mild differences in the hydrogen mass. For a given kinetic energy, the lower the hydrogen mass the higher the minimum velocity of the ejected hydrogen. Similarly, the spectroscopic differences between typical SNe Ic and SNe Ib could be caused mainly by moderate differences in the helium mass. For example, Matheson et al. (2001) found higher blueshifts of the O I $\lambda$7773 line in SNe Ic than in SNe Ib. For a given kinetic energy, the lower the helium mass the higher the minimum velocity of the ejected helium, and therefore the higher the velocity of the ejected oxygen. These suggestions are not original to this paper, but they are strengthened by our finding that the H I and He I optical depths in SNe Ib are not very high. These suggestions also are not inconsistent with arguments, based on light curves, for the existence of different physical classes of hydrogen–poor events that cut across the conventional spectroscopic types (e.g., Clocchiatti & Wheeler 1997). The number of SNe Ib for which good spectral coverage is available is still relatively small. More events should be observed to explore the degree of the spectral homogeneity and to find out whether there is a continuum of hydrogen line strengths. Also needed are detailed non–LTE spectrum calculations for supernova models having radially stratified compositions, — to determine the the hydrogen and helium masses and the distribution of the $^{56}$Ni that is required to excite the helium. The possibility that nonthermally excited Ne I can produce spectral features strong enough to be seen needs to be investigated. Detailed non–LTE calculations for parameterized SN Ib models, using the PHOENIX code (e.g., Baron et al. 1999), are underway. This material is based upon work supported by the National Science Foundation under Grants No. AST–9986965 and AST–9731450 at Oklahoma and AST–9987438 at Berkeley. A.V.F. is grateful to the Guggenheim Foundation for a Fellowship. Barbon, R., Buondi, V., Cappellaro, E., & Turatto, M. 1999, A&AS, 139, 531 Baron, E., Branch, D., Hauschildt, P. H., Filippenko, A. V., & Kirshner, R. P. 1999, ApJ, 527, 739 Benetti, S., Cappellaro, E., Turatto, M., & Pastorello, A. 2000, IAU Circ. 7375 Blaylock, M., Branch, D., Casebeer, D., Millard, J., Baron, E., Richardson, D., & Ancheta, C. 2000, PASP, 112, 1439 Branch, D. 1972, A&A, 16, 247 Branch, D. 2001, in Supernovae and Gamma–Ray Bursts: The Largest Explosions in the Universe, ed. M. Livio (Cambridge: Cambridge University Press), in press Casebeer, D., Branch, D., Blaylock, M., Millard, J., Baron, E., Richardson, D., & Ancheta, C. 2000, PASP, 112, 1433 Clocchiatti, A. & Wheeler, J. C. 1997, in Thermonuclear Supernovae, ed. P. Ruiz–Lapuente, R. Canal, & J. Isern (Dordrecht: Kluwer), p. 863 Deng, J. S., Qiu, Y. L., Hu, J. Y., Hatano K., & Branch D. 2000, ApJ, 540, 452 Deng, J., Qiu, Y., & Hu, J. 2001, preprint Eck, C. R. 1998, PhD Thesis, University of Oklahoma Filippenko, A. V. 1997, ARAA, 35, 309 Filippenko, A. 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P. Ruiz–Lapuente, R. Canal, & J. Isern (Dordrecht: Kluwer), p. 821 [lrrcrr]{} 1954A & NGC 4214 & 291 & April 21\ 1983N & NGC 5236 & 513 & July 17\ 1984L & NGC 991 & 1534 & August 22\ 1991ar & IC 49 & \*4520 & \*September 2\ 1996N & NGC 1398 & 1491 & \*March 12\ 1997dc & NGC 7678 & 3480 & \*August 5\ 1998T & NGC 3690 & \*3080 & \*March 3\ 1998dt & NGC 945 & \*4580 & September 12\ 1999di & NGC 776 & \*4920 & July 27\ 1999dn & NGC 7714 & 2700 & August 31\ 2000H & IC 454 & 3894 & February 11\ [lrrcrrrr]{} 1983N & 0703 & -14 & 17000 &&&&\ 1983N &0706 &-11 & 13000 &&&&\ 1999dn& 0821& -10 & 14000 & 2.0 & 14000 & 1.5& 18000\ 1983N & 0713 & -4 & 11000 &&&&\ 1983N & 0717 & 0 & 11000 & 2.5& 11000 & 0.8 & 15000\ 2000H & 0211 & 0 & 11000 & 1 & 11000 & 6.0 & 13000\ 1999dn &0831 & 0 & 10000 & 5.0 &11000 & 1.0 & 14000\ 1983N & 0719 & 2 & 10000 & 4.0 & 10000& 0.7 & 14000\ 2000H & 0216 & 5 & 8000 & 2.0 & 9000 & 2.5 & 13000\ 1984L& 0830 & 8 & 9000 & 2.0 & 10000 & 0.6 & 15000\ 1998dt& 0920 & 8 & 9000 & 4.0 &11000 &&\ 1984L & 0831& 9 & 8000 & 1.5 & 10000 & 0.6& 15000\ 1983N & 0727 & 10 & 7000 & 7.0 & 8000 & 0.6 & 12000\ 1999dn & 0910 & 10 & 7000 & 3.0 & 8000 &&\ 1984L & 0903 & 12 & 7000 & 1.0& 9000 & 0.5& 14000\ 1999dn& 0914 &14 & 6000 & 10 & 7000 &&\ 1999dn &0917 &17 &6000 & 10 & 8000 &&\ 2000H &0302 &19 &6000 & 5.0& 8000& 2.0& 13000\ 1999di &0817 &21 &7000 & 10 &8500 & 4.0 & 12000\ 1984L &0919 &28 &5000 &&&&\ 2000H &0313 &30 &5000 & 3.0& 7000& 1.5 &13000\ 1984L &0923 &32 &5000 & 5.0& 7000 &&\ 1998dt &1015 & 33 &9000 & 10 & 9000 &&\ 1984L &0928 &37 &5000 & 10 & 6000 &&\ 1999dn &1008 &38 & 6000 & 10 & 7000&&\ 1999di &0910 & 45& 6000 & 10 & 7000 & 2.0 & 12000\ 2000H &0330 &47 &5000 & 1.0& 7000 & 0.5& 12000\ 1999di &0917 &52 &6000 & 10 & 7000& 1.0& 12000\ 2000H &0408 &56 &4000 &&&&\ 1984L &1018 &57 &4000 &&&&\ [^1]: These spectra are partially based on observations collected at the European Southern Observatory, Chile, ESO N$^0$65.H–0292, and at the Asiago Observatory

--- abstract: | We consider planar cubic maps, i.e. connected cubic graphs embedded into plane, with marked spanning tree and marked directed edge (not in this tree). The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is Catalan number. author: - Yury Kochetkov title: 'On enumeration of tree-rooted planar cubic maps' --- Introduction ============ Plane triangulation is a planar map, where the perimeter of each face is three. The corresponding dual graph is *cubic*, i.e. the degree of each vertex is three. A plane triangulation will be called *proper*, if each edge is incident to exactly *two* faces. Otherwise it will be called *improper*. $$\begin{picture}(265,70) \put(15,5){\small proper triangulation} \put(165,5){\small improper triangulation} \put(25,20){\line(1,0){60}} \put(25,20){\circle*{2}} \put(85,20){\circle*{2}} \put(55,65){\circle*{2}} \put(25,20){\line(2,3){30}} \put(85,20){\line(-2,3){30}} \put(215,45){\oval(40,40)} \put(180,45){\circle*{2}} \put(195,45){\circle*{2}} \put(210,45){\circle*{2}} \put(180,45){\line(1,0){30}} \end{picture}$$ The corresponding dual graphs are presented below: $$\begin{picture}(160,50) \put(20,25){\oval(40,40)} \put(20,5){\circle*{2}} \put(20,45){\circle*{2}} \put(20,5){\line(0,1){40}} \put(60,22){¨} \put(100,25){\oval(30,30)} \put(160,25){\oval(30,30)} \put(115,25){\circle*{2}} \put(145,25){\circle*{2}} \put(115,25){\line(1,0){30}} \end{picture}$$ A connected graph with marked directed edge will be called *edge-rooted*. Proper edge-rooted triangulations where enumerated by Tutte in the work [@Tu]: the number $T_n$ of proper planar triangulations with $2n$ faces and marked directed edge is $$T_n=\frac{2\,(4n-3)!}{n!\,(3n-1)!}.$$ A combinatorial proof of Tutte formula see in [@PS] (see also [@AP]). Let $F_n$ be the number of planar edge-rooted cubic graphs with $2n$ vertices, i.e. the number of planar edge-rooted triangulations (proper and improper) with $2n$ faces. Let us define numbers $f_n$, $n\geqslant -1$, in the following way: - $f_{-1}=1/2$; - $f_0=2$; - $f_n=(3n+2)F_n$, $n>0$. In [@GJ] a recurrent relation for numbers $f_n$ was proposed: $$f_n=\frac{4(3n+2)}{n+1}\sum_{\scriptsize \begin{array}{c} i\geqslant -1,\, j\geqslant -1\\ i+j=n-2\end{array}} f(i)f(j).\eqno(1)$$ From (1) it follows that $F_1=4$. Indeed, there are four ways to choose a root edge in a planar cubic map with two vertices: $$\begin{picture}(340,40) \put(15,20){\oval(30,30)} \put(15,5){\vector(0,1){30}} \put(70,20){\oval(30,30)} \put(120,20){\oval(30,30)} \put(85,20){\vector(1,0){20}} \put(175,20){\oval(30,30)} \put(225,20){\oval(30,30)} \put(190,20){\line(1,0){20}} \put(210,15){\vector(0,1){5}} \put(275,20){\oval(30,30)} \put(290,20){\line(1,0){20}} \put(325,20){\oval(30,30)} \put(310,25){\vector(0,-1){5}} \end{picture}$$ Also we have that $F_2=32$. Indeed, there are six cubic maps with 4 vertices (and 6 edges): $$\begin{picture}(300,50) \put(40,25){\oval(40,40)} \put(40,5){\line(0,1){40}} \put(60,25){\line(1,0){20}} \put(95,25){\oval(30,30)} \put(5,20){\small 1)} \put(155,25){\oval(30,30)} \put(215,25){\oval(30,30)} \qbezier(155,40)(185,55)(215,40) \qbezier(155,10)(185,-5)(215,10) \put(125,20){\small 2)} \put(280,25){\oval(40,40)} \put(280,25){\line(0,1){20}} \put(245,20){\small 3)} \put(280,25){\line(3,-2){17}} \put(280,25){\line(-3,-2){17}} \end{picture}$$ $$\begin{picture}(310,50) \put(20,35){\circle{10}} \put(20,5){\circle{10}} \put(40,20){\line(-4,3){16}} \put(40,20){\line(-4,-3){16}} \put(40,20){\line(1,0){20}} \put(65,20){\circle{10}} \put(0,17){\small 4)} \put(110,20){\oval(20,20)} \put(120,20){\line(1,0){20}} \put(150,20){\oval(20,20)} \put(160,20){\line(1,0){20}} \put(190,20){\oval(20,20)} \put(85,17){\small 5)} \put(240,20){\oval(20,20)} \put(250,20){\line(1,0){20}} \put(290,20){\oval(40,40)} \put(290,20){\oval(20,20)} \put(300,20){\line(1,0){10}} \put(215,17){\small 6)} \end{picture}$$ [Figure 1]{} Group of automorphisms of the first map is trivial, of the second has order 4, of the third has order 12, of the forth has order 3, of the fifth and the sixth has order 2. Thus, there are 12 ways to choose a root edge in the first map, 3 — in the second, 1 — in the third, 4 — in the forth, 6 — in the fifth and the sixth. All this gives us 32 edge-rooted maps. However, this formula does not seem to have a geometrical/combinatorial explanation. In [@Mu] a nice formula was proposed for the number tree-rooted planar maps, i.e. edge-rooted planar maps with distinguished spanning tree: the number of such maps with $n$ edges is $C_n\cdot C_{n+1}$, where $C_k$ is $k$-th Catalan number. An elegant proof of this formula see in [@Be]. There are four planar maps with two edges: $$\begin{picture}(215,40) \multiput(0,25)(15,0){3}{\circle*{3}} \put(0,25){\line(1,0){30}} \put(13,2){\small 1} \put(70,25){\oval(20,20)} \put(80,25){\circle*{3}} \put(95,25){\circle*{3}} \put(80,25){\line(1,0){15}} \put(68,2){\small 2} \put(125,25){\circle*{3}} \put(145,25){\circle*{3}} \put(135,25){\oval(20,20)} \put(134,2){\small 3} \put(195,25){\circle*{3}} \put(185,25){\oval(20,20)} \put(205,25){\oval(20,20)} \put(194,2){\small 4} \end{picture}$$ - There is one way to choose a spanning tree in the first map and two ways to choose a directed edge. - There is one way to choose a spanning tree in the second map and four ways to choose a directed edge. - There is one way to choose a spanning tree in the third map and two ways to choose a directed edge. - There is no spanning trees in the forth map and two ways to choose a directed edge. Thus we have $10=C_2\cdot C_3$ tree rooted planar maps with two edges. We will study tree-rooted cubic maps with additional property: a root edge *does not* belong to the spanning tree. **Theorem.** *The number of such tree-rooted cubic maps with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is $k$-th Catalan number.* The main construction: from map to curve ======================================== By tree-rooted plane cubic map we will understand a cubic graph imbedded into plane (sphere) with - marked spanning tree; - marked directed edge that *does not* belong to the spanning tree. Let $G$ be a tree-rooted pane cubic map with $2n$ vertices. We draw triangles, one triangle for each vertex, in such way that: - triangles are disjoint; - each vertex is inside the corresponding triangle; - each side of triangle intersect one outgoing edge of corresponding vertex. $$\begin{picture}(260,60) \qbezier[25](0,30)(0,50)(20,50) \qbezier[25](0,30)(0,10)(20,10) \qbezier[25](20,10)(40,10)(40,30) \qbezier[20](60,30)(60,45)(75,45) \qbezier[20](60,30)(60,15)(75,15) \qbezier[20](75,45)(90,45)(90,30) \qbezier[20](75,15)(90,15)(90,30) \linethickness{0.5mm} \put(20,10){\line(0,1){40}} \put(40,30){\line(1,0){20}} \qbezier(20,50)(40,50)(40,30) \put(20,7){$\to$} \put(110,28){$\Rightarrow$} \thinlines \qbezier[25](140,30)(140,50)(160,50) \qbezier[25](140,30)(140,10)(160,10) \qbezier[25](160,10)(180,10)(180,30) \qbezier[20](200,30)(200,45)(215,45) \qbezier[20](200,30)(200,15)(215,15) \qbezier[20](215,45)(230,45)(230,30) \qbezier[20](215,15)(230,15)(230,30) \linethickness{0.5mm} \put(160,10){\line(0,1){40}} \put(180,30){\line(1,0){20}} \qbezier(160,50)(180,50)(180,30) \thinlines \put(152,42){\line(1,0){16}} \put(152,42){\line(1,2){8}} \put(168,42){\line(-1,2){8}} \put(152,18){\line(1,0){16}} \put(152,18){\line(1,-2){8}} \put(168,18){\line(-1,-2){8}} \put(188,22){\line(0,1){16}} \put(188,22){\line(-2,1){16}} \put(188,38){\line(-2,-1){16}} \put(192,22){\line(0,1){16}} \put(192,22){\line(2,1){16}} \put(192,38){\line(2,-1){16}} \put(160,7){$\longrightarrow$}\end{picture}$$ Thick lines above mark spanning tree and an arrow indicates the direction of the root edge. Two triangles will be called adjacent, if the corresponding vertices are adjacent and the edge, that connects them, belongs to the spanning tree. The sides of adjacent triangles that intersect this edge also will be called adjacent. We construct a polygon $P$ by glewing adjacent triangles by adjacent sides. This polygon has $2n+2$ sides and is divided into $2n$ triangles. Each edge of the cubic map, that does not belong to the spanning tree, intersects two sides of $P$ and we will say that these sides constitute a pair. Polygon $P$ has a marked side: the marked edge of the cubic map intersects it in direction from inside $P$ to outside. Continuation of Example. $$\begin{picture}(160,200) \put(20,70){\line(2,-3){40}} \put(20,70){\line(1,0){80}} \put(20,70){\line(2,3){80}} \put(60,10){\line(2,3){80}} \put(60,130){\line(2,-3){40}} \put(60,130){\line(1,0){80}} \put(100,190){\line(2,-3){40}} \linethickness{0.6mm} \put(60,50){\line(0,1){40}} \qbezier(60,90)(80,100)(100,110) \put(100,110){\line(0,1){40}} \thinlines \put(60,50){\line(-2,-1){40}} \put(60,50){\vector(2,-1){40}} \put(60,90){\line(-2,1){40}} \put(100,110){\line(2,-1){40}} \put(100,150){\line(2,1){40}} \put(100,150){\line(-2,1){40}} \put(65,5){\small A} \put(105,65){\small B} \put(145,125){\small C} \put(10,68){\small F} \put(50,128){\small E} \put(90,188){\small D} \end{picture}$$ Here $EF$ and $FA$, $AB$ and $BC$, $CD$ and $DE$ are pairs and $AB$ is the marked edge. If we identify sides that are in pairs (i.e. $EF$ with $FA$, $AB$ with $BC$ and $CD$ with $DE$), then we will obtain a triangulated genus 0 curve. The main construction: from curve to map ======================================== Let $P$ be a $2n$-gon with marked side $M$ and triangulated by non-intersecting diagonals into $2n-2$ triangles. Sides of $P$ are divided into pairs in such way, that the identification of sides in each pair gives us a genus 0 curve. We will construct a plane tree-rooted cubic map with root edge (not in the spanning tree) in the following way. - We put a vertex $v_i$ inside each triangle $\triangle_i$ and connect vertices in adjacent triangles — the spanning tree is constructed. - Let sides $L$ and $L'$ be in pair. $L$ and $L'$ are sides of triangles $\triangle_i$ and $\triangle_j$, respectively (these triangles may coincide). We draw an arc that connect $v_i$ and $v_j$ in the following way: going from $v_i$ the arc intersects $L$. Its next part lies in the exterior of $P$ and connects $L$ and $L'$. After intersecting $L'$ the arc goes to $v_j$. - An arc, that intersects $M$ will be the root edge. At intersection point it is directed from inside $P$ to outside. $$\begin{picture}(320,120) \put(10,40){\line(0,1){40}} \put(10,40){\line(1,-1){30}} \put(10,40){\line(1,1){70}} \put(10,80){\line(1,1){30}} \qbezier(10,40)(25,75)(40,110) \put(40,110){\line(1,0){40}} \put(40,10){\vector(1,0){40}} \put(40,10){\line(2,5){40}} \qbezier(40,10)(75,25)(110,40) \put(80,110){\line(1,-1){30}} \qbezier(80,110)(95,75)(110,40) \put(110,40){\line(0,1){40}} \put(80,10){\line(1,1){30}} \put(0,36){\small H} \put(0,76){\small G} \put(36,0){\small A} \put(36,113){\small F} \put(79,0){\small B} \put(79,113){\small E} \put(115,36){\small C} \put(115,76){\small D} \put(150,55){$\Rightarrow$} \put(200,40){\line(0,1){40}} \put(200,40){\line(1,-1){30}} \put(200,40){\line(1,1){70}} \put(200,80){\line(1,1){30}} \qbezier(200,40)(215,75)(230,110) \put(230,110){\line(1,0){40}} \put(230,10){\vector(1,0){40}} \put(230,10){\line(2,5){40}} \qbezier(230,10)(265,25)(300,40) \put(270,110){\line(1,-1){30}} \qbezier(270,110)(285,75)(300,40) \put(300,40){\line(0,1){40}} \put(270,10){\line(1,1){30}} \put(190,36){\small H} \put(190,76){\small G} \put(226,0){\small A} \put(226,113){\small F} \put(269,0){\small B} \put(269,113){\small E} \put(305,36){\small C} \put(305,76){\small D} \linethickness{0.5mm} \put(210,80){\line(1,0){20}} \qbezier(230,80)(235,65)(240,50) \put(240,50){\line(1,0){30}} \put(270,50){\line(0,-1){30}} \qbezier(270,50)(280,65)(290,80) \thinlines \put(270,20){\vector(-3,-2){30}} \put(270,20){\line(2,-1){20}} \put(290,80){\line(2,-3){20}} \put(290,80){\line(0,1){20}} \put(240,50){\line(-1,-1){35}} \put(230,80){\line(2,3){25}} \put(210,80){\line(-2,-3){20}} \put(210,80){\line(0,1){25}} \end{picture}$$ Here sides $AB$ and $FG$, $BC$ and $CD$, $DE$ and $EF$, $GH$ and $HA$ constitute pairs and $AB$ is the marked side. Thus, we must connect the arc that intersects $AB$ with the arc that intersects $FG$, the arc that intersects $BC$ with the arc that intersects $CD$, the arc that intersects $DE$ with the arc that intersects $EF$ and the arc that intersects $GH$ with the arc that intersects $HA$. An arrow in the arc that intersects $AB$ indicates the direction of the root edge of the cubic graph. The cubic graph itself and its “simplification” are presented in the figure below. $$\begin{picture}(300,100) \linethickness{0.6mm} \put(0,30){\line(1,0){80}} \put(60,30){\line(0,1){10}} \thinlines \put(20,30){\line(0,1){10}} \put(20,30){\oval(40,40)[b]} \put(50,30){\oval(100,100)[t]} \put(40,40){\oval(40,40)[t]} \put(60,30){\oval(40,20)[tr]} \put(90,30){\oval(20,20)[b]} \put(78,23){$\downarrow$} \put(140,50){$\Rightarrow$} \linethickness{0.6mm} \put(200,20){\line(0,1){60}} \put(200,20){\line(1,0){80}} \put(280,20){\line(0,1){30}} \qbezier(280,20)(310,50)(280,80) \thinlines \qbezier(200,20)(170,50)(200,80) \put(200,80){\line(1,0){80}} \put(200,50){\line(1,0){80}} \put(280,50){\line(0,1){30}} \put(280,80){\vector(-1,0){15}} \end{picture}$$ We can draw above mentioned arcs in such way, that they do not intersect in the exterior of $P$. Let us connect midpoints of all sides in pairs by segments inside $P$. As the identification of sides in pairs generates a genus zero curve, then these segments do not intersect. The polygon $P$ is embedded into sphere, so we can interchange its interior and exterior domains. Main statement ============== The number of tree-rooted cubic maps with $2n$ vertices and a marked edge, that does not belong to the spanning tree, is $C_{2n}\cdot C_{n+1}$, where $C_k$ is $k$-th Catalan number. Our theorem follows from two statements. 1. A convex $n$-gon with a marked side can be divided into triangles by non-intersecting diagonals in $C_{n-2}$ ways [@St]. 2. There are $C_n$ ways to define a pairwise identification of sides of a convex $2n$-gon with a marked side to obtain a genus 0 curve [@LZ]. According to theorem, we have $C_4\cdot C_3=70$ tree-rooted cubic maps with $4$ vertices. In what follows a map with a marked spanning tree will be called *t-map*. The first cubic map in Figure 1 generates six t-maps. $$\begin{picture}(330,50) \qbezier[20](0,25)(0,45)(20,45) \qbezier[20](0,25)(0,5)(20,5) \qbezier[20](60,25)(60,40)(75,40) \qbezier[20](60,25)(60,10)(75,10) \qbezier[20](75,40)(90,40)(90,25) \qbezier[20](75,10)(90,10)(90,25) \qbezier[30](20,5)(20,25)(20,45) \qbezier[20](180,25)(180,40)(195,40) \qbezier[20](180,25)(180,10)(195,10) \qbezier[20](195,40)(210,40)(210,25) \qbezier[20](195,10)(210,10)(210,25) \qbezier[20](120,25)(120,45)(140,45) \qbezier[20](120,25)(120,5)(140,5) \qbezier[20](140,5)(160,5)(160,25) \qbezier[20](300,25)(300,40)(315,40) \qbezier[20](300,25)(300,10)(315,10) \qbezier[20](315,40)(330,40)(330,25) \qbezier[20](315,10)(330,10)(330,25) \qbezier[20](260,5)(280,5)(280,25) \qbezier[30](260,5)(260,25)(260,45) \linethickness{0.5mm} \qbezier(20,45)(40,45)(40,25) \qbezier(20,5)(40,5)(40,25) \put(40,25){\line(1,0){20}} \put(160,25){\line(1,0){20}} \put(280,25){\line(1,0){20}} \qbezier(140,45)(160,45)(160,25) \put(140,5){\line(0,1){40}} \qbezier(240,25)(240,45)(260,45) \qbezier(240,25)(240,5)(260,5) \qbezier(260,45)(280,45)(280,25) \end{picture}$$ $$\begin{picture}(210,50) \qbezier[20](0,25)(0,45)(20,45) \qbezier[20](0,25)(0,5)(20,5) \qbezier[20](60,25)(60,40)(75,40) \qbezier[20](60,25)(60,10)(75,10) \qbezier[20](75,40)(90,40)(90,25) \qbezier[20](75,10)(90,10)(90,25) \qbezier[20](20,45)(40,45)(40,25) \qbezier[20](180,25)(180,40)(195,40) \qbezier[20](180,25)(180,10)(195,10) \qbezier[20](195,40)(210,40)(210,25) \qbezier[20](195,10)(210,10)(210,25) \qbezier[20](140,45)(160,45)(160,25) \put(140,5){\line(0,1){40}} \qbezier[20](140,5)(160,5)(160,25) \qbezier[30](140,5)(140,25)(140,45) \linethickness{0.5mm} \put(20,5){\line(0,1){40}} \put(40,25){\line(1,0){20}} \qbezier(20,5)(40,5)(40,25) \qbezier(120,25)(120,45)(140,45) \qbezier(120,25)(120,5)(140,5) \put(160,25){\line(1,0){20}} \qbezier(140,5)(160,5)(160,25) \end{picture}$$ In each case we have six ways to choose a marked edge, that does not belong to the tree. Thus, the first map generates $30$ tree-rooted maps. The second cubic map in Figure 1 generates four t-maps. $$\begin{picture}(310,80) \put(0,37){\small 1)} \qbezier[20](40,60)(60,60)(60,40) \qbezier[20](40,20)(60,20)(60,40) \qbezier[20](120,60)(100,60)(100,40) \qbezier[20](120,20)(100,20)(100,40) \qbezier[20](120,60)(140,60)(140,40) \qbezier[20](120,20)(140,20)(140,40) \put(40,20){\circle*{3}} \put(40,60){\circle*{3}} \put(120,20){\circle*{3}} \put(120,60){\circle*{3}} \put(170,37){\small 2)} \qbezier[20](210,60)(230,60)(230,40) \qbezier[20](210,20)(230,20)(230,40) \qbezier[20](290,60)(270,60)(270,40) \qbezier[20](290,20)(270,20)(270,40) \put(210,20){\circle*{3}} \put(210,60){\circle*{3}} \put(290,20){\circle*{3}} \put(290,60){\circle*{3}} \qbezier[60](210,60)(250,90)(290,60) \linethickness{0.5mm} \qbezier(40,60)(20,60)(20,40) \qbezier(40,20)(20,20)(20,40) \qbezier(40,20)(80,-10)(120,20) \qbezier(40,60)(80,90)(120,60) \qbezier(210,60)(190,60)(190,40) \qbezier(210,20)(190,20)(190,40) \qbezier(210,20)(250,-10)(290,20) \qbezier(290,60)(310,60)(310,40) \qbezier(290,20)(310,20)(310,40) \end{picture}$$ $$\begin{picture}(310,80) \put(0,37){\small 3)} \qbezier[20](40,60)(60,60)(60,40) \qbezier[20](40,20)(60,20)(60,40) \qbezier[20](120,60)(140,60)(140,40) \qbezier[20](120,20)(140,20)(140,40) \put(40,20){\circle*{3}} \put(40,60){\circle*{3}} \put(120,20){\circle*{3}} \put(120,60){\circle*{3}} \qbezier[60](40,60)(80,90)(120,60) \put(170,37){\small 4)} \qbezier[20](210,60)(190,60)(190,40) \qbezier[20](210,20)(190,20)(190,40) \qbezier[20](290,60)(270,60)(270,40) \qbezier[20](290,20)(270,20)(270,40) \put(210,20){\circle*{3}} \put(210,60){\circle*{3}} \put(290,20){\circle*{3}} \put(290,60){\circle*{3}} \qbezier[60](210,60)(250,90)(290,60) \linethickness{0.5mm} \qbezier(40,60)(20,60)(20,40) \qbezier(40,20)(20,20)(20,40) \qbezier(40,20)(80,-10)(120,20) \qbezier(120,60)(100,60)(100,40) \qbezier(120,20)(100,20)(100,40) \qbezier(210,60)(230,60)(230,40) \qbezier(210,20)(230,20)(230,40) \qbezier(210,20)(250,-10)(290,20) \qbezier(290,60)(310,60)(310,40) \qbezier(290,20)(310,20)(310,40) \end{picture}$$ The first two t-maps have trivial groups of automorphisms. Thus, they generate six tree-rooted cubic maps each. But the group of automorphisms of the third and the of forth t-maps has order two. Thus, they generate three tree-rooted cubic maps each and the second cubic map generates $18$ tree-rooted maps. The third cubic map in Figure 1 generates three t-maps. $$\begin{picture}(270,95) \put(0,30){\circle*{3}} \put(70,30){\circle*{3}} \put(35,50){\circle*{3}} \put(35,90){\circle*{3}} \qbezier[30](0,30)(35,5)(70,30) \qbezier[30](0,30)(0,70)(35,90) \qbezier[30](70,30)(70,70)(35,90) \put(33,2){\small 1} \put(100,30){\circle*{3}} \put(170,30){\circle*{3}} \put(135,50){\circle*{3}} \put(135,90){\circle*{3}} \qbezier[30](100,30)(100,70)(135,90) \qbezier[30](135,50)(135,70)(135,90) \qbezier[30](135,50)(152,40)(170,30) \put(133,2){\small 2} \put(200,30){\circle*{3}} \put(270,30){\circle*{3}} \put(235,50){\circle*{3}} \put(235,90){\circle*{3}} \qbezier[30](270,30)(270,70)(235,90) \qbezier[30](235,50)(235,70)(235,90) \qbezier[30](200,30)(217,40)(235,50) \put(233,2){\small 3} \linethickness{0.5mm} \qbezier(0,30)(17,40)(35,50) \qbezier(35,50)(52,40)(70,30) \put(35,50){\line(0,1){40}} \qbezier(100,30)(117,40)(135,50) \qbezier(100,30)(135,5)(170,30) \qbezier(170,30)(170,70)(135,90) \qbezier(200,30)(235,5)(270,30) \qbezier(200,30)(200,70)(235,90) \qbezier(235,50)(252,40)(270,30) \end{picture}$$ The group of automorphisms of the first of them has order $3$ and of the second and the third — order $2$. Thus they generate $2+3+3=8$ tree-rooted cubic maps. The forth cubic map in Figure 1 generates one t-map with order three group of automorphisms. Thus, it generates $2$ tree-rooted cubic maps. The fifth cubic map in Figure 1 also generates one t-map with trivial group of automorphisms. Thus, it generates $6$ tree-rooted cubic maps. The sixth cubic map in Figure 1 generates two t-maps $$\begin{picture}(240,80) \put(10,40){\oval(20,20)} \qbezier(40,40)(40,70)(70,70) \qbezier(70,70)(100,70)(100,40) \put(70,40){\oval(20,20)} \put(150,40){\oval(20,20)} \qbezier(180,40)(180,10)(210,10) \qbezier(210,10)(240,10)(240,40) \put(210,40){\oval(20,20)} \linethickness{0.6mm} \put(20,40){\line (1,0){20}} \put(80,40){\line(1,0){20}} \qbezier(40,40)(40,10)(70,10) \qbezier(70,10)(100,10)(100,40) \put(160,40){\line (1,0){20}} \put(220,40){\line(1,0){20}} \qbezier(180,40)(180,70)(210,70) \qbezier(210,70)(240,70)(240,40) \end{picture}$$ with order two group of automorphisms each. Thus, they generate $3+3=6$ tree-rooted cubic maps. So, we have $$30+18+8+2+6+6=70$$ tree-rooted cubic maps, as expected. [99]{} Albenque M. and Poulalhon D., *Generic method for bijections between blossoming trees and planar maps*, arXiv: 1305.1312. Bernardi O., *Bijective counting of tree-rooted maps and shuffles of parenthesis systems*, arXiv:math/0601684. Goulden I.P. and Jackson D.M., *The KP hierarchy, branched covers, and triangulations*, arXiv: 0803.3980. Lando S. and Zvonkin A., *Graphs on surfaces and their applications. Encyclopedia of mathematical sciences* **141**, Springer-Verlag, Berlin, 2004. Mullin R.C., *On the enumeration of tree-rooted maps*, Canad. J. Math., 1967, 19, 174-183. Poulalhon D. and Schaeffer G., *Optimal coding and sampling of triangulations*, Algorithmica, 2006, 46(3), 505-527. Stanley R.P., *Enumerative combinatorics, volume 2*, Wadsworth & Brooks, 1999. Tutte W.T., *A census of planar triangulations*, Canad. J. Math., 1962, 14, 21-38.

--- author: - 'M. Mori$^1$, N. Afzal Shooshtary$^1$, T. Tohyama$^2$, S. Maekawa$^{1,3}$' bibliography: - 'NQR.bib' title: 'Nuclear Quadrupole Resonance Frequency in Multi-Layered Cuprates' --- Much study has aimed the higher superconducting transition temperature, $T_c$. Regarding the high-[$T_{\rm c}$]{} cuprates, maximum [$T_{\rm c}$]{} depends on materials, e.g., [$T_{\rm c}$]{}$\sim$40 K for La-family and [$T_{\rm c}$]{}$\sim$90 K for Y-family, even if carrier densities are optimized. In particular, among hole doped cuprates, such a material dependence closely correlates with an energy difference between in-plane and apical oxygens, [$\mathit{\Delta}\varepsilon _{\rm A}$]{}[@Ohta90]. Actually, a spatial variation of the pairing gap due to a modulation of the apical oxygen, [O$_{\rm A}$]{}, is observed by the scanning tunneling microscopy [@Slez08; @Mori08]. In addition, it is known that disorder around the apical site is much more harmful on [$T_{\rm c}$]{} than other randomness [@Eisa06]. On the other hand, $T_c$ of multi-layered cuprates correlates with the number of [CuO$_2$]{} planes in a unit cell, $n$, and shows a maximum at $n$=3[@Karp99; @Iyo07]. One of the reasons why $T_c$ is suppressed for $n>$3 is the charge imbalance, i.e., a hole concentration in the pyramidally-coordinated-outer-planes (OP’s) is different from that in the square-coordinated-inner-planes (IP’s) [@Karp99; @Trok91; @Juli96; @Toku00; @Kote01]. The nuclear magnetic resonance (NMR) study has reported that the hole concentration in OP, [$\delta_{\rm OP}$]{}, is larger than that in IP, [$\delta_{\rm IP}$]{}[@Kote01]. The magnitude of charge imbalance, [$\mathit{\Delta}\delta$]{}$=$ [$\delta_{\rm OP}$]{}$-$ [$\delta_{\rm IP}$]{}, increases with $n$, and becomes 0.10 at most for $n$=3[@Kote01]. Such a charge imbalance induces some interesting phases, e.g., two kinds of superconducting gap[@Toku00] and coexistence of superconducting and anti-ferromagnetic states[@Kote04; @Muku06; @Muku06JPSJ; @Muku08; @Mori05; @Gan08]. To estimate the local charge density, the nuclear quadrupole resonance (NQR) frequency, [$\nu_{\rm Q}$]{}, is a useful quantity [@Zheng95; @Zheng96], since [$\nu_{\rm Q}$]{} is determined by a local charge distribution in the ground state. In the NQR measurement for a three-layered [Tl$_2$Ba$_2$Ca$_2$Cu$_3$O$_{10}$]{} (Tl2223)[@Zheng96], the charge imbalance has been observed as two separated peaks with about 6.6 MHz difference. Here, we note that the Knight shift, $K_c$, is also proportional to a local charge density on a measured site. However, one needs to examine not only the ground state but also the excited states and the neighboring spin-spin interactions, since $K_c$ is proportional to the susceptibility and the transferred hyperfine couplings[@Mila89]. In addition, one needs to measure not only $^{63}$Cu site but also $^{17}$O site to estimate the hole densities in [CuO$_2$]{} plane. However, the NMR measurement on $^{17}$O site is known to be rather difficult. Hence, an empirical relation between $K_c$ on $^{63}$Cu at room temperature and the hole density estimated by the NQR measurement has been used in general[@Kote01]. Previous theoretical studies on the NQR showed material dependence of [$\nu_{\rm Q}$]{} by neglecting a contribution from [O$_{\rm A}$]{}[@Hanz90; @Schw90; @Ohta92]. By following Ohta [*et al.*]{} [@Ohta92], we tried to estimate the charge imbalance observed in the NQR measurement by assuming [$\mathit{\Delta}\delta$]{}$=$0.10 and no difference of charge transfer energy, $\Delta$, between IP and OP. However, our estimated value of the difference was only 2.8 MHz at most. The experimentally observed 6.6 MHz difference is hard to be explained without taking account of [O$_{\rm A}$]{}[@note1]. To obtain more accurate value of charge imbalance, which correlates with [$T_{\rm c}$]{}, one cannot ignore [O$_{\rm A}$]{}. In this Letter, [$\nu_{\rm Q}$]{} on a Cu site is numerically studied in a cluster model including the 2$p_z$ orbital of [O$_{\rm A}$]{} by the exact diagonalization method. We carefully study effects of [$\mathit{\Delta}\varepsilon _{\rm A}$]{} and $\Delta$ on [$\nu_{\rm Q}$]{}, and show the doping dependence of [$\nu_{\rm Q}$]{} in IP and OP. To make a link between our theoretical result and real multi-layered material, we calculate the Madelung potential of ionic model for several different series of multi-layered cuprates. We predict a large enhancement of the splitting of [$\nu_{\rm Q}$]{} between OP and IP in Tl-based cuprates with more than three layers. The hole densities on the Cu site, which determine [$\nu_{\rm Q}$]{}, are numerically calculated by the exact diagonalization method in the cluster model as shown in Fig. \[cluster\](a) for OP. On the other hand, for IP, we adopt another cluster, in which 2$p_z$ and 4$p_z$ orbitals are omitted from the cluster in Fig. \[cluster\](a)[@Ohta92]. ![Schematic pictures of (a) CuO$_5$ cluster with Cu 4$p_\alpha$ orbitals for OP, and (b) its energy level scheme for holes. For IP, CuO$_4$ cluster with Cu 4$p_{x(y)}$ orbitals is adopted, and then 2$p_z$ and 4$p_z$ orbitals are omitted[@Ohta92]. []{data-label="cluster"}](cluster.eps){width="8.5cm"} We assume that the on-site Coulomb interactions $U_d$=8.5 eV, $U_{p}$=$U_{p_z}$=4.1 eV and $U_{4p}$=$U_{4p_z}$=2.0 eV are material independent[@Ohta91; @Ohta92], and the hopping integrals $T_{pd}$=1.06 eV, $T_{pp}$=0.46 eV, $T_{pp_z}$=0.22 eV, $T'_{p4p}$=1.07 eV and $T'_{p_z4p_z}$=0.38 eV are adjusted to the case of Tl2223 (ref. 24) by the standard bond-length dependence of $T_{pd}\propto d^{-4}$, $T_{pp}\propto d^{-3}$ with the parameters given in refs. 1 and 21. The notations, $p$, $p_z$, 4$p$, 4$p_z$ and $d$, denote 2$p_{x,y}$, 2$p_z$, 4$p_{x,y}$, 4$p_z$ and $d_{x^2-y^2}$, respectively. Figure \[cluster\](b) is the schematic picture of level separations; [$\mathit{\Delta}\varepsilon _{\rm A}$]{} between $p_z$ and $p$, $\Delta$ between $p$ and $d$, and $\Delta'$ between $d$ and 4$p$. We assume that 4$p_{x,y}$ and 4$p_z$ are degenerate. For both IP and OP, $\Delta'$ is fixed to 4.8 eV[@Ohta92]. Regarding the doping rate, [$\delta_{\rm p}$]{} (p=OP, IP), we calculate two cases of hole number in the cluster: 7 and 8 holes (5 and 6 holes) in OP (IP), where 7 (5) holes are associated with the half-filling, [$\delta_{\rm OP}$]{}=0 ([$\delta_{\rm IP}$]{}=0). The case of 8 (6) holes in OP (IP) corresponds to the hole-doped case with [$\delta_{\rm OP}$]{}=1 ([$\delta_{\rm IP}$]{}=1). Since the cluster size is small, we assume that an intermediate case of [$\delta_{\rm p}$]{} could be linearly interpolated between [$\nu_{\rm Q}$]{}’s of [$\delta_{\rm p}$]{}=0 and 1: \_[Q]{}(\_[p]{})=(1-\_[p]{})\_[Q]{}(\_[p]{}=0)+\_[Q]{}(\_[p]{}=1)\_[p]{}, \[nuqdP\] where the notations, [$\nu_{\rm Q}$]{}([$\delta_{\rm IP}$]{}) and [$\nu_{\rm Q}$]{}([$\delta_{\rm OP}$]{}), are used to denote the NQR frequency in IP with [$\delta_{\rm IP}$]{} and OP with [$\delta_{\rm OP}$]{}, respectively. On a $^{63}$Cu nucleus with the quadrupole moment $Q=-0.211$ barn and the spin angular moment $I=3/2$, [$\nu_{\rm Q}$]{} in OP is calculated by &&\_[Q]{}= { n\_[3d\_[x\^2-y\^2]{} ]{} M\_[3d]{} + (4 - n\_[4p\_x]{} - n\_[4p\_y]{} )M\_[4p]{}\ &&    - 2(2 - n\_[4p\_z]{} )M\_[4p]{}}. \[nqr\] The hole density on a Cu site is denoted by $n_i$ ($i$ = $3d_{x^2-y^2}, 4p_x, 4p_y, 4p_z$). Since the third term in eq. (\[nqr\]) represents the [O$_{\rm A}$]{} contribution to OP, [$\nu_{\rm Q}$]{} in IP is given by the sum of the first and the second terms. We use the values $M_{3d}$=$-47\times10^{21}$ V/m$^2$ and $M_{4p}$=125$\times10^{21}$ V/m$^2$[@Ohta92]. The $M_{4p}$-terms in eq. (\[nqr\]) arise from 4$p_\alpha$ ($\alpha$=$x$, $y$, and $z$) orbitals on a Cu site, which are partial-wave components of the 2$p_\alpha$ wave function in the neighboring O sites[@Schw90; @Ohta92]. The $z$-axis is perpendicular to the plane, and the isotropy of $x$ and $y$ is assumed. ![The calculated NQR frequency [$\nu_{\rm Q}$]{} as a function of $\Delta$. Square and triangle symbols are the results for IP and OP, and the open and closed symbols mean the undoped ([$\delta_{\rm p}$]{}=0) and doped ([$\delta_{\rm p}$]{}=1) cases, respectively. These symbols are fitted by a quadratic form $\nu_{\rm Q}=A_0+A_1\Delta+A_2\Delta^2$. []{data-label="nuq"}](nuq.eps){width="8.5cm"} In Fig. \[nuq\], [$\nu_{\rm Q}$]{} is plotted as a function of $\Delta$ for a fixed value of [$\mathit{\Delta}\varepsilon _{\rm A}$]{}=2.5 eV. We note that moderate variation of [$\mathit{\Delta}\varepsilon _{\rm A}$]{} does not change the results so much. The symbols shown in Fig. \[nuq\] are fitted by a quadratic form, $\nu_{\rm Q}=A_0+A_1\Delta+A_2\Delta^2$. We find that the values of $A_1$ and $A_2$ between OP and IP are similar for a given [$\delta_{\rm p}$]{} [@note2]. [$\nu_{\rm Q}$]{} increases monotonically with $\Delta$, because of the increase of $n_{3d_{x^2-y^2}}$ as reported in Ref. 21. The large enhancement of [$\nu_{\rm Q}$]{} ($\sim25$ MHz) from [$\delta_{\rm p}$]{}=0 to [$\delta_{\rm p}$]{}=1 is also attributed to the increase of $n_{3d_{x^2-y^2}}$. For a given [$\delta_{\rm p}$]{}, [$\nu_{\rm Q}$]{} at OP is larger by $\sim2.5$ MHz than [$\nu_{\rm Q}$]{} at IP. This is due to the contribution of the third term in eq. (\[nqr\]). It is worth emphasizing that Fig. \[nuq\] is available for evaluating one of the three quantities ($\nu_{\rm Q}$, $\Delta$, and [$\delta_{\rm p}$]{} (see eq. (\[nuqdP\])) when two of them are provided. As an example, let us evaluate the charge imbalance $\Delta\delta$ for Tl2223 to which the 6.6 MHz difference of [$\nu_{\rm Q}$]{}’s between OP and IP has been reported[@Zheng96]. First of all, we need to know $\Delta$ at OP and IP. Since it is very difficult to evaluate $\Delta$ experimentally, we have to rely on theoretical results. As will be discussed below, both $\Delta$’s at OP and IP are estimated to be $\sim$2.5 eV when $n\le 3$. Putting the estimated $\Delta$ into the fitted equation of [$\nu_{\rm Q}$]{}, we find \_[Q]{}(\_[OP]{})-\_[Q]{}(\_[IP]{})3.0+24.0. \[chargeimbalance\] This is a useful relation between the frequency difference and charge imbalance for the case of $n\le3$. The 6.6 MHz difference of [$\nu_{\rm Q}$]{} gives $\Delta\delta$=0.15, which is slightly larger than 0.10 estimated by the empirical relation used in the NMR study[@Kote01]. One possibility of this discrepancy is that the linear interpolation eq. (\[nuqdP\]) is too simple. Another one is that the empirical relation may need to be modified by considering the coordination number of oxygens around Cu site. We can see in Fig. \[nuq\] that [$\nu_{\rm Q}$]{} sensitively changes with $\Delta$, i.e., 0.1 eV increment of $\Delta$ provides 5$\sim$6 MHz enhancement in [$\nu_{\rm Q}$]{}. Therefore, to study the charge imbalance in the multi-layered cuprates, we need to estimate $\Delta$ in each plane. Below, the following three series of multi-layered cuprates are examined (1) Tl-12 series, [TlBa$_2$Ca$_{n-1}$Cu$_n$O$_{2n+3}$]{}[@Moro88; @parkin1988tcn; @ihara1988nht; @Naka89], ([$T_{\rm c}$]{}=48K, 110K, 123K, 112K, 107K from $n$=1 to $n$=5), (2) Tl-22-series, [Tl$_2$Ba$_2$Ca$_{n-1}$Cu$_n$O$_{2n+4}$]{}[@Izum92; @Subr88; @Cox88; @Kiku89; @Naka89], ([$T_{\rm c}$]{}=95K, 118K, 125K, 112K, 105K from $n$=1 to $n$=5), (3) Hg-12-series, [HgBa$_2$Ca$_{n-1}$Cu$_n$O$_{2n+2}$]{}[@Haung95; @hunt94; @paran96; @Haung95-1], ([$T_{\rm c}$]{}=98K, 128K, 135K, 127K, 111K from $n$=1 to $n$=5). The distance between Cu and [O$_{\rm A}$]{}, [$d_{\rm A}$]{}, is plotted as a function of $n$ in Fig. \[dA\]. ![ The bond length between Cu and [O$_{\rm A}$]{}, [$d_{\rm A}$]{}, versus the number of CuO$_2$ layer, $n$, for three series of multi-layered cuprates. The atomic coordinates are found in the following references: Refs. [@Moro88; @parkin1988tcn; @ihara1988nht; @Naka89] for Tl-12 family, Refs. [@Izum92; @Subr88; @Cox88; @Kiku89; @Naka89] for Tl-22 family, Refs. [@Haung95; @hunt94; @paran96; @Haung95-1] for Hg-12 family. []{data-label="dA"}](dA.eps){width="7.5cm"} Obviously, [$d_{\rm A}$]{} decreases for $n>3$ except for Hg-family. This may be a common feature of Tl-based cuprates[@note3]. The suppression of [$d_{\rm A}$]{} may be relevant to the suppression of [$T_{\rm c}$]{}, since the apical oxygen position directly correlates with [$\mathit{\Delta}\varepsilon _{\rm A}$]{} and the paring gap[@Ohta90; @Mori08]. To estimate [$\mathit{\Delta}\varepsilon _{\rm A}$]{} and $\Delta$, we calculate the Madelung potential $V({\rm A})$ (A=Cu, [O$_{\rm A}$]{}, and O in the plane) in the ionic model. In Fig. \[va\], [$\mathit{\Delta}\varepsilon _{\rm A}$]{}$\equiv$ ($V$([O$_{\rm A}$]{})$-V$(O))/$\epsilon_0$ is plotted with $n$ for each family. ![ Potential difference between apical oxygen and in-plane oxygen, [$\mathit{\Delta}\varepsilon _{\rm A}$]{}$\equiv (V(O_{\rm A})-V({\rm O}))/\epsilon_0$, versus $n$ for three different series of multi-layered cuprates. []{data-label="va"}](va.eps){width="7.5cm"} Note that the screening effect by the long-ranged Coulomb interaction can be effectively expressed by dividing the Madelung potentials by a constant $\epsilon_0$, whose value is assumed to be $\epsilon_0=3.5$[@Ohta90; @Maek04]. On the whole, [$\mathit{\Delta}\varepsilon _{\rm A}$]{} does not strongly depend on $n$, while each family shows different trend. In Tl-12 families, [$\mathit{\Delta}\varepsilon _{\rm A}$]{} slightly decreases with $n$ ($>2$), while [$\mathit{\Delta}\varepsilon _{\rm A}$]{} in Hg-family increases up to $n=4$. Since the moderate variation of [$\mathit{\Delta}\varepsilon _{\rm A}$]{} shown in Fig. \[va\] does not change [$\nu_{\rm Q}$]{} so much, we can assume [$\mathit{\Delta}\varepsilon _{\rm A}$]{}=2.5 eV in calculating [$\nu_{\rm Q}$]{}. ![ The charge transfer energy in the outer plane, $\Delta$(OP), versus $n$ for three series of multi-layered cuprates. The charge transfer energy in the inner plane, $\Delta$(IP), is almost independent of $n$ as, $\Delta$(IP)$\sim$2.5 eV. []{data-label="D"}](D.eps){width="8cm"} On the other hand, one cannot ignore the variation of $\Delta$ shown in Fig. \[D\], where $\Delta$ in OP, $\Delta$(OP), is plotted as a function of $n$. We used the relation, $\Delta\equiv(V({\rm O})-V({\rm Cu}))/\epsilon_0-10.9$[@Ohta90]. It is noticed that $\Delta$(OP) of Tl-families increases particularly for $n$=4 and 5. The deviation of Tl-family or the flatness of Hg-family must be due to their characteristic local structure as shown in Fig. \[dA\]. Note that $\Delta$ in IP, $\Delta$(IP), is always about 2.5 eV independent of family. Let us consider Tl1245, i.e., $n$=5 in the Tl-12 family. Taking $\Delta$(OP)=3.4 eV and $\Delta$(IP)=2.5 eV, we can predict from Fig. \[nuq\] a large NQR splitting, [$\nu_{\rm Q}$]{}(OP)$-$[$\nu_{\rm Q}$]{}(IP)$\sim$25 MHz, for the case of no carrier ([$\delta_{\rm OP}$]{}=[$\delta_{\rm IP}$]{}=0). Since real Tl1245 has carriers and [$\delta_{\rm OP}$]{}$>$[$\delta_{\rm IP}$]{}, the NQR splitting for Tl1245 must be more enhanced than 25 MHz. Such a large NQR splitting has not been observed so far, but careful NQR measurements may discover a large enhancement of the NQR splitting in the Tl-families with $n$=4 and 5. When we try to estimate the charge imbalance from observed [$\nu_{\rm Q}$]{}, we should be careful about the difference of $\Delta$(OP) and $\Delta$(IP). For example, for Tl2234 or Tl2245 one should take $\Delta$(OP)$\approx3.0$ eV and $\Delta$(IP)$\approx2.5$ eV according to Fig. \[D\] in obtaining a relation between the splitting and the charge imbalance. The relation reads $\nu_{\rm Q}(\delta_{\rm OP})-\nu_{\rm Q}(\delta_{\rm IP}) = 17.64+24.0(\delta_{\rm OP}-\delta_{\rm IP})-1.5\delta_{\rm OP}\sim 17.64+24.0\Delta\delta$. If one observed a NQR splitting, for instance, with 20 MHz, one might obtain [$\mathit{\Delta}\delta$]{}=0.10. To conclude, we studied the NQR frequency, [$\nu_{\rm Q}$]{}, in the cluster model including the contribution from $p_z$ orbital of apical oxygen by the exact diagonalization method. The charge imbalance in the three-layered Tl2223 is estimated by comparing our result and the experimental [$\nu_{\rm Q}$]{}. Our estimate on the charge imbalance [$\mathit{\Delta}\delta$]{}$=$[$\delta_{\rm OP}$]{}$-$[$\delta_{\rm IP}$]{} is 0.15, which is larger than the NMR result, [$\mathit{\Delta}\delta$]{}$=$0.10. The Madelung potential in several types of multi-layered cuprates is calculated in each plane to show the $n$-dependences of energy difference between apical and in-plane oxygens, [$\mathit{\Delta}\varepsilon _{\rm A}$]{}, and charge transfer energy, $\Delta$. It is found that a large splitting of [$\nu_{\rm Q}$]{} between OP and IP can be expected in the Tl-12 and Tl-22 families. Recently, the angle-resolved photoemission spectroscopy (ARPES) study in [Bi$_2$Ba$_2$Ca$_2$Cu$_3$O$_{10}$]{} has reported that $\delta({\rm OP})$=0.26 and $\delta({\rm IP})$=0.06, which is a quite large charge imbalance, [$\mathit{\Delta}\delta$]{}=0.20[@Idet09]. In the ARPES study, the hole densities are estimated by the ratio of the Fermi surface volume and the first Brillouin zone. The ARPES result is larger than both ours and the NMR result. These discrepancies among theory and experiments must be solved in near future. We would like to thank H. Mukuda, Y. Kitaoka, A. Iyo, H. Eisaki, and A. Fujimori for useful discussions. This work was supported by the Grand-in-Aid for Scientific Research and Nanoscience Program of Next Generation Supercomputing Project from MEXT. The authors thank the Supercomputer Center, Institute for Solid State Physics, University of Tokyo for the use of facilities.

--- abstract: 'The limits of the Second Law of Thermodynamics, which reigns undisputed in the macroscopic world, are investigated at the mesoscopic level, corresponding to spatial dimensions of a few microns. An extremely simple isolated system, modeled after Callen’s adiabatic piston, [@Callen60] can, under appropriate conditions, be described as a self-organizing Brownian motor and shown to exhibit a perpetuum mobile behavior.' author: - Bruno Crosignani - Paolo Di Porto - Claudio Conti title: 'Callen’s Adiabatic Piston and the Limits of the Second Law of Thermodynamics' --- Introduction ============ As first clearly stated by Schroedinger “a living organism tends to approach the dangerous state of maximum entropy, which is death. It can only keep aloof from it, i.e., alive, by continually drawing from its environment negative entropy. What an organism fed upon is negative entropy".[@Schroedinger45] Is it possible for an organism immersed in a thermal bath to be insulated, and thus avoid thermalization, and still be able to reduce its entropy? Unfortunately, the above scenario contradicts one of the most cherished laws of physics, that is the Second Law of Thermodynamics, which has victoriously resisted all the attempts aimed at finding a particular case where it could be violated. However, one has to consider that the Second Law has been formulated in the context of macroscopic physics and it is in this context that it has been successfully applied and verified. While the Second Law, being inherently statistical in its nature, cannot carry over to microscopic cases where the number of involved particles is too small, its limits of validity are not well understood in the transition region bridging the macroscopic to the microscopic world, where the system dimensions are small but still the number of particles large enough to justify the use of Thermodynamics. In particular, some intermediate mesoscopic regime, like, e.g., the biological one associated with typical cell dimensions (about one micron), is still *terra incognita* and open to possible surprises. Recent years have indeed witnessed a growing interest in the Thermodynamics of small-scale non-equilibrium devices, especially in connection with the operation and the efficiency of Brownian motors.[@Reinmann02] These are devices aimed at extracting useful work out of thermal noise and are the descendents of the famous ratchet mechanism popularized by Feynman [@FeynmanBook] in order to show the impossibility of violating the Second Law of Thermodynamics. From a quantitative point of view, they are usually described in terms of a variable $x(t)$, which can be identified as the trajectory of a “Brownian particle” of given mass $M$ obeying Newton’s equation of motion. The forces acting on the “particle” are those resulting from some kind of prescribed potential $V(x)$, from the viscous-friction term $-\gamma \dot{x}(t)$ and from the randomly fluctuating force (Langevin’s force) associated with thermal noise. Even in the presence of a spatially asymmetric (ratchet-type) potential $V(x)$, no preferential direction of motion is possible and no perpetuum mobile mechanism leading to a violation of the second law.[@FeynmanBook] In this paper, we show that a very fundamental thermodynamic system, the so-called “adiabatic piston”, first introduced by Callen,[@Callen60] can be actually described as a self-organizing Brownian motor. Under specific conditions, the system can be analytically modeled as a harmonic oscillator undergoing Brownian motion. ![ The adiabatic piston: an insulating cylinder divided into two regions by a movable, frictionless and insulating piston. \[f1\]](figure1.eps){width="50.00000%"} The relevant variable of the system is the position $x(t)$ of the moving wall with respect to its central position $L/2$, the harmonic potential being not prescribed but naturally emerging, as well as the friction term, as the result of the internal dynamics of the system. We recall that the adiabatic-piston problem deals with a system consisting (see Fig.1) of an adiabatic cylinder of length $L$, divided into two sections A and B, containing the same number $n$ of gas moles in different equilibrium states, by a fixed frictionless adiabatic piston . At a given instant t=0, the piston is let free to move and the problem is to predict the final equilibrium state of the system. This deceivingly simple question, which cannot be actually answered in the frame of equilibrium thermodynamics, has been firstly solved in 1996 in the frame of a simple kinetic model [@Crosignani96] and has since then given rise to a number of papers of increasing sophistication. Both analytic [@Piasecki99; @Chernov02; @Munakata01; @Gruber03] and molecular dynamics approaches [@Kestemont00; @Renne01; @White02; @Mansour05] reveal the existence of two successive stages of evolution, the first one leading the system to a state of mechanical equilibrium (where the pressures in both sections are equal) and the second, intrinsically random and occurring over a much longer time scale, to a final state of both mechanical and thermal equilibrium (where the average position of the piston is equal to L/2 and the average gas temperatures in the two sections become equal). The above conclusions have been basically drawn by investigating the average behavior of the significant variables describing the system evolution but less attention has been devoted to the fluctuations around these average quantities, which are somehow assumed to be small. By extending and clarifying the results of a previous paper [@Crosignani01], we wish to show how this is not always the case. More precisely, if one starts from specific initial conditions, then for spatial dimensions of the cylinder corresponding to the mesoscopic range (about 1 micron) and for values of the ratio $\mu=M/M_g$ between the piston and the gas mass lesser than one, fluctuations around equilibrium position can be a sizeable fraction of the average quantities, larger by a factor $\sqrt{M/m}$ ($m$ being the mass of the single gas molecule) than the ones expected if the system were in thermal equilibrium. This makes rather questionable the statement that in this case a final equilibrium configuration can be actually reached. Actually, the evolution of the system from an initial state in which both $x(t)=X(t)-L/2$ and $\dot{x}(t)$ are zero and the temperatures are equal in the two sections exhibits unexpected features: the system fails to settle down to a well-defined equilibrium state since the piston never comes to a stop but keeps wandering symmetrically around the initial position,performing oscillations whose mean-square value (evaluated over an ensemble of many replica) is much larger than that pertaining to standard thermal fluctuations. According to this behavior, the system appears to deserve the name of [*perpetuum mobile*]{}, even if there is no preferential direction of motion ($\langle x\rangle =\langle \dot{x}\rangle =0$). This unusual behavior can be associated with a possible challenge to the Second Law. In fact, a suitable process can be conceived through which the entropy of the [*isolated*]{} adiabatic piston system turns out to undergo a significant decrease (see Sections IV-V). In practice, as we shall see below, no significant entropy decrease occurs outside a mesoscopic regime corresponding to spatial dimensions of a few microns; in fact, the time interval $t_{as}$ over which the significant entropy decrease occurs depends on the fourth power of $L$ ($t_{as} \propto L^4$): thus, any deviation from the above spatial scale gives rise to such large $t_{as}$ as to affect the validity of our model. We note that the coincidence between the spatial scales over which the model is valid and a typical biological dimension is definitely intriguing. The stochastic adiabatic piston =============================== The investigations of the adiabatic-piston dynamics appeared in the last ten years (see, e.g., [@Crosignani96; @Piasecki99; @Chernov02; @Munakata01; @Gruber03]) have placed into evidence two distinct stages of temporal evolution. The first one occurs, on a relatively short time scale, when the system is let free to evolve from an initial configuration in which the two sections are in thermal equilibrium at different pressures, and leads to a final mechanical equilibrium (equal pressures) configuration that depends on the initial conditions. The second one is characterized by a stochastic dynamic evolution during which the common pressure on the two sides remains essentially constant, and may eventually lead to a final equilibrium configuration in which also the temperatures attain a common value and the piston is in the central position. In this section, we confine ourselves to the second stage by investigating the piston dynamics starting from a specific initial configuration characterized by $x(t=0)=\dot{x}(t=0)=0$ and by a common temperature $T_0$ in the two sections. In particular, we wish to focus our attention on the fluctuations of the piston positions around $L/2$. In effect, while $\langle x(t) \rangle$ and $\langle \dot{x}(t) \rangle$ turn out to vanish at all times, as [*a priori*]{} dictated by symmetry, there is a range of values of the system parameters, corresponding to a mesoscopic regime where the mean-square value $\langle x^2(t)\rangle^{1/2}$ can be a not negligible fraction of the piston half-length $L/2$, a quite surprising result which seems to indicate that no effective final equilibrium position can be reached. We start from the equation describing the deterministic piston motion in the presence of a finite pressure difference, that is [@Crosignani96] $$\label{eq1} \ddot{X}=\displaystyle \frac{n R }{M}\left(\frac{T_A}{X}-\frac{T_B}{L-X} \right)-\sqrt{\frac{8 n R M_g}{\pi M^2}} \left( \frac{\displaystyle\sqrt{T_A}}{X}+\frac{\displaystyle\sqrt{T_B}}{L-X}\right)\dot{X}+\frac{M_g}{M} \left(\frac{1}{X}-\frac{1}{L-X}\right)\dot{X}^2$$ where $R$,$M_g$ and $M$ are the molar gas constant, the common values of the gas masses in $A$ and $B$, and the piston mass, respectively, while the dot stands for derivation with respect to time. It has been derived by assuming that the velocity distributions of the ideal gas in the two sections are Maxwellian, i.e., that thermal equilibrium is continuously restored on a time scale much shorter than the characteristic time scale of the piston motion, an assumption which has to be checked [*a posteriori*]{}. We observe that Eq.(1) is identical to the one obtained by Munakata and Ogawa (Eq. (22)of Ref. [@Munakata01])but for the extra-term containing $\dot{X}^2$. This term, which is negligible for finite pressure-difference in the two sides, turns out to play a fundamental role when studying the piston dynamics for equal pressures, i.e., in the situation investigated in the present paper. In this case, Eq.(\[eq1\]) has to be suitably modified. In effect, besides setting equal to zero the first term on its R.H.S. associated with pressure difference, we have to explicitly take into account the random nature of the hits suffered by the piston because of the collisions with the gas molecules. To this aim, we resort to Langevin’s approach (see, e.g. [@Crosignani01]) based on the introduction of an [*ad hoc*]{} Langevin acceleration $a(t)$, so that Eq.(\[eq1\]) is superseded by $$\label{eq2} \ddot{X}=-\frac{N}{M}\sqrt{\frac{8 k m }{\pi }}\left(\frac{\displaystyle\sqrt{T_A}}{X}+ \frac{\displaystyle\sqrt{T_B}}{L-X} \right)\dot{X}+ \frac{ N m}{M}\left(\frac{1}{X}-\frac{1}{L-X}\right)\dot{X}^2 +a(t)\text{,}$$ where $k$ is the Boltzmann constant and $N=M_g/m$ is the number of molecules of mass $m$ on each side. By taking again advantage of the equal pressure condition, i.e., $T_A(t)/X(t)=T_B(t)/[L-X(t)]=2T_0/L$ Eq.(\[eq2\]) can be rewritten as $$\label{eq3} \ddot{ X} + \frac{4N}{N}\sqrt{\frac{k T_0 m}{\pi L}} \left( \frac{1}{\sqrt{X}} + \frac{1}{\sqrt{L-X}} \right) \dot{X} + \frac{2 N m}{M} \left[ \frac{X-L/2}{X(L-X)} \right] \dot{X}^2= a(t)\text{.}$$ Determining the correct expression for $a(t)$ is a delicate task since the standard Langevin approach does not in general carry over to nonlinear dynamical systems [@VanKampenBook], as the one described by Eq.(\[eq3\]). In order to be able to take advantage of the Langevin method, we linearize the above equation: a) by considering small displacements around the starting position of the piston, that is $|(X-L/2)|/(L/2)<<1$ and b) by approximating the square $\dot{X}^2$ of the piston velocity by the average thermal value $k T_0/M$ ; both hypotheses have to be proved consistent [*a posteriori*]{}. More precisely, by resorting to assumption b) and by using Langevin’s method , it is possible to identify two characteristic times, $t_{th}$ and $t_{as}>>t_{th}$ (see Eq.(\[eq10bis\])). The first one represents the temporal scale over which the piston velocity thermalizes, while the second one is the characteristic time over which the large piston-position fluctuations occur. The relation $t_{th}<<t_{as}$ justifies a posteriori the application of the fluctuation-dissipation theorem. After introducing the new variable $x(t)=X(t)-L/2$, the linearized version of Eq.(\[eq3\]) reads $$\label{eq4} \ddot{x} + \frac{4 N}{\sqrt{ \pi}}\frac{v_0}{L}\left(\frac{m}{M}\right) \dot{x} + 4 N\frac{v_0^2}{L^2}\left(\frac{m}{M}\right)^2 x = a(t)\text{,}$$ $v_0=(2k T_0/m)^{1/2}$ being the most probable velocity of the corresponding gas Maxwellian distribution. [@SommerfeldBook] At this point, a straightforward application of the fluctuation-dissipation theorem [@Pathria] yields $$\langle a(t)a(t')\rangle =\frac{16 N}{\sqrt{\pi} L} v_0^3 \left(\frac{m}{M}\right)^2\delta(t-t')\text{.}$$ According to the above results, the variable $x(t)$, describing the evolution of the state of our system, obeys an equation formally similar to that of a “Brownian particle” in one dimension. However, the potential term and the viscous drag have naturally emerged out of the internal dynamics of the system and do not correspond to an external active force or to a phenomenological friction term as in the case of Brownian motors. If we define $$\label{eq5} \beta=\frac{4 N}{\sqrt{\pi}}\frac{v_0}{L}\frac{m}{M} \text{,}\hspace{1cm}\omega^2=4 N \frac{v_0^2}{L^2}\left(\frac{m}{M}\right)^2$$ Eq.(\[eq4\]) is formally identical to that describing the Brownian motion of a harmonically-bound particle of mass $M$, that is $$\label{eq6} \ddot{x}+\beta\dot{x}+\omega^2 x=a(t)$$ where $a(t)$ is the Langevin acceleration, a case which has been thoroughly investigated in the literature [@Chandrasekhar43]. The piston as a harmonically-bound Brownian particle ==================================================== The solution of the stochastic differential equation (\[eq6\]) governing the piston motion is readily obtained in terms of the general results provided in [@Chandrasekhar43] for the Brownian motion of a harmonically-bound particle. Equation (\[eq6\]),together with Eq.(\[eq5\]), describe an “overdamped” case (in fact, $\beta/\omega=\sqrt{16/\pi}\sqrt{N}>>1$, since $N$ is typically a large number).For this situation, one has [@Chandrasekhar43] $$\label{chandraeq} \begin{array}{l} \langle x(t)\rangle=x_0 e^{-\beta t /2}\left( \cosh \frac{\beta_1 t}{2}+\frac{\beta}{\beta_1} \sinh\frac{\beta_1 t}{2} \right)+\frac{2u_0}{\beta_1}e^{-\beta t/2} \sinh \frac{\beta_1 t}{2}\text{,}\\ \langle u(t)\rangle=u_0 e^{-\beta t /2}\left( \cosh \frac{\beta_1 t}{2}-\frac{\beta}{\beta_1} \sinh\frac{\beta_1 t}{2} \right)-\frac{2 x_0 \omega^2}{\beta_1}e^{-\beta t/2} \sinh \frac{\beta_1 t}{2}\text{,}\\ \langle x(t)^2 \rangle=\langle x(t) \rangle^2 +\frac{k T_0}{M \omega^2}\left[1-e^{-\beta t} \left(2\frac{\beta^2}{\beta_1^2}\sinh^2 \frac{\beta_1 t}{2} +\frac{\beta}{\beta_1} \sinh \beta_1 t +1\right)\right]\text{,}\\ \langle u(t)^2 \rangle=\langle u(t) \rangle^2 +\frac{k T_0}{M }\left[1-e^{-\beta t} \left(2\frac{\beta^2}{\beta_1^2}\sinh^2 \frac{\beta_1 t}{2} -\frac{\beta}{\beta_1} \sinh \beta_1 t +1\right)\right]\text{,} \end{array}$$ where $u(t)=dx/dt$ , $x_0=x(t=0)$, $u_0=u(t=0)$ and $$\beta_1=\sqrt{\beta^2-4\omega^2}\cong \beta-2\omega^2/\beta\text{.}$$ In our case $x_0=0$, $u_0=0$, so that, as expected, $\langle x\rangle=\langle u \rangle=0$ at all times (see Eqs. (\[chandraeq\])$_{1,2}$). By taking the temporal asymptotic limit of the above equations, it is possible to recognize the existence of two characteristic time scales $t_{th}$ and $t_{as}$ , such that $t_{th}<<t_{as}$, given by $$\label{eq10bis} \begin{array}{l} t_{th}=\frac{1}{2\beta}\text{,}\\ t_{as}=\frac{\beta}{2\omega^2}\text{.} \end{array}$$ They respectively represent the time over which the piston velocity thermalizes, i.e., its mean-square attains the value $$\label{eq10} \langle \dot{x}^2 \rangle_{as}=\frac{k T_0}{M}\text{,}$$ and the time over which the mean-square value of the piston position reaches the asymptotic value $$\langle x^2 \rangle_{as}=\frac{k T_0}{M \omega^2}\text{.}$$ By using Eq.(\[eq5\]) we have $$\label{eq12} t_{as}=\frac{1}{2\sqrt{\pi}}\frac{L}{v_0}\frac{M}{m}=\frac{1}{2\sqrt{\pi}}\frac{L}{v_0}N\mu$$ and $$\label{eq14tth} t_{th}=\frac{\pi}{4 N} t_{as}.$$ Since $N>>1$, the asymptotic time $t_{as}$ is much larger than the thermalization time $t_{th}$, which justifies the replacement in Eq.(\[eq3\]) of $\dot{X}^2$ with its thermal value given by Eq.(\[eq10\]). It is worth noting that the difference between the time scales of $\langle x^2 \rangle$ and $\langle \dot{x}^2 \rangle$ (i.e. $t_{as}>>t_{th}$) is associated with the presence of the sign “plus” and “minus” in the r.h.s. of Eq. (\[chandraeq\])$_3$ and (\[chandraeq\])$_4$, respectively. In fact, it is the minus sign in Eq. (\[chandraeq\])$_4$ which is responsible for the cancellation of the slowly-decaying asymptotic terms in $\langle u^2 \rangle$, while these terms, present in Eq. (\[chandraeq\])$_3$, dictate the long time behavior of $\langle x^2 \rangle_{as}$. The asymptotic mean-square displacement $\langle x^2\rangle_{as}$ of the piston from its central equilibrium position $x=0$ is, according to Eqs. (\[chandraeq\]) and (\[eq5\]), $$\label{eq16} \langle x^2\rangle_{as} =\frac{\mu}{2}\left( \frac{L}{2}\right)^2=\frac{1}{2N}\frac{M}{m}\left(\frac{L}{2}\right)^2\text{.}$$ In the limit $\mu<<1$ (that is, small piston mass with respect to gas mass), the root-mean square displacement $\langle x^2\rangle_{as}^{1/2}$ is much smaller than $L/2$, so that both assumptions justifying the linearization of Eq.(\[eq6\]), and thus the application of Langevin’s approach, are verified. These fluctuations can be compared with those pertaining to a diathermal piston, associated with a system identical with the one described above but for the presence of a thermally conducting piston instead of an adiabatic one. To this end, let us assume the piston in Fig.1 to be a good heat conductor, so that both sections possess the common constant temperature $T_0$. Any displacement $x$ of the piston from the central position results in an unbalance $p(x)$ between the pressures $p_A(x)$ and $p_B(x)$ in the two sections, i.e., $$\Delta p(x)=p_A(x)-p_B(x)=\frac{N k T_0}{\mathcal{A}(L/2+x)}-\frac{N k T_0}{\mathcal{A}(L/2-x)}\text{,}$$ where $\mathcal{A}$ is the transverse section area, so that the force $F(x)$ exerted by the gas on the piston reads $$\label{eq16F} F(x)=\frac{N k T_0}{L/2 + x}-\frac{N k T_0}{L/2-x}\cong - \frac{2 N k T_0}{(L/2)^2}x\text{,}$$ having assumed $|x|<<L/2$, as verified [*a posteriori*]{}. Therefore, the piston feels both the thermal bath at temperature $T_0$ and the harmonic potential $$V(x)=\frac{N k T_0}{(L/2)^2}x^2\text{,}$$ and, as a consequence, its position probability-density $P(x)$ reads $$P(x)=\sqrt{\frac{N}{\pi (L/2)^2}}\exp\left[-\frac{ N}{(L/2)^2}x^2\right]\text{.}$$ The associated mean square value $\langle x^2 \rangle$ is $$\label{eq19} \langle x^2 \rangle =\frac{1}{2 N}\left(\frac{L}{2}\right)^2\text{,}$$ smaller than the corresponding mean square displacement of the adiabatic piston (see Eq.(\[eq16\])) by a factor $m/M$. Note that Eq.(\[eq19\]) agrees with the standard result obtained in the frame of equilibrium-thermodynamic microscopic fluctuations. [@NewCallen] Our derivation clarifies the basic difference between adiabatic and diathermal situation. In the first case, the restoring force acting on the piston is given by $-(8 N k T_0 m/L^2M)x$ (see Eq.(\[eq4\])), while, in the second it reads $-(8 N k T_0/L^2)x$ (see Eq.(\[eq16F\])), so that the restoring force acting on the adiabatic piston is $m/M$ times smaller than that acting on the diathermal one. We stress that the anomalously large fluctuations of the adiabatic piston are not limited to the macroscopic regime, but are also present in the mesoscopic regime, provided the number of molecules $N$ is large enough, as in our case, to justify the use of ordinary concepts of pressure and temperature. This accounts for the peculiarity of the adiabatic piston system. The large random displacements of the piston are not trivially related to the system dimensions, but to its adiabatic nature: the diathermal piston exhibits a mean-square-value $\langle x^2\rangle=(m/M)\langle x^2\rangle_{as}$ (see Eq.(\[eq19\]) and Eq.(\[eq16\])) smaller by a factor $m/M$ than the corresponding factor for the adiabatic piston. Variation of the entropy ======================== The peculiar behavior of the adiabatic-piston system discussed above has a natural counterpart in its entropy variations. In order to clarify this point, we note that the entropy behavior of our system is not uniquely defined. First, let us consider an ensemble of identical systems, in each of which the piston is let indefinitely free to move starting from the central position $x=0$ at time $t=0$. After a time interval reasonably larger than $t_{as}$, the probability distribution of the piston position will stay unchanged. Accordingly, it is possible to define an entropy of the system (gas+piston) ensemble whose value turns out to be larger than the initial one pertaining to the one in which all pistons are fixed in the central position. The above entropy does not allow for a simple statistical interpretation like Boltzmann’s one. The latter requires a single system in thermal equilibrium, and this is not the case for the wandering adiabatic piston. Thus, to give meaning to Boltzmann’s entropy, we consider a single system and stop the piston either at a given time of the order of $t_{as}$ or at a given prefixed position of the order $\langle x^2 \rangle_{as}^{1/2}$. The two procedures are conceptually quite different. In fact, halting the piston at a given time requires a device comparable in size with the piston-position fluctuations it is trying to harness: in other words, the stopping mechanism seems to behave like a demon device while it rectifies the large piston fluctuations, thus forbidding any violation of the Second Law. Similar arguments go back to Smoluchowsky (see ref. [@Smolu]), and an interesting example is connected with the Feynman’s pawl and ratchet device. [@FeynmanBook] Vice versa, in principle, if the halting device is placed at a position $\bar x$ chosen a priori (e.g., $\bar x=+\langle x^2 \rangle_{as}^{1/2}$), the piston stopping operation takes place without the intervention of relevant statistical fluctuations, i.e., without any demon mechanism. Of course, due to the random nature of the piston wandering, we do not know when the halting process will take place. However, if we wait long enough (e.g., for a time of the order of a few $t_{as}$), there is a high likelihood that the piston has come to a halt at the prefixed position. The above argument highlights a peculiar feature of the adiabatic piston system. Unlike other systems in which unusually large thermal fluctuations cannot be rectified without introducing comparable external fluctuations, the piston wandering can be conveniently exploited. If the piston is stopped at a prefixed position, the Boltzmann gas entropy significantly decreases without any entropy increase of the environment. To clarify this point, we consider the entropy change $\Delta S$ undergone by the system when passing from the equilibrium state in which the piston is held in the central position to the final one in which the piston is stopped at the position $x=\bar x$. Recalling that the pressure does not change during the process (see Sect.II), we have $$\label{DeltaXX} \Delta S=\Delta S_A+\Delta S_B=n c_p \ln \frac{L/2+\bar x}{L/2}+n c_p \ln \frac{L/2-\bar x}{L/2}= n c_p \ln \left[1-\left( \frac{\bar x}{L/2}\right)^2 \right]$$ where $c_p$ is the molar heat at constant pressure and $n$ the common number of molecules in sections A and B. By recalling Eq.(\[eq16\]) and that $\bar x^2/(L/2)^2=\langle x^2\rangle_{as}/(L/2)^2$ is a small number, Eq. (\[DeltaXX\]) yields $$\label{eq22bis} \frac{\Delta S}{k}=\frac{n c_p}{k} \ln \left[1-\left(\frac{\bar x}{L/2}\right)^2 \right]\cong -\frac{n c_p}{k} \left(\frac{\bar x}{L/2}\right)^2=-\frac{c_p}{2 R}N \mu\text{,}$$ or, by comparing it with the standard entropy fluctuations $\Delta S_{th}=(k n c_p)^{1/2}$ [@Landau] $$\label{deltaS3} \frac{\Delta S}{\Delta S_{th}}=-\frac{1}{2}\sqrt{\frac{c_p}{R}}\sqrt{N} \mu\text{.}$$ This corresponds, whenever $\mu \sqrt{N}>>1$, i.e. $M/m>>\sqrt{N}$, to a sensible negative entropy decrease, and constitutes a violation of the Second Law, consistent with the *perpetuum mobile* behavior of the system. In particular, if the system is embedded in a thermal bath at temperature $T_0$, it is possible to devise a cyclic process by means of which the work $$\label{workeq} W\cong k T_0 N \mu$$ can be extracted from the environment (see Appendix). Actually, according to the accumulated experimental evidence confirming the validity of the Second Law, we do not expect these results to hold true in standard macroscopic situations. In fact, our model imposes strict limitations on the spatial scale over which our results can be trusted. In order to assess these limitations, we refer to the specific case of a gas at standard conditions of temperature and pressure. In this case, expressing hereafter $L$ in microns, and assuming the volume of the piston to be $L^3$, $N=3 \times 10^7 L^3$ it follows from Eq.(\[deltaS3\]) that $$\label{deltaS4} \frac{ \Delta S }{\Delta S_{th}}=-5\times 10^3 L^{3/2} \mu\text{,}$$ while from Eq.(\[eq12\]) we have, taking $v_0=4\times10^8$ microns/sec (molecular oxygen), $$\label{deltaS5} t_{as}=5\times10^{-2} \mu L^4 \text{sec.}$$ We now observe that, if we reasonably require the work $W$ extracted in a cycle to scale with the system dimensions, i.e., with $N$, the factor $\mu$ has to be kept constant when varying the system dimensions. Therefore, Eq. (\[deltaS5\]) reveals an extremely sensitive fourth-power dependence of $t_{as}$ on the linear dimension of the system, which severely limits the applicability of our model to values of $L$ up to a few microns. In fact, beyond this mesoscopic scale, $t_{as}$ becomes so large as to invalidate our results. As an example, for $\mu=0.01$ and $L=1cm$, one gets $t_{as}=5\times10^{12} sec$, that is about 1000 centuries! Conversely, by taking $L=1\mu m$, we get for $t_{as}$ a reasonable value of $5\times10^{-4}$ sec, while $ \Delta S /\Delta S_{th}= -50$ is still quite large. It is important to stress that this specific mesoscopic regime, around $1\mu m$, has emerged spontaneously from the adiabatic-piston problem, without the introduction of any *ad hoc* parameter. The Second Law, stating that *the entropy of an isolated system cannot decrease*, is actually violated in the specific example provided above. The associated spatial scale appears to set a borderline between the macroscopic realm and the mesoscopic one. ![ Normalized asymptotic standard deviation $\eta$ of the piston position as a function of $\mu$ and $N=3\times10^4$ (averaged over 1000 realizations). The continuous line represents the predicted behavior in the linearized regime (see Eq. (\[eq16\])). \[f5\]](figure2.eps){width="8.3cm"} Beyond the linear regime ======================== The above approach has allowed us to deal with the situation $\mu \ll 1$. In order to have an insight into the behavior of our process in the more general case $\mu \lesssim 1$, we assume Langevin’s approach to be approximately valid also in this moderately nonlinear regime, and use the nonlinear Eq.(\[eq3\]) with the same stochastic acceleration $a(t)$ worked out in the linear case. After introducing the dimensionless units $\xi =X/L$ and $\tau=t/t_o$, where $t_o=4 \sqrt{2} t_{th}$, Eq.(\[eq3\]) reads $$\label{eq20} \frac{d^2 \xi}{d \tau^2}+ \left( \frac{1}{\sqrt{\xi}}+\frac{1}{\sqrt{1-\xi}} \right) \frac{d\xi}{d\tau}- \frac{1}{\mu} \left( \frac{1}{\xi}-\frac{1}{1-\xi} \right) \left(\frac{d\xi}{d\tau}\right)^2 = \sigma \alpha (\tau),$$ where $a(\tau)$ is a unitary-power white noise process and $\sigma^2 =(\pi/2\sqrt{2})(\mu/N)$. This last equation can be numerically integrated by adopting a second-order leap-frog algorithm as the one developed in reference [@Qiang00]. In particular, we can evaluate the asymptotic value of the mean-square root deviation $\eta\equiv\langle (\xi-1/2)^2\rangle^{1/2}$ as a function of $\mu$ and, for a given value of $\mu$ , the time evolution of $\langle \left(d\xi/d\tau\right)^2\rangle$. Fig.2 shows that the asymptotic mean-square root displacement of the piston increases as $\sqrt{\mu}$ for $\mu\lesssim 0.3$ , in fairly good agreement with the linearized theory (see Eq.(\[eq16\])), while it tends to saturate for larger values of $\mu$. Fig. 3 confirms that the piston velocity attains its thermal value in a time much shorter than the asymptotic time $t_{as}$, a necessary condition for the application of Langevin’s approach. As far as the entropy change $\Delta S$ is concerned, the linear dependence on $\bar x^2 =\langle x^2 \rangle_{as}$ predicted by Eq.(\[eq22bis\]) implies as well its saturation for increasing $\mu$ . The above results show that, for fixed N, both piston asymptotic displacement and entropy decrease saturate with increasing piston mass $M$. ![ Time evolution of the r.m.s. of the normalized piston velocity $u=d\xi/d\tau$ as a function of $\tau'=10^{-4} t/t_0$, for $\mu=0.5$ and $N=3\times 10^4$, averaged over $1000$ realizations. \[fvelocity\]](figure3.eps){width="8.3cm"} Conclusions =========== In the present paper, we have examined the dynamic evolution of the adiabatic piston system starting from both mechanical and thermal equilibrium. Obviously, in this case $\langle x\rangle=0$ and $\langle \dot x \rangle=0$ for symmetry reasons, so that the significant quantity is $\langle x^2(t)\rangle^{1/2}$. This situation can be analyzed by generalizing the deterministic kinetic model developed in [@Crosignani96] with the introduction of Langevin’s force. The resulting equation for $x(t)$ turns out to be, whenever the mass $M$ of the piston is small compared to the gas mass $M_g$, completely identical to that describing the Brownian motion of an harmonically-bound particle of mass $M$. This allows us to provide an analytic expression for $\langle x^2(t)\rangle^{1/2}$ , which can be a sizeable fraction of the cylinder length $L$ (see Eq. (\[eq16\])), thus implying that the piston never reaches an equilibrium position but keeps oscillating around its initial position $x=0$. This peculiar behavior is restricted to systems possessing specific spatial dimensions pertaining to a mesoscopic realm, where the laws of thermodynamics are not necessarily valid. The fact that the piston never stops is strictly related to the finite length of the cylinder: if we allow the length $L$ of the cylinder to go to infinity (as assumed, for example, in [@Gruber03]), the ratio $M/M_g$ goes to zero and so does $\langle x(t)^2 \rangle^{1/2}/(L/2)$ (see Eq.(\[eq16\])). At the same time, $t_{as}$ becomes extremely large, so that, in this limit, the piston practically does not move. In view of this, the violation of the Second Law does not appear that startling since it concerns a regime outside the macroscopic thermodynamic limit. However, it can be interpreted as the signature of something remarkable happening at the borderline between the macroscopic and mesoscopic realms, signaling that a new way of looking at standard thermodynamic concepts may be needed if we wish to continue to apply them outside the boundaries within which they were first introduced. In order to have an insight into the regime of mild nonlinearity occurring when $\mu=M/M_g$ approaches unity, we have numerically solved the stochastic Eq.(\[eq20\]) with the standard initial conditions adopted in our model. The results of the linearized approach are in good agreement with those associated with the numerical solution of Eq.(\[eq20\]). The same problem of time evolution and approach to equilibrium of the system has been also considered by using molecular dynamics simulations.[@Kestemont00; @Renne01; @White02; @Mansour05] These typically involve a considerable number of point particles (around $500$) which model the gas inside the cylinder and are separated by a frictionless movable piston, without internal degrees of freedom, against which they undergo perfectly elastic collisions. Some general features emerge from the numerical analysis, which reveal, consistently with our results, a very slow approach to the final equilibrium state. In particular, Ref [@White02] presents the molecular dynamics simulations of a system evolving from an initial state in which the piston is fixed and the number of particles and the temperatures in both sections are equal, which is precisely the case we are dealing with in the present paper. Our analytic approach predicts a relaxation time (see Eq.(\[eq12\])) which, in the normalized units adopted in [@White02] ($v_0= 2$, $m=1$ and $L=60$), reads $t_{as}= (30/\sqrt{\pi})(M/m)\cong 17 M$. This is in remarkably good agreement with the molecular dynamics simulations reported in [@White02] (see Fig.5 of Ref.[@White02]) where, for $M<100$ (corresponding to $\mu=M/N=100/250=0.4$), $t_{as}=16M$. ![ \[figcyclic\] Cyclic process through which work can be extracted from a thermal bath at temperature $T_0$.\[f7\] ](figure4.eps){width="8.3cm"} In conclusion, our simple kinetic analytic description has revealed that Callen’s adiabatic piston can exhibit, when starting from a particular mechanical and thermal equilibrium state, large fluctuations, a very intriguing feature which presents the characteristics of [*perpetuum mobile*]{}. This occurs for very specific spatial dimensions of the system (around $1\mu m$), pertaining to the mesoscopic regime. Whether this is actually challenging the limits of validity of the Second Law seems a legitimate question, whose answer may shed some light on the understanding of small-scale nonequilibrium devices. We wish to thank Noel Corngold for his constant encouragement and many useful comments. [0]{} H. B. Callen, [*Thermodynamics*]{} (Wiley, New York, 1960), p. 321. E. Schroedinger, [*What is life?*]{} (Cambridge University Press, Cambridge, 1945), Ch. VI. R. Reinmann and P. Hanggi, Appl.Phys. A [**175**]{}, 169. (2002). R. P. Feynman, R. Leighton, and M. Sands, *The Feynman Lectures on Physics I* (Addison-Wesley, Reading, Mass, 1965), ch. 39. B. Crosignani, P. Di Porto, and M. Segev, Am. J. Phys. [**64**]{}, 610 (1996). J. Piasecki and Ch. Gruber, Physica A [**264**]{}, 463 (1999). N. I. Chernov, J. L. Lebowitz and Sinai Ya. G., Russian Mathemathical Surveys [**57**]{}, 1045 (2002). T. Munakata and H. Ogawa, Phys. Rev. E [**64**]{}, 036119 (2001). Ch. Gruber, S. Pache and A. Lesne, J. Stat. Phys. [**112**]{}, 1177 (2003). E. Kestemont, C. Van den Broeck, and M. Malek Mansour, Europhys. Lett. [**49**]{}, 143 (2000). M. Renne, M. Ruijgrok, and T. Ruijgrok, Acta Physica Polonica [**32**]{}, 4183 (2001). J. A. White, F. L. Roman, A. Gonzalez, and S. Velasco, Europhys. Lett. [**59**]{}, 479 (2002). M. Malek Mansour, C. Van den Broeck, and E. Kestemont, Europhys. Lett. [**69**]{}, 510 (2005). B. Crosignani and P. Di Porto, Europhys. Lett. [**53**]{}, 290 (2001). N. Van Kampen N, [*Stochastic Process in Physics and Chemistry*]{} (North-Holland, Amsterdam, 1992). A. Sommerfeld, [*Thermodynamics and Statistical Mechanics*]{} (Academic Press, New York, 1956), p. 177. See, e.g., R. K. Pathria, *Statistical Mechanics*, 2nd ed. (Buttherworth-Heineman, Oxford,1996), p. 481. See, e.g., S. Chandrasekhar, Rev. Mod. Phys. [**15**]{}, 1 (1943). H. B. Callen, [*Thermodynamics and an Introduction to Thermostatistics*]{} (Wiley, New York, 1985, second edition), pag. 428. M. Smoluchowski, Phys. Zeit. [**12**]{}, 1069 (1912). L. Landau and E. Lifshitz, [*Statistical Physics*]{} (Pergamon Press, Oxford) 1969. J. Qiang and S. Habib, , 7430 (2000). Appendix {#appendix .unnumbered} ======== If our system is embedded in a thermal bath at temperature $T_0$ and pressure $P_0$, the following process, sketched in Fig. \[figcyclic\] , can be used for extracting work from the environment. In the first panel, the system is in the initial state. The second panel shows the system after the entropy decrease $\Delta S$ has occurred. At this point, we separate the two sections (panel 3) and drive the two gases back to the initial temperature $T_0$ by means of two reversible adiabatic processes, the relation between the new volumes $V'_1$ and $V'_2$ and the previous volumes $V_1$ ,$V_2$ and temperatures $T_1$,$T_2$ being $$\label{eqA1} V'_1=V_1 (T_1/T_0)^{\frac{1}{\gamma-1}}\text{,}\hspace{1cm}V'_2=V_2 (T_2/T_0)^{\frac{1}{\gamma-1}}\text{.}$$ This is followed (panel 4) by two reversible isothermal processes in which the two sections are in contact with the thermal bath at temperature $T_0$, which take the two sections back to the initial temperature $T_0$ and volume $V_0/2$ (frame 5). The total work extracted in the process is $$\label{eqA2} W=W_1+W_2= n R T_0 \ln(\frac{V_0}{V'_1})+n R T_0 \ln (\frac{V_0}{V'_2})= n R T_0 \ln [\frac{V_0^2 (T_0^2)^{\frac{1}{\gamma-1}}}{V_1 V_2 (T_1 T_2)^{\frac{1}{\gamma-1}}}]\text{.}$$ Since $V_1+V_2=2 V_0$ and $T_1+T_2=2 T_0$, we obviously have $T_0^2>T_1 T_2$ and $V_0^2>V_1 V_2$, so that a work $W$ larger than zero has been obtained at the expenses of the heat $Q=W$ extracted by a single source (the thermal bath). It can be easily checked that $W=-T_0 \Delta S\cong k T_0 N \mu$, that is each gas molecule contributes with a fraction $\mu k T_0/2$ to the work extracted in the process.

--- abstract: 'We report the Suzaku/XIS results of the Galactic oxygen-rich supernova remnant (SNR), G292.0$+$1.8, a remnant of a core-collapse supernova. The X-ray spectrum of G292.0$+$1.8 consists of two type plasmas, one is in collisional ionization equilibrium (CIE) and the other is in non-equilibrium ionization (NEI). The CIE plasma has nearly solar abundances, and hence would be originated from the circumstellar and interstellar mediums. The NEI plasma has super-solar abundances, and the abundance pattern indicates that the plasma originates from the supernova ejecta with a main sequence of 30–35[*[M]{}*]{}$_{\odot}$. Iron K-shell line at energy of 6.6keV is detected for the first time in the NEI plasma.' author: - 'Fumiyoshi <span style="font-variant:small-caps;">Kamitsukasa</span>, Katsuji <span style="font-variant:small-caps;">Koyama</span>, Hiroshi <span style="font-variant:small-caps;">Tsunemi</span>, Kiyoshi <span style="font-variant:small-caps;">Hayashida</span>, Hiroshi <span style="font-variant:small-caps;">Nakajima</span>, Hiroaki <span style="font-variant:small-caps;">Takahashi</span>, Shutaro <span style="font-variant:small-caps;">Ueda</span>, Koji <span style="font-variant:small-caps;">Mori</span>, Satoru <span style="font-variant:small-caps;">Katsuda</span>, and Hiroyuki <span style="font-variant:small-caps;">Uchida</span>' title: 'Suzaku Discovery of Fe K-Shell Line from the O-Rich SNR G292.0$+$1.8' --- Introduction ============ X-ray spectra of optically-thin hot plasmas in supernova remnants (SNRs) provide key information on the nucleosynthesis during the stellar evolution and the supernova (SN) explosion. Since iron (Fe) is the final product of the major nuclear reaction network in a massive star and a SN, it is particularly important element. So far, Fe abundances of some SNRs have been estimated using the Fe L-shell lines, which are dominant in the low energy band around 1keV. This energy band is complex with other strong emission lines such as oxygen (O) and neon (Ne). Moreover the Fe L-lines consist of many emission lines in this narrow energy band, and hence conventional X-ray detectors such as X-ray CCD cannot resolve these many lines. The Fe K-shell lines at 6.4–6.7keV are simpler and their emission model is more reliable than those of the L-shell lines. However, due to the limited line fluxes and the sensitivity in the energy band above 6keV, the Fe K-shell lines have been reported from only a limited number of X-ray bright SNRs. Core-collapse (CC) SNe eject less Fe than those of Type Ia SNe (@Nomoto1984; @Iwamoto1999, and references therein). Most of the Fe in the center of the progenitor star is collapsed into a central compact object (neutron star or black hole). Thus observationally, Fe K-shell lines would become weaker than those in Type Ia. The flux of Fe may come from the combined effect of the elemental abundances and the thermal state of the gas. Thus the differences between Type Ia and CC SNRs may be related to the initial condition of the circumstellar medium (CSM) and an explosion mechanism. Fe K-shell lines have been reported from other candidates of CC SNRs (e.g., W49B: @Ozawa2009; IC443: @Yamaguchi2009; G349.7+0.2: @Slane2002 and G350.1-0.3: @Gaensler2008; 3C397: @Chen1999). However, some of them are controversial whether they are CC SNRs or not. The most reliable criteria of CC SN are the presence of a neutron star (pulsar), a pulsar wind nebula (PWN), and oxygen-rich (O-rich) knots, because O is largely enhanced in the ejecta of a CC SN. At present, three O-rich remnants, Cas A, PuppisA and G292.0$+$1.8, have been reported in our Galaxy (see @Vink2012, and references therein). Cas A is the most luminous, and hence the Fe K-shell line was firstly detected at 6.6keV (@Tsunemi1986). The spatial distribution is not a simple stratified structure (@Willingale2002). Thus the SN explosion would be highly asymmetric. PuppisA has also the interesting distribution of the ejecta, in that the ejecta is found only in the east, mostly north-east portion (@Winkler1985; @Hwang2008; @Katsuda2010), while a neutron star is propelled in the opposite direction (e.g., @Becker2012 and references therein). Such a recoil between SN ejecta and a neutron star is expected in a recent SN explosion model (@Scheck2006), and would be examined by asymmetric Fe distribution. However, no Fe K-shell line has been observed from PuppisA. For the study of the Fe K-shell line in CC SNRs, we observed G292.0$+$1.8. The discovery of a pulsar and a PWN in G292.0$+$1.8 is further confirmation of a CC SNR (@Hughes2001, [-@Hughes2003]; @Camilo2002; @Gaensler2003). The pulsar is located about 0.9southeast from the geometrical center of the SNR. The distance of G292.0$+$1.8 is estimated to be 6kpc (@Gaensler2003), and the age is likely 2990$\pm$60years (@Winkler2009). The morphology of G292.0$+$1.8 consists of many small knots and the central belt-like filaments running from the east to the west (@Park2002). The central filaments have a normal solar-type composition, suggesting that these are the shocked CSM. @Lee2010 reproduced the intensity profile of the outer CSM region by a slow wind from a red supergiant (RSG) star with the total mass of the wind of 15–40[*[M]{}*]{}$_\odot$. The implied progenitor mass ($M >$ 20[*[M]{}*]{}$_\odot$) was in plausible agreement with previous estimates (@Hughes1994; @Gonzalez2003; @Park2004). The knots have an enhanced metallicity; Si is enhanced in north-northeast, O is enhanced primarily in southeast, Ne is in northwest and southeast, and Mg is in northwest (@Park2002). These knots are probably ejecta origin. The asymmetric distribution of the ejecta elements is interpreted to be non-uniform thermodynamic conditions of the X-ray-emitting ejecta (@Park2007). In spite of these extensive studies, no significant Fe K-shell line has been detected. @Park2004 proposed that the ejecta are strongly stratified by composition and the reverse shock has not propagated to the Fe rich-zone yet. However, the X-ray spectra reported so far are faint in the hard band (except for that of PWN), and hence observed lines have been limited up to sulfur (S) K-shell line. In this paper, we report the Suzaku discovery of an Fe K-shell line in a high temperature plasma ($kT_e$ = 2–3keV) extending to $E$ = 10keV. K-shell lines of argon (Ar) and calcium (Ca) are also reported. Based on the wide band spectral analysis, we discuss the nature of G292.0$+$1.8. We adopt the solar abundances of @Anders1989. Unless otherwise specified, all errors represent 1$\sigma$ confidence levels. Observation and Data Reduction ============================== The Suzaku satellite (@Mitsuda2007) observed G292.0$+$1.8 on 2011 July 22-23 (ObsID: 506062010, PI: K. Koyama) with the X-ray Imaging Spectrometer (XIS, @Koyama2007). The XIS consists of four X-ray CCD cameras placed on the focal plane of the X-Ray Telescope (XRT). All four XRTs are co-aligned to image the same region of the sky. The field of view (FOV) of the XIS is $\times$. Details of Suzaku, the XIS and the XRT are given in @Mitsuda2007, @Koyama2007 and @Serlemitsos2007, respectively. Three of the XIS (XIS0, XIS2, and XIS3) have front-illuminated (FI) CCDs, sensitive in the 0.4–14keV energy band, and the other (XIS1) has a back-illuminated (BI) CCD, with high sensitivity down to 0.2keV. XIS2 has been out of function from 2006 November 9 and a small fraction of the XIS0 area has not been available from 2009 June 23, both due to the damage by micro-meteorites. Data reduction and analysis were performed by the HEAsoft version 6.9. The XIS data were processed with the Suzaku pipe-line software version 2.7. We combined the 3$\times$3 and 5$\times$5 event files. The response functions were generated by using the CALDB 2012-10-09. After removing hot and flickering pixels, we compiled the data using the ASCA-grade 0, 2, 3, 4, and 6 data. We excluded the data obtained at the South Atlantic Anomaly, during the earth occultation, at the elevation angle from the earth rim below 5$^{\circ}$ (night earth) and 20$^{\circ}$ (day earth). The exposure time after these screenings was 44ks. The spectral resolution has been degraded due to the radiation of cosmic particles 5 years after the launch, and restored by the spaced-row charge injection (SCI) technique; the charge traps are filled by the artificially injected electrons through CCD readouts. Details of the SCI technique are given in @Nakajima2008 and @Uchiyama2009. Analysis and Result =================== Combined Analysis of SNR and PWN -------------------------------- Figure \[xis\_image\] (a)-(c) shows X-ray images in the 0.3–0.8, 0.8–6 and 6–8keV energy bands. In the high energy band above 6keV (figure \[xis\_image\] (c)), we see a compact X-ray source at ($\alpha , \delta$) = (, ). This source corresponds to the pulsar/PWN (@Hughes2001, [-@Hughes2003]; @Camilo2002; @Gaensler2003). We make two source spectra, one is from the solid circle (radius of ) as shown figure \[xis\_image\] (c) (here PWN region). The other is from the solid circle (radius of ) excluding the dashed circle (radius of ) as shown in figure \[xis\_image\] (b) (here SNR region). In figure \[xis\_image\] (a), we show the background (BG) region: the whole FOV of the XIS (solid square), excluding the dashed circle (radius of ). This larger radius than that of the SNR region is employed to avoid the contamination from the SNR. We also exclude the region of calibration sources, which are shown by the dashed circles in the XIS corner. For all the regions of PWN, SNR and BG, we separately make the non X-ray background (NXB) spectra using [*[xisnxbgen]{}*]{} in the HEAsoft package (@Tawa2008). We make an X-ray background (XB) spectrum from the BG region by subtracting the relevant NXB. The spectra from the SNR and PWN regions are also made by subtraction of relevant NXBs for these regions. From these spectra, we subtract the XB spectrum assuming the uniform distribution within the FOV of the XIS after the correction of the vignetting effect. The resulting spectra of the SNR and the PWN regions in figure \[SNR-PWN\_spec\] show many emission lines. From the center energies of these lines, we identify them to be O Ly$\alpha$, Ne He$\alpha$, Ne Ly$\alpha$, Mg He$\alpha$, Mg Ly$\alpha$, Si He$\alpha$ and S He$\alpha$. Thus the spectra should be composed of an optically thin hot plasma with the temperature $kT_e \sim$1keV (@Gonzalez2003; @Park2004; @Lee2010). In the spectrum of the SNR region, we find line-like features at 3.1, 3.9 and 6.6 keV, which are likely K-shell transition lines of Ar, Ca and Fe. The Fe line at 6.6 keV is particularly clear. Therefore, in addition to the 1keV plasma (low-$kT_e$ plasma), a higher temperature plasma (high-$kT_e$ plasma) to emit K-shell lines of Ar, Ca and Fe should be prevailing in the SNR region. We call these two plasmas the SNR components. The X-rays of the SNR-components distribute not only in the SNR region but also in the PWN region. On the other hand, according to the observation with Chandra, the PWN is compact of – (@Gonzalez2003), and the spectrum is fitted with a power-law model of $\Gamma\sim$ 1.7 (@Hughes2001), indicating its non-thermal nature. We call this power-law emission the PWN-component. The spectrum extracted from the PWN region contains both the SNR-components and the PWN-component. The spectrum from the SNR region, on the other hand, is contaminated by X-rays of the PWN component due to the large point spread function (beam size) of the Suzaku XRT. We hence simultaneously fit the spectra in the SNR and the PWN regions with the combined model, SNR plus PWN-components. The ancillary response files (ARFs) employed in the fit are generated with [*[xissimarfgen]{}*]{} (@Ishisaki2007). The ARF for the PWN-component is generated from the Chandra image in the 4–7keV band, while that for the SNR-component is made using the thermal emission of the Chandra image (0.6–2.0keV), where the emissions of the PWN-component are excluded. The energy ranges of the PWN and the SNR regions are 1–9keV and 0.6–9keV, respectively. The former energy band is selected because the contamination of the SNR-component in the PWN region becomes large below 1keV. Considering the background level, we also ignore the energy band upper than 9keV for the FI, while 8keV for the BI. @Park2004 and @Gonzalez2003 reported that the spectra are significantly different from position to position. Therefore, the integrated spectrum from the entire SNR cannot be described by any single component model fit. We thus search for a many-components model, starting from one-component model then adding another component one by one, monitoring how much $\chi^2$ is reduced. We use VPSHOCK (@Borkowski2001) to represent multi-$n_e t$ non equilibrium ionization (NEI) plasma, where $n_e$ and $t$ are the plasma density and elapsed time after the shock heating. In order to fine-tune the calibration errors, between XIS0, 1 and 3, the gains and normalizations are set to be independent parameters for each XIS. A 1-VPSHOCK model fails with extreme large $\chi^2 / $d.o.f. of 15006/2071 = 7.25. A 2-VPSHOCK model is largely improved the fitting with $\chi^2 / $d.o.f. of 5400/2060 = 2.62, but still unacceptable. We thus add the third VPSHOCK component (VPSHOCK 1, 2 and 3), then $\chi^2 / $d.o.f. is improved to 5076/2049 = 2.48. Although the decrease of $\chi^2 / $d.o.f. is only $\delta$ = 0.14, the decrease of $\chi^2$ is 324, which is statistically highly significant. In fact, we check the significance using an F-test tool in the Xspec package, then this process is significant with better than 0.01% level. Although errors are large, the best-fit abundances in VPSHOCK 1 and 2 are the same with each other. We hence link the abundances in these two VPSHOCK components. Also abundances of Ni and Ca are linked to Fe and Ar, respectively. Since $n_e t$ of VPSHOCK 3 is 10$^{12-13}$ cm$^{-3}$s$^{-1}$, we replace this model by an APEC model (collisional ionization equilibrium plasma model; CIE). The $\chi^2 / $d.o.f. of this fit is 5184/2062 = 2.51, leaving large residuals in the low energy band. We thus added another APEC component linking the abundances to the APEC component in the 3-component model. This another APEC improve the $\chi^2 / $d.o.f. to 4838/2060 = 2.35, the F-test statistical significant is even better than the previous process. Though this $\chi^2 / $d.o.f is still large from a statistical point of view, its value would be due to non-negligible systematic errors. In fact, we find line-like residuals at about 0.82, 1.2, 1.3 and 1.8keV for both FI and BI, and 1.7keV for BI. The 1.7 and 1.8keV line structures are due to the well-known problem of the response function near the neutral Si K-edge energy at 1.84keV (@Yamaguchi2009). The other line structures would be due to the incompleteness of the VPSHOCK model code. The lines at 0.82, 1.2 and 1.3keV correspond to Fe-L complex (@Uchida2013; @Nakashima2013). We thus added extra 5 Gaussians to compensate these line-like residuals. The normalization factors of these Gaussians are linked between FI and BI, but that of the 1.7 and 1.8keV lines are treated as an independent parameters between FI and BI (@Suchy2011). The calibration errors of the contamination on the optical blocking filter has some problems in the low energy band (http://www.astro.isas.jaxa.jp/suzaku/doc/suzaku\_td/). For a fine-tuning of the cross errors between the FI and BI CCDs, we allow the $N_{\rm H}$ value in the BI CCD to be independent from the FI CCDs. The results of the combined fits by this model are shown in figure \[SNR-PWN\_spec\]. The best-fit parameters are given in table \[SNR-PWN\_para\]. We finally improve the $\chi^2 / $d.o.f. to 2872/2059 = 1.39. The F-test statistical significant is better than 0.01%. Thus, we regard that this model (2-VPSHOCK $+$ 2-APEC $+$ PL $+$ 5-Gaussians) is a reasonable approximation for the SNR and PWN spectra, and apply in the following analysis and discussion. Spatial Analysis of SNR ----------------------- In order to examine spatial asymmetry of the elements in the SNR, we make spatially-resolved spectra. Since the spatial resolution of Suzaku is limited compared to the size of G292.0$+$1.8 ($\sim$$\times$), we crudely divide the SNR into 3: the center, north and south regions as shown in figure \[N-C-S\_region\] by the solid lines. The spectra are given in figure \[each\_region\_spec\]. We find no significant differences, except a hint of Fe K$\alpha$ flux variations among the 3 regions. For quantitative estimate, we fit with the same model and the method given in subsection 3.1, but $N_{\rm H}$s are fixed to the best-fit values in table \[SNR-PWN\_para\]. We obtain nice fit with $\chi^2 / $d.o.f. of 1283/973 = 1.32, 1391/1055 = 1.32 and 1380/1022 = 1.35, for the north, center and south regions, respectively. The best-fit spectra are given in figure \[each\_region\_spec\] by the solid histograms. The best-fit parameters, including abundances, show no spatial-difference within their large errors (typical errors for the abundances are $\sim$50%). The only spatial-difference is found in the Fe abundances of the NEI plasma (2-VPSHOCK), which are 1.7$\pm$0.5, 0.75$\pm$0.22 and 1.0$\pm$0.4, for the north, center and south regions, respectively. The Fe abundance in the NEI plasma may be affected by the flux of the power-law component (PWN component), because the continuum emission of the ejecta is equal or even less than the power-law emission except the north region (see figure \[each\_region\_spec\]). We therefore re-fit the spatially-resolved spectra, changing the normalizations of the PWN component by $\pm$5% of the value in table \[SNR-PWN\_para\], and fixing the photon index of $\Gamma$ = 1.91. The fit gives no essential changes of the Fe abundances from those of the original value. Discussion ========== In the plasma evolution in SNRs, the X-ray emissions have two different components: the CSM (plus ISM) heated by the blast wave, and the ejecta from the progenitor star heated by the reverse shock. In the spectral fitting of G292.0$+$1.8, we find two type plasmas in CIE and NEI conditions; the 2-APEC ($kT_e\sim$0.2 and 0.7keV) and the 2-VPSHOCK ($kT_e\sim$1 and 2.5keV) plasmas. We call these two type plasmas, the low-$kT_e$ plasma and the high-$kT_e$ plasma, respectively. Since the low-$kT_e$ has nearly solar abundances for all elements and the high-$kT_e$ has super-solar abundances (see table \[SNR-PWN\_para\]), these would be the CSM plus ISM and the ejecta origin of a CC SN, respectively. Chandra spectra from many selected regions of bright small spots are described by 1-VPSHOCK model with super-solar abundances (@Park2004), while those from the faint outer-most shell are 1 or 2-VPSHOCK model with sub-solar to solar abundances (@Gonzalez2003; @Lee2010). These results of “no-CIE” plasma are in contrast to the existence of CIE ($\sim$ solar) components in the Suzaku spectra. Since the Chandra results are from selected spots or filaments and those of Suzaku are from the entire SNR region, we suspect that our CIE plasmas are prevailing over the entire SNR except the outer-most shell, while the bright spots are dominated by the VPSHOCK plasma. We discover Fe K-shell line at 6.6 keV in the eject plasma for the first time. The energy indicates that ionization state of Fe is around B-like. This medium ionization state is similar to another young CC SNR, Cas A, but is in contrast to nearly Ne-like states in young well known Type Ia SNRs, Tycho, Kepler, and SN1006. Figure \[ejecta\_abundance\_pattern\] is the abundance plot of the ejecta for O, Ne, Mg, Si, S, Ar and Fe relative to Si (from table \[SNR-PWN\_para\]) together with those of the CC SN model in various progenitor masses (@Woosley1995). We see that the observed abundance pattern is globally in agreement with the model of 30–35[*[M]{}*]{}$_\odot$. These mass range of the progenitor star confirm the previous report of 30–40[*[M]{}*]{}$_\odot$ (@Gonzalez2003), which was estimated based on the limited information of non-detection of the explosively synthesized heavy elements such as Ar, Ca and Fe. One may argue that CC SN of a massive progenitor 30–35[*[M]{}*]{}$_\odot$ may leave a black hole rather than a neutron star. However, other observations show that a neutron star can be still formed from even these massive progenitor stars (@Muno2006). Although @Park2004 and @Gonzalez2003 reported significant spatial variations in the sub-arcmin scale, we find no significant and systematic variations in the spatial scale over arcmin. In fact, the best-fit abundances of most of the heavy elements show no variations within their typical errors of 50%. Nevertheless, we find marginal evidence of spatial variation of Fe in the ejecta; the north region is enhanced compared to that of the center region. Since the position of the neutron star (PWN) is off-set to southeast from the geometrical SNR center (@Park2007), it would be conceivable that Fe from the core region would be ejected to the opposite northwest direction. Our observational result of the Fe variation is marginal to support this off-set effect due to large errors. To establish this kick-off scenario, we need higher quality observations. The best-fit spectral parameters of the PWN, the photon index and unabsorbed flux (4–8keV) are 1.91$\pm$0.03 and (3.80$\pm$0.18)$\times$10$^{-4}$ photons s$^{-1}$ cm$^{-2}$, respectively. The photon index is steeper than that of the pulsar (1.6–1.7, @Hughes2001, [-@Hughes2003]). Probably the index increases as the distance from the central pulsar increases (e.g. @Slane2000). The PWN flux is 52% of the total flux (4–8keV) from the whole SNR ((7.30$\pm$0.06)$\times$10$^{-4}$ photons s$^{-1}$ cm$^{-2}$). This ratio is slightly smaller than 66%, determined with the high spatial resolution observation of Chandra (Hughes et al. 2001). This difference, however, would be within uncertainty range due possibly to the NXB and CXB subtraction[^1], and/or other systematic cross errors including different data reduction processes between Suzaku and Chandra. Thus our simultaneous fitting analysis properly estimates the flux and spectra of both the SNR and the PWN, although the spatial resolution of Suzaku is limited to completely separate the emissions from these two sources. Summary ======= We have analyzed Suzaku/XIS data obtained from G292.0$+$1.8. The results are summarized as follows: 1. We confirm that the thermal X-ray emission from G292.0$+$1.8 consists of two type plasmas in CIE and NEI conditions. 2. The NEI plasma includes K-shell line from B-like Fe, with super solar abundances for O, Ne, Mg, Si, S, Ar, and Fe. Therefore this plasma is likely the ejecta origin of the CC SNR. 3. Using the abundance pattern of the ejecta, we confirm the progenitor mass to be 30–35[*[M]{}*]{}$_{\odot}$. 4. The CIE plasma has nearly solar abundances for all the relevant elements, and hence is likely the CSM and ISM origin. Acknowledgments {#acknowledgments .unnumbered} =============== We thank all members of the Suzaku operation and calibration teams. This work is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 24540229 (Katsuji Koyama), 23000004 (Hiroshi Tsunemi), 23340071 (Kiyoshi Hayashida), 24684010 (Hiroshi Nakajima), 12J01194 (Hiroaki Takahashi), 12J01190 (Shutaro Ueda), 24740167 (Koji Mori), 25800119 (Satoru Katsuda), 11J00535 (Hiroyuki Uchida). S.K. is also supported by the Special Postdoctoral Researchers Program in RIKEN. (160mm,100mm)[figure1.eps]{} (80mm,50mm)[figure2a.eps]{} (80mm,50mm)[figure2b.eps]{} (80mm,50mm)[figure2c.eps]{} (80mm,50mm)[figure2d.eps]{} [@l@l@c@]{} Component & Parameter & Value\ Absorption & $N_{\rm H}$ (10$^{21}$ cm$^{-2}$) & 4.44$\pm$0.19 (FI)\ & & 4.11$\pm$0.19 (BI)\ Power-Law & photon index & 1.91$\pm$0.03\ & Absorbed flux& 3.71$\pm$0.17\ & Unabsorbed flux& 3.80$\pm$0.18\ APEC 1 & $kT_e$ (keV) & 0.17$\pm$0.04\ & O & 0.58$\pm$0.30\ & Ne & 0.74$\pm$0.56\ & Mg & 1.69$\pm$0.27\ & Si & 0.80$\pm$0.16\ & S & 0.83$\pm$0.50\ & Ar (=Ca) & $<$ 1.21\ & Fe (=Ni) & 0.36$\pm$0.09\ & VEM (10$^{11}$ cm$^{-5}$)& 230$\pm$160\ APEC 2 & $kT_e$ (keV) & 0.72$\pm$0.01\ & VEM (10$^{11}$ cm$^{-5}$)& 105$\pm$15\ VPSHOCK 1 & $kT_e$ (keV) & 1.07$\pm$0.19\ & O & 8.5$\pm$3.5\ & Ne & 17.8$\pm$6.1\ & Mg & 6.3$\pm$2.4\ & Si & 3.1$\pm$1.0\ & S & 2.9$\pm$1.4\ & Ar (=Ca) & 5.2$\pm$2.4\ & Fe (=Ni) & 1.7$\pm$0.5\ & $n_e t$ (10$^{11}$ cm$^{-3}$ s) & 3.0$\pm$2.6\ & VEM (10$^{11}$ cm$^{-5}$)& 6.3$\pm$2.3\ VPSHOCK 2 & $kT_e$ (keV) & 2.67$\pm$0.41\ & $n_e t$ (10$^{11}$ cm$^{-3}$ s) & 0.86$\pm$0.19\ & VEM (10$^{11}$ cm$^{-5}$)& 5.2$\pm$2.0\ $\chi^2 / $d.o.f. & & 1.39 (2872/2059)\ (53mm,53mm)[figure3.eps]{} (80mm,50mm)[figure4a.eps]{} (80mm,50mm)[figure4b.eps]{} (80mm,50mm)[figure4c.eps]{} (80mm,80mm)[figure5.eps]{} [^1]: The 4-8 keV band fluxes of the BG region are about 8% and 70% of the whole SNR regions for the data of Suzaku and Chandra, respectively

--- abstract: 'Assuming $V=L(\mathbb{R})+AD$, using methods from inner model theory, we give a new proof of the strong partition property for $\utilde{\d}^2_1$. The result was originally proved in [@KKMW].' author: - | Grigor Sargsyan [^1]\ Department of Mathematics\ Rutgers University\ Hill Center for the Mathematical Sciences\ 110 Frelinghuysen Rd.\ Piscataway, NJ 08854 USA\ http://math.rutgers.edu/$\sim$gs481\ grigor@math.rutgers.edu bibliography: - 'pertitionproperty.bib' date: title: 'An inner model proof of the strong partition property for $\utilde{\delta}^2_1$ [^2] [^3]' --- The main theorem of this note is the following special case of Theorem 1.1 of [@KKMW] originally due to Kechris-Kleinberg-Moschovakis-Woodin. \[main theorem\] Assume $V=L(\mathbb{R})+AD$. Then $\utilde{\delta}^2_1$ has the strong partition property, i.e., $\utilde{\delta}^2_1\rightarrow (\utilde{\delta}^2_1)^{\utilde{\delta}^2_1}$ holds. Our proof uses techniques from inner model theory and resembles Martin’s proof of strong partition property for $\omega_1$ (see [@Jackson]). We expect that it will have other applications and in particular, can be used to show that under $AD^+$, if $\Gamma$ is any $\Pi^1_1$-like [^4] scaled pointclass and $\d=\d(\Gamma)$ then $\d$ has the strong partition property. Our motivation to find a new proof of [Theorem \[main theorem\]]{} comes from a desire to prove Kechris-Martin like results for $\Pi^1_1$-like scaled pointclasses which will settle Question 19 of [@OpenProblems] and most likely, several other questions in the same neighborhood. We are optimistic that inner model theoretic techniques will settle this question and our optimism comes from the fact that the literature is already full of descriptive set theoretic results that have been proved using methods from inner model theory (for instance, see [@Hjorth01], [@GeneralizedHjorth] and [@OIMT]). More importantly for us, recently, Neeman, in [@KMN], found a proof of the Kechris-Martin theorem for $\Pi^1_3$ using techniques from inner model theory. Finally, we believe that our proof can be used to prove the strong partition property for many cardinals $\d=\d(\Gamma)$ where $\Gamma$ has strong closure properties. In fact, we expect that it can be used to prove Theorem 1.1 of [@KKMW] but we certainly haven’t done so. We now start proving [Theorem \[main theorem\]]{}.\ Let $\kappa=\utilde{\delta}^2_1$. By Martin’s theorem (see Theorem 2.31 and Definition 2.30 of [@Jackson]), it is enough to show that $\k$ is $\k$-reasonable, i.e., there is a non-selfdual pointclass $\utilde{\Gamma}$ closed under $\exists^{\mathbb{R}}$ and a map $\phi$ with domain $\mathbb{R}$ satisfying: 1. $\forall x (\phi(x)\subseteq \k\times \k)$, 2. $\forall F: \k\rightarrow \k$, $\exists x \in \mathbb{R} ( \phi(x)=F)$, 3. $\forall \b<\k$, $\forall {\gamma}<\k$, $R_{\b, {\gamma}}\in \utilde{\Delta}$ where $x\in R_{\b, {\gamma}} \iff \phi(x)(\b, {\gamma}) \wedge \forall {\gamma}^\prime <\k (\phi(x)(\b, {\gamma}^\prime) \rightarrow {\gamma}^\prime={\gamma})$ 4. Suppose $\b<\l$, $A\in \exists^{\mathbb{R}}\utilde{\Delta}$, and $A\subseteq R_\b=\{ x : \exists {\gamma}<\k R_{\b, {\gamma}}(x)\}$. Then $\exists {\gamma}_0<\k$ such that $\forall x\in A\exists {\gamma}<{\gamma}_0 R_{\b, {\gamma}}(x)$. Let $\Gamma=\Sigma^2_1$. We claim that $\utilde{\Gamma}$ is as desired and spend the rest of the proof to argue for it. In what follows, we will freely use the terminology developed for analyzing $\H$ of models of $AD^+$. This terminology has been exposited in many places including [@GeneralizedHjorth], [@StrengthPFA1], [@ATHM], [@CMI], [@OIMT] and more recently in [@SSW]. In particular, recall the definitions of suitable premouse, short tree, maximal tree, short tree iterable and etc. Given a suitable premouse $\P$, we let $\d_\P$ be its Woodin cardinal and $\l_\P$ be the least cardinal which is $<\d_\P$-strong in $\P$. Suppose $a\in HC$. We say an $a$-premouse $\Q$ is *good* if 1. $\Q$ is $(\omega, \omega_1)$-iterable, 2. $\Q{\vDash}ZFC-Powerset$+“there are no Woodin cardinals" +“there is a largest cardinal" 3. $\Q$ is full, i.e., for every cutpoint $\xi$ of $\Q$, $Lp(\Q|\xi)\inseg\Q$. If $\Q$ is good then it has a unique $(\omega, \omega_1)$-iteration strategy with Dodd-Jensen property. We let $\Sigma_\Q$ be this strategy. Also, let $\eta_\Q$ be the largest cardinal of $\Q$. Given an iteration tree $\T$ on $\Q$ according to $\Sigma_{\Q}$ with last model $\R$ such that $\pi^{\T}$ exists, we let $\pi_{\Q, \R}:\Q\rightarrow \R$ be the iteration embedding. Notice that because $\Sigma_\Q$ has the Dodd-Jensen property, $\pi^\T$ is independent of $\T$. We say $\Q$ is *excellent* if whenever $\R$ is a $\Sigma_\Q$-iterate of $\Q$ such that $\pi_{\Q, \R}$ is defined $\R$ is good. In this case, we also say that $\Sigma_\Q$ is fullness preserving. Suppose now $\a<\k$ is such that it ends a weak gap (see [@Scales]). We then let $\mathcal{F}(\a, a)=\{ \Q: J_\a(\mathbb{R}){\vDash}``\Q$ is an excellent $a$-premouse"$\}$. Given $a$-premouse $\P$ such that $J_\a(\mathbb{R}){\vDash}``\P$ is suitable and short tree iterable" we let $\mathcal{F}(\a, a, \P)$ be the set of $\Q$ such that in $J_\a(\mathbb{R})$, there is a correctly guided short tree $\T$ on $\P$ with last suitable model $\P^*$ such that for some $\P^*$-cardinal $\eta\leq \l_{\P^*}$, $\Q=\P^*|(\l_{\P^*}^+)^{\P^*}$. Suppose $\a<\k$ ends a weak gap, $a\in HC$ and $\P$ is an $a$-premouse such that $J_\a(\mathbb{R}){\vDash}``\P$ is suitable and short tree iterable". Then $\mathcal{F}(\a, a, \P)\subseteq \mathcal{F}(\a, a)$. Fix $\Q\in \mathcal{F}(\a, a, \P)$. Work in $J_\a(\mathbb{R})$. Let $\T$ be a correctly guided short tree on $\P$ with last suitable model $\P^*$ such that for some $\P^*$-cardinal $\eta< \l_{\P^*}$, $\Q=\P^*|(\l_{\P^*}^+)^{\P^*}$. Because $\P$ is short tree iterable, we have that $\Q$ is $(\omega, \omega_1)$-iterable via a unique iteration strategy $\Sigma$. As the iterations of $\Q$ can also be viewed as iterations of $\P^*$, we have that $\Sigma$ is fullness preserving, implying that $\Q$ is excellent. Notice that if $\b>\a$ is such that $\b$ ends a weak gap and $J_\b(\mathbb{R}){\vDash}``\P$ is suitable and short tree iterable $a$-premouse", then there could be $\Q\in \mathcal{F}(\b, a, \P)$ which is not in $\mathcal{F}(\a, a, \P)$. However, we always have the following easy lemma. \[inclusion\] Suppose $\a<\b<\k$ are two ordinals which end weak gaps and such that $J_\a(\mathbb{R})$ and $J_\b(\mathbb{R})$ both satisfy that $\P$ is suitable and short tree iterable. Then $\mathcal{F}(\a, a, \P)\subseteq \mathcal{F}(\b, a, \P)$. The lemma follows because any iteration tree on $\P$ which is correctly guided and short in the sense of $J_\a(\mathbb{R})$ is also correctly guided and short in the sense of $J_\b(\mathbb{R})$. Next we define $\leq_{\a, a}$ on $\mathcal{F}(\a, a)$ by setting $\Q\leq_{\a, a}\R$ iff there is an iteration tree $\T$ on $\Q$ according to $\Sigma_\Q$ with last model $\S$ such that $\pi^\T$ exists, $\S\inseg\R$ and $\S=\R|(\eta_\S^+)^\R$. Also, let $\leq_{\a, a, \P}=\leq_{\a, a}{\restriction}\mathcal{F}(\a, a)$. As usual, we have that \[directedness\] $\leq_{\a, a}$ and $\leq_{\a, a, \P}$ are directed, and $\leq_{\a, a, \P}$ is dense in $\leq_{\a, a}$. Let then $\M_\infty(\a, a)$ be the direct limit of $(\mathcal{F}(\a, a), \leq_{\a, a})$ under the iteration embeddings $\pi_{\Q, \R}$. Also, let $\M_\infty(\a, a, \P)$ be the direct limit of $(\mathcal{F}(\a, a, \P), \leq_{\a, a, \P})$ under the iteration embeddings $\pi_{\Q, \R}$. It follows from [Lemma \[directedness\]]{} that \[equality of direct limits\] $\M_\infty(\a, a)=\M_\infty(\a, a, \P)$. We let $\pi_{\Q, \infty}:\Q\rightarrow \Q^*\inseg\M_\infty(\a, a, \P)$ be the direct limit embedding[^5]. We can now define $\phi$. First let $S$ be the set of those reals $x$ which code a pair $(y_x, \P_x)$ such that 1. $y_x\in \mathbb{R}$, 2. for some $\a<\k$ ending a weak gap, $J_\a(\mathbb{R}){\vDash}``\P_x$ is suitable and short tree iterable $y_x$-premouse". Clearly $S$ is $\Sigma^2_1$. Also let $f:\kappa^2\rightarrow \k$ be the function given by: for all $(\b, {\gamma})\in \kappa^2$, $f(\b, {\gamma})$ is the least ordinal $\a$ such that $\alpha$ ends a weak gap and $J_\a(\mathbb{R}){\vDash}\max(\b, {\gamma})<\utilde{\d}^2_1$. Notice that $f$ is $\Delta^2_1$ in codes. We define $\phi$ as follows. \[definition of phi\] If $x\not \in S \cap \mathbb{R}$ then let $\phi(x)=\emptyset$. Suppose now $x\in S$. Let $(y_x, \P_x)$ be the pair coded by $x$. Given $\b, {\gamma}<\k$, we let $(\b, {\gamma})\in \phi(x)$ iff letting $\P=\P_x$ and $f(\b, {\gamma})=\a$ then for some $a\in \P$ the following holds in $J_\a(\mathbb{R})$: 1. $\P$ is suitable and short tree iterable, 2. $a$ is the collapse of $x(0)$, 3. $a\subseteq \l_\P\times \l_\P$, 4. there is a correctly guided short tree $\T$ with last model $\S$ such that $\pi_{\P, \S}$ exists and an $\S$-cardinal $\eta$ such that 1. $(\eta^+)^\S<\l^\S$, 2. if $\Q=\S|(\eta^+)^\S$ and $a^\Q=\pi_{\P, \S}(a){\restriction}\eta$ then $(\b, {\gamma})\in \pi_{\Q, \infty}(a^\Q)\cap rng(\pi_{\Q, \infty})$. Given $\a<\Theta$ we let $S_a$ and $\phi_\a$ be what the above definitions give over $J_\a(\mathbb{R})$. The following lemmas establish that $\phi$ is as desired. We start with the following easy lemma. \[easy lemma\] For each $x\in \mathbb{R}$, $\phi(x)=\cup_{\a<\k} \phi_\a(x)$. Suppose $(\b, {\gamma})\in \phi(x)$. Then letting $\a=f(max(\b, {\gamma}), f(\b, {\gamma}))$. Then $(\b, {\gamma})\in \phi_\a(x)$. The other direction is similar. For every $x\in \mathbb{R}$, $\phi(x)\subseteq \kappa\times \kappa$. The claim follows from the fact that for every $\a$ and $a$, $\M_{\infty}(\a, a)\subseteq J_\a(\mathbb{R})$. Suppose $F:\kappa\rightarrow \kappa$. Then there is $x\in dom(\phi)$ such that $\phi(x)=F$. Fix $y$ such that $F\in \H_y$. There is then a suitable $\P$ over $y$ such that $F\in rng(\pi_{\P, \emptyset, \infty})$[^6] Notice that $\pi_{\P, \emptyset, \infty}(\l_\P)=\k$ (see Chapter 8 of [@OIMT]). Let then $a\subseteq \l_\P\times \l_\P$ be such that $\pi_{\P, \emptyset, \infty}(a)=F$ and let $x$ code the pair $(y, \P)$ such that $x(0)=a$. It is then easy to see that $\phi(x)=F$ (use [Lemma \[easy lemma\]]{}). Suppose $\b, {\gamma}<\k$. Let $x\in R_{\b, {\gamma}} \iff \phi(x)(\b, {\gamma}) \wedge \forall {\gamma}^\prime <\k (\phi(x)(\b, {\gamma}^\prime) \rightarrow {\gamma}^\prime={\gamma})$. Then $R_{\b, {\gamma}}$ is $\utilde{\Delta}^2_1$. We have that the following are equivalent: 1. $x\in R_{\b, {\gamma}}$. 2. There is $\a>f(\b, {\gamma})$ such that $J_{\a}(\mathbb{R}){\vDash}``x\in dom(\phi_\a)$ and ${\gamma}$ is the unique ordinal such that $(\b, {\gamma})\in \phi_\a(x)"$, 3. For all $\a>f(\b, {\gamma})$, $J_{\a}(\mathbb{R}){\vDash}``x\in dom(\phi_\a)$ and ${\gamma}$ is the unique ordinal such that $(\b, {\gamma})\in \phi_\a(x)"$ Clearly 1 implies 2 and 3. Also, that 3 implies 1 is rather straightforward. We show that 2 implies 1. Fix then $\a>f(\b, {\gamma})$ such that $J_{\a}(\mathbb{R}){\vDash}``x\in dom(\phi_\a)$ and ${\gamma}$ is the unique ordinal such that $(\b, {\gamma})\in \phi_\a(x)"$. Let $(y, \P)$ be the pair coded by $x$ and $a\in \P$ the transitive collapse of $x(0)$. Working in $J_\a(\mathbb{R})$, let $\T$ be a correctly guided short tree on $\P$ with last model $\S$ such that $\pi_{\P, \S}$ exists and an $\S$-cardinal $\eta$ such that 1. $(\eta^+)^\S<\l^\S$, 2. if $\Q=\S|(\eta^+)^\S$ and $a^\Q=\pi_{\P, \S}(a){\restriction}\eta$ then $(\b, {\gamma})\in \pi_{\Q, \infty}(a^\Q)\cap rng(\pi_{\Q, \infty})$. Suppose now there is some $\xi$ such that for some ${\gamma}^\prime$, $(\b, {\gamma}^\prime)\in \phi_\xi(x)$. Working in $J_\xi(\mathbb{R})$, let $\T^*$ be a correctly guided short tree on $\P$ with last model $\S^*$ such that $\pi_{\P, \S}$ exists and an $\S^*$-cardinal $\nu$ such that 1. $(\nu^+)^\S<\l^\S$, 2. if $\R=\S|(\nu^+)^\S$ and $a^\R=\pi_{\P, \S^*}(a){\restriction}\nu$ then $(\b, {\gamma}^\prime)\in \pi_{\R, \infty}(a^\R)\cap rng(\pi_{\R, \infty})$. Without loss of generality assume that $\xi>\a$. We then have that $J_\xi(\mathbb{R}){\vDash}``\S$ and $\S^*$ are suitable and short tree iterable". Work now in $J_\xi(\mathbb{R})$. We can then find $\S^{**}$ which is a suitable correct iterate of both $\S$ and $\S^*$. Notice that since $\S^{**}$ is suitable, the iteration embeddings $i : \S|(\l_\S^+)^\S\rightarrow \S^{**}|(\l_{\S^{**}}^+)^{\S^{**}}$ and $j: \S^*|(\l_{\S^*}^+)^{\S^*}\rightarrow \S^{**}|(\l_{\S^{**}}^+)^{\S^{**}}$ exists. Suppose now that ${\gamma}\not ={\gamma}^\prime$. Let $(\bar{\b}, \bar{{\gamma}}, \bar{{\gamma}}^\prime)\in \S^{**}$ be such that letting $\zeta=max(i(\eta_\Q), j(\eta_\R))$ and $\W=\S^{**}|(\zeta^+)^{\S^{**}}$, $\pi_{\W, \infty}(\bar{\b}, \bar{{\gamma}}, \bar{{\gamma}}^\prime)=(\b, {\gamma}, {\gamma}^\prime)$. It then follows that $(\bar{\b}, \bar{{\gamma}})\in i(\pi^\T_{\P, \S}(a))$ and $(\bar{\b}, \bar{{\gamma}}^\prime)\in j(\pi^{\T^*}_{\P, \S^*}(a))$. However, $i\circ \pi^\T_{\P, \S}=j\circ \pi^{\T^*}_{\P, \S^*}$, implying that $i(\pi^\T_{\P, \S}(a))=j(\pi^{\T^*}_{\P, \S^*}(a))$ and that $S^{**}{\vDash}(\bar{b}, \bar{{\gamma}})\in i(\pi^\T_{\P, \S}(a)) \wedge (\bar{b}, \bar{{\gamma}}^\prime)\in i(\pi^\T_{\P, \S}(a))$. Let now $(\tau, \tau^*)\in \Q$ be such that $\pi_{\Q, \infty}(\tau, \tau^*)=(\b, {\gamma})$. By elementarity of $i$, we then get that $\S{\vDash}``$there is $\tau^{**}\not =\tau^*$ such that $(\tau, \tau^{**})\in \pi_{\P, \S}(a)"$. Fix such a $\tau^{**}$ and let $\varsigma \in (\tau^{**}, \l_\S)$ be an $\S$-cardinal. Then letting $\Q^*=\S|(\varsigma^+)^\S$ we have that $(\b, \pi _{\Q^*, \infty}(\tau^{**}))\in \phi_\a(x)$ and $\pi_{\Q^*, \infty}(\tau^{**})\not = {\gamma}$, contradiction. The next lemma finishes the proof. Suppose $\b<\l$, $A\in \utilde{\Delta}^2_1$ and $A\subseteq R_\b=\{ x : \exists {\gamma}<\k R_{\b, {\gamma}}(x)\}$. Then $\exists {\gamma}_0<\k$ such that $\forall x\in A\exists {\gamma}<{\gamma}_0 R_{\b, {\gamma}}(x)$. Let $f:A\rightarrow \k$ be defined by $f(x)=\nu$ if $\nu$ is the least such that $\nu$ ends a weak gap and $J_\nu(\mathbb{R}){\vDash}x\in R_\b$. Then $f$ is $\utilde{\Sigma}_1$ over $J_{\k}(\mathbb{R})$ and hence, as $\k$ is $\mathbb{R}$-admissible, $f$ is bounded. [^1]: This material is partially based upon work supported by the National Science Foundation under Grant No DMS-0902628. Part of this paper was written while the author was a Leibniz Fellow at the Mathematisches Forschungsinstitut Oberwolfach. [^2]: 2000 Mathematics Subject Classifications: 03E15, 03E45, 03E60. [^3]: Keywords: Mouse, inner model theory, descriptive set theory, hod mouse. [^4]: i.e., closed under $\forall^{\mathbb{R}}$ and non-selfdual [^5]: We drop $\a$ and $a$ from our notation as the embedding doesn’t depend on them [^6]: Recall the direct limit construction that converges to $\H|\Theta$. Here $\pi_{\P, \emptyset, \infty}$ is the direct limit embedding given by $\emptyset$-iterability embeddings. For more details see either of the aforementioned papers.

--- abstract: 'We consider stability properties of spherically symmetric spacetimes of stars in metric $f(R)$ gravity. We stress that these not only depend on the particular model, but also on the specific physical configuration. Typically configurations giving the desired $\gamma_{\rm PPN} \approx 1$ are strongly constrained, while those corresponding to $\gamma_{\rm PPN} \approx 1/2$ are less affected. Furthermore, even when the former are found strictly stable in time, the domain of acceptable static spherical solutions typically shrinks to a point in the phase space. Unless a physical reason to prefer such a particular configuration can be found, this poses a naturalness problem for the currently known metric $f(R)$ models for accelerating expansion of the Universe.' author: - Kimmo Kainulainen - Daniel Sunhede title: 'On the stability of spherically symmetric spacetimes in metric $f(R)$ gravity' --- Introduction {#sec:Intro} ============ The observation that the expansion of the Universe appears to be accelerating [@astier; @spergel] has provoked discussion of a number of models for extended gravity involving nonlinear interactions in the Ricci scalar $R$: $$S = \frac{1}{2\kappa} \int {\rm d}^{4}x \sqrt{-g} [R + f(R)] + S_{\rm m} \,. \label{eq:action}$$ Here $\kappa \equiv 8\pi G$, $S_{\rm m}$ is the usual matter action and $f(R)$ describes the new physics in the gravity sector; setting $f(R) = -2\Lambda$ corresponds to the canonical Einstein-Hilbert action in General Relativity (GR) with a cosmological constant $\Lambda$. The idea is that if cosmological data could be fitted by the use of some nontrivial function $f(R)$, one might avoid the theoretical difficulties and fine-tuning issues related to a pure cosmological constant. However, it has been shown that when understood as a [*metric*]{} theory, the action (\[eq:action\]) can lead to predictions that are not consistent with Solar System measurements [@chiba; @erickcek; @Kainulainen:2007bt]. While observations require a parameter $|\gamma_{\rm PPN}-1|\lesssim 10^{-4}$ [@obsongamma] in the Parametrized Post-Newtonian (PPN) formalism, the value predicted in metric $f(R)$ theories is typically $\gamma_{\rm PPN} \approx 1/2$. This is certainly the case [@chiba; @erickcek; @Kainulainen:2007bt] for the first simple $f(R)$ models suggested in the literature [@vollick; @carroll]. It is however difficult to make a completely generic prediction of this result and there have been many arguments both for [@chiba; @erickcek; @Kainulainen:2007bt; @metricFail] and against [@metricPass; @Zhang:2007ne; @Hu:2007nk; @Nojiri:2007as; @Nojiri:2007cq; @Clifton:2008jq] metric $f(R)$ gravity failing Solar System tests. In particular, more complicated $f(R)$ functions have since been suggested which claim to yield $\gamma_{\rm PPN} \approx 1$ [@Zhang:2007ne; @Hu:2007nk; @Nojiri:2007as; @Nojiri:2007cq]. In this paper we set up the conditions which the function $f(R)$ must fulfill, so that a solution to the field equations which is compatible with Solar System observations exists, in particular with $\gamma_{\rm PPN} \approx 1$. However, we will also argue that the mere existence of such a solution does not imply that a model is consistent with observations. Since metric $f(R)$ gravity is a fourth-order theory, spacetime geometry and matter are not in as strict a correspondence as in General Relativity; depending on the boundary conditions on the metric, a given matter distribution can be consistent with different static spacetimes and with different values of $\gamma_{\rm PPN}$. Moreover, no physical principle tells us that only the boundary conditions corresponding to $\gamma_{\rm PPN} \approx 1$ solutions should be acceptable. The question is then, which solutions are the most natural ones? How plausible is it that the collapse of a protostellar dust cloud leads to the formation of the spacetime observed in the Solar System? To answer these questions one would ideally like to study the full dynamical collapse, and given a domain of reasonable initial conditions, determine the attractor in the configuration space of possible solutions. This computation is beyond the scope of this paper however. We will instead approach the problem by studying how the time stability argument constrains the phase space of configurations with the desired properties. The conditions that a generic metric $f(R)$ model should satisfy in order to yield acceptable solutions are: first, the Ricci scalar should closely follow the trace of the energy-momentum tensor inside a changing matter distribution, where at the same time the dimensionless quantities $f/R$ and $F \equiv \partial f/\partial R$ should remain much smaller than 1 at regions of high density. Second, the effective mass term $m^2_R$ for a perturbation in the Ricci scalar should be positive in order to assure that the GR-like $\gamma_{\rm PPN} \approx 1$ configurations are stable in time. Third, the mass $m_R^2$ should remain small so that a finite domain of static, GR-like configurations exist. This is guaranteed if $m_R^2 \lesssim 1/r_{\odot}^2$, where $r_{\odot}$ is the radius of the Sun. If this last condition is not fulfilled, the domain of GR-like configurations shrinks to essentially a point in the phase space, while a continuum of equally good, but observationally excluded, solutions still exists. In such a case the credibility of the theory requires an argument as to why the particular GR-like configuration should be preferred. None of the models so far proposed in the literature, including Refs. [@Hu:2007nk; @Nojiri:2007as; @Nojiri:2007cq], satisfy all of these constraints, and we also failed to construct a model that would. Largely this failure comes from the difficulty to keep both the function $F$ and $m_R^2 \sim 1/(3F_{,R})$ small simultaneously when the Ricci scalar follows the matter distribution, $R \approx \kappa\rho$. It should be noted that the above considerations only apply for the [*desired*]{} GR-like $\gamma_{\rm PPN} \approx 1$ configurations when a model is tuned to mimic a (very small) cosmological constant. Metric models with a true cosmological constant plus some additional sufficiently small $f(R)$ correction can be perfectly fine. Thus the above arguments do not exclude generic $f(R)$ modifications, such as might arise from quantum corrections, to the Einstein-Hilbert action. One should also note that it is in general easy to construct stable attractor solutions yielding $\gamma_{\rm PPN} \approx 1/2$ in metric $f(R)$ theory. It is the precision data from the Solar System which makes these solutions unacceptable. The paper is organized as follows. We start by reviewing the Solar System constraints in Sec. \[sec:ppn\]. We consider the Dolgov-Kawasaki time instability [@Dolgov:2003px] in Sec. \[sec:timestab\] and discuss the corresponding stability criterion for spherically symmetric configurations. Section \[sec:paltrack\] considers static configurations and the possibility for metric $f(R)$ gravity to follow stable, GR-like solutions that are compatible with Solar System constraints. We find that the condition for finding a finite *domain* of boundary conditions giving rise to a GR-like metric is nearly orthogonal to the time stability condition. Finally, Sec. \[sec:summary\] contains our conclusions and discussion. Solar System constraints and the Palatini track {#sec:ppn} =============================================== Let us begin by reviewing the main constraints from the Solar System observations on static solutions in metric $f(R)$ gravity. Varying the action (\[eq:action\]) with respect to the metric gives the equation of motion: $$\begin{aligned} (1+F) R_{\mu \nu} - \frac{1}{2} (R+f) g_{\mu \nu} && \nonumber \\ - \nabla_\mu\nabla_\nu F+ g_{\mu \nu}\Box F & = & \kappa T_{\mu \nu} \,, \label{eq:eom}\end{aligned}$$ where $F \equiv f_{,R} = \partial f/\partial R$ and $\Box = g^{\mu\nu}\nabla_\mu\nabla_\nu$. Taking the trace of this equation one finds: $$\Box F - \frac{1}{3}(R(1-F) + 2f) = \frac{1}{3}\kappa T \,. \label{eq:trace}$$ If $F\rightarrow 0$ and $f\rightarrow R$ this equation reduces to the standard algebraic GR relation between the Ricci scalar and the trace of the energy-momentum tensor $T$. In a generic metric $f(R)$ theory $R$ is a dynamical variable however, and the theory may exhibit an instability which we will discuss in the next section. Assuming a static, spherically symmetric metric $g_{\mu \nu}$, $$ds^2 \equiv g_{\mu \nu} x^{\mu} x^{\nu} = -e^{A(r)}{\rm d}t^2 + e^{B(r)}{\rm d}r^2 + r^2{\rm d}\Omega^2 \,, \label{eq:metric}$$ the full field equations (\[eq:eom\]) reduce to the following source equations for the metric functions $A$ and $B$ in the weak field limit (to first order in small quantities): $$\begin{aligned} (rB)' & \approx & \kappa \rho r^2 \bigg( 1 - \frac{1}{3}\bigg[ \frac{1+3F}{1+F} - \frac{R}{\kappa\rho}\frac{1 + \frac{F}{2}+\frac{f}{2R}}{1+F} \bigg] \bigg) \nonumber \\ & & - \gamma rA' \,, \label{eq:sourceB} \\ A' & \approx & \frac{1}{1+\gamma} \left( \frac{B}{r} - \frac{r}{2(1+F)}\bigg[ FR - f + \frac{4}{r}F' \bigg] \right) \,,\qquad \label{eq:sourceA}\end{aligned}$$ where $\gamma \equiv rF'/2(1+F)$, a prime refers to a derivative with respect to $r$, and we have neglected pressure so that $T \approx -\rho$. The parameter $\gamma$ and the terms in the square brackets highlight the deviation from General Relativity. The value of $\gamma_{\rm PPN} \approx - B/A$ far away from a gravitational source depends on the continuous evolution of $A$ and $B$ throughout the Sun. It is particularly sensitive to the evolution through the core where the density is the highest. Hence, to obtain $\gamma_{\rm PPN} \approx 1$ and the correct gravitational strength in the Solar System, the only solution, not obviously dependent on an enormous amount of fine-tuning [^1], is that the extra terms in Eqns. (\[eq:sourceB\]-\[eq:sourceA\]) must remain small throughout the interior of the Sun. Now, if the extra terms can be neglected in the $B'$ equation, one finds that $B \lesssim 10^{-6}$ throughout the interior of the Sun [@Kainulainen:2007bt]. It then becomes clear from the $A'$ equation that $f/R$, $F$, and $rF'$ need to be very small compared to 1. However, to make the correction vanish in the $B'$ equation one in addition needs to require that the Ricci scalar traces the matter density $R/\kappa \rho \approx 1$. So, barring perhaps some fantastic fine-tunings, the only possibility is that one finds a configuration for which: $$F \ll 1 \,, \quad f/R \ll 1 \,, \quad {\rm and} \quad R \approx \kappa \rho \,. \label{eq:conditions}$$ Note that this limit was also discussed in Ref. [@Hu:2007nk]. Here we see that the above conditions are a necessary requirement for fulfilling the local gravity constraints. In GR the equation $R = -\kappa T$ is of course exact (remember that we are neglecting pressure throughout so that $T\approx -\rho$), but this is in general very difficult to arrange in a metric $f(R)$ model. The problem lies in the dynamical nature of the Ricci scalar in metric $f(R)$ gravity. To see this, consider the static trace equation (\[eq:trace\]) in the weak field limit: $$\begin{aligned} F'' + \frac{2}{r}F' & = & \frac{1}{3} \big(R - \kappa \rho - FR + 2f \big) \nonumber \\ & \equiv & \frac{1}{3} \big(\Sigma(F) - \kappa \rho \big) \,, \label{eq:traceWeak}\end{aligned}$$ where we have again assumed that pressure is negligible. Assume now that $F=F_0$ at the center of the Sun. If the nonlinear term $\Sigma(F)$ is small compared to $\kappa \rho$, then the solution for $F$ becomes: $$F(r) = - \int_0^r {\rm d}r' \frac{2G m(r')}{3r'^2} + F_0 \,, \label{eq:Fsol}$$ where $m(r) \equiv \int_0^r {\rm d}r' 4\pi r'^2 \rho$. In the case of the Sun this implies that $F$ evolves only little: $|F(r)-F_0| \lesssim 10^{-6}$ [@Kainulainen:2007bt]. The solution (\[eq:Fsol\]) is in general not compatible with $R \approx \kappa \rho$ as required by the conditions (\[eq:conditions\]). Let us now set $\Sigma(F_0) = \kappa \rho_0$ at $r = 0$. If the gradients somehow remain small throughout the evolution, then the solution follows the Palatini trace equation: $$\Sigma(F) = R(1-F) + 2f = 8\pi G \rho \,. \label{eq:tracePal}$$ For small $F$ and $f/R$ this evolution [*would*]{} be consistent with the condition $R\approx \kappa \rho$ and, since we are following the Palatini track, give $\gamma_{\rm PPN} \approx 1$ (see [*e.g. *]{}[@solarPal]). Whether such a solution actually exists is more difficult to prove. However, one can study under which conditions such a solution, if it exists, would be an attractor and whether it would also be sufficiently stable in time. Finally, let us note that if $\Sigma(F) \ll \kappa \rho$ so that $F$ is given by Eqn. (\[eq:Fsol\]), small $F$ and $f/R$ result in a different class of solutions with $R/\kappa \rho \ll 1$. In this case the field Eqns. (\[eq:sourceB\]-\[eq:sourceA\]) reduce to $$\begin{aligned} (rB)' & \approx & \frac {2}{3}\kappa \rho r^2 \,, \label{eq:sourceB2} \\ A' & \approx & \frac{B}{r} - 2F' \,. \label{eq:sourceA2}\end{aligned}$$ It is easy to show that together with Eqn. (\[eq:Fsol\]) these give $\gamma_{\rm PPN} \approx 1/2$. Time stability {#sec:timestab} ============== Let us first consider the time stability of spherically symmetric configurations in generic $f(R)$ models. Perturbing around some arbitrary configuration, $R(r) \rightarrow \widetilde{R}(r,t) = R(r) + \delta R(r,t)$, and expanding to first order in $\delta R$, $\delta R'$ and $\dot{\delta R}$, where the prime refers to a derivative with respect to $r$ and the dot to a derivative with respect to $t$, one can write the trace equation (\[eq:trace\]) in the following form in the weak field limit: $$\begin{aligned} (\partial_t^2 - \vec{\nabla}^2) \delta R & = & - m_R^2\delta R + 2\frac{F_{,RR}}{F_{,R}}R'\delta R' \nonumber \\ & & + \vec{\nabla}^2 R + \frac{1}{3F_{,R}}\Delta + \frac{F_{,RR}}{F_{,R}} (R')^2 \,, \quad \label{eq:tracePert}\end{aligned}$$ where $\Delta \equiv -\kappa T - R(1-F) - 2f$ and $$m_R^2 \equiv \frac{1}{3F_{,R}}(1-F-\varepsilon) \,, \label{eq:mR}$$ with $$\begin{aligned} \varepsilon \phantom{.} \delta R & \equiv & R(\widetilde{F}-F) - 2(\widetilde{f} - f) + \bigg(1-\frac{\widetilde{F}_{,R}}{F_{,R}} \bigg)\Delta \nonumber \\ && {}+ 3 F_{,RR}\bigg(\frac{\widetilde{F}_{,RR}}{F_{,RR}} - \frac{\widetilde{F}_{,R}}{F_{,R}}\bigg)(R')^2 \,. \label{eq:epsilon}\end{aligned}$$ Here a tilde is used to denote that a quantity is perturbed, [*i.e. *]{}it is a function of $\widetilde{R}(r,t)$ as opposed to the background value $R(r)$. Assuming that the configuration $R(r)$ we are perturbing around is a solution to the static equation, the second line in Eqn. (\[eq:tracePert\]) drops out. Moreover, for most cases the gradient term proportional to $\delta R'$ is completely negligible inside a stellar object and can be dropped as well. See Fig. \[fig:gradient\] for some examples. The behavior of the perturbation around a static, spherically symmetric solution is thus governed by the equation ![The gradient term proportional to $\delta R'$ in Eqn. (\[eq:delR\]), normalized to $m_R^2/r_{\odot}$, for various $f(R)$ models ($R = \kappa \rho$): $-\mu^4/R$ (solid blue), $-\mu^4/R + \alpha R^2/\mu^2$ (dashed green), Hu & Sawicki (dot-dashed red) [@Hu:2007nk], and $\alpha R \log{(R/\mu^2)}$ (solid black). The actual density profile used in all figures corresponds to the known density profile of the Sun with a central density of $150$ g/cm$^3$ and with a roughly exponential dependence on $r$. We have also superimposed a constant dark matter distribution on the profile of the Sun with $\rho_{\rm DM} = 0.3$ GeV/cm$^3$.[]{data-label="fig:gradient"}](Figs/figGrad.eps){width="8cm"} ![The parameter $F$ given as a function of the radius for various $f(R)$ models ($R = \kappa \rho$): $-\mu^4/R$ (solid blue), $-\mu^4/R + \alpha R^2/\mu^2$ (dashed green), Hu & Sawicki (dot-dashed red) [@Hu:2007nk], and $\alpha R \log{(R/\mu^2)}$ (solid black).[]{data-label="fig:F"}](Figs/figF.eps){width="8cm"} $$\begin{aligned} (\partial_t^2 - \vec{\nabla}^2) \delta R & = & - m_R^2 \delta R \,. \label{eq:delR}\end{aligned}$$ Note that the mass $m_R^2$ only depends on the background value of the Ricci scalar $R(r)$. Table \[table1\] and \[table2\] show the components of $m_R^2$ in some particular models and the corresponding parameter values used in all figures are displayed in Table \[table3\]. As discussed in the previous section, $F$ needs to be small compared to one for GR-like configurations (see Fig. \[fig:F\], we will discuss this constraint further below). When this is the case one typically finds that also $\varepsilon \ll 1$, so that $m_R^2 \approx 1/3F_{,R}$. It then follows that if $F_{,R} < 0$, then $m_R^2 < 0$ and the coefficient of $\delta R$ is negative for the configuration in question, so that system exhibits an instability. This is the instability first found by Dolgov and Kawasaki in the context of an $f(R) = -\mu^4/R$ model [@Dolgov:2003px] (see Ref. [@Faraoni:2006sy] for a more general case). It is important to note that the instability depends not only on the model, but also on the particular configuration. Certain configurations in a given model are more stable than others and the instability may even vanish in some cases. The nature of the instability is most transparent in the special case with constant curvature. Then $m_R$ is a constant and one can obtain an exact solution for $\delta R(r,t)$. Expanding $\delta R$ in Fourier modes, one finds that a mode with wave vector $\vec{k}$ has the time dependence $$\begin{aligned} \delta R_k(\vec{k},t) \sim e^{\pm i\sqrt{k^2 + m_R^2}t} \,, \label{eq:delRsol}\end{aligned}$$ so that for negative $m_R^2 \sim 1/3F_{,R}$, all modes with $k < |m_R|$ are unstable. This does not necessarily rule out a model however. If for example $|m_R| \sim H_0$, then the instability time is much longer than the lifetime of the Solar System and the model is safe. Moreover, whenever $|m_R|^{-1}$ is much larger than the size of the physical system under consideration, only modes corresponding to scales much larger than the system are unstable and this can not alter its local geometry. Now, assume that we have a GR-like solution, such that $R \sim -\kappa T \approx 8\pi G \rho$. One then has $$\frac{R}{\mu^2} \sim 10^{29} \left(\frac{\Lambda}{\mu^2} \right) \left(\frac{\rho}{{\rm g}/{\rm cm}^3}\right) \,, \label{eq:Rmagn}$$ where we have used $\Lambda \approx 0.73 \kappa \rho_{\rm crit}$. Hence, for a pure $-\mu^4/R$ model the mass squared is on the order of $$m_R^2 \sim -(10^{-26} \textrm{ s})^{-2}\left(\frac{\Lambda}{\mu^2} \right)^2 \left(\frac{\rho}{{\rm g}/{\rm cm}^3}\right)^3 \,.$$ This system is violently unstable at normal densities for all scales larger than $\sim 10^{-18}{\rm~m}$, if $\mu$ is fixed to account for the present accelerating expansion of the Universe. As was pointed out by Dick [@Dick:2003dw] and later discussed by Nojiri & Odintsov [@Nojiri:2003ft], adding a conformal term $\alpha R^2/\mu^2$ can stabilize this system; for $f(R) = -\mu^4/R + \alpha R^2/\mu^2$ the previous approximation for the mass reads: $$m_R^2 \sim -\frac{R^3}{6\mu^4}\left(\frac{1}{1-\alpha R^3/\mu^6}\right) \sim \frac{\mu^2}{\alpha} \,, \label{eq:mRmagn}$$ where the last step assumes that $\alpha R^3/\mu^6 > 1$. If $\alpha \sim 1$ this may be true even for $R \sim \mu^2$ so that one always finds a very small positive mass $m_R^2 \sim \mu^2/\alpha \sim (10^{18}\textrm{ s})^{-2}$. However, the above stabilization mechanism runs into problems with the conditions in Eqn. (\[eq:conditions\]). Indeed, for $\alpha \sim 1$ and $R \approx \kappa \rho \gg \mu^2$ one has $$F = \frac{\mu^4}{R^2} + \alpha \frac{R}{\mu^2} \gg 1\,,$$ so that the configuration would clearly not be GR-like [^2]. The problem is that changing $F$ modifies the effective strength of the gravitational constant $G_{\rm eff} = G/(1+F)$, which controls the buildup of the gravitational potential inside the star. In fact, for $\alpha \sim 1$ the effect is so strong that it would weaken the gravitational force so much as to prohibit the growth of any density contrasts much above the critical density. This argument can be turned around to a constraint: in order for the function $F$ to remain small inside the densest objects we have reasonably accurate information on, the neutron stars, one has to have $\alpha \kappa \rho_{\rm nucl}/\mu^2 \ll 1$, where $\rho_{\rm nucl}$ is the nuclear density [^3]. Since $\mu^2 \sim \kappa \rho_{\rm crit}$ we find that $$\alpha \lesssim \frac{\rho_{\rm crit}}{\rho_{\rm nucl}} \sim 10^{-45} \,.$$ This is quite a stringent constraint, but it does not rule out the model based on the required time stability. Indeed, for example with $\alpha = 10^{-47}$ one has $\alpha (\kappa \rho )^3/\mu^6 = 1$ when $\rho \sim 10^{16}\rho_{\rm crit}$. For any density higher than this value, the system is stable in time with a very [*large*]{} positive mass squared given by the formula in Eqn. (\[eq:mRmagn\]): $m_R^2 \sim \mu^2/\alpha \sim (10^{-6} {\rm~s})^{-2}$. There is a caveat to this argument however, since for these parameters the gradient term proportional to $\delta R'$ becomes very large inside the Sun (see Fig. (\[fig:gradient\])) and the simplified equation (\[eq:delR\]) can no longer be trusted. The complete mass squared function for the model with a fine-tuned conformal $\alpha R^2$ term, using the exact expression (\[eq:mR\]), is shown in Fig. \[fig:mR2\] (dashed green curve). The lower panel displays the absolute value $|m_R^2|$ and the upper panel the sign of $m_R^2$. In the model at hand the mass would remain large and *positive* throughout the entire interior of the Sun, which is the necessary condition for time stability. The GR-like configuration does become unstable at low densities, but this would not necessarily change the value of $\gamma_{\rm PPN}$ in the Solar System. ![The mass squared $m_R^2$ in units $1/r_{\odot}^2$ for various $f(R)$ models ($R = \kappa \rho$): $-\mu^4/R$ (solid blue), $-\mu^4/R + \alpha R^2/\mu^2$ (dashed green), Hu & Sawicki (dot-dashed red) [@Hu:2007nk], and $\alpha R \log{(R/\mu^2)}$ (solid black). The horizontal solid gray line corresponds to the limit $1/r_{\odot}^2$. For the $-\mu^4/R$ and $\alpha R\log{(R/\mu^2)}$ models we have also plotted $1/3F_{,R}$ with dotted blue and dotted black lines, respectively. The upper panel displays the corresponding sign of the mass squared where we have excluded the $-\mu^4/R$ model for which $m_R^2$ is strictly negative.[]{data-label="fig:mR2"}](Figs/figM2.eps){width="8cm"} $\displaystyle f(R)$ Parameter values ------------------------------------------------------------------ -- -- ------------------------------------------------------------------------------------------- $\displaystyle -\frac{\mu^4}{R}$ $\displaystyle \mu^2 = 4\Lambda/\sqrt{3}$ $\displaystyle -\frac{\mu^4}{R} + \alpha \frac{R^2}{\mu^2}$ $\displaystyle \mu^2 = 4\Lambda/\sqrt{3} \,, \quad \alpha = 10^{-47}$ $\displaystyle -\mu^2 \frac{c_1(R/\mu^2)^n}{c_2(R/\mu^2)^n + 1}$ $\displaystyle \mu^2 = (8315 {\rm~Mpc})^{-1} \,, \quad n = 1\,,$ $\displaystyle c_1/c_2 = 6 \times 0.76/0.24 \,,$ $\displaystyle \phantom{\Bigg|}c_1/c_2^2 = 10^{-6}\times41^{n+1}/n $ $\displaystyle \alpha R \log{\frac{R}{\mu^2}}$ $\displaystyle \mu^2 = 4\Lambda e^{(1-\alpha)/\alpha} \,, \quad \alpha = 1/\log{10^{32}}$ : Chosen parameter values for the different $f(R)$ models in Figs. \[fig:gradient\]-\[fig:mR2\], where $\Lambda = 0.73 \kappa \rho_{\rm crit}$. For the third model, originally suggested in Ref. [@Hu:2007nk], we have used values given in the original publication. Note that a value $n=4$, which was also discussed in [@Hu:2007nk], would result in an even larger value of $m_R^2$ in this scenario.[]{data-label="table3"} Fig. \[fig:mR2\] also displays $m_R^2$ for several other models (for parameter values used in each model see Table \[table3\]): the solid blue line represents the simple $-\mu^4/R$ model, which has a very large negative mass inside the Sun, and the dash-dotted red curve shows the mass function in a model by Hu & Sawicki (HS) [@Hu:2007nk]. The HS model fulfills the conditions (\[eq:conditions\]) by construction, and its very large positive mass guarantees time stability. The fact that $m_R^2$ becomes negative around $r \sim 6r_{\odot}$ in the HS model is caused by the $\varepsilon$ term in the complete expression (\[eq:mR\]), but this does not necessarily have any effect on $\gamma_{\rm PPN}$. Moreover, this feature is sensitive to the particular form of the exterior density profile (where we have neglected for example the Solar wind) and it is not important for our main results. Overall one sees that the expression $m_R^2 \approx 1/3F_{,R}$ is a very good approximation for the first three models described in Table \[table1\], except for a small region around $r \sim 6r_{\odot}$ where the $\varepsilon$ term may come into play. For the logarithmic model this approximation is only good for very low densities and we will discuss this in more detail in section \[sec:paltrack\]. We can summarize this section as follows: the stability of a static, spherically symmetric GR-like configuration with $R \approx \kappa \rho$ is predominantly governed by the mass term $m_R^2 \sim 1/3F_{,R}$. If $F_{,R} < 0$, all perturbations with wavelengths larger than $1/|m_R|$ will be unstable so that for a large mass $|m_R|$, the curvature inside a stellar object will evolve rapidly before some nonperturbative effect stabilizes the system. Hence, in order for a model to be stable, $m_R^2$ should be positive throughout the Sun for GR-like configurations [^4]. Both the model with a fine-tuned conformal $\alpha R^2$ term (apart from the caveat mentioned above) and the HS scenario do satisfy all constraints discussed so far. An upper bound on $m_R^2\,$? {#sec:paltrack} ============================ As mentioned in the introduction, a given matter distribution can be consistent with many different static geometries, depending on how the boundary conditions are defined at the center of the star. One always requires that the exterior metric is asymptotically flat and so different solutions are characterized by different values of $\gamma_{\rm PPN}$. There appears to be no [*a priori*]{} preference of one solution to another and indeed the question is: what is the most probable configuration to arise through gravitational collapse? Lacking a dynamical calculation we are restricted here to study how special the GR-like solutions are in the phase space. Consider a static solution $R(r) = R_T(r) + \delta(r)$ where $R_T$ is the solution to the Palatini trace equation and $\delta/R_T \ll 1$ so that $R$ remains very close to the Palatini track. Note that the function $\delta(r)$ is not a true perturbation since $R_T(r)$ is not a solution to the complete metric trace equation (\[eq:trace\]). However, one can easily obtain the equation governing $\delta$ via Eqn. (\[eq:tracePert\]), giving $$\begin{aligned} \vec{\nabla}^2 \delta & = & \frac{1}{3F_{,R}}(1-F-\varepsilon)\delta - 2\frac{F_{,RR}}{F_{,R}}R'_T\delta' \nonumber \\ & & - \vec{\nabla}^2 R_T - \frac{F_{,RR}}{F_{,R}} (R'_T)^2 \,, \label{eq:del}\end{aligned}$$ where $F$ and its derivatives are functions of the “background” value $R_T(r)$. Similarly, the “perturbed” quantities in the definition for $\varepsilon$ are functions of $R(r) = R_T(r) + \delta(r)$. In analogy with the above analysis for $\delta R(r,t)$, let us consider a constant density object so that $R_T = {\rm const.}$, giving $$\begin{aligned} \vec{\nabla}^2 \delta & = & m_R^2 \delta \,. \label{eq:delConst}\end{aligned}$$ The mass term $m_R^2$ is of course the same mass that appears in the equation for $\delta R(r,t)$. Now, for $m_R^2 < 0$, the solution for $\delta(r)$ is decaying so that the Palatini track acts as an attractor for the solution $R(r)$. This is exactly the behavior that was demonstrated by solving the full field equations in Ref. [@Kainulainen:2007bt] for a $f(R) = -\mu^4/R$ model. However, if $m_R^2 > 0$, the solution for $\delta(r)$ will also contain a growing component: $$\delta(r) = \frac{C_1}{r}e^{+m_R r} + \frac{C_2}{r}e^{-m_R r}\,. \label{eq:delSol}$$ The fine-tuning problem we have to face is manifest from this equation: the time stability argument of the preceding section requires that $m_R^2>0$, so that Eqn. (\[eq:delSol\]) with its growing mode is what describes the deviation away from GR-like solutions. From Sec. \[sec:timestab\] we already know that setting up a GR-like configuration requires fine-tuning at the center of the star. However, what Eqn. (\[eq:delSol\]) implies is worse: starting from $r=0$ we could always choose a boundary condition $(F_0,F'_0)$ that kills the growing mode, but as any perturbation around this solution would be exponentially enhanced, the boundary condition must be set with an incredible precision when $m_R^2$ is large. Numerically such a solution can always be found by use of a differentiation method that kills the growing mode as was done in Ref. [@Hu:2007nk]. However, different choices of boundary conditions would lead to other physically, but not observationally, acceptable spacetimes. For example, if one starts from a point a little off from the GR track, $\delta$ initially grows exponentially pulling the solution away from $R \approx \kappa \rho$. Then the nonlinear terms typically become negligible in Eqn. (\[eq:traceWeak\]) and $R(F)$ starts to approach the solution of Eqn. (\[eq:Fsol\]). For $R/\kappa\rho \ll 1$ this limit corresponds to the evolution of $A$ and $B$ given by Eqns. (\[eq:sourceB2\]-\[eq:sourceA2\]), which leads to $\gamma_{\rm PPN} \approx 1/2$. Thus, for a large $m_R^2$ the nearly singular static GR-like solution is surrounded by a continuum of equally acceptable configurations, however with observationally excluded values for $\gamma_{\rm PPN}$. Hence, given no physical reason to prefer a given set of boundary conditions, it would appear more natural to expect that the metric around a generic star would correspond to $\gamma_{\rm PPN}$ different from one. As stated above, to make a definitive statement would require solving the dynamical problem of collapse, but this is beyond the scope of the present work. Nevertheless we believe that we have identified a potential problem for metric $f(R)$ gravity models: for an $f(R)$ model to be credible, it is not sufficient to provide a mere proof of existence of a GR-like solution, but one should also give an argument as to why this particular solution is preferred. The situation would be ameliorated if the growing mode is not excluded, but the length scale dictating the growth of the perturbations, $1/m_R$, is small enough. Roughly one should have $$m_R^2 \lesssim \frac{1}{r_{\odot}^2} \,, \label{eq:constraint}$$ throughout the Sun. However, both the HS scenario and the fine-tunded $f(R) = -\mu^4/R + \alpha R^2/\mu^2$ model fail this constraint by a large margin, as can be seen from Fig. \[fig:mR2\]. This is also the case for the model in Ref. [@Zhang:2007ne] where a stabilizing conformal term creates a behavior very similar to the $\alpha R^2$ model discussed here. The same argument also applies to more recent models introduced in Refs. [@Nojiri:2007as; @Nojiri:2007cq]. These scenarios behave very similar to the HS model at late times, but were designed to also account for inflation at very high energies. For example, for the model suggested in Ref. [@Nojiri:2007cq], $$f(R) = \frac{\alpha R^{m+l} - \beta R^n}{1 + \gamma R^l} \,, \label{eq:NojOdmodel}$$ where the authors chose $m = l = n$ for simplicity, and $n \ge 2$, one can show that the mass squared is given by [@Nojiri:2007cq] $$m_{R}^2 \sim + \frac{R_I}{3n(n-1)}\left( \frac{R_I}{R} \right)^{n-1} \,. \label{eq:NojOdmass}$$ Here $R_I \sim (10^{15} {\rm~GeV})^2$ is set to the scale of inflation, and so this mass is enormous in comparison with the bound (\[eq:constraint\]) inside the Sun. The generic problem is that a small value of $m_R^2 \sim 1/3F_{,R}$ requires a large value for $F_{,R}$. However, at the same time one also needs $F \ll 1$ in order to obtain a reasonable gravitational potential. This tension is what makes it difficult to find a suitable function $f(R)$. Let us illustrate the problem further by trying to construct an explicit model by Þrst making sure that the toughest requirement is satisfied. At the center of Sun $R \approx \kappa\rho \sim 10^{31} \Lambda \sim 10^{-3}/r_{\odot}^2$, which is much smaller than the upper limit on $m_R^2$. Thus we can take $$m_R^2 \sim +R \,.$$ Using this together with the formula $F_{,R} \sim 1/m_R^2$, we can construct a candidate model: $$f(R) = \alpha R\log{\frac{R}{\mu^2}} \,, \label{eq:logmodel}$$ where $\mu^2 = 4\Lambda e^{(1-\alpha)/\alpha}$ in order to obtain the desired accelerating expansion of the Universe at present. Furthermore, demanding that $F \ll 1$ in the interior of the Sun yields $\alpha \lesssim 0.01$ so that $\mu^2 \gtrsim e^{100} \Lambda$ [^5]. So, curiously enough the Solar System constraints would force this model to create the desired accelerating expansion without an extremely small energy scale $\sim \sqrt{\Lambda}$. Unfortunately, there is a flaw in the above argumentation, since we implicitly assumed that $\varepsilon \ll 1$ by assuming $m_R^2 \sim 1/3F_{,R}$. This assumption was fine for the discussion in the previous sections, but it fails here. Indeed, when $F_{,R}$ is large, the gradient term proportional to $[3F_{,R}](R'/R)^2$ in $\varepsilon$ may also become large (see Table \[table2\]). We can estimate the size of this term using an exponential density profile for the Sun $\rho \sim \rho_0/(1 + e^{\xi(r-r_{\odot})})$, where $\xi \sim 10 r_{\odot}^{-1}$: $$\Big(\frac{R'}{R} \Big)^{\! 2} \approx \Big(\frac{\rho'}{\rho} \Big)^{\! 2} \sim \xi^2 \sim 100 \frac{1}{r_{\odot}^2} \,.$$ Now, since $3F_{,R} \gtrsim r_{\odot}^2$ one gets $[3F_{,R}](R'/R)^2 \gg 1$ and our simple estimate for the mass fails. A more careful estimate in the model (\[eq:logmodel\]) finds that the mass is dominated by the gradient term and one has $$m_R^2 \approx m_R^2\big|_{\rm grad} = -\Big(\frac{R'}{R} \Big)^{\! 2} \sim -100 \frac{1}{r_{\odot}^2} \,. \label{eq:m2RlogR}$$ This mass squared is actually negative, so the fine-tuning problem is no longer an issue. However, we have recreated a time instability corresponding to the characteristic length scale $\xi^{-1}$ of the system. This behavior is clearly visible in Fig. \[fig:mR2\] where we have plotted both the full mass squared $m_R^2$ (solid black) and the bare function $1/3F_{,R}$ (dotted black) for the $\alpha R\log{(R/\mu^2)}$ model. Summary and Discussion {#sec:summary} ====================== We have shown in this paper that attempts to find stable static solutions with $\gamma_{\rm PPN} \approx 1$ in metric $f(R)$ models, designed to also account for the accelerating expansion of the Universe, lead to a string of constraints on the model parameters. One must find a configuration for which simultaneously $F \equiv \partial f/\partial R$ and $f/R$ remain small compared to one in the interior of the star, where the strength of the gravitational field is built up, while the Ricci scalar traces the matter distribution: $R\approx \kappa \rho$. (See also Ref. [@Hu:2007nk]). In addition, for the configuration to be stable in time, the effective mass term $m_R^2$ for a perturbation in the Ricci scalar needs to be either positive [@Dolgov:2003px], or if negative, $|m_R|^{-1}$ must be much larger than the size of the physical system under consideration. Furthermore, we showed that unless $m_R^2 \lesssim 1/r_{\odot}^2$, the domain of GR-like static configurations shrinks to essentially a point in the phase space, while for example a continuum of solutions corresponding to $\gamma_{\rm PPN} \approx 1/2$ exists. Hence, in particular for large positive $m_R^2$, it would appear more natural to expect that the metric around a generic star would correspond to $\gamma_{\rm PPN}$ different from one. To make a more definitive statement one should solve the dynamical gravitational collapse of a protostellar dust cloud, which is beyond the scope of this paper however. Moreover, to a degree the problem with the boundary conditions would merely be translated to setting the initial conditions for the collapse. Nevertheless, we believe that we have identified a potential problem in that to make a given metric $f(R)$ model credible, one should give an argument as to why the GR-like configurations should be preferred. Otherwise, if the PPN and stability constraints are supplemented by our fine-tuning argument, it seems unlikely that any $f(R)$ model can pass the test – unless one gives up the hope that the theory is also responsible for the accelerating expansion of the Universe. This is because the condition $m_R^2 \lesssim 1/r_{\odot}^2$, combined with $m_R^2 \sim 1/3F_{,R}$, implies that $F_{,R}$ needs to be large. However, since at the same time one needs $F \ll 1$, a tension is created that makes finding a suitable function $f(R)$ very difficult. Let us finally note that while completely GR-like configurations are hard to construct in $f(R)$ models, it does not mean that such theories would be somehow fundamentally ill. In fact many metric $f(R)$ theories could describe gravitational physics quite well in most situations; it is the very precise information on $\gamma_{\rm PPN}$ from Solar System experiments which eventually forces one to set $R \approx \kappa \rho$. Indeed, if one looks even at the simplest model with $f(R) = -\mu^4/R$, one finds that setting $R$ essentially to any other value than $\kappa \rho$ gives $\gamma_{\rm PPN} \approx 1/2$ [@Kainulainen:2007bt]. 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[^1]: By this we mean that it is in principle possible that an evolution of $B/A$ quite different from the one in GR, could actually lead to the same value for the exterior of a star. However, away from the GR track the outcome becomes sensitive to the form of the density profile, which leads to even more uncertainties as to how the actual dynamical gravitational collapse would proceed. [^2]: In general, $F>1$ implies that the leading approximation in Eqn. (\[eq:mRmagn\]) is no longer valid. However, it still holds for $f(R) = -\mu^4/R + \alpha R^2/\mu^2$, since the contribution of the conformal term in $F$ will cancel with the leading term in $\varepsilon$ where additional terms remain small compared to one. [^3]: The loophole to this argument is if one only follows $R \approx \kappa \rho$ inside the Sun, but that the Ricci scalar is allowed to significantly deviate from this relation inside neutron stars. In such a case one could maintain $F \ll 1$ for much larger values of $\alpha$, resulting in a smaller $m_R^2$ for the Sun. This scenario seems very contrived however. [^4]: Configurations with a small enough negative $m_R^2$ can be accepted as well. The minimum requirement is then that at least all perturbations with wavelengths smaller than the size of the physical system under consideration should remain stable. [^5]: However, note that since $F = \alpha(\log{(R/\mu^2)} + 1) = \alpha\log{(R/4\Lambda) - 1 + 2\alpha}$, there is some tension between getting $F \ll 1$ inside *both* the Sun and a neutron star. Although this may indeed prove relevant in further considerations, it is not of importance for the discussion at hand.

--- abstract: 'We analyze Regge quantum gravity coupled to SU(2) gauge theory on $4^3\times 2$, $6^{3}\times 4$ and $8^{3}\times 4$ simplicial lattices. It turns out that the window of the well-defined phase of the gravity sector where geometrical expectation values are stable extends to negative gravitational couplings as well as to gauge couplings across the deconfinement phase transition. We study the string tension from Polyakov loops, compare with the $\beta$-function of pure gauge theory and conclude that a physical limit through scaling is possible.' address: - 'Department of Physics and SCRI, Florida State University, Tallahassee, FL 32306, USA' - 'Institut für Kernphysik, Technische Universität Wien, A-1040 Vienna, Austria' author: - 'B.A. Berg[^1]' - 'W. Beirl$^*$, B. Krishnan[^2], H. Markum, and J. Riedler' title: 'Phase diagram of Regge quantum gravity coupled to SU(2) gauge theory' --- Introduction ============ Regge Calculus [@R] provides a nonperturbative way for investigations of Euclidean quantum gravity on a simplicial lattice and offers the possibility to construct a unified theory by coupling gauge fields to the skeleton. One remarkable finding was the discovery of an “entropy dominated”, well-defined phase, where the expectation values with respect to the pure Regge-Einstein action are stable [@BB1; @HH1; @BGM]. The next question addressed was the physical relevance of this regime which has been tested by coupling a non-Abelian gauge field to gravity [@BKK]. If one assumes that the world without gravity is described by a grand unified asymptotically free theory, these numerical studies investigate the relation of the hadronic scale to the Planck scale. In particular, it has already been shown that confinement exists in the coupled system [@BBKMRpl]. In this work, we perform a nonperturbative analysis of the phase diagram of Regge quantum gravity coupled to SU(2) gauge fields on several lattice sizes in four spacetime dimensions. The stability and boundary of the well-defined phase is investigated on lattices of sizes up to $6^3\times 4$, considerably extending the previously available data. Within this phase we find that the confinement-deconfinement transition of conventional lattice gauge theory is still present. We extract values for the string tension and gain some evidence from the $\beta$-function that the window of the well-defined phase extends to large $\beta$ values corresponding to small lattice spacings $a$ in the order of the Planck scale. Entropy Dominated Phase ======================= In Regge Calculus the edge lengths are considered to be the dynamical degrees of freedome of the discretized spacetime manifold. In $d=4$ dimensions the geometry is Euclidean inside of a $d$-simplex and the curvature is concentrated at the $d-2$-subset of the lattice, the triangles. Quantization proceeds via the path integral although the choice of the gravitational measure is an unresolved issue. Investigations on regular triangulations do not favor any of them [@BGM], but for irregular triangulations a preference for scale-invariant measures was found [@BMR]. It may be that measures are divided into universality classes such that identical physics is obtained within one class. In our simulations we chose a hypercubic triangulation with $N_{s}^{3}\times N_{t}$ vertices and the scale-invariant measure: $$D[\{l^{2}\}]\,=\,\prod_{l}\frac{dl^{2}}{l^{2}}\,, \label{eq:measure}$$ where $l$ is used to denote the link label as well as its length. The system of SU(2) gauge fields coupled to quantum gravity has the action [@BKK] $$\label{coupact} S\,=\,2m_{p}^{2}\sum_{t}A_{t}\alpha_{t} - \frac{\beta}{2}\sum_{t}W_{t}\,{\rm Re}[ {\rm Tr}(1\,-\,U_{t})]\; , \label{eq:action}$$ with the first term being the Regge-Einstein part composed of the bare Planck mass $m_{p}$ and $A_{t}$, $\alpha_{t}$ the area and the deficit angle of the triangle $t$. The addition of the SU(2) gauge term is straightforward and follows ordinary lattice gauge theory by assigning SU(2) matrices to the links. The elementary plaquettes become triangles on the Regge skeleton. $\beta$ corresponds to the inverse gauge coupling and the weights , with a 4-volume $V_t$ assigned to each triangle, describe the coupling of gravity to the gauge field. $U_{t}$ is the ordered product of SU(2) matrices around $t$. In contrast to the flat lattice, the unit matrix in the action is important because the weight factors are dynamical. They are constructed such that the correct continuum limit is ensured in the limit of vanishing lattice spacing [@CL]. We present Monte Carlo (MC) results concerning the boundary and stability of the well-defined phase. For this purpose large statistics simulations were performed on $4^3\times 2$ and $6^3\times 4$ lattices, covering a variety of $(m_p^2,\beta)$ values. The accumulated statistics is summarized in Tables \[table1\] and \[table2\]. The stability of the coupled system was analyzed from MC-time histories of the Regge action. Examples are presented in Fig. \[his\]. Long runs on the larger lattice show that it is difficult to decide whether the well-defined phase is stable or just metastable. In the latter case an additional curvature term of higher order [@HeHa] may stabilize the system. The deconfinement transition was studied from the behavior of the Polyakov loops . For $4^3\times 2$ and $6^3\times 4$ lattices Fig. \[poli\] shows MC-time histories of $P$ and $l$ in the confined and deconfined phase for gravitational couplings in the well-defined phase. The link lengths are largely independent of fluctuations of the order parameter. Notable is the long equilibration time in the deconfinement phase. Extracted phase diagrams of the gauge-gravity system are displayed in Fig. \[phadia\]. The dotted lines are to guide the eyes and rely on the depicted stable versus unstable data points. From the $4^3\times 2$ lattices we have numerical evidence that the stable phase extends to $\beta=3.0$ (Table \[table1\]). The $6^3\times 4$ lattices indicate a glitch in the well-defined–ill-defined boundary when passing from confinement to deconfinement. However, the present runs do not decide the question conclusively. It may be accidental that at $\beta = 1.6$ the $m_p^2=0.025$ and $m_p^2=0.023$ data did not run away. For $\beta=1.55$ tunneling into the ill-defined phase happened for these $m_p^2$ values only after more than 100k sweeps. Our systems exhibit a small shift of the deconfinement phase transition from $\beta_c(N_t=2)=1.525$ to $\beta_c(N_t=4)=1.575$ with error bars less than 0.025. String Tension ============== The Polyakov loop $P(R)$ in the short extent $L_{t}$ of the lattice describes the propagation of a static quark. We introduce a quark source and a sink separated by a distance $R$ and calculate the correlations of the Polyakov loops at these points. As in conventional SU(2) lattice gauge theory at finite temperature $T$ [@MS; @KPS], we extract from the correlation function $$\langle P(0) P^{\dagger}(R) \rangle\,=\,\exp[- \frac{1}{T} V(R)] , \label{eq:correl}$$ the quantity $V(R)$ corresponding to the potential between the static quark-antiquark pair. In the confinement phase $V(R)$ should grow linearly for large $R$ due to an infinite free energy of isolated quarks: $$\label{Vc} V_c(R)=\frac{-\alpha}{R}+\sigma R+C ,$$ where $\alpha$ is the Coulomb parameter, $\sigma$ the string tension, and $C$ a constant. The correct distance between two points should be measured using geodesic distances. We take the distance $R$ between the source and sink to be equal to the index distance along the main axes of the skeleton. This seems a reasonable approximation in the well-defined phase with small curvature fluctuations. Using scalar field propagation [@HH2] one may calculate corrections which are expected to be small for our purposes. To extract a reliable value for the string tension, we simulated the system on an $8^{3}\times 4$ lattice. Our data rely on 30000 measurements after equilibration for the coupled system as well as for the pure gauge system on a flat simplicial lattice. Figure \[pots\](a) presents the data points for the confinement potentials for several gauge couplings in the presence of gravity with $m_{p}^{2}=0.005$ in the “entropy dominated” phase, while Fig. \[pots\](b) depicts the situation with gravity switched off. The dotted lines correspond to fits to the correlations in Eq. (\[eq:correl\]) according to the potential of Eq. (\[Vc\]) with the Coulomb parameter fixed to $\alpha=\frac{\pi}{12}$ and a mirror term included. Since we have extracted string tensions for several $\beta$ values, we are in a position to study its scaling behavior. For pure gauge theory $\beta$-functions are derived in the literature also for a simplicial lattice [@DM]. We fit our string-tension data to the function $$\label{betafunc} \sigma_{\rm flat}=\frac{\sigma_{\rm phys}}{\Lambda^2_{\rm flat}} \left(\frac{6\pi^2\sqrt{5}\beta}{11}\right)^{102/121} \exp{\left(-\frac{6\pi^2\sqrt{5}\beta}{11}\right)}$$ and obtain a value for $\Lambda_{\rm flat}=0.0102(1)\sqrt{\sigma_{\rm phys}}$. This is in good agreement with an analysis of Wilson-loop ratios in the $T=0$ case, yielding . For the fluctuating lattice to our knowledge a $\beta$-function is not worked out, we are aware only of a recent study for the pure gravity case within dynamical triangulation [@B]. Thus, we used as a starting point the above SU(2) function Eq. (\[betafunc\]). In Fig. \[scal\] we compare the string-tension scaling for the flat and the fluctuating lattices. To have both systems on the same scale, we rescaled the inverse gauge coupling of the fluctuating system. We find $\Lambda_{\rm grav} = 0.0090(1)\sqrt{\sigma_{\rm phys}}$ which is very similar to the flat case. Assuming that the pure-gauge $\beta$-function is a reasonable approximation for the full renormalization relation, our data show that the scaling window opens already at the $\beta$ values considered, similar to the pure SU(2) case [@DM]. As a consequence, the continuum limit could be performed along $\beta\to\infty$, eventually stopping at a finite value corresponding to a lattice spacing equal to the Planck length. The existence of a corridor to large $\beta$’s is indicated in the $(m_p, \beta)$ diagrams of Fig. \[phadia\]. Conclusion ========== The boundary between the well-defined and the ill-defined phase of Regge quantum gravity coupled to SU(2) gauge theory was studied with the largest so far available statistics on $6^3\times 4$ lattices. These lattices are already very CPU time intensive, because the gravitational dynamics is very slow. Altogether, evidence was gained that the well-defined phase is stable with the increase from $4^3\times 2$ to $6^3\times 4$. Within the well-defined phase we find that the confinement mechanism from the non-Abelian gauge fields is not spoiled by quantum gravitational effects. This is not trivial, because it is was not clear how quantum gravity affects a gauge theory. In the confined phase we observed a potential linearly rising with $R$. Extracting string tension values for both the coupled system and for the pure gauge theory on a simplicial lattice without gravity we found a very similar scaling behavior. It indicates that gravity effects do not destroy the physics of conventional asymptotically free field theories, even if the gravity-gauge coupling is large as in our situation. This gives hope that the physical limit can be approached through scaling from the investigated region towards the Planck length. Additionally, to reproduce Newton’s law one should demonstrate a diverging gravitational correlation length of graviton propagators when approaching the boundary of the well-defined phase. 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B [**356**]{}, 466 (1995). $m_p^2 \setminus \beta $ 0.0 $m_p^2 \setminus \beta $ 1.0 1.5 1.55 1.6 1.7 1.8 3.0 -------------------------- ----- -------------------------- ----- ----- ------ ----- ----- ----- ----- $+0.0225 $ 200 $+0.05 $ 100 $+0.02 $ 200 $+0.04 $ 40 $+0.0 $ 200 $+0.0350 $ 167 $-0.005 $ 40 $+0.03 $ 200 100 100 $-0.01 $ 40 $+0.025 $ 100 200 140 200 $-0.025 $ 40 $+0.0225 $ 30 200 200 200 200 $-0.03 $ 40 $+0.005 $ 100 $-0.04 $ 40 $+0.0 $ 35 $-0.05 $ 40 $-0.0025 $ 200 200 $-0.06 $ 40 $-0.005 $ 60 200 200 100 200 $-0.06375$ 100 $-0.0075 $ 100 $-0.06425$ 200 $-0.008 $ 100 $-0.065 $ 18 $-0.009 $ 200 $ $ $-0.0105 $ 100 : \[table1\] Statistics for the $4^3\times 2$ lattices in units of 1k sweeps. $m_p^2 \setminus \beta$ 0.0 0.073 0.5 0.8 1.0 1.2 1.4 1.55 1.6 1.65 ------------------------- ----- ------- ----- ----- ----- ----- ----- ------ ----- ------ -- $+0.03 $ 24 152 88 $+0.0275$ 72 64 $+0.026 $ 48 120 208 $+0.025 $ 136 200 $+0.023 $ 208 144 208 $+0.02 $ 56 208 $+0.0175$ 23 $+0.015 $ 208 56 48 $+0.01 $ 56 $-0.005 $ 208 $-0.01 $ 24 40 160 40 24 40 $-0.015 $ 32 $-0.02 $ 24 40 16 $-0.04 $ 40 $-0.05 $ 144 $-0.06 $ 40 : \[table2\] Statistics for the $6^3\times 4$ lattices in units of 1k sweeps. [^1]: Work supported in part by DOE under Contracts DE-FG05-87ER40319 and DE-FC05-85ER2500. [^2]: Lise-Meitner Postdoctoral Research Fellow sponsored by FWF under Project M212-PHY.

[**Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime**]{} [**Claudio Dappiaggi$^{1,a}$**]{}, [**Valter Moretti$^{2,3,b}$**]{}, [**Nicola Pinamonti$^{1,c}$**]{}\ $^1$ II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany. $^2$ Dipartimento di Matematica, Università di Trento and Istituto Nazionale di Fisica Nucleare – Gruppo Collegato di Trento, via Sommarive 14 I-38050 Povo (TN), Italy. $^3$ Istituto Nazionale di Alta Matematica “F.Severi”– GNFM E-mail: $^a$claudio.dappiaggi@desy.de, $^b$moretti@science.unitn.it, $^c$nicola.pinamonti@desy.de\ . The discovery of the radiation properties of black holes prompted the search for a natural candidate quantum ground state for a massless scalar field theory on Schwarzschild spacetime, here considered in the Eddington-Finkelstein representation. Among the several available proposals in the literature, an important physical role is played by the so-called Unruh state which is supposed to be appropriate to capture the physics of a black hole formed by spherically symmetric collapsing matter. Within this respect, we shall consider a massless Klein-Gordon field and we shall rigorously and globally construct such state, that is on the algebra of Weyl observables localised in the union of the static external region, the future event horizon and the non-static black hole region. Eventually, out of a careful use of microlocal techniques, we prove that the built state fulfils, where defined, the so-called Hadamard condition; hence, it is perturbatively stable, in other words realizing the natural candidate with which one could study purely quantum phenomena such as the role of the back reaction of Hawking’s radiation.\ From a geometrical point of view, we shall make a profitable use of a bulk-to-boundary reconstruction technique which carefully exploits the Killing horizon structure as well as the conformal asymptotic behaviour of the underlying background. From an analytical point of view, our tools will range from Hörmander’s theorem on propagation of singularities, results on the role of passive states, and a detailed use of the recently discovered peeling behaviour of the solutions of the wave equation in Schwarzschild spacetime. Introduction ============ In the wake of Hawking’s discovery of the radiating properties of black holes [@Hawking], several investigations on the assumptions leading to such result were prompted. In between them, that of Unruh [@Unruh] caught the attention of the scientific community, since he first emphasised the need to identify a physically sensible candidate quantum state which could be called the vacuum for a quantum massless scalar field theory on the Schwarzschild spacetime. This is especially true when such spacetime is viewed as that of a real black hole obtained out of the collapse of spherically symmetric matter. If we adopt the standard notation ([*e.g.*]{}, see [@Wald2]), this spacetime can be identified with the union of the regions I and III in the Kruskal manifold including the future horizon, though we must omit the remaining two regions together with their boundaries [@Wald; @Wald2]. To the date, in the literature, three candidate background states are available, going under the name of [*Boulware*]{} (for the external region), [*Hartle-Hawking*]{} (for the complete Kruskal manifold) and [*Unruh*]{} state (for the union of both the external and black hole region, including the future event horizon). The goal of this paper is to focus on the latter, mostly due to its remarkable physical properties. As a matter of fact, earlier works (see for example [@Candelas; @Balbinot; @Balbinot2]) showed that such a state could be employed to compute the expectation value of the regularised stress-energy tensor for a massless scalar field in the physical region of Schwarzschild spacetime, above pointed out. The outcome is a regular expression on the future event horizon while, at future null infinity, it appears an outgoing flux of radiation compatible with that of a blackbody at the black hole temperature. As pre-announced, this result, together with Birkhoff’s theorem, lead to the conjecture that the very same Unruh state, say $\om_U$, as well as its smooth perturbations, is the natural candidate to be used in the description of the gravitational collapse of a spherically symmetric star. However, to this avail, one is also lead to assume that $\om_U$ fulfils the so-called [*Hadamard property*]{} [@KW; @Wald2], a prerequisite for states on curved background to be indicated as physically reasonable. As a matter of fact, in between the many properties, it is noteworthy to emphasise that such condition assures the existence of a well-behaved averaged stress energy tensor [@Wald2]. Therefore, from a heuristic point of view, this condition is tantamount to require that the ultraviolet behaviour mimics that of the Minkowski vacuum, leading to a physically clear prescription on how to remove the singularities of the averaged stress-energy tensor; this comes at hand whenever one needs to compute the back-reaction of the quantum matter on the gravitational background through Einstein’s equations. The relevance of the Hadamard condition is further borne out by the analysis in [@Fredenhagen], where the description of the gravitational collapse of a spherically symmetric star is discussed and, under the assumption of the existence of suitable algebraic states of Hadamard form, it is shown that the appearance of the Hawking radiation, brought, at large times, by any of the said states, is precisely related to the scaling-limit behaviour of the underlying two-point function of the state computed on the $2$-sphere determined by the locus where the star radius crosses the Schwarzschild one. It is therefore manifest the utmost importance to verify whether $\om_U$ satisfies or not the Hadamard property, a condition which appears reasonable to assume at least in the static region of Schwarzschild spacetime also in view both of the former analysis in [@Candelas] and of the general results achieved in [@SV00] applied to those in [@DimockKay]. Indeed such a check is one of the main purposes to write this paper. Our goals are, however, broader, as we shall make a novel use of the Killing and conformal structure of Schwarzschild spacetime in order to construct rigorously and unambiguously the Unruh state, contemporary in the static region, inside the internal region and on the future event horizon. To this avail, we shall exploit some techniques which in the recent past have been successfully applied to manifolds with Killing horizons, asymptotically flat spacetimes (see also the recent [@Schroer]) as well as cosmological backgrounds [@MP05; @DMP; @Moretti06; @Da08a; @Da08; @Moretti08; @DMP2; @DMP3]. Within this respect, it is also important to mention that, although, for different physical goals, a mathematically similar technology was employed in [@Ho00] including a proof of the Hadamard property of the relevant states.\ From the perspective of this manuscript, the above cited paper are most notable for their underlying common “philosophy”. To wit, as a first step, one always identifies a preferred codimension $1$ null submanifolds of the background, one is interested in. Afterwards, the classical solutions of the bulk dynamical system, one wishes to consider, are projected on a suitable function set living on the chosen submanifold. The most notable property of this set is that one can associate to it a Weyl algebra of observable, which carries a corresponding distinguished quantum algebraic state which can be pulled-back to bulk via the above projection map. On the one hand this procedure induces a state for the bulk algebra of observables and, on the other hand, such new state enjoys several important physical properties, related both with the symmetries of the spacetime and with suitable notions of uniqueness and energy positivity. Particularly, although at a very first glance, one would be tempted to conclude that the Hadamard property is automatically satisfied as a consequence of the construction itself and of the known results for the microlocal composition of the wave front sets, actually we face an harsher reality. To wit, this feature has to be verified via a not so tantalising case by case analysis since it is strictly intertwined to the geometrical details of the background. Unfortunately the case, we analyse in this paper, is no exception and, thus, we shall be forced to use an novel different procedure along the lines below outlines.\ As a starting point, we shall remark that, in the Schwarzschild background, the role of the distinguished null codimension one hypersurface, on which to encode the bulk data, will be played by the union of the complete Killing past horizon and of null past infinity. Afterwards, as far as the state is concerned, it will be then defined on the selected hypersurfaces just following the original recipe due to Unruh: a vacuum defined with respect to the affine parameter of the null geodesics forming the horizon and a vacuum with respect to the Schwarzschild Killing vector $\partial_t$ at past null infinity. At a level of two-point function, the end point of our construction takes a rather distinguished shape whenever restricted to the subalgebra smeared by compactly supported functions, which coincides with the one already noticed in [@Sewell; @DimockKay; @KW]. Nonetheless, from our perspective, the most difficult technical step will consist of the extension of the methods employed in our previous papers, the reason being that the full algebra both on the horizon and on null infinity is subject to severe constraints whose origin can be traced back to some notable recent achievements by Dafermos and Rodnianski [@DR08]. To make things worse, a similar problem will appear for the state constructed for the algebra at the null infinity. Nonetheless we shall display a way to overcome both potential obstructions and the full procedure will ultimately lead to the implementation of a fully mathematically coherent Unruh state, $\om_U$ for the spacetime under analysis.\ Despite these hard problems, the bright side of the approach, we advocate, lies in the possibility to develop a global definition for $\om_U$ for the spacetime which encompasses the future horizon, the external as well as the internal region. Furthermore our approach will be advantageous since it allows to avoid most of the technical cumbersomeness, encountered in the earlier approaches, the most remarkable in [@DimockKay] (see also [@KayD]), where the Unruh state was defined via an S-matrix out of the solutions of the corresponding field equation of motion in asymptotic Minkowski spacetimes. Alas, the definition was established only for the static region and the Hadamard condition was not checked, hence leaving open several important physical questions.\ Differently, our boundary-to-bulk construction, as pre-announced, will allow us to make a full use of the powerful techniques of microlocal analysis, thus leading to a verification of the Hadamard condition using the global microlocal characterisation discovered by Radzikowski [@Rada; @Radb] and fruitfully exploited in all the subsequent literature (see also [@BFK]). Differently from the proofs of the Hadamard property presented in [@Moretti08] and [@DMP3] here we shall adopt a more indirect procedure, which has the further net advantage to avoid potentially complicated issues related to the null geodesics reaching $i^-$ from the interior of the Schwarzschild region. The Hadamard property will be first established in the static region making use of an extension of the formalism and the results presented in [@SV00] valid for [*passive states*]{}. The black hole region together with the future horizon will be finally encompassed by a profitable use of the Hörmander’s propagation of singularity theorem joined with a direct computation of the relevant remaining part of wavefront set of the involved distributions, all in view of well-established results of microlocal analysis.\ From a mathematical point of view, it is certainly worth acknowledging that the results we present in this paper are obtainable thanks to several remarkable achievements presented in a recent series of papers due to Dafermos and Rodnianski [@DR03; @DR07; @DR08; @DR05], who discussed in great details the behaviour of a solution $\varphi$ of the Klein-Gordon equation in Schwarzschild spacetime improving a classical result of Kay and Wald [@KW2]. Particularly we shall benefit from the obtained peeling estimates for $\varphi$ both on the horizons and at null infinity, thus proving the long-standing conjecture known as Price law [@DR03].\ In detail, the paper will be divided as follows.\ In section 2.1, we recall the geometric properties of Schwarzschild spherically symmetric solution of Einstein’s equations. Particularly, we shall introduce, characterise and discuss all the different regions of the background which will play a distinguished role in the paper.\ Subsequently, in section 2.2 and 2.3, we shall define the relevant Weyl C$^*$-algebras of observables respectively in the bulk and in the codimension $1$ submanifolds, we are interested in, namely the past horizon and null infinity.\ Eventually, in section 2.4, we shall relate bulk and boundary data by means of an certain isometric $*$-homomorphism whose existence will be asserted and, then, discussed in detail.\ Section 3 will be instead devoted to a detailed analysis on the relation between bulk and boundary states. Particularly we shall focus on the state defined by Kay and Wald for a (smaller) algebra associated with the past horizon $\cH$ [@KW], showing that that state can be extended to the (larger) algebra relevant for our purposes.\ The core of our results will be in section 4 where we shall first define the Unruh state and, then, we will prove that it fulfils the Hadamard property. Eventually we draw some conclusions.\ Appendix A contains further geometric details on the conformal structure of Schwarzschild spacetime, while Appendix C encompasses the proofs of most propositions. At the same time Appendix B is noteworthy because it summarises several different definitions of the KMS condition and their mutual relation is briefly sketched.\ Notation, mathematical conventions ---------------------------------- \[secgauge\] Throughout, $A\subset B$ (or $A \supset B$) includes the case $A=B$, moreover $\bR_+\doteq [0,+\infty)$, $\bR^*_+ \doteq (0,+\infty)$, $\bR_-\doteq (-\infty,0]$, $\bR^*_- \doteq (-\infty,0)$ and $\bN\doteq \{1,2,\ldots\}$. For smooth manifolds $\mM,\mN$, $C^\infty(\mM;\mN)$ is the space of smooth functions $f: \mM\to \mN$. $C^\infty_0(\mM;\mN)\subset C^\infty(\mM;\mN)$ is the subspace of compactly-supported functions. If $\chi : \mM\to \mN$ is a diffeomorphism, $\chi^*$ is the natural extension to tensor bundles (counter-, co-variant and mixed) from $\mM$ to $\mN$ (Appendix C in [@Wald]). A [**spacetime**]{} $(\mM,g)$ is a Hausdorff, second-countable, smooth, four-dimensional connected manifold $\mM$, whose smooth metric has signature $-+++$. We shall also assume that a spacetime is [*oriented*]{} and [*time oriented*]{}. The symbol $\Box_g$ denotes the standard [**D’Alembert operator**]{} associated with the unique metric, torsion free, affine connection $\nabla_{(g)}$ constructed out of the metric $g$. $\Box_g$ is locally individuated by $g_{ab}\nabla_{(g)}^a \nabla_{(g)}^b$. We adopt definitions and results about causal structures as in [@Wald; @ON], but we take recent results [@BS03; @BS06] into account, too. If $(\mM,g)$ and $(\mM',g')$ are spacetimes and $S\subset \mM\cap \mM'$, then $J^\pm(S;\mM)$ ($I^\pm(S;\mM)$) and $J^\pm(S;\mM')$ ($I^\pm(S;\mM')$) indicate the [**causal**]{} (resp. [**chronological**]{}) [**sets**]{} generated by $S$ in the spacetime $\mM$ or $\mM'$, respectively. An (anti)symmetric bilinear map over a real vector space $\sigma : V \times V \to \bR$ is [**nondegenerate**]{} when $\sigma(u,v)=0$ for all $v\in V$ entails $u=0$.\ Quantum Field theories - bulk to boundary relations =================================================== Schwarzschild-Kruskal spacetime ------------------------------- In this paper we will be interested in the analysis of a Klein-Gordon scalar massless field theory on Schwarzschild spacetime and, therefore, we shall first recall the main geometric properties of the background we shall work with. Within this respect, we shall follow section 6.4 of [@Wald] and we will focus on the [*physical region*]{} $\mM$ of the full Kruskal manifold $\mK$ (represented in figure 2 in the appendix), associated with a black hole of mass $m>0$.\ $\mM$ is made of the union of three pairwisely disjoint parts, $\mW,\mB$ and $\cH_{ev}$ which we shall proceed to describe. According to figure 1 (and figure 2 in the appendix), we individuate ${\mW}$ as the (open) [**Schwarzschild wedge**]{}, the (open) [**black hole**]{} region is denoted by ${\mB}$ while their common boundary, the [**event horizon**]{}, is indicated by $\cH_{ev}$.\ The underlying metric is easily described if we make use of the standard [**Schwarzschild coordinates**]{} $t,r,\theta,\phi$, where $t\in \bR$, $r\in (r_S,+\infty)$, $(\theta,\phi) \in \bS^2$ in ${\mW}$, whereas $t\in \bR$, $r\in (0,r_S)$, $(\theta,\phi) \in \bS^2$ in ${\mB}$. Within this respect the metric in both ${\mW}$ and ${\mB}$ assumes the standard Schwarzschild form: \[Schw\] -(1-) dt dt + (1-)\^[-1]{} dr dr + r\^2 h\_[\^2]{}(,), where $h_{\bS^2}$ is the standard metric on the unit $2$-sphere. Here, per direct inspection, one can recognise that the locus $r=0$ corresponds to proper metrical singularity of this spacetime, whereas $r=r_S=2m$ individuates the apparent singularity on the event horizon.\ It is also convenient to work with the [**Schwarzschild light**]{} or [**Eddington-Finkelstein coordinates**]{} [@KW; @Wald2] $u,v,\theta,\phi$ which cover ${\mW}$ and ${\mB}$ separately, such that $(u,v)\in \bR^2$, $(\theta,\phi)\in \bS^2$ and $$\begin{aligned} &u \doteq t-r^* \mbox{ in ${\mW}$,} \quad u\doteq-t-r^* \mbox{ in $\mB$,} \\ &v \doteq t+r^* \mbox{ in ${\mW}$,} \quad v\doteq t-r^* \mbox{ in $\mB$,} \\ &r^* \doteq r + 2m \ln \left|\frac{r}{2m}-1 \right| \in \bR\:. $$ A third convenient set of [**global null coordinates**]{} $U,V,\theta,\phi$ can be introduced on the whole Kruskal spacetime [@Wald]: &U = -e\^[-u/(4m)]{}, V = [e\^[ v/(4m)]{}]{}\ &U = e\^[ u/(4m)]{}, V = [e\^[ v/(4m)]{}]{} .In this frame, $$\begin{gathered} \mW\equiv \{ (U,V, \theta,\phi) \in \bR^2 \times \bS^2\:| \: U<0,V>0\}\:,\\ \mB \equiv \{ (U,V, \theta,\phi) \in \bR^2 \times \bS^2\:| \:UV<1 \:, U, V > 0\}\:,\\ {\mM} \doteq \mW \cup \mB \cup \cH_{ev} \equiv \{ (U,V, \theta,\phi) \in \bR^2 \times \bS^2\:| \: UV<1\:, V > 0\}\:. \end{gathered}$$ Each of the three mentioned regions, seen as independent spacetimes, is globally hyperbolic. (150,130)(0,0) (0,10)[{height="5cm"}]{} (142,98)[$\mW$]{} (65,125)[$\mB$]{} (178,35)[$\scrim$]{} (178,120)[$\scri$]{} (215,78)[$i_0$]{} (136,0)[$i^-$]{} (136,157)[$i^+$]{} (21,104)[$\cH^+$]{} (58,70)[$\cB$]{} (88,37)[$\cH^-$]{} (120,120)[$\cH_{ev}$]{} (135,72)[ $\Si'$]{} (125,50)[ $\Si$]{} The event horizon of $\mW$, $\cH_{ev}$ is one of the two horizons we shall consider. The other is the complete [**past horizon**]{} of $\mM$, $\cH$ which is part of the boundary of $\mM$ in the Kruskal manifold. These horizons are respectively individuated by: $$\cH_{ev} \equiv \{ (U,V, \theta,\phi) \in \bR^2 \times \bS^2\:| \: U = 0, V>0\} \:, \quad \cH \equiv \{ (U,V, \theta,\phi) \in \bR^2 \times \bS^2\:| \: V = 0, U\in \bR\} \:.$$ For future convenience, we decompose $\cH$ into the [*disjoint*]{} union $\cH = \cH^- \cup \cB \cup \cH^+$ where $\cH^\pm$ are defined according to $U>0$ or $U <0$ while $\cB$ is the [**bifurcation surface**]{} at $U=0$, [*i.e.*]{}, the spacelike $2$-sphere with radius $r_S$ where $\cH$ meets the closure of $\cH_{ev}$.\ The metric on $\mM$ (and in the whole Kruskal manifold) takes the form: g = - e\^[-]{} (dU dV + dV dU) + r\^2 h\_[\^2]{}(,), \[g\] where the apparent Schwarzschild-coordinate singularity on both $\cH$ and $\cH_{ev}$ has disappeared. It coincides with the radial Schwarzschild coordinate in both ${\mW}$ and ${\mB}$, hence taking the constant value $r_s$ on $\cH_{ev} \cup \cB$; at the same time, the metric singularity, located at $r=0$, corresponds to $UV=1$.\ Let us now focus on the Killing vectors structure. Per direct inspection of either or , one realizes that there exists a space of Killing vectors generated both by all the complete Killing fields associated with the spherical symmetry – $\partial_\phi$ for every choice of the polar axis $z$ – and by a further smooth Killing field $X$. It coincides with $\partial_t$ in both ${\mW}$ and ${\mB}$, although it is timelike and complete in the former static region, while it is spacelike in the latter. Moreover $X$ becomes light-like and tangent to $\cH$ and $\cH_{ev}$ (as well as to the whole completion of $\cH_{ev}$ in the Kruskal manifold) while it vanishes exactly on $\cB$, giving rise to the structure of a [*bifurcate Killing horizon*]{} [@KW]. It is finally useful to remark that the coordinates $u$ and $v$ are respectively well defined on both $\cH_{ev}$ and $\cH^\pm$ where it turns out that: $$X = \mp \partial_u \mbox{ on $\cH^\pm$,} \quad X = \partial_v \mbox{ on $\cH_{ev}$.}$$ To conclude this short digression on the geometry of Kruskal-Schwarzschild spacetime, we notice that, by means of a conformal completion procedure, outlined in Appendix \[geometry\], one can coherently introduce the notion of future and past [**null infinity**]{} ${\Im}^\pm$. Along the same lines (see again figure 1 and figure 2 in the appendix), we also shall refer to the formal [*points at infinity*]{} $i^\pm$, $i^0$, often known as [**future**]{}, [**past**]{} and [**spatial infinity**]{} respectively. The Algebra of field observables of the spacetime ------------------------------------------------- \[observables\] We are interested in the quantisation of the free massless scalar field $\vphi$ [@KW; @Wald2] on the globally hyperbolic spacetime $(\mN,g)$. The real field $\vphi$ is supposed to be smooth and to satisfy the massless [**Klein-Gordon**]{} equation in $(\mN,g)$: \[KG\] P\_g=0, P\_g -\_g + R\_g. Since we would like to use conformal techniques, we have made explicit the conformal coupling with the metric, even if it has no net effect for the case $\mN=\mM$, since the curvature $R_g$ vanishes therein. Nonetheless, this allows us to make a profitable use of the discussion in Appendix \[geometry\] when $\mN = \mM$ and $\widetilde{\mM}\supset \mM$. Here $\widetilde{\mM}$ stands for the conformal extension (see also figure 2 in the appendix) of the previously introduced physical part of Kruskal spacetime ${\mM}$, equipped with the metric $\widetilde{g}$ which coincides with $g/r^2$ in $\mM$. In such case, if the smooth real function $\widetilde{\vphi}$ solves the Klein-Gordon equation in $\widetilde{\mM}$ (where now $R_{\widetilde{g}} \neq 0$): \[tildeKG\] P\_ =0, P\_ -\_ + R\_, $\vphi \doteq \frac{1}{r} \widetilde{\vphi}\spa\rest_{\mN}$ solves (\[KG\]) in $\mM$.\ Generally we shall focus our attention to the class $\sS(\mN)$ of real smooth solutions of (\[KG\]) which have compact support when restricted on a (and thus on every) spacelike smooth Cauchy surface of a globally hyperbolic spacetime $(\mN,g)$. This real vector space becomes a symplectic one $(\sS(\mN), \sigma_\mN)$ when equipped with the non-degenerate, $\Sigma$-independent, symplectic form [@KW; @Wald2; @BGP], for $\vphi_1,\vphi_2 \in \sS(\mN)$, \[symp\] \_(\_1,\_2)\_[\_]{} (\_2\_n \_1-\_1\_n \_2) d\_g(\_) . Here $\Sigma_\mN$ is any spacelike smooth Cauchy surface of $\mN$ with the metric induced measure $\mu_g(\Sigma_\mN)$ and future-directed normal unit vector $n$.\ Furthermore, for any $\mN' \subset \mN$ such that $(\mN',g\spa\rest_{\mN'})$ is globally hyperbolic, the following inclusion of symplectic subspaces holds $$(\sS({\mN'}),\sigma_{{\mN'}}) \subset (\sS(\mN),\sigma_{\mN})\:.$$ Such statement can be proved out of both and the independence from the used smooth spacelike Cauchy surface. To this avail, it is crucial that every compact portion of a spacelike Cauchy surface of $\mN'$ can be viewed as that of a second smooth spacelike, hence acausal, Cauchy surface of $\mN$ as shown in [@BS06] (though, for acausality, one should also refer to Lemma 42 in Chap. 14 of [@ON]). The quantisation procedure within the algebraic approach goes along the guidelines given in [@KW; @Wald2] as follows: the elementary observables associated with the field $\vphi$ are the (self-adjoint) elements of the [**Weyl**]{} ($C^*$-) [**algebra**]{} $\cW(\sS(\mN))$ [@Haag; @BR2; @KW; @Wald2] whose generators will be denoted by $W_\mN(\vphi)$, $\vphi \in \sS(\mN)$, as discussed in the Appendix \[algebras\]. In order to interpret the elements in $\cW(\sS(\mN))$ as [*local observables*]{} smeared with functions of $C_0^\infty(\mN; \bR)$, we introduce some further technology. Generally, globally hyperbolicity of the underlying spacetime, as in the case of $(\mN, g)$, entails the existence of the [**causal propagator**]{}, $E_{P_g}: A_{P_g}-R_{P_ g} : C_{0}^{\infty}(\mN; \bR) \to \sS(\mN)$ associated to $P_g$ and defined as the difference of the advanced and retarded fundamental solution [@Wald2; @BGP]. Furthermore $E_{P_g}: C_{0}^{\infty}(\mN; \bR) \to \sS(\mN)$ is linear, surjective with $Ker E_{P_g} = P_g(C_0^\infty(\mN; \bR))$ and it is continuous with respect to the natural topologies of both $C_{0}^{\infty}(\mN; \bR)$ and $C^{\infty}(\mN; \bR)$. Finally, given $\psi \in \sS(\mN)$ and any open neighbourhood $\mN'$ of any fixed smooth spacelike Cauchy surface of $\mN$, there exists $f_\psi \in C_{0}^{\infty}(\mN'; \bR)$ with $E_{P_g} f_\psi = \psi$. Consequently, $supp\; \psi \subset J^+(supp\; f_\psi; \mN) \cup J^-(supp\; f_\psi; \mN)$. The standard Hilbert space picture, where the generators $W_\mN(E_{P_g}f)$ are interpreted as exponentials of standard field operators, $e^{i\Phi(f)}$, can be introduced in the [*GNS representation*]{}, $(\gH_\omega, \Pi_\omega, \Psi_\omega)$, of any fixed algebraic state $\omega : \cW(\sS(\mN)) \to \bC$ [@Haag; @Wald2], such that the unitary one-parameter group $\bR \ni t \mapsto \Pi_\omega\left(W_\mN(t E_\mN f)\right)$ is strongly continuous. The [**field operators**]{} $\Phi_\omega(f)$ which arise as the self-adjoint generators of those unitary one-parameter groups, $\Pi_\omega\left(W_\mN( E_\mN t f)\right) = \exp\{i t\Phi_\omega(f)\}$, enjoy all the standard properties of usual quantisation procedure of Klein-Gordon scalar field based on CCR [@KW; @Wald2]. A different but equivalent definition is presented in the Appendix \[algebras\]. A physically important point, which would deserve particular attention, is the choice of physically meaningful states, but we shall just come back later to such issue. Algebras on $\cH$ and $\Im^\pm$ ------------------------------- \[secN\] Let us consider the case $\mN = \mM$, the latter being the physical part of Kruskal spacetime beforehand introduced. The null $3$-surfaces $\cH$, $\Im^\pm$, as well as, with a certain difference, $\cH_{ev}$ and $\cH^\pm$, can be equipped with a Weyl algebra of observables along the guidelines given in [@DMP2] and references therein. These play a central role in defining physically interesting states for $\cW(\sS(\mM))$ in the bulk. To keep the paper sufficiently self-contained, we briefly sketch the construction. Let $\cN$ be any $3$-submanifold of a spacetime – either $(\mM,g)$ or its conformal completion $(\widetilde{\mM},\widetilde{g})$ –, whose metric, when restricted to $\cN$, takes the [**complete Bondi form**]{}: c\_(- d d- dd+ h\_[\^2]{}(,) ) where $c_\cN$ is a non vanishing constant, while $(\ell, \Omega, \theta,\phi)$ defines a coordinate patch in a neighbourhood of $\cN$ seen as the locus $\Omega=0$ though such that $d\Omega\spa \rest_\cN \neq 0$. Out of this last condition we select $\ell\in \bR$ as a complete parameter along the integral lines of $(d\Omega)^a$ and, in view of the given hypotheses, $\cN$ turns out to be a null embedded codimension $1$-submanifold diffeomorphic to $\bR \times \bS^2$.[^1] It is possible to construct a symplectic space $(\sS(\cN),\si_\cN)$, where $\sS(\cN)$ is a real linear space of smooth real-valued functions on $\cN$ which includes $C_0^\infty(\cN; \bR)$ and such that the right-hand side of \[siN\] \_(,’) c\_\_ (’ - ) dd\^2, ,’ () can be interpreted in the sense of $L^1(\bR \times \bS^2; d\ell \wedge d\bS^2)$, where $d\bS^2$ is the standard volume form on $\bS^2$. Similarly to what it has been done in the bulk, since the only structure of symplectic space is necessary, one may define the Weyl algebra $\cW(\sS({\cN}))$, since the assumption that $C_0^\infty(\cN; \bR) \subset \sS(\cN)$ entails that $\sigma_\cN$ is non-degenerate, hence $\cW(\sS({\cN}))$ is well-defined.\ An interpretation of $\si_\cN $ can be given thinking of $\psi,\psi'$ as boundary values of fields $\vphi,\vphi'\in \sS(\mM)$. The right hand side of (\[siN\]) can then be seen as the integral over $\cN$ of the $3$-form $\eta[\vphi,\vphi']$ associated with $\vphi,\vphi'\in \sS(\mK)$ \[eta\] (\^a ’ - ’ \^a ) \_[abcd]{}dx\^b dx\^b dx\^c , where $\epsilon_{abcd}$ is totally antisymmetric with $\epsilon_{1234} =1$ and where $\psi\doteq \vphi \spa \rest_\cN$, $\psi'\doteq \vphi' \spa \rest_\cN$. Furthermore, in order to give a sense to the integration of $\eta[\vphi,\vphi']$ over $\cN$, we assume that $\cN$ is positively oriented with respect to its future-directed normal vector. The crucial observation is now that, integrating $\eta[\vphi,\vphi']$ over a spacelike Cauchy surface $\Sigma \subset \mM$, one gets exactly the standard symplectic form $\sigma_\mM(\vphi,\vphi')$ in (\[symp\]) (or that appropriate for the globally hyperbolic spacetime containing $\cN$). In view of the validity of the Klein-Gordon equation both for $\vphi$ and $\vphi'$, the form $\eta[\vphi,\vphi']$ satisfies $d \eta[\vphi,\vphi']=0$. Therefore one expects that, as a consequence of Stokes-Poincaré theorem it can happen that $\sigma_\mM(\vphi,\vphi') =\si_\cN (\vphi\spa\rest_\cN,\vphi'\spa\rest_\cN)$. If this result is valid, it implies the existence of an identification of $\cW(\sS(\mM))$ (or some relevant sub algebra) and $\cW(\sS(\cN))$. This is nothing but the idea we want to implement shortly with some generalisations. In the present case we shall consider the following manifolds $\cN$ equipped with the Bondi metric and thus the associated symplectic spaces $(\sS({\cN}),\sigma_{\cN})$: \(a) $\cH$ with $\ell \doteq U$ where $c_\cN = r_S^2$, $r_S$ being the Schwarzschild radius, \(b) $\Im^\pm$ with $\ell\doteq u$ or, respectively, $\ell \doteq v$ where $c_\cN= 1$. In the cases (b), the metric restricted to $\cN$ with Bondi form is the conformally rescaled and extended Kruskal metric $\widetilde{g}$, with $\widetilde{g}\spa\rest_\mM= g/r^2$, defined in the conformal completion $\widetilde{\mM}$ of $\mM$, as discussed in the Appendix \[geometry\].\ It is worth stressing that $\ell$ in Eq. (\[siN\]) can be replaced, without affecting the left-hand side of (\[siN\]), by any other coordinate $\ell' = f(\ell)$, where $f: \bR \to (a,b) \subset \bR$ is any smooth diffeomorphism. This allows us to consider the further case of symplectic spaces $(\sS({\cN}),\sigma_{\cN})$ where $\cN$ is: \(c) $\cH^\pm$ with $\ell\doteq u$ and $c_\cN = r_S^2$, independently from the fact that, in the considered coordinates, the metric $g$ over $\cH^\pm$ does not take the Bondi form. Injective isometric $*$-homomorphism between the Weyl algebras -------------------------------------------------------------- To conclude this section, as promised in the introduction, we establish the existence of some injective (isometric) $*$-homomorphisms which map the Weyl algebras in the bulk into Weyl subalgebras defined on appropriate subsets of the piecewise smooth null $3$-surfaces $\Im^-\cup \cH $. To this end we have to specify the definition of $\sS(\cH)$, $\sS(\cH^\pm)$ and $\sS(\Im^\pm)$. From now on, referring to the definition of the preferred coordinate $\ell$ as pointed out in the above-mentioned list and with the identification of $\cH$, $\cH_{ev}$, $\cH^\pm$ and $\Im^{\pm}$ with $\bR \times \bS^2$ as appropriate: \[SW\] () {C\^(\^2; )| M\_&gt;1, C\_,C’\_0 |(,,)| &lt; .. ,\ . |\_(,,)|&lt; ||&gt;M\_, (,) \^2 }, where $\ell = U$ on $\cH$, and $$\begin{aligned} \label{Sscri} \sS(\Im^\pm) \doteq &\left\{\psi \in C^\infty(\bR \times \bS^2; \bR)\:\left| \: \psi(\ell) = 0 \:\mbox{in a neighbourhood of $i^0$ and } \: \exists\exists C_\psi,C'_\psi \geq 0 \mbox{ with } \nonumber \right.\right.\\ & \left.\left. |\psi(\ell,\theta,\phi)| < \frac{C_\psi}{\sqrt{1+ |\ell|}} \right.\right. \:, \left. \left|\partial_\ell\psi(\ell,\theta,\phi)\right|< \frac{C'_\psi}{1+|\ell|} \:,\quad (\ell, \theta,\phi) \in \bR\times \bS^2 \right\}\:,\end{aligned}$$ where $\ell = u$ on $\Im^+$ or $\ell= v$ on $\Im^-$, and, finally, $$\begin{aligned} \label{SHpm} \sS(\cH_{ev})\:, \sS(\cH^\pm) \doteq &\left\{\psi \in C^\infty(\bR \times \bS^2; \bR)\:\left| \: \psi(\ell) = 0 \:\mbox{in a neighbourhood of $\cB$ and } \exists\exists C_\psi,C'_\psi \geq 0 \mbox{ with }\nonumber \right.\right.\\ & \left.\left. |\psi(\ell,\theta,\phi)| < \frac{C_\psi}{1+|\ell|} \right.\right. \:, \left. \left|\partial_\ell\psi(\ell,\theta,\phi)\right|< \frac{C'_\psi}{1+|\ell|}\:,\quad (\ell, \theta,\phi) \in \bR\times \bS^2 \right\}\:,\end{aligned}$$ where $\ell = v$ on $\cH_{ev}$ and $\ell = u$ on $\cH^\pm$ .\ It is a trivial task to verify that the above defined sets are real vector spaces; they include $C_0^\infty(\bR \times \bS^2; \bR)$ and, if $\psi$ belongs to one of them, $\psi \pa_\ell \psi \in L^1(\bR \times \bS^2, d\ell \wedge d\bS^2)$ as requested. Furthermore the above definitions rely upon the fact that the restrictions of the wavefunctions of $\sS(\mM)$ to the relevant boundaries of $\mM$ satisfy the fall-off conditions in (\[SW\]), (\[Sscri\]), (\[SHpm\]) while approaching $i^\pm$, a fact which will shortly play a crucial role.\ To go on, notice that, given two real symplectic spaces (with nondegenerate symplectic forms) $(\sS_1, \sigma_1)$ and $(\sS_2,\sigma_2)$, we can define the direct sum of them, as the real symplectic space $(\sS_1\oplus \sS_1, \sigma_1 \oplus \sigma_2)$, where the nondegenerate symplectic form $\sigma_1 \oplus \sigma_2 : (\sS_1\oplus \sS_2) \times (\sS_1\oplus \sS_2) \to \bR $ is \[ss\] \_1\_2 ((f,g), (f’,g’)) \_1(f,f’) + \_2(g,g’), If we focus on the Weyl algebras $\cW(\sS_1)$, $\cW(\sS_2)$, $\cW(\sS_1\oplus\sS_2)$, it is natural to identify the $C^*$-algebra $\cW(\sS_1\oplus\sS_2)$ with $\cW(\sS_1) \otimes \cW(\sS_2)$ providing, in this way, the algebraic tensor product of the two $C^*$-algebras with a natural $C^*$-norm (there is no canonical $C^*$-norm for the tensor product of two generic $C^*$-algebras). This identification is such that $W_{\sS_1\oplus \sS_2}((f_1,f_2))$ corresponds to $W_{\sS_1}(f_1)\otimes W_{\sS_2}(f_2)$ for all $f_1\in \sS_1$ and $f_2 \in \sS_2$. We are now in place to state and to prove the main theorems of this section, making profitable use of the results achieved in [@DR05]. Most notably, we are going to show that $\cW(\sS(\mM))$ is isomorphic to a sub $C^*$-algebra of $\cW(\sS(\cH))\otimes \cW(\sS(\Im^-))$. As a starting point, let us notice that, if $\vphi$ and $\vphi'$ are solutions of the Klein-Gordon equation with compact support on any spacelike Cauchy surface $\Sigma$ of $\mM$, the value of $\sigma_\mM(\vphi,\vphi' )$ is independent on the used $\Sigma$ and, therefore, we can deform it preserving the value of $\sigma_\mM( \vphi,\vphi')$. A tricky issue arises if one performs a limit deformation where the final surface tends to $ \cH \cup \Im^-$ since \_(,’) = \_( \_,’\_) + \_[\^-]{}( \_[\^-]{},’\_[\^-]{}), \[fres\] where the arguments of the symplectic forms in the right-hand side (which turns out to belong to the appropriate spaces (\[SW\]), (\[Sscri\])) are obtained either as restrictions to $\cH$ or as (suitably rescaled) limit values towards $\Im^-$ of both $\vphi$ and $\vphi'$. As the map $\vphi \mapsto (\vphi_{\cH}, \vphi_{\Im^-})$ is linear and the sum of the above symplectic forms is the symplectic form $\sigma$ on $\sS \doteq \sS(\cH) \oplus \sS(\Im^-)$, this entails that we have built up a symplectomorphism from $\sS(\mM)$ to $\sS$, $\vphi \mapsto (\vphi_{\cH}, \vphi_{\Im^-})$ which must be injective. In view of known theorems [@BR2], this entails the existence of an isometric $*$-homomorphism $\imath : \cW(\sS(\mM)) \to \cW(\sS(\cH))\otimes \cW(\sS(\Im^-)) $. Our goal now is to formally state and to prove the result displayed in (\[fres\]).\ \[Main1\] [*For every $\vphi \in \sS(\mM)$, let us define $$\vphi_{\Im^-} \doteq \lim_{\to \Im^-} r\vphi\:, \quad \mbox{and}\quad \vphi_{\cH} \doteq \vphi\rest_{\cH}\:.$$ Then the following facts hold.*]{} .2cm [*[**(a)**]{} The linear map $$\Gamma : \sS(\mM) \ni \vphi \mapsto (\vphi_{\Im^-},\vphi_{\cH})\:,\quad$$ is an injective symplectomorphism of $\sS(\mM)$ into $\sS(\Im^-) \oplus \sS(\cH)$ equipped with the symplectic form, such that, for $\vphi,\vphi' \in \sS(\mM)$: \[sigmas\] \_(,’) \_[\^-]{}( \_[\^-]{},’\_[\^-]{}) + \_[\_]{}( \_,’\_). There exists a corresponding injective isometric $*$-homomorphism $$\imath : \cW(\sS(\mM))\spa \to \cW(\sS(\Im^-))\otimes \cW(\sS(\cH))\:,$$ which is unambiguously individuated by $$\imath\left(W_\mM(\vphi)\right) = W_{\Im^-}\left(\vphi_{\Im^-}\right) \otimes W_{\cH} \left(\vphi_{\cH} \right).$$*]{} [*Proof*]{}. Let us start from point (a). If $\varphi \in \sS(\mM)$, we can think of it as a restriction to $\mM$ of a solution $\varphi'$ of the Klein-Gordon equation in the whole Kruskal manifold. To this end one should also notice that the initial data of $\varphi$ on a spacelike Cauchy surface of $\mM$ can also be seen as initial data on a spacelike Cauchy surface of the whole Kruskal manifold. This is a direct application of the results in [@BS06] and [@ON]. Therefore $\vphi_{\cH} \doteq \vphi'\rest_{ \cH}$ is well-defined and smooth. Similarly, the functions $\vphi_{\Im^-} \doteq \lim_{\to \Im^-} r\vphi$ are well defined, smooth and vanish in a neighbourhood of the relevant $i^0$ in view of the following lemma whose proof is sketched in the Appendix \[Appendixproofs\]. \[lemma1\] [*If $\vphi \in \sS(\mM)$, $r\vphi$ uniquely extends to a smooth function $\widetilde{\vphi}$ defined in $\mM$ joined with open neighbourhoods of $\Im^+$ and $\Im^-$ included in the conformal extension $\widetilde{\mM}$ of $\mM$ discussed in the Appendix \[geometry\]. Furthermore, there are constants $v^{(\vphi)}, u^{(\vphi)} \in (-\infty,\infty)$ such that $\widetilde{\vphi} $ vanishes in $\mW$ if $u<u^{(\vphi)}, v> v^{(\vphi)}$ and thus, per continuity, it vanishes in the corresponding limit regions on $\Im^+ \cup \Im^-$.*]{}\ Since the map $\Gamma$ is linear by construction, it remains to prove that (i) $\vphi_{\cH} \in \sS(\cH)$ and $\vphi_{\Im^\pm}\in \sS(\Im^\pm)$ as defined in (\[SW\]) and (\[Sscri\]), and that (ii) $\Gamma$ preserves the symplectic forms, [*i.e.*]{}, \[sigmas2\] \_(\_1, \_2) = \_[() (\^-)]{}( \_1, \_2). Notice that, since $\sigma_{\mM}$ is nondegenerate, the above identity implies that the linear map $\Gamma$ is injective. Let us tackle point (i): since the behaviour of $\vphi_{\Im^-}$ in a neighbourhood of $i^0$ is harmless, we only need to establish the vanishing of both $\vphi_{\cH}$ and $\vphi_{\Im^-}$ as they approach $i^-$, with a peeling-off rate consistent with that of definitions (\[SW\]) and (\[Sscri\]). Such a result is a consequence of the following proposition whose proof, in Appendix \[Appendixproofs\], enjoys a lot from [@DR05]. \[PropDR\] [*Let us fix $\hat R > r_S$, then the following facts hold:\ [**(a)**]{} If $\vphi \in \sS(\mM)$ and $\widetilde{\vphi}$ extends $r\vphi$ across $\Im^\pm$ as stated in Lemma \[lemma1\], there exist constants $C_1,C_2\geq 0$ depending on both $\vphi$ and $C_3,C_4$ depending on $\vphi$ and $\hat{R}$, such that the following pointwise bounds hold in both $\mW\cup \cH_{ev}$ and $\mW\cup \cH^-$: \[stime1\] || ,|X()| , and, respectively, \[stime1agg\] || ,|X()|. Similarly, if one assumes also $r \geq \hat R$ and $t > 0$ (including the points on $\Im^+$), \[stime2\] || , |X()|, or, if $r \geq \hat R$ but $t < 0$ (including the points on $\Im^-$), $$|\widetilde{\vphi}|\leq \frac{C_3}{\sqrt{1 +|v|}} \;, \qquad |X(\widetilde{\vphi})|\leq \frac{C_4}{1+ |v|}.$$ $X$ is the smooth Killing vector field on the conformally extended Kruskal spacetime with $X =\partial_t$ in $\mW$, $X = \partial_v$ on $\cH_{ev}$, $X = \partial_u$ on $\cH^-$, $X = \partial_u$ on $\Im^+$ and $X = \partial_v$ on $\Im^-$.\ [**(b)**]{} If the Cauchy data $(\vphi\spa \rest_{\Sigma}, \nabla_n \vphi\spa \rest_\Sigma)$ on $\Sigma\hookrightarrow \mK$ of $\vphi$ tend to $0$ in the sense of the test function (product) topology on $C^\infty_0(\Sigma; \bR)$, then the associated constants $C_i$ tend to $0$, for $i=1,3$.\ If the Cauchy data $(\vphi'\spa \rest_{\Sigma}, \nabla_n \vphi'\spa \rest_\Sigma)$ on $\Sigma\hookrightarrow \mK$ of $\vphi' \doteq X(\vphi)$ tend to $0$ in the sense of the test function (product) topology on $C^\infty_0(\Sigma; \bR)$, then the associated constants $C_i$ tend to $0$, for $i=2,4$.*]{}\ It is noteworthy to emphasise that, during the final stages of the realization of this paper, a new result on the peeling-off behaviour of the solutions of the wave equation in Schwarzschild black-hole was made public [@Luk]. Particularly the decay rate on the horizon has been improved; nonetheless, to our purposes, the original one obtained by Dafermos and Rodnianski suffice. Since $\vphi$ and $\widetilde{\vphi}$ are smooth, $X(\vphi)=\partial_u\vphi$ on $\cH^-$ and $X(\widetilde{\vphi})= \partial_v\widetilde\vphi$ on $\Im^-$, it comes out, per direct inspection, that $\vphi_{\cH} \in \sS(\cH)$ and $\vphi_{\Im^-}\in \sS(\Im^-)$ since the definitions (\[SW\]) and (\[Sscri\]) are fulfilled, for $\ell = U$ and $\ell =v$ respectively; furthermore, in view of the last statement of the above proposition, it holds $\vphi_{\Im^+}\in \sS(\Im^+)$.\ In order to conclude, let us finally prove item (ii), that is (\[sigmas2\]), making use once more of Proposition \[PropDR\]. Let us consider $\vphi,\vphi' \in \sS(\mM)$ and a spacelike Cauchy surface $\Si_\mM$ of $\mM$ so that, $$\sigma_{\mM}(\vphi,\vphi') = \int_{\Si_\mM} \left(\vphi' \nabla_n\vphi - \vphi \nabla_n\vphi'\right)\: d\mu_g(\Si_\mM),$$ where $n$ is the unit normal to the surface $\Si_\mM$ and $\mu_g(\Si_\mM)$ is the metric induced measure on $\Si_\mM$ and, in the following, we shall write $d\mu_g$ in place of $d\mu_g(\Sigma_\mM)$ and $\Sigma$ in place of $\Sigma_{\mM}$ . Since both $\vphi$ and $\vphi'$ vanish for sufficiently large $U$, we can use the surface $\Sigma$, defined as the locus $t=0$ in $\mW$, and, out of the Poincaré theorem (employing the $3$-form $\eta$ as discussed in Sec. \[secN\]), we can write \[symplecticT\] \_(,’) = \_ ’ X() - X(’) d\_g + r\_S\^2 \_[\^+]{} ( ’ \_U- \_U ’) dU d\^2, where we have used the fact that $\cB \cap \Sigma$ has measure zero. We shall prove that, if one restricts the integration to $\mW$, \_ ’ X() - X(’) d\_g = r\_S\^2 \_[\^-]{} ( ’ \_U- \_U ’) dU d\^2\ + \_[\^-]{}( \_v- \_v)du d\^2. \[fine1\] Since, with the same procedure, one gets an analogous statement for the portion of the initial integration taken in $\mW$, though with the integration in $dU$ extended over $\bR^-$ and the remaining one on $\Im^-$, this will conclude the proof.\ In order to prove the identity (\[fine1\]), we notice at first that: $$\begin{gathered} \int_{\Sigma\cap \mW} \vphi' X(\vphi) - \vphi X(\vphi') d\mu_g = \int_{[r_S, +\infty) \times\bS^2} \left.\frac{r^2 \left( \vphi' X(\vphi) - \vphi X(\vphi')\right)}{1-2m/r}\right|_{(t=0,r,\theta,\phi)} dr \wedge d\bS^2(\theta,\phi) \:. \end{gathered}$$ Afterwards, we break the integral on the right-hand side into two pieces with respect to the coordinate $r^*$: $$\begin{gathered} \int_{[r_S, +\infty) \times\bS^2} \left.\frac{r^2 \left( \vphi' X(\vphi) - \vphi X(\vphi')\right)}{1-2m/r}\right|_{(t=0,r,\theta,\phi)} dr \wedge d\bS^2 \nonumber \\ = \int_{(-\infty, \hat R^*) \times\bS^2} r^2 \left.\left( \vphi' X(\vphi) - \vphi X(\vphi')\right)\right|_{(t=0,r^*,\theta,\phi)} dr^* \wedge d\bS^2 \nonumber\\ + \int_{[\hat R^*, +\infty) \times\bS^2} \left.\left( r\vphi' X(r\vphi) - r\vphi X(r\vphi')\right)\right|_{(t=0,r^*,\theta,\phi)} dr^* \wedge d\bS^2\:.\end{gathered}$$ We started assuming $\Sigma$ as the surface $t=0$ in ; however, the value of $t$ is immaterial, since we can work, with a different surface $\Sigma_t$ obtained by evolving $\Sigma$ along the flux of the Killing vector $X$. We remind that $X = \partial_t$ in $\mW$ and $X=0$ exactly on $\cB$, which, as a consequence, is a fixed submanifold of the flux. Furthermore we also know that the symplectic form $\sigma_{\mM}(\vphi,\vphi')$ is constructed in such a way that its value does not change varying $t$, by construction. Since $\cB$ is fixed under the flux of $X$, per direct application of Stokes-Poincaré theorem, one sees that this invariance holds also for the integration restricted to $\mW$. In other words, for every $t>0$: $$\begin{gathered} \int_{\Sigma\cap \mW} \sp\sp\sp\vphi' X(\vphi) - \vphi X(\vphi') d\mu_g = \int_{\Sigma_t\cap \mW} \sp\sp\sp\vphi' X(\vphi) - \vphi X(\vphi') d\mu_g = \int_{\bR \times\bS^2} \sp\sp r^2\left.\left(\vphi' X(\vphi) - \vphi X(\vphi')\right)\right|_{(t,v-t, \theta,\phi)} \sp dv \wedge d\bS^2 \nonumber \\ + \int_{\bR \times\bS^2}\sp \sp\;\left.\left(\widetilde{\vphi'} X(\widetilde\vphi) - \widetilde\vphi X(\widetilde{\vphi'})\right)\right|_{(t,t-u,\theta,\phi)} \sp du \wedge d\bS^2\nonumber \:,\end{gathered}$$ where we have also changed the variables of integration from $r^*$ either to $v= t+r^*$ or to $u=t-r^*$. Hence \_ ’ X() - X(’) d\_g = \_[t-]{}\_[\^2]{} .r\^2(’ X() - X(’)) |\_[(t,t-u,,)]{} du d\^2\ + \_[t-]{}\_[\^2]{}.( X() - X())|\_[(t,v-t,,)]{} dv d\^2\[dec\] . The former limit should give rise to an integral over $\cH^-$, whereas the latter to an analogous one over $\Im^-$. Let us examine them separately and we start from the latter. To start with we notice that, in view of (a) in Lemma \[lemma1\], the integration in $v$ can be performed in $(-\infty,v_0]$ for some constant $v_0 \in \bR$, without affecting the integral for every $t<0$. Therefore $$\begin{gathered} \lim_{t\to -\infty}\int_{\bR \times\bS^2}\sp\left.\left(\widetilde{\vphi'} X(\widetilde\vphi) - \widetilde \vphi X(\widetilde{\vphi'})\right)\right|_{(t,v-t,\theta,\phi)} \sp dv \wedge d\bS^2 =\\ \lim_{t\to -\infty} \int_{(-\infty,v_0] \times\bS^2}\sp\left.\left(\widetilde{\vphi'} X(\widetilde\vphi) - \widetilde\vphi X( \widetilde{\vphi'})\right)\right|_{(t,v-t,\theta,\phi)} \sp dv \wedge d\bS^2\:.\end{gathered}$$ In view of the uniform bounds, associated with the constants $C_3$ and $C_4$, given by $v$-integrable functions in $(\infty,v_0]$, as stated in Proposition \[PropDR\], we can now apply Lebesgue’s dominated convergence theorem to the limit in the right-hand side: \[intone\] \_[t-]{}\_[\^2]{}.( X() - X())|\_[(t,v-t,,)]{} dv d\^2 = \_[\^-]{}( \_v- \_v) dv d\^2. Let us now consider the remaining integral on the right-hand side of (\[dec\]). To this end, let us fix $u_0 \in \bR$ and the following decomposition $$\begin{gathered} \int_{\bR \times\bS^2} \sp\sp r^2\left.\left( \vphi' X(\vphi) - \vphi X(\vphi')\right)\right|_{(t,t-u,\theta,\phi)} \sp du \wedge d\bS^2 =\int_{\Sigma^{(u_0)}_t} \vphi' X(\vphi) - \vphi X(\vphi') d\mu_g\nonumber \\ + \int_{(-\infty,u_0] \times\bS^2}\sp\sp r^2\left.\left(\vphi' X(\vphi) - \vphi X(\vphi')\right)\right|_{(t, t-u,\theta,\phi)} \sp\; du \wedge d\bS^2\:.$$ Here we have used the initial expression for the first integral, which is performed over the compact subregion $\Sigma^{(u_0)}_t$ of $\Sigma_t\cap \mW$ which contains the points with null coordinate $U$ included in $[-\exp\{-u_0/(4m)\} ,0]$. It is noteworthy that such integral is indeed the one of the smooth $3$-form $\eta \doteq \eta [\vphi, \vphi']$ defined in (\[eta\]) and, furthermore, in view of Klein-Gordon equation, $d\eta = 0$. Thus, by means of an appropriate use of the Stokes-Poincaré theorem, this integral can be re-written as an integral of $\eta$ over two regions. The first is a compact subregion of $\cH^+$ which can be constructed as the points with coordinate $U\in [U_0,0]$, where $U_0 \doteq -e^{-u_0/(4m)}$; the second, instead is the compact null $3$-surface $S^{(u_0)}_t$ formed by the points in $\mM$ with $U=U_0$ and lying between $\Sigma_t$ and $\cH^-$. To summarise: $$\int_{\Sigma^{(u_0)}_t} \vphi' X(\vphi) - \vphi X(\vphi') d\mu_g = \int_{\cH^- \cap \{U_0\leq U \leq 0\}}\eta + \int_{S^{(u_0)}_t}\eta \:.$$ If we adopt coordinates $U,V,\theta,\phi$, the direct evaluation of the first integral on the right-hand side produces: $$\int_{\Sigma^{(u_0)}_t} \vphi' X(\vphi) - \vphi X(\vphi') d\mu_g = r_S^2 \int_{\cH^- \cap \{U_0\leq U \leq 0\}}\left( \vphi' \partial_U\vphi - \vphi \partial_U \vphi' \right) dU \wedge d\bS^2 + \int_{S^{(u_0)}_t}\eta \:.$$ We have obtained \_[t-]{}\_[\^2]{} r\^2.( ’ X() - X(’))|\_[(t,t-u,,)]{} du d\^2 = r\_S\^2 \_[\_R\^+ {U\_0U 0}]{}( ’ \_U- \_U ’ ) dU d\^2\ + \_[t-]{} \_[S\^[(u\_0)]{}\_t]{}+ \_[t-]{} \_[(-,u\_0\] \^2]{} r\^2.( ’ X() - X(’))|\_[(t,t-u,,)]{} du d\^2.\[agg\] If we perform the limit as $t\to -\infty$, one has $\int_{S^{(u_0)}_t}\eta \to 0$, because it is the integral of a smooth form over a vanishing surface (as $t\to -\infty$), whereas $$\begin{gathered} \lim_{t\to -\infty} \int_{(-\infty,u_0] \times\bS^2} \sp\sp r^2\left.\left( \vphi' X(\vphi) - \vphi X(\vphi')\right)\right|_{(t,t-u,\theta,\phi)} \sp\; du \wedge d\bS^2 = \int_{\cH^- \cap \{u_0 \geq u\}} r_S^2\left( \vphi' \partial_u \vphi - \vphi \partial_u\vphi' \right) du \wedge d\bS^2 \nonumber \\ = r^2_S \int_{\cH^- \cap \{U_0\geq U\}}\left( \vphi' \partial_U\vphi - \vphi \partial_U \vphi' \right) dU \wedge d\bS^2,\end{gathered}$$ where we stress that the final integrals are evaluated over $\cH^-$ and we have used again Lebesgue’s dominated convergence theorem thanks to the estimates associated with the constants $C_1$ and $C_2$ in Proposition \[PropDR\]. Inserting the achieved results in the right-hand side of (\[agg\]), we find that: $$\lim_{t\to -\infty} \int_{\bR \times\bS^2} \sp\sp \left.r^2\left( \vphi' X(\vphi) - \vphi X(\vphi')\right)\right|_{(t,t-u,\theta,\phi)} \sp du\; \wedge d\bS^2 = r_S^2 \int_{\cH^-}\left( \vphi' \partial_U\vphi - \vphi \partial_U \vphi' \right) dU \wedge d\bS^2_{ \bS^2}\:.$$ Such identity, brought in (\[dec\]), yields, together with (\[intone\]), (\[fine1\]), hence concluding the proof of (a).\ Item (b) can be proved as follows. In the following $\sS\doteq \sS(\cH) \oplus \sS(\Im^-)$ and $\sigma$ is the natural symplectic form on such space. Let us consider the closure of the sub $*$-algebra generated by all the generators $W_{\sS}(\Gamma \vphi) \in \cW(\sS)$ for all $\vphi\in \sS(\mM)$. This is still a $C^*$-algebra which, in turn, defines a realization of $\cW(\sS(\mM))$ because $\Gamma$ is an isomorphism of the symplectic space $(\sS(\mM), \sigma_{\mM})$ onto the symplectic space $(\Gamma(\sS(\mM)), \sigma\spa\rest_{\Gamma(\sS(\mM)) \times \Gamma(\sS(\mM))})$. As a consequence of Theorem 5.2.8 in [@BR2], there is a $*$-isomorphism, hence isometric, between $\cW(\sS(\mM))$ and the other, just found, realization of the same Weyl algebra, unambiguously individuated by the requirement $\imath_{\mM}(W_{\mM})(\vphi)\doteq W_{\sS}(\Gamma \vphi)$. This isometric $*$-isomorphism individuates an injective $*$-homomorphism of $\cW(\sS(\mM))$ into $\cW(\sS,\sigma)\equiv \cW(\sS(\cH))\otimes \cW(\sS(\Im^-))$. $\Box$\ As a byproduct and a straightforward generalisation, the proof of the above theorem also establishes the following: \[Main3\][With the same definitions as in Theorem \[Main1\] and defining, for $\vphi \in \sS(\mW)$, $\vphi_{\cH^-} \doteq \lim_{\to \cH^-} \vphi$ and $\vphi_{\cH_{ev}} \doteq \lim_{\to \cH_{ev}} \vphi$, the linear maps $$\Gamma_- : \sS(\mW) \ni \vphi \mapsto (\vphi_{\cH^-},\vphi_{\Im^-}) \in \sS(\cH^-) \oplus \sS(\Im^-)\:,\quad \Gamma_+ : \sS(\mW) \ni \vphi \mapsto (\vphi_{\cH_{ev}},\vphi_{\Im^+}) \in \sS(\cH_{ev}) \oplus \sS(\Im^+)$$ are well-defined injective symplectomorphisms. As a consequence, there exists two corresponding injective isometric $*$-homomorphisms: $$\imath^- : \cW(\sS(\mW)) \to \cW(\sS(\cH^-)) \otimes \cW(\sS(\Im^-))\:,\quad \imath^+ : \cW(\sS(\mW)) \to \cW(\sS(\cH_{ev})) \otimes \cW(\sS(\Im^+)) \:,$$ which are respectively unambiguously individuated by the requirements for $\vphi\in\sS(\mW)$ $$\quad\imath^-\left(W_{\mW}(\vphi)\right) = W_{\cH^-} \left(\vphi_{\cH^-} \right)\otimes W_{\Im^-}\left(\vphi_{\Im^-}\right)\:, \quad \quad\imath^+\left(W_{\mW}(\vphi)\right) = W_{\cH_{ev}} \left(\vphi_{\cH_{ev}} \right)\otimes W_{\Im^+}\left(\vphi_{\Im^+}\right) .$$]{}\ Before the conclusion of the present section, we would like to stress that a result similar to the one presented in Theorem \[Main1\] and in Theorem \[Main3\] can be obtained for the algebra of observables defined on the whole Kruskal extension $\mK$ of the Schwarzschild spacetime. In such case, an injective isometric $*$-homomorphisms $ \imath_\mK : \cW(\sS(\mK)) \to \cW(\sS(\Im^+_L)) \otimes \cW(\sS(\cH)) \otimes \cW(\sS(\Im^-)) \: $ can be constructed out of the projection $ \Ga_\mK: \sS(\mK) \ni \vphi \mapsto (\vphi_{\Im^+_L},\vphi_{\cH},\vphi_{\Im^-}) \in \sS(\Im^+_L)\oplus \sS(\cH) \oplus \sS(\Im^-) $ from the requirement $ \quad\imath_\mK\left(W_{\mK}(\vphi)\right) = W_{\Im^+_L} \left(\vphi_{\Im^+_L} \right)\otimes W_{\cH} \left(\vphi_{\cH} \right)\otimes W_{\Im^-}\left(\vphi_{\Im^-}\right) $ where $\Im^+_L$ stands for the future null infinity of the left Schwarzschild wedge in the Kruskal spacetime $\mK$. Interplay of bulk states and boundary states. ============================================= Bulk states induced form boundary states by means of the pullback of $\imath$ and $\imath^-$ -------------------------------------------------------------------------------------------- In this section we construct the mathematical technology to induce algebraic states (see Appendix \[algebras\]) on the algebras $\cW(\sS(\mM))$ and $\cW(\sS(\mW))$ from those defined, respectively, on $\cW(\sS(\cH)) \otimes \cW(\sS(\Im^-))$ and $\cW(\sS(\cH^-)) \otimes \cW(\sS(\Im^-))$. A bit improperly, we shall call [**bulk states**]{} those with respect to $\cW(\sS(\mM))$ and on the other subalgebra defined above while [**boundary states**]{} will be called those on $\cW(\sS(\cH))\otimes \cW(\sS(\Im^-))$. To this end, the main tools are Theorem \[Main1\] and \[Main3\].\ Let us consider the case of $\cW(\sS(\mM))$ as an example. If the linear functional $\omega : \cW(\sS(\cH))\otimes \cW(\sS(\Im^-)) \to \bC$ is an algebraic state, the isometric $*$-homomorphism $\imath$ constructed in Theorem \[Main1\] gives rise to $\omega_{\mM}:\cW(\sS(\mM))\to\bC$ defined as \[inductedstate\] \_ \^\*(), (\^\*())(a) ((a) ) , A similar conclusion can be drawn using $\imath^-$ for the corresponding algebra. The situation will now be specialised to [*quasifree states*]{} and, as discussed in Appendix \[algebras\], one of these can be unambiguously defined on $\cW(\sS(\cH))\otimes \cW(\sS(\Im^-))$, just requiring that $$\omega_\mu\left(W_{\cH \cup \Im^-}(\psi)\right) = e^{-\mu(\psi,\psi)/2} \:, \quad \mbox{for all $\psi \in \sS(\cH) \oplus \sS(\Im^-)$\:,}$$ where $\mu: (\sS(\cH) \oplus \sS(\Im^-)) \times (\sS(\cH) \oplus \sS(\Im^-)) \to \bR$ is a real scalar product satisfying (\[sm\]). Furthermore the “quasi-free”-property is stable under pull-back, [*i.e.*]{}, if (\[inductedstate\]) is quasifree, then $\omega_{\mM}$ is such. Therefore, we can simply turn our attention to quasifree states defined on the boundaries $\cW(\sS(\Im^\pm))$, $\cW(\sS(\cH))$, and on the possible [*composition*]{} of such states in view of the following proposition. \[propstates\] [*Let $(\sS_1,\sigma_1)$, $(\sS_2,\sigma_2)$ be symplectic spaces and $\omega_1$, $\omega_2$ be two quasifree algebraic states on $\cW(\sS_1,\sigma_1)$ and $\cW(\sS_2,\sigma_{2})$, induced respectively by the real scalar products $\mu_1 : \sS_1 \times \sS_1 \to \bR$ and $\mu_2: \sS_2 \times \sS_2 \to \bR$. Then the scalar product $\mu_1 \oplus \mu_2 : (\sS_1\oplus \sS_2) \times (\sS_1\oplus \sS_2) \to \bR$ defined by: $$\mu_{\sS_1 \oplus \sS_2}((\psi_1,\psi_2), (\psi'_1,\psi'_2)) \doteq \mu_1(\psi_1,\psi'_1) + \mu_2(\psi_2,\psi'_2)\:,\quad \mbox{for all $(\psi_1,\psi_2), (\psi'_1, \psi'_2) \in \sS_1 \oplus \sS_2$}\:,$$ uniquely individuates a quasifree state $\omega_1\otimes \omega_2$ on $\cW(\sS_1) \otimes \cW(\sS_2)$ as $$\omega_1 \otimes \omega_2\left(W_{\sS_1}(\psi_1)\otimes W_{\sS_2}(\psi_2)\right) = e^{-\mu_1 \oplus \mu_2((\psi_1,\psi_2), (\psi_1',\psi'_2))/2} \:, \quad \mbox{for all $(\psi_1,\psi_2)\in \sS_1 \oplus \sS_2$}\:.$$*]{} [*Proof*]{}. Only the validity of (\[sm\]) for $\mu_1\oplus \mu_2$ has to be proved with respect to $\sigma_1\oplus \sigma_2$ defined in (\[ss\]). This fact immediately follows from the definition of $\mu_1\oplus \mu_2$ and making use of (2) in remark \[remarkstates\]. $\Box$\ We can iterate the procedure in order to consider the composition of three (or more) states on corresponding three (or more) Weyl algebras. Hence, in view of the established proposition we may study separately the quasifree states on the Weyl algebras $\cW(\sS(\cN))$ associated to the null surfaces $\cN$ (a)-(c) listed in Sec. \[secN\]. The Kay-Wald quasifree state on $\cW(\cH)$ ------------------------------------------ We remind the reader that, if $\mu$ individuates a quasifree state over $\cW(\sS,\sigma)$, its [**two-point function**]{} is defined as $\lambda_\mu(\psi_1,\psi_2) \doteq \mu(\psi_1,\psi_2) - \frac{i}{2}\sigma(\psi_1,\psi_2)$ (see Appendix \[algebras\]). When one focuses on the one-particle space structure $(K_\mu, \sH_\mu)$ (see Appendix \[algebras\]) one has $\lambda_\mu (\psi_1,\psi_2) = \langle K_\mu \psi_1, K_\mu \psi_2 \rangle_\mu$, where $\langle \cdot,\cdot \rangle_\mu$ is the scalar product in $\sH_\mu$. The two-point function of a quasifree state on a given Weyl algebra brings in the same information as the scalar product $\mu$ itself since the symplectic form is known [*a priori*]{}; thus the two-point function individuates the state completely.\ In [@KW], some properties are discussed for a particular state on $\cW(\sS(\mK))$, where $\mK$ is the whole Kruskal extension of the Schwarzschild spacetime. If existent, such state was proved to be unique with respect to certain algebras of observables and to satisfy the KMS property when one works on a suitable algebra of observables in $\mW$. From a physical perspective, this is nothing but the celebrated Hartle-Hawking state when the background is the whole Kruskal spacetime. It is important to remark that, in [@KW], general globally-hyperbolic spacetimes with bifurcate Killing horizon are considered, whereas our work only focuses on $\mM$. As an intermediate step, Kay and Wald also showed that the two-point function $\lambda_{KW}$ of the state has a very particular form when restricted to the horizon $\cH$, more precisely \[lKW\] \_[KW]{}(\_1,\_2) = \_[0\^+]{}- \_[\^2]{} dU\_1 dU\_2 d\^2. $\vphi_1\spa\rest_{\cH},\vphi_2\spa\rest_{\cH} \in C_0^\infty(\bR\times \bS^2; \bR)$. It is important to stress that the above expression is valid when $\vphi_1\spa\rest_{\cH}$ and $\vphi_2\spa\rest_{\cH}$ have compact support on $\cH$. Actually, the same two-point function was already found both in [@Sewell], while discussing the physical consequences of the Bisognano-Wichmann theorem, and in [@DimockKay], while analysing the various states in the right Schwarzschild wedge $\mW$ of the Kruskal manifold with an [*$S$-matrix*]{} point of view. In the latter paper the two-point function in (\[lKW\]) was referred to the Killing horizon in the [*two-dimensional Minkowski spacetime rather than Kruskal one*]{}. In such a case there are smooth solutions of the Klein-Gordon equation, for $m\geq 0$, and with compactly supported Cauchy data, which intersect the horizon in a compact set. These solutions of the [*characteristic Cauchy problem*]{} can be used in the right-hand side of (\[lKW\]) when the discussion is referred to Minkowski spacetime instead of the Kruskal one. These “Minkowskian solutions”, at least in the case $m=0$ where asymptotic completeness was proved to hold, are related to the corresponding solutions ([*i.e.*]{}, $\vphi_1,\vphi_2$) in Schwarzschild spacetime by means of a relevant Møller operator. Unfortunately, in the proper Schwarzschild space, the wavefunctions $\vphi_1$ and $\vphi_2$ with compact support on $\cH$ fail to be smooth in general, since they are [*weak solutions*]{} of the characteristic problem [@DimockKay], hence they do [*not*]{} belong to the space $\sS(\mK)$ in general, making difficult the direct use of $\lambda_{KW}$. This is a potential issue in [@KW] which has minor consequences for the validity of the KMS property discussed below (see also the [*Note added in proof*]{} in [@KW] for more details).\ We shall now prove that, actually, such form of the two-point function can be extended in order to work on elements of $\sS(\cH)$ and, with this extension, it defines a quasifree state on $\cW(\sS(\cH))$. This result is by no means trivial, because the space $\sS(\cH)$ contains the restrictions to the horizon of the various elements of $\sS(\mK)$, that is all the smooth wavefunctions with compact support on spacelike Cauchy surfaces. Our result, which is valid for the particular case of the Kruskal spacetime and for $m=0$, is obtained thanks to the achievements recently presented in [@DR05]. At the same time the space $\sS(\cH)$ is just the one used in the hypotheses of theorem \[Main1\], which assures the existence of the $*$-homomorphism $\imath$. As remarked at the end of the previous section, the procedure can be generalised in order to individuate an injective $*$-homomorphism from the algebra of observables on the whole Kruskal space to the algebras on $\Im_L^+$, $\cH$ and $\Im^-$, that is $\imath_\mK : \cW(\sS(\mK)) \to \cW(\sS(\Im_L^+)) \otimes \cW(\sS(\cH)) \otimes \cW(\sS(\Im^-))$. Therefore, the state on $\sS(\mK)$ could be used, together with a couple of states on $\cW(\Im^-)$ and on $\cW(\Im^+_L)$ to induce a further one on the whole algebra of observables $\cW(\sS(\mK))$. This should provide an [*existence theorem*]{} for the Hartle-Hawking state on the whole Kruskal manifold $\mK$. However we shall not attempt to give such an existence proof here and we rather focus attention on another physically interesting state, the so called [*Unruh vacuum*]{} defined only in the submanifold $\mM$. Nevertheless, even in this case we have to tackle the problem of the extension of the two-point function (\[lKW\]) to the whole space $\sS(\cH)$. We shall prove the existence of such an extension that individuates, moreover, a pure quasifree state on $\cW(\sS(\cH))$, and which turns out to be KMS at inverse Hawking’s temperature when restricting on a half horizon $\cW(\sS(\cH^\pm))$ with respect to the Killing displacements given by $X\spa \rest_{\cH}$. The way we follow goes on through several steps. As a first step we introduce a relevant Hilbert space which we show later to be the one-particle space of the quasifree state we wish to define on $\cW(\sS(\cH))$. The proof of the following proposition is in Appendix \[Appendixproofs\]. From now on, $$\mF(\psi) (K,\theta,\phi) \doteq \int_{\bR}\frac{e^{iKU}}{\sqrt{2\pi}} \psi(U,\theta,\phi) dU\:,$$ indicates the $U$-Fourier transform of $\psi$, also in the $L^2$ (Fourier-Plancherel) sense or even in distributional sense if appropriate. For all practical purposes, the properties are essentially the same as for the standard Fourier transform[^2]. \[PropMain4\] *Let $\overline{\left(C_0^\infty(\cH; \bC), \lambda_{KW} \right)}$ be the Hilbert completion of the complex vector space $C_0^\infty(\cH; \bC)$ equipped with the Hermitian scalar product: \[lKW2\] \_[KW]{}(\_1,\_2) \_[0\^+]{} - \_[\^2]{}dU\_1 dU\_2 d\^2 . where $\cH \equiv \bR \times \bS^2$ adopting the coordinate $(U,\theta,\phi)$ over $\cH$. Denote by $\widehat{\psi}_+\doteq\mF(\psi)\spa\rest_{\{K\geq 0,\theta,\phi \in \bS^2\}}$ the restriction to positive values of $K$ of the $U$-Fourier transform of $\psi \in C_0^\infty(\cH; \bC)$. The following facts hold.* [**(a)**]{} The linear map $$C_0^\infty(\cH; \bC) \ni \psi \mapsto \widehat{\psi}_+(K,\theta,\phi) \in L^2(\bR_+\times \bS^2, 2KdK \wedge r_S^2d\bS^2) \doteq \sH_{\cH}$$ is isometric and uniquely extends, by linearity and continuity, to a Hilbert space isomorphism of $$F_{(U)}: \overline{\left(C_0^\infty(\cH; \bC), \lambda_{KW} \right)} \to \sH_{\cH}\:.$$ [**(b)**]{} If one switches to $\bR$ in place of $\bC$ $$\overline{F_{(U)}\left(C_0^\infty(\cH; \bR) \right)} = \sH_{\cH}\:.$$ As a second step we should prove that there is a natural way to densely embed $\sS(\cH)$ into the Hilbert space $\overline{(C_0^\infty(\cH; \bC), \lambda_{KW})}$, that is into $\sH_{\cH}$, as the definition of quasifree state requires. However, this is rather delicate because the most straightforward way, computing the $U$-Fourier transform of $\psi \in \sS(\cH)$ and checking that it belongs to $L^2(\bR_+\times \bS^2, 2KdK \wedge r_S^2d\bS^2) = \sH_{\cH}$, does not work. The ultimate reason lies in the too slow decay of $\psi$ as $|U| \to +\infty$ obtained in [@DR05] and embodied in the definition of $\sS(\cH)$ itself. As a matter of fact, the idea we intend to exploit is, first, to decompose every $\psi \in \sS(\cH)$ as a sum of three functions, one compactly supported and the remaining ones supported in $\cH^+$ and $\cH^-$ respectively and, then, to consider each function separately. The following proposition, whose proof is in Appendix \[Appendixproofs\], analyses the features of the last two functions. It also introduces some results, which will be very useful later when dealing with the KMS property of the state $\lambda_{KW}$.\ In the following $H^1(\cH^\pm)_u$ are the Sobolev spaces of the functions $\psi: \bR\times \bS^2 \to \bC$, referred to the coordinate $(u,\theta,\phi)\in \bR\times \bS^2$ on $\cH^\pm$, which lie in $L^2(\bR \times \bS^2, du \wedge d\bS^2)$ together with their first (distributional) $u$ derivative. If one follows the same proof as that valid for $C_0^\infty(\bR; \bC)$ and $H^1(\bR)$ along the line of Theorem VIII.6 in [@Bre] (employing sequences of regularising functions which are constant in the angular variables), one establishes that $C_0^\infty(\cH^\pm; \bC)$ is dense in $H^1(\cH^\pm)_u$. Every $\psi \in \sS(\cH^\pm)$ is an element of $H^1(\cH^\pm)_u$ as it follows immediately from the definition of $\sS(\cH^\pm)$. \[propbastarda\] [*The following facts hold, where $u\doteq 2r_S \ln(U) \in \bR$ and $u\doteq -2r_S \ln(-U) \in \bR$ are the natural global coordinate covering $\cH^+$ and $\cH^-$, respectively, while $\mu(k)$ is the positive measure on $\bR$: $$d\mu(k)\doteq 2r^2_S \frac{k e^{2\pi r_S k}}{e^{2\pi r_S k}-e^{-2\pi r_S k}} dk\:.$$*]{} [*[**(a)**]{} If $\widetilde{\psi} = (\mF(\psi))(k,\theta,\phi) = \widetilde{\psi}(k,\theta,\phi)$ denotes the $u$-Fourier transform of either $\psi \in C_0^\infty(\cH^+; \bC)$ or $\psi \in C_0^\infty(\cH^-; \bC)$ the maps $$C_0^\infty(\cH^\pm; \bC) \ni \psi \mapsto \widetilde{\psi} \in L^2(\bR \times \bS^2, d\mu(k) \wedge d\bS^2)$$ are isometric when $C_0^\infty(\cH^\pm; \bC)$ is equipped with the scalar product $\lambda_{KW}$. It uniquely extends, per continuity, to the Hilbert space isomorphisms: \[Fv\] F\^[()]{}\_[(u)]{} : L\^2(\^2, d(k) d\^2), where $\overline{C_0^\infty(\cH^\pm; \bC)}$ are viewed as Hilbert subspaces of $\overline{\left(C_0^\infty(\cH; \bC), \lambda_{KW} \right)}$.*]{} [*[**(b)**]{} The spaces $\sS(\cH^\pm)$ are naturally identified with real subspaces of $\overline{C_0^\infty(\cH; \bC)}$ in view of the following.\ If either $\{\psi_n\}_{n\in \bN}, \{\psi'_n\}_{n\in \bN} \subset C_0^\infty(\cH^+; \bR)$ or $\{\psi_n\}_{n\in \bN}, \{\psi'_n\}_{n\in \bN} \subset C_0^\infty(\cH^-; \bR)$ and, according to the case, both sequences $\{\psi_n\}_{n\in \bN}, \{\psi'_n\}_{n\in \bN}$ converge to the same $\psi \in \sS(\cH^\pm)$ in $H^1(\cH^\pm)$, then both sequences are of Cauchy type in $\overline{\left(C_0^\infty(\cH; \bC), \lambda_{KW} \right)}$ and $\psi_n-\psi'_n \to 0$ in $\overline{\left(C_0^\infty(\cH; \bC), \lambda_{KW} \right)}$.\ The subsequent identification of $\sS(\cH^\pm)$ with real subspaces of $\overline{C_0^\infty(\cH; \bC)}$ is such that: F\^[()]{}\_[(u)]{}\_[(\^)]{} = \_[(\^)]{},\[FF\] where $\mF: L^2(\bR \times \bS^2, du \wedge d\bS^2) \to L^2(\bR \times \bS^2, dk \wedge d\bS^2)$ stands for the standard $u$-Fourier-Plancherel transform.*]{}\ We are finally in place to specify how $\sS(\cH)$ is embedded in $\sH_{\cH}$. Let us consider a compactly supported smooth function $\chi\in C^\infty(\cH)$, such that $\chi=1$ in a neighbourhood of the bifurcation sphere $\cB\in\cH$. Every $\psi \in \sS(\cH)$ can now be decomposed as the sum of three functions: = \_- + \_0 + \_+ , \[dec0\] , Now let us define the map $\sK_{\cH} : \sS(\cH) \to \sH_{\cH} = L^2(\bR_+\times \bS^2, dK \wedge d\bS^2)$ as \[mapid\] \_ : () F\_[(U)]{}(\_-) + F\_[(U)]{}(\_0) + F\_[(U)]{}( \_+) \_, where $F_{(U)}(\psi_\pm)$ makes sense in view of the identification of $\sS(\cH)$ with a real subspace of $\overline{\left(C_0^\infty(\cH; \bC), \lambda_{KW} \right)}$ as established in (b) of Proposition \[propbastarda\]. The following proposition yields that $\sK_{\cH}$, in particular, is well-defined and injective and, thus, it identifies $\sS(\cH)$ with a subspace of $\sH_{\cH}$. Such identification enjoys a nice interplay with the symplectic form $\sigma_{\cH}$. Furthermore we prove that, $\sK : \sS(\cH) \to \sH_{\cH}$ is continuous if viewing $\sS(\cH)$ as a normed space equipped with the norm \[norme\] \_\^= (1-) \_[H\^1(\^-)\_u]{} + \_[H\^1()\_U]{} + (1-)\_[H\^1(\^+)\_u]{} where $\| \cdot \|_{H^1(\cH^\pm)_u}$ and $\| \cdot \|_{H^1(\cH)_U}$ are the norms of the Sobolev spaces $H^1(\cH^\pm)_u$ and $H^1(\cH)_U$ respectively.\ Notice that, $\|\cdot\|^\chi_{\cH}$ and $\|\cdot\|^{\chi'}_{\cH}$, defined with respect of different decompositions generated by $\chi$ and $\chi'$, are equivalent, in the sense that there are two positive real numbers $C_1$ and $C_2$ such that $ C_1 \| \psi \|_{\cH}^\chi \leq \| \psi \|_{\cH}^{\chi'} \leq C_2 \| \psi \|_{\cH}^\chi $ for all $\psi \in \sS(\cH)$. The proof of such an equivalence is based on the decomposition of the various integrals appearing in the mentioned norms with respect to both the partitions of the unit $\chi, 1-\chi$ and $\chi', 1-\chi'$. Afterwards one employs iteratively the triangular inequality and the fact that the norms $\| \cdot \|_{H^1(\cH^\pm)_u}$ and $\| \cdot\|_{H^1(\cH)_U}$ are equivalent when evaluated on smooth functions whose support is compact and does not include zero, because the Jacobian of the change of coordinates in the lone variable $U$ is strictly positive and bounded. To conclude the proof one should notice that $(\chi-\chi') $ is a compactly supported smooth function on the disjoint union of a pair of fixed compact sets $J\times\bS^2\subset \cH$, that do not contain ${0}$. Due to such an equivalence, we will often write $\| \psi \|_{\cH}$ in place of $\| \psi \|_{\cH}^\chi$. \[Propembedding\] *The linear map $\sK_{\cH} : \sS(\cH) \to \sH_{\cH}$ in (\[mapid\]) verifies the following properties:* [**(a)**]{} it is independent from the choice of the function $\chi$ used in the decomposition (\[dec0\]) of $\psi \in \sS(\cH)$; [**(b)**]{} it reduces to $F_{(U)}$ when restricting to $C_0^\infty(\cH; \bR)$; [**(c)**]{} it satisfies \_(,’) = -2 Im \_(), \_(’) \_[\_]{}, \[ultimabastarda1\] [**(d)**]{} it is injective; [**(e)**]{} it holds $\overline{\sK_{\cH}(\sS(\cH))} = \sH_{\cH}$; [**(f)**]{} it is continuous with respect to the norm $\|\cdot\|_{\cH}$ defined in [(\[norme\])]{} for every choice of the function $\chi$. Consequently, there exists $C>0$ such that $$|\langle \sK_{\cH}(\psi), \sK_{\cH}(\psi') \rangle_{\sH_{\cH}}| \leq C^2 \| \psi \|_{\cH} \cdot \| \psi' \|_{\cH} \; \quad \mbox{if $\psi,\psi' \in \sS(\cH)$.}$$ The proof is in Appendix \[Appendixproofs\]. Collecting all the achievements and presenting some further result, we can now conclude stating the theorem about the state individuated by $\lambda_{KW}$. \[Main4\][The following facts hold referring to $(\sH_{\cH},\sK_{\cH})$.]{} *[**(a)**]{} The pair $(\sH_{\cH},\sK_{\cH})$ is the one-particle structure for a quasi-free pure state $\omega_{\cH}$ on $\cW(\sS(\cH))$ uniquely individuated by the requirement that its two-point function coincides to the right-hand side of (\[lKW2\]) under restriction to $C_0^\infty(\cH; \bR)$.* [**(b)**]{} The state $\omega_{\cH}$ is invariant under the natural action of the one-parameter group of $*$-automorphisms generated by $X\spa\rest_{\cH}$ and of those generated by the Killing vectors of $\bS^2$. [**(c)**]{} The restriction of $\omega_{\cH}$ to $\cW(\sS(\cH^\pm))$ is a quasifree state $\omega^{\beta_H}_{\cH^\pm}$ individuated by the one particle structure $(\sH^{\beta_H}_{\cH^\pm},\sK^{\beta_H}_{\cH^\pm})$ with: $$\sH^{\beta_H}_{\cH^\pm} \doteq L^2(\bR\times \bS^2, d\mu(k) \wedge d\bS^2)\quad\mbox{and $\sK^{\beta_H}_{\cH^ \pm} \doteq \mF\spa\rest_{\sS(\cH^\pm)} = F^{(\pm)}_{(u)}\spa\rest_{\sS(\cH^\pm )}$.}$$ [**(d)**]{} The states $\omega^{\beta_H}_{\cH^\pm}$ satisfy the KMS condition with respect to one-parameter group of $*$-automorphisms generated by, respectively, $\mp X\spa\rest_{\cH}$, with Hawking’s inverse temperature $\beta_H = 4\pi r_S$. [**(e)**]{} If $\{\beta^{(X)}_\tau\}_{\tau \in \bR}$ denotes the pull-back action on $\sS(\cH^-)$ of the one-parameter group generated by $X\spa\rest_\cH$, that is $(\beta_\tau(\psi))(u,\omega) = \psi(u-\tau,\omega)$, for every $\tau\in \bR$ and every $\psi \in \sS(\cH^-)$ it holds: $$\sK^{\beta_H}_{\cH^-} \beta_\tau^{(X)}(\psi) = e^{i\tau \hat{k}}\sK^{\beta_H}_{\cH^-} \psi$$ where $\hat{k}$ is the $k$-multiplicative self-adjoint operator on $L^2(\bR\times \bS^2, d\mu(k) \wedge d\bS^2)$. An analogous statement holds for $\cH^+$. \ [*Proof*]{}. (a) In view of Proposition \[proposition2\] (and Lemma \[lemma1A\]), the wanted state is the one uniquely associated with the real scalar product over $\sS(\cH)$ \_(,’) Re \_ , \_’ \_[\_]{}, \[statemuHR\] and the one-particle structure is just $( \sH_{\cH},\sK_{\cH})$. This holds true provided two conditions are fulfilled, as required in Proposition \[proposition2\]. The first one asks for |\_(,’)|\^2 4 \_(,)\_(’,’). \[ultimabastarda\] This fact is an immediate consequence of (c) in Proposition \[Propembedding\]. The second condition to be satisfied is that $\overline{\sK_{\cH}(\sS(\cH)) + i \sK_{\cH}(\sS(\cH))} = \sH_{ \cH}$ and, actually, a stronger fact holds: $\overline{\sK_{\cH}(\sS(\cH))} = \sH_{\cH}$, because of (e) in Proposition \[Propembedding\]. As a consequence, the state $\omega_{\cH}$ is pure for (d) in Proposition \[proposition2\].\ (c) We only consider the case of $\cH^+$, the other case being analogous. The state $\omega^{\beta_H}_{\cH^+}$, which is the restriction of $\omega_{\cH}$ to $\cW(\sS(\cH^+))$, is by definition completely individuated out of the requirement that $$\omega^{\beta_H}_{\cH^+}\left(W_{\cH^+}(\psi) \right) = e^{-\mu_{\cH}(\psi,\psi)/2} \quad \mbox{for $\psi \in \sS(\cH^+)$.}$$ One can also prove the following three facts. (i) If $\psi,\psi' \in \sS(\cH^+)$, then: $$\begin{aligned} \mu_{\cH}(\psi,\psi') &= Re \lambda_{KW}(\psi,\psi') = Re \langle F^{(+)}_{(u)}\psi, F^{(+)}_{(u)}\psi' \rangle_{\sH^{\beta_H}_{\cH^+}} = Re \langle \widetilde{\psi}, \widetilde{\psi'} \rangle_{L^2(\bR\times \bS^2, d\mu(k)\wedge d\bS^2)} \nonumber \\ &= Re \langle \sK^{\beta_H}_{\cH^+} \psi, \sK^{\beta_H}_{\cH^+}\psi' \rangle_{\sH^{\beta_H}_{\cH^+}}\:, \nonumber \end{aligned}$$ due to (a) and (b) in Proposition \[propbastarda\]. (ii) Condition (\[ultimabastarda\]) is valid also under restriction to $\sS(\cH^+)$ if one notices that $\sigma_{\cH^+}= \sigma_{\cH}\spa \rest_{\sS(\cH^+) \times \sS(\cH^+)}$. (iii) One has $\overline{\sK^{\beta_H}_{\cH^+}(\sS(\cH^+)) + i \sK^{\beta_H}_{\cH^+}(\sS(\cH^+))} = \sH^{\beta_H}_ {\cH^+}$ by (a) and (b) of Proposition \[propbastarda\], if one bears in mind that $\sS(\cH^+)+ i\sS(\cH^+) \supset C_0^\infty(\cH^+; \bC)$. This concludes the proof because (i), (ii) and (iii) entail that $( \sH^{\beta_H}_{\cH^+},\sK^{\beta_H}_{ \cH^+})$ is the one-particle structure of $\omega^{\beta_H}_{\cH^+}$ in view of Proposition \[proposition2\] (and Lemma \[lemma1A\]).\ (b) If $\psi \in \sS(\cH)$, the $1$-parameter group of symplectomorphisms $\beta^{(X)}_\tau$ generated by $X$ individuates $\beta^{(X)}_\tau (\psi) \in \sS(\cH)$ such that $\beta^{(X)}_\tau (\psi)(U,\theta,\phi) = (\psi)\left(e^{ \tau/(4m)}U,\theta,\phi\right)$. This is an obvious consequence of $X= -\partial_u$ on $\cH^+$, $X= \partial_u$ on $\cH^-$ and $X=0$ on the bifurcation at $U=0$. Since $\beta^{(X)}$ preserves the symplectic form $\sigma_{\cH}$, there must be a representation $\alpha^{(X)}$ of $\beta^{(X)}$, in terms of $*$-automorphisms of $\cW(\sS(\cH))$. We do not need now the explicit form of $\alpha^{(X)}$, rather let us focus on $\beta^{(X)} $ again. If $\psi \in C_0^\infty(\cH; \bR)$, one has immediately, from the definition of $F_{(U)}$, which coincides with that of $\sK_{\cH}$ in the considered case, that $\sK_{\cH}(\beta^{(X)}_\tau (\psi))(K,\theta,\phi) = e^ {-\tau/ (4m)}\sK_\cH(\psi)\left(e^{-\tau/(4m)}K,\theta,\phi\right)$. This result generalises to the case where $\psi \in \sS(\cH)$ has support in the set $U>0$ (or $U<0)$ as it can be proved along the lines of the proof of (b) of Proposition \[propbastarda\]. Here. if one employs a sequence of smooth functions $\psi_n$ supported in $U>0$ (resp. $U<0$) which converges to $\psi$ in the Sobolev topology of $H^1(\cH^\pm,du)$ (see the mentioned proof), and uses the fact that $\beta^{(X)}_\tau (\psi_n)$ converges to $\beta^{(X)}_\tau (\psi)$ in the same topology. Summing up, from definition (\[mapid\]), one gets that $\sK_\cH(\beta^{(X)}_\tau (\psi))(K,\theta,\phi) = \left(U^{(X)}_\tau \psi\right)(U,\theta,\phi) \doteq e^{-\tau/(4m)}\sK_\cH(\psi)\left(e^{-\tau/(4m)}K, \theta,\phi\right)$ for every $\psi \in \sS(\cH)$ without further restrictions. Since $U^{(X)}_\tau$ is an isometry of $L^2(\bR_+\times \bS^2, KdK \wedge d\bS^2)$, in view of the definition of $\omega_{\cH}$ it yields that $\omega_{\cH}(W_{\cH}(\beta^{(X)}_\tau\psi)) = \omega_{\cH}(W_{\cH}(\psi))$ for all $\psi \in \sS(\cH)$, and, per continuity and linearity, this suffices to conclude that $\omega$ is invariant under the action of the group of $*$-automorphisms $\alpha^{(X)}$ induced by $X$. The proof for the Killing vectors of $\bS^2$ is similar.\ (d) and (e) In $\sS(\cH^-)$, the natural action of the one parameter group of isometries generated by $X\spa\rest_{\cH^-}$ is $\beta^{(X)}_\tau : \psi \mapsto \beta^{(X)}_\tau(\psi)$ with $\beta^{(X)}_\tau(\psi)(u,\theta,\phi) \doteq \psi(u-\tau,\theta,\phi)$, for all $u,\tau, \in \bR$, $(\theta,\phi) \in \bS^2$ and for every $\psi \in \sS(\cH^-)$. As previously, this is an obvious consequence of $X= \partial_u$ on $\cH^-$. Since $\beta^{(X)}$ preserves the symplectic form $\sigma_{\cH^-}$, there must be a representation $\alpha^{(X)}$ of $\beta^{(X)}$, in terms of $*$-automorphisms of $\cW(\sS(\cH^-))$. Let us prove that $\alpha^{(X)}$ is unitarily implemented in the GNS representation of $\omega^{\beta_H}_{\cH^-}$. To this end, we notice that $\beta$ is unitarily implemented in $\sH_{\cH^-}$, the one-particle space of $\omega^{\beta_H}_{\cH^-}$ out of the strongly-continuous one-parameter group of unitary operators $V_{\tau}$ such that $\left(V_\tau \widetilde{\psi}\right)(k, \theta,\phi) = e^{i k \tau}\widetilde{\psi}(k,\theta,\phi)$. This describes the time displacements with respect to the Killing vector $\partial_u$. Thus the self-adjoint generator of $V$ is $h : Dom(\hat{k}) \subset L^2(\bR\times \bS^2, d\mu(k) \wedge d\bS^2) \to L^2(\bR\times \bS^2, d\mu(k) \wedge d\bS^2)$ with $\hat{k}(\phi)(k,\theta,\phi) = k \phi(k,\theta,\phi)$ and $$Dom(\hat{k}) \doteq \left\{ \phi \in L^2(\bR\times \bS^2, d\mu(k) \wedge d\bS^2) \:\left| \: \int_{\bR\times \bS^2} |k\phi(k,\theta,\phi)|^2 d\mu(k) \wedge d\bS^2 <+ \infty\right.\right\}\:.$$ Per direct inspection, if one employs the found form for $V$ and exploits $$\omega^{\beta_H}_{\cH^-} \left(W_{\cH^-}(\psi)\right) = e^{-\frac{1}{2}\langle \widetilde{\psi}, \widetilde{\psi}\rangle_{L^2(\bR\times \bS^2, d\mu(k)\wedge d\bS^2)}}\:,$$ one sees that $\omega^{\beta_H}_{\cH^-}$ is invariant under $\alpha^{(X)}$, so that it must admit a unitary implementation [@Araki]. In order to establish that the $\alpha^{(X)}$-invariant quasifree state $\omega^{\beta_H}_{\cH^-}$ over the Weyl algebra $\cW(\sS(\cH^-))$ is a KMS state with inverse temperature $\beta_H = 4\pi r_S$ with respect to $\alpha^{(X)}$ which, in turn, is unitarily implemented by $V = \{\exp\{i \tau \hat{k}\}\}_{\tau\in \bR}$ in the one particle space $\sH^{\beta_H}_{\cH^-}$, on can use proposition \[kkms\] in the appendix and prove that $\sK^{\beta_H}_{\cH^-} (\sS(\cH^-)) \subset Dom\left( e^{-\frac{1}{2}\beta \hat{k}}\right)$ while $\langle e^{i\tau \hat{k}} \sK^{\beta_H}_{\cH^-} \psi, \sK^{\beta_H}_{\cH^-} \psi' \rangle = \langle e^{-\beta_H \hat{k}/2}\sK^{\beta_H}_{\cH^-} \psi', e^{-\beta_H \hat{k}/2} e^{i\tau \hat{k}} \sK^{\beta_H}_{\cH^-} \psi \rangle$. Luckily these requirements hold per direct inspection since $\sK^{\beta_H}_{\cH^-}(\psi) = \widetilde{\psi}\in L^2(\bR \times \bS^2, d\mu(k) \wedge d\bS^2)$. Here we used the explicit form of the measure $\mu(k)$ and the identity $\overline{\widetilde{\psi}(- k,\omega)} = \widetilde{\psi}(k,\omega)$ if $\psi \in \sS(\cH^-)$ because $\psi$ is real-valued. The case of $\cH^+$ is strongly analogous, the only difference being $X\spa\rest_{\cH^+} = -\partial_u$. $\Box$\ We conclude stating without proof (straightforward in this case) the following proposition which concerns the natural $X$-invariant vacuum states of $\cH^-$ and $\cH_{ev}$(actually, a [*quasifree regular ground states*]{} in the sense of [@KW]). \[PropMain4.5\] [*If $\sK_{\cH^-}: \sS(\cH^-) \to \sH_{\cH^-}\doteq L^2(\bR_+ \times \bS^2, 2kdk\wedge d\bS^2)$ denotes the standard $u$-Fourier-Plancherel transform, followed by the restriction to $\bR_+\times \bS^2$, the following facts hold.*]{} [*[**(a)**]{} The pair $(\sH_{\cH^-},\sK_{\cH^-})$ is the one-particle structure for a quasi-free pure state $\omega_{\cH^-}$ on $\cW(\sS(\cH^-))$.*]{} [*[**(b)**]{} The state $\omega_{\cH^-}$ is invariant under the natural action of the one-parameter group of $*$-automorphisms generated by $X\spa\rest_{\cH^-}$ and those generated by the Killing vectors of $\bS^2$.\ If one replaces the $u$-Fourier-Plancherel transform with the $v$-Fourier-Plancherel one, an analogous state $\omega_{\cH_{ev}}$ can be defined, which is invariant under the natural action of the one-parameter group of $*$-automorphisms generated by $X\spa\rest_{\cH_{ev}}$ and those generated by the Killing vectors of $\bS^2$.*]{} The vacuum state $\omega_{\scrim}$ on $\cW(\Im^-)$ -------------------------------------------------- We now introduce a relevant vacuum state $\omega_{\Im^-}$ on $\cW(\Im^-)$ which is invariant with respect to $u$-displacements and under the isometries of $\bS^2$. The idea is, in principle, the same as for $\omega_{\cH}$, [*i.e.*]{}, one starts from a two-point function similar to $\lambda_{KW}$, with the important difference that the coordinate $U$ is now replaced by $v$. As a starting point we state the following proposition whose proof is, [*mutatis mutandis*]{}, identical to that of proposition \[PropMain4\]. \[PropMain5\] *Consider the Hilbert completion $\overline{\left(C_0^\infty(\Im_R^-; \bC), \lambda_{\Im^-} \right)}$ of the complex vector space $C_0^\infty(\Im^-; \bC)$ equipped with the Hermitian scalar product: \[lscri\] \_[\^-]{}(\_1,\_2) \_[0\^+]{} - \_[\^2]{}dv\_1 dv\_2 d\^2 , where $\scrim \equiv \bR \times \bS^2$ adopting the coordinate $(v,\theta,\phi)$ over $\scrim$. The following facts hold.* [**(a)**]{} If $\widehat{\psi}_+(k,\theta,\phi) \doteq \mF(\psi)\spa\rest_{\{k\geq 0, (\theta,\phi) \in \bS^2\}}(k,\theta,\phi)$ denotes the $v$-Fourier transform of $\psi \in C_0^\infty(\scrim; \bC)$ restricted to $k\in \bR_+$ (see the Appendix C in [@Moretti08]), the map $$C_0^\infty(\scrim; \bC) \ni \psi \mapsto \widehat{\psi}_+(k,\theta,\phi) \in L^2(\bR_+\times \bS^2, 2kdk \wedge d\bS^2) =: \sH_{\scrim}$$ is isometric and it uniquely extends, per continuity, to a Hilbert space isomorphism of \[Fu\] F\_[(v)]{}: \_. [**(b)**]{} If one replaces $\bC$ with $\bR$: \[scriaggdens\] = \_. We have now to state and to prove the corresponding of the Proposition \[Propembedding\], which establishes that there exists a state $\omega_\scrim$ which is completely determined by $\lambda_\scrim$ and it is such that the one-particle space coincides with $\sH_\scrim$. The delicate point is to construct the corresponding of the $\bR$-linear map $\sK_{\cH}$, which now has to be thought of as $\sK_{\scrim}: \sS(\scrim) \to \sH_\scrim$. Let us notice that $\sK_{\scrim}$ cannot be defined as the $v$-Fourier transform (neither the Fourier-Plancherel transform), since the elements of $\sS(\scrim)$ do not decay rapidly enough. Similarly to what done before, a suitable extension with respect to the topology of $\overline{\left(C_0^\infty(\scrim; \bC), \lambda_{\scrim} \right)}$ is necessary. To this end, we are going to prove that the real subspace of the functions of $\sS(\scrim)$ supported in the region $v>0$ can be naturally identified with a real subspace of $\overline{\left(C_0^\infty(\scrim; \bC), \lambda_{\scrim} \right)}$. This is stated in the following proposition whose proof is in the Appendix \[Appendixproofs\]. In the following, we pass to the coordinate over $\bR$ defined by $x \doteq \sqrt{v}$ if $v\geq 0$ and $x\doteq -\sqrt{-v}$ if $v\leq0$. Then, if we adopt the coordinate $x$ over the factor $\bR$ of $\scrim \equiv \bR \times \bS^2$, the Sobolev space $H^1(\scrim)_x$, is that of the functions which belong to $L^2(\bR \times \bS^2, dx \wedge d\bS^2)$ with their (distributional) first $x$ derivative. Notice that, in view of the very definition of $\sS(\scrim)$, if $\psi$ is supported in the subset of $\scrim$ with $v<0$ ([*i.e.*]{}, $x<0)$ and $\psi \in \sS(\scrim)$, then $\psi \in H^1(\scrim)_x$. \[propidscri\] *If $\psi \in \sS(\scrim)$ and $\supp(\psi) \subset \bR^*_- \times \bS^2$ (where $\bR^*_-\doteq (-\infty,0)$), the following holds.* [**(a)**]{} Every sequence $\{\psi_n\}_{n\in \bN} \subset C_0^\infty(\bR^*_- \times \bS^2; \bR)$ such $\psi_n \to \psi$ as $n\to +\infty$ in $H^1(\scrim)_x$ is necessarily of Cauchy type in $\overline{\left(C_0^\infty(\scrim; \bC), \lambda_{\scrim} \right)}$. [**(b)**]{} There is $\{\psi_n\}_{n\in \bN} \subset C_0^\infty(\bR^*_- \times \bS^2; \bR)$ such $\psi_n \to \psi$ as $n\to +\infty$ in $H^1(\scrim)_x$ and, if $\{\psi'_n\}_{n\in \bN} \subset C_0^\infty(\bR^*_- \times \bS^2; \bR)$ converges to the same $\psi$ in $H^1(\scrim)_x$, then $\psi'_n-\psi_n\to 0$ in $\overline{\left(C_0^\infty(\scrim; \bC), \lambda_{\scrim} \right)}$.\ As a consequence every $\psi \in \sS(\scrim)$ with $\supp(\psi) \subset \bR^*_+ \times \bS^2$ can be naturally identified with a corresponding element of $\overline{\left(C_0^\infty(\scrim; \bC), \lambda_{\scrim} \right)}$, which we indicate with the same symbol $\psi$.\ With this identification it holds F\_[(v)]{} \_[()]{} = \_[()]{}, \[FumF\] and, for $\psi,\psi'\in \sS(\scrim)$, \_(,’) = \_[\_+\^2]{} (( I + C) ())(h,) 2hdhd\^2(),\[antilin\] where, $\Theta(h)=0$ if $h\leq 0$ and $\Theta(h)=1$ otherwise. Here $\mF : L^2(\bR \times \bS^2, dx \wedge d\bS^2) \to L^2(\bR \times \bS^2, dh \wedge d\bS^2)$ is the $x$-Fourier-Plancherel transform ($x\doteq -\sqrt{-v}$ if $v\leq 0$ and $x\doteq \sqrt{v}$ if $v\geq 0$) while $C$ stands for the standard complex conjugation. \ We are in place to define the map $\sK_{\scrim}$ along the lines followed for $\sK_{\cH}$. Let $\chi$ be a non-negative smooth function on $\scrim$ whose support is contained in $\bR^*_-\times\bS^2$, and such that $\eta(v,\theta,\phi)=1$ for $v<v_0<0$. Consider $\psi \in \sS(\scrim)$ and decompose it as: = \_0 + \_-, \[dec0scri\] Obviously, $\psi_0 \in C_0^\infty(\scrim;\bR)$ and $\supp(\psi_-) \subset \bR^*_-\times \bS^2$, where $\bR^*_-$ is referred to the coordinate $v$ on $\bR$. Finally, let us define \_() F\_[(v)]{}(\_0) + F\_[(v)]{}(\_-), (), \[mapidscri\] where, $\psi_-$ in the second term is considered an element of $\overline{\left(C_0^\infty(\scrim; \bC), \lambda_{\scrim} \right)}$ in view of Proposition \[propidscri\]. The map $\sK_\scri : \sS(\scrim) \to \sH_\scrim$ is continuous when the domain is equipped with the norm \[normeScri\] \_\^= \_- \_[H\^1()\_x]{} + \_0 \_[H\^1()\_v]{} where $\| \cdot \|_{H^1(\scrim)_x}$ and $\| \cdot \|_{H^1(\scrim)_v}$ are the norms of the Sobolev spaces $H^1(\scrim)_x$ and $H^1(\scrim)_v$ respectively, the latter hence with respect to the $v$-coordinate. Let us remark that, as before, different $\eta$ and $\eta'$ produce equivalent norms $\|\cdot\|^\eta_{ \scrim}$ and $\|\cdot\|^{\eta'}_{\scrim}$; for this reason we shall drop the index $\eta$ in $\|\cdot\|^\eta_\scrim$ if not strictly necessary. The following proposition states that the definition of $\sK_{\scrim}$, given above, is meaningful; its proof, which will be discussed in the Appendix \[Appendixproofs\], relies on Propositions \[PropMain5\] and \[propidscri\] and it is very similar to that of Proposition \[Propembedding\]. \[Propembeddingscri\] *The linear map $\sK_{\scrim} : \sS(\scrim) \to \sH_{\scrim}$ in (\[mapidscri\]) enjoys the following properties:* [**(a)**]{} it is well-defined, i.e., it is independent from the chosen decomposition (\[dec0scri\]) for a fixed $\psi \in \sS(\scrim)$; [**(b)**]{} it reduces to $F_{(v)}$ when restricting to $C_0^\infty(\scrim; \bR)$; [**(c)**]{} it satisfies: $$\sigma_{\scrim}(\psi,\psi') = -2 Im \langle \sK_{\scrim}(\psi), \sK_{\scrim}(\psi') \rangle_{\sH_{\scrim}}\:, \quad \mbox{if $\psi,\psi' \in \sS(\scrim)$;}$$ [**(d)**]{} it is injective; [**(e)**]{} it holds $\overline{\sK_{\scrim}(\sS(\scrim))} = \sH_{\scrim}$; [**(f)**]{} it is continuous with respect to the norm $\|\cdot\|_{\scrim}$ defined in [(\[normeScri\])]{} for every choice of the function $\eta$. Consequently, there exists a constant $C>0$ such that: $$|\langle \sK_{\scrim}(\psi), \sK_{\scrim}(\psi') \rangle_{\sH_{\scrim}}|\leq C^2 \| \psi \|_{\scrim} \cdot \| \psi' \|_{\scrim} \; \quad \mbox{if $\psi,\psi' \in \sS(\scrim)$.}$$ We can now define the state $\omega_\scrim$ collecting all the achieved results. \[Main4scri\][The following facts hold referring to $(\sH_{\scrim},\sK_{\scrim})$.]{} *[**(a)**]{} The pair $(\sH_{\scrim},\sK_{\scrim})$ is the one-particle structure for a quasi-free pure state $\omega_{\scrim}$ on $\cW(\sS(\scrim))$ which is uniquely determined by the requirement that its two-point function coincides with the right-hand side of (\[lscri\]) under the restriction to $C_0^\infty(\scrim; \bR)$.* [**(b)**]{} The state $\omega_{\scrim}$ is invariant under the natural action of the one-parameter group of $*$-automorphisms generated both by $X\spa\rest_{\scrim}$ and by the Killing vectors of $\bS^2$. [**(c)**]{} If $\{\beta^{(X)}_\tau\}_{\tau \in \bR}$ denotes the pull-back action on $\sS(\scrim)$ of the one-parameter group generated by $X\spa\rest_\scrim$ that is $(\beta_\tau(\psi))(v,\omega) = \psi(v-\tau,\omega)$, for every $\tau\in \bR$ and every $\psi \in \sS(\scrim)$ it holds: $$\sK_{\scrim} \beta_\tau^{(X)}(\psi) = e^{i\tau \hat{h}}\sK_{\scrim} \psi$$ where $\hat{h}$ is the $h$-multiplicative self-adjoint operator on $\sH_\scrim = L^2(\bR\times \bS^2, 2hdh \wedge d\bS^2)$.\ Analogous statements hold for $\cW(\sS(\Im^{\pm}_{L}))$ and for $\cW(\sS(\Im^{+}))$, hence there exists the corresponding states $\omega_{\Im^{\pm}_{L}}$ and $\omega_{\Im^+}$ exist .\ [*Proof*]{}. The proof of (a) and (b) is essentially identical to that of the corresponding items in Theorem \[Main4\]. Particularly, the proof of item (b) is a trivial consequence of Lemma \[ULTIMOlemma\]. $\Box$ The extended Unruh state $\omega_U$. ==================================== When a spherically-symmetric black hole forms, the metric of the spacetime outside the event horizon, as well as that inside the region containing the singularity away from the collapsing matter, must be of Schwarzschild type due to the Birkhoff theorem (see [@WR; @Wald2] for a more mathematically detailed discussion). A model of this spacetime can be realized selecting a relevant subregion of $\mM$ in the Kruskal manifold, [*i.e.*]{}, the so called regions I and II of the Kruskal diagram as depicted in chapter 6.4 of [@Wald]. A quantum state that accounts for Hawking’s radiation in such background was heuristically defined by Unruh in $\mM$, who employed a mode decomposition approach [@Unruh; @Candelas; @Wald2]. A rigorous, though indirect, definition of $\omega_U$, restricted to $\mW$, has been subsequently proposed by Kay and Dimock in terms of an $S$-matrix interpretation, though under the assumption of asymptotic completeness, which was proved to hold in the massless case [@DimockKay]. It is imperative to stress that, in the last cited papers, the restriction to the static region $\mW$ was crucial to employ the mathematical techniques used to describe the scattering in stationary spacetimes and, as a byproduct, the algebras $\cW(\sS(\cH))$ and $\cW(\sS(\Im^-))$ were introduced and used with some differences with respect to our approach.\ The states $\omega_U$, $\omega_B$ and their basic properties ------------------------------------------------------------ We are in place to give a rigorous definition of the Unruh state by means of the technology previously introduced. Our definition is valid for the whole region $\mM$ and it does not require any $S$-matrix interpretation, nor formal manipulation of distributional modes as in the more traditional presentations (see [@Candelas]). Our prescription is a possible rigorous version of Unruh original idea according to which the state is made of thermal modes propagating in $\mM$ from the white hole and of vacuum modes entering $\mM$ from $\scrim$. Together the Unruh state $\omega_U$ on $\cW(\sS(\mM))$ we also define the [*Boulware vacuum*]{}, $\omega_B$ on $\cW(\sS(\mW))$, since it will be useful later. \[omegaHdef\] [Consider the states $\omega_\scri$, $\omega_\scrim$, $\omega_{\cH}$ and $\omega_{\cH_{ev}}$ as in Theorem \[Main4scri\], Theorem \[Main4\], Proposition \[PropMain4.5\]. The [**Unruh state**]{} is the unique one $\omega_U : \cW(\sS(\mM))\to\bC$ such that: \_U( W\_()) = \_( W\_(\_) ) \_(W\_(\_) ) \[omegaH\] The [**Boulware vacuum**]{} is the unique state $\omega_B : \cW(\sS(\mW)) \to \bC$ such that: \_B( W\_()) = \_[\_[ev]{}]{}( W\_[\_[ev]{}]{}(\_[\_[ev]{}]{}) ) \_(W\_(\_) ) \[boulware\]In other words $ \omega_U\doteq\left(\imath\right)^* \left( \omega_{\cH}\otimes \omega_{\scrim} \right) $ and $ \omega_B\doteq\left(\imath^+\right)^* \left( \omega_{\cH^+}\otimes \omega_{\scrim} \right) $.]{}\ We study now the interplay between $\omega_U$, $\omega_B$ and the action of $X$. The Killing field $X$ individuates a one-parameter group of (active) symplectomorphisms $\{\beta^{(X)}_t\}_{t \in\bR}$ on $\sS(\mM)$ which leaves $\sS(\mM)$ and $\sS(\mW)$ invariant. As $X$ is defined on the whole manifold $\widetilde{\mM}$, similarly, a one-parameter group of (active) symplectomorphisms are induced on $\sS(\Im^\pm)$, $\sS(\cH)$, $\sS(\cH^-)$, $\sS(\cH_{ev})$ and, henceforth, we shall use the same symbol $\{\beta^{(X)}_t\}_{t\in \bR}$ for all these groups. In turn, $\{\beta^{(X)}_t\}_{t\in \bR}$ induces a one-parameter group of $*$-automorphisms, $\{\alpha_t^{(X)}\}_{t\in \bR}$, on $\cW(\mM)$ unambiguously individuated by the requirement: \^[(X)]{}\_t( W\_())W\_(\^[(X)]{}\_t() ), \[ad\] Whenever $\{\alpha_t^{(X)}\}_{t\in \bR}$ acts on $\cW(\sS(\mM))$ and $\cW(\sS(\mW))$, it leaves these algebras fixed and the second one in particular represents the time-evolution, with respect to the Schwarzschild time, of the observables therein. Analogous one-parameter groups of $*$-automorphisms, indicated with the same symbol, are defined on $\cW(\Im^\pm)$, $\cW(\cH)$, $\cW(\cH^-)$, $\cW(\cH_{ev})$ by $X$. The following relations hold true, for all $t\in\bR$ and $\varphi\in\sS(\mM)$: (\^[(X)]{}\_t()) = (\^[(X)]{}\_t(\_), \^[(X)]{}\_t(\_)).\[cad\] The same result is valid if one replaces $\mM$ with $\mW$, $\cH$ with $\cH^-$ or $\cH_{ev}$ and, in the second case, $\scrim$ with $\scri$, so that $\Gamma$ is substituted by $\Gamma_-$ or $\Gamma_+$ respectively, while $\imath$ by $\imath^-$ or $\imath^+$ correspondingly. The proof is a consequence of the invariance of the Klein-Gordon equation under $\beta^{(X)}$. Similar identities hold concerning the remaining Killing $\bS^2$-symmetries of both $\mM$ and $\mW$. \[omegaHinvariance\] [*The following facts hold,\ [**(a)**]{} $\omega_U$ and $\omega_B$ are invariant under the action of $\{\alpha_t^{(X)}\}_{t\in \bR}$ and under that of the remaining Killing $\bS^2$-symmetries of the metric of $\mM$ and $\mW$ respectively.\ [**(b)**]{} $\omega_B$ is a [**regular quasifree ground state**]{}, [*i.e.*]{}, the unitary one-parameter groups which implements $\{\alpha_t\}_{t\in\bR}$ are strongly continuous and the self-adjoint generators have positive spectrum with no zero eigenvalues in the one-particle spaces. Hence it coincides to the analogous vacuum state defined with respect to the past null boundary of $\mW$, [*i.e.*]{}, $\omega_B = \left(\imath^- \right)^*(\omega_{\cH^-}\otimes \omega_{\Im^-})$.*]{}\ [*Proof*]{}. (a) If one bears in mind the same statement for the region $\mW$, the one under analysis follows from (\[ad\]), (\[cad\]) together with the definitions (\[omegaH\]) and (\[boulware\]). One must also take into account that the states $\omega_{\cH}$, $\omega_\scrim$, $\omega_{\cH_{ev}}$, $\omega_{\Im^+}$, are invariant under the action of both $\{\alpha_t^{(X)}\}_{t\in \bR}$ and the remaining Killing symmetries, as established in theorems \[Main4\], \[Main4scri\] and proposition \[PropMain4.5\].\ (b) By direct inspection one sees that, in the GNS representation space of the quasifree states, $\omega_B$ and $ \left(\imath^-\right)^*(\omega_{\cH^-}\otimes \omega_{\Im^-})$ are quasifree regular ground states with respect to $\{\alpha_t\}_{t\in\bR}$. Thus [*Kay’s uniqueness theorem*]{} [@KayU] implies that $\omega_B=\left(\imath^-\right)^*(\omega_{\cH^-}\otimes\omega_{\Im^-})$. $\qed$\ If $\varphi,\varphi'\in\sS(\mW)$, the function $F_{\varphi,\varphi'}(t)\doteq\omega_{U} \left(W_{\mW}(\varphi)\alpha_t^{(X)}\left(W_{\mW}(\varphi)\right)\right)$ decomposes in a product $$F_{\varphi,\varphi'}(t) = F^{(\beta_H)}_{\varphi,\varphi'}(t)F^{(\infty)}_{\varphi,\varphi'}(t)\:.$$ If one refers to the Schwarzschild-time evolution, the first factor fulfils the KMS requirements (see definition \[KMSdef\]), whereas the second factor enjoys the properties of a ground state two-point function: it can be extended to an analytic functions for $Im t >0$ which is continuous and bounded in $Im\; t \geq 0$ and tends to $1$ as $\bR\ni t\to\pm\infty$. The term $F^{(\beta_H)}_{\varphi,\varphi'}(t)$, which evaluates only the part $\varphi_{\cH^-}$ and $\varphi'_{\cH^-}$ of the wavefunctions, represents the components of the wavefunction which brings the thermal radiation entering $\mW$ through the white hole. The latter, which evaluates only the components $\varphi_{\Im^-}$ and $\varphi'_{\Im^-}$ of the wavefunctions, represents the part of the wavefunction associated with the Boulware vacuum.\ On the Hadamard property ------------------------ Let us consider a quasifree state $\omega$ on the Weyl algebra of the real Klein-Gordon scalar field $\cW( \mN)$ for a globally hyperbolic spacetime $(\mN, g)$ and let $(\sH_\omega,\sK_\omega)$ be its one-particle structure which determines the Fock GNS representation $(\gH_\omega, \Pi_\omega, \Psi_\omega)$ of $\omega$. Finally introduce the field operators ${\Phi}_\omega(f)$ as discussed in sec. \[observables\]. The [**two-point function**]{} of $\omega$ is the bilinear form $\lambda : \sS(\mN)\times \sS(\mN) \to \bC$ where $\lambda_\omega(\psi,\psi') \doteq \langle \sK_\omega \psi, \sK_\omega \psi' \rangle_{\sH_\omega}$. Equivalently, if one follows Sec. \[observables\] and the Appendix \[algebras\], it turns out that $$\lambda_\omega (\psi,\psi') = \langle \Psi_\omega, {\Phi}_\omega(f) {\Phi}_\omega(f') \Psi_\omega \rangle\:, \quad \psi = E_{P_g} f\:,\quad \psi' = E_{P_g}f'\:,$$ where the expectation value of the product of two field operators ${\Phi}_\omega(f)$ and ${\Phi}_\omega (f')$ is computed with respect to the cyclic vector $\Psi_\omega$ of the GNS representation of $\omega$ and where $E_{P_g}: C_0^\infty(\mN; \bC) \to \sS(\mN)$ is the causal propagator. Therefore a [**smeared two-point function**]{} can equivalently be defined as a bilinear map $\Lambda_\omega : C^\infty(\mN; \bR) \times C^\infty(\mN; \bR) \to \bC$ associated with the formal integral kernel $\Lambda_\omega(x,x')$ with $$\Lambda_\omega(f,g) \doteq \int_{\mN\times \mN} \Lambda_\omega(x,x') f(x) g(x') d\mu_g(x) d\mu_g(x')\: \doteq \langle \Psi_\omega,{\Phi}_\omega(f) {\Phi}_\omega(f') \Psi_\omega \rangle \:.$$ Furthermore $$\Lambda_\omega(f,g) = \lambda_\omega(E_{P_g}f, E_{P_g}g) \quad \mbox{if $f,g \in C^\infty(\mN;\bR)$}\:.$$ In this framework, the state $\omega$ is said to satisfy the local [**Hadamard property**]{} when, in a geodetically convex neighbourhood of any point the two-point (Wightmann) function $\omega(x,x')$ of the state has the structure $$\Lambda_\omega(x,x') = \frac{\Delta(x,x')}{ \sigma(x,x')} + V(x,x') \ln \sigma(x,x') + w(x,x')\:,$$ where $\Delta(x,x')$ and $V(x,x')$ are determined by the local geometry, $\sigma(x,x')$ is the signed squared geodetical distance of $x$ and $x'$ , while $w$ is a smooth function determining the quasifree state. The precise definition, also at global level and up to the specification of the regularisation procedure enclosed in the definition of $\sigma$, was stated in [@KW]. The knowledge of the singular part of the two-point function and, thus, of all $n$-point functions in view of Wick expansion procedure, allows the definition of a suitable renormalisation procedure of several physically interesting quantities such as the stress energy tensor, to quote just one of the many examples [@Wald; @Mo03; @HW04]. It has thus been the starting point of a full renormalisation procedure in curved spacetime as well as other very important developments of the general theory [@BFK; @BF00; @HW01; @BFV03]. A relevant technical achievement was obtained by Radzikowski [@Rada; @Radb] who, among other results, proved the following: if one refers to the Klein-Gordon scalar field, the global Hadamard condition for a quasifree state $\omega$ whose two-point function is a distribution $\Lambda_\omega \in \mD'(\mN \times\mN)$, where $(\mN,g)$ is globally hyperbolic and time-oriented, is equivalent to the following constraint on the [*wavefront set*]{} [@Hormander] of $\Lambda_\omega$. \[WFgen\] WF(\_) = (x,y,k\_x,k\_y)T\^\*(){0} | (x,k\_x)\~(y,-k\_y), k\_x0 , that is usually referred to [**microlocal spectrum condition**]{} - see [@Sanders] for recent developments -. One should notice that, above, $0$ denotes the null section of $T^*(\mN \times \mN)$ and $(x,k_x)\sim(z,k_z)$ means that there exists a light-like geodesic $\ga$ connecting $x$ to $z$ with $k_x$ and $k_z$ as (co)tangent vectors of $\ga$ respectively at $x$ and at $z$. Particularly if $x=z$, it must hold that $k_x=k_z$, $k_z$ being of null type. The symbol $\triangleright$ indicates that $k_x$ must lie in the future-oriented light cone. The aim of this subsection is to prove that the two-point function associated to the state on $\cW(\mM)$ fulfils the Hadamard property by means of the microlocal approach based on condition (\[WFgen\]). To this avail, the general strategy, we shall follow, consists of combining in a new non trivial way the results presented in [@SV00] and in [@Moretti08; @DMP3]. Since we interpret the two-point function as a map from $C^\infty_0(\mM; \bC) \times C^\infty_0(\mM; \bC) \to \bC$ a useful tool is the map $\Ga :\sS(\mM)\to\sS(\cH)\oplus\sS(\scrim)$ introduced in the statement of Theorem \[Main1\]. We shall combine it with the causal propagator to obtain (\^f\_,\^f\_)E\_[P\_g]{} f. \[BCN\] We can now state the following proposition, whose ultimate credit is to allow us to check microlocal spectrum condition (\[WFgen\]) since the two-point function of $\omega_U$ determines a proper distribution of $\mD'(\mM\times \mM)$. \[distrib\][The smeared two-point function $\Lambda_U :C^\infty_0(\mM; \bR) \times C^\infty_0(\mM; \bR) \to\bC$ of the Unruh state $\om_U$ can be written as the sum \_U = \_+\_[\^-]{}, \[LULHLI\] with $\Lambda_{\cH}$ and $\Lambda_{\Im^-}$ defined out of the following relations for $\la_{\cH}$ and $\la_\scrim$ as in and : $$\Lambda_{\cH}(f, g)\doteq \lambda_{\cH}(\varphi^f_{\cH}, \varphi^g_{\cH})\;, \qquad \Lambda_{\Im^-}(f, g)\doteq \lambda_{\Im^-}(\varphi^f_{\scrim}, \varphi^g_{\scrim})\;, \quad \mbox{for every $f,g \in C^\infty_0(\mM; \bR)$,}$$ Separately, $\Lambda_\cH$, $\Lambda_\scrim$ and $\Lambda_U$ individuate elements of $\mD'(\mM\times\mM)$ that we shall indicate with the symbols $\Lambda_\cH$, $\Lambda_\scrim$ and $\Lambda_U$. These are uniquely individuated by $\bC$-linearity and continuity under the assumption (\[LULHLI\]) as \[twopointseparately\] \_(f g)\_(\^f\_, \^g\_), \_[\^-]{}(f g)\_[\^-]{}(\^f\_, \^g\_), ]{} The proof is in Appendix \[Appendixproofs\]. .2cm In the remaining part of this section we shall prove one of the main theorems of this paper, namely that $\Lambda_U$ satisfies the microlocal spectral condition (\[WFgen\]) and thus $\omega_U$ is Hadamard. \[maxt\][The two-point function $\Lambda_U\in \mD'(\mM \times \mM)$ associated with the Unruh state $\om_U$ satisfies the microlocal spectral condition: \[WF2\] WF(\_U) = (x,y,k\_x,k\_y)T\^\*() {0}, (x,k\_x)\~(y,-k\_y), k\_x 0 , consequently $\omega_U$ is of Hadamard type.]{} [*Proof*]{}. As it is often the case with identities of the form , the best approach, to prove them, is to show that two inclusions $\supset$ and $\subset$ hold separately, hence yielding the desired equality. Nonetheless, in this case, we should keep in mind that $\Lambda_U$ is a two-point function, hence it satisfies in a weak sense the equation of motion with respect to $P_g$, a properly supported, homogeneous of degree 2, hyperbolic operator of real principal part and the antisymmetric part of $\Lambda_U$ must correspond to the causal propagator $E_{P_g}$ introduced in subsection \[observables\]. In this framework, all the hypothesis to apply the theorem of propagation of singularities (PST), as in Theorem 6.1.1 in [@DH], are met. Hence one has all the ingredients necessary to proceed as in the proof of Theorem 5.8 in [@SV01], to conclude that the inclusion $\supset$ holds true once $\subset$ has been established. Therefore, in order to prove (\[WF2\]), it is enough to establish only the inclusion $\subset$. This will be the goal of the remaining part of the proof and we shall divide our reasoning in two different sequential logical steps. In the first part, below indicated as [*part 1*]{}, we shall prove that the microlocal spectrum condition is fulfilled in the static region $\mW$. In the second, displayed as [*part 2*]{}, we apply this result extending it to the full $\mM$, mostly by means of the PST which strongly constraints the form of $WF(\Lambda_U)$ in the full background. The left-over terms, which are not fulfilling , are eventually excluded by means of a case-by-case analysis. .3cm [*Part 1*]{}. In order to establish the validity of the microlocal spectral condition in $\mW$, our overall idea is to restrict $\Lambda_U$ to a distribution in $\mD'(\mW\times \mW)$ and to apply/adapt to our case the result on the wave-front set of the two-point function of passive quantum states, as devised in [@SV00].\ As a starting point, let us remind that $\mW$ is a static spacetime with respect to the Schwarzschild Killing vector $X$, and that the state $\Lambda_U$ is invariant under the associated time translation, as established in Proposition \[omegaHinvariance\]. However, despite this set-up, $\Lambda_U$ is not passive in the strict sense given in [@SV00] and, hence, we cannot directly conclude that the Hadamard property is fulfilled in $\mW$, [*i.e.*]{}, in other words, Theorem 5.1 in [@SV00] does not straightforwardly go through. Nonetheless, luckily enough, a closer look at the proof of the mentioned statement reveals that it can be repeated slavishly with the due exception of the step 2) in which the passivity condition is explicitly employed. Yet, this property is not used to its fullest extent and, actually, a weaker one suffice to get the wanted result; in other words, the mentioned “step 2)”, or more precisely formula (5.2) in the last mentioned paper, can be recast as the following lemma for $\Lambda_U$. \[KV\][The wave front set of the restriction to $\mD(\mW\times \mW)$ of $\Lambda_U$, satisfies the following inclusion $$WF(\Lambda_U\spa \rest_{\mD(\mW\times \mW)})\subset \ag (x,y,k_x,k_y) \in T^*(\mW\times \mW)\setminus \{0\}, \; k_x(X)+k_y(X)=0 , \; k_y(X) \geq 0 \cg \;,$$ where $X$ is the generator of the Killing time translation.\ ]{} [*Proof.*]{} As a first step we recall the invariance of $\Lambda_U$, as well as of $\Lambda_\scrim$ and $\Lambda_{\cH}$, under the action of $X$, an assertion which arises out of part (b) of both Theorem \[Main4\] and \[Main4scri\]. Furthermore, out of , it is manifest that both $\Lambda_\scrim$ and $\Lambda_{\cH}$ satisfy in a weak sense and in both entries the equation of motion, since they are constructed out of the causal propagator [(\[twopointseparately\])]{}. Yet their antisymmetric part does not correspond to the causal propagator and this lies at the heart of the impossibility to directly apply the proof of theorem 5.1 as it appears in [@SV00].\ Nevertheless, if we still indicate by $\beta^{(X)}_t$ ($t\in\bR)$ the pull-back action of one-parameter group of isometries generated by $X$ on elements in $C^\infty_0(\mW;\bR)$, we can employ , as well as the definition of both $\la_\scrim$ and $\la_{\cH}$, to infer the following: $\Lambda_\scrim$, which we shall refer as [**vacuum like**]{}, fulfils formula (A1) in [@SV00]: $$\int\limits_\bR\widehat{f}(t) \Lambda_\scrim(h_1\otimes \beta^{(X)}_t(h_2))dt =0,\quad h_1,\;h_2\in C^\infty_0(\mW;\bR)$$ for all $\widehat{f}(t)\doteq\int_\bR e^{-ikt} f(k) dk$ such that $f\in C^\infty_0 (\bR^*_-;\bC)$. At the same time $\Lambda_{\cH}$ fulfils formula (A2) in the same mentioned paper, which implies that it is [**KMS like**]{} at inverse temperature $\beta_H$, [*i.e.*]{}, $$\int\limits_\bR\widehat{f}(t) \Lambda_{\cH}(h_1\otimes \beta^{(X)}_t(h_2)) dt = \int\limits_\bR\widehat{f}(t+ i\beta_H) \Lambda_{\cH}(\beta^{(X)}_t(h_2)\otimes h_1)dt,\quad h_1,\;h_2\in C^\infty_0( \mW;\bR)\:,$$ for every $f\in C^\infty_0(\bR; \bR)$. The former identity arises out of the Fubini-Tonelli’s theorem and of basic properties of the Fourier-Plancherel transform. To wit, if one bears in mind the definition of $\Lambda_\scrim$, $\omega_\scrim$, the explicit expression of $\sH_{\scrim} = L^2(\bR_+ \times \bS^2; 2kdk \wedge d\bS^2)$ as well as part (c) of Theorem \[Main4scri\]: $$\Lambda_\scrim(h_1\otimes \beta^{(X)}_t(h_2)) = \int_{\bS^2} d\bS^2(\omega) \int_0^{+\infty} \overline{\psi_1(k ,\omega)} e^{itk} \psi_2(k,\omega) 2k dk,$$ for suitable functions $\psi_1$ and $\psi_2 \in L^2(\bR\times \bS^2; 2kdk \wedge d\bS^2)$ which corresponds to $h_1$ and $h_2$ . We also stress that the $k$ integration is only extended to the [*positive real axis*]{}, whereas the support of $f$ is contained in $\bR_-$. If one notices that, if $h\in C_0^\infty(\mW; \bR)$, then $\varphi^h_\cH \in \sS(\cH^-)$, then the second identity follows similarly from Theorem \[Main4\]. Here the key ingredients are the definition of $\Lambda_{\cH}$, $\omega^{\beta_H}_{\cH^-}$ and the explicit expression of the measure $\mu(k)$ in $\sH^{\beta_H}_{\cH^-}=L^2(\bR \times \bS^2; \mu(k) \wedge d\bS^2)$, and point (e) of Theorem \[Main4\].\ The validity of this pair of identities suffices to establish the statement of Proposition 2.1 in [@SV00], whose proof can be slavishly repeated with our slightly weaker assumptions, though one should mind the different conventions in our definition of the Fourier transform. From this point onwards, one can follow, in our framework and step by step, the calculations leading to the second point in the proof of Theorem 5.1 in [@SV00], which is nothing but the statement of our lemma. We shall not reproduce all the details here, since it would lead to no benefit for the reader. $\qed$\ Equipped with the proved lemma, and following the remaining steps of the proof of Theorem 5.1 in [@SV00] the last statement in the thesis of Theorem 5.1 in [@SV00] can be achieved in our case, too. As remarked immediately after the proof of the mentioned theorem in [@SV00], that statement entails the validity of the microlocal spectrum condition for the considered two-point function. Thus we can claim that \[propWFW\][The two-point function $\Lambda_U\in\mD'( {\mM\times \mM})$ of the Unruh state, restricted on $C_0^\infty(\mW\times \mW; \bC)$, satisfies the microlocal spectral condition (\[WFgen\]) with $\mN= \mW$ and thus $\omega_U\spa\rest_{\cW(\mW)}$ is a Hadamard state.]{} .3cm [*Part 2*]{}. Our goal is now to establish that the microlocal spectrum condition for $\Lambda_U(x,x')$ holds true also considering pairs $(x,x') \in \mM\times \mM$ which do not belong to $\mW\times \mW$. The overall strategy, we shall employ, mainly consists of a careful use of the propagation of singularity theorem which shall allow us to divide our analysis in simpler specific subcases.\ To this avail, we introduce the following bundle of null cones $\mN_g\subset T^*\mM\setminus\left\{0\right\}$ constructed out of the principal symbol of $P_g$, as in : $$\mN_g \doteq\ag(x,k_x)\in T^*\mM\setminus\left\{0\right\}\;,\;\; g^{\mu\nu}(x)(k_{x})_{\mu}(k_{x})_{\nu}=0 \cg\;.$$ We define the [**bicharacteristic strips**]{} generated by $(x,k_x) \in \mN_g$ $$B(x,k_x)\doteq\ag(x',k_x')\in \mN_g\; |\; (x',k_{x'})\sim (x,k_x)\cg,$$ where $\sim$ was introduced in . The operator $P_g$ is such that we can apply to the weak-bisolution $\Lambda_U$ the theorem of propagation of singularities (PST), as devised in Theorem 6.1.1 of [@DH]. This guarantees that, on the one hand: WF(\_U)({0}\_g)({0}\_g), \[PST1\] while, on the other hand, B(x,k\_x)B(y,k\_[y]{})WF(\_U). \[PST2\] A pair of technical results, we shall profitably use in the proof, are given by the following lemma and proposition whose proofs can be found in appendix \[Appendixproofs\].\ The proposition characterises the decay property, with respect to $p\in T^*_x\mM$, of the distributional Fourier transforms even though one should notice that in [@Hormander] the opposite convention concerning the sign in front of $i\langle p,\cdot\rangle$ is adopted: $$\varphi^{f_{p}}_\scrim\doteq \lim_{\to \scrim} E_{P_g} (f e^{i\langle p,\cdot\rangle } ) \;,\qquad \varphi^{f_{p}}_\cH\doteq E_{P_g} (f e^{i\langle p,\cdot\rangle } ) \spa \rest_\cH$$ where we have used the complexified version of causal propagator, which enjoys the same causal and topological properties as those of the real one. Henceforth $\langle \cdot , \cdot \rangle$ denotes the standard scalar product in $\bR^4$ and $|\cdot|$ the associated norm, computed after the choice of normal coordinates. From now on we also shall assume to fix a coordinate patch whenever necessary, all the results being independent from such a choice, as discussed after Theorem 8.2.4 in [@Hormander]. We remind the reader that, given a function $F: \bR^n \to \bC$, an element $k\in \bR^n \setminus\{0\}$ is said to be of [**rapid decrease**]{} for $F$ if there exists an open conical set $V_k$, [*i.e.*]{}, an open set such that, if $p\in V_k$ then $\lambda p\in V_k$ for all $\lambda >0$, such that, $V_k \ni k$ and, for every $n=1,2, \ldots$, there exists $C_n \geq 0$ with $|F(p)|\leq C_n/(1+ |p|^{n})$ for all $p \in V_{k}$. \[RapidDecay\][Let us take $(x,k_x) \in \mN_g$ such that (i) $x\in\mM\setminus \mW$ and (ii) the unique inextensible geodesic $\ga$ (co-)tangent to $k_x$ at $x$ intersects $\cH$ in a point whose $U$ coordinate is nonnegative. Let us also fix $\chi'\in C^\infty(\cH; \bR)$ with $\chi' =1$ in $U\in (-\infty, U_0]$ and $\chi' = 0$ if $U\in [U_1,+\infty)$ for a constant value of $U_0 < U_1 < 0$.\ For any $f\in C^\infty_0(\mM)$ with $f(x)=1$ and sufficiently small support, $k_x$ is a direction of rapid decrease for both $p\mapsto \|\varphi^{f_{p}}_\scrim\|_\scrim $ and $p\mapsto \|\chi' \varphi^{f_{p}}_{\cH}\|_\cH$.]{}\ The pre-announced lemma has a statement which closely mimics an important step in the analysis of the Hadamard form of two-point functions, first discussed in [@SV01]. It establishes the the right-hand side of (\[PST1\]) can be further restricted. The next step in our proof consists of the analysis of $WF(\Lambda_U)$, in order to establish the validity of with $=$ replaced by $\subset$. We shall tackle the cases which are left untreated by the statement of Proposition \[propWFW\] in particular. As previously discussed, this suffices to conclude the proof of the Hadamard property for $\omega_U$.\ The remaining cases amount to the points in $WF(\Lambda_U)$ such that, in view of Lemma \[Nozero\], $(x,y,k_x,k_y) \in \mN_g\times \mN_g$ with either $x$, either $y$ or both in $\mM\setminus \mW$. Therefore, we shall divide the forthcoming analysis in two parts, [**case A**]{}, where only one point is in $\mM\setminus \mW$, and [**case B**]{}, where both lie in $\mM\setminus \mW$. .2cm [**Case A**]{}. Let us consider an arbitrary $(x,y,k_x,k_y) \in \mN_g \times \mN_g$ which belongs to $WF(\Lambda_U)$ and such that $x\in\mM\setminus\mW$ and $y\in\mW$, the symmetric case being treated analogously. If a representative of the equivalence class $B(x,k_x)$ has its basepoint in $\mW$, (\[PST2\]) entails that the portion of $B(x,k_x) \times B(y,k_y)$ enclosed in $T^*(\mW\times\mW)$ must belong to $WF(\Lambda_U\spa\rest_{C_0^\infty(\mW\times \mW;\bC)})$ and, thus, it must have the shape stated in Proposition \[propWFW\]. Thanks to the uniqueness of a geodesic which passes through a point with a given (co-)tangent vector, it implies that $(x, k_x) \sim (y, -k_y)$ and $k_x\triangleright 0$ as wanted.\ Let us consider the remaining subcase where no representative of $B(x,k_x)$ has a basepoint in $\mW$. Our goal is to prove that, in this case, $(x,y, k_x, k_y) \not \in WF(\Lambda_U)$ for every $k_y$. This will be established showing that there are two compactly supported smooth functions $f$ and $g$ with $f(x)=1$ and $g(y)=1$ such that $(k_x, k_y)$ individuate directions of rapid decrease of $(p_x,p_y) \mapsto \Lambda_U((fe^{i\langle p_x, \rangle} \otimes h e^{i\langle p_y, \rangle})$.\ If $B(x,k_k)$ does not meet $\mW$, there must exist $(q,k_q)\in B(x,k_x)$, such that $q\in\cH$ and the Kruskal null coordinate $U=U_q$ is nonnegative. Let us consider, then, the two-point function $$\Lambda_U(f\otimes h)= \Lambda_{\cH}(f \otimes h)+\Lambda_\scrim(f \otimes h),\quad f,h\in C^\infty_0(\mM; \bR),$$ where $\Lambda_{\cH}$ and $\Lambda_\scrim$ are as in . If the supports of the chosen $f$ and $h$ are sufficiently small, we can always engineer a function $\chi\in C^\infty_0(\cH)$ in such a way that $\chi(U_q,\theta,\phi)=1$ for all $(\theta,\phi)\in\bS^2$ and $\chi=0$ on $J^-(supp\; h)$ and $\cH$. Furthermore, if we use a coordinate patch which identifies an open neighbourhood of $supp(f)$ with $\bR^4$ and we set $\chi' \doteq 1-\chi$, we can arrange a conical neighbourhood $\Ga_{k_x} \in \bR^4 \setminus \{0\}$ of $k_x$ such that all the bicharacteristics $B(s,k_s)$ with $s \in supp(f)$ and $k_s \in \Ga_{k_x}$ do not meet any point of $supp \chi'$ on $\cH$. If we refer to (\[BCN\]), we can now divide $\Lambda_{\cH}(f \otimes h)$ as: $$\Lambda_{\cH}(f \otimes h)=\la_{\cH}(\chi \varphi^{f}_{\cH},\varphi_{\cH}^{h})+\la_{\cH}(\chi' \varphi^{f}_{ \cH},\varphi_{\cH}^{h}),$$ and we separately analyse the behaviour of the following three contributions at large $(k_x,k_y)$ : \[distributions\] \_(\^[f\_[k\_x]{}]{}\_,\_\^[h\_[k\_y]{}]{}) , \_(’ \^[f\_[k\_x]{}]{}\_,\_\^[h\_[k\_y]{}]{}) \_(\_\^[f\_[k\_x]{}]{},\_\^[h\_[k\_y]{}]{}). Each of these should be seen as the action of a corresponding distribution in $\mD'(\mM\times \mM)$. The scenario, we face, is less complicated than it looks at first glance since we know that neither $(x,y,k_x,0)$ nor $(x,y,0,k_y)$ can be contained in $WF(\La_U)$, as Lemma \[Nozero\] yields. Hence this implies that, in the splitting we are considering in , we can focus only on the points $(x,y,k_x,k_y)$ where both $k_x$ and $k_y$ are not zero. If we were able to prove that these points are not contained in the wave front set of any of the three distributions , we could conclude that they cannot be contained in the wave front set of the their sum $\La_U$, because the wave front set of the sum of distributions is contained in the union of the wave front set of the single component. At the same time, the second and third distribution in the right-hand side of turn out to be dominated by $ C \|\chi' \varphi^{f_{-k_x}}_{\cH}\|_{\cH} \| \varphi^{h_{k_y}}_{\cH}\|_{\cH}$ and $C' \|\varphi^{f_{-k_x}}_{\scrim}\|_\scrim \| \varphi^{h_{k_y}}_{\scrim}\|_\scrim$, respectively, $C$ and $C' $ being suitable positive constants, whereas $\|\cdot\|_{\cH}$ and $\|\cdot\|_{\scrim}$ stand for the norm and . This is a by-product of the continuity property presented in points (f) of both propositions \[Propembedding\] and \[Propembeddingscri\], here adapted for complex functions, too. Furthermore, per Proposition \[RapidDecay\], both $\|\chi' \varphi^{f_{k_x}}_{\cH}\|_{\cH}$ and $\|\varphi^{f_{k_x}}_{\scrim}\|_\scrim$ are rapidly decreasing in $k_x \in T^*\mM\setminus\{0\}$ for an $f$ with sufficiently small support and if $k_x$ is in a open conical neighbourhood of any null direction. The remaining two terms $\|\varphi^{h_{k_y}}_{\cH}\|_{\cH}$ and $\|\varphi^{h_{k_y}}_{\scrim}\|_\scrim$, in , can at most grow polynomially in $k_y$. The last property can be proved as follows: if one starts from the bounds for the behaviour of the wavefunctions restricted to on $\cH^-$ and $\scrim$, as per Proposition \[PropDR\], one can estimate the norms $\|\varphi^{h_{k_y}}_{\scrim}\|_\scrim$, $\|\varphi^{h_{k_y}}_{\cH}\|_\cH$ embodying the dependence on $k_y$ in the explicit expression of the coefficients $C_i$ which appear in Proposition \[PropDR\]. Then, out of an argument similar to the one exploited in the proof of Proposition \[RapidDecay\], for fixed $k_y$ and $h$, those coefficients can be bounded by $C \sqrt{|\tilde{E}_5( \varphi^{h_{k_y}})|}$ as in [(\[E2\])]{}, where $\tilde{E}_5(\varphi^{h_{k_y}})$ is the integral of a polynomial of derivatives of $\varphi^{h_{k_y}}$ on a suitable Cauchy surface $\Sigma\subset\mW$. Notice that $\varphi^{h_{k_y}}(z) = (E_{P_g}(h_{k_y}))(z)$ is smooth, has compact support when restricted on a Cauchy surface, and together with the compact supports of its derivatives are contained in a common compact subset $K\subset \Si$. One can exploit the continuity of the causal propagator $E_{P_g}$, to conclude that, for every fixed multi-index $\alpha$, $\sup_K| \partial^\alpha E_{P_g}(h_{k_y})|$ is bounded by a corresponding polynomial in the absolute values of the components of $k_y$. The coefficients are the supremum of derivatives of $h\in C_0^{\infty}(\mM;\bR)$ up to a certain order. This implies immediately that $\tilde{E}_5(\varphi^{h_{k_y}})$, as well as $\|\varphi^{h_{k_y}}_{\scrim}\|_ \scrim$, $\|\varphi^{h_{k_y}}_{\cH}\|_\cH$ are polynomially bounded in $k_y$, also because the computation of $\tilde{E}_5(\varphi^{h_{k_y}})$ has to be performed on a compact set $K\subset\Sigma$.\ We now remind the reader that we have identified $\mK \times \mK$ with $\bR^4\times \bR^4$ by means of a suitable pair of coordinate frames. Hence cotangent vectors at different points $x$ and $y$ can be thought of as elements of the same $\bR^4$ and, hence, compared. This allows us to define the following open cone in $\bR^4$, $\Ga\subset \bR^4\times \bR^4$, [*i.e.*]{}, with $0<\epsilon< 1$, \[cono\] \_[k\_x]{}=(p\_x,p\_y)\^4\^4 | |p\_x| &lt; |p\_y| &lt; |p\_x| . , -p\_x U\_[-k\_x]{}where $U_{k_x}$ is an open cone around the null vector $k_x\neq 0$ where $p \mapsto \|\chi' \varphi^{f_{p_x}}_{\cH}\|^\eta_{\cH^-}$ and $p\mapsto \|\varphi^{f_{p_x}}_{\scrim}\|_\scrim$ decrease rapidly. Hence, per construction, for any direction $(k_x,k_y)$ with both $k_x \neq 0$ and $k_y\neq 0$ of null type, there is a cone $\Ga_{k_x}$ containing it. Moreover all the directions contained in $\Ga_{k_x}$ are of rapid decrease for both $\la_{\cH}(\chi \varphi^{f_{k_x}}_{\cH},\varphi_{\cH}^{h_{k_y}})$ and $\la_{\cH}(\chi' \varphi^{f_{k_x}}_{\cH},\varphi_{\cH}^{h_{k_y}})$ because, just in view of the shape of $\Gamma_{k_x}$, the rapid decrease of $\|\chi' \varphi^{f_{-p_x}}_{\cH} \|_{\cH}$ and $\|\varphi^{f_{-p_x}}_{\scrim}\|_\scrim$ controls the polynomial growth in $|p_y|$ of $\| \varphi^{h_{p_y}}_{\cH}\|_{\cH}$ and $\|\varphi^{h_{p_y}}_{\scrim}\|_\scrim$ respectively.\ We are thus left off only with the first term in and, also in this case, if the support of $f$ and $h$ are chosen sufficiently small, $\la_{\cH}(\chi \varphi^{f_{k_x}}_{\cH},\varphi_{\cH}^{ h_{k_y}})$ can be shown to be rapidly decreasing in both $k_x$ and $k_y$. To this end, let us thus choose $\chi''\in C^\infty(\cH; \bR )$ such that both $\chi''(p)=1$ for every $p$ in $\supp(\varphi^{h_{k_y}}_{\cH})$ (also for every $k_y$) and $\chi''\cap \chi =\emptyset$. We can write $$\la_{\cH}(\chi \varphi^{f_{k_x}}_{\cH},\varphi_{\cH}^{ h_{k_y}}) = \int_{\cH\times \cH} \sp\sp \chi(x') \left(E_{P_g}(f_{kx})\right)(x')\: T(x',y') \chi''(y') \varphi^{h_{k_y}}_{\cH}(y')\quad dU_{x'}d\bS^2(\theta_{x'},\phi_{x'})\:dU_{y'}d\bS^2(\theta_{y'},\phi_{y'})$$ Theorem 8.2.14 of [@Hormander] guarantees us that $$(x',y',k_{x'},k_{y'})\not\in WF\left((T\chi'')\circ (\chi E_{P_g}\spa \rest_{\cH})\right)\;\qquad \forall (y',k_{y'})\in T^*\mM,$$ where $T$ is the integral kernel of $\la_{\cH}$ seen as a distribution on $\mD'(\cH\times \cH)$, while $\circ$ stands for the composition on $\cH$ [*with $E_{P_g}$ on the left of $T$*]{}. Finally $E_{P_g}\spa \rest_{\cH}$ means that the left entry of the causal propagator has been restricted on the horizon $\cH$, an allowed operation thanks to theorem 8.2.4 in [@Hormander]. One can convince himself, out of a direct construction, that the set of normals associated to the map embedding $\cH$ in $\mW$ does not intersect the wave front set of $E_{P_g}$. The integral kernel of $(\chi T\chi'')(x',y')$, with the entry $x'$ restricted on the support of $\chi$ and the entry $y'$ restricted on that of $\chi''$, moreover, is always smooth and, if one keeps $x'$ fixed, it is dominated by a smooth function whose $H^1$-norm in $y'$ is, uniformly in $x'$, finite. This also yields that, the $H^1(\cH)_U$-norm of $\|(T \chi'' )\circ \chi E_{P_g} f_{k_x}\|_{H^1(\cH)_U}$ is dominated by the product of two integrals one over $x'$ and one over $y'$. The presence of the compactly supported function $\chi$ and the absence of points of the form $(x,y,k_x,0)$ and $(y,x,0,k_y)$ in $WF(E_{P_g})$ assures that the integral kernel of $\chi T\chi''$ is rapidly decreasing in $k_x$. Summing up we have that \[aggI\] |\_(\^[f\_[k\_x]{}]{}\_,\_\^[h\_[k\_y]{}]{})| C ((T” )(E\_[P\_g]{})) (f\_[-k\_x]{})\_[H\^1()\_U]{} \_\^[h\_[k\_y]{}]{} \_ , where the second norm in the right-hand side is given in . This bound proves that, for a fixed $k_y$, $k_x \to \la_{\cH}(\chi \varphi^{f_{k_x}}_{\cH},\varphi_{\cH}^{h_{k_y}})$ is rapidly decreasing.\ To conclude, if we look again at and if we introduce a cone as in , out of Lemma \[Nozero\], we can control the (at most) polynomial growth of $\| \varphi_{\cH}^{h_{k_y}} \|_{\cH}$ using the rapidly decreasing map $k_x \mapsto \|\left((T\chi'' )\circ (\chi E_{P_g})\right) (f_{-k_x})\|_{H^1(\cH)_U} $. Hence we establish that $(k_x,k_y)$ is a direction of fast decreasing of $\la_{ \cH}(\chi \varphi^{f_{k_x}}_{\cH},\varphi_{\cH}^{h_{k_y}})$.\ .2cm [**Case B**]{}. We shall now tackle the case in which we consider an arbitrary but fixed $(x,y,k_x, k_y)\in \mN_g\times \mN_g$, with both $x$ and $y$ lying $\mM\setminus \mW$.\ If one assumes that $(x,y,k_x, k_y) \in WF(\Lambda_U)$ we have to prove that both $(x,k_x)\sim(y,-k_y)$ and $k_x\triangleright 0$ have to be valid. If $B(x,k_x)$ and $B(y,k_y)$ are such that both admit representatives in $\mW$, we make use of both (\[PST2\]) and of the fact that elements in the wavefront set of the restriction of $\Lambda_U$ to $\mW$ fulfils $(x',k'_{x'})\sim(y',-k'_{y'})$ and $k'_{x'}\triangleright 0$. Hence one extends this property to $(x, y, k_x, k_y)$ following the same reasoning as the one at the beginning of the [ *Case A*]{}. If, instead, only one representative, either of $B(x,k_x)$ or of $B(y,k_y)$ lies in $\mW $, then we fall back in [*Case A*]{} studied above again thanks to (\[PST2\]). Thus, we need only to establish the wanted behaviour of the wave front set when it is possible to find representatives of both $B(x,k_x)$ and $B(y,k_y)$ which intersect $\cH$ at a nonnegative value of $U$. We shall follow a procedure similar to the one already employed in [@Moretti08].\ In this framework, let us consider the following decomposition of $\Lambda_U(\varphi^{f_{k_x}}\otimes \varphi^{h_{k_y}})$: $$\la_U(\varphi^{f_{k_x}},\varphi^{h_{k_y}})=\la_{\cH}(\varphi^{f_{k_x}}, \varphi^{h_{k_y}})+\la_\scrim(\varphi^{f_{k_x}},\varphi^{h_{k_y}}),$$ where $f, h\in C^\infty_0(\mM)$ and they attain the value $1$ respectively at the point $x$ and $y$.\ As before, we start decomposing the first term in the preceding expression by means of a partition of unit $\chi, \chi'$ on $\cH$, where $\chi, \chi' \in C^\infty_0(\cH)$ satisfy $\chi+\chi'=1 :\cH \to \bR$. We obtain $$\begin{gathered} \la_{\cH}(f_{k_x},h_{k_y}) = \la_{\cH}(\chi \varphi^{f_{k_x}}_{\cH}, \chi \varphi^{h_{k_y}}_{\cH})+ \la_{\cH}(\chi' \varphi^{f_{k_x}}_{\cH}, \chi \varphi^{h_{k_y}}_{\cH})+\notag\\ \la_{\cH}(\chi\varphi^{f_{k_x}}_{\cH}, \chi' \varphi^{h_{k_y}}_{\cH})+ \la_{\cH}(\chi'\varphi^{f_{k_x}}_{\cH}, \chi'\varphi^{h_{k_y}}_{\cH}).\label{decomp}\end{gathered}$$ Furthermore, the above functions $\chi, \chi' $ can be engineered in such a way that the inextensible null geodesics $\ga_x$ and $\ga_y$, which starts respectively at $x$ and $y$ with cotangent vectors $k_x$ and $k_y$, intersect $\cH$ in $u_x$ and $u_y$ (possibly $u_x=u_y)$, respectively, included in two corresponding open neighbourhoods $O_x$ and $O_y$ (possibly $O_x=O_y$) where $\chi'$ vanishes. Let us start from the first term in the right hand side of and, particularly, we shall focus on the wave front set of the unique extension of $f\otimes g \mapsto \la_{\cH}(\chi \varphi^{f}_ {\cH},\chi\varphi^{h}_{\cH})$ to a distribution in $\mD'(\mM\times \mM)$. If we indicate as $T$ the integral kernel of $\la_{\cH}$, interpreted as distribution of $\mD'(\cH\times\cH)$, we notice that, as an element in $\mD'(\mM\times \mM)$, $\la_{\cH}$ can be written as: $$\la_{\cH}(\chi \varphi^{f}_{\cH}, \chi \varphi^{h}_{\cH})\doteq\;\chi T\chi\left(E_{P_g}\rest_{\cH}\otimes E_{P_g}\rest_{\cH}\at f\otimes h\ct\right),$$ where $E_{P_g}\rest_{\cH}$ is the causal propagator with one entry restricted on the horizon $\cH$ and $\chi T \chi\in\cE'(\cH\times\cH)$. Thanks to the insertion of the compactly supported smooth functions $\chi$, and with the knowledge that $WF(E_{P_g}\otimes E_{P_g})_{\cH\times\cH}=\emptyset$ (see [@Moretti08]), we can make sense of the previous expression as an application of Theorem 8.2.13 in [@Hormander], of which we also employ the notation. The wave front set of $T$ has been already explicitly written in Lemma 4.4 of [@Moretti08] and, hence, still Theorem 8.2.13 in [@Hormander] guarantees us that if $(x,y,k_x,k_y)$ is contained in the wave front set of the resulting distribution then $(x,k_x)\sim(y,-k_y)$ and $k_x\triangleright 0$ hold.\ If we come back to the remaining terms in , it is possible to show that all of them, together with $\lambda_\scrim$ are rapidly decreasing in both $k_x$ and $k_y$, provided that $f$ and $h$ have sufficiently small support. Hence they give no contribution to $WF(\Lambda_U)$.\ Here we analyse in details only the second term in [(\[decomp\])]{} since the others can be treated exactly in the same way. To start with, notice that, due to (f) in proposition \[Propembedding\] $|\la_{\cH}(\chi' \varphi^{f_{k_x}}_{\cH }, \chi \varphi^{h_{k_y}}_{\cH})|$ is bounded by $C \|\chi' \varphi^{f_{k_x}}_{\cH}\|_{\cH^- } \|\chi \varphi^{h_{k_y}}_{\cH}\|_{\cH}$, where $\|\cdot\|_{\cH}$ is the norm introduced in and $C>0$ is a constant. Due to Proposition \[RapidDecay\], $\|\chi' \varphi^{f_{k_x}}_{\cH}\|_{\cH}$ is rapidly decreasing in $k_x$ for some $f$ with sufficiently small support. Finally, the rapid decrease of $\|\chi' \varphi^{f_{-k_x}}_{\cH}\|_{\cH}$ can control the at-most polynomial growth of $\|\chi \varphi^{h_{k_y}}_{\cH}\|_{\cH}$, as discussed above in the analysis of using the fact that $(x,y,k_x,0)$ and $(x,y,0,k_y)$ cannot be contained in the wave front set of $\Lambda_U$; this leads to the construction of the open cone $\Gamma_{k_x}$.\ If we collect all the pieces of information we have got about the shape of $\Lambda_U$, we can state that: $$WF(\Lambda_U) \subset \ag (x,y,k_x,k_y)\in T^*(\mM\times \mM)\setminus \{0\},\; (x,k_x)\sim (y,-k_y),\; k_x \triangleright 0 \cg$$ and this concludes the proof. $\qed$ .3cm To conclude this section we have an important remark on the physical interpretation of $\omega_U$, that arises if one combines the results presented above, on the Hadamard property fulfilled by $\om_U$ in the [*full region*]{} $\mM$, together with the known achievements due to Fredenhagen and Haag [@Fredenhagen] (see also the discussion Sec VIII.3 in [@Haag]). In such paper the authors showed that, whenever a state $\om'$ is vacuum like far away from the black hole, its two-point function $\La'(x,x')$ tends to zero when the spatial separation between $x$ and $x'$ tends to infinity and whenever it is of Hadamard form in a neighborhood of $\cH_{ev}$, then, towards future infinity, the Hawking radiation appears. More precisely, if $h$ is a compactly supported smooth function supported far away from the black hole, they show that, for positive large values of $T$, the expectation value of $\La'(\beta_T^{X}(h),\beta_T^{X}(h))$, interpreted as the response of a detector, is composed by two contributions. One relates the signals received by outward directed detectors (looking away from the collapsed star) and such contribution is completely due to the Boulware vacuum. The other one takes, instead, the approximated form \_[l,m]{} \_ d |\_[lm]{}()|\^2 \[LAST\] valid at positive large $T$ and assuming that the support of $h$ stays in a region $r>> R >0$. Above $|D_\ell(\epsilon)|^2$ is the (gravitational) barrier penetration factor at energy $\epsilon$. Furthermore $|\widetilde{h}_{lm}(\epsilon)|^2$ is the sensitivity of the detector to quanta of energy $\epsilon$ and to angular momentum individuated by the quantum numbers $l$ and $m$, found employing an approximated mode decomposition as displayed in equation (VIII.3.45) of [@Haag]. Formula shows that the asymptotic counting rate is the one produced by an outgoing flux of radiation at temperature $1/\beta_H$ modified by the barrier penetration effect.\ We stress that, as $\omega_U$ is of Hadamard form on $\mM$, the Fredenhagen-Haag’s result can be applied to the Unruh state, proving that it describes the appearance of the Hawking radiation near $\Im^+$ as described in . Here, the splitting of $\La_U(\beta_T^{(X)}(h),\beta_T^{(X)}(h))$ in the two contributions $\La_{\cH}(\beta_T^{(X)}(h),\beta_T^{(X)}(h))$, due to $\cH$ and $\La_{\scrim}(\beta_T^{(X)}(h),\beta_T^{(X)}(h))$ due to $\scrim$ is already embodied in the very construction of $\omega_U$ as $\omega_\cH\otimes \omega_\scrim$. Furthermore, they coincide separately with the two terms surviving in the limit $T\to+\infty$ according to the analysis given in [@Fredenhagen]. This last extent can be shown by invariance of $\omega_U$ under $\beta^{(X)}$, hence moving towards the past the Cauchy surface used in [@Fredenhagen], instead of moving the support of $h$ towards the future. In the limit $T\to+\infty$, the contribution due to $\cH$, gives rise to the expression (4.13) in [@Fredenhagen] in particular. Finally, one can follow almost slavishly the very steps of [@Fredenhagen], which are based on the asymptotic behaviour of the solutions of the Klein-Gordon equation in $\mW$, to get (\[LAST\]).\ As a last comment, we emphasize that this result is stable under perturbations of the state $\om_U$ that involve only modification of $\omega_{\cH}$ in such a way that its integral kernel, seen as an hermitian map on $\sS(\cH)$, differs from the one of $\la_\cH$ in by a smooth function on $\cH\times\cH$, integrable in the product measure $(dU \wedge d\bS^2)\wedge (dU' \wedge d\bS'^2)$. Let us indicate by $\om''$ the perturbed state and by $\La''$ its two-point function, which is supposed to be a well defined distribution in $\mD'(\mM\times\mM)$; per direct application of Lebesgue’s dominated convergence theorem, one finds that the contribution due to the perturbation vanishes at large $T$ under the action of $\beta^{(X)}_T$, so that: $$\lim_{T\to\infty} \La''(\beta^{(X)}_T f\otimes \beta^{(X)}_T h)=\La_U(f\otimes h) \;. \qquad f, h\in C^\infty_0(\mW)$$ This is tantamount to claim that, for large positive values of $T$, $\om''$ tends weakly to the Unruh state. In other words, far in the future, the effects seen in $\om''$ coincide with those shown in $\om_U$, hence the Hawking radiation also appears in the perturbed state. Conclusions =========== In this paper we employed a bulk-to-boundary reconstruction procedure to rigorously and unambiguously construct and characterise on $\mM$ ([*i.e.*]{}, the static joined with the black hole region of Schwarzschild spacetime, event horizon included) the so-called Unruh state $\om_U$. Such state plays the role of natural candidate to be used in the quantum description of the radiation arising during a stellar collapse. Furthermore we proved that $\om_U$ fulfils the so-called Hadamard condition, hence it can be considered a genuine ground state for a massless scalar field theory living on the considered background. Overall, the achieved result can be seen as a novel combination of earlier approaches [@DMP; @Moretti08; @DMP2; @DMP3] with the theorems proved in [@SV00] as well as with the powerful results obtained by Rodnianski and Dafermos in [@DR03; @DR07; @DR05]. Therefore we can safely claim that it is now possible to employ the Unruh vacuum in order to use the analysis in [@Fredenhagen] as a starting point, to study quantum effects such as the role of the back reaction of Hawking’s radiation, a phenomenon which was almost always discarded as negligible. At the same time it would be certainly interesting to try to enhance the results of this paper since, as one can readily infer from the main body of the manuscript, $\om_U$ has been here constructed only on $\mM$. It is worth stressing that it is, however, possible to extend $\omega_U$ to the whole Kruskal manifold following our induction procedure, defining a further part of the state on $\Im_L^+$. The obtained state on the whole Weyl algebra $\cW(\mK)$ would be invariant under the group of Killing isometries generated by $X$ and without zero modes, if one refers to the one-parameter group of isometries. The problem with this extension is related with the Hadamard constraint. Indeed, we do not expect that this extension is of Hadamard form on $\cH$, due to a theoretical obstruction beyond Candelas’ remarks [@Candelas]. In view of the uniqueness and KMS-property theorem proved in [@KW] for a large class of spacetimes including the Kruskal one, the validity of the Hadamard property on the whole spacetime together with the invariance under $X$ and the absence of zero-modes imply that the state is unique on a certain enlarged algebra of observables $\mA$ on $\mK$. Furthermore it coincides with a KMS state with respect to the Killing vector $X$ at the Hawking temperature, [*i.e.*]{} it must be the Hartle-Hawking state for a certain subalgebra of observables $\mA_{0I} \subset \mA$ supported in the wedge $\mW$. These algebras are obtained out of a two steps approach. At first one enlarges $\cW(\mK)$ to $\mA$, whose Weyl generators are smeared with both the standard solutions of KG equation with compactly supported Cauchy data in $\mK$ and a certain class of weak solutions of the same equation. Afterwards one restricts this enlarged algebra to a certain subalgebra of observables $\mA_{0I}$ supported in $\mW$, in a suitable sense related with the properties of the supports of the smearing distributions across the Killing horizon. With respect to our state we know that the KMS property is not verified in a neighbourhood of $\Im^-$, so we do not expect that any extension of that state satisfies there the KMS property. Nonetheless the issue is not completely clear since the extension we are discussing and the failure of the KMS condition are both referred to $\cW(\mW)$ rather than $\mA_{0I}$. Hence further investigations in such direction would be desirable.\ A further and certainly enticing possible line of research consists of using the very same approach discussed in this paper in order both to rigorously define the very Hartle-Hawking state and to prove its Hadamard property; although, from a physical perspective, this is certainly a very interesting problem, from a mathematical perspective, it amounts to an enhancement of the peeling behaviours for the solutions of the Klein-Gordon equation discussed by Rodnianski and Dafermos, also beyond what recently achieved in [@Luk]. Although there is no proof that the obtained ones are sharp conditions, the high degree of mathematical specialisation, needed to obtain the present results, certainly makes the proposed programme a challenging line of research, which we hope to tackle in future papers.\ As an overall final remark, it is important stress that all our results are only valid for the massless case, since the massive one suffers of a potential sever obstruction which is the same as the one pointed out in [@DMP]. To wit it appears impossible to directly project on null infinity a solution of the massive wave equation and, hence, the problem must be circumvented with alternative means, as it has been done, for example, in Minkowski spacetime in [@Da08a]. A potential solution of this puzzle in the Schwarzschild background would be certainly desirable. Acknowledgements. {#acknowledgements. .unnumbered} ================= The work of C.D. is supported by the von Humboldt Foundation and that of N.P. has been supported by the German DFG Research Program SFB 676. We are grateful to S. Hollands for the useful discussions and for having pointed out [@DR05]. We also deeply acknowledge B.S. Kay for the useful clarifications of the various equivalent definitions of KMS states as well as for having pointed out some relevant references. Further details on the geometric setup. ======================================= \[geometry\] In this paper, the extension of the underlying background to include null infinities as well as a region beyond them, plays a pivotal role and we shall now dwell into a few more details. To this end, one purely follows [@SW83] and rescales the global metric $g$ in (\[g\]) by a factor $1/r^2$ after which one can notice that the obtained manifold $(\mM, g/r^2)$ admits a smooth larger extension $(\widetilde{\mM},\widetilde{g})$. We have to notice that, in this case, the singularity present at $r=0$ in $(\mM,g)$ is pushed at infinity in the sense that the non-null geodesics takes an infinite amount of affine parameter to reach a point situated at $r=0$. The extension of $(\widetilde{\mM},\widetilde{g})$ obtained in this way does not cover the sets indicated as $i^\pm$ and $i_0$ in figure 1 and figure 2, though it includes the boundaries ${\Im}^{\pm}$, called [**future**]{} and [**past null infinity**]{} respectively. These represent subsets of $\widetilde{\mM}$ which are null $3$-submanifold of $\widetilde{\mM}$ formally localised at $r=+\infty$. Let us now examine the form of the rescaled extended metric restricted to the Killing horizon $\cH$ as well as to the null infinities ${\Im}^\pm$. Per direct inspection, one finds that, if one fixes $\Omega \doteq 2V$, which vanishes on $\cH$, $$\widetilde{g}\spa\rest_{\cH} = r_S^2\left( - d\Omega \otimes dU - dU \otimes d \Omega + h_{\bS^2}(\theta,\phi)\right)\:.$$ In this case $V\in \bR $ is the complete affine parameter of the null $\widetilde{g}$-geodesics generating $\cH$ and $\cH$ itself is obtained setting $U=0$. This form of the metric is called [**geodetically complete Bondi form**]{}.\ The same structure occurs on $\Im^+$, formally individuated by $v= +\infty$ and on $\Im^-$, formally individuated by $u= -\infty$, where the metric $\tilde{g}$ has still a [**geodetically complete Bondi form**]{}, namely $$\widetilde{g}\spa\rest_{\Im^+} = - d\Omega \otimes du - du \otimes d\Omega + h_{\bS^2}(\theta,\phi)\:,$$ where $\Omega \doteq -2/v$ individuates $\scri$ for $\Omega=0$. Similarly $$\tilde{g}\spa\rest_{\Im^-} = - d\Omega \otimes dv - dv \otimes d\Omega + h_{\bS^2}(\theta,\phi)\:,$$ where $\Omega \doteq -2/u$ individuates $\Im^+$ for $\Omega =0$ $$X = \partial_u \mbox{ on $\Im^+$,} \quad X = \partial_v \mbox{ on $\Im^-$}.$$ (150,200)(0,0) (0,0)[{height="8.5cm"}]{} (205,80)[$\scrim$]{} (205,155)[$\scri$]{} (205,155)[$\scri$]{} (172,50)[$i^-$]{} (172,185)[$i^+$]{} (172,115)[$I$]{} (52,115)[$II$]{} (110,150)[$III$]{} (110,85)[$IV$]{} In both cases the coordinates $u$ and $v$ are well defined and they coincide with the complete affine parameters of the null $\widetilde{g}$-geodesics forming $\Im^+$ and $\Im^-$ respectively.\ With respect to Killing symmetries, we notice that the $g$-Killing vector $X$ is also a Killing vector for $\widetilde{g}$ and it extends to a $\widetilde{g}$-Killing vector $X$ defined on $\widetilde{\mM}$. Particularly, in $\partial \mM$ it satisfies $$X=\partial_u\;\;\textrm{on}\;\Im^+,\qquad X=\partial_v\;\;\textrm{on}\;\Im^-.$$ Weyl algebras, quasifree states, KMS condition. {#algebras} =============================================== A $C^*$-algebra $\cW(\sS)$ is called [**Weyl algebra**]{} associated with a (real) symplectic space $(\sS,\sigma)$ (the symplectic form $\sigma$ being nondegenerate) if it contains a class of non-vanishing elements $W(\psi)$ for all $\psi \in \sS$, called [**Weyl generators**]{}, which satisfy [**Weyl relations**]{}[^3]: $$(W1)\quad\quad W(-\psi)= W(\psi)^*\:,\quad\quad\quad\quad (W2)\quad\quad W(\psi)W(\psi') = e^{i\sigma(\psi,\psi')/2} W(\psi+\psi') \:.$$ $\cW(\sS)$ coincides with the closure of the $*$-algebra (finitely) generated by Weyl generators. As a consequence of (W1) and (W2), one gets: $W(0) = \bI$ (the unit element), $W(\psi)^*= W(\psi)^{-1}$, $||W(\psi)||=1$ and, out of the non degenerateness of $\sigma$, $W(\psi)=W(\psi')$ iff $\psi=\psi'$ .\ $\cW(\sS)$ is [*uniquely*]{} determined by $(\sS,\sigma)$ (theorem 5.2.8 in [@BR2]): Two different realizations admit a unique $*$-isomorphism which transform the former into the latter, preserving Weyl generators, and the norm on $\cW(\sS)$ is unique, since $*$ isomorphisms of $C^*$-algebras are isometric. This result implies that every GNS $*$-representation of a Weyl algebra is always faithful and isometric. It is also worth mentioning that, per construction, any GNS $*$-representation of a Weyl algebra is such that the generators are always represented by unitary operators, but it is not the case for other $*$-representations in Hilbert spaces.\ $\cW(\sS)$ can always be realized in terms of bounded operators on $\ell^2(\sS)$, viewing $\sS$ as a Abelian group and defining the generators as $(W(\psi)F)(\psi')\doteq e^{-i\sigma(\psi,\psi')/2}F(\psi+\psi')$ for every $F\in \ell^2(\sS)$. In this realization (and thus in every realization) it turns out that the generators $W(\psi)$ are [*linearly independent*]{}. A state $\omega$ on $\cW(\sS)$, with GNS triple $(\gH_\omega, \Pi_\omega, \Omega_\omega)$, is called [**regular**]{} if the maps $\bR\ni t\mapsto \Pi_\omega(W(t\psi))$ are strongly continuous. In general, strong continuity of the unitary group implementing a $*$-automorphism representation $\beta$ of a topological group $G \ni g \mapsto \beta_g$ for a $\beta$-invariant state $\omega$ on a Weyl algebra $\cW(\sS)$, is equivalent to $\lim_{g\to \bI} \omega(W(-\psi)\beta_g W(\psi)) = 1$ for all $\psi \in \sS$. The proof follows immediately from $||\Pi_\omega\left(\beta_{g'} W(\psi)\right)\Omega_\omega - \Pi_\omega\left(\beta_{g} W(\psi)\right)\Omega_\omega||^2 = 2- \omega\left(W(-\psi)\beta_{g'^{-1}g}W(\psi)\right) - \omega\left(W(-\psi)\beta_{g^{-1}g'}W(\psi)\right)$ and $\overline{\Pi_\omega(\cW(\sS))\Omega_\omega} = \gH_\omega$.\ If $\omega$ is regular, in accordance with Stone theorem, one can write $\Pi_\omega(W(\psi)) = e^{i\sigma(\psi,\Phi_\omega)}$, $\sigma(\psi,\Phi_\omega)$ being the (self-adjoint) [**field operator symplectically-smeared**]{} with $\psi$.\ When $\cW(\sS)= \cW(\sS(\mN))$ is the Weyl algebra on the space of Klein-Gordon equation solutions as in Sec. \[observables\], the field operator $\Phi_\omega(f)$ introduced in that section, smeared with smooth compactly supported functions $f \in C_0^\infty(\mN;\bR)$, is related with $\sigma(\psi,\Phi_\omega)$ by $$\Phi_\omega(f) \doteq \sigma(E_{P_g}(f),\Phi_\omega)\quad \mbox{for all $f \in C_0^\infty(\mN;\bR)$,}$$ where we exploit the notations used in Sec. \[observables\]. In this way, the field operators enter the theory in the Weyl algebra scenario. At a formal level, Stone theorem together with (W2) imply both $\bR$-linearity and the standard CCR: $$(L)\quad \sigma(a\psi + b \psi', \Phi_\omega) = a \sigma(\psi,\Phi_\omega) + b \sigma(\psi',\Phi_\omega)\:, \quad\:\: (CCR) \quad \mbox{$[$}\sigma(\psi,\Phi_\omega), \sigma(\psi',\Phi_\omega)\mbox{$]$} = -i\sigma(\psi,\psi')I\:,$$ for $a,b\in \bR$ and $\psi,\psi' \in \sS$. Actually (L) and (CCR) hold rigorously in an invariant dense set of analytic vectors by Lemma 5.2.12 in [@BR2] (it holds if $\omega$ is quasifree by proposition \[proposition2\]).\ In the standard approach of QFT, based on bosonic real scalar field operators $\Phi$, [*either a vector or a density matrix*]{} state are [*quasifree*]{} if the associated $n$-point functions satisfy (i) $\langle\sigma(\psi,\Phi) \rangle =0$ for all $\psi\in \sS$ and (ii) the $n$-point functions $\langle \sigma(\psi_1,\Phi)\cdots \sigma(\psi_n,\Phi)\rangle$ are determined from the functions $\langle \sigma(\psi_i,\Phi)\sigma(\psi_j,\Phi) \rangle$, with $i,j=1,2,\cdots, n$, using standard Wick’s expansion. A technically different but substantially equivalent definition, completely based on the Weyl algebra was presented in [@KW]. It relies on the following three observations: (a) if one works formally with (i) and (ii), one finds that it holds $\langle e^{i\sigma(\psi,\Phi)} \rangle = e^{-\langle \sigma(\psi,\Phi)\sigma(\psi,\Phi) \rangle/2}$. In turn, at least formally, such identity determines the $n$-point functions by Stone theorem and (W2). (b) From (CCR) it holds $\langle \sigma(\psi,\Phi)\sigma(\psi',\Phi)\rangle = \mu(\psi,\psi') - (i/2) \sigma(\psi,\psi')$, where $\mu(\psi,\psi')$ is the symmetrised two-point function $(1/2)(\langle \sigma(\psi,\Phi)\sigma(\psi',\Phi)\rangle + \langle \sigma(\psi',\Phi)\sigma(\psi,\Phi) \rangle)$ which defines a symmetric positive-semidefined bilinear form on $\sS$. (c) $\langle A^\dagger A\rangle\geq 0$ for elements $A\doteq [e^{i\sigma(\psi,\Phi)} -I]+i[e^{i\sigma(\psi,\Phi)}-I]$ entails: |(,’)|\^2 4(,)(’,’), \[sm\], which, in turn, implies that [*$\mu$ is strictly positive defined*]{} because $\sigma$ is non degenerate. If one reverses the procedure, the general definition of quasifree states on Weyl algebras is the following. \[defquasifree\] [*Let $\cW(\sS)$ be a Weyl algebra and $\mu$ a real scalar product on $\sS$ satisfying (\[sm\]). A state $\omega_\mu$ on $\cW(\sS)$ is called the [**quasifree state**]{} associated with $\mu$ if $$\omega_\mu(W(\psi)) \doteq e^{-\mu(\psi,\psi)/2} \:, \quad \mbox{for all $\psi\in \sS$.}$$*]{} The following technical lemma is useful to illustrate the GNS triple of a quasifree state as established in the subsequent theorem. The last statement in the lemma arises out of the Cauchy-Schwarz inequality and the remaining part out of Proposition 3.1 in [@KW]. \[lemma1A\] *Let $\sS$ be a real symplectic space with $\sigma$ non degenerate and $\mu$ a real scalar product on $\sS$ fulfilling (\[sm\]). There exists a complex Hilbert space $\sH_\mu$ and a map $K_\mu: \sS \to \sH_\mu$ with:* \(i) $K_\mu$ is $\bR$-linear with dense complexified range, i.e. $\overline{K_\mu(\sS) + i K_\mu(\sS)}= \sH_\mu$, \(ii) for all $\psi,\psi' \in \sS$, $\langle K_\mu\psi , K_\mu\psi'\rangle= \mu(\psi,\psi') - (i/2) \sigma(\psi,\psi')$.\ Conversely, if the pair $(\sH,K)$ satisfies (i) and $\sigma(\psi,\psi')= -2 Im \langle K\psi , K\psi'\rangle_\sH$, with $\psi,\psi' \in \sS$, the unique real scalar product $\mu$ on $\sS$ satisfying (ii) verifies (\[sm\]). \ An existence theorem for quasifree states can be proved using the lemma above with the following proposition relying on Lemma A.2, Proposition 3.1 and a comment on p.77 in [@KW]). \[proposition2\] [*For every $\mu$ as in definition \[defquasifree\] the following hold.\ [**(a)**]{} There exists a unique quasifree state $\omega_\mu$ associated with $\mu$ and it is regular.\ [**(b)**]{} The GNS triple $(\gH_{\omega_\mu}, \Pi_{\omega_\mu}, \Omega_{\omega_\mu})$ is determined as follows with respect to $(\sH_\mu,K_\mu)$ as in lemma \[lemma1A\]. (i) $\gH_{\omega_\mu}$ is the symmetric Fock space with one-particle space $\sH_\mu$. (ii) The cyclic vector $\Omega_{\omega_\mu}$ is the vacuum vector of $\gH_\omega$. (iii) $\Pi_{\omega_\mu}$ is determined by $\Pi_{\omega_\mu}(W(\psi)) = e^{i\overline{\sigma(\psi,\Phi_{\omega_\mu})}}$, the bar denoting the closure, where[^4] $$\sigma(\psi,\Phi_{\omega_\mu}) \doteq ia(K_\mu\psi) -ia^\dagger(K_\mu\psi)\:, \quad \mbox{for all $\psi\in \sS$}$$ $a(\phi)$ and $a^\dagger(\phi)$, $\phi\in \sH_\mu$, being the usual annihilation (antilinear in $\phi$) and creation operators defined in the dense linear manifold spanned by the states with finite number of particles.\ [**(c)**]{} A pair $(\sH,K)\neq (\sH_\mu,K_\mu)$ satisfies (i) and (ii) in lemma \[lemma1A\] for $\mu$, thus determining the same quasifree state $\omega_\mu$, if and only if there is a unitary operator $U: \sH_\mu\to \sH$ such that $UK_\mu=K$.\ [**(d)**]{} $\omega_\mu$ is pure, i.e., its GNS representation is irreducible if and only if $\overline{K_\mu(\sS)} = \sH_\mu$. In turn, this is equivalent to $4\mu(\psi',\psi') = \sup_{\psi\in \sS\setminus\{0\}} |\sigma(\psi,\psi')|/\mu(\psi,\psi)$ for every $\psi'\in \sS$.*]{} \[remarkstates\]\ [**(1)**]{} $K_\mu$ is always injective due to (ii) and non degenerateness of $\sigma$.\ [**(2)**]{} Consider the real Hilbert space obtained by taking the completion of $\sS$ with respect to $\mu$. The requirement (\[sm\]) is equivalent to the fact that there is is a bounded operator $S$ everywhere defined over the mentioned Hilbert space, with $S=-S^*$, $||S|| \leq 1$ and such that $\frac{1}{2}\sigma(\psi,\psi') = \mu(\psi, S \psi')$, for all $\psi,\psi' \in \sS$.\ [**(3)**]{} The pair $(\sH_\mu,K_\mu)$ is called the [**one-particle structure**]{} of the quasifree state $\omega_\mu$.\ Let us pass to discuss the KMS condition [@hug; @Haag; @BR2]. KMS state are the algebraic counterpart, for infinitely extended systems, of thermal states of standard statistic mechanics. There are several different equivalent definitions of KMS states, see [@BR2] for a list of various equivalent definitions. While bearing in mind Definition 5.3.1 and Proposition 5.3.7 in [@BR2], we adopt the following one:\ \[KMSdef\] A state $\omega$ on a $C^*$-algebra $\mA$ is said to be a [**KMS state at inverse temperature $\beta\in \bR$**]{} with respect to a one-parameter group of $*$-automorphisms $\{ \alpha_t\}_{t\in \bR}$ which represents, from the algebraic point of view, some notion of time-evolution if, for every pair $A,B \in \mA$, and with respect to the function $\bR \ni t \mapsto \omega\left(A\alpha_t(B)\right)=: F^{(\omega)}_{A,B}(t)$, the following facts hold.\ (a) $F^{(\omega)}_{A,B}$ extends to a continuous complex function $F^{(\omega)}_{A,B} = F^{(\omega)}_{A,B}(z)$ with domain $$\overline{D_\beta} \doteq \{z \in \bC\:|\: 0 \leq Im z \leq \beta\}\quad \mbox{if $\beta\geq 0$, or}\quad \overline{D_\beta} \doteq \{z \in \bC\:|\: \beta \leq Im z \leq 0\} \quad \mbox{if $\beta\leq 0$,}$$ (b) $F^{(\omega)}_{A,B} = F^{(\omega)}_{A,B}(z)$ is analytic in the interior of $\overline{D_\beta}$;\ (c) it holds, and this identity is – a bit improperly – called the [**KMS condition**]{}: $$F^{(\omega)}_{A,B}(t+i\beta) = \omega\left(\alpha_t(B)A \right)\:, \quad \mbox{for all $t \in \bR$.}$$ With the given definition, an $\{ \alpha_t\}_{t\in \bR}$-KMS state $\omega$ turns out to be invariant under $\{ \alpha_t\}_{t\in \bR}$ [@BR2]; the function $\overline{D_\beta} \ni z \mapsto F^{(\omega)}_{A,B}(z)$ is uniquely determined by its restriction to real values of $z$ (by the “edge of the wedge theorem”) and $\sup_{\overline{D_\beta} } |F^{(\omega)}_{A,B}| = \sup_{\partial D_\beta} |F^{(\omega)}_{A,B}|$ (by the “three lines theorem”) [@BR2].\ Equivalent definitions of KMS states are obtained by the following propositions, the second for quasifree states, due to Kay [@Kay12; @KW] and relying upon earlier results by Hugenholtz [@hug]. We sketch the proofs since they are very spread in the literature. \[KMSkay\] [*An algebraic state $\omega$, on the $C^*$-algebra $\mA$, which is invariant under the one-parameter group of $*$-automorphisms $\{ \alpha_t\}_{t\in \bR}$ is a KMS state at the inverse temperature $\beta\in \bR$ if and only if its GNS triple $(\cH_\omega, \Pi_\omega, \Omega_\omega)$ satisfies the following three requirements.\ (1) The unique unitary group $\bR \ni t \mapsto U_t$ which leaves $\Omega_\omega$ invariant and implements $\{ \alpha_t\}_{t\in \bR}$ – i.e. $\Pi_\omega\left(\alpha_{t}(A)\right) = U_t \Pi_\omega(A) U^*_t\quad \mbox{for all $A\in \mA$ and $t\in \bR$}$ – is strongly continuous, so that $U_t = e^{itH}$ for some self-adjoint operator $H$ on $\cH_\omega$.\ (2) $\Pi_\omega\left(\mA\right) \Omega_\omega\subset Dom\left(e^{-\beta H/2}\right)$.\ (3) There exists an antilinear operator $J: \cH_\omega \to \cH_\omega$ with $JJ=I$ such that: $$Je^{-it H} = e^{-it H}J \quad \mbox{for all $t\in \bR$, and}\quad e^{-\beta H/2} \Pi_\omega(A)\Omega_\omega = J \Pi_\omega(A^*)\Omega_\omega \quad \mbox{for all $A\in \mA$.}$$*]{} [*Proof*]{}. A $\{ \alpha_t\}_{t\in \bR}$-KMS state with inverse temperature $\beta$ is $\{ \alpha_t\}_{t\in \bR}$-invariant and fulfils the conditions (1), (2) and (3) due to Theorem 6.1 in [@hug]. Conversely, consider an $\{ \alpha_t\}_{t\in \bR}$-invariant state $\omega$ on $\mA$ which fulfils the conditions (1), (2) and (3). When $A$ and $B$ are entire analytic elements of $\mA$ (see [@BR2]), $\bR \ni t \mapsto F^{(\omega)}_{A,B}(t)$ uniquely extends to an analytic function on the whole $\bC$ and thus (a) and (b) in def. \[KMSdef\] are true. (1), (2), (3) and $e^{z H} \Omega_\omega = \Omega_\omega$, for all $z\in \overline{D_\beta}$ (following from (2) and (3)) also entail (c): $$\omega(\alpha_t(B) A) = \langle\Omega_\omega,\: U_t \Pi_\omega(B) U^*_t \Pi_\omega(A) \Omega_\omega \rangle = \langle \Pi_\omega(B^*)\Omega_\omega,\: U_t^* \Pi_\omega(A) \Omega_\omega\rangle = \langle J U_t^* \Pi_\omega(A)\Omega_\omega ,\: J\Pi_\omega(B^*) \Omega_\omega\rangle$$ $$= \langle U_t^* e^{-\beta H/2}\Pi_\omega(A^*) \Omega_\omega ,\: e^{-\beta H/2} \Pi_\omega(B) \Omega_\omega\rangle = \langle\Omega_\omega,\: \Pi_\omega(A) e^{i(t+i\beta)H} \Pi_\omega(B)e^{-i(t+i\beta)H} \Omega_\omega\rangle = F_{A,B}^{(\omega)}(t+i\beta)\:.$$ The validity of conditions (a), (b) and (c) for entire analytic elements $A,B \in \mA$ implies the validity for all $A,B\in \mA$, as established in [@BR2] (compare Definition 5.3.1 and Proposition 5.3.7 therein). $\Box$ \[kkms\] *Consider a quasifree algebraic state $\omega_\mu$ on the Weyl-algebra $\cW(\sS)$, with one-particle structure $(\sH_\mu,\sK_\mu)$. Assume that $\omega_\mu$ is invariant under the one-parameter group of $*$-automorphisms $\{ \alpha_t\}_{t\in \bR}$, which is implemented by the strongly continuous unitary one-parameter group $\bR \ni t \mapsto V_t \doteq e^{i \tau h}$ in the one particle space $\sH_\mu$ so that $U_t$ in proposition \[KMSkay\] is the tensorialization of $V_{t}$. The following facts are equivalent.* [**(a)**]{} $\omega_\mu$ is a KMS at the inverse temperature $\beta \in \bR$ with respect to $\{ \alpha_t\}_{t\in \bR}$. [**(b)**]{} There is an anti-unitary operator $j: \sH_\mu \to \sH_\mu$ with $jj = I$ and the following facts hold:\ (i) $\sK_\mu (\sS) \subset Dom\left( e^{-\frac{1}{2}\beta h}\right)$, (ii) $[j, V_t] =0$ for all $t\in \bR$, (iii) $e^{-\frac{1}{2}\beta h} \sK_\mu \psi = -j \sK_\mu \psi$ for all $\psi \in \sS(\cH_\mu)$. [**(c)**]{} $\sK_\mu (\sS) \subset Dom\left( e^{-\frac{1}{2}\beta h}\right)$ and $\langle e^{-it h} x, y \rangle = \langle e^{-\beta h/2} y , e^{-it h} e^{-\beta h/2} x\rangle$ if $x,y \in \sK_\mu(\sS)$ and $t\in \bR$. \ [*Proof*]{}. (a) is equivalent to (b) as proved on pages 80-81 in [@KW]. (b) entails (c) straightforwardly. If one assumes (c) and exploits (i) of lemma \[lemma1A\], $j:\sH_\mu\to \sH_\mu$, which fulfils (b), is completely individuated by continuity and anti-linearity under the request that $j \sK_\mu \psi = -e^{-\frac{1}{2}\beta h}\sK_\mu\psi$ when $\psi \in \sS$. $\Box$ Proofs of some propositions. {#Appendixproofs} ============================ [**Proof of Lemma \[lemma1\]**]{}. As in Appendix \[geometry\], let us consider the conformal extension $(\widetilde{\mM},\widetilde{g})$ of the spacetime $(\mM,g)$ determined in [@SW83] where $\widetilde{g} = g/r^2$ in $\mM$ (see figure 2). In view of the previously illustrated properties of $E_{P_g}$, if $\varphi \in \sS(\mM)$, there is a smooth function $f_\varphi$ with support contained in $\mM$ and such that $\varphi = E_{P_g}f_\varphi$, and $supp \varphi \in J^+(supp f_\varphi ; \mM) \cup J^-(supp f_\varphi ; \mM)$. Since $J^\pm(supp f_\varphi ; \mM) \subset J^\pm(supp f_\varphi ; \widetilde{\mM})$, the very structure of $\widetilde{\mM}$ (see figure 2) guarantees that, if the smooth extension $\widetilde{\varphi}$ of $r\varphi$ in a neighbourhood of $\Im^\pm\subset \widetilde{\mM}$ exists, it must have support bounded by constants $v^{(\vphi)}, u^{(\vphi)} \in (-\infty,\infty)$. Here we adopt the relevant null coordinates in the considered neighbourhood: $(\Omega, u, \theta, \phi)$ or $(\Omega, v, \theta, \phi)$ respectively, where $\Omega = 1/r$ in $\mW$. Furthermore, in view of the shape of $J^\pm(supp f_\varphi ; \widetilde{\mM})$, the analogous property holds true for the support of $\varphi$ in $\mW$. The existence of $\widetilde{\varphi}$ can be established examining the various possible cases. To start with, let us assume that $supp f_\vphi \subset \mW$. Let $p\in \mW$ be in the chronological past of $supp f_{\varphi}$ sufficiently close to $i^-$. Afterwards, let us consider a second point $q$ beyond $\Im^+$, though sufficiently close to $\Im^+$ so that the closure of $\mN_{p,q} \doteq I^+(p;\widetilde{\mM}) \cap I^-(q; \widetilde{\mM})$ does not meet the timelike singularity in the conformal extension of $\mM$ on the right of $\Im^+$. Let us consider $\mN_{p,q} $ as a spacetime equipped with the metric $\tilde{g}$. It is globally hyperbolic since, per direct inspection, one verifies that the diamonds $J^+(r; \mN_{p, q} ) \cap J^-(s; \mN_{p,q} )$ are empty or compact for $r,s \in \mN_{p, q}$ while the spacetime itself is causal. Hence $E_{P_{\widetilde{g}}}$ is well defined and individuates a solution $\widetilde{\varphi} \doteq E_{P_{\widetilde{g}}} f_\varphi$ of the Klein-Gordon equation associated with $P_{\widetilde{g}}$, as in (\[tildeKG\]), with $\widetilde{g} = g/r^2$. Thanks to the properties of the Klein-Gordon equation under conformal rescaling [@Wald], one has $\widetilde{\varphi} = r \varphi$ in $\mM$ because $\widetilde{g} = g/r^2$ therein. If we keep $p$ fixed while moving $q$ in a parallel way to $\Im^+$ towards $i^+$, one obtains an increasing class of larger globally-hyperbolic spacetimes $\mN_{p, q}$ and, correspondingly, a class of analogous extensions $\widetilde{\varphi}$ on corresponding $\mN_{p, q}$. Furthermore, if one considers two of these extensions, they coincide in the intersections of their domains (see figure 3). (150,170)(0,0) (0,0)[{height="6cm"}]{} (102,22)[$i^-$]{} (98,175)[$q$]{} (83,123)[$supp(f^\varphi)$]{} (142,18)[$p$]{} (165,130)[$\mM$]{} (200,110)[$\scrim$]{} (10,110)[$\cH$]{} In order to draw these conclusions, we exploited of the uniqueness of the solution of a Cauchy problem as well as the property according to which any compact portion of a spacelike Cauchy surface of a globally hyperbolic spacetime can be extended to a smooth spacelike Cauchy surface of any larger globally hyperbolic spacetime [@BS06]. Hence the initial data can be read on the larger spacelike Cauchy surface, also thanks to its acausal structure (Lemma 42 from Chap. 14 in [@ON]). Accordingly, a smooth extension of $r\varphi$ turns out to be defined in a neighbourhood of $\Im^+$ and an almost slavish procedure yields the analogous extension on $\Im^-$. Let us now suppose that $supp f_{\varphi} \subset \mB$. In such case $\varphi$ cannot reach $\Im^+$ and, thus, the only extension of $r\varphi$ concerns $\Im^-$. The employed procedure is similar to the one above, though the class of globally hyperbolic spacetime is constructed as follows. Let us take a point $p$ beyond $\Im^-$ sufficiently close to $i^-$ and let us consider the intersection $\mN_p \doteq I^+(p;\widetilde{\mM})\cap I^-(\mM;\widetilde{\mM})$. If one moves $p$ parallelly to $\Im^-$ drawing closer to $i^-$, one obtains an increasing class of globally hyperbolic spacetimes. equipped with the metric $\widetilde{g} = g/r^2$, and, correspondingly, a class of solutions of the rescaled Klein-Gordon equation. These define the smooth extension $\widetilde{\varphi}$ of $r\varphi$ in an open neighbourhood of $\Im^-$. Let us now consider the case where $supp f_{\varphi}$ is concentrated in an arbitrarily shrunk open neighbourhood of $\cH_{ev}$. While the behaviour of $\varphi$ in a neighbourhood of $\Im^-$, mimics the previously examined one, that around $\Im^+$ deserves a closer look mostly with reference to the construction of the relevant globally hyperbolic spacetimes. To this end, let us fix a point $p\in \mW$ in the chronological past of $supp f_{\varphi}$ sufficiently close to $\cH$. Afterwards, let us consider a smooth spacelike surface $\Sigma$ in the chronological future of $supp f_{\varphi}$, which lies in the past of $i^+$ in $\widetilde{\mM}$ and it intersects $\Im^+$ for some $u=u_\Sigma$. The relevant class of globally hyperbolic spacetimes is now made of the sets $\mN_{p, \Sigma} \doteq J^-(\Sigma; \widetilde{\mM}) \cap J^+(p; \widetilde{\mM})$ when $\Sigma$ moves towards $i^+$. It remains to consider the case where $supp f_{\varphi}$ intersect $\cH_{ev}$, but it is not confined in a small neighbourhood of $\cH_{ev}$. In this case, if one takes into account the linearity of both the causal propagator and $P_{\widetilde{g}}$, we can reduce ourselves to a combination of the three above considered cases. If one decomposes the constant function $1$ in $\mW$ as the sum of three non-negative smooth functions $1= f_1+f_2+f_3$, with $f_1$ supported in $\mB$, $f_2$ supported in $\mW$ and $f_3$ supported in an arbitrarily shrunk open neighbourhood of $\cH_{ev}$, we have $f_\varphi = f_\varphi\cdot f_1+f_\varphi\cdot f_2+f_\varphi\cdot f_3$. If we fix $r\varphi_i\doteq r E_{P_g} (f_\varphi \cdot f_i)$, $i=1,2,3$, each wavefunction can be treated separately as discussed above, hence yielding corresponding extensions $\widetilde{\varphi}_i$ to $\Im^+$. The sum of these extensions is, per construction, the wanted one $\widetilde{\varphi}$ of $r\varphi$. The same procedure applies to the case of $\Im^-$. $\Box$\ [**Proof of Proposition \[PropDR\].**]{} (a) We consider the proof for the case of $t>0$, [*i.e.*]{}, the behaviour of the wavefunctions about $\cH_{ev}$ and $\Im^+$ only), the remaining case being then an immediate consequence of the symmetry $X \to -X$ of the Kruskal geometry.\ To start with, it is worth noticing that each of our coordinates $u,v$ amounts to twice the corresponding one defined in [@DR05] and the difference of our $r^*$ and that defined in [@DR05] is $3m +2m\ln m$.\ The bounds concerning the constants $C_1$ and $C_3$ are proved in Theorem 1.1 of [@DR05]. Here, sufficiently regular solutions of the massless Klein-Gordon equation are considered and initial data are assigned on a smooth complete spacelike Cauchy surfaces of the full Kruskal extension of $\mM$ which is asymptotically flat at spatial infinity. Furthermore it is imperative that the said data vanish fast enough at space infinity. In our case these requirements are fulfilled because the elements of $\sS(\mM)$ are smooth and have compact support on every smooth spacelike, hence acausal, Cauchy surface of $\mM$; therefore we can employ the results in [@BS06] to view these as Cauchy data on a smooth spacelike Cauchy surface of the full Kruskal extension. The bound which concerns $C_2$ has the same proof as that for $C_1$ because, when $\vphi\in\sS(\mM)$, $X(\vphi)\in\sS(\mM)$, $X$ itself being a smooth Killing vector field. To conclude the proof, it is enough to show the last bound, related with the constant $C_4$. To this end, let us fix $\vphi \in \sS(\mM)$ and re-define, if necessary, the origin of the killing time $t$ in $\mW$ in order that $u^{(\vphi)} \geq 2$, where $u^{(\vphi)}$ is the constant defined in Lemma \[lemma1\]. Now we focus on the proof contained in sec. 13.2 of [@DR05] and particularly on the part called “[ *decay in $r\geq \hat{R}$”*]{} which concerns the bound associated with $C_3$. We want to adapt such proof to our case, replacing the solution $\phi$ there considered with our $X(\vphi)$, so that also $r\phi$ is replaced by $rX(\vphi) = \widetilde{X}(r\vphi)$ in $\mW$. Furthermore it smoothly extends to $X(\widetilde{\vphi})$ on $\mW \cup \Im^+$. It is remarkable that it suffices to prove the bound in the region $\{r>\hat R\}\cap \{t>0\}$ in $\mW$, since it would then hold on $\Im^+$ per continuity.\ One should notice that only the region $\{r \geq \hat R\} \cap \{t>0\} \cap \{ u \geq 2\}$ has to be considered. Indeed, in the set $\{ u < 2\} \cap \{v > v^{(\vphi)}_0\}$ for some $v^{(\vphi)}_0 \in \bR$, $X( \widetilde{\vphi})$ vanishes due to Lemma \[lemma1\]. Hence $X(\widetilde{\vphi})$ vanishes in $\{r\geq\hat R\} \cap \{t>0\}\cap \{ u < 2\} \cap \{v > v^{(\vphi)}_0\}$, trivially satisfying the wanted bound. The region individuated by $\{r \geq \hat R\} \cap \{t\geq 0\} \cap \{ u \leq 2\} \cap \{v\leq v^{(\vphi)}_0\}$ is, moreover, compact thus $X(\widetilde{\vphi})$ is bounded therein and it also satisfies the looked-for bound.\ In the region $\{r \geq \hat R\} \cap \{t>0\} \cap \{ u \geq 2\}$, along the lines of p.916-917 in [@DR05], though with $\phi$ replaced by $\varphi' \doteq {\widetilde X}(\vphi)$, we achieve, out of a Sobolev inequality on the sphere $$r^2 \left|\varphi'(u,v,\theta,\phi)\right|^2 \leq C \int_{\bS^2} r^2 \left|\varphi'\right|^2 d\bS^2 + C \int_{\bS^2} \left|r \spa \not{\nabla} \varphi'\right|^2 r^2 d\bS^2 + C \int_{\bS^2}\left|r^2\not{\nabla} \spa \not{\nabla} \varphi'|^2\right) r^2 d\bS^2\:,$$ where $\not{\nabla}$ denotes the covariant derivative with respect to metric induced on the sphere of radius $r$, while $d\bS^2$ is the volume form on the unit sphere. If the squared angular momentum operator is denoted as $\Omega^2\doteq r^2\not{\nabla}\spa\not{\nabla}$ the above inequality can be re-written as: \[inter\] r\^2 |’(u,v,,)|\^2 C \_[\^2]{} |\^0 ’|\^2 r\^2 d\^2+ C \_[\^2]{} |\^0 ’| | \^1 ’| r\^2 d\^2 + C \_[\^2]{}| \^2 ’|\^2 r\^2 d\^2.To conclude it is sufficient to prove that, for $k=0,1,2$ and if $r\geq \hat R$, $u\geq 2$, $t>0$: \[middle\] \_[\^2]{} |\^k ’|\^2 r\^2 d\^2 B\_k/u\^2, for some constants $B_k \geq 0$. Let us notice that, in view of Cauchy-Schwartz inequality, the second integral in the right hand side of (\[inter\]) is bounded by the product of the square root of the integrals with $k=0$ and $k=1$ in the left-hand side of (\[middle\]). If we follow [@DR05] and if we pass to the coordinates $(t,r^*, \theta,\phi)$ (see Section \[geometry\]), and for some constant $D\geq 0$: \[inter2\] \_[\^2]{} |\^k ’|\^2 r\^2(t,r\^\*,,) d\^2\_[\^2]{} |\^k ’|\^2 r\^2(t,\^\*,,) d\^2\ + D \_[\^\*]{}\^[r\^\*]{}\_[\^2]{} |\_\^k ’||\^k’| r\^2(t,,,) dd\^2 + D \_[\^\*]{}\^[r\^\*]{}\_[\^2]{} |\^k ’|\^2 r(t,,,) dd\^2. If we stick to [@DR05], the parameter $\tilde{r}^*\geq \hat{R}^*$ can be fixed so that the first integral in the right-hand side satisfies \[bastarda0\] \_[\^2]{} |\^k ’|\^2 r\^2(t,\^\*,,) d\^2 |[E]{}\_2/t\^[2]{} |[E]{}\_2/ u\^2, where the constant $\bar{E}_2$ was defined in [@DR05] and it depends on $\Omega^k \varphi'\in \sS(\mM)$. Here we have also used the fact that $u = t - r^* \geq 2$ with $r^*>0$ and $t>0$; hence that $u \leq t$. From now on, our procedure departs form that followed in [@DR05]. With respect to the third integral in the right-hand side of (\[inter2\]), it can be re-written $$\int_{\tilde{r}^*}^{r^*}\int_{\bS^2} (\partial_t\Omega^k \vphi)^2 r(t,\rho,\theta,\phi) d\rho d\bS^2 \leq const. \int_{\tilde{r}^*}^{r^*}\int_{\bS^2} (\partial_t\Omega^k \vphi)^2 r^2(t,\rho,\theta,\phi) d\rho d\bS^2\leq F(\cS),$$ where $\cS$ is the achronal hypersurface individuated by the fixed time $t$, the interval $[\tilde{r}^*, r^*]$ and the coordinates $(\theta,\phi)$ which vary over $\bS^2$. $F(\cS)$ is then the flux of energy through $\cS$ associated with the Klein-Gordon field $\Omega^k \vphi$. Theorem 1.1 in [@DR05] assures now that, for some constant $C'$, which depends on $\vphi$, $$F(\cS) \leq C'/v_+(\cS)^2 + C'/u_+(\cS)^2\:,$$ where $v_+(\cS) = \max\{\inf_\cS v, 2\}$ and $u_+(\cS) = \max\{\inf_\cS u, 2\}$. In our case, per construction, we have $\max\{\inf_\cS v, 2\}\geq t+\hat{R}^*$ and $\max\{\inf_\cS u, 2\} = u(t,r^*,\phi,\theta)$. For $t>0$, $r\geq \hat R$, $u>2$, one can conclude: $$\int_{\tilde{r}^*}^{r^*}\int_{\bS^2} |\Omega^k \varphi'|^2 r(t,\rho,\theta,\phi) d\rho d\bS^2 \leq \frac{C'}{(t + \hat{R}^*)^2} + \frac{C'}{u^2} = \frac{C'}{(u + r^*+ \hat{R}^*)^2} + \frac{C'}{u^2} \leq \frac{2C'}{u^2}\:,$$ and thus \[bastarda1\] \_[\^\*]{}\^[r\^\*]{}\_[\^2]{} |\^k ’|\^2 r(t,,,) dd\^2 const. \_[\^\*]{}\^[r\^\*]{}\_[\^2]{} |\^k ’|\^2 r(t,,,)\^2 dd\^2 . Let us finally consider the second integral in the right-hand side of (\[inter2\]). We notice that $$\int_{\tilde{r}^*}^{r^*}\int_{\bS^2} |\partial_\rho \Omega^k \varphi'|^2 r(t,\rho,\theta,\phi)^2 d\rho d\bS^2 \leq F'(\cS),$$ where $F'(\cS)$ is the flux of energy through $\cS$ associated with the Klein-Gordon field $\Omega^k \varphi'$. If we deal with it as before, we obtain the bound \[bastarda2\] \_[\^\*]{}\^[r\^\*]{}\_[\^2]{} |\_\^k ’|\^2 r(t,,,)\^2 d d\^2 . The Cauchy-Schwartz inequality, together with (\[bastarda1\]) and (\[bastarda2\]), leads to \[bastarda3\] \_[\^\*]{}\^[r\^\*]{}\_[\^2]{} |\_\^k ’||\^k’| r(t,,,)\^2 dd\^2 . If one puts all together in the right-hand side of (\[inter2\]), the bounds (\[bastarda0\]), (\[bastarda1\]) and (\[bastarda3\]) yield (\[middle\]).\ (b) Let us fix $\Sigma$ as any smooth spacelike Cauchy surface of $\mM$. Notice that if the sequence of initial data converge to zero in the test function topology on $\Sigma$, there is a compact set $C\subset\Sigma$ which, per definition, contains all the supports of the initial data of the sequence. In view of [@BS06], we can construct a smooth spacelike Cauchy surface $\Sigma'$ of the complete Kruskal manifold $\mK$, which includes that compact. Thus, the sequence of initial data tends to $0$ in the test function topology of $\Sigma'$ as well. Such data on $\Sigma$ can be read on $\Sigma'$ since the supports of the solutions cannot further intersect $\Sigma'$ as it is acausal. From standard results of continuous dependence from compactly-supported initial data of the smooth solutions of hyperbolic equations in globally hyperbolic spacetimes (see Theorem 3.2.12 in [@BGP]), if the initial data on a fixed spacelike Cauchy surface $\Sigma'$ tend to $0$ in the test function topology, then also the solution tends to $0$ in the topology of $C^\infty(\mK; \bR)$. At the same time, as one can prove out of standard results on the topology of causal sets ([*e.g.*]{}, see [@Wald] and particularly theorems 8.3.11 and 8.3.12 in combination with the fact that the open double cones form a base of the topology) $J^+(C; \mK) \cup J^-(C; \mK)$ has compact intersection with every spacelike Cauchy surface of $\mK$, since $C$ is compact in $\Sigma'$. So all initial data on $\Sigma''$ of the considered sequence of solutions are contained in a compact, too. From these results we conclude that, if the initial data tend to $0$ in the test function topology on $\Sigma'$, the associated solution, whenever restricted on any other Cauchy surface $\Sigma''\subset\mK$ yields, per restriction, new initial data, which also tend to $0$. For convenience, we fix $\Sigma''$ as an extension of the spacelike Cauchy surface of $\mW$ (whose closure intersects $\cB$) individuated in $\mW$ as the locus $t=1$. If we refer to (a), one sees that the coefficients $C_i$ are obtained as the product of universal constants and integrals of derivatives of the compactly supported Cauchy data of both $\vphi$, and, where appropriate, $X(\vphi)$ over $\Sigma'' \cap \overline{\mW}$. This is explained in Theorem 1.1, Theorem 7.1 as well as in the formulae appearing in sec. 4 of [@DR05], though one should reformulate them with respect both to $r^*$ and to the global coordinates $U$ and $V$ instead of $u$, $v$ and $r$). From these formulas it follows immediately that the constants $C_i$ vanish provided that the Cauchy data tend to $0$ in the test function topology on $\Sigma''$, and this requirement is valid in our hypotheses. $\Box$\ [**Proof of Proposition \[PropMain4\]**]{}. (a) To start with, we notice that, by direct inspection, as shown in [@KW] and [@Moretti08], though the angular coordinates $(\theta,\phi)$ substituted by the complex ones $(z,\bar{z})$ obtained out of stereographic projection, it turns out that: $$\begin{gathered} \langle \widehat{\psi}_+, \widehat{\psi'}_+ \rangle_{\sH_{\cH}} \doteq \int_{\bR_+\times \bS^2} \overline{\widehat{\psi}_+}(K, \theta,\phi) \widehat{\psi'}_+(K,\theta,\phi)\: 2K dK \wedge r^2_S d\bS^2\nonumber \\ = \lim_{\epsilon \to 0^+} -\frac{r_S^2}{\pi} \int_{\bR \times \bR \times \bS^2}\spa \frac{\overline{\psi(U_1,\theta,\phi)} \psi'(U_2,\omega)} {(U_1-U_2 -i\epsilon)^2} dU_1 \wedge dU_2 \wedge d\bS^2,\end{gathered}$$ for $\psi,\psi' \in C_0^\infty(\cH; \bC)$.\ As a consequence we have obtained that the map $M: C_0^\infty(\cH; \bC) \ni \psi \mapsto \widehat{\psi}_+(K,\omega) \in \sH_{\cH}$ is isometric and thus, per continuity, it uniquely extends to a Hilbert space isomorphism $F_{(U)}$ of $\overline{\left(C_0^\infty(\cH;\bC), \lambda_{KW} \right)}$ onto the closed Hilbert space $\overline{M(C_0^\infty(\cH; \bC))} \subset \sH_{\cH}$. To conclude the proof of the first statement in (a), it is enough to establish that $\overline{M(C_0^\infty(\cH; \bC))} = \sH_{\cH}$. This immediately follows from the two lemma proved below. \[Lemma1TeoMain4\] [*$\overline{M(C_0^\infty(\cH; \bC))}$ includes the space $\mS_0$ whose elements $f = f(K,\omega)$ are the restrictions to $\bR_+\times \bS^2$ of the functions in $\mS(\bR\times \bS^2)$ and they vanish in a neighbourhood of $K=0$ depending on $f$.*]{}\ Above and henceforth $\mS(\bR\times \bS^2)$ denotes the complex [**Schwartz space**]{} on $\bR \times \bS^2$, [*i.e.*]{} the space of complex-valued smooth functions on $\bR\times \bS^2$ which vanish, with all their $K$-derivatives of every order, as $|K|\to +\infty$ uniformly in the angles and faster than every inverse power of $|K|$. This space can be equipped with the usual topology induced by seminorms (see Appendix C of [@Moretti08]). \[Lemma2TeoMain4\] [*$\mS_0$ is dense in $\sH_{\cH}$.*]{}\ Concerning (b), we notice that, if $f\in \mS_0$, $if \in \mS_0$ and that both vanish in a neighbourhood of $K=0$. Therefore, it is possible to arrange two [*real*]{} functions in $\mS(\cH)$, $g_1$ and $g_2$ such that $\widehat{g_1}_+ = f$ and $\widehat{g_2}_+ = if$. With the same proof of Lemma \[Lemma1TeoMain4\] one can establish that $g_i$ are the the limits, in the topology of $\lambda_{KW}$, of sequences $\{f_{(i) n}\} \subset C_0^\infty(\cH; \bR)$. We have obtained that every complex element of the dense subspace $\mS_0 \subset \sH_{\cH}$ is the limit of elements of $F_{(U)}\left(C_0^\infty(\cH; \bR)\right)$. $\Box$\ [**Proof of Lemma \[Lemma1TeoMain4\]**]{}. Let us take $f \in \mS_0$. As a consequence, it can be written as the restriction to $\bR_+\times \bS^2$ of $F \in \mS(\bR\times \bS^2)$. In turn, $F= \mF_+(g)$ for some $g \in \mS(\bR\times \bS^2)$, since the Fourier transform is bijective from $\mS(\bR\times \bS^2)$ onto $\mS(\bR\times \bS^2)$ (see Appendix C of [@Moretti08]). Since $C_0^\infty(\bR \times \bS^2; \bC)$ is dense in $\mS(\bR\times \bS^2)$ in the topology of the latter, there is a sequence $\{g_n\} \subset C_0^\infty(\bR \times \bS^2;\bC)$ with $g_n \to g$ in the sense of ${\mS(\bR\times \bS^2)}$. Since the Fourier transform is continuous with respect to that topology, we conclude that $\mF_+(g_n) \to F$ in the sense of ${\mS(\bR\times \bS^2)}$. By direct inspection one finds that the achieved result implies that $\mF_+(g_n)\rest_{\bR_+\times \bS^2} \to F\rest_{\bR_+\times \bS^2}$ in the topology of every $L^2(\bR_+\times \bS^2, c K^n dK \wedge r_S^2 d\bS^2)$ for every power $n=0,1,2,\ldots$ and $c>0$. Particularly it happens for $n=c =2$. We have found that, for every $f \in \mS_0$, there is a sequence in $M(C_0^\infty(\bR \times \bS^2;\bC))$ which tends to $f$ in the topology of $\sH_{\cH_R}$ and thus $\mS_0 \subset \overline{M(C_0^\infty(\bR \times \bS^2;\bC))}$. $\Box$\ [**Proof of Lemma \[Lemma2TeoMain4\].**]{} In this proof $\bR^*_+ \doteq (0,+\infty)$ and $\bN^* = \{1,2,\ldots\}$. A well-known result is that $C_0^\infty((a,b); \bC)$ is dense in $L^2((a,b), dx)$ so that, particularly, $C_0^\infty((1/n,n); \bC)$ is dense in $L^2((1/n,n),dx)$, and, thus, if we introduce the new variable $K = \sqrt{x}$, it turns out that the space $C^\infty_0((1/\sqrt{n},\sqrt{n}); \bC)$ is dense in $L^2((1/\sqrt{n},\sqrt{n}),2KdK)$.\ Since, in the sense of the Hilbertian direct sum, $\oplus_{n\in \bN} L^2((1/\sqrt{n},\sqrt{n}),2KdK) = L^2(\bR^*_+, 2KdK)$ (for instance making use of Lebesgue’s dominated convergence theorem), we conclude that $C_0^\infty(\bR_+^*; \bC) = \cup_{n\in \bN^*}C_0^\infty((1/n,n);\bC)$ is dense in $L^2(\bR^*_+, 2KdK)= L^2(\bR_+, 2KdK)$ and, thus, there must exist a Hilbert base $\{f_n\}_{n\in \bN} \subset C_0^\infty(\bR_+^*; \bC)$.\ By standard theorems on Hilbert spaces with product measure, we know that a Hilbert base of the space $L^2(\bR_+ \times \bS^2, 2KdK\wedge r^2_Sd\bS^2)$ is $\{f_{n} Y_{m}\}_{n,m \in \bN}$, provided that $\{Y_m\}_{m\in \bN}$ and $\{f_n\}_{n\in\bN}$ are respectively one for $L^2(\bS^2,r^2_Sd\bS^2)$ and for $L^2(\bR_+, 2KdK)$. The elements $Y_m$ can be chosen as harmonic functions so that they are smooth and compactly supported. Therefore, if $\{f_n\}_{n\in \bN}\subset C_0^ \infty((0,+\infty); \bC)$, it holds that $\{f_{n} Y_{m}\}_{n,m \in \bN} \subset C_0^\infty(\bR^*_+\times \bS^2; \bC)$ and, thus, trivially, the space $C_0^\infty(\bR^*_+\times \bS^2;\bC)$ is dense in $L^2(\bR_+ \times \bS^2, 2KdK\wedge r^2_Sd\bS^2)$. Since it holds $C_0^\infty(\bR^*_+\times\bS^2; \bC)\subset\mS_0$, the achieved result proves the thesis. $\Box$\ [**Proof of Proposition \[propbastarda\].**]{} We only consider the case of $\cH^+$, the proof for $\cH^-$ being identical.\ (a) If $\psi_1, \psi_2 \in C_0^\infty(\cH^+; \bC)$, then: \[lKWs\] \_[KW]{}(\_1,\_2) = \_[0\^+]{} - \_[\^2]{} du\_1 du\_2 d\^2. It follows from $\lambda_{KW}$ (\[lKW\]), though working with the coordinates $u_1,u_2$ and making an appropriate use of Sokhotsky’s formula $1/(x-i0^+)^2 = 1/x^2 - i \delta'(x)$ (where $1/x^2$ is the derivative of the distribution $-1/x$ interpreted in the sense of the principal value) that a bounded strictly positive factor, which appears in front of $\epsilon$, cancels out. Let us notice that the required boundedness arises form the fact that the used test functions are supported in $\cH^+$, so that they have compact support in the variables $(u,\theta,\phi)\in \bR \times \bS^2$. In spite of the different relation between the coordinate $U$ and $u$, the same result arises referring to $\cH^-$ instead of $\cH^+$. We notice that the $u$-Fourier transform of the distribution $-\frac{1}{4\pi}\frac{1}{\left[\sinh\left(\frac{u}{4r_S}\right)-i0^+\right]^2}$ turns out to be just $\frac{1}{\sqrt{2\pi}}\frac{d\mu(k)}{dk}$. Hence, the limit as $\epsilon \to 0^+$ of the integral in the right-hand side of (\[lKWs\]) can be interpreted as the $L^2(\bR\times \bS^2, dv\wedge d\bS^2)$ scalar product of $\psi_1$ and the the $L^2(\bR\times \bS^2, dk\wedge d\bS^2)$ function obtained by the $u$-convolution of the Schwartz distribution $const. /\left[\sinh\left(\frac{u}{4r_S}\right) -i0^+\right]$ with the compactly-supported function $\psi_2$. The convolution makes sense if one interprets $\psi_2$ as a distribution with compact support; it produces a distribution which is the antitransform of $\widetilde{\psi_2} d\mu/dk$ which, in turn, belongs to the Schwartz space by construction. Hence, up to an antitransformation, the said convolution has to be an element of $L^2(\bR\times \bS^2, du\wedge d\bS^2)$ as previously stated. In this sense we can apply first the convolution theorem for Fourier transforms and, afterwards, the fact that the Fourier transform is an isometry, achieving: $$\lim_{\epsilon \to 0^+} -\frac{1}{4 \pi} \int_{\bR \times \bR \times \bS^2}\spa \frac{\overline{\psi_1(u_1,\theta,\phi)} \psi_2(u_2,\theta,\phi)}{ \left[\sinh\left(\frac{u_1-u_2}{4r_S}\right) -i\epsilon\right]^2} du_1 \wedge du_2\wedge d\bS^2 = \int_{\bR\times \bS^2} \overline{\widetilde{\psi}(k,\theta,\phi)} \widetilde{\psi}(k,\theta,\phi) \frac{d\mu}{dk} dk\wedge d\bS^2 \:,$$ which implies that the map $C_0^\infty(\cH^+;\bC) \ni \psi \mapsto \widetilde{\psi} \in L^2(\bR\times \bS^2, d\mu(k)\wedge d\bS^2)$ is isometric, when the domain is equipped with the scalar product $\lambda_{KW}$. The fact that this map extends to a Hilbert space isomorphism $F^{(+)}_{(u)}:\overline{C_0^\infty(\cH^+;\bC)} \to L^2(\bR\times \bS^2, d\mu(k)\wedge d\bS^2)$ is very similar to the proof of the analogue for $F_{(U)}$ and the details are left to the reader.\ (b) Let us indicate by $\widetilde{\psi}\equiv\mF(\psi)$ the Fourier-Plancherel transform of $\psi$, computed with respect to the coordinate $u$. Per definition, if $\psi \in \sS(\cH^+)$, one has $\psi, \partial_u\psi \in L^2(\bR\times \bS^2, du\wedge d\bS^2)$, so that $\psi$ belongs to the Sobolev space $H^1(\cH^+)_u$ and, equivalently, $\widetilde{\psi} \in L^2(\bR\times \bS^2, dk\wedge d\bS^2) \cap L^2(\bR\times \bS^2, k^2dk\wedge d\bS^2)$. The last inclusions also implies that $\widetilde{\psi}$ belongs to $L^2(\bR\times \bS^1,|k|dk\wedge d\bS^2)$ and $L^2(\bR\times \bS^1, d\mu\wedge d\bS^2)$. Since $C_0^\infty(\cH^+; \bC)$ is dense in $H^1(\cH^+)_u$, if $\psi \in \sS(\cH^+)$, there is a sequence of functions $\psi_n \in C_0^\infty(\cH^+; \bR)$ with $F^{(+)}_{(u)}(\psi_n) = \mF(\psi_n) \to \widetilde{\psi}$, in the topology of both $L^2(\bR\times \bS^2, dk\wedge d\bS^2)$ and $L^2(\bR\times \bS^2, k^2dk\wedge d\bS^2)$. In turn this implies the convergence in the topology of $L^2(\bR\times \bS^2, d\mu\wedge d\bS^2)$. Since $L^2(\bR\times \bS^1, d\mu\wedge d\bS^2)$ is isometric to $\overline{C_0^\infty(\cH^+; \bC)}$, the sequence $\{\psi_n\}$ is of Cauchy type in $\overline{(C_0^\infty(\cH; \bC),\lambda_{KW})}$. For the same reason, any other $\{\psi'_n\} \in C_0^\infty(\cH^+; \bR)$ which converges to the same $\psi$, is such that $\psi_n -\psi'_n \to 0$ in $\overline{(C_0^\infty(\cH; \bC),\lambda_{KW})}$. Therefore $\psi$ is naturally identified with an element of $\overline{C_0^\infty(\cH^+; \bC)}$, which we shall denote with the same symbol $\psi$. With this identification, for $\psi \in \sS(\cH^+)$, the fact that $F^{(+)}_{(u)}(\psi_n) \to \widetilde{\psi} = \mF(\psi)$ in the topology of $L^2(\bR\times \bS^2, kdk\wedge d\bS^2)$ implies that $F^{(+)}_{(u)}(\psi) = \mF(\psi)$ by continuity of $F_{(u)}$. $\Box$\ [**Proof of Proposition \[Propembedding\].**]{} The map $\sK_{\cH}$ is per construction linear. Let us prove that (a) is valid, [*i.e.*]{}, $\sK_{\cH}$ does not depend on the particular decomposition (\[dec0\]) for a fixed $\psi \in \sS(\cH)$. Consider a different analogous decomposition $\psi = \psi'_- + \psi'_0 + \psi'_+$. We have that the two definitions of $\sK_{\cH}\psi$ coincides because their difference is: $$\begin{aligned} & F_{(U)}(\psi_-) - F_{(U)}(\psi'_-) + F_{(U)}(\psi_0) -F_{(U)}(\psi'_0)+ F_{(U)}(\psi_+)- F_{(U)}(\psi'_+)\nonumber \\ &= F_{(U)}(\psi_-- \psi'_-) + F_{(U)}(\psi_0 -\psi'_0)+ F_{(U)}(\psi_+- \psi'_+) = \widehat{\psi_-- \psi'_-} + \widehat{\psi_0-\psi'_0} + \widehat{\psi_+- \psi'_+}\nonumber\\ & = \widehat{\psi-\psi} =0 \:,\end{aligned}$$ Here we have used the fact that, per construction, $\psi_\pm -\psi'_\pm$ and $\psi_0-\psi_0$ belongs to $C_0^\infty(\cH; \bR)$ and thus $F_{(U)}$, acting on each of them, produces the standard $U$-Fourier transform indicated by $\widehat{\cdot}$.\ (b) The statement is valid per definition of $\sK_{\cH}$. Let us thus prove (c). From now on we write $\sigma$ instead of $\sigma_{\cH}$. Let us take $\psi,\psi' \in \sS(\cH)$ and decompose them as $\psi= \psi_1+\psi_2+\psi_3$ and as $\psi'= \psi'_1+\psi'_2+\psi'_3$ where $\psi_1,\psi'_1 \in \sS(\cH^+)$, $\psi_2,\psi'_2 \in C_0^\infty(\cH; \bR)$ and $\psi_3,\psi'_3 \in \sS(\cH_R^-)$. In this way we have: $$\begin{gathered} \sigma(\psi,\psi') = \:\sigma(\psi_1,\psi'_1)+\sigma(\psi_2,\psi'_2)+\sigma(\psi_3,\psi'_3) + \sigma(\psi_1,\psi'_2)+\sigma(\psi_1,\psi'_3)\nonumber\\ + \sigma(\psi_2,\psi'_1)+\sigma(\psi_2,\psi'_3) + \sigma(\psi_3,\psi'_1)+\sigma(\psi_3,\psi'_2)\:.\end{gathered}$$ Let us examine each term separately. Consider $\sigma(\psi_1,\psi'_1)$. From now on $\widetilde{\psi}\equiv \mF(\psi)$ is the Fourier-Plancherel transform of $\psi$, computed with respect to the coordinate $u$. Notice that $\widetilde{\psi}_1(-k,\theta,\phi) = \overline{\widetilde{\psi}_1(k, \theta,\phi)}$ since $\psi_1$ and $\psi'_1$ are real. By direct inspection, if one uses these ingredients and the definition of $d\mu(k)$, one gets immediately the first identity: $$\begin{gathered} \sigma(\psi_1,\psi'_1) = - 2 Im \langle \widetilde{\psi}_1, \widetilde{\psi'}_1\rangle_{L^2(\bR\times \bS^1, d\mu\wedge d\bS^2)}= \nonumber \\ -= 2 Im \langle F_{(U)}\circ (F^{(+)}_{(u)})^{-1}(\widetilde{\psi_1}), F_{(u)}\circ (F^{(+)}_{(u)})^{-1}(\widetilde{\psi'_1}) \rangle_{L^2(\bR_+\times \bS^1, dK\wedge d\bS^2)} =- 2 Im \langle \sK_{\cH} \psi_1, \sK_{\cH} \psi'_1\rangle_{\sH_{\cH}}\:, \label{p1}\end{gathered}$$ The second identity arises form the fact that $ F_{(U)}\circ (F^{(+)}_{(u)})^{-1}$ is an isometry as follows from (b) in Proposition \[propbastarda\] and (a) in Proposition \[PropMain4\]. The last identity is nothing but the definition of $\sK_{\cH}$. With the same procedure we similarly have $$\sigma(\psi_3,\psi'_3) = - 2 Im \langle \sK_{\cH} \psi_1, \sK_{\cH} \psi'_1\rangle_{\sH_{\cH}}\:.$$ If we refer to $\sigma(\psi_2,\psi'_2)$, we can employ the coordinate $U$ taking into account that the support of those smooth functions is compact when referred to the coordinates $(U,\theta, \phi)$ over $\cH$.\ Hence, $\psi_2,\psi'_2, \partial_U\psi_2,\partial_U\psi'_2 \in L^2(\bR \times \bS^2, dU \wedge d\bS^2)$ so that, at a level of $U$-Fourier transforms, it holds $\widehat{\psi}_{2+},\widehat{\psi'}_{2+} \in L^2(\bR_+ \times \bS^2, dK \wedge d\bS^2)\cap L^2(\bR_+ \times \bS^2, K dK \wedge d\bS^2)$. Finally, in the considered case, directly by the definition, $\sK_{\cH} \psi'_2 = \widehat{\psi'}_{2+}$ and $\sK_{\cH} \psi_2 = \widehat{\psi}_{2+}$. If one uses the fact that $\widehat{\psi}_2(-K,\theta,\phi)=\overline{\widehat{\psi}_2(K, \theta,\phi)}$ since $\psi_2$ and $\psi'_2$ are real, one straightforwardly achieves the first identity: $$\begin{gathered} \sigma(\psi_2,\psi'_2) = - 2 Im \langle \widehat{\psi}_{2+}, \widehat{\psi'}_{2+} \rangle_{L^2(\bR_+\times\bS^2, 2K dK\wedge d\bS^2)} = \nonumber \\ = - 2 Im \langle F_{(U)}{\psi}_{2+}, F_{(U)}{\psi'}_{2+} \rangle_{L^2(\bR_+\times\bS^2, 2K dK\wedge d\bS^2)} = - 2 Im \langle \sK_{\cH} \psi_2, \sK_{\cH} \psi'_2\rangle_{\sH_{\cH}}\:, \end{gathered}$$ The remaining identities follow from the definition of $F_{(U)}$ and $\sK_{\cH}$. As a further step we notice that $$\sigma(\psi_1,\psi'_3) = 0 = - 2 Im \langle \sK_{\cH} \psi_1, \sK_{\cH} \psi'_3\rangle_{\sH_{\cH}}\quad \sigma(\psi_3,\psi'_1) = 0 = - 2 Im \langle \sK_{\cH} \psi_3, \sK_{\cH} \psi'_1\rangle_{\sH_{\cH}}\:.$$ Let us focus on the first identity, the second being analogous; it holds true because the functions have disjoint supports, whereas $\langle \sK_{\cH} \psi_3, \sK_{\cH} \psi'_1\rangle_{\sH_{\cH}}=0$ since, per direct application of (b) in Proposition \[propbastarda\], $\psi_1 \in \sS(\cH^+)$ is the limit of a sequence of real smooth functions $f^{(1)}_n$ with support in $\cH^+$ whereas $\psi'_3 \in \sS(\cH^+)$ is the limit of a sequence of real smooth functions $f^{(3)}_n$ with support in $\cH^-$. Hence $$\begin{aligned} &Im \langle \sK_{\cH} f^{(1)}_n, \sK_{\cH} f^{(2)}_m\rangle_{\sH_{\cH}} = Im \lambda_{KW}(f^{(1)}_m, f^{(2)}_n) \nonumber \\ &= -\frac{r_S^2}{\pi} Im \int_{\bR \times \bR \times \bS^2}\spa \frac{{f^{(1)}_n(U_1,\theta,\phi)} f^{(2)}_m(U_2,\theta,\phi)}{(U_1-U_2 -i0^+)^2} dU_1 \wedge dU_2 \wedge d\bS^2(\theta,\phi)\nonumber \\&= -\frac{r_S^2}{\pi} Im \int_{\bR \times \bR \times \bS^2}\spa \frac{\partial_{U_1}{f^{(1)}_n(U_1,\theta,\phi)} f^{(2)}_m(U_2,\theta,\phi)}{U_1-U_2 -i0^+} dU_1 \wedge dU_2 \wedge d\bS^2(\theta,\phi)\nonumber\\ &= -r_S^2 \int_{\bR \times \bR \times \bS^2}\spa \partial_{U_1}{f^{(1)}_n(U_1,\theta,\phi)} f^{(2)}_m(U_2,\theta,\phi) \delta(U_1 -U_2) dU_1 \wedge dU_2 \wedge d\bS^2(\theta,\phi) =0, \nonumber\end{aligned}$$ since $f^{(1)}_n$ and $f^{(2)}_m$ have disjoint support. Let us examine the term $\sigma(\psi_1,\psi'_2)$: in this case we decompose $\psi_1 = f_1 + g_1$ where $f_1 \in C_0^\infty(\cH^+; \bR)$ and $g_1 \in \sS(\cH^+)$, but $\supp(g_1) \cap \supp(\psi'_2) = \emptyset$. We have: $$\sigma(\psi_1,\psi'_2) = \sigma(f_1,\psi'_2) + \sigma(g_1,\psi'_2)\:.$$ At the end of this proof we shall also prove that: (\_1,’\_2) = 0 = - 2 Im \_ \_1, \_ ’\_2\_[\_]{} \[laaast\].Conversely $\sigma(f_1,\psi'_2) = - 2 Im \langle \sK_{\cH} f_1, \sK_{\cH} \psi'_2\rangle_{\sH_{\cH}}$, exactly as in the case $\sigma(\psi_2,\psi'_2)$ examined above. If we sum up, per $\bR$-linearity: $$\sigma(\psi_1,\psi'_2) = - 2 Im \langle \sK_{\cH} \psi_1, \sK_{\cH} \psi'_2\rangle_{\sH_{\cH}}\:.$$ With an analogous procedure we also achieve: $$\sigma(\psi_2,\psi'_1) = - 2 Im \langle \sK_{\cH} \psi_1, \sK_{\cH} \psi'_2\rangle_{\sH_{\cH}} \quad \sigma(\psi_2,\psi'_3) = - 2 Im \langle \sK_{\cH} \psi_2, \sK_{\cH} \psi'_3\rangle_{\sH_{\cH}},$$ and (\_3,’\_2) = - 2 Im \_ \_3, \_ ’\_2\_[\_]{} \[p7\]. The identities (\[p1\])-(\[p7\]), per $\bR$-linearity, yield the thesis: (,’) = - 2 Im \_ , \_ ’\_[\_]{} . The proof ends provided we demonstrate (\[laaast\]). We only sketch the argument leaving the details to the reader. The proof is based on the following result. If $\psi \in \sS(\cH)$ and $T\in \bR$, let us denote by $\psi_T \in \sS(\cH)$ the function such that $\psi_T(U,\theta,\phi) \doteq \psi(U-T,\theta,\phi)$. It is possible to prove that \[notte\] (\_(\_T))(K,,)=e\^[-iKT]{}(\_())(K,,). The proof of (\[notte\]) is straightforward when $\psi \in C_0^\infty(\cH; \bR)$, since, in such case $\sK_{\cH}$ is the positive frequency part of the $U$-Fourier transform of $\psi$. If $\psi \not\in C_0^\infty(\cH; \bR)$, we can decompose it as $\psi_- + \psi_0 +\psi_+$, as in the definition of $\sK_{\cH}$, fixing $\psi_-$ and $\psi_+$ in order that $(\psi_{\pm})_T$ are still supported in $(-\infty,0)$ and $(0,+\infty)$ respectively if $|T'| \leq T$. If one uses the fact, which can be proved by inspection, that – up to a re-definition of the initially taken $\psi_n$ – $C_0(\cH^+; \bR) \ni (\psi_{n})_T \to (\psi_+)_T$ in $H^1(\cH^+)_u$ if $C_0(\cH^+; \bR) \ni \psi_n \to \psi_+$, one gets that (\[notte\]) is valid for $\psi_+$. The very same argument applies also $\psi_-$. The very definition of $\sK_{\cH}$ entails the validity of (\[notte\]) for every $\psi \in \sS(\cH)$, which, in turn, yields (\[laaast\]) immediately, because, in the examined case, $$\sigma(\psi_1,\psi'_2) = 0 = - 2 Im \langle \sK_{\cH} \psi_1, \sK_{\cH} \psi'_2\rangle_{\sH_{\cH}}.$$ The left hand side vanishes as $\psi_1,\psi'_2$ have disjoint supports, whereas the right-hand side can be re-written as: $$\begin{gathered} - 2 Im \int_{\bR \times \bS^2} \overline{e^{-iTK} \left(\sK_{\cH} g_1\right)(K,\theta,\phi)} e^{-iTK} \left(\sK_{\cH} \psi'_2\right)(K,\theta,\phi)\: 2KdK \wedge d\bS^2(\theta,\phi) =\\ -2 Im \left\langle \sK_{\cH}((g_1)_T), \sK_{\cH}((\psi'_2)_T) \right\rangle\:. \end{gathered}$$ Such term is also vanishing, because we can fix $T$ so that $\supp((g_1)_T) \subset \cH^-$ and $\supp ((\psi'_2)_T) \subset \cH^+$, hence reducing to the case $\sigma(\psi_1,\psi_3')=0 = - 2 Im \langle \sK_{\cH} \psi_1, \sK_{\cH} \psi'_3\rangle_{\sH_{\cH}}$ examined beforehand.\ (d) is a trivial consequence of (c) : if $\sK_{\cH} \psi =0$, then $Im \langle \sK_{\cH} \psi, \sK_{\cH} \psi'\rangle =0$ and thus $\sigma_{\cH}(\psi,\psi')=0$ for every $\psi' \in \sS(\cH)$. Since $\sigma_{\cH}$ is nondegenerate, it implies $\psi=0$. Let us prove (e). As $C_0^\infty(\cH;\bR)) \subset \sS(\cH)$, $$\sH_{\cH} = \overline{F_{(U)}(C_0^\infty(\cH;\bR))}= \overline{\sK_{\cH}(C_0^\infty(\cH;\bR))} \subset \overline{\sK_{\cH}(\sS(\cH))}\subset \sH_{\cH}\quad \mbox{and thus $\overline{\sK_{\cH}(\sS(\cH))} = \sH_{\cH}$.}$$ The first identity arises out of (a) in Proposition \[PropMain4\], the second out of (b) in Proposition \[Propembedding\].\ We can now conclude proving (f). The continuity of $\sK_\cH$ with respect to the considered norm holds for the following reason. If $\{\psi_n\}_{n\in \bN} \subset \sS(\cH)$ and $||\psi_n||_\cH^\chi \to 0$, then, if we decompose $\psi_n = \psi_{0n}+ \psi_{+n}+ \psi_{-n}$, separately, $\psi_{0 n}$ and $\psi_{\pm n} \to 0$ in the respective Sobolev topologies. In turn $\sK_\cH(\psi_{0n}) = F_U(\psi_{0n})\to 0$ because the Sobolev topology is stronger than that of $L^2(\bR \times \bS^2; dU\wedge d\bS^2)$ and $\sK_\cH(\psi_{\pm n})\to 0$ for (b) in Proposition \[propbastarda\]. Per definition of $\sK_\cH$, it hence holds $\sK_\cH(\psi_{n}) \to 0$. Thus the linear map $\sK_\cH:\sS(\cH) \to sH_\cH$ is continuous it being continuous in $0$. Particularly, we conclude, that there exists $C_\chi >0$ (the value $0$ is not allowed since $\sK_\cH$ cannot be null function) with $||\sK_\cH(\psi)||_{\sH_\cH} \leq C_\chi ||\psi||_\cH^\chi$ for every $\psi \in \sS(\cH)$. The Cauchy-Schwartz inequality implies the one displayed in (f). $\Box$\ [**Proof of Proposition \[propidscri\].**]{} Let us define $v=x^2$ if $x\geq 0$ and $v= -x^2$ if $v<0$ .Per direct inspection one sees that, if $\psi,\psi'\in C^\infty_0(\bR^*_-\times \bS^2; \bR)$, $$\begin{gathered} \int_{\bR^2\times \bS^2} \frac{\psi(v,\theta,\phi)\psi'(v',\theta,\phi)}{(v-v'-i0^+)^2} dv \wedge dv'\wedge d\bS^2(\theta,\phi) = \int_{\bR^2\times \bS^2} \frac{\psi(v(x),\theta,\phi)\psi'(v(x'),\theta,\phi)}{(x'-x-i0^+)^2} dx\wedge dx' \wedge d\bS^2(\theta,\phi) \notag \\ +\int_{\bR^2\times \bS^2} \frac{\psi(v(x),\theta,\phi)\psi'(v(-x'),\theta,\phi)}{(x'-x-i0^+)^2} dx\wedge dx' \wedge d\bS^2(\theta,\phi)\label{AGG}\:.\end{gathered}$$ At a level of $x$-Fourier transform, let us denote as $\dot{\psi}= \dot{\psi}(h,\theta,\phi)$, with $h\in \bR$ and $(\theta,\phi) \in \bS^2$ the $x$-Fourier transform of $\psi(v(x))$. Let us also define $\dot{\psi}_+ \doteq \dot{\psi}\spa \rest_{\bR_+ \times \bS^2}$, then, out of the fact that if $\phi$ is real valued, as it happens for $\psi$ and $\psi'$, then $\overline{\dot{\phi}_+(h,\theta,\phi)}$ is the $x$ Fourier transform of $x \mapsto \phi(-x,\theta,\phi)$), can be re-written as $$\begin{gathered} \lambda_{\scri}\left(\psi, \psi'\right) = \int_{\bR_+\times \bS^2} \overline{\dot{\psi'}_+(h,\theta,\phi)} \dot{\psi}_+(h,\theta,\phi) 2hdh \wedge d\bS^2 +\notag\\ \int_{\bR_+\times \bS^2} \overline{\dot{\psi'}_+(h,\theta,\phi)}\left(C\dot{\psi}_+\right)(h,\theta,\phi) 2hdh \wedge d\bS^2\:,\label{ddd} \end{gathered}$$ where the operator $C : L^2(\bR_+\times \bS^2, 2hdh) \to L^2(\bR_+\times \bS^2, 2hdh) $ is anti-unitary and it is nothing but the complex conjugation. Now let us take $\psi \in \sS(\scrim)$ which is completely supported in $\bR_+\times \bS^2$. Per definition of $\sS(\scrim)$, the function $\psi=\psi(v(x) )$ and its $x$-derivative belong to $L^2(\bR\times \bS^2, dx \wedge d\bS^2)$ and, thus, $\psi$ belongs to $H^1(\scri)_x$. A sequence of functions $\psi_n \in C^\infty(\bR^*_-\times \bS^2; \bR)$ which converges to $\psi$ in $H^1(\scrim)_x$ can be constructed as $\psi_n = \chi_n \cdot \psi$, where $\chi_n(x) \doteq \chi(x/n)\geq 0$ with $\chi(x) = 1$ if $x\in (-1, +\infty)$ and $\chi(x) = 0$ for $x\leq -2$. Per direct inspection and thanks to Lebesgue’s dominated convergence theorem, one achieves that $C^\infty(\bR^*_-\times\bS^2;\bR)\ni\psi_n\to\psi$ in $H^1(\scrim)_x$ as $n\to +\infty$. Consequently $\dot{\psi_n}\to \dot{\psi}$ both in $L^2(\bR\times \bS^2, dh)$ and in $L^2(\bR_+\times \bS^2, h^2dh)$. Therefore $\dot{\psi_n}_+\to \dot{\psi}_+$ in the topology of $L^2(\bR_+\times \bS^2, 2hdh)$. Finally, in view of (\[ddd\]) and on account of the continuity of $C$, the sequence $\{\psi_n\}_{n\in \bN}$ is of Cauchy type with respect to $\lambda_{\scrim}$. The same argument shows that, if $C^\infty(\bR^*_-\times \bS^2; \bR) \ni \psi'_n \to \psi$ in $H^1(\scrim)$ as $n\to +\infty$, then $\lambda_{\scrim}(\psi_n-\psi'_n, \psi_n-\psi'_n) \to 0$ as $n\to+ \infty$. Hence (\[FumF\]) and (\[antilin\]) are trivial consequences of what proved, which is tantamount to verify (a) and (b). $\Box$\ [**Proof of Proposition \[Propembeddingscri\].**]{} The proofs of items (a),(b),(d),(e) as well as (f) are very similar to those of the corresponding items in Proposition \[Propembedding\], so they will be omitted. We instead focus our attention on (c), whose proof is similar to the same point (c) of Proposition \[Propembedding\], though with some relevant differences. Let us ake $\psi,\psi' \in \sS(\scrim)$ and let us decompose them as $\psi= \psi_0 + \psi_1$, $\psi'= \psi'_0+ \psi'_1$ where $\psi_0,\psi_1 \in C_0^\infty(\scrim; \bR)$ while $\psi'_0,\psi'_1$ are supported in $(0,+\infty) \times \bS^2$. Since $\sigma\doteq \sigma_\scrim$ and $\langle, \rangle = \langle,\rangle_\scri$, it holds $$\sigma(\psi,\psi') = \sigma(\psi_0,\psi'_0)+\sigma(\psi_0,\psi'_1)+\sigma(\psi_1,\psi'_0)+\sigma(\psi_1,\psi'_1)\:.$$ Exactly as in (c) of the Proposition \[Propembedding\], we conclude that \[ONE\] (\_0,’\_0)= -2Im \_\_0,\_’\_0 . With reference to $\sigma(\psi_1,\psi'_1)$, we have instead: $$-2Im \langle \sK_\scrim \psi_1,\sK_\scrim \psi'_1 \rangle = -2Im \langle F_{(v)}(\psi_1), F_{(v)}(\psi'_1) \rangle = -2 Im \lambda_\scrim(\psi_1,\psi'_1)\:.$$ If we make both use of (\[antilin\]), and of the fact that the above identity can be used for $\psi_1,\psi_1'$ as established in (b) of Proposition \[PropMain5\], we have $$-2Im \langle \sK_\scrim \psi_1,\sK_\scrim \psi'_1 \rangle = -2 Im \int_{\bR_+\times \bS^2} \overline{\dot{\psi_1}} \dot{\psi_1'} 2hdh \wedge d\bS^2 -2 Im \int_{\bR_+\times \bS^2} \overline{\dot{\psi_1} \dot{\psi_1'}} 2hdh \wedge d\bS^2\:,$$ where $\dot{\psi}(h,\theta,\phi)$ is the $x$-Fourier-Plancherel transform of $\psi=\psi(v(x),\theta,\phi)$. The last term in the right-hand side can be omitted for the following reason. If we look at (\[AGG\]), we see that $-i0^+$ can be replaced by $+i0^+$ without altering the result, since the functions in the numerator have disjoint supports. This is equivalent to say that, in the right-hand side of (\[ddd\]), the last term can be replaced with its complex conjugation without affecting the final result. Finally, this means that the identity written above can be equivalently recast as $$-2Im \langle \sK_\scri \psi_1,\sK_\scrim \psi'_1 \rangle = -2 Im \int_{\bR_+\times \bS^2} \overline{\dot{\psi_1}} \dot{\psi_1} 2hdh \wedge d\bS^2 -2 Im \overline{\int_{\bR_+\times \bS^2} \overline{\dot{\psi_1} \dot{\psi_1}} 2hdh \wedge d\bS^2}\:.$$ As a consequence the last term can be dropped, so that: \[TWO\] -2Im \_\_1,\_’\_1 = -4 Im \_[\_+\^2]{} h dh d\^2 = 2i \_[\^2]{} h dh d\^2 = (\_1,’\_1).In the last passage we have used that $-ih\dot{\psi'_1}$ is the $x$-Fourier transform of $\partial_x\psi'_1$, and that the integration in $\sigma(\psi_1,\psi'_1)$ can be performed in the variable $x$ since the singularity of the coordinates at $x=0$ is irrelevant, the supports of $\psi_1$ and $\psi'_1$ being away from there. One should also notice that these functions are real so that $\overline{\dot{\psi_i}(h,\theta,\phi)} = \dot{\psi_i}(-h,\theta,\phi)$. Let us consider the term $\sigma(\psi_0,\psi'_1)$ the other, $\sigma(\psi_1,\psi'_0)$ can be treated similarly. To this end, let us decompose $\psi'_1 = \phi'_0 + \phi'_1$ in order that $\phi'_0 \in C_0^\infty(\scrim; \bR)$ and the support of $\phi'_1$ is disjoint from that of $\psi_0$. Therefore: $\sigma(\psi_0,\psi'_1) = \sigma(\psi_0,\phi'_0) + \sigma(\psi_0,\phi'_1) = -2Im \langle \sK_\scri \psi_1,\sK_\scrim \phi'_1 \rangle + \sigma(\psi_0,\phi'_1)\:.$ Since we shall prove that: \[ULTIMA\] (\_0,’\_1) = 0 = -2Im \_\_0,\_’\_1 , we also have that $$\sigma(\psi_0,\psi'_1) = -2Im \langle \sK_\scrim \psi_0,\sK_\scrim \psi'_1 \rangle\:, \quad \mbox{and similarly,}\quad \sigma(\psi_1,\psi'_0) = -2Im \langle \sK_\scrim \psi_1,\sK_\scrim \psi'_0 \rangle\:,$$ which, together (\[ONE\]) and (\[TWO\]) implies the validity of (c) by bi-linearity: $$\sigma(\psi,\psi') = -2Im \langle \sK_\scrim \psi,\sK_\scrim \psi' \rangle\:.$$ To conclude, it is enough to prove (\[ULTIMA\]). The left-hand side vanishes since the supports of the functions $\psi_0,\phi'_1$ are disjoint by construction. Hence it remains to verify that $Im \langle \sK_\scri \psi_0,\sK_\scrim \phi'_1 \rangle =0$. If it were $\supp(\psi_0) \subset (-\infty, 0)\times \bS^2$ and $\supp(\phi'_1) \subset (0, +\infty)\times \bS^2$, one would achieve $Im \langle \sK_\scrim \psi_0,\sK_\scrim \phi'_1 \rangle =0$ through the same argument used in the corresponding case (that of $\sigma(\psi_1,\psi'_3)$) in the proof of (c) of the Proposition \[Propembedding\]. To wit, one should employ a sequence of real smooth functions which tends to $\phi'_1$ in the topology of $\lambda_\scrim$ and with compact supports all enclosed in $(0, +\infty)\times \bS^2$. Such a sequence exists in view of Proposition \[PropMain5\]. As a matter of fact, we can focus our attention to the lone case $\supp(\psi_0)\subset (-\infty, 0)\times \bS^2$ and $\supp(\phi'_1)\subset (0, +\infty)\times\bS^2$, thanks to the following lemma which will also play a pivotal role in the proof of (b) of Theorem \[Main4scri\]. \[ULTIMOlemma\] [*For $\psi \in \sS(\scrim)$ and $L\in \bR$, let $\psi_L \in \sS(\scrim)$ denote the function with $\psi_L(v,\theta,\phi) \doteq \psi(v-L,\theta,\phi)$ for all $v\in \bR$ and $\theta,\phi\in \bS^2$. With the given definition for $\sK_\scrim : \sS(\scrim) \to \sH_\scrim$, it holds: \[ULTIMOlemmaEq\] (\_\_L)(k,,) = e\^[-iLk]{} (\_)(k,,), (k,,) \_+\^2.* ]{} [*Proof of Lemma \[ULTIMOlemma\]*]{}. Per definition, if $\psi\in \sS(\scrim)$ is fixed, $\sK_\scrim \psi = F_{(v)} \psi_0 + F_{(v)} \psi_-$, where $\psi = \psi_0+ \psi_-$ with $\psi_0 \in C_0^\infty(\scrim; \bR)$ and $\psi_-\in \sS(\scrim)$ with $\supp(\psi_-)\subset (-\infty,0) \times \bS^2$. Let us fix $L \in \bR$ and let us notice that, the very definition of $F_{(v)}$ on $C_0^\infty(\scrim;\bR)$ yields $$F_{(v)} (\psi_0)_L = e^{-iLk} F_{(v)}\psi_0\:.$$ To conclude it is sufficient to establish that it also holds: \[stanco\] F\_[(v)]{} (\_-)\_L = e\^[-iLk]{} F\_[(v)]{}\_-. Since the definition of $\sK_\scrim \psi$ does not depend on the chosen decomposition $\psi = \psi_0+ \psi_-$, we can fix $\psi_0$ and $\psi_-$ such that the support of $(\psi_-)_L$ is still included in $(-\infty, 0)$. This holds true for every $\psi_-$ if $L\leq 0$, but it is not straightforward for $L > 0$ and, in this case, the support of $\psi_-$ has to be fixed sufficiently far from $0$). To establish (\[stanco\]), let us use the coordinate $x=-\sqrt{-v}$ for $v<0$. The singularity at $v=0$ does not affect the procedure since the supports of all the involved functions do not include it. We know, thanks to the proof of Proposition \[PropMain5\], that there exists a sequence $C^\infty_0(\scrim; \bR) \ni \psi_n \to \psi_-$, all supported in $\supp(\psi_-) \subset (-\infty,0)\times \bS^2$; the convergence is here meant both in in the topology of $H^1(\scrim)$ and in that of $\lambda_{\scrim}$. Per direct inspection, one sees that, for the above-mentioned sequence it holds $\supp (\psi_n)_L \subset \supp(\psi_L) \subset (0,+\infty)\times \bS^2$ and $\psi_n \to \psi_-$ entails $(\psi_n)_L \to (\psi_-)_L$ for $n\to +\infty$ in the topology of $H^1(\scrim)_x$. According to (b) in Proposition \[PropMain5\], this also implies that the convergence holds in the topology of $\lambda_\scrim$. Since $F_{(v)}$ is continuous with respect to the last mentioned topology, we get, as $n\to +\infty$: $$e^{-iLk} F_{(v)}\psi_n = F_{(v)} (\psi_n)_L \to F_{(v)} (\psi_-)_L\:.$$ On the other hand, since $F_{(v)}\psi_n \to F_{(v)}\psi_-$ in $L^2(\bR_+\times \bS^2, kdk \wedge d\bS^2)$, it trivially holds as $n\to +\infty$: $$e^{-iLk} F_{(v)}\psi_n \to e^{-iLk} F_{(v)}\psi_-\:,$$ and thus $$e^{-iLk} F_{(v)}\psi = F_{(v)} (\psi_-)_L\:,$$ which implies (\[stanco\]), concluding the proof. $\Box$\ To conclude the proof of (c), we note that, in view of (\[ULTIMOlemmaEq\]), it must hold $-2Im \langle \sK_\scrim \psi_0,\sK_\scri \phi'_1 \rangle = -2Im \langle \sK_\scrim (\psi_0)_L,\sK_\scrim (\phi'_1)_L \rangle$ for every $L \in \bR$. Therefore, we can fix $L$ so that $\supp (\psi_0)_L \subset (-\infty, 0)\times \bS^2$ and $\supp((\phi'_1))_L) \subset (0, +\infty)\times\bS^2$, obtaining, as said before, $$-2Im \langle \sK_\scrim \psi_0,\sK_\scrim \phi'_1 \rangle = -2Im \langle \sK_\scrim (\psi_0)_L,\sK_\scrim(\phi'_1)_L \rangle=0\:.$$ This implies (\[ULTIMA\]) and, hence, (c). $\Box$\ [**Proof of Proposition \[distrib\].**]{} The first assertion arises per direct inspection of definition \[omegaHdef\] and, thus, we need only to prove that the (complexified) functionals on $C^\infty_0(\mM; \bC)\times C^\infty_0(\mM; \bC)$, $\Lambda_{\cH}$ and $\Lambda_{\Im^-}$ are separately distributions in $\mD'(\mM\times \mM)$. To this end, it suffices to show that the maps $f\mapsto\Lambda_i(f,\cdot)$ and $g\mapsto\Lambda_i(\cdot,g)$ are weakly continuous, [*i.e.*]{}, they tend to $0$ when tested with any sequence of functions $h_j\in C^ \infty_0(\mM; \bC)$ which converges to $0$ in the topology of test functions. Here and hereafter the subscript $i$ stands either for $\cH$ or for $\Im^-$. According to theorem 2.1.4 of [@Hormander], such statement entails that both $\Lambda_i(f,\cdot)$ and $\Lambda_i(\cdot,g)$ are distributions in $\mD'(\mM)$, hence they are sequentially continuous. Once established, one can, therefore, invoke the Schwartz’ integral kernel theorem to conclude that $\Lambda_i\in \mD'(\mM\times \mM)$. In view of the complexification procedure, it is sufficient to consider only the case of real valued test functions.\ Let us start with $\Im^-$; in this case $\la_{\Im^-}$ has the explicit form in proposition \[propidscri\] and, thus, one can take into account a generic decomposition generated by a smooth function $\eta$ supported on $\bR^*_-\times\bS^2$ and equal to one for $v<v_0<0$. Due to the continuity property, discussed at point (f) of Proposition \[Propembedding\], $$| \Lambda_{\Im^-}(f,h)|=|\la_{\scrim}(\varphi^f_\scrim,\varphi^h_\scrim)| \leq C\; \| \varphi^f_\scrim \|_\scrim^\eta \; \| \varphi^h_\scrim \|_\scrim^\eta\;.$$ We recall that, for every $\varphi\in\sS(\scrim)$, $\|\varphi\|_\scrim^\eta$ is defined in as the sum $$\|\varphi\|_\scrim^\eta=\|\eta \varphi\|_{H^1(\scrim)_x}+\|(1-\eta) \varphi\|_{H^1(\scrim)_v}\;.$$ Continuity is tantamount to show that, for $f,g\in C_0^\infty(\mM;\bR)$, if $f\to 0$ with a fixed $g$ (or $g\to 0$ with a fixed $f$) in the topology of $C^\infty_0(\mM; \bC)$, both Sobolev norms, above written, tend to zero. Let us start from the second one. For a given compact $K\subset\mM$, as in the proof of Lemma \[lemma1\], let us fix a sufficiently large globally hyperbolic spacetime $\mN\subset \widetilde{\mM}$ which is equipped with the metric $\widetilde{g}$ and which extends $\mM$ partly around $\Im^-$. Furthermore $\mN$ must include $K$ and $\mN\cap\Im^-$ should encompass all the points with $v \geq v_0$, reached by the closure in $\mM \cup \scrim$ of $J^-(K;\mM)$. If we notice that the causal propagator $E_{P_{ \widetilde{g}}}$ is a continuous map from $C^\infty_0(\mM; \bR)\to C^\infty (\mN;\bR)$, one has that $(1-\eta)\varphi^f_\scrim$ and all the $v$-derivatives uniformly vanishes as $f \to 0$ in the topology of $C^\infty_0(\mM; \bC)$. Since, per constriction, all the functions $(1-\eta)\varphi^f_\scrim$ have support in a common compact of $\bR \times \bS^2$ determined by $\eta$ and $J^-(K;\mM))$, also $\|(1-\eta) \varphi_\scrim^f\|_{H^1(\scrim)_v}$ tends to zero in view of the integral expression of the Sobolev norm.\ To conclude, in order to deal with the contribution $\|\eta \varphi_\scrim^f\|_{H^1(\scrim)_x}$, let us notice that, according to proposition \[PropDR\] (point (b) in particular), the restriction of a solution of the D’Alembert wave equation on $\Im^-$ decays on null infinity, for $|v|$ greater than a certain $|v_0|$, as $\frac{C_f}{\sqrt{1+ |v|}}$, while its $v$-derivative as $\frac{C_f}{1+ |v|}$, where $C_f$ tends to $0$ as $f\to 0$ in the topology of $C^\infty_0(\mM;\bC)$. Hence per direct inspection, if we work with the coordinate $x$, also$\|\eta \varphi_\scrim^f\|_{H^1(\scrim)_ x}$ vanishes as $f$ tends to zero in the topology of $C^\infty_0(\mM;\bC)$.\ The case of $\cH$ can be dealt with in the same way using the continuity presented in point $(f)$ of proposition $\ref{Propembedding}$ and the appropriate decay estimates of the wave functions presented in proposition \[PropDR\]. As before, one can reach the conclusion that both $\Lambda_{\cH}(f,\cdot)$ and $\Lambda_{\cH}(\cdot,g)$ lie in $\mD'( \mM)$. $\qed$\ [**Proof of Proposition \[RapidDecay\].**]{} Let us start considering $\|\varphi^{f_{p}}_\scrim \|_\scrim$ as defined in for some generic decomposition based on the choice of the function $\eta$). Here $f\in C^\infty_0(\mM)$ and $p$ lie in a conic neighbourhood $V_{k_x}$ of $k_x$, we are going to specify. The procedure we shall employ can be similarly used also for $\|\chi'\varphi^{f_{p}}_{\cH}\|_ {\cH^-}$ to show that it is rapidly decreasing in $p$. Furthermore we recall that $f_{p}\doteq f e^{i \langle p,\cdot\rangle}$, while $\varphi^{f_{p}}_\scrim$ is the smooth limit towards of $\scrim$ of $E_{P_g}f_{p}$, where $E_{P_g}$ is the causal propagator of $P_g$ as in . Furthermore $\varphi^{f_{p}}_\scrim $, together with its derivative along the global null coordinate $v$ is known to decay at $-\infty$ according to the estimates in Proposition \[PropDR\], in turn based on the work of [@DR05], [ *i.e.*]{}, $$|\varphi^{f_{p}}_\scrim |\leq \frac{C_3}{\sqrt{1+|v|}}\;, \qquad |X(\varphi^{f_{p}}_\scrim ) |\leq \frac{C_4}{{1+|v|}}\;.$$ Here $X$ still stands for the smooth Killing vector field on the conformally extended Kruskal spacetime, coinciding with $\partial_v$ on $\scrim$. They yields that the norm $\|\varphi^{f_{p}}_\scrim \|_\scrim$, defined as in , is controlled by the above coefficients $C_3$ and $C_4$ which depend on $\varphi^{f_{p}}$, since the norms of the remaining universal functions smoothed about $i^-$ are finite. Hence, we shall analyse them explicitly and we notice that all the relevant results in \[PropDR\] can be straightforwardly extended to the complex case. Our goal is to establish that the coefficients $C_3$ and $C_4$ are rapidly decreasing in $p$ when computed for $\varphi^{f_p}$ in the given hypotheses about $x$ and $k_x$.\ As a starting point, let us consider the case in which $x\in I^+(\cB; \mM)$, $\cB$ being the bifurcation. In order to study this, as well as all other scenarios, we make use of the results and of the techniques available in [@DR05] of which we shall adopt nomenclatures and conventions. In this last cited paper it is manifest, that, up to a term depending on the support of initial data, the dependence on the wave function in $C_3$ and $C_4$ is factorised in the square root of the so-called coefficient $\tilde{E}_5$, namely formula (5.4) in [@DR05]. After few formal manipulations, the relevant expression can be (re)written as an integral over the constant time surface $\Si_1\subset\mW$, unambiguously individuated, in the coordinates $(t,r,\theta,\phi)$, as the locus $t=-1$. Here we consider $t=-1$ because we are interested in the decay property in a neighbourhood of $i^-$. Hence $$\begin{gathered} \tilde{E}_5(\varphi)= \sum_{i=1..2} \int_{\Si_1} T_{\mu\nu}(\Om^i\varphi)\; n^\mu n^\nu d\mu (\Si_1)+ \sum_{i=1..4} \int_{\Si_1} T_{\mu\nu}(\Om^i\varphi)\; K^\mu n^\nu d\mu (\Si_1)+ \nonumber \\ + \sum_{i=1..5} \int_{\Si_1} T_{\mu\nu}(\Om^i\varphi)\; X^\mu n^\nu d\mu (\Si_1), \label{E2}\end{gathered}$$ where $n$ is the vector orthogonal to $\Si_1$, pointing towards the past, and normalised as $g^{\mu\nu}n_\mu n_\nu=-1$, $K\doteq v^2\frac{\pa}{\pa v}+u^2\frac{\pa}{\pa u}$ is the so-called [*Morawetz vector field*]{}, $X$ is the timelike Killing vector field $\frac{\pa}{\pa t}$, whereas $d\mu (\Si_1)$ is the metric induced measure on $\Si_1$. Furthermore, $$T_{\mu\nu}(\varphi) = \frac{1}{2}\left(\pa_\mu \overline{\varphi} \pa_\nu \varphi + \pa_\nu \overline{\varphi} \pa_\mu \varphi \right) -\frac{1}{2} g_{\mu\nu} \at \pa_\la \overline{\varphi} \pa^\la \varphi\ct,$$ stands for the stress-energy tensor computed with respect of the solution $\varphi$, while $\Om^2\doteq r^2\displaystyle{ \not}{\nabla}\displaystyle{\not}{\nabla}$ is the squared angular momentum operator, $\displaystyle{\not}{\nabla}$, being the covariant derivative induced by the metric , normalised with $r=1$, on the orbits of $SO(3)$ isomorphic to $\bS^2$. We remark both that the above expression can be found in theorem 4.1 in [@DR08] and, more important to our purposes, that the integrand is a (hermitian) quadratic combination of a finite number of derivatives of $\varphi^{f_{p}}$ on $\Si_1$. Furthermore, since $J^-(supp(f_p); \mM)\cap \Si_1$ is compact, the integrand in does not vanish at most on a compact set and, thus, the overall integral can be bounded by a linear combination of products of the sup of the absolute value of derivatives of $\varphi^{f_{p}}$ up to a certain order, all evaluated on $\Si_1$. Let us notice that, all the remaining functions in the integrand which define $\tilde{E}_5$, barring the said products of derivatives, are continuous and, thus, bounded on the compact set where $\varphi^{f_{p}}$ does not vanish on $\Sigma_1$.\ Let us thus focus on $\varphi^{f_{p}}$ itself as well as on both the initially chosen $x\in I^+(\cB;\mM)$ and $k_x$. If one uses global coordinates, we identify an open relatively compact set $\cO$ which contains both the support of $f$ and that of the function $\rho$ we shall introduce in $\bR^4$ by means of a local coordinate patch. In this way every vector $p \in \bR^4$ can be viewed as an element of the cotangent space at any point in that set. It is also always possible to select $f\in C_0^\infty(\mM;\bR)$ with $f(x)=1$ and with a sufficiently small support, such that every inextensible geodesic starting from $supp (f)$, with cotangent vector equal to $k_x$, intersects $\cH$ in a point with coordinate $U>0$. Hence, we can always fix $\rho\in C^\infty_0(\mK;\bR)$ such that (i) $\rho =1$ on $J^-(supp (f); \mM)\cap \Si_1$ and (ii) the null geodesics emanating from $supp(f)$ with $k_x$ as cotangent vector do not meet the support of $\rho$. Furthermore, on account of the form of the wave front set of $E_{P_g}(z,z')$, now thought of in the whole Kruskal spacetime $\mK$, whose elements $(z,z',k_z, k_{z'})$ have always to fulfil $(z,k_z)\sim (z',-k_{z'})$, we realize that, with $(x,k_x)$ fixed as above and with the given definitions of $f$ and $\rho$, $$\left\{(x_1,x_2,k_1,k_x)\in T^*(\mM\times \mM)\:|\; x_1\in\supp(\rho),\; x_2\in \supp(f),\; k_1\in\bR^4 \right\} \cap WF(E)=\emptyset\;.$$ If we employ this result and if we remember the definition of wavefront set we can use Lemma 8.1.1 in [@Hormander], though working in the coordinate frame initially fixed on the compact $\overline{\cO}$, to further adjust $\rho$, $f$ while preserving the constraints already stated. In this way there exists an open conical neighbourhood $V_{k_x}$ of $k_x$ in $T^*_x\mM$ such that for all $n,n' = 1,2,\dots$, one can find two nonnegative constants $C_n$ and $C'_n$ which fulfil |(k\_1,p)| , \[Li\] uniformly for $(k_1,p)\in(\bR^4\setminus\{0\})\times V_{k_x}$. The searched bounds on the behaviour at large $|p|$ for $C_3$ and $C_4$, computed for $\varphi^{f_p}$ with $p$ in a open conical neighbourhood of $k_x$, arise in term of corresponding bounds of the derivatives $|\partial_x^a \overline{\varphi^{f_{p}}(x)} \partial_x^b \varphi^{f_{p}}(x)|$. One must take into account the explicit expression of both $C_3$ and $C_4$ as integrals over the relevant portion of $\Sigma_1$, which has finite measure because it is compact. Each factor $\partial_x^a \varphi^{f_{p}}(x)$ coincides with the inverse $k_1$-Fourier transform of $\widehat{\rho E f }(k_1,p)$ multiplied with powers of the components of $k_1$ up to a finite order which depends on the considered degree of the derivative. As a last step, to get rid of the $k_1$ dependence, one needs to integrate the absolute value over $k_1$, but the right hand side of (\[Li\]) grants us that the overall procedure yields that the supremum of the integrand in is of rapid decrease in $p$ for all $p\in V_{k_x}$.\ Nonetheless, the result is not yet conclusive since we still need to analyse the case in which the point $x$ lies in $\partial J^+(\cB; \mK)\cap \mM$, that is $x \in \cH_{ev}$. In such case, for every open cone $\Ga\in T^*_x\mM$ containing $k_x$, there exists $p\in \Gamma$ such that the inextensible geodesic which starts form $x$ and it is tangent to $p$, meets the closure of $\Si_1$, hence reaching $\cB$. Therefore, in order to apply the same argument as before, we need to modify the form of $\Si_1$ in the computation of in a neighbourhood of $\cB$. Therefore we need a slightly more refined estimate of the decay-rate of the solutions of on $\scrim$. This can be achieved if we adapt the proof of Theorem 1.1 in [@DR05] under the assumption that we modify the form of $\Si_1$, used to compute into that of another spacelike hypersurface, say $\Si'_1$, contained in $\overline{\mW}$ and such that it intersects $\cH$ at some negative value of the Kruskal null coordinate $U$. Hence it differs from $\Si_1$ only in a neighbourhood of $\cB$. In the forthcoming discussion, we shall briefly review the arguments given in [@DR05] in order to show that it is really possible to deform the initial surface $\Si_1$ on which the value of $\tilde{E}_5$ is computed, preserving at the same time the decay estimates presented above as well as in . To this end we shall follow the discussion and the notation introduced in [@DR05] in order to obtain the decay estimates in the neighbourhood of $i^+$. The desired estimates towards $i^-$ could be obtained out of the time reversal symmetry. Let us start noticing that a central role in the analysis performed in [@DR05] is played by the flux generated by the Morawetz vector field $K= v^2\frac{\pa}{\pa v}+u^2\frac{\pa}{\pa u}$. Moreover, as explained in Section 9 of [@DR05], the crucial estimates, are obtained out of the divergence, [*a.k.a*]{}, Stokes-Poincaré theorem, applied to the current $J^K_\mu(\varphi)$: $$J^K_\mu(\varphi)= K^\nu T_{\mu\nu}(\varphi) + |\varphi|^2 \nabla_\mu \psi - \psi \nabla_\mu |\varphi|^2 \;, \qquad \psi = \frac{t r^*}{4 r}\at 1-\frac{2m}{r} \ct$$ which is generated by $K$ though with a modification due to total derivatives. If we follow such way of reasoning, we can compute the mentioned flux between two spacelike smooth surfaces $\Si_1$ and $\Si_2$ in $\mW$, identified respectively as the loci with fixed time coordinate $\{t=t_1\}$ and $\{t=t_2\}$, though with $t_2>t_1$. The end point is $$\hat{E}^{K}_\varphi{(t_2)}=\hat{E}^{K}_\varphi{(t_1)}+ \hat{I}^K_\varphi(\cP),$$ where $\hat{E}^{K}_\varphi{(t_2)}$ is the boundary term computed on $\Si_2$ and $\hat{I}^K_\varphi(\cP)$ is the the volume term computed in the region $\cP\doteq J^+(\Si_1) \cap J^-(\Si_2)$. Let us notice that the integrand of the boundary terms $\hat{E}^K_\varphi(t_1)$ are everywhere positive, while, as it can be seen from Proposition 10.7 of [@DR05], the one of the volume element $\hat{I}^K_\varphi(\cP)$ is negative everywhere, but in the region $\cP\cap\{ r_0<r< R \}$ where the constant $r_0$ and $R$ (with $2m < r_0 < 3m <R$) are defined in section 6 of [@DR05]. For our later purposes, since we would like to eventually deform both $\Si_1$ and $\Si_2$ in a neighbourhood of $\cB$, one should notice that the integrand is negative on such a neighbourhood if chosen in the region $r<r_0$. Since the pointwise decay estimate towards $i^+$ on $\scri$ can be obtained from $\hat{E}^K_\varphi(t)$, the problem boils down to control the bad positive volume term in $\hat{I}^K_\varphi(\cP)$. Luckily enough, the positive part of $\hat{I}^K_{\varphi}(\cP)$ can be tamed by $t_2$ times $\hat{I}^{\bf X}_\varphi( \cP)+\hat{I}^{\bf X}_{\Om \varphi}(\cP)$ where $\hat{I}^{\bf X}_\varphi(\cP)$ is the sum of the volume terms. These arise out of the divergence theorem applied to the modified current generated by vectors like $X_\ell = f_\ell(r^*)\frac{\pa}{\pa r^*}$ acting separately on an angular mode decomposition[^5]. We refer to Section 7 of [@DR05] for further details on the construction of $\hat{I}^{\bf X}_\varphi(\cP)$ and to [@DR07] for recent results that do not require a decomposition in modes. Notice that, as discussed in proposition 10.2 of [@DR05] the boundary terms $| E^{\bf X}_{\varphi} (t) |$ are always smaller then a constant $C$ times the conserved flux of energy $E_\varphi(t)$, with respect to the Killing time $\frac{\pa}{\pa t}$. Hence, if we collect all these results, it is possible to write \[EK\] \^[K]{}\_ \^[K]{}\_+ (t-t\_1) C E\_[\^]{}(t\_1)+ E\_[\^]{}(t\_1) , where $\Om$ is the square root of the angular momentum while $\varphi^\chi$ is a solution of the equation of motion coinciding with $\varphi$ on $(t_1,t)\times (r_0,R)\times \bS^2$. This vanishes in a neighbourhood of $\cB$, as the one constructed in the proof of Proposition 10.12 in [@DR05]. More precisely, for $t$ sufficiently close to $t_1$, $\varphi^\chi$ can be chosen as the solution generated by the following compactly supported Cauchy data on $\Si_{t_1}$: $\varphi^\chi(t_1,r^*)=\chi(2 r^*/t_1) \varphi(t_1,r^*)$ and $\pa_t\varphi^\chi(t_1r^*)=\chi(2 r^*/t_1)\pa_t \varphi(t_1,r^*)$, where $\chi$ is a compactly supported smooth function on $\bR$ equal to 1 on $[-1,1]$ and vanishing outside $[-1.5,1.5]$. As explained in Section 12.1 of [@DR05], it can be shown that, if $t_2=1.1 t_1$ and $t_1$ is sufficiently large, then $E_{\varphi^\chi}(t_2) \leq C\; t_2^{-2}\; \hat{E}^{K}\varphi {(t_2)}$ and this allows to obtain a better estimate then [(\[EK\])]{}, namely it yields \[sommaintervalli\] t\_2 \^[**X**]{}\_() \^[K]{}\_+ C E\_[\^]{}(t\_1)+ E\_[\^]{}(t\_1) , which is valid for $t_2=1.1 t_1$ in particular. The estimate for a generic interval $t-t_1$ can be obtained, along the lines of Section 12.1 of [@DR05], dividing $t-t_1$ in sub interval $t_{i+1}= 1.1 t_{i}$ and eventually summing the estimates over $i$. In such a way it is possible to obtain $$t \hat{I}^{\bf X}_\varphi(\cP) \leq C\hat{E}^{K}_\varphi{(t_1)}+ C \log(t) \at E_{\varphi^\chi}(t_1)+ E_{\Om \varphi^\chi}(t_1) \ct,$$ for a generic interval. As a final step, if we apply the same reasoning for $t\hat{I}^{\bf X}_{\Omega \varphi}(\cP)$ and if we use both of them to control $\hat{I}^K_\varphi$, we obtain a better estimate for $\hat{E}^K$ then the one [(\[EK\])]{}, namely \[EK2\] \^[K]{}\_=C \^[K]{}\_[(t\_1)]{}+C\^[K]{}\_[(t\_1)]{}+ C (t) E\_ [\^]{}(t\_1)+ E\_[\^]{}(t\_1) + E\_[\^]{}[(t\_1)]{}, where $t$ in [(\[EK\])]{} is substituted by $\log(t)$, the price to pay in order to consider higher angular derivatives. The $\log(t)$ can eventually be removed once again out of the same line of reasoning, using [(\[EK2\])]{} in place of [(\[EK\])]{} to improve [(\[sommaintervalli\])]{}. The end point is $$\hat{E}^{K}_\varphi{(t_2)} \leq C \at \sum_{n=0..3}\hat{E}_{\Om^n\varphi}{(t_1)} + \sum_{n=0..2}\hat{E}^{K}_{ \Om^n\varphi}{(t_1)}\ct \leq \tilde{E}_5(\Si_1).$$ We would like to stress that, since the integrand $I^K_\varphi(\cO)$ is positive whenever $\cO$ is a small neighbourhood of $\cB$, the very same results can be obtained out of a modification of the surfaces $\Si_1$ and $\Si_2$ in such a way that they are still spacelike while they intersect the horizon $\cH_{ev}$ at positive $V$ equal to $V_0$; in this new framework the form of $\tilde E_5(\Si_1')$ is left unaltered with respect to , though it is computed on a modified surface $\Si_1'$. The decay estimate towards $i^+$ on $\scri$ can eventually be obtained as in Section 13.2 of [@DR05]. At this point, out of time reversal, we can employ a similar argument as before in order to get the rapid decrease in $p$ of $\|\varphi^{f_{p}}_\scrim \|^\eta_\scrim$. The horizon case can be dealt in a similar way and, in such case, the pointwise decay on $\cH^-$ can be shown to be controlled by an integral similar to the one defining $\tilde E_5$, though here it is again computed on the modified surface. In order to establish the mentioned peeling off rate, it is, however, necessary to consider another flux, namely that generated by a vector field $Y$ which approaches $\frac{1}{1- \frac{2m}{r}}\pa_u$ on the horizon $\cH$, as described in Section 8 of [@DR05]. In this framework, even if the integrand of the volume term $\hat{I}^Y_\varphi$, associated with $Y$, is negative in a region formed by the compact interval $[\hat r_0,R]$, it can be controlled in a similar way as previously discussed for $\hat{I}^{\bf X}_\varphi$. $\qed$\ [**Proof of Lemma \[Nozero\].**]{} As a starting point, let us recall that $\Lambda_U$ is a weak-bisolution of , whose antisymmetric part is nothing but the causal propagator $ E_{P_g}$ in $\mM$. The wave-front set of $E_{P_g}$ is well-known [@Rada] and it contains only pair of non-vanishing light-like covectors, so that: $$(x,y,k_x,0)\notin WF(E_{P_g}) \;,\qquad (x,y,0,k_y)\notin WF(E_{P_g})\;.$$ Therefore, whenever $(x,y,k_x,0)\in WF(\Lambda_U)$, also $(y,x,0,k_x)$ must lie in $WF(\Lambda_U)$ and [*vice versa*]{}; otherwise the wavefront set of the antisymmetric part of $\Lambda_U$, which is nothing but $E_{P_g}$, would contain a forbidden element $(x,y,k_x,0)$. This allow us to focus only on an arbitrary, but fixed $(x,y,k_x,0)\in T^*(\mM\times \mM)\setminus\{0\}$ and we need to show that it does not lie in $WF(\Lambda_U)$. Furthermore we know, thanks to [*Part 1*]{} of the proof of theorem \[maxt\], that $\Lambda_U$ is of Hadamard form in $\mW$ and, thus, the statement of this lemma holds if $x,y \in \mW$. We shall hence focus on the case of $x\in\mM \setminus \mW$ and $y\in\mW$, the remaining ones will be treated later. In this scenario, it suffices to consider only those $k_x$ such that there are no representatives of $B(x,k_x)$ lying in $\mW$, otherwise we would be falling in the already discussed case using a propagation of singularities argument. This restriction yields, however, that a representative $(q,k_q)\in B(x,k_x)$ exists such that $q\in \cH^+\cup \cB$. Summarising, we are going to prove that $(x,y,k_x,0)$ is a direction of rapid decreasing for $\Lambda_U(f_{k_x},h)$, for some functions $f,g\in C_0^\infty(\mM; \bR)$ with $f(x)=h(y)=1$, provided that both $x\in\mM \setminus \mW$, $y\in\mW$ and a representative $(q,k_q)\in B(x,k_x)$ exists such that $q\in \cH^+\cup \cB$. As before, $f_{k_x}\doteq fe^{i\langle k_x,\cdot\rangle}$ and $\vphi^h \doteq Eh$.\ In this scenario, let us pick a partition of unit $\chi+\chi'=1:\cH\to\bR$ where $\chi\in C^\infty_0(\cH; \bR)$ and $\chi= 1$ in a neighbourhood of $q$. Hence \[dec2\] \_[U]{}(f\_[k\_x]{},h)=\_ (\^[f\_[k\_x]{}]{}\_,\^h\_)+\_(’\^[ f\_[k\_x]{}]{}\_,\^h\_)+\_(\^[f\_[k\_x]{}]{}\_,\^h\_). The second and third terms are rapidly decreasing in $k_x$ because they are respectively dominated by $C \|\chi' \varphi^{f_{k_x}}_{\cH} \|_{\cH^-} \cdot \| \varphi^h_{\cH}\|_{\cH^-}$ and $C' \|\varphi^{f_{k_x}}_{\scrim} \|_\scrim \cdot \| \varphi^h_{\scrim}\|_\scrim$, $C$ and $C'$ being positive constants, which, in turn, are rapidly decreasing in $k_x$ due to Proposition \[RapidDecay\]. The norms $\|\cdot\|_{\cH}$ and $\|\cdot\|_\scrim$ are those respectively defined in and in . Therefore, we need only to establish that $k_x$ is of rapid decreasing for $\la_{\cH} (\chi \varphi^{f_{k_x}}_{\cH},\varphi^h_{\cH})$. This can be done by the same procedure as that used at the end of the [**case A**]{} in the proof of Theorem \[maxt\] leading to (\[aggI\]), to prove the rapid decrease of $k_x \mapsto \la_{\cH} (\chi \varphi^{f_{k_x}}_{\cH},\varphi^{h_{k_y}}_{\cH})$ for a fixed $k_y$ and assuming $k_y=0$ there (that part of the proof is independent form the lemma we are proving here, while this lemma is used elsewhere therein).\ Let us now treat the case $y\in\mM \setminus \mW$ and $x\in\mW$, and let us prove that $(x,y,k_x,0)\not\in WF (\La_U)$ following procedures analogous to those exploited in [@SV00] . To this end we adopt an overall frame where a coordinate, indicated by $t$, is tangent to $X$ and the remaining three coordinates are denoted as $\overline{x}$. In this setting, the pull-back action of the one-parameter group generated by $X$ acts trivially as $(\beta_\tau f)(t,\overline{x}) = f(t-\tau,\overline{x})$. To start with, let us notice that, due to the restriction [(\[PST1\])]{}, the cases of $k_x$ spacelike or timelike can be immediately ruled out, so we are left to consider $k_x\in T^*_x(\mM\setminus\mW) \equiv \bR^4$ of null type. Hence we can exploit the splitting $k_x=({k_x}_t,\overline{k_x})$, where we have isolated the $t$-component from the three remaining ones $\overline{k_x}$.\ For $k_x$ as before, let us consider the two non-null and non-vanishing covectors $q=(0,\overline{k_x})$ and $q'=(-{k_x}_t,0)$. In view of $(x,y,q,q')\not\in WF(\La_U)$, hence, out of (c) of Proposition 2.1 of [@Verch], there exists an open neighbourhood $V'$ of $(q,q')$, as well as a function $\psi'\in C^\infty_0(\bR^4\times\bR^4; \bC)$ with $\psi'(0,0)=1$, such that for all $n\geq 1$, \[bastardissimo\] \_[k,k’V’]{} |dd’ d ’ d’ ’(x’,y’) e\^[i \^[-1]{}(k\_t+ ’)]{} e\^[i \^[-1]{}(k\_t’’ + [’]{} ’)]{} \_[U]{}(\_\_[’]{} (F\^[(p)]{}\_[(’, ),]{}))| C\_n \^n, which holds for every $ 0<\la<\la_n$, where both $C_n\geq 0$ and $\lambda_n>0$ are suitable constants. In the preceding expression we have employed the notation $x'=(\tau,\overline{x}')$, and $y'=(\tau',\overline{y}')$ where we have highlighted the $t-$component. Moreover $F^{(p)}_{(\overline{x}', \overline{y'})\lambda}(z,u)$ is defined as follows $$F^{(p)}_{(\overline{x}',\overline{y}'),\lambda}(z,u)\doteq F(x+\lambda^{-p}(z-\overline{x}'-x), y+\lambda^{-p}(u-\overline{y}'-y) )\;, \qquad F \in C^\infty_0(\mM\times \mM; \bC)\;,\qquad \widehat{F}(0,0)=1\:,$$ $ \widehat{F}$ being the standard Fourier transform. At this point we can make a clever use of the translation invariance of $\Lambda_U$ under the action of $\beta_{-\tau-\tau'}\otimes\beta_{-\tau-\tau'}$ in order to infer that $\Lambda_{U}(\beta_\tau\otimes \beta_{\tau'}(F^{(p)}_{(\overline{x}', \overline{y}'),\la}))$ is equal to $\Lambda_{U}(\beta_{-\tau'}\otimes \beta_{-\tau}(F^{(p)}_{(\overline{x}', \overline{y}'),\la}))$, Hence, from , it arises that, for all $p\geq 1$: $$\sup_{k,k'\in V} \left|\int d\tau d\tau' d \overline{x}' d\overline{y}' \; \psi'(x',y') \; e^{i \la^{-1}(k_t\tau + {\overline{k}} \overline{x}')} e^{i \la^{-1}(k_t'\tau' + {\overline{k}'} \overline{y}')} \Lambda_{U}(\beta_{-\tau'} \otimes\beta_{-\tau} (F^{(p)}_{(\overline{x}', \overline{y}),\la}))) \right| \leq C_n \la^n,$$ if $0<\lambda<\lambda_n$. The found result implies that [(\[bastardissimo\])]{} also holds if one replaces (i) $\psi'$ with $\psi(x',y')\doteq \psi((\tau',\overline{x}'),(\tau,\overline{y}'))$ and (ii) $V'$ with $V=\{ (-k'_t, \overline{k}),(-k_t, \overline{k'}) ) \in \bR^4\times \bR^4 \:|\: ((k_t, \overline{k}),(k'_t, \overline{k'})) \in V' \}$. This is an open neighbourhood of $(k_x,0)$ as one can immediately verify since $V' \ni (q,q')$, so that both $V \ni (k_x,0)$, and the map $ \bR^4\times \bR^4 \ni ((k_t, \overline{k}),(k'_t, \overline{k'})) \mapsto (-k'_t, \overline{k}),(-k_t, \overline{k'}) ) \in \bR^4 \times \bR^4$ is a homeomorphism, it being linear and bijective. If one exploit once more proposition 2.1 of [@Verch], it yields that $(x,y,k_x,0) \not \in WF(\La_U)$ as desired.\ In order to conclude the proof, we need to analyse the last possible case, namely both $x, y\in \mM \setminus \mW$. If a representative of either $B(x,k_x)$ or $B(y,k_y)$ lies in $T^*\mW$, we fall back in the previous analysis. 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--- abstract: 'The scenario of constant-roll inflation in the frame work of a non-canonical inflaton model will be studied. Both of these modifications lead to appearance of some differences in the slow-roll parameters besides the Friedmann equations resulted in a better justification of theoretical predictions comparing to the observation. Phenomenologically, by assuming a constant $\eta$, i.e. second slow roll parameter, and recalculating the related perturbation equations obviously there should appear some modification in the scalar spectral index and amplitude of scalar perturbations. It will be shown that finding an exact solution for Hubble parameter is one of the main advantages and triumphs of this approach. Also, whereas making a connection between sub-horizon and super-horizon regions has a crucial role in inflationary studies the main perturbation parameters will be obtained at the horizon crossing time. To examine the accuracy of our results we shall consider the Planck 2018 results as a confident criterion. To do so by virtue of the $r-n_s$ diagram, the acceptable ranges of the free parameters of the model will be illustrated. As a result it will be found out the second slow-roll parameter should be a positive constant and smaller than unity. By constraining the free parameters of the model, also the energy scale of inflation will be estimated that is of order $10^{-2}$. Even more, by investigating the attractor behavior of the model it will be cleared that the aforementioned properties could be appropriately satisfied.' author: - 'Abolhassan Mohammadi$^{1,2}$' - Khaled Saaidi$^1$ - 'Haidar Sheikhahmadi$^{3,4}$' title: 'Constant-roll approach to non-canonical inflation' --- Introduction ============ The first ideas about the very early [Universe]{} to cope with (at least) three fundamental problems of the standard Big Bang model dates back to 1981 and the paper proposed by Guth [@Guth], called “old inflation”. But this model faced a problem to justify the smooth exit of the end of inflation. To solve this fatal problem many models of inflation have been put forward such as new inflation [@Linde; @Albrecht], chaotic inflation [@Lindea], k-inflation [@Armendariz-Picon; @Garriga], brane inflation [@Maartens; @Golanbari], G-inflation [@Abolhasani; @Maeda; @Alexander; @Tirandari] warm inflation [@Berera; @Bereraa; @Taylor; @Hall; @Bastero; @Sayar; @Akhtari], etc. Inflation is assumed to be a phase of a very rapid accelerated expansion of the super high energy Universe. In single or multiple field scenarios of inflation usually the Universe evolved in the presence of scalar fields [@Chen:2009we; @Chen:2017ryl]. Nonetheless, there are some models in which instead of scalar field for instance the gauge fields play the main role [@Golovnev:2008cf; @Adshead:2012kp].\ Confidently, the inflationary models almost are based on the slow-roll assumption, in which the scalar field undergoes and rolls down slowly from the top of its potential to the bottom of the hill. The flatness of the potential provides a condition for having a quasi-de Sitter expansion, weak dependency on time of Hubble parameter, especially at the beginning of the inflation. Besides the aforementioned necessary parts to run inflation we need the enough smallness of slow-roll parameters to cope with the well known three problems of the hot Big Bang theory [@Lindeb]. The definition of the first slow-roll parameter is based on the time evolution of Hubble parameter divided by its square, i.e. $\epsilon=-{\dot{H} / H^2}$, and the condition $\epsilon < 1$ is required to have an accelerated expansion phase at the initiation eras ($\ddot{a}>0$). The second slow-roll parameter is defined as $\eta = {\ddot{\phi} / H\dot{\phi}}$ in which holds out the rate of the time derivative of scalar field during a Hubble time [@weinberg]. Smallness of the latter parameter states that $\dot\phi$ should vary very slowly, and therefore guarantees an appropriate behaviour of inflation.\ Whereas the single field canonical versions of inflation could not able to cover all results risen by observations one might seek a modification in inflationary models [@PhysRevLett112011302; @Emami:2013lma; @Sheikhahmadi:2019xkx]. One possible extension for instance is quasi-single field inflation or maybe multiple field inflation[@Baumann:2011nk; @Chen:2012ge; @Sefusatti:2012ye], etc. Another proposal of scalar field where the Lagrangian is a function of the scalar field $\phi$ and the kinetic term $X={1 \over 2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi $; i.e. $\mathcal{L}(\phi,X)$ is the k-essence model [@Armendariz-Picon; @Garriga; @Baumann]. In this model, the sound speed is no longer equal to the light speed, and it could be smaller which leads to a smaller tensor-to-scalar ratio to meet the observational range. One type of aforementioned Lagrngian are those with modified kinetic term of the canonical scalar field Lagrangian. In the present work, the modification to the kinetic term is restricted to the time derivative of the scalar field so that it is possible to receive the canonical one. This type of the Lagrangian has received attention to be applied for cosmological studies, for instance one can see [@1a]. In this reference, the authors studied the dynamical system of the Universe when it is filled with a barotropic fluid and a scalar field with modified kinetic term. Also, in [@2a], a non-canonical scalar field with a modified Lagrangian has bee taken as a candidate for dark energy. It has claimed that for a simple choice of the modified kinetic term, this model can be considered as a unified model of dark matter and dark energy. Besides, these models which usually are based on modification in kinetic portion have received more attention recently cause of their ability to justify the inflation behaviour. In [@3a], the authors have shown that for specific choices of the free parameters the tensor-to-scalar ratio could increase which originated from enhancement of the sound speed. It has indicated that other consequences of this model can be enumerates as the larger energy scales and higher temperatures for the reheating [@5a]. In the latter reference one can find some good clues about the intermediate inflation for non-canonical model with power-law kinetic term. Additionally the authors in [@6a] have approved that despite the canonical intermediate inflation, non-canonical model could properly satisfy the observational data in their case for chaotic inflation. The quantities of the model are derived, and the existence of an inflationary attractor is confirmed. Even more, the authors have shown that for a steep potential, the model is able to properly describe inflation. In addition, results of [@7a] have clarified that the power-law inflation in standard model of inflation leads to a tensor-to-scalar ratio which is out of range compared to observational data. However, by using a non-canonical scalar field there could be a new power-law inflation which its predictions are in consistency with Planck results. In [@8a] the scenario of warm inflation is extended to the non-canonical case, where it means the kinetic term will be modified again. The new, but still scale invariant, curvature spectrum is obtained and it is demonstrated that the tensor-to-scalar ratio is insignificant for strong regime and significant for weak regime.\ Such models have a better consistency with observational data compared to canonical scalar field models. Some interesting features of non-canonical scalar field model can be addressed as follows [@Unnikrishnan] - Steep potentials like $v(\phi) \propto \phi^{-n}$, which are known as the dark energy potential, could give a better inflaton potential in non-canonical scalar field compared to the canonical ones. - The consistency relation $r = -8n_T$ is violated in non-canonical scalar field model of inflation. - In the canonical scalar field models of inflation, the exponential potential roughly stands in an acceptable range of data. However, this type of potential in the non-canonical model of inflation could show a better agreement with data. The slow-roll features of inflation almost is provided by a potential with a flat part. It rises to this question what happens if the potential is exactly flat?. This question for the first time was considered in [@Kinney]. From the equation of motion of scalar field, it is determined that for a flat potential the second slow-roll parameters becomes $\eta=-3$, which is not smaller than unity, actually it is of order one. After that, in [@Namjoo] the authors have studied the non-Guassianity of the case and it was specified that the non-Gaussianity is not ignorable anymore and it could be of order one [@Namjoo]. The idea of having flat potential was generalized in [@Martin], where the authors assumed $\eta$ could be a constant. They found an approximate solution for the model and obtained a scalar perturbation amplitude that could even varies on superhorizon scales, and also for some choices it could be scale invariant. In [@Motohashi], where for the first time the name “constant-roll” was addressed, the same model was reconsider by using Hamilton-Jacobi formalism [@Salopek; @Liddle; @Kinneya; @Guo; @Aghamohammadi; @Saaidi; @Sheikhahmadi] and they found an exact solution for Hubble parameter which possesses the attractor behavior as well. Also, it was concluded that the power spectrum could remain scale invariant for specific choices of the constant. The scenario of constant-roll inflation in modified gravity was studied in [@Motohashi-b; @Nojiri; @Odintsov-c; @Oikonomou-a; @karam], and the generalized version of this approach, known as smooth-roll inflation could be found in [@Odintsov-a; @Odintsov-b; @Oikonomou-b].\ The interesting feature and application of non-canonical scalar field in slow-roll inflationary scenarios motivated us to investigate this model for constant-roll inflation. The perturbation equations will be reconsidered and we will find the modified amplitude of scalar perturbations, and also the correction to the scalar spectral index are determined which are of the second order of $\eta$. Comparing the result with observational data showed that for some specific values of $\eta$ one could have obtain a scale invariant perturbation on super horizon scale.\ The paper is organized as follow: In Sec.II, the general formulae of the frame work of non-canonical scalar field will be obtained and they are rewritten for specific choices of the Kinetic term. The slow-roll parameters and the differential equation for the Hubble parameter will be addressed in Sec.III, where the constant-roll approach is applied on the equations. The scalar and tensor perturbations will be discussed in Sec.IV, and the power spectrum of perturbations are derived. And finally We conclude and discuss our results in Section in Sec.V.\ Non-canonical scalar field model ================================ The action is assumed to be $$\label{action} S = {-1 \over 16\pi G} \; \int d^4x \sqrt{-g} \; R + \int d^4x \sqrt{-g} \mathcal{L}(\phi,X)$$ where $X = g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi /2$, and $\mathcal{L}(\phi,X)$ is the Lagrangian of non-canonical scalar field that in general is an arbitrary function of scalar field $\phi$ and $X$. Variation of at action with respect to the metric comes to the field equation of the model $$\label{fieldequation} R_{\mu\nu} - {1 \over 2}\; g_{\mu\nu} R = 8\pi G \left( {\partial \mathcal{L} \over \partial X}\; \partial_\mu \phi \partial_\nu \phi - g_{\mu\nu} \mathcal{L} \right)$$ and also variation of the action with respect to the field come to the following equation of motion $$\label{EoM} {\partial \mathcal{L} \over \partial \phi} - {1 \over \sqrt{-g}}\; \partial_\mu\left( \sqrt{-g} \; {\partial \mathcal{L} \over \partial \big( \partial_\mu \phi \big)} \right) = 0.$$ It is assumed that the geometry of the universe is describe by a spatially flat FLRW metric $$ds^2 = dt^2-a^{2}(t)\; \left(dx^2 + dy^2 + dz^2 \right)$$ Since the term in parenthesis on the right hand of the field equation (\[action\]) is the energy-momentum tensor of scalar field, in a comparison to the energy-momentum of a perfect fluid $T_{\mu\nu} = (\rho + p) u_\mu u_\nu - p g_{\mu\nu}$ it is concluded that the energy density, pressure and four velocity of the field are $$\label{energypressure} \rho = 2X {\partial \mathcal{L} \over \partial X} - \mathcal{L}, \qquad p = \mathcal{L}, \qquad u_\mu = {\partial_\mu \phi \over \sqrt{2X}}.$$ Substituting this metric into the field equation (\[fieldequation\]), one arrives at the Friedmann equations $$\label{friedmann} H^2 = {8\pi G \over 3} \; \rho, \qquad \dot{H} = -4\pi G \Big( \rho + p \Big).$$ Also, the equation of motion (\[EoM\]) for this geometry is read as [@Unnikrishnan] $$\label{eoms} \left( {\partial \mathcal{L} \over \partial X} + 2X {\partial^2 \mathcal{L} \over \partial X^2} \right) \ddot{\phi} + \left( 3H {\partial \mathcal{L} \over \partial X} + \dot{\phi}\; {\partial \mathcal{L} \over \partial X \partial \phi} \right) \dot{\phi} - {\partial \mathcal{L} \over \partial \phi} = 0$$ In the present work, we are going to work with a specific type of k-essence Lagrangian which contains a modification in the kinetic term. We notice this model comes to interesting results working out inflation besides the studies of dynamical system of the Universe and dark energy. The Lagrangian is assumed to be $$\label{Lagrangian} \mathcal{L} = X\; \left( {X \over M^4} \right)^{\alpha-1} - V(\phi),$$ where $\alpha$ is a dimensionless constant and $M$ is a constant with mass dimension. Using this definition for scalar field Lagrangian, its energy density and pressure are expressed by $$\label{rhopressure} \rho = (2\alpha - 1) X \left( {X \over M^4} \right)^{\alpha-1} + V(\phi), \qquad p = X\; \left( {X \over M^4} \right)^{\alpha-1} - V(\phi).$$ Then, the equation of motion (\[eoms\]) is reduced to $$\label{eomphi} \ddot{\phi} + {3H \over 2\alpha-1} \; \dot\phi + \left( {2 M^4 \over \dot\phi^2} \right)^{\alpha-1} {V_\phi(\phi) \over \alpha(2\alpha-1)} =0.$$ Introducing the Hubble parameters as a function of scalar field, and using Eq.(\[friedmann\]), (\[rhopressure\]) and relation $\dot{H}=\dot{\phi} H_\phi(\phi)$, the time derivatives of the scalar field is given by $$\dot\phi^{2\alpha-1} = {-2^\alpha M_p^{2(2\alpha-1)} \mu^{4(\alpha-1)} \over \alpha}\; H_{,\phi}(\phi),$$ in which $M_p^2 = 1/8\pi G$ and $\mu \equiv M/M_p$. Then substituting this in the Friedmann equation (\[friedmann\]), and using Eq.(\[rhopressure\]) the potential of scalar field is read as $$\label{potential} V(\phi) = 3M_p^2 H^2(\phi) - {(2\alpha - 1)M_p^{4\alpha} \over 2^\alpha M^{4(\alpha-1)}} \left[ {-2^\alpha \mu^{4(\alpha-1)} \over \alpha}\; H_{,\phi}(\phi) \right]^{2\alpha \over 2\alpha-1}$$ For the rest of the paper, the reduced mass planck is taken as $M_p=1$ for more convenience, and also we defined the new parameter $\lambda = M^{4(\alpha-1)}$. Non-canonical inflation ======================= During the inflation, the universe undergoes an extreme expansion in short period of time. Here, it is assumed that the inflation is caused by a non-canonical scalar field. Usually, inflation is describe by using the slow-roll parameters. The first slow-roll parameter indicates the rate of the Hubble parameter during a Hubble time as [@weinberg] $$\label{sr01} \epsilon=- {\dot{H} \over H^2},$$ so that to have a quasi-de Sitter expansion this parameter should be much smaller than unity [@weinberg]. Using this equation and Eq.(\[friedmann\]), one could obtain the time derivative of the scalar field in terms of the Hubble parameter and the slow-roll parameters $\epsilon$ as [@Unnikrishnan] $$\label{phidot} \dot\phi^{2\alpha} = {2^\alpha \lambda \over \alpha} \; \epsilon \; H^2.$$ Assuming the Hubble parameter as a function of scalar field, $H:=H(\phi)$, one has $\dot{H}=\dot\phi H_{,\phi}$. Then, the slow-roll parameter $\epsilon$ is read as $$\label{epsilon} \epsilon = \left( {2^\alpha \lambda \over \alpha} \right)^{1 \over 2\alpha - 1} \; {H_{,\phi}^{2\alpha \over 2\alpha - 1} \over H^2}$$ The second slow-roll parameter is defined as $\eta= \ddot{\phi} / \dot{\phi}H$, where in constant-roll inflation it is assumed as a constant $\eta = \beta$. Using this definition and also Eq.(\[phidot\]) and (\[epsilon\]), we arrive at the following differential equation for the Hubble parameter $$\label{hubblede} H_{,\phi}^{2-2\alpha \over 2\alpha - 1} H_{,\phi\phi} = C_0 \; H, \qquad C_0 \equiv (2\alpha-1) \; \beta \left( {\alpha \over 2^\alpha \lambda } \right)^{1 \over 2\alpha - 1},$$ which comes to the same differential equation for the Hubble parameter as in [@Motohashi], where the authors consider the constant-roll inflation for the canonical scalar field. Since $H_{,\phi} = dH/d\phi$ there is $d\phi=dH / H_{,\phi}$. Therefore, for the second derivative of the Hubble parameter in terms of the scalar field we have $H_{,\phi\phi}= dH_{,\phi} / d\phi = H_{,\phi}dH_{,\phi} / dH$. Substituting this into the above differential equation, one arrives at $$\label{Hphi} H_{,\phi}^{2\alpha \over 2\alpha -1} = {\alpha C_0 \over 2\alpha -1 } \; H^2 + C_1;$$ where $C_1$ is the constant of integration. Taking another integration from Eq.(\[Hphi\]), the scalar field could be expressed in terms of the Hubble parameter as $$\label{phiH} \phi = \phi_0 + {H \over C_1} \; {}_2F_1\Big[{1 \over 2},1-{1 \over 2\alpha},{3 \over 2},{C_0 \alpha \over C_1(1-2\alpha)}\;H^2 \Big],$$ in which $\phi_0$ is constant of integration too. From Eqs.(\[Hphi\]) and (\[phiH\]), it seems that every quantity could be expressed in terms of the Hubble parameter, such as the slow-roll parameter $$\label{epsilonH} \epsilon = \left( {2^\alpha \lambda \over \alpha} \right)^{1 \over 2\alpha - 1} \; { \left( {\alpha C_0 \over 2\alpha -1 } \; H^2 + C_1 \right) \over H^2 },$$ and also for the number of e-folds there is $$\begin{aligned} \label{efold} N & = & \int_{t_i}^{t_e} H dt = \int_{H_i}^{H_e} {H \over \dot{H}} dH = \int_{H_i}^{H_e} {-1 \over \epsilon \; H} \; dH \nonumber \\ & = & -\left( \alpha \over 2^\alpha \lambda \right)^{1 \over 2\alpha-1} \int_{H_i}^{H_e} {H \; dH \over {\alpha C_0 \over 2\alpha -1} \; H^2 + C_1}\end{aligned}$$ and by taking integrate one arrives at $$\label{efoldN} N = -\left( \alpha \over 2^\alpha \lambda \right)^{1 \over 2\alpha-1} \; {2\alpha-1 \over 2\alpha C_0} \; \ln\left[ {\alpha C_0 \over 2\alpha -1} \; H^2 + C_1 \right]\Bigg|_{H_i}^{H_e}$$ where $H_e$ is the Hubble parameter at the end of inflation, and $H_i$ is the Hubble parameter at the horizon exit time. From Eq.(\[potential\]) the potential of the scalar field could be derived only as a function of the Hubble parameter too $$\label{potH} V = 3 M_p^2 H^2 \left[1 - {(2\alpha-1) \over 3\alpha} \; \left( {2^\alpha \lambda \over \alpha} \right)^{1 \over 2\alpha - 1} { \left( {\alpha C_0 \over 2\alpha -1 } \; H^2 + C_1 \right) \over H^2 } \right]$$ Perturbations of non-canonical scalar field =========================================== Assume a small inhomogeneity of scalar field as $\phi(t,\mathbf{x}) = \phi_0(t) + \delta\phi(t,\mathbf{x})$. This perturbation with respect to the background $\phi_0(t)$ (which from now on we omit the subscript “0”) induce a perturbation to the metric because from the field equation it is clear that geometry and matter are tightly coupled together. The metric in longitudinal gauge is written as $$ds^2 = \big(1 + 2\Phi(t,\mathbf{x}) \big) dt^2 - a^2(t) \big(1 - 2\Psi(t,\mathbf{x})\big)\delta_{ij} dx^i dx^j$$ where by assuming a diagonal tensor for the spatial part of energy-momentum tensor (i.e. $\delta T^i_j \propto \delta^i_j$) there is $\Psi(t,\mathbf{x})=\Phi(t,\mathbf{x})$.\ The action for linear scalar perturbation is derived as [@Garriga; @Mukhanov] $$\label{vaction} S = {1 \over 2} \int \left( v'^2 + c_s^2 v (\nabla v)^2 + {z'' \over z} v \right) d\tau d^3\mathbf{x}.$$ in which $v \equiv z \zeta$, and $\zeta$ is the curvature perturbation given by $\zeta = \Phi + H \; {\delta\phi \over \dot\phi}$, and prime denotes derivative with respect to the conformal time $\tau$, $ad\tau=dt$. Also, $c_s$ is the sound speed of the model which is stated as $$\label{cs} c_s = {1 \over \sqrt{2\alpha -1}}$$ which is a constant. To get a physical sound speed, the constant $\alpha$ should be always positive and bigger than $0.5$, and also to not exceed the speed of light it should be bigger than one.\ The quantity $z$ is known as the Mukhanov variable that for our model it is defined as [@Garriga; @Mukhanov] $$z^2 \equiv {2\alpha a X \over c_s^2 H^2 } \; \left( {X \over M^4} \right)^{\alpha-1}.$$ From Eq.(\[vaction\]), the equation for perturbation quantity $v$ becomes $$\label{vequation} {d^2 \over d\tau^2} v(\tau,\mathbf{x}) - c_s^2 \nabla^2 v(\tau,\mathbf{x}) - {z'' \over z} \; v(\tau,\mathbf{x})=0.$$ and by utilizing the Fourier mode, one arrives at $$\label{vkequation} {d^2 \over d\tau^2} v_k(\tau) +\left( c_s^2 k^2 - {1 \over z}\;{d^2z \over d\tau^2} \right) \; v_k(\tau)=0.$$ Before discussing the solution of the above equation, we try to compute the term $z''/z$. From the definition of $z$ and the slow-roll parameters of the previous section, there is $$z \equiv \sqrt{2\alpha \over 2^\alpha \lambda} \; {a \dot\phi^2 \over c_s H}$$ $${dz \over d\tau} = z \big( aH \big) \Big[ 1 + \alpha \eta + \epsilon \Big]$$ To calculate the second derivative, we first need to obtain the derivative of the slow-roll parameters with respect to the conformal time $${d\epsilon \over d\tau} = \big( aH \big) \Big[ 2\alpha \eta\epsilon + \epsilon^2 \Big],$$ Then, the second derivative of the quantity $z$ with respect to the conformal time is read by $${1 \over z}{d^2z \over d\tau^2} = \big( aH \big)^2 \Big[ 2 + 2\epsilon + 3\eta + (\alpha+2\alpha\theta) \eta \epsilon + \alpha^2 \eta^2 \Big].$$ Here $\theta = \pm 1$, and appears through the time derivative of the scalar field (From Eqs.(\[phidot\]) and (\[epsilon\]), $\dot\phi^2$ are derived in terms of the scalar field. On the other hand, time derivative of $\epsilon$ could be expressed as $\dot{\epsilon} = \dot\phi \epsilon_{,\phi}$. Then, we need $\dot\phi$ which in general is $\dot\phi = \theta \; \sqrt{\mathcal{O}}$. )\ At subhorizon scale, where $c_sk \gg aH$ (i.e. $c_sk \gg z''/z$), the quantity $v_k$ is obtained as $$\label{subv} {d^2 \over d\tau^2} v_k(\tau) + c_s^2 k^2 v_k(\tau)=0, \quad \Rightarrow \quad v_k(\tau)={1 \over \sqrt{2c_s k}} e^{ic_s k \tau}.$$ In order to find the general solution of equation (\[vkequation\]), we make the variable changes $v_k=\sqrt{-\tau} f_k$ and $x\equiv -c_sk\tau$. After some manipulation, the find the following Bessel’s differential equation $$\label{bessel} {d^2 f_k \over dx^2} + {1 \over x}{d f_k \over dx} + \Big( 1 - {\nu^2 \over x^2} \Big)f_k=0, \qquad {z'' \over z} = {\nu^2 - {1 \over 4} \over \tau^2 }$$ where we use $a^2H^2 = (1+\epsilon)^2 / \tau^2$, and keeping only first order of the slow-roll parameter $\epsilon$, the parameter $\nu$ is acquired as $$\label{nu} \nu^2 = {9 \over 4} + 6\epsilon + 3\alpha \; \eta + (7 \alpha + 2\alpha\theta) \; \epsilon \; \eta + (1+2\epsilon)\; \alpha^2 \eta^2.$$ The general solution for the above differential equation is the first and second kind of Henkel function. However to have a constancy with the solution in subhorizon scale, the second kind of Henkel function should be ignored, therefore the final solution is acquired as $$\label{vksolution} v_k(\tau) = {\sqrt{\pi} \over 2}\; e^{i{\pi \over 2}(\nu + {1 \over 2})} \sqrt{-\tau} \; H_\nu^{(1)}(-c_sk\tau).$$ The spectrum of curvature perturbation is defined as $$\mathcal{P}_s = {k^3 \over 2\pi^2}\; \left| \zeta \right|^2 = {k^3 \over 2\pi^2}\; \left| {v_k \over z} \right|^2.$$ On superhorizon scale, the asymptotic behavior of Henkel function is $$\lim_{-k\tau \rightarrow \infty}H_\nu^{(1,2)}(x) = \sqrt{{2 \over \pi}} \; {1 \over \sqrt{2c_s k}} e^{\mp (ic_s k\tau + \delta)}, \qquad \delta={1 \over 2} \big( \nu + {1 \over 2} \big).$$ Then, the spectrum of curvature perturbation on superhorizon scale will be $$\label{psspectrum} \mathcal{P}_s = \left( 2^{\nu-{3 \over 2}} \Gamma(\nu) \over \Gamma(3/2) \right)^2 \; \left( H^2 \over 2\pi \sqrt{c_s(\rho+p)} \right)^2 \; \left( c_s k \over a H \right)^{3-2\nu}$$ Consistency with observation ============================ An advantage of the inflationary scenario is the prediction of the quantum perturbations including scalar, vector and tensor ones. Amongst them the scalar perturbations are considered as the primary seeds for large structure formation and the tensor perturbation is interpreted as the source for the gravitational waves. During inflation these perturbations are stretched out the horizon and remain invariant. On the other hand the observational data almost indicates an scale invariant spectrum for curvature perturbations. This scale invariant feature is described by scalar spectral index, so that it is defined as $\mathcal{P}_s = A_s^2 \left( c_s k \over a H \right)^{n_s-1}$, where $A_s^2$ is the amplitude of scalar perturbation at horizon crossing $c_sk = aH$. For $n_s=1$ the amplitude of scalar perturbation is exactly scale invariant, however the latest observational data implies that $\ln\left( A_s \times 10^{10} \right) = 3.044 \pm 0.014 $ and $n_s=0.9649 \pm 0.0042$ expressing an almost scale invariant perturbations [@planck]. To measure the tensor perturbations the usual procedure is indirectly through the parameter $r$ known as the tensor-to-scalar ratio, $r=\mathcal{P}_t/\mathcal{P}_s$. The data originated by Planck-2018 show there is only an upper bound on this parameter, $r < 0.064$, and it is still not measured exactly [@planck].\ Consistency of the presented model with observational data is the main goal of this section. At the end of inflation the first slow-roll parameter reaches unity[^1] in which it happens for $H=H_e$ $$\label{epsilon=1} \epsilon(H_e)= \left( {2^\alpha \lambda \over \alpha} \right)^{1 \over 2\alpha - 1} \; { \left( {\alpha C_0 \over 2\alpha -1 } \; H^2 + C_1 \right) \over H^2 }=1,$$ which leads to the following value for the Hubble parameter at the end of inflation $$\label{He} H_e^2 = \left( {2^\alpha \lambda \over \alpha} \right)^{1 \over 2\alpha-1} \; {-C_1 \over \alpha\eta - 1}.$$ To guarantee the positiveness of $H_e^2$, the term $-C_1 \lambda / (\alpha\eta -1)$ should always be positive. At the horizon exit the slow-roll parameter $\epsilon$ is smaller than unity, and the corresponding Hubble parameter could be determine through the number of e-fold equation (\[efold\]) $$\label{hini} H^2_\star(N) = \left( {2^\alpha \lambda \over \alpha} \right)^{1 \over 2\alpha-1} \; {-C_1 \over \alpha\eta - 1} \; \left( {e^{2\alpha \eta N} + \alpha \eta - 1 \over \alpha \eta} \right)$$ which is expressed in term of the number of e-fold parameter $N$. The right hand side should always be positive, so beside above restriction, the term on the parenthesis should be always positive. Note that, in order to overcome the horizon and flatness problems we need about $60-70$ number of e-fold [@Baumann].\ From Eqs.(\[friedmann\]), (\[energypressure\]) and (\[phidot\]) that the time derivatives of the Hubble parameter is negative, so by passing time and approaching to the end of inflation, the Hubble parameter decreases. On the other hand, since the $\epsilon=1$ is taken as the end of inflation, it is expected that the first slow-roll parameter to be smaller than one for bigger values of the Hubble parameter, namely the horizon crossing occurs for bigger values of the Hubble parameter. This point is illustrated in Fig.\[epsilonH\], where by passing time and decreasing the Hubble parameter the parameter $\epsilon$ approaches one, indicating the end of inflation (the selected values for the constant in the figure is based on the results that will be determined in next lines).\ ![The figures shows the behavior of the first slow-roll parameter $\epsilon$ in terms of the Hubble parameter for different values of $\alpha$ and $\eta$.[]{data-label="epsilonH"}](epsilonH.eps){width="10cm"} Using Eq.(\[hini\]), the main perturbations parameters such as the scalar spectral index, amplitude of scalar perturbation, and tensor-to-scalar ratio could be obtained in terms of $N$ as $$\begin{aligned} % \nonumber % Remove numbering (before each equation) \epsilon(N) &=& {\alpha \beta e^{2\alpha \beta N} \over e^{2\alpha \beta N} + \alpha \beta - 1}, \\ n_s(N) &=& 4 - 2\nu(N) \\ \nu^2(N) &=& {9 \over 4} + 3 \alpha \beta + \alpha^2 \beta^2 + \Big( 6 + 7 \alpha + 2\alpha \theta + 2 \alpha^2 \beta^2\Big) \; \epsilon(N) \\ \mathcal{P}_s(N) &=& {1 \over 8\pi^2} \; \left( 2^{\nu(N)-{3 \over 2}} \Gamma\big(\nu(N)\big) \over \Gamma(3/2) \right)^2 \; {H^2(N) \over c_s \epsilon(N) } \\ r(N) &=& 16 \; \left( \Gamma(3/2) \over 2^{\nu(N)-{3 \over 2}} \Gamma\big(\nu(N)\big) \right)^2 c_s \epsilon(N)\end{aligned}$$ It is clear that the scalar spectral index and the tensor-to-scalar ratio only depend on the constant $\alpha$ and $\eta$. Using the Planck $r-n_s$ diagram, we could obtained a range of these constants that put the model predictions about $n_s$ and $r$ in the range of data. Fig.\[aetaplot\] shows this area in which the light blue area illustrated the range related to the $95\%$ CL and the dark blue color determines values of $\alpha$ and $\eta$ that put $n_s$ and $r$ in $68\%$ CL. It shows that the second slow-roll parameter should be positive and there is no consistent result for negative $\eta$. The second slow-roll parameter could be of order $10^{-3}$, but the best results occurs for smaller values. By increasing the constant $\eta$ the range of $\alpha$ deceases.\ The other constants $M$ and $C_1$ appear in the amplitude of curvature perturbation and also in the potential of the scalar field through the Hubble parameter. Imposing the observational data for the amplitude of curvature perturbations determines that there should be $$\label{MC1} {-C_1 M^{4(\alpha-1) \over 2\alpha-1} \over \alpha \beta -1} = \left( \alpha \over 2^\alpha \right)^{1 \over 2\alpha -1} \; {\mathcal{P}_s^\star \over 8\pi^2 c_s \epsilon^\star} \; \left( \Gamma(3/2) \over 2^{\nu^\star - {3 \over 2}} \Gamma(\nu^\star) \right)^2 \; {\alpha \eta \over e^{2\alpha \eta N} + \alpha \eta -1}$$ The amplitude of the curvature perturbation is clear from data, so by choosing some values for the constants $\alpha$ and $\eta$ from Fig.\[aetaplot\], the right hand side would be clear. Table.\[MC1table\] expresses this terms for different choices of $\alpha$ and $\eta$.\ Also, the term on the left hand side of Eq.(\[MC1\]) appears in the potential of the scalar field, in which at the time of horizon crossing, one has $$\label{MC1Vstar} V^\star = 3 \left( 2^\alpha \over \alpha \right)^{1 \over 2\alpha-1} \; {-C_1 M^{4(\alpha-1) \over 2\alpha-1} \over \alpha \beta -1} \; {e^{2\alpha \eta N} \over \epsilon^\star} \; \left[ 1 - {(2\alpha-1) \over 3\alpha} \; \epsilon^\star \right]$$ Then, the determined values of $\alpha$ and $\eta$, the inflation energy scale is determined, that has been implied in Table.\[MC1table\]. It is clearly seen that the energy scale of inflation is about $10^{-2}$. $\alpha$ $\quad \eta$ $\ \ -C_1 M^{4(\alpha-1) \over 2\alpha-1}$ $V^\star$ ---------- -------------- -------------------------------------------- ----------------------- $3$ $0.0002$ $-4.56 \times 10^{-11}$ $2.26 \times 10^{-8}$ $3$ $0.001$ $-3.30 \times 10^{-11}$ $1.92 \times 10^{-8}$ $5$ $0.0003$ $-5.73 \times 10^{-11}$ $2.94 \times 10^{-8}$ $5$ $0.0005$ $-4.69 \times 10^{-11}$ $2.66 \times 10^{-8}$ $7$ $0.0002$ $-6.73 \times 10^{-11}$ $3.53 \times 10^{-8}$ $7$ $0.0004$ $-5.58 \times 10^{-11}$ $3.21 \times 10^{-8}$ : Determining the constant term $C_1 M^{4(\alpha-1) \over 2\alpha-1}$ by using Fig.\[aetaplot\], and data for the curvature perturbations. Then, the energy scale of inflation is illustrated foe these values of the constants. The result have been derived for number of e-fold $N=65$.[]{data-label="MC1table"} Attractor Behavior ================== Considering the attractor behavior of the solution will be performed following the same process as mentioned in [@lyth; @Odintsov-a; @Odintsov-b] which is a simple approach and also effective as one works in Hamilton-Jacobi formalism. In this formalism it is assumed that there is a homogenous perturbation to the Hubble parameter as $H=H_0+\delta H$. Now by introducing it into the Hamilton-Jacobi equation (\[potential\]), and keeping the terms up to the first order of $\delta H$, one arrives at $${\delta H ' \over \delta H} = 3 \left( {2^\alpha \lambda \over \alpha} \right) \; {H_0 \over \dot\phi_0^{2\alpha}} \; H'_0,$$ By taking integrate from the above equation, we have $$\label{attractor-eq} \delta H = \delta H_i \exp\left( - 3 \int_{H_i}^{H} {H \over \dot{H}} \; dH \right).$$ where $\delta H_i$ is the perturbation at initial time. From above equation, it is realized that by approaching to the end of inflation the term within the power of the exponential term grows, and the negative sign brings out this fact that the perturbation $\delta H$ decreases continuously. This feature is determined in Fig.\[attractor\], where the integral has been plotted for different values of $\alpha$, $\beta$ and $M$ and for all cases the term deceases during the inflation. Therefore, it is resulted that the model satisfies the attractor behavior. ![The power of the exponential term in Eq.(\[attractor-eq\]) during the inflation. By passing time and approaching to the end of inflation, the power increases with negative sign. Then, the perturbation $\delta H$ exponentially decreases during inflation and the solution is attractive. []{data-label="attractor"}](attractor.eps){width="10cm"} Conclusion ========== This work was relied on a scenario of constant-roll evolution of the initial Universe. It has been supposed that inflation has derived considering a modification in the kinetic term of the Lagrangian called the non-canonical model of the scalar field. In the constant-roll scenario it has assumed that, the second slow-roll parameter, i.e. $\eta$, has to be a constant and therefore we had to recalculate necessary inflationary parameters. By calculating the slow-roll parameters in term of the Hubble parameter we could derive a differential equation for the Hubble parameter which could lead to the corresponding differential equation for canonical scalar field model by taking $\alpha=1$. On the other side, by taking into account the assumption of constant-roll method the perturbation equation should be recalculate again. Doing so obviously there are some modifications in the amplitude of scalar perturbation and scalar spectral index lead to the appearance of the second order of $\eta$ in our investigation.\ By finding an exact solution, which is one of the main advantages and triumphs of this approach, it was realized that every parameter of the model can be expressed in terms of the Hubble parameter. Therefore, comparing our approach with the usual Hamilton-Jacobi formalism of inflation the Hubble parameter here, behaves as same as that scalar field in that approach. The importance of this result goes back to this fact that there is no need to introducing an ansatz for Hubble parameter based on the scalar field. The condition $\epsilon = 1$ clarified the Hubble parameter at the end of inflation, i.e. $H_e$, and using the number of e-fold the Hubble parameter has determined in terms of the number of e-fold and other constant parameters of the model. Utilizing this result, we have expressed the main perturbation parameters of the model in terms of the number of e-fold to obtain better estimations comparing to observationas.\ It has been shown that the scalar spectral index and the tensor-to-scalar ratio at the horizon exit can be obtained in terms of two free parameters of the model namely $\alpha$ and $\eta$. Then, by making use of the $r-n_s$ diagram originated from the Planck 2018, the proper ranges based on these two parameters have been estimated and depicted in Fig.\[aetaplot\]. It has been noticed for any values of $\alpha$ and $\eta$ in the best estimated range both the scalar spectral index and tensor-to-scalar ration are in good agreement with observational data. The results show that the second slow-roll parameter $\eta$ should be constant, as it should, and is of order $10^{-4}$, and by increasing the constant $\alpha$ the range of $\eta$ becomes smaller. On the other hand, applying the data for the amplitude of curvature perturbation, the constant $C_1 M^{4(\alpha-1) \over 2\alpha-1}$ has determined, and imposing this result in the potential of the scalar field it has found out that the energy scale of inflation is around $V^{\star {1/4}} \propto 10^{-2}$.\ Ultimately, the attractor behavior of the model by assuming a homogenous perturbation to the Hubble parameter has been investigated. By introducing into the Hamilto-Jacobi equation, it leads to a differential equation of the homogenous perturbation which from Fig.\[attractor\] it was obvious that the perturbation decreases exponentially by approaching to the end of inflation, that indicates the attractor behavior of solution. AM would like to thanks the “Ministry of Science, Research and Technology” of Iran for financial support, and Prof. R. Casadio for hospitality during the visit. HS thanks A. 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--- abstract: 'This paper studies how to verify the conformity of a program with its specification and proposes a novel constraint-programming framework for bounded program verification (CPBPV). The CPBPV framework uses constraint stores to represent the specification and the program and explores execution paths nondeterministically. The input program is partially correct if each constraint store so produced implies the post-condition. CPBPV does not explore spurious execution paths as it incrementally prunes execution paths early by detecting that the constraint store is not consistent. CPBPV uses the rich language of constraint programming to express the constraint store. Finally, CPBPV is parametrized with a list of solvers which are tried in sequence, starting with the least expensive and less general. Experimental results often produce orders of magnitude improvements over earlier approaches, running times being often independent of the variable domains. Moreover, CPBPV was able to detect subtle errors in some programs while other frameworks based on model checking have failed.' author: - 'Hélène Collavizza, Michel Rueher, Pascal Van Hentenryck' title: | CPBPV: A Constraint-Programming Framework\ For Bounded Program Verification --- Introduction ============ This paper is concerned with software correctness, a critical issue in software engineering. It proposes a novel constraint-programming framework for bounded program verification (CPBPV), i.e., when the program inputs (e.g., the array lengths and the variable values) are bounded. The goal is to verify the conformity of a program with its specification, that is to demonstrate that the specification is a consequence of the program. The key idea of CPBPV is to use constraint stores to represent the specification and the program, and to non-deterministically explore execution paths over these constraint stores. This non-deterministic constraint-based symbolic execution incrementally refines the constraint store, which initially consists of the precondition. Non-determinism occurs when executing conditional or iterative instructions and the non-deterministic execution refines the constraint store by adding constraints coming from conditions and from assignments. The input program is partially correct if each constraint store produced by the symbolic execution implies the post-condition. It is important to emphasize that CPBPV considers programs with complete specifications and that verifying the conformity between a program and its specification requires to check (explicitly or implicitly) all executables paths. This is not the case in model-checking tools designed to detect violations of some specific property, e.g., safety or liveness properties. The CPBPV framework has a number of fundamental benefits. First, contrary to earlier work using constraint programming or SMT [@ABM07; @CoR06; @CoR07], CPBPV does not use predicate abstraction or explore spurious execution paths, i.e., paths that do not correspond to actual executions over inputs satisfying the pre-condition. CPBPV incrementally prunes execution paths early by detecting that the constraint store is not consistent. Second, CPBPV uses the rich language of constraint programming to express the constraint store, including arbitrary logical and threshold combination of constraints, the [*element*]{} constraint, and global/combinatorial constraints that express complex relationships on a set of variables. Finally, CPBPV is parametrized with a list of solvers which are tried in sequence, starting with the least expensive and less general. The CPBPV framework was evaluated experimentally on a series of benchmarks from program verification. Experimental results of our (slow) prototype often produce orders of magnitude improvements over earlier approaches, and indicate that the running times are often independent of the variable domains. Moreover, CPBPV was able to found subtle errors in some programs that some other verification frameworks based on model-checking could not detect. The rest of the paper is organized as follows. Section \[motivation\] illustrates how CPBPV handles constraints store on a motivating example. Section \[formalization\] formalizes the CPBPV framework for a small programming language and Section \[implementation\] discusses the implementation issues. Section \[experimental\] presents experimental results on a number of verification problems, comparing our approach with state of the art model-checking based verification frameworks. Section \[related\] discusses related work in test generation, bounded program verification and software model checking. Section \[conclusion\] summarizes the contributions and presents future research directions. The Constraint-Programming Framework at Work {#motivation} ============================================ This section illustrates the CPBPV verifier on a motivating example, the binary search program. CPBPV uses Java programs and JML specifications for the pre- and post-conditions, appropriately enhanced to support the expressivity of constraint programming. Figure \[BsearchFig\] depicts a binary search program to determine if a value $v$ is present in a sorted array $t$. (Note that $\backslash$[result]{} in JML corresponds to the value returned by the program). To verify this program, our prototype implementation requires a bound on the length of array $t$, on its elements, and on $v$. We will verify its correctness for specific lengths and simply assume that the values are signed integers on a number of bits. /*@ requires (\forall int i; i>=0 && i<t.length-1;t[i]<=t[i+1]) @ ensures @ (\result != -1 ==> t[\result] == v) && @ (\result == -1 ==> \forall int k; 0 <= k < t.length ; t[k] != v) @*/ 1 static int binary_search(int[] t, int v) { 2 int l = 0; 3 int u = t.length-1; 4 while (l <= u) { 5 int m = (l + u) / 2; 6 if (t[m]==v) 7 return m; 8 if (t[m] > v) 9 u = m - 1; 10 else 11 l = m + 1; } // ERROR else u = m - 1; 12 return -1; } The initial constraint store of the CPBPV verifier, assuming an input array of length 8, is the precondition[^1] $c_{pre} \equiv \forall 0 \leq i < 7: t^0[i] \leq t^0[i+1]$ where $t^0$ is an array of constraint variables capturing the input. The constraint variables are annotated with a version number as CPBPV performs a SSA-like renaming [@CFR91] on the fly since each assignment generates constraints possibly linking the old and the new values of the assigned variable. The assignments in lines 2–3 add the constraints $l^0 = 0 \wedge u^0 = 7$. CPBPV then considers the loop instruction. Since $l^0 \leq u^0$, it enters the loop body, adds the constraint $m^0 = (l^0 + u^0)/2$, which simplifies to $m^0 = 3$, and considers the conditional statement on line 6. The execution of the statement is nondeterministic: Indeed, both $t^0[3] = v^0$ and $t^0[3] \neq v^0$ are consistent with the constraint store, so that the two alternatives, which give rise to two execution paths, must be explored. Note that these two alternatives correspond to actual execution paths in which $t[3]$ in the input is equal to, or different from, input $v$. The first alternative adds the constraint $t^0[3] = v^0$ to the store and executes line 7 which adds the constraint $result = m^0$. CPBPV has thus obtained an execution path $p$ whose final constraint store $c_p$ is: $ c_{pre} \; \wedge \; l^0 = 0 \wedge u^0 = 7 \; \wedge \; m^0 = (l^0 + u^0)/2 \; \wedge \; t^0[m^0] = v^0 \; \wedge \; result = m^0 $\ CPBPV then checks whether this store $c_p$ implies the post-condition $c_{post}$ by searching for a solution to $c_p \; \wedge \; \neg c_{post}$. This test fails, indicating that the computation path $p$, which captures the set of actual executions in which $t[3] = v$, satisfies the specification. CPBPV then explores the other alternatives to the conditional statement in line 6. It adds the constraint $t^0[m^0] \neq v^0$ and executes the conditional statement in line 8. Once again, this statement is nondeterministic. Its first alternative assumes that the test holds, generating the constraint $t^0[m^0] > v^0$ and executing the instruction in line 9. Since $u$ is (re-)assigned, CPBPV creates a new variable $u^1$ and posts the constraint $u^1 = m^0 - 1 =2$. The execution returns to line 4, where the test now reads $l^0 \leq u^1$, since CPBPV always uses the most recent version for each variable. Since the constraint stores entails $l^0 \leq u^1$, the only extension to the current path consists of executing line 5, adding the constraint $m^1 = (l^0 + u^1)/2$, which actually simplifies to $m^1 = 1$. Another complete execution path is then obtained by executing lines 6 and 7. Consider now a version of the program in which line 11 is replaced by [u = m-1]{}. To illustrate the CPBPV verifier, we specify partial execution paths by indicating which alternative is selected for each nondeterministic instruction. For instance, $\langle T_4,F_6,T_8,T_5,T_6\rangle$ denotes the last execution path discussed above in which the true alternative is selected for the first execution of the instruction in line 4, the false alternative for the first execution of instruction 6, the true alternative for the first instruction of instruction 8, the true alternative of the second execution of instruction 5, and the true alternative of the second execution of instruction 6. Consider the partial path $\langle T_4,F_6,F_8 \rangle$ and let us study how it can be extended. The partial path $\langle T_4,F_6,F_8,T_4,T_6 \rangle$ is not explored, since it produces a constraint store containing\ $ c_{pre} \; \wedge \; t^0[3] \neq v^0 \; \wedge \; t^0[3] \leq v^0 \; \wedge \; t^0[1] = v^0 $\ which is clearly inconsistent. Similarly, the path $\langle T_4,F_6,F_8,T_4,F_6,T_8\rangle$ cannot be extended. The output of CPBPV on this incorrect program when executed on an array of length 8 (with integers coded on 8-bits to make it readable) produces, in 0.025 seconds, the counterexample:\ $ v^0 = -126 \ \wedge \ t^0 = [-128,-127,-126,-125,-124,-123,-122,-121] \ \wedge \ result = -1. $\ This example highlights a few interesting benefits of CPBPV. 1. The verifier only considers paths that correspond to collections of actual inputs (abstracted by constraint stores). The resulting execution paths must all be explored since our goal is to prove the partial correctness of the program. 2. The performance of the verifier is independent of the integer representation on this application: it only requires a bound on the length of the array. 3. The verifier returns a counter-example for debugging the program. Note that $CBMC$ and $ESC/Java 2$, two state-of-the-art model checkers fail to verify this example as discussed in Section \[experimental\]. Formalization of the Framework {#formalization} ============================== \[semantics\] This section formalizes the CPBPV verifier on a small abstract language using a small-step SOS semantics. The semantics primarily specifies the execution paths over constraint stores explored by the verifier. It features `assert` and `enforce` constructs which are necessary for modular composition. #### **Syntax** Figure \[syntax-c\] depicts the syntax of the programs and the constraints generated by the verifier. In the following, we use $s$, possibly subscripted, to denote elements of a syntactic entity $S$. $$\begin{aligned} \begin{array}{l} L: \mbox{\it list of instructions}; I: {\it instructions}; B: \mbox{\it Boolean expressions} \\ E: \mbox{\it integer expressions}; A: {\it arrays}; V: \mbox{\it variables} \\ \\ L ::= I ; L \; | \; \epsilon \\ I ::= A[E] \leftarrow E \; | \; V \leftarrow E \; | \; {\bf\it if} \ B \ I \; | \; {\bf\it while} \ B \ I \; | \; {\bf\it assert(B)} \; | \; {\bf\it enforce(B)} \; | \; {\bf\it return} \ E \; | \; \{ L \} \\ B ::= true \;|\; false \;|\; E > E \;| \; E \geq E \;| \; E = E \;| \; E \neq E \;| \; E \leq E \;| \; E < E \\ B ::= \neg B \;| \; B \wedge B \; | \; B \vee B \;| \; B \Rightarrow B \\ E ::= V \;|\; A[E] \;|\; E + E \;|\; E - E \;|\; E \times E \;|\; E / E \;|\; \\ \\ C: \mbox{\it constraints} \hspace*{2cm} E^+: \mbox{\it solver expressions} \\ V^+ = \{ v^i \ | \ v \in V \ \& \ i \in {\cal N}\}: \mbox{\it solver variables} \\ A^+ = \{ a^i \ | \ a \in A \ \& \ i \in {\cal N}\}: \mbox{\it solver arrays} \\ \\ C ::= true \;|\; false \;|\; E^+ > E^+ \;| \; E^+ \geq E^+ \;| \; E^+ = E^+ \;| \; E^+ \neq E^+ \;| \; E^+ \leq E^+ \;| \; E^+ < E^+ \\ C ::= \neg C \;| \; C \wedge C \; | \; C \vee C \;| \; C \Rightarrow C \\ E^+ ::= V \;|\; A[E^+] \;|\; E^+ + E^+ \;|\; E^+ - E^+ \;|\; E^+ \times E^+ \;|\; E^+ / E^+ \;|\; \end{array}\end{aligned}$$ #### **Renamings** CPBPV creates variables and arrays of variables “on-the-fly” when they are needed. This process resembles an SSA normalization but does not introduce the join nodes, since the results of different execution paths are not merged. Similar renamings are used in model checking. The renaming uses mappings of type $V \cup A \rightarrow {\cal N}$ which maps variables and arrays into a natural numbers denoting their current “version numbers”. In the semantics, the version number is incremented each time a variable or an array element is assigned. We use $\sigma_{\bot}$ to denote the uniform mapping to zero (i.e., $\forall x \in V \cup A: \sigma_{\bot}(x) = 0$) and $\sigma[x/i]$ the mapping $\sigma$ where $x$ now maps to $i$, i.e., $\sigma[x/i](y) = {\it if} x = y \mbox{ {\it then} } i \mbox{ {\it else} } \sigma(y).$ These mappings are used by a polymorphic renaming function $\rho$ to transform program expressions into constraints. For example, $\rho \ \sigma \ b_1 \oplus b_2 = (\rho \ \sigma \ b_1) \oplus (\rho \ \sigma \ b_2) (\mbox{where } \oplus \in \{\wedge,\vee,\Rightarrow\})$ is the rule used to transform a logical expression. #### **Configurations** The CPBCV semantics mostly uses configurations of the type $\langle l, \sigma, c \rangle$, where $l$ is the list of instructions to execute, $\sigma$ is a version mapping, and $c$ is the set of constraints generated so far. It also uses configurations of the form $\langle \top, \sigma, c \rangle$ to denote final states and configurations of the form $\langle \bot, \sigma, c \rangle$ to denote the violation of an assertion. The semantics is specified by rules of the form $ \frac{\mbox{conditions}} {\gamma_1 \longmapsto \gamma_2} $ stating that configuration $\gamma_1$ can be rewritten into $\gamma_2$ when the conditions hold. #### **Conditional Instructions** The conditional instruction ${\bf\it if} \ b \ i$ considers two cases. If the constraint $c_b$ associated with $b$ is consistent with the constraint store, then the store is augmented with $c_b$ and the body is executed. If the negation $\neg c_b$ is consistent with the store, then the constraint store is augmented with $\neg c_b$. Both rules may apply, since the store may represent some memory states satisfying the condition and some violating it. $$\frac{c \wedge (\rho \ \sigma \ b) \mbox{ is satisfiable}} {\langle {\bf\it if} \ b \ i \; ; \; l, \sigma, c \rangle \longmapsto \langle i \; ; \; l, \sigma, c \wedge (\rho \ \sigma \ b) \rangle}$$ $\;\;\; \;\;\;$ $$\frac{c \wedge \neg (\rho \ \sigma \ b) \mbox{ is satisfiable}} {\langle {\bf\it if} \ b \ i \; ; \; l, \sigma, c \rangle \longmapsto \langle l, \sigma, c \wedge \neg (\rho \ \sigma \ b) \rangle}$$ #### **Iterative Instructions** The while instruction ${\bf\it while} \ b \ i$ also considers two cases. If the constraint $c_b$ associated with $b$ is consistent with the constraint store, then the constraint store is augmented with $c_b$, the body is executed, and the while instruction is reconsidered. If the negation $\neg c_b$ is consistent with the constraint store, then the constraint store is augmented with $\neg c_b$. $$\frac{c \wedge (\rho \ \sigma \ b) \mbox{ is satisfiable}} {\langle {\bf\it while} \ b \ i \; ; \; l, \sigma, c \rangle \longmapsto \langle i;while \ b \ i \; ; \; l, \sigma, c \wedge (\rho \ \sigma \ b) \rangle}$$ $$\frac{c \wedge \neg (\rho \ \sigma \ b) \mbox{ is satisfiable}} {\langle {\bf\it while} \ b \ i \; ; \; l, \sigma, c \rangle \longmapsto \langle l, \sigma, c \wedge \neg (\rho \ \sigma \ b) \rangle}$$ #### **Scalar Assignments** Scalar assignments create a new constraint variable for the program variable to be assigned and add a constraint specifying that the variable is equal to the right-hand side. A new renaming mapping is produced. $$\frac{\sigma_2 = \sigma_1[v/\sigma_1(v)+1] \;\; \& \;\; c_2 \equiv (\rho \ \sigma_2 \ v) = (\rho \ \sigma_1 \ e)} {\langle v \leftarrow e \; ; \; l, \sigma_1, c_1 \rangle \longmapsto \langle l, \sigma_2, c_1 \wedge c_2 \rangle}$$ #### **Assignments of Array Elements** The assignment of an array element creates a new constraint array, add a constraint for the index being indexed and posts constraints specifying that all the new constraint variables in the array are equal to their earlier version, except for the element being indexed. Note that the index is an expression which may contain variables as well, giving rise to the well-known [*element*]{} constraint in constraint programming [@VanHentenryck89]. $$\frac{\begin{array}{l} \sigma_2 = \sigma_1[a/\sigma_1(a)+1] \\ c_2 \equiv (\rho \ \sigma_2 \ a) [\rho \ \sigma_1 \ e_1] = (\rho \ \sigma_1 \ e_2) \\ c_3 \equiv \forall i \in 0..{\it a.length}: (\rho \ \sigma_1 \ e_1) \neq i \; \Rightarrow \; (\rho \ \sigma_2 \ a) [i] = (\rho \ \sigma_1 \ a) [i] \end{array}} {\langle a[e_1] \leftarrow e_2, \sigma_1\; ; \; l, c_1 \rangle \longmapsto \langle l, \sigma_2, c_1 \wedge c_2 \wedge c_3 \rangle}$$ #### **Assert Statements** An assert statement checks whether the assertion is implied by the control store in which case it proceeds normally. Otherwise, it terminates the execution with an error. $$\frac{ c \Rightarrow (\rho \ \sigma \ b) } {\langle {\bf\it assert} \; b \; ; \; l, \sigma, c \rangle \longmapsto \langle l, \sigma, c \rangle}$$ $\;\;\;$ $$\frac{ c \wedge \neg (\rho \ \sigma \ b) \mbox{ is satisfiable} } {\langle {\bf\it assert} \; b \; ; \; l, \sigma, c \rangle \longmapsto \langle \bot, \sigma, c \rangle}$$ #### **Enforce Statements** An enforce statement adds a constraint to the constraint store if it is satisfiable. $$\frac{ c \wedge (\rho \ \sigma \ b) \mbox{ is satisfiable} } {\langle {\bf\it enforce} \; b \; ; \; l, \sigma, c \rangle \longmapsto \langle l, \sigma, c \wedge (\rho \ \sigma \ b) \rangle}$$ #### **Block Statements** Block statements simply remove the braces. $${\langle \{ l_1 \}\; ; \; l_2, \sigma, c \rangle \longmapsto \langle l_1:l_2, \sigma, c \rangle}$$ #### **Return Statements** A return statement simply constrains the [*result*]{} variable. $$\frac{ c_2 \equiv (\rho \ \sigma_1 \ result) = (\rho \ \sigma_1 \ e) } {\langle {\bf\it return} \ e \; ; \; l, \sigma_1, c_1 \rangle \longmapsto \langle \sigma_1, c_1 \wedge c_2 \rangle}$$ #### **Termination** Termination also occurs when no instruction remains. $${\langle \epsilon, \sigma, c \rangle \longmapsto \langle \top, \sigma, c \rangle}$$ #### **The CPBPV Semantics** Let ${\cal P}$ be program $b_{pre} \; l \; b_{post}$ in which $b_{pre}$ denotes the precondition, $l$ is a list of instructions, and $b_{post}$ the post-condition. Let $\stackrel{*}{\longmapsto}$ be the transitive closure of $\longmapsto$. The final states are specified by the set $${\it SFN}(b_{pre},{\cal P}) = \{ \ \langle f, \sigma, c \rangle | \langle i, \sigma_{\bot}, \rho \ \sigma_{\bot} \ b_{pre} \rangle \stackrel{*}{\longmapsto}{*} \langle f , \sigma, c \rangle \; \wedge \; f \in \{\bot,\top\} \ \}$$ The program violates an assertion if the set $${\it SFE}(b_{pre},{\cal P},b_{post}) = \{ \langle \bot, \sigma, c \rangle \in {\it SFN}(b_{pre},{\cal P}) \}$$ is not empty. It violates its specification if the set $${\it SFE}(b_{pre},{\cal P},b_{post}) = \{ \top, \sigma, c \rangle \in {\it SFN}(b_{pre},{\cal P}) \ | \ c \ \wedge \ (\rho \ \sigma \ \neg b_{post}) \mbox{ satisfiable} \}$$ is not empty. It is partially correct otherwise. Implementation issues {#implementation} ===================== The CPBPV framework is parametrized by a list of solvers $(S_1,\ldots,S_k)$ which are tried in sequence, starting with the least expensive and less general. When checking satisfiability, the verifier never tries solver $S_{i+1},\ldots,S_{k}$ if solver $S_i$ is a decision procedure for the constraint store. If solver $S_i$ is not a decision procedure, it uses an abstraction $\alpha$ of the constraint store $c$ satisfying $c \Rightarrow \alpha$ and can still detect failed execution paths quickly. The last solver in the sequence is a constraint-programming solver (CP solver) over finite domains which iterates pruning and searching to find solutions or prove infeasibility. When the CP solver makes a choice, the earlier solvers in the sequence are called once again to prune the search space or find solutions if they have become decision procedures. Our prototype implementation uses a sequence $(MIP,CP)$, where MIP is the mixed integer-programming tool ILOG CPLEX[^2] and CP is the constraint-programming tool Ilog JSOLVER. Our Java implementation also performs some trivial simplifications such as constant propagation but is otherwise not optimized in its use of the solvers and in its renaming process whose speed and memory usage could be improved substantially. Practically, simplifications are done on the fly and the MIP solver is called at each node of the executable paths. The CP solver is only called at the end of the executable paths when the complete post condition is considered. Currently, the implementation use a depth-first strategy for the CP solver, but modern CP languages now offer high-level abstractions to implement other exploration strategies. In practice, when CPBPV is used for model checking as discussed below, it is probably advisable to use a depth-first iterative deepening implementation. Experimental results {#experimental} ==================== In this section, we report experimental results for a set of traditional benchmarks for program verification. We compare CPBVP with the following frameworks: - ESC/Java is an Extended Static Checker for Java to find common run-time errors in JML-annotated Java programs by static analysis of the code and its annotations. See http://kind.ucd.ie/products/opensource/ESCJava2/. - CBMC is a Bounded Model Checker for ANSI-C and C++ programs. It allows for the verification of array bounds (buffer overflows), pointer safety, exceptions, and user-specified assertions. See http://www.cprover.org/cbmc/. - BLAST, the Berkeley Lazy Abstraction Software Verification Tool, is a software model checker for C programs. See http://mtc.epfl.ch/software-tools/blast/. - EUREKA is a C bounded model checker which uses an SMT solver instead of an SAT solver. See http://www.ai-lab.it/eureka/. - Why is a software verification platform which integrates many existing provers (proof assistants such as Coq, PVS, HOL 4,...) and decision procedures such as Simplify, Yices, ...). See http://why.lri.fr/. Of course, neither the expressiveness nor the objectives of all these systems are the same as the one of CPBPV. For instance, some of them can handle CTL/LTL constraints whereas CPBPV dos not yet support this kind of constraints. Nevertheless, this comparison is useful to illustrate the capabilities of CPBPV. All experiments were performed on the same machine, an Intel(R) Pentium(R) M processor 1.86GHz with 1.5G of memory, using the version of the verifiers that can be downloaded from their web sites (except for EUREKA for which the execution times given in [@ABM07; @AMP06] are reported.) For each benchmark program, we describe the data entries and the verification parameters. In the tables, “UNABLE” means that the corresponding framework is unable to validate the program either because a lack of expressiveness or because of time or memory limitations, “NOT\_FOUND” that it does not detect an error, and “FALSE\_ERROR” that it reports an error in a correct program. Complete details of the experiments, including input files and error traces, can be found in [@CRV08]. #### **Binary search** We start with the binary search program presented in figure \[BsearchFig\]. ESC/Java is applied on the program described in Figure \[BsearchFig\]. ESC/Java requires a limit on the number of loop unfoldings, which we set to $log(n)+1$ which is the worst case complexity of binary search algorithm for an array of length $n$. Similarly, CBMC requires an overestimate of the number of loop unfoldings. Since CBMC does not support first-order expressions such as JML $\setminus forall$ statement, we generated a C program for each instance of the problem (i.e., each array length). For example, the postcondition for an array of length $8$ is given by (result!=-1 && a[result]==x)|| (result==-1 && (a[0]!=x&&a[1]!=x&&a[2]!=x&&a[3]!=x&&a[4]!=x&&a[5]!=x&&a[6]!=x&&a[7]!=x) For the Why framework, we used the binary search version given in their distribution. This program uses an assert statement to give a loop invariant. Note that CPBPV does not require any additional information: no invariant and no limits on loop unfoldings. During execution, it selects a path by nondeterministically applying the semantic rules for conditional and loop expressions. Table \[tabsearch\] reports the experimental results. Execution times for CPBPV are reported as a function of the array length for integers coded on 31 bits.[^3] Our implementation is neither optimized for time or space at this stage and times are only given to demonstrate the feasibility of the CPBPV verifier. The “Why” framework [@FiM07] was unable to verify the correctness without the loop invariant; 60% of the proof obligations remained unknown. The CBMC framework was not able to do the verification for an instance of length 32 (it was interrupted after 6691,87s). ESC/Java was unable to verify the correctness of this program unless complete loop invariants are provided [^4]. [[|c||l|c|c|c|c|c|c|]{}]{} \*[CPBPV]{} & array length & 8 & 16 & 32 & 64 & 128 & 256\ & time & 1.081s & 1.69s & 4.043s & 17.009s & 136.80s& 1731.696s\ \*[CBMC]{} & array length & 8 & 16 & 32 & 64 & 128 & 256\ & time & 1.37s & 1.43s & UNABLE & UNABLE &UNABLE &UNABLE\ \*[Why]{} & with invariant &\ & without invariant &\ ESC/Java &\ BLAST &\ #### **An Incorrect Binary search** Table \[tabsearchKO\] reports experimental results for an incorrect [*binary search*]{} program (see Figure \[BsearchFig\], line 11) for CPBPV, ESC/Java, CBMC, and Why using an invariant. The error trace found with CPBPV has been described in Section \[motivation\]. The error traces provided by CBMC and ESC/Java only show the decisions taken along the faulty path can be found in [@CRV08]. In contrast to CPBPV, they do not provide any value for the array nor the searched data. Observe that CPBPV provides orders of magnitude improvements in efficiency over CBMC and also outperforms ESC/Java by almost a factor 8 on the largest instance. CPBPV ESC/Java CBMC WHY with invariant BLAST ------------ -------- ---------- --------- -------------------- -------- -- length 8 0.027s 1.21 s 1.38s NOT\_FOUND UNABLE length 16 0.037s 1.347 s 1.69s NOT\_FOUND UNABLE length 32 0.064s 1.792 s 7.62s NOT\_FOUND UNABLE length 64 0.115s 1.886 s 27.05s NOT\_FOUND UNABLE length 128 0.241s 1.964 s 189.20s NOT\_FOUND UNABLE : Experimental Results for an Incorrect Binary Search[]{data-label="tabsearchKO"} #### **The Tritype Program** The tritype program is a standard benchmark in test case generation and program verification since it contains numerous non-feasible paths: only 10 paths correspond to actual inputs because of complex conditional statements in the program. The program takes three positive integers as inputs (the triangle sides) and returns 2 if the inputs correspond to an isosceles triangle, 3 if they correspond to an equilateral triangle, 1 if they correspond to some other triangle, and 4 otherwise. The [ tritype]{} program in Java with its specification in JML can be found in[@CRV08]. Table \[tabTritype\] depicts the experimental results for CPBPV, ESC/Java, CBMC, BLAST and Why. BLAST was unable to validate this example because the current version does not handle linear arithmetic. Observe the excellent performance of CPBPV and note that our previous approach using constraint programming and Boolean abstraction to abstract the conditions, validated this benchmark in $8.52$ seconds when integers were coded on 16 bits [@CoR07]. It also explored 92 spurious paths. CPBPV ESC/Java CBMC Why BLAST ------ -------- ---------- ------- ------- -------- time 0.287s 1.828s 0.82s 8.85s UNABLE : Experimental Results on the Tritype Program[]{data-label="tabTritype"} #### **An Incorrect Tritype Program** Consider now an incorrect version of [*Tritype*]{} program in which the test [*“if ((trityp==2)&&(i+k$>$j))”*]{} in line 22 (see [@CRV08]) is replaced by [*“if ((trityp==1)&&(i+k$>$j))”*]{}. Since the local variable [ *trityp*]{} is equal to [*2*]{} when [*i==k*]{}, the condition [ *(i+k)$>$j*]{} implies that [*(i,j,k)*]{} are the sides of an isosceles triangle (the two other triangular inequalities are trivial because j$>$0). But, when [*trityp=1*]{}, [*i==j*]{} holds and this incorrect version may answer that the triangle is isosceles while it may not be a triangle at all. For example, it will return [*2*]{} when [ *(i,j,k)=(1,1,2)*]{}. Table \[tabTritypeKO\] depicts the experimental results. Execution times correspond to the time required to find the first error. The error found with CPBPV corresponds to input values $(i,j,k)=(1,1,2)$ mentioned earlier. Once again, observe the excellent behavior of CPBPV compared to the remaining tools. [^5] CPBPV ESC/Java CBMC WHY ------ ---------- ---------- ------------ ------------ time 0.056s s 1.853s NOT\_FOUND NOT\_FOUND : Experimental Results for the Incorrect Tritype Program[]{data-label="tabTritypeKO"} #### **Bubble Sort with initial condition** This benchmark (see [@CRV08]) is taken from [@ABM07] and performs a bubble sort of an array $t$ which contains integers from $0$ to $t.length$ given in decreasing order. Table \[tabbuble\] shows the comparative results for this benchmark. CPBPV was limited on this benchmark because its recursive implementation uses up all the JAVA stack space. This problem should be remedied by removing recursion in CPBPV. CPBPV ESC/Java CBMC EUREKA ----------- -------- ---------- -------- -------- length 8 1.45s 3.778 s 1.11s 91s length 16 2.97s UNABLE 2.01s UNABLE length 32 UNABLE UNABLE 6.10s UNABLE length 64 UNABLE UNABLE 37.65s UNABLE : Experimental Results for Bubble Sort[]{data-label="tabbuble"} #### **Selection Sort** We now present a benchmark to highlight both modular verification and the [element]{} constraint of constraint programming to index arrays with arbitrary expressions. The benchmark described in [@CRV08]. Assume that function `findMin` has been verified for arbitrary integers. When encountering a call to `findMin`, CPBPV first checks if its precondition is entailed by the constraint store, which requires a consistency check of the constraint store with respect to the negation of the precondition. Then CPBPV replaces the call by the post-condition where the formal parameters are replaced by the actual variables. In particular, for the first iteration of the loop and an array length of 40, CPBPV generates the conjunction $ 0 \leq k^0 < 40 \; \wedge \; t^0[k^0] \leq t^0[0] \; \wedge \; \ldots \; \wedge \; t^0[k^0] \leq t^0[39] $ which features [element]{} constraint [@VanHentenryck89]. Indeed, $k^0$ is a variable and a constraint like $t^0[k^0] \leq t^0[0]$ indexes the array $t^0$ of variables using $k^0$. The modular verification of the selection sort explores only a single path, is independent of the integer representation, and takes less than $0.01s$ for arrays of size 40. The bottleneck in verifying selection sort is the validation of function `findMin`, which requires the exploration of many paths. However the complete validation of selection sort takes less than 4 seconds for an array of length 6. Once again, this should be contrasted with the model-checking approach of Eureka [@ABM07]. On a version of selection sort where all variables are assigned specific values (contrary to our verification which makes no assumptions on the inputs), Eureka takes 104 seconds on a faster machine. Reference [@ABM07] also reports that CBMC takes 432.6 seconds, that BLAST cannot solve this problem, and that SATABS [@CKS05] only verifies the program for an array with 2 elements. #### **Sum of Squares** Our last benchmark is described in [@CRV08] and computes the sum of the square of the $n$ first integers stored in an array. The precondition states that $n$ is the size of the array and that $t$ must contain any possible permutation of the $n$ first integers. The postcondition states that the result is $n\times(n+1)\times(2\times n+1)/6$. The benchmark illustrates two functionalities of constraint programming: the ability of specifying combinatorial constraints and of solving nonlinear problems. The `alldifferent` constraint[@Reg94] in the pre-condition specifies that all the elements of the array are different, while the program constraints and postcondition involves quadratic and cubic constraints. The maximum instance that we were able to solve with CPBPV was an array of size 10 in 66.179s. CPLEX, the MIP solver, plays a key role in all these benchmarks. For instance, the CP solver is never called in the Tritype benchmark. For the Binary search benchmark, there are length calls to the CP solver but almost 75% of the CPU time is spent in the CP solver. Since there is only path in the Buble sort benchmark, the CP solver is only called once. In the Sum of squares example, 80% of the CPU time is spent in the CP solver. Discussion and Related Work {#related} =========================== We briefly review recent work in constraint programming and model checking for software testing, validation, and verification. We outline the main differences between our CPBPV framework and existing approaches. #### **Constraint Logic Programming** Constraint logic programming (CLP) was used for test generation of programs (e.g., [@GBR98; @JaV00; @SyD01; @GLM08]) and provides a nice implementation tool extending symbolic execution techniques [@BGM06]. Gotlieb et al. showed how to represent imperative programs as constraint logic programs and used predicate abstraction (from model checking) and conditional constraints within a CLP framework. Flanagan [@Fla04] formalized the translation of imperative programs into CLP, argued that it could be used for bounded model checking, but did not provide an implementation. The test-generation methodology was generalized and applied to bounded program verification in [@CoR06; @CoR07]. The implementation used dedicated predicate abstractions to reduce the exploration of spurious execution paths. However, as shown in the paper, the CPBPV verifier is significantly more efficient and often avoids the generation of spurious execution paths completely. #### **Model Checking** It is also useful to contrast the CPBPV verifier with model-checking of software systems. SAT-based bounded model checking for software[@CBR01] consists in building a propositional formula whose models correspond to execution paths of bounded length violating some properties and in using SAT solvers to check whether the resulting formula is satisfiable. SAT-based model-checking platforms [@CBR01] have been widely popular thanks to significant progress in SAT solvers. A fundamental issue faced by model checkers is the state space explosion of the resulting model. Various techniques have been proposed to address this challenge, including generalized symbolic execution (e.g., [@KPV03]), SMT-based model checking, and abstraction/refinement techniques. SMT-based model checking is the idea of representing and checking quantifier-free formulas in a more general decidable theory (e.g. [@GHN04; @DuM06; @NOR07]). These SMT solvers integrate dedicated solvers and share some of the motivations of constraint programming. Predicate abstraction is another popular technique to address the state space explosion. The idea consists in abstracting the program to obtain an abstract program on which model checking is performed. The model checker may then generate an abstract counterexample which must be checked to determine if it corresponds to a concrete execution path. If the counterexample is spurious, the abstract program is refined and the process is iterated. A successful predicate abstraction consists of abstracting the concrete program into a Boolean program (e.g., [@BPR01; @CKL04; @CKS04]). In recent work [@AMP06; @ABM07], Armando & al proposed to abstract concrete programs into linear programs and used an abstraction of sets of variables and array indices. They showed that their tool compares favourably and, on some of the programs considered in this paper, outperforms model checkers based on predicate abstraction.\ Our CPBPV verifier contrasts with SAT-based model checkers, SMT-based model checkers and predicate abstraction based approaches: It does not abstract the program and does not generate spurious execution paths. Instead it uses a constraint-solver and nondeterministic exploration to incrementally construct abstractions of execution paths. The abstraction uses constraint stores to represent sets of concrete stores. On many bounded verification benchmarks, our preliminary experimental results show significant improvements over the state-of-the-art results in [@ABM07]. Model checking is well adapted to check low-level C program and hardware applications with numerous Boolean constraints and bitwise operations: It was successfully used to compare an ANSI C program with a circuit given as design in Verilog [@CKL04]. However, it is important to observe that in model checking, one is typically interested in checking some specific properties such as buffer overflows, pointer safety, or user-specified assertions. These properties are typically much less detailed than our post-conditions and abstracting the program may speed up the process significantly. In our CPBPV verifier, it is critical to explore all execution paths and the main issue is how to effectively abstract memory stores by constraints and how to check satisfiability incrementally. It is an intriguing issue to determine whether an hybridization of the two approaches would be beneficial for model checking, an issue briefly discussed in the next section. Observe also that this research provides convincing evidence of the benefits of Nieuwenhuis’ challenge [@NOR07] aiming at extending SMT[^6] with CP techniques. Perspectives and Future Work {#conclusion} ============================ This paper introduced the CPBPV framework for bounded program verification. Its novelty is to use constraints to represent sets of memory stores and to explore execution paths over these constraint stores nondeterministically and incrementally. The CPBPV verifier exploits the fact that, when variables and arrays are bounded, the constraint store can always be checked for feasibility. As a result, it never explores spurious execution path contrary to earlier approaches combining constraint programming and predicate abstraction [@CoR06; @CoR07] or integrating SMT solvers and the abstraction/refinement approach from model checking [@ABM07]. We demonstrated the CPBPV verifier on a number of standard benchmarks from model checking and program checking as well as on nonlinear programs and functions using complex array indexings, and showed how to perform modular verification. The experimental results demonstrate the potential of the approach: The CPBPV verifier provides significant gain in performance and functionalities compared to other tools. Our current work aims at improving and generalizing the framework and implementation. In particular, we would like to include tailored, light-weight solvers for a variety of constraint classes, the optimization of the array implementation, and the integration of Java objects and references. There are also many research avenues opened by this research, two of which are reviewed now. Currently, the CPBPV verifier does not check for variable overflows: the constraint store enforces that variables take values inside their domains and execution paths violating these constraints are thus not considered. It is possible to generalize the CPBPV verifier to check overflows as the verification proceeds. The key idea is to check before each assignment if the constraint store entails that the value produced fits in the selected integer representation and generate an error otherwise. (Similar assertions must in fact be checked for each subexpression in the right hand-side in the language evaluation order. Interval techniques on floats [@BGM06] may be used to obtain conservative checking of such assertions. An intriguing direction is to use the CPBPV approach for properties checking. Given an assertion to be verified, one may perform a backward execution from the assertion to the function entry point. The negation of the assertion is now the pre-condition and the pre-condition becomes the post-condition. This requires to specify inverse renaming and executions of conditional and iterative statements but these have already been studied in the context of test generation. #### **Acknowledgements** Many thanks to Jean-François Couchot for many helps on the use of the [*[Why]{}*]{} framework. [10]{} Aït-Kaci H., Berstel B., Junker U., Leconte M., Podelski A. : Satisfiability Modulo Structures as Constraint Satisfaction : An Introduction. Procs of JFLA 2007. Armando A., Benerecetti M., and Montovani J. Abstraction Refinement of Linear Programs with Arrays. Proceedings of TACAS 2007, LNCS 4424: 373–388. Armando A., Mantovani J., and Platania L. Bounded Model Checking of C Programs using a SMT solver instead of a SAT solver. Proc. SPIN’06. LNCS 3925, Pages 146-162. Botella B., Gotlieb A., Michel C. Symbolic execution of floating-point computations. Software Testing, Verification and Reliability. 16:2:97–121.2006. Thomas Ball, Andreas Podelski, Sriam K. Rajamani Boolean and Cartesian Abstraction for Model Checking C Programs. Proc. of TACAS 2001. E. Clarke, A. Biere, R. Raimi, and Y. Zhu. Bounded Model Checking using Satisfiability Solving. FMSD, 19(1):7–34, 2001. Clarke E., Kroening D., Lerda F. : A Tool for Checking ANSI-C programs. Tacas 2004, LNCS 2988, pp 168-176, 2004 Clarke E., Kroening D., Sharygina N., Yorav K. : Predicate abstraction of ANSI-C Programs using SAT. FMSD, 25:105–127, 2004 Clarke E., Kroening D., Sharygina N., Yorav K. : SATABS: SAT-Based Predicate Abstraction for ANSI-C. TACAS’05, 570–574, 2005. Cytron R., Ferrante J., Rosen B., Wegman M., and Zadeck K. : fficently [C]{}omputing [S]{}tatic [S]{}ingle [A]{}ssignment [F]{}orm and the [C]{}ontrol [D]{}ependence [G]{}raph. , 13(4):451–490, October 1991. Collavizza H. and Rueher M. : Software Verification using Constraint Programming Techniques. Procs of TACAS 2006, LNCS 3920: 182-196. Collavizza H. and Rueher M. : Exploring different constraint-based modelings for program verification Procs of CP 2007, LNCS 3920: 182-196 Collavizza H. Rueher M., Van Hentenryck P. : Comparison between CPBPV with ESC/Java, CBMC, Blast, EUREKA and Why. http://www.i3s.unice.fr/\~rueher/verificationBench.pdf Bruno Dutertre and Leonardo Mendon¸ca de Moura. A fast linear-arithmetic solver for DPLL(T). CAV 2006, pages 81–94. LNCS 4144. Cormac Flanagan, “Automatic software model checking via constraint logic” (2004). Science of Computer Programming. 50 (1-3), pp. 253-270. Filliâtre J.C., Claude Marché.The Why/Krakatoa/Caduceus Platform for Deductive Program Verification Proc. CAV’2007, LNCS 4590. pp 173-177. Gotlieb A., Botella B. and Rueher M : Automatic Test Data Generation using Constraint Solving Techniques. Proc. ISSTA 98, ACM SIGSOFT (2), 1998. Ganzinger H., Hagen G., Nieuwenhuis R.,Oliveras A., and Tinelli C.: DPLL(T): Fast Decision Procedures. Proc. of CAV 2004, 175-188, 2004. P. Godefroid, M. Y. Levin, D. Molnar: Automated Whitebox Fuzz Testing, NDSS 2008, Network and Distributed System Security Symposium. Daniel Jackson and Mandana Vaziri, Finding Bugs with a Constraint Solver, ACM SIGSOFT Symposium on Software Testing and Analysis, 14–15, 2000. Khurshid, S., Pasareanu, C.S., and Vissser, W. “Generalized Symbolic Execution for Model Checking and Testing”, in TACAS 2003, Warsaw, Poland. R. Nieuwenhuis, A. Oliveras, E. Rodríguez-Carbonell and A. Rubio: Challenges in Satisfiability Modulo Theories. Invited Talk. RTA 2007, LNCS 4533, pp 2-18. J-C. Régin. A filtering algorithm for constraints of difference in CSPs. AAAI-94, Seattle, WA, USA, pp 362–367, 1994. Sy N.T. and Deville Y.: Automatic Test Data Generation for Programs with Integer and Float Variables. Proc of. 16th IEEE ASE01, 2001. VanHentenryck P. (1989) Constraint Satisfaction in Logic Programming, MIT Press. Numerica: A Modeling Language for Global Optimization Pascal Van Hentenryck, Laurent Michel, Yves Deville. MIT Press, 1997. [^1]: We omit the domain constraints on the variables for simplicity. [^2]: See http://www.ilog.com/products. [^3]: The commercial MIP solver fails with 32-bit domains because of scaling issues. [^4]: a version with loop invariants that allows to show the correctness of this program has been written by David Cok, a developper of ESC/Java, after we contacted him. [^5]: For CBMC, we have contacted D. Kroening who has recommended to use the option CPROVER\_assert. If we do so, CBMC is able to find the error, but we must add some assumptions to mean that there is no overflow into the sums, in order to prove the correct version of tritype with this same option. [^6]: See also [@ABJ07] for a study of the relations between constraint programming and Satisfiability Modulo Theories (SMT)

--- abstract: | In the limit of small couplings in the nearest neighbor interaction, and small total energy, we apply the resonant normal form result of a previous paper of ours to a finite but arbitrarily large mixed Fermi-Pasta-Ulam Klein-Gordon chain, i.e. with both linear and nonlinear terms in both the on-site and interaction potential, with periodic boundary conditions. An existence and orbital stability result for Breathers of such a normal form, which turns out to be a generalized discrete Nonlinear Schrödinger model with exponentially decaying all neighbor interactions, is first proved. Exploiting such a result as an intermediate step, a long time stability theorem for the true Breathers of the KG and FPU-KG models, in the anti-continuous limit, is proven. author: - 'Simone Paleari[^1] [^2] and Tiziano Penati$\null^*$ [^3]' title: | Long time stability of small amplitude Breathers\ in a mixed FPU-KG model --- Introduction and statement of the results {#s:1} ========================================= We consider a mixed Fermi-Pasta-Ulam Klein-Gordon model (FPU-KG) as described by the following Hamiltonian $$\begin{aligned} \label{e.H} H(x,y) &= \frac12\sum_{j=1}^N \left[ y^2_j + x^2_j + a(x_{j+1}-x_j)^2 \right] + \frac14\sum_{j=1}^N{{\left[x_j^4 + {b}(x_{j+1}-x_j)^4\right]}}\ ,\\ x_0 &= x_N \ , \qquad y_0 = y_N \ ,\end{aligned}$$ i.e. a finite chain of $N$ degrees of freedom and periodic boundary conditions, where $a>0$ and $b\geq 0$ are the linear and nonlinear coupling coefficients. It can be remarked that the classical KG model ($a\neq0$, $b=0$) is included as a particular case, but the pure FPU one is clearly not covered[^4]. According to a previous result of ours, for any $r\geq 1$, provided the coupling parameters $a$ and $b$ are correspondingly small enough, there exists a canonical transformation $T_\Chi$ which puts the Hamiltonian into an extensive resonant normal form of order $r$ $${H^{(r)}_{}} = H_\Omega + {\mathcal Z}+ {P^{(r+1)}}\ , \qquad\qquad \{H_\Omega,{\mathcal Z}\}=0 \ .$$ with $H_\Omega$ a system of $N$ identical oscillators whose frequency $\Omega$ turns out to be the average of the linear spectrum of the original Hamiltonian, ${\mathcal Z}$ a non-homogeneous polynomials of order $2r+2$, $P^{(r+1)}$ a remainder of order at least $2r+4$ (see Theorem \[prop.gen\] in Section \[s:2\]). Strictly speaking, the original statement is formulated in the case $b=0$, see [@PalP14], but holds also for $b\neq0$ and small. The above normal form was indeed shown in [@PalP14] to be well defined in a small ball $B_R(0)$ both in euclidean and in supremum norm, i.e. both in a regime of finite total and respectively specific[^5] energy; one of the key points was indeed to consider finite but arbitrarily large systems (along a direction we followed also, e.g., in [@PP12]), with estimates uniform in the size of the chain, hence valid in the limit $N\to+\infty$. However, only in the case of the euclidean norm the almost invariance of $H_\Omega$ over times $|t|\sim R^{-r-1}$ was granted, due to the equivalence between $H_\Omega$ and the selected norm. Moreover, looking at the structure of ${\mathcal Z}$, the normal form $H_\Omega+{\mathcal Z}$ appears as a generalized discrete nonlinear Schrödinger (GdNLS[^6]) chain: it is characterized by all neighbors couplings, with exponential decay of the coefficients with the distance between sites, both in the linear and nonlinear terms, the last ones being of order $2r+1\geq 3$. Since the Hamiltonian of such a normal form is given by an expansion both in energy, through the degree of the polynomials, and in coupling, it is actually cumbersome and somewhat useless to give here a complete and explicit formulation; the following are the leading terms, in the transformed variables $(\tilde x,\tilde y)$ $$\begin{aligned} H_{\rm GdNLS} = & \sum_{j=1}^N \bigg[\frac{\Omega}2 ( {\tilde y}_j^2 + {\tilde x}_j^2 ) \; + && H_\Omega \\ +\; & \mathcal{O}(c) ({\tilde x}_j{\tilde x}_{j+1} + {\tilde y}_j{\tilde y}_{j+1}) + \mathcal{O}(c^2) ({\tilde x}_j{\tilde x}_{j+2} + {\tilde y}_j{\tilde y}_{j+2}) + \mathcal{O}(c^3) \; + &&{\mathcal Z}\text{: quadratic part} \\ +\; & \mathcal{O}(c^0) ({\tilde x}_j^2+{\tilde y}_j^2)^2 + \mathcal{O}(c) ({\tilde x}_j^2+{\tilde y}_j^2)({\tilde x}_j{\tilde x}_{j\pm 1}+ {\tilde y}_j{\tilde y}_{j\pm 1}) + \mathcal{O}(c^2) \Big] + &&{\mathcal Z}\text{: quartic part} \\ +\; &\ldots &&{\mathcal Z}\text{: higher orders}\end{aligned}$$ where we have introduced the collective coupling constant $$\label{e.c} c:=\max\{a,b\} \ .$$ If one truncates the above Hamiltonian, using only $H_\Omega$ and the first term of both the quadratic and quartic part of ${\mathcal Z}$, it is possible to recognize the usual dNLS, here written in real coordinates. In this work we exploit the invariance of $H_\Omega$ in the above resonant normal form part, in order to get some stability results about true or approximated Breather solution in the model . The results we are going to present hold in the small total energy $E<E_*(r)$ regime and for $c<c_*(E,r)$ small enough, hence in the anti-continuous limit. For a more precise formulation, let us give some notation. We denote with ${\mathcal P}$ the phase space $\RR^{2N}$ endowed by the usual euclidean norm ${\left\|z\right\|}$, where $z=(x,y)$ is the generic element of ${\mathcal P}$. Given $z\in{\mathcal P}$, let us also denote by ${{{\mathsf O}}}(z)$ the orbit through $z$. We also need to introduce a suitable “orbital” distance. We use the Hausdorff distance $d_H$, which is a metric once restricted to the subset of non empty and compact sets: since we consider periodic orbits and subsets of orbits parametrized by a closed interval of time, $d_H$ satisfies all the relevant properties we need. We recall the definition: given two subsets $A$ and $B$ of ${\mathcal P}$, $$\label{e.dist.def} \begin{aligned} d(A,B) &:= \sup_{a\in A}\inf_{b\in B}{\left\|a - b\right\|} \ , \\ d_H(A,B) &:= \max\{d(A,B),d(B,A)\} \ . \end{aligned}$$ Let us denote[^7] by $\Psi_{a,b}$ and ${{{\mathsf O}}}(\Psi_{a,b})$ respectively the Breather initial profile and its orbit for our FPU-KG model . The existence of such an object in the anti-continuous limit (i.e. as a family in $(a,b)$ emerging from the trivial one-site excitation solution available when $(a,b)=(0,0)$) has been obtained originally in [@MacA94] (strictly speaking in the case $b=0$). In the formulation below of our result, the long stability time $T_{\eps,r,R}$ (see ) scales as $$\label{e.appr.tscale} T_{\eps,r,R} \simeq \eps^2 {\frac{(Rr)^{-2r}}{R^4}} \ ,$$ where $r$ is the aforementioned normal form order, $R$ control the small energy of the objects involved, and $\eps$, sufficiently smaller than $R$, is the (tunable part of the) radius of the stability neighborhood. Finally, to unambiguously fix the two-parameter family $\Psi_{a,b}$, we require it to emerge, in the anti-continuous limit, from the single-site oscillator with prescribed amplitude ${\left\|\Psi_{0}\right\|}=R/6$, where here and in the following, we will use a single sub-scripted $0$ to indicate the values $(a,b) = (0,0)$ (see  and  for explicit definitions). \[t.b.kg\] Fix an arbitrary integer $r\geq1$. Then there exists $R_*(r)<1$ such that for all $R<R_*$ and $0<\eps \ll R^2$ there exist $c_*(r,R,\eps)$ and $\delta(\eps)$, such that for all $c<c_*$ the (piece of) orbit ${{{\mathsf O}}}(\phi):=\{\phi(t) \ :\ |t| \leq T_{\eps,r,R} \,,\ \phi(0)=\phi \}$, solution of , satisfies $$\label{e.main} {\left\|\phi-\Psi_{a,b}\right\|}<\delta \quad\Longrightarrow\quad d_H{{\left({{{\mathsf O}}}(\phi),{{{\mathsf O}}}(\Psi_{a,b})\right)}} < \eps \ .$$ A first comment on the above statement pertains the coupling threshold $c_*$ and its dependence on the relevant parameters. It depends on $r$ because of the normal form construction: the larger the transformation steps number required, the smaller the perturbation parameter, i.e. the coupling. The dependence from $R$ comes both from the normal form procedure, when we need to control the size of the transformation domains and the smallness of the remainder, and from the existence of Breathers solutions of the GdNLS (this will be shown to be a necessary intermediate step). The $\eps$ dependence appears instead in the last part of the proof, when the distance between the Breather of the full system and that of the normal form must be controlled. Let us add some more details. As we said, our proof of Theorem \[t.b.kg\] is based on the long time stability result of Theorem \[p.orb.contr\]. We indeed first show the expected existence and stability of a Breather for the normal form (GdNLS), respectively by a continuation from the anti-continuous limit and exploiting the second conserved quantity of the GdNLS. Let us denote by ${{{\mathsf O}}}(\psi_{a,b})$ the orbit of such a Breather, emerging from the same single-site oscillator $\Psi_0$ introduced above. We remark that the closed trajectory ${{{\mathsf O}}}(\psi_{a,b})$ represents an approximated solution for . We then use the small remainder given by the normal form transformation to translate the infinite time stability of the GdNLS dynamics around the GdNLS Breather ${{{\mathsf O}}}(\psi_{a,b})$ into a long time stability of the FPU-KG dynamics around the same object. This concludes the sketch of the proof of Theorem \[p.orb.contr\], where a stability control of the FPU-KG dynamics can be obtained in the form $${\left\|\phi-\psi_{a,b}\right\|}<\delta \quad\Longrightarrow\quad d_H{{\left({{{\mathsf O}}}(\phi),{{{\mathsf O}}}(\psi_{a,b})\right)}} < \eps \ ,$$ for $|t|\lesssim T_{\eps,r,R}$ and $c<c_*$, in this case with $c_*(r,R)$ independent of $\eps$. Thus, at fixed $r$ and $R$, one can play[^8] with $\eps$ to strengthen the stability control without further requirements on the couplings. To get the result of Theorem \[t.b.kg\] one eventually exploits the closeness of the FPU-KG Breather ${{\mathsf O}}(\Psi_{a,b})$ to the GdNLS Breather ${{\mathsf O}}(\psi_{a,b})$: both the objects emerge in the anti-continuous limit from the same configuration $\Psi_{0}$, thus using the continuity in their (common) continuation parameter $c$ one gets a (weak) closeness of order $\sqrt[4]{c}$. Here enters the dependence of $c_*$ also on $\eps$: this is needed in order to ensure that the true Breather configuration $\Psi_{a,b}$ lies well within the stability basin of the approximated Breather orbit ${{\mathsf O}}(\psi_{a,b})$. Furthermore, in order to transfer the stability of ${{\mathsf O}}(\psi_{a,b})$ to the stability of ${{\mathsf O}}(\Psi_{a,b})$ the triangular inequality of the Hausdorff metric $d_H$ is also needed. We would like to stress that our result resembles, in its formulation and strategy,  Theorem 2.1 of [@Bam96], which is the first, and actually one of the few, result of long time stability of Breathers for weakly coupled oscillators (see also [@Bam98]): indeed, although Nekhoroshev-type stability was expected since the earliest papers (see, e.g. [@MacA94]), most of the literature on the stability of Breathers (and of their multi-site generalizations, called Multibreathers) deals with the linear stability (see [@Aub97; @PelKF05; @PelS12; @Yos12]). There are however some differences with [@Bam96], that we would like to underline here. The first one is that in [@Bam96] the closeness to the Breather solutions was obtained with a “local” normal form around a generic an-harmonic oscillator (the system being infinite), using only the linear coupling $a$ as small parameter (since it treats the model with $b=0$). As a consequence, it is valid also for arbitrary large amplitudes and not only in the small energy regime, like Theorem \[t.b.kg\]. Our normal form is instead “more global”, in the sense that it holds in a whole neighborhood of the origin. Hence, within its limit of validity given by the smallness of the energy, it can be used to capture the main features of any Cauchy Problem. Moreover, and differently from our Theorem \[t.b.kg\], in [@Bam96] the small parameter $a$ is used also in order to fix the domain of stability: indeed, in [@Bam96], the corresponding of our radius $\delta$ of the stability basin vanishes as ${a}\to 0$. This is a consequence of the way the “local” normal form Theorem (ref. Theorem 4.1 in [@Bam96]) has been used, choosing $\sqrt{a}$ as the size of the domain of validity, and it seems in contrast with the intuition that by approaching the uncoupled system ($a=0$ in that case, $c=0$ in the present one), the Breather should be increasingly stable, not only in terms of time scale but also in terms of domain. With respect to this aspect, our result is more flexible: as already pointed, at fixed time scale (i.e. fixing $r$ and $E$) we are allowed to arbitrarily decrease the coupling $a$ without shrinking the stability basin. Concerning instead the dependence on the coupling of the stability time scale, the result in [@Bam96] appears to be as strong as one could hope, i.e. one has an exponential dependence of the form $T_a \simeq \exp(a^{-1/6})$. Our result, on the contrary, seems to fail completely in the expected growth of the time scale as the couplings vanish, since neither $a$ nor $b$ appear explicitly in . However our result is indeed somewhat similar once the implicit dependence on the couplings is taken into account: the formulation of Theorem \[t.b.kg\] provides a stability time $T_{\eps,r,R}$ which scales as a power of $(Rr)^{-1}$, which is large provided the “amplitude” $R$ is sufficiently small with respect to $1/r$ (see also condition ), with an exponent $r$ which can be arbitrarily increased by sufficiently decreasing the coupling $c$. In the parameters plane $(r,c)$, the allowed region has a border[^9] roughly described by $cr^4=\text{const}$. Thus one can either formulate the statement, as we do, assuming an arbitrary $r$, provided $c$ is smaller than something scaling as $1/r^4$; or one could fix $c$ (sufficiently small for independent reasons) and let $r$ up to $1/\sqrt[4]{c}$. In the latter case, provided $R$ vanishes at least as $\sqrt[4]{c}$, the stability time scale resemble very closely the exponential one of [@Bam96]. The price to be paid is that, the smaller is the amplitude $R$, the smaller has to be the stability domain parameter $\eps$. There is a last comment in the comparison of our results with the reference paper [@Bam96]. The stability in [@Bam96], as we said, is obtained through a normal form around the an-harmonic oscillator which is going to constitute the core of the Breather, actually by removing the dominant part of the coupling of such an oscillator $(J,\varphi)$ with the rest of the chain: this typically requires a Diophantine non resonance condition for the true frequency $\omega(J)$ of the Breather with respect to its linear frequency $\omega_0$. However, the smaller is $J$, the closer is $\omega(J)$ to $\omega_0$ and thus proportionally smaller must be the parameter $\nu$ in the Diophantine condition $|k_1\omega+k_2\omega_0|\geq \nu/|k|^2$. And this affects the small coupling interval $(0,a_*)$ for which the result in [@Bam96] applies: indeed, since the normal form construction needs $\sqrt{a}/\nu<1$ in order to be performed, the threshold $a_*$ has to decrease at least like $\nu^2$, which means $a_*\lesssim R^4$ in terms of small amplitude $R$. Our result is instead completely free of any Diophantine condition on the Breather frequency, implicitly requiring only non-resonance and non-degeneracy of the frequency in order to have the existence of the Breather. And we stress that, even though we also require the coupling $c$ (and then $a$) to be small enough with respect to the amplitude, as a sufficient condition for the variational continuation from the anti-continuous limit, our smallness condition is weaker, being of the order $c_*\ll R^2$. We conclude this Introduction by remarking that, since our strategy is strongly based on a normal form construction for the quadratic part of the Hamiltonian (see also [@GioPP12; @GioPP13]), it can be applied also to different local nonlinearities, like for example the Morse or the cubic potential in the DNA models [@PeyB89; @DauPW92]. Indeed, even the FPU-KG model presented here is an easy extension with respect to the classical KG one, and we included it here both to give a more general result and because we were motivated by recent papers like [@KarSKC13; @Yos12], where a nonlinear quartic interaction is taken into account. Moreover, the perturbation approach we exploited here, even simplified in its preliminary step involving the quadratic part, can be applied to those model where the coupling is purely nonlinear ($a=0$), thus justifying the long time stability of compact-like Breathers (see [@TchR99; @RosS05]). The paper is organized as follows. In Section \[s:2\] we reformulate the normal form result (and the main ideas related) discussed in [@PalP14]. In Section \[s:3\] we present and comment the two results concerning the long time orbital stability of the approximated and true breather solution. A short Appendix contains the proofs of the existence and stability of the GdNLS breather. Background: an extensive resonant normal form Theorem {#s:2} ===================================================== The aim of this Section is to present the resonant normal form Theorem obtained in [@PalP14], with a slightly different formulation which is necessary to deduce Theorem \[t.b.kg\]. At variance with respect to the original paper, we here decided to select $r$, the order of the normal form, as the main parameter used to express the thresholds of validity of the construction, instead of the small couplings. Such a different choice fixes the order of the normal form, hence its non linear terms, leaving the small couplings as “free” parameters for the continuation procedure from the anti-continuous limit. For the above reasons, and in order to introduce some definitions and remarks necessary for the comprehension of the perturbation part, in this Section we also repeat, and slightly extend, some ingredients of [@PalP14]. Extensivity {#ss:form} ----------- The perturbation construction developed in [@GioPP13; @PalP14] is strongly based on the property of extensivity typical of a class of Hamiltonian like : physically speaking, in all these models the extensivity results from both the translation invariance and the short interaction potentials. In particular, the extensivity allows to sharply manage the dependence on the size of the system $N$ in the estimates involved in the perturbation approach. We here recall a possible formalization of this property, by means of the cyclic symmetry, which has been already introduced, widely analyzed and then exploited in [@GioPP12; @GioPP13; @PalP14]. We denote by $x_j,\,y_j$ the position and the momentum of a particle, with $x_{j+N} = x_j$ and $y_{j+N}=y_j$ for any $j$. #### Cyclic symmetry. {#p:cyclic} We formalize the translation invariance by using the idea of *cyclic symmetry*. The *cyclic permutation* operator $\tau$, acting separately on the variables $x$ and $y$, is defined as $$\label{e.perm} \tau(x_1,\ldots,x_N) = (x_2,\ldots,x_N,x_1)\ ,\quad \tau(y_1,\ldots,y_N) = (y_2,\ldots,y_N,y_1)\ .$$ We extend its action on the space of functions as $$\bigl(\tau f\bigr)(x,y) = f(\tau(x,y)) = f(\tau x,\tau y) \ .$$ \[d.cs\] We say that a function $F$ is *cyclically symmetric* if $\tau F = F$. We introduce now an operator, indicated by an upper index $\oplus$, acting on functions: given a function $f$, a new function $F= f^{\oplus}$ is constructed as $$F= f^{\oplus} := \sum_{l=1}^{N} \tau^l f \ . \label{e.cycl-fun}$$ We shall say that $f^{\oplus}(x,y)$ is generated by the *seed* $f(x,y)$. We shall often use the convention of denoting cyclically symmetric functions with capital letters and their seeds with the corresponding lower case letter. #### Polynomial norms. {#p:polinorms} Let $f(x,y)=\sum_{|j|+|k|=s} f_{j,k} x^j y^k$ be a homogeneous polynomial of degree $s$ in $x,\,y$. Given a positive $R$, we define its polynomial norm as $$\label{e.polinorm} \|f\|_R := R^s \sum_{j,k} |f_{j,k}|\ .$$ #### Norm of an extensive function. {#p:norm-ext} Assume now that we are equipped with a norm for our functions ${\left\|\cdot\right\|}$, e.g. the above defined polynomial norm. We introduce a corresponding norm ${\left\|\cdot\right\|}^\oplus$ for an extensive function $F=f^\oplus$ by defining $$\label{e.norm-germ} \bigl\|F\bigr\|^{\oplus} = \|f\|\ ,$$ i.e. by actually measuring the norm of the seed. Although the norm so defined depends explicitly on the choice of the seed, this is harmless in the perturbation estimates since $$\|F\| \le N \bigl\|F\bigr\|^{\oplus} = N {\left\|f\right\|}\ ,$$ for any $f$ such that $F=f^\oplus$. #### Circulant matrices. {#p:circulant} Dealing with particular functions which are quadratic forms, the cyclic symmetry coming from extensivity assumes a particular form. Let us thus restrict our attention to the harmonic part of the Hamiltonian: it is a quadratic form represented by a matrix $A$ $$\label{e.Aintro} H_0(x,y) = \frac12 y\cdot y + \frac12 Ax\cdot x.$$ If the Hamiltonian $H$ is extensive, then $H_0=h_0^\oplus$. This implies that $A$ commutes with the matrix $\tau$ representing the cyclic permutation  $$\label{e.tau} \tau_{ij}= \begin{cases} 1\quad {\rm if}\ i=j+1\>({\rm mod}\,N)\>,\\ 0\quad \rm{otherwise}. \end{cases}$$ We remark that the matrix $\tau$ is orthogonal and generates a cyclic group of order $N$ with respect to the matrix product. In our problem the cyclic symmetry of the Hamiltonian implies that the matrix $A$ of the quadratic form is circulant. Obviously it is also symmetric, so that the space of matrices of interest to us has dimension ${{\left[\frac{N}2\right]}}+1$. Indeed, a circulant and symmetric matrix is completely determined by ${{\left[\frac{N}2\right]}}+1$ elements of its first line. \#1[[*\#1*]{}]{} #### Interaction range {#ss.int.range} We give here a formal characterization of finite and short range interaction, pointing out some properties that will be useful in the rest of the paper. We restrict our analysis to the set of polynomial functions. We start with some definitions. Let us label the variables as $x_l,y_l$ with $l\in\ZZ$, and consider a monomial $x^jy^k$ (in multiindex notation). We define the *support* $S(x^jy^k)$ of the monomial and the *interaction distance* $\ell(x^jy^k)$ as follows: considering the exponents $(j,k)$ we set $$\label{e.supp} S(x^jy^k) = \{l\>:\>j_l\neq0 {\rm\ or\ } k_l\neq0 \}\ ,\quad \ell(x^jy^k) = \diam\bigl(S(x^jy^k)\bigr) \ .$$ We say that the monomial is *left aligned* in case $S(x^jy^k)\subset \{0,\ldots,\ell(x^jy^k)-1\}$. The definition above is extended to a homogeneous polynomial $f$ by saying that $S(f)$ is the union of the supports of all the monomials in $f$, and that $f$ is left aligned if all its monomials are left aligned. The relevant property is that if $\tilde f$ is a seed of a cyclically symmetric function $F$, then there exists also a left aligned seed $f$ of the same function $F$: just left align all the monomials in $\tilde f$. #### Short range (exponential decay of) interaction. {#p:shortrange} For the seed $f$ of a function consider the decomposition $$\label{e.decomp} f(z) = \sum_{m\ge 0} f^{(m)}(z)\ ,\quad f^{(m)}(z) = \sum_{\ell(k)\le m} f_k z^k\ ,$$ assuming that every $f^{(m)}$ is left aligned. \[d.class\] The seed $f$ (of an extensive function) is said to be of class ${\mathcal{D}}(C_f,\sigma)$ if $$\label{dcdm.5} {\left\|f^{(m)}\right\|}_1 \le C_f e^{-\sigma m}\ ,\quad C_f\gt 0\,,\> \sigma\gt 0\ .$$ #### Continuity of extensive polynomials. We add here some regularity properties, which are absent in [@PalP14]. \[l.cont.pol\] Any $F=f^\oplus$, polynomial of degree $m$, with $f\in{\mathcal{D}}(C_f,\sigma)$ is of class ${\cal C}^m(\RR^{2N},\RR)$, with $$\label{e.cont.pol} |F(z)| \leq {\left\|F\right\|}^\oplus{\left\|z\right\|}^m\ .$$ Since $$f(z) = \sum_{|k|=m} f_k z^k\ ,$$ we have also $$|F(z)| \leq \sum_{j=0}^{N-1}\sum_{|k|=m}|f_k| |z^k\circ \tau^j| \leq \sum_{|k|=m}|f_k| {{\left(\sum_{j=0}^{N-1}|z^k\circ \tau^j|\right)}} \leq {\left\|F\right\|}^\oplus{\left\|z\right\|}^m\ .$$ Any polynomial $F$ is represented by a symmetric ($m$) multilinear operator $\hat F$, such that $$\hat F(z,\ldots,z) = F(z)\ ,$$ hence gives $$\sup_{{\left\|z\right\|}\leq 1,\,z\not=0} |\hat F(z,\ldots,z)|\leq {\left\|F\right\|}^\oplus<\infty\ ,$$ which is the continuity of $\hat F$. The continuity of the differentials follows immediately from $\hat F$ being multilinear. #### Hamiltonian vector fields We consider, as an Hamiltonian, an extensive function $F$ with seed $f$; we will make use of the common notation[^10] $X_F=(X_1, \ldots,X_N,X_{N+1}, \ldots, X_{2N})$ to indicate the associated Hamiltonian vector field $J\nabla F$, with $J$ given by the Poisson structure. The first easy, but important, result is that also the Hamiltonian vector field inherits, in a particular form, the cyclic symmetry; a possible choice for the equivalent of the seed turn out to be the couple $(X_{1}, X_{N+1})$, i.e. the first and the $(N+1)^{\rm th}$ components of the vector. This fact, which will be more clear thanks to the forthcoming Lemma \[l.seme.campo\], allows us to define in a reasonable and consistent way the following norm $$\label{e.def1} {\Big\|X_F\Big\|^\oplus}_R := {\left\|X_1\right\|}_R+{\left\|X_{N+1}\right\|}_R \ .$$ \[l.seme.campo\] Given $F=f^\oplus$, for the components of its Hamiltonian vector field $X_F$ we have[^11] $$\label{e.seme.campo} \begin{aligned} X_j &= \tau^{j-1} X_1 \cr X_{N+j} &= \tau^{j-1} X_{N+1} \end{aligned} \qquad\qquad j=1,\ldots,N \ .$$ Moreover, it holds $$\label{e.xxx} {\Big\|X_F\Big\|^\oplus}_R = \sum_{l=1}^{2N}{\left\|{{\frac{\partial f}{\partial z_l}}}\right\|}_R.$$ Resonant normal form -------------------- In this part we recall, with a slightly different statement more based on the parameter $r$, the resonant normal form result of [@PalP14]. Although the model has an additional nonlinear term, its main Theorem, here formulated as Theorem \[prop.gen\] still apply, since it requires some decay properties of the seeds of the Hamiltonian which are true also for . Indeed, we first recall the splitting of the Hamiltonian  as a sum of its quadratic and quartic parts $H=H_0+H_1$, i.e. $$\label{e.H.dec} H_0(x,y) := \frac12\sum_{j=1}^N {{\left[y^2_j + x^2_j + a(x_j-x_{j-1})^2\right]}} \ , \qquad H_1(x,y) := \frac14\sum_{j=1}^N {{\left[x_j^4+ b(x_{j+1}-x_j)^4\right]}} \ .$$ ### Discussion on the small parameters {#sss:small.par} Since both $0<a<1$ and $0\leq b<1$ have to be considered as small parameters, we define $$\label{e.mu} \mu:=\sqrt[4]{\frac{2c}{1+2c}}\ ,$$ where $c$ has been introduced in : the new parameter $\mu$ will play the role of main perturbation parameter together with the small radius $R$. Moreover, in order to deal with exponentially decaying interactions (and to explain why we defined $\mu$ as we did), let us introduce the following parameters, which will serve as decay rates in the sense of Definition \[d.class\] $$\label{e.sigma.ab0} \sigma_a := \ln{{{\left(\frac{1+2a}{2a}\right)}}}\ ,\qquad \sigma_b := \ln{{{\left(\frac{1+2b}{2b}\right)}}}\ ,\qquad \sigma_0:=\min\{\sigma_a,\sigma_b\}\ .$$ As a consequence of and , one has $$\label{e.mu.sigma0} \sigma_0 = \ln{{{\left(\frac{1+2c}{2c}\right)}}}\ ,\qquad\qquad \mu = e^{-\sigma_0/4}\ .$$ It is important to notice that $H_1=h_1^\oplus$ with $h_1\in{\mathcal{D}}(C_{h_1},\sigma_b)$. Indeed by definition $$H_1(x) = h_1^\oplus\ ,\qquad h_1:=\pm\frac{1}4x_0^4+ \frac{b}4(x_{1}-x_0)^4\ ;$$ however by rearranging the monomials, it is possible to select $h_1$ as $$h_1=h_1^{(0)} + h_1^{(1)}\ ,\qquad h_1^{(0)}=\frac{\pm1+2b}4x_0^4 \ ,\qquad h_1^{(1)}= \frac32b x_0^2x_1^2 - bx_0x_1(x_0^2+x_1^2)\ .$$ with $${\left\|h_1^{(l)}\right\|}\leq C_{h_1} e^{-\sigma_b l}\ ,\qquad l=0,1\ ,\qquad C_{h_1}:=\frac74(1+2b)\ ,$$ ### Preliminary linear transformation {#sss:lin.tr} [We start performing the normalization process with]{} an *exponentially localized* linear transformation (see Proposition 2 of [@PalP14], and also [@GioPP12; @GioPP13]) to put the quadratic part into a resonant normal form. Rewrite the matrix $A$, introduced in , as $$\label{e.def-A} A= (1+2a){{\left[\Id - \frac{e^{-\sigma_a}}2(\tau + \tau^{\top})\right]}} \ ,$$ which is clearly circulant and symmetric, and gives a finite range interaction, in the form of a $e^{-\sigma_a}$ small perturbation of the identity. Let the constant frequency $\Omega$ be the average of the square roots of the eigenvalues of $A$, and take any $\sigma_1\in (0,\sigma_0)$. We have (see mainly Proposition 3.1 in [@GioPP13] and the related Proposition 1 in [@GioPP12]) \[p.1\] The canonical linear transformation $q=A^{1/4} x$, $p=A^{-1/4}y$ gives the Hamiltonian $H_0$ the particular resonant normal form $$\label{e.dec.H0} H_0 = H_\Omega + Z_0 \ , \qquad {\{H_\Omega,Z_0\}}=0$$ with $H_\Omega$ and $Z_0$ cyclically symmetric with seeds $$h_\Omega = \frac\Omega2({\tilde x}_1^2+{\tilde y}_1^2) \ , \qquad \zeta_0\in{\mathcal{D}}\bigl(C_{\zeta_0}(a),\sigma_0\bigr) \ ,$$ and transforms $H_1$ into a cyclically symmetric function with seed $$h_1\in{\mathcal{D}}\bigl(C_{h_1}(a),\sigma_1\bigr) \ .$$ Some remarks are in order. 1. We first recall that it is exactly the above linear transformation which introduces the interaction among all sites, with an exponential decay with respect to their distance. 2. The original claim in [@GioPP13] would actually give $\zeta_0\in{\mathcal{D}}(C_{\zeta_0}(a),\sigma_a)\subseteq {\mathcal{D}}(C_{\zeta_0}(a),\sigma_0)$, since $\sigma_a\geq \sigma_0$. The choice of taking $\zeta_0\in{\mathcal{D}}(C_{\zeta_0}(a),\sigma_0)$, thus loosing a bit of the exponential decay, is useful to simplify the control of the decay in the whole normal form construction. 3. As in the proof of Lemma 3.4 of [@GioPP13], the decay rate $\sigma_1$ of the seed $h_1$ cannot be equal to that of the linear transformation, but can be chosen arbitrarily close, i.e. one has that $h_1\in{\mathcal{D}}\bigl(C_{h_1}(a),\sigma_1\bigr)$ for any $\sigma_1<\sigma_0$. This is especially true when $\sigma_b>\sigma_a=\sigma_0$. We nevertheless make the following choice for $\sigma_1$ $$\label{e.sigma.1} \sigma_1 := \frac12\sigma_0\ ,$$ once again, in order to simplify some calculations. ### Normal form Theorem {#sss:nft} From now on we will simply indicate with ${\mathsf{C}}$ any constant which does not depend on the relevant parameters, i.e. $R$, $r$ and $c$. Consider the extensive Hamiltonian $H$ in the new “exponentially localized” coordinates $(\tilde x,\tilde y)$, introduced by the previous linear transformation $$H = H_\Omega + Z_0 + H_1\ ;$$ we have (see Theorem 1 in [@PalP14]): \[prop.gen\] Consider the Hamiltonian $H=h^{\oplus}_{\Omega}+\zeta^{\oplus}_0 + h^{\oplus}_1$ with seeds $h_{\Omega}=\frac{\Omega}{2}(x_0^2+y_0^2)$, the quadratic term $\zeta_0$ of class ${\mathcal{D}}(C_{\zeta_0},\sigma_0)$ with $\zeta_0^{(0)}=0$, and the quartic term $h_1$ of class ${\mathcal{D}}(C_{h_1},\sigma_1\,)$. Pick a positive $\sigma_0/4\leq\sigma_*<\sigma_1$, then there exist positive $\gamma$, $r_{*}$ and $C_*$ such that for any positive integer $r$ satisfying $$\label{e.muperr} r\lt r_*\ ,$$ there exists a finite generating sequence $\Chi=\{\chi^{\oplus}_1,\ldots,\chi^{\oplus}_r\}$ of a Lie transform such that $T_{\Chi}{H^{(r)}_{}} = H$ where ${H^{(r)}_{}}$ is an extensive function of the form $$\label{e.Ham.r} {H^{(r)}_{}} = H_\Omega + {\mathcal Z}+ {P^{(r+1)}}\ , \qquad\qquad \begin{aligned} {\mathcal Z}:&= Z_0 + \dots + Z_r \\ \lie{\Omega}Z_s&=0 \ , \quad \forall s\in\{0,\ldots,r\} \ , \end{aligned}$$ with $Z_s$ of degree $2s+2$ and $P^{(r+1)}$ a remainder starting with terms of degree equal or bigger than $2r+4$. Moreover, defining $C_r := 64r^2C_*$ and $\sigma_j := \sigma_1 -\frac{j-1}{r}(\sigma_1-\sigma_*)$, the following statements hold true: 1. the seed $\chiph_s$ of $\Chi_s$ is of class ${\mathcal{D}}(C_r^{s-1} \frac{C_{h_1}}{\gamma s}, \sigma_s)$; 2. the seed $\zeta_s$ of $Z_s$ is of class ${\mathcal{D}}(C_r^{s-1}\frac{C_{h_1}}{s}, \sigma_s)$; 3. with the choice $\sigma_* = \sigma_0/4$, if it is satisfied the smallness condition on the total energy $$\label{e.R.sm1} R<R^*:= \sqrt{\frac2{3(1+e)C_r}}\ ,$$ then the generating sequence $\Chi$ defines an analytic canonical transformation on the domain $B_{\frac23 R}$ with the properties $$B_{R/3}\subset T_\Chi B_{\frac23 R} \subset B_R\qquad\qquad B_{R/3}\subset T_\Chi^{-1} B_{\frac23 R} \subset B_R\ .$$ Moreover, the deformation of the domain $B_{\frac23 R}$ is controlled by $$\label{e.def.Tchi} z\in B_{\frac23 R}\qquad\Rightarrow\qquad {\left\|T_\Chi(z)-z\right\|}\leq {\mathsf{C}}C_* R^3\ ,\qquad {\left\|T^{-1}_\Chi(z)-z\right\|}\leq {\mathsf{C}}C_* R^3\ .$$ 4. with the choice $\sigma_*=\sigma_0/4$, if it is satisfied , the remainder is an analytic function on $B_{\frac23 R}$, and it is represented by a series of extensive homogeneous polynomials ${H^{(r)}_{s}}$ of degree $2s+2$ $$\label{e.rem.r} P^{(r+1)} = \sum_{s\geq r+1}{H^{(r)}_{s}}\qquad {H^{(r)}_{s}} = {{\left(h^{(r)}_s\right)}}^{\oplus}\ ,$$ and the seeds $h^{(r)}_s$ are of class ${\mathcal{D}}(2\tilde C_r^{s-1}C_{h_1},\sigma_*)$ with $\tilde C_r = \frac32 C_r$. In the following, for the same reasons bringing to the choice , we will assume $\sigma_*=\sigma_0/4$, as in the last two sub-points of Theorem \[prop.gen\]. Hence, from and the previous setting of $\sigma_*$ one gets the relation $$\label{e.mu.sigma*} \mu = e^{-\sigma_*}\ ,$$ and it is possible to give the following values of some of the constants involved in the above Theorem: $$\label{const.prop.gen} \begin{aligned} r_* &= \frac{\Omega} {24C_{\zeta_0}}f(\mu)\ ,\qquad f(\mu):=\frac{(1-\mu^4)(1-\mu^3)}{\mu^2} \\ \gamma &= 2\Omega\Bigl(1-\frac{r}{2r_*}\Bigr) \ , \\ C_* &= \frac{3 C_{h_1}}{\gamma(1-\mu^4)(1-\mu^3)} \ . \end{aligned}$$ By noticing that condition implies $$\Omega<\gamma<2\Omega\ ,$$ we obtain that $C_*$ [*essentially*]{} depends on $\mu$, through $\sigma_0$ and ${\frac{C_{h_1}}{\Omega}}$ $$\frac{3 C_{h_1}}{2\Omega(1-\mu^4)(1-\mu^3)} < C_* < \frac{3 C_{h_1}}{\Omega(1-\mu^4)(1-\mu^3)} \ ;$$ and this provides $C_r=C_r(r,\mu)$ and $R^* = R^*(r,\mu)$ with $$\frac{\partial C_r}{\partial r}>0 \ , \qquad \frac{\partial C_r}{\partial \mu}>0 \ , \qquad\qquad \frac{\partial R^*}{\partial r}<0 \ , \qquad \frac{\partial R^*}{\partial \mu}<0 \ .$$ In the forthcoming application, developed in Section \[s:3\], instead of fixing $\mu$ as the main (small) parameter like in Theorem \[prop.gen\], we decide to pick the order $r\geq 1$ of the normal form, and thus the length of the time scale, as the principal parameter. As a consequence, by inverting the function $f(\mu)$ in the first of , the normal form holds for all $\mu<\mu^*(r)$, with $$\label{e.mu.star} \mu^*(r):= f^{-1}{{\left(\frac{24 C_{\zeta_0}r}{\Omega}\right)}}\ .$$ Thus, for any $\mu<\mu^*$ we have $R^*(r,\mu) > R^*(r,\mu^*)$. We then take a threshold $R_*(r)$ for the norm which is uniform with $\mu<\mu^*$ $$\label{e.R.star} R_{*}(r) := R^*(r,\mu^*(r))\ .$$ We summarize the new conditions on the parameters as follows $$\begin{aligned} \label{e.all.sm} r &\geq 1\ ,\nonumber\\ \mu &< \mu_1^*(r)\ ,\qquad\Leftrightarrow\qquad c < c_1^*(r)\ ,\\ R &< R_{*}(r)\ .\nonumber\end{aligned}$$ The normal form Theorem \[prop.gen\] immediately gives the almost invariance of $H_\Omega$ and ${\mathcal Z}$, which we here formulate [(see [@PalP14], proof of Corollary 1)]{} in the transformed variables $\tilde z={\mathcal T}_\Chi(z)$ \[c.Hom.K.var\] Let us take $\tilde z(0)\in B_{\frac49R}$ and let $\tau>0$ be the escape time of the orbit $\tilde z(t)$ from $B_{\frac23 R}$. Then, for all times $|t|<\tau$, the approximate integrals of motion $H_\Omega$ and ${\mathcal Z}$ fulfill $$\begin{aligned} |H_\Omega(\tilde z(t))-H_\Omega(\tilde z(0))| &\leq {\mathsf{C}}\qquad\;\; \frac{C_{h_1} \Omega}{(1 - \mu)^2} \qquad\;\; \;R^4 {{\left(\frac23 R^2 C_r\right)}}^r \; |t|\ , \\ |{\mathcal Z}(\tilde z(t))-{\mathcal Z}(\tilde z(0))| &\leq {\mathsf{C}}\; \frac{C_{h_1} (\mu C_{\zeta_0} + C_{h_1} R^2)}{(1 - \mu)^2} \;R^4 {{\left(\frac23 R^2 C_r\right)}}^r \; |t|\ .\end{aligned}$$ Stability of true and approximated FPU-KG breathers {#s:3} =================================================== Let us know denote the normal form part of $H^{(r)}$ – see  – as $$\label{e.K} K:=H_\Omega + {\mathcal Z}\ ,\qquad\qquad {\{H_\Omega,{\mathcal Z}\}}=0\ ,$$ so that the transformed Hamiltonian $H^{(r)}$ can be split as $H^{(r)} = K + P^{(r+1)}$, and the corresponding Hamilton equations are $$\label{e.Ham.eq} \dot z = X_K(z) + X_{P^{(r+1)}}(z)\ .$$ The Hamiltonian $K$ (the normal form) looks naturally as the Hamiltonian of a Generalized discrete Non Linear Schrödinger equation (GdNLS), with $H_\Omega$ in the role of the additional conserved quantity; an explicit expression of the leading terms of $K$ is available in the Introduction. As a first, and intermediate, application of such a normal form, we give an approximation result for the original system : we show that for sufficiently small couplings its dynamics stays close for long times to a closed trajectory in the phase space, provided its initial datum is also close enough to such an object. This trajectory is not an orbit of the original system, but it is a breather of the GdNLS model. The theorem we formulate actually contains, as a first point, and then exploits, an existence and stability result for the GdNLS breather itself with respect to the GdNLS dynamics. Such a first part, despite the generalized nature of the model, is not unexpected in the anti-continuous limit. The other point, actually the long time control for the (FPU-)KG model, is less trivial and indeed it is strongly based on our normal form result. As a second application, we obtain a result of stability of the true breather for the original system , based on the observation that in the anti-continuous limit there always exists a true breather which is close enough, with respect to the greatest parameter $c$, to the approximated one. Thus, the stability we get is actually due to the stability of the GdNLS orbit. Stability of approximated FPU-KG breathers {#ss:gdnls.br} ------------------------------------------ Since we base the existence part on the anti-continuous limit, let us denote by $\tilde\psi_0$ the $0^{\rm th}$-site excitation in the transformed coordinates $({\tilde x}_j,{\tilde y}_j)$, i.e. [ $$\label{e.def.psi0} \tilde\psi_0 := \{({\tilde x}_j,{\tilde y}_j)_{j=0,\ldots,N-1} \ \colon\ \ \ {\tilde x}_0=\rho ,\ \ {\tilde y}_0=0,\ \ {\tilde x}_j={\tilde y}_j=0 \ \forall j\neq0\} \ ,$$ ]{} which is indeed the profile of an initial datum belonging to a periodic orbit [${{\mathsf O}}(\tilde\psi_0)$]{} (trivially a breather) for the uncoupled system with $a=b=\mu=0$ (see  below), and for every fixed value of $\rho$. A consistent choice for the values of $\rho$ will be made later. \[p.orb.contr\] Given $r$ and $R$ fulfilling , there exists $c_*(r,R)$ such that, for any $c<c_*$: 1. there exist a profile $\tilde\psi_{a,b}$ and a frequency $\lambda_{a,b}$ such that $\tilde\psi_{a,b} e^{\Im \lambda_{a,b} t}$ is a Breather solution for the GdNLS with ${\left\|\tilde\psi_{a,b}\right\|}=R/6$ and $$\label{e.ex.psi.ab} {\left\|\tilde\psi_{a,b}-\tilde\psi_0\right\|} \leq {\mathsf{C}}\,\mu\ .$$ 2. let us define $$\label{e.inv.psi.ab} \psi_{a,b}:={\mathcal T}_{\Chi}^{-1}{\tilde\psi_{a,b}}\ .$$ For any $0<\eps\ll R^2$ there exists $\delta(\eps)$ such that the (piece of) orbit ${{{\mathsf O}}}(\phi):=\{\phi(t) \ :\ |t| \leq T_{\eps,r,R} \,,\ \phi(0)=\phi \}$, solution of , satisfies $$ {\left\|\phi-\psi_{a,b}\right\|}<\delta \quad\Longrightarrow\quad d_H{{\left({{{\mathsf O}}}(\phi),{{{\mathsf O}}}(\psi_{a,b})\right)}} < \eps \ .$$ where $$\label{e.times} T_{\eps,r,R} := C_T \frac{\eps^2}{{R^4}} {{{\left(C_{**}Rr\right)}}^{-2r}}\ ,$$ with $C_T$ a suitable constant independent on $\eps,\,r$ and $R$, [and $C_{**}:=8\sqrt{\frac{2C_*}{3}}$.]{} We could rephrase the result as follows: for small but non vanishing coupling $\mu$, if we start close enough to the trajectory ${{{\mathsf O}}}( \psi_{a,b})$ of a GdNLS Breather, we stay close to it (actually in a small tubular neighborhood of it) for long times. The proof of the Theorem is made of three steps, which are discussed in the following subsections: first the existence of a breather for the GdNLS with a continuation from the $\mu=0$ limit, then its orbital stability exploiting the exact conservation of $H_\Omega$ (or equivalently of ${\mathcal Z}$) for the Hamiltonian $K$, and as a last step the control of the time scale needed to see the effect of the remainder $P^{(r+1)}$ once the dynamics taken into account is that of the original system. ### Existence of Breather solutions for the GdNLS We denote with ${{{\mathcal S}}}\subset{\mathcal P}$ the sphere ${{{\mathcal S}}} := \{z\in{\mathcal P}\,\big|\,H_\Omega(z)=\rho^2\}$ of (small) radius $\rho<R_*$. The proof of the first part of Theorem \[p.orb.contr\] is given by the Proposition below, setting $\rho=R/6$. \[p.exist.b\] Given $\rho<R_*$, there exists a threshold $c_2^*(\rho)$ and a function $G:(a,b) \mapsto \tilde\psi_{a,b}$, which belongs to ${\mathcal C}^1([0,c_2^*)\times [0,c_2^*),{{\mathcal S}})$, such that $G(0,0)=\tilde\psi_0$ and $$\label{e.sol.b} d{\mathcal Z}\big|_{{{{\mathcal S}}}} (\tilde\psi_{a,b},a,b) = 0\ .$$ Moreover, $\tilde\psi_{a,b}$ is close to $\tilde\psi_0$ $$\label{e.close.b} {\left\|\tilde\psi_{a,b}-\tilde\psi_0\right\|} \leq {\mathsf{C}}\, \mu\ .$$ A formal proof of Proposition \[p.exist.b\] is deferred to the Appendix. As we said above, the idea for existence and localization is to exploit a continuation from the uncoupled limit. If $\mu=0$, the model $K$ reduces to a system of $N$ uncoupled an-harmonic oscillators which admits [$\tilde\psi_0$]{} (see  above) as a local extremizer of the constrained problem $$\label{e.dec.constr} \lambda X_{H_\Omega}(z) = X_{{\mathcal Z}}(z)\ ,\qquad {\mathcal Z}:=Z_1+\ldots+Z_r\ ,$$ where $$\zeta_s(\tilde x,\tilde y) := c_s({\tilde x}_0^2+{\tilde y}_0^2)^{s+1}\ .$$ Indeed, for $\mu=0$ (which means $a=b=0$), the first linear transformation becomes the identity and all the resonant normal form construction reduces to $N$ identical Birkhoff normal forms for a single an-harmonic oscillator. This means that $\sigma_0=\sigma_*=\infty$ and $\gamma=\Omega=1$ and $c_s$ fulfill $$\label{e.cs} \begin{cases} |c_1| = C_{h_1}\\ |c_s| \leq \frac{C_{h_1}}{s}C_r^{s-1}\ ,\qquad s=2,\ldots,r \end{cases}\ .$$ Moreover, [from its definition in Theorem \[prop.gen\],]{} $C_r = {\cal O}{{\left(r^2C_{h_1}\right)}}$. The uncoupled constrained problem reads explicitly $$\label{e.constr} \begin{cases} 2\lambda_0 {\tilde x}_j = 4 {\tilde x}_j \sum_{s=1}^r s c_s ({\tilde x}_j^2+{\tilde y}_j^2)^{s} \\ 2\lambda_0 {\tilde y}_j = 4 {\tilde y}_j \sum_{s=1}^r s c_s ({\tilde x}_j^2+{\tilde y}_j^2)^{s} \\ \sum_j {{\tilde x}_j^2+{\tilde y}_j^2}=\rho^2 \end{cases}\ , \qquad\Rightarrow\qquad \tilde\psi_0(t) = \tilde\psi_0 e^{\Im \lambda_0 t} \ ,$$ with the Lagrange multiplier $\lambda_0 = 2 \sum_{s=1}^r s c_s \rho^{2s}$ satisfying $$\label{e.lambda0.est} |\lambda_0|\geq 2 C_{h_1}\rho^2 - 2 \sum_{s=2}^r s |c_s| \rho^{2s} > \frac57 C_{h_1} \rho^2\ ,$$ where and the condition have been used. The constrained Hessian $M$ is a block-diagonal matrix with blocks $M_j$; when evaluated on $\tilde\psi_0$, for any $j\not=0$ the block is $M_j(\tilde\psi_0)=-2\lambda_0\Id$, while for $j=0$ its spectrum is $\sigma(M_0(\tilde\psi_0))=\{0,{\cal O}(C_{h_1} \rho^2)\} = \{0,4\lambda_0\}$ with associated eigenvectors $(-{\tilde y}_0,{\tilde x}_0)$ for the Kernel and $({\tilde x}_0,{\tilde y}_0)$ for the positive direction. This easily proves that $M$ is definite in all the directions transverse to the orbit generated by the Hamiltonian field $X_{H_\Omega}=\Omega(-{\tilde y}_0,{\tilde x}_0,0,\ldots,0)$. Then[^12] for $\mu/\rho^2$ small enough the Implicit Function Theorem (IFT) can be applied to uniquely continue $\tilde\psi_0$ to a solution $\tilde\psi_{a,b}$ of ${\mathcal Z}$ constrained to the level surfaces of $H_\Omega$ which generates a breather evolution ${{{\mathsf O}}}{{\left(\tilde\psi_{a,b}\right)}}$ $$\label{e.Br.gdnls} \begin{cases} \lambda_{a,b}\nabla H_\Omega = \nabla {\mathcal Z}\\ \sum_j {{\tilde x}_j^2+{\tilde y}_j^2}=\rho^2 \end{cases}\ , \qquad\Rightarrow\qquad {{{\mathsf O}}}{{\left(\tilde\psi_{a,b}\right)}} = \tilde\psi_{a,b} e^{\Im \lambda_{a,b} t}\ .$$ Concerning the exponential localization, we first remark that it is a meaningful property also in finite dimension since we aim at estimates uniform in $N$. From a technical point of view, in the infinite dimensional case ($N=\infty$) the exponential decay of the amplitudes ${\tilde x}_j^2+{\tilde y}_j^2$ of can be obtained for example with the IFT on the (Hilbert) phase space $\ell^2_\sigma\times\ell^2_\sigma$ of square summable sequences with an exponential decay. Alternatively, it can be obtained by homoclinic orbits, as in [@QinX07], or by some properties of the inverse of a Tridiagonal linear operator (still in the IFT framework), as in [@MacA94]. In the finite case ($N<\infty$), we can speak of an asymptotic exponential decay as $N$ grows arbitrarily large. For the sake of simplicity, we prefer showing the proof in the energy norm. However, we stress here that the proof holds also when using a norm with exponential weights, i.e. the finite dimensional subspace of $\ell^2_\sigma$. In this case, the coercivity constant of the constrained extremizer is proportional to $e^{-2\sigma}$, hence the threshold $\mu^*(\rho,\sigma)$ will be a decreasing function of the decay rate $\sigma$. ### Orbital stability of the GdNLS breather \[p.orb.stab\] Let $\tilde\psi_{a,b}$ be given by Proposition \[p.exist.b\]. For any positive $\eps\ll \rho^2$ there exists a positive $\delta(\eps)$ such that the orbit ${{{\mathsf O}}}(\tilde\phi):=\{\tilde\phi(t) \ :\ t\in\RR \,,\ \tilde\phi(0)=\tilde\phi \}$ of the GdNLS (see ), satisfies $$ {\left\|\tilde\phi-\tilde\psi_{a,b}\right\|}<\delta \quad\Longrightarrow\quad d_H{{\left({{{\mathsf O}}}(\tilde\phi),{{{\mathsf O}}}(\tilde\psi_{a,b})\right)}} < \eps \ .$$ Given the above use of a constrained critical point formulation to get the existence of the Breather, the Lyapunov stability in the energy norm of such an orbit follows once we verify that the Breather is still an extremizer of ${\mathcal Z}$ in all the constrained transverse directions and we exploit the fact that $H_\Omega$ and ${\mathcal Z}$ are exact constants of motion for $K$. The detailed proof is deferred to the Appendix. ### Orbital stability of the approximated FPU-KG breather Here[^13] we want to use the existence and stability of $\tilde\psi_{a,b}$ of Propositions \[p.exist.b\] and \[p.orb.stab\] in order to prove the second part of Theorem \[p.orb.contr\]. For this reason, we start taking $$c_*(r,R) := \min\{c_1^*,c_2^*\}\ ,$$ where $c_{1,2}^*$ are introduced in and in Proposition \[p.exist.b\]. We initially remark that the result is first formulated for the transformed Hamiltonian . In order to give the corresponding statement in the original variables, one has to recall that the canonical transformation ${\mathcal T}_{\Chi}$ is a perturbation of the identity , hence $${\mathcal T}_{\Chi}(z) = z + w(z)\qquad\qquad {\left\|w(z)\right\|} = {\cal O}({\left\|z\right\|}^3)\ ,$$ which implies that ${\mathcal T}_{\Chi}$ (and its inverse) is locally Lipschitz in any ball $B_R$ sufficiently close to the origin, with a constant $L = {\cal O}(1)$. Thus the control in the transformed variables can be transferred to the original ones using the Lipschitz constant of $T_\Chi^{-1}$ $$d(\phi(t),{{{\mathsf O}}}(\psi_{a,b})) = \inf_{\tilde w\in{{{\mathsf O}}}(\tilde\psi_{a,b})}{\left\|{\mathcal T}_{\Chi}^{-1} \tilde \phi(t) - {\mathcal T}_{\Chi}^{-1} \tilde w\right\|} \leq L d(\tilde\phi(t),{{\mathsf O}}(\tilde\psi_{a,b}))\ .$$ Let us work, then, in the transformed variables $\tilde z$. Our original system is in the normal form , thus $H_\Omega$ and ${\mathcal Z}$ are only approximate integrals of motion. Hence the drift from the tubular neighborhood of ${{{\mathsf O}}}(\tilde\psi_{a,b})$ is bounded by the variation of $H_\Omega$ and ${\mathcal Z}$. We assume $\phi(0)\in B_{R/3}$ and ${\left\|\tilde\psi_{a,b}\right\|}=R/6$ (hence $\tilde\psi_{a,b}\in B_{\frac23 R}$) where $R$ satisfies . We define $$\tilde\phi(t) := {\mathcal T}_\Chi\phi(t)\ ,\qquad\qquad \tilde\phi(0) := {\mathcal T}_\Chi\phi(0)\ .$$ The variations of $H_\Omega$ and ${\mathcal Z}$ in the transformed variables are controlled by Corollary \[c.Hom.K.var\], at least as long as $\tilde\phi(t)\in B_{\frac23R}$. The idea is to work as in the proof of Proposition  in the Appendix. Let us assume that $\tilde\phi(t)\in{\mathcal U}$ (which is true for $t<T_{{\mathcal U}}$ for some $T_{{\mathcal U}}$ if $\tilde\phi(0)$ is sufficiently close to $\tilde\psi_{a,b}$), where ${\mathcal U}$ is the tubular neighborhood (defined in the above mentioned proof). In such a neighborhood the orbital distance from $\tilde\psi_{a,b}$ can be related to the variations of $H_\Omega$ and ${\mathcal Z}$ as in . Then it follows immediately $$\begin{aligned} d(\tilde\phi(t),{{\mathsf O}}(\tilde\psi_{a,b})) &< \sqrt{c_3|H_\Omega(\tilde\phi(t))-H_\Omega(\tilde\phi(0))| + \frac{c_4}{C_\mu}|{\mathcal Z}(\tilde\phi(t)) - {\mathcal Z}(\tilde\phi(0))|} +\\ &+\sqrt{c_3|H_\Omega(\tilde\phi(0))-H_\Omega(\tilde\psi_{a,b})| + \frac{c_4}{C_\mu}|{\mathcal Z}(\tilde\phi(0)) - {\mathcal Z}(\tilde\psi_{a,b})|}=:A+B\ .\end{aligned}$$ with $C_\mu$ defined at the beginning of the proof of Proposition \[p.orb.stab\]. By using both $(C_{\zeta_0}\mu+C_{h_1} R^2)/C_\mu={\cal O}(1)$ and Corollary \[c.Hom.K.var\], we obtain $$\label{e.dist.2} A^2 \leq {\mathsf{C}}\frac{C_{h_1} \Omega}{(1-\mu)^2} {\;R^4 {{\left(\frac23 R^2 C_r\right)}}^r \; |t|\ ,}$$ which gives $$A<\frac{\eps}{2L}\ ,\qquad\qquad |t|\leq T_{\eps,r,R}\ ,$$ with a suitable choice of $C_T$. On the other hand, the distance in the original coordinates can be bounded by exploiting the (local) Lipschitz constant $L$ $$d(\tilde\phi(0),\tilde\psi_{a,b}) \leq L d(\phi(0),\psi_{a,b} )\leq L\delta(\eps)\ ,$$ thus for any $\eps$ there exists $\delta(\eps)$ such that $$d(\phi(0),\psi_{a,b} )<\delta \qquad\Rightarrow\qquad B<\frac{\eps}{2L}\ .$$ We can collect all the previous estimates to get $$d(\tilde\phi(t),{{\mathsf O}}(\tilde\psi_{a,b})) < \frac{\eps}{L}\ ,$$ which ensures $\tilde\phi(t)\in{\mathcal U}$ and yields to $$\label{e.dist.3} d(\phi(t),{{{\mathsf O}}}(\psi_{a,b})) < \eps \ll R^2\ ,$$ for all $|t|\leq T_{\eps,R,r}$, i.e. we have $d({{\mathsf O}}(\phi),{{{\mathsf O}}}(\psi_{a,b})) < \eps$. The same arguments of the final part of the proof of Proposition \[p.orb.stab\] apply here, so we can get also $d({{{\mathsf O}}}(\psi_{a,b}),{{\mathsf O}}(\phi)) < \eps$ and conclude the estimate. As a final comment, we observe that for the time scale considered, the orbits ${{\mathsf O}}(\phi)$ and ${{\mathsf O}}(\tilde\phi)$ remain in the domains of definition of the normal form. Indeed since $${\left\|{\mathcal T}_{\Chi}^{-1}{{{\mathsf O}}}(\tilde\psi_{a,b})\right\|} = {\left\|\tilde\psi_{a,b}\right\|} + {\cal O}(R^3)=\frac{R}6+ {\cal O}(R^3)\ ,$$ we obtain $${\left\|\phi(t)\right\|} < \frac{R}3\ ,\qquad\qquad {\left\|\tilde\phi(t)\right\|}<\frac23R\ .$$ [[-18pt$\square$]{}]{} Orbital stability of the true FPU-KG Breather: proof of Theorem \[t.b.kg\] -------------------------------------------------------------------------- We recall again that when $c=0$ the normal form transformation ${\mathcal T}_\Chi$ corresponds to the common Birkhoff change of coordinates replicated for all the identical an-harmonic oscillators. We denote by $$\label{e.def.Psi0} \Psi_0 := {{\mathcal T}}_{\Chi}^{-1} \tilde\psi_0 \ ,$$ the one-site excitation in the Birkhoff coordinates, with $\tilde\psi_0$ given by . Since $${\left\|\Psi_0 - \tilde\psi_0\right\|} \approx {\left\|\tilde\psi_0\right\|}^3\ ,$$ the new amplitude will be a $R^3$ deformation of the original one $${\left\|\Psi_0\right\|} = \frac{R}6 + {\cal O}(R^3)\ .$$ At fixed small amplitude, when the coupling parameters $a,\,b$ are switched on, the one-site periodic orbit $\Psi_0$ can be continued to (form) a family $\Psi_{a,b}$, provided $c<c_3^*(R)$, as originally proved in [@MacA94]. On the other hand, the reference solution $\tilde\psi_0$ can be continued to (form) a family $\tilde\psi_{a,b}$ of orbitally stable Breather solutions for the normal form , as claimed in Proposition \[p.orb.stab\]. We recall that in we have denoted by $\psi_{a,b}$ the inverse image of such a family of [*approximated Breather solution*]{} for the original FPU-KG model, in the sense of Proposition \[p.orb.contr\]. Due to our initial choice for $\Psi_0$, the two families initially coincide. Hence, there exists $c_4^*(r,R)<c_3$, such that for $c<c_4^*$ the two families are $\mu$-close $$\label{e.a-close} {\left\|\Psi_{a,b} -{\psi_{a,b}}\right\|}<{\mathsf{C}}\,\mu\ .$$ With this kind of control we are actually able to close the proof by the use of the triangle inequality (and this is ultimately the reason to use the Hausdorff distance). Indeed we control the distance between ${{\mathsf O}}(\Psi_{a,b})$ and ${{\mathsf O}}(\phi)$ triangulating via ${{\mathsf O}}(\psi_{a,b})$. We proceed as follow. We exploit the stability of the GdNLS breathers to control both ${{\mathsf O}}(\phi)$ and ${{\mathsf O}}(\Psi_{a,b})$. We thus apply twice the second part of Theorem \[p.orb.contr\], first to ${{\mathsf O}}(\phi)$: $$\label{e.tri.1} \exists \delta(\eps/2)\ :\ {\left\|\phi-\psi_{a,b}\right\|}<\delta \qquad\Longrightarrow\qquad d_H{{\left({{\mathsf O}}(\psi_{a,b}),{{\mathsf O}}(\phi)\right)}}<\frac{\eps}2 \ ,$$ and then to ${{\mathsf O}}(\Psi_{a,b})$: $$\label{e.tri.2} \exists \delta(\eps/2)\ :\ {\left\|\Psi_{a,b}-\psi_{a,b}\right\|}<\delta \qquad\Longrightarrow\qquad d_H{{\left({{\mathsf O}}(\psi_{a,b}),{{\mathsf O}}(\Psi_{a,b})\right)}}<\frac{\eps}2 \ .$$ Concerning this second estimate we remark that the period of the Breather ${{\mathsf O}}(\Psi_{a,b})$ is of order 1 and surely shorter[^14] that then stability time $T_{\eps,r,R}$; the Breather itself is thus entirely contained in the tubular neighborhood. In order to use implications  and  one has to ensure the control on the distance of the initial datum from $\psi_{a,b}$: for $\Psi_{a,b}$ we use , and for $\phi$ we triangulate again, this time around $\Psi_{a,b}$. More precisely $${\left\|\phi-\psi_{a,b}\right\|} \leq {\left\|\phi-\Psi_{a,b}\right\|} + {\left\|\Psi_{a,b}-\psi_{a,b}\right\|} < \delta(\eps/2) \ ,$$ where the first addendum is the one whose smallness we are free to impose in the statement of the Theorem, and the second can be made as small as we wish again using . Thus, provided $c$ is small enough to effectively use , and we are close enough to $\Psi_{a,b}$ with our initial datum $\phi$, estimates  and  hold, so that $$d_H{{\left({{\mathsf O}}(\phi),{{\mathsf O}}(\Psi_{a,b})\right)}} \leq d_H{{\left({{\mathsf O}}(\phi),{{\mathsf O}}(\psi_{a,b})\right)}} + d_H{{\left({{\mathsf O}}(\psi_{a,b}),{{\mathsf O}}(\Psi_{a,b})\right)}} < \eps.$$ [[-18pt$\square$]{}]{} Appendix {#s:6} ======== Proof of Proposition \[p.exist.b\] ---------------------------------- Differently from the mostly used technique of Lagrange multipliers (see e.g. [@Wei99]), we here prefer working locally on the constraint, thus we make use of a local parametrization of the manifold with its tangent space. In this way, we still have a functional defined over a linear euclidean space. After this preliminary operation, the problem is treated with the usual IFT (see [@AmbP95] and [@KolF89]). The geometric part of the proof is trivial since the phase space is of finite dimension $2N$. However, the estimates are uniform with $N$. We consider the tangent space in a point ${\tilde z}\in{{{\mathcal S}}}$, as defined by $T_{\tilde z}{{{\mathcal S}}} := \{Y\in{\mathcal P}\,\big|\,\sum_j {\tilde z}_j Y_j=0\} = {{\langle{\tilde z}\rangle}}^\perp$, where ${{\langle{\tilde z}\rangle}}$ represents the linear space generated by ${\tilde z}$. Since $X_{H_\Omega}({\tilde z})\in T_{\tilde z}{{{\mathcal S}}}$, the set $V:=\{Y\in T_{\tilde z}{{{\mathcal S}}}\, \big| \,\sum_j Y_j (X_{H_\Omega}({\tilde z}))_j=0\}\subset T_{\tilde z}{{{\mathcal S}}}$ is a linear subspace of ${\mathcal P}$ (of dimension $2N-2$). Take ${\tilde z}=\tilde\psi_0$ as in . The phase space ${\mathcal P}$ can be decomposed into the direct sum of the tangent space $T_{\tilde\psi_0}{{{\mathcal S}}}$ and its orthogonal direction $\tilde\psi_0$, and also the tangent space itself can be decomposed into the field direction $X_{H_\Omega}(\tilde\psi_0)$ and its orthogonal complement $V$ $${\mathcal P}= T_{\tilde\psi_0}{{{\mathcal S}}}\oplus{\tilde\psi_0}\ , \qquad T_{\tilde\psi_0}{{{\mathcal S}}} = V\oplus X_{H_\Omega}(\tilde\psi_0)\ .$$ This gives the characterization $$V={\{(\tilde x,\tilde y)\in{\mathcal P}\,\big|\,{\tilde x}_0={\tilde y}_0=0\}}\ .$$ Let us work locally on a neighborhood of $\tilde\psi_0\in{\cal S}$. There exist ${{\mathcal U}}(\tilde\psi_0)\subset{{{\mathcal S}}}$ and a function $f:{\cal W}\subset T_{\tilde\psi_0}{{{\mathcal S}}}\rightarrow{{\langle\tilde\psi_0\rangle}}$ such that, for any ${\tilde z}\in{{\mathcal U}}$ there exists $h\in T_{\tilde\psi_0}{{{\mathcal S}}}$ satisfying $${\tilde z} = P(h):= \tilde\psi_0+h+f(h)\ ;$$ in rough words, locally the sphere is the graph of a function $f$ defined on the tangent space. The above map is a ${\mathcal C}^2({\mathcal W},{\mathcal U})$ diffeomorphism. From the previous decomposition of $T_{\tilde\psi_0}{{{\mathcal S}}}$, it is locally well defined the submanifold $$\label{e.subman.M} {\mathcal M}:= \{{\tilde z}\in{{\mathcal U}}\, \big| \, {\tilde z}=P(h),\, h\in V\cap{\mathcal W}\}\ .$$ By construction we have $T_{\tilde\psi_0}{\mathcal M}= V$. Since $H_\Omega$ is a preserved quantity for ${\mathcal Z}$, the flow of $X_{H_\Omega}$ is a symmetry and then $$d{\mathcal Z}\big|_{{{{\mathcal S}}}} ({\tilde z},a,b) = 0\qquad\Leftrightarrow\qquad d{\mathcal Z}\big|_{{\mathcal M}}(\tilde z,a,b)=0\ .$$ We are interested in the problem $$d {\mathcal Z}\big|_{{\mathcal M}} ({\tilde z},a,b) = 0 \ ,$$ which has the solution $\tilde\psi_0$ for $a=b=\mu=0$. From the local linear representation of ${\mathcal M}$, we can consider ${\mathcal Z}$ on the linear space $V$ $${\mathcal Z}(h,a,b) := {\mathcal Z}(P(h),a,b) = {\mathcal Z}\big|_{{\mathcal M}} ({\tilde z},a,b) \qquad h\in V\cap{\mathcal W}\ ,$$ which is at least ${\mathcal C}^2(V,\RR)$. We look for a map $$\label{e.g} g:(a,b)\in[0,c^*)\times[0,c^*)\mapsto h=g(a,b)\in V\ ,\qquad g(0,0)=0\ ,$$ such that ${\mathcal Z}'_h(g(a,b),a,b) =0$; we already know that ${\mathcal Z}'_h(0,0,0) = 0$. We set the operator $F$ $$\label{e.F} F(h,a,b) := {\mathcal Z}_h'(h,a,b) = {\mathcal Z}_\psi'(P(h),a,b)P'(h)\ ,$$ which, due to Lemma \[l.cont.pol\], is ${\mathcal C}^1$ from $V\times\RR^2$ to $V^*:=L(V,\RR)$ and satisfies $F(0,0,0)={\mathcal Z}_\psi'(\tilde\psi_0,0,0)\Id=0$. The differential $F_h'(0,0,0) = {\mathcal Z}_\psi''(\tilde\psi_0,0,0)\Id$, which maps $V$ to its dual $V^*$, has the inverse bounded by the constant $1/C_{h_1} \rho^2$; indeed, we already observed [in subsection 3.1.1]{} that, [when $a=b=0$]{}, the whole orbit generated by $\tilde\psi_0$ is a constrained strong extremizer, hence the constrained Hessian is coercive in all the directions transverse to $X_{H_\Omega}(\tilde\psi_0)$, with coercivity constant $2 C_{h_1} \rho^2$ in the euclidean norm. A direct computation, which is based on the explicit computations developed in subsection 3.1.1, shows indeed that one has $$\label{e.coerc} |{\mathcal Z}_\psi''(\tilde\psi_0,0,0)\Id{{\left[Y,Y\right]}}| \geq C_2 \sum_{j\not=0}Y_j^2 \geq {C_2}{\left\|Y\right\|}^2\ ,\qquad C_2 := 2C_{h_1} \rho^2\ ,$$ hence $$\label{e.inv.F1} {\left\|{{\left[F_h'(0,0,0)\right]}}^{-1}\right\|}_{L(V^*,V)} \leq 1/C_2\ .$$ Then there exist $\mu_2^*(\rho)$, and hence from a $c_2^*=c_2^*(\rho)$, and a function $g\in{\mathcal C}^1([0,c_2^*)\times[0,c_2^*),V)$ as in such that $F(g(a,b),a,b)=0$ with $${\left\|g(a,b)\right\|}< \mu\ ,\qquad\qquad |\mu|<\mu_2^*\ .$$ Furthermore, we recall that the IFT is based on the contraction Theorem on a closed $\eps$-ball $B_\eps\subset V$ for the operator $A_{a,b}(h):= h - {{\left[F_h'(0,0,0)\right]}}^{-1}F(h,a,b):V\mapsto V$. The requirements of being a contraction and surjective on $B_\eps$, implies that $\mu_2^*$ is bounded by the coercive constant $C_2$ in . From the property $f(h)=o({\left\|h\right\|})$ of the parametrization $P$, it immediately follows that the solution $\tilde\psi_{a,b} := P(g(a,b)) =: G(a,b)$ is $\mu$ close to $\tilde\psi_0$ $${\left\|\tilde\psi_{a,b}-\tilde\psi_0\right\|} \leq {\mathsf{C}}\, \mu\ .$$ [[-18pt$\square$]{}]{} Proof of Proposition \[p.orb.stab\] ----------------------------------- From the continuity of ${\mathcal Z}''$ we deduce that $\tilde\psi_{a,b}$ is still a strong extremizer in the direction $V$, with a coercive constant $C_\mu = \mathcal{O}(\rho^2)$. Hence the orbit generated by he flow of $X_{H_\Omega}(\psi)$ is orbitally Lyapunov stable with ${\mathcal Z}$ being the Lyapunov function (see [@BamN98; @Bam96; @Wei99; @Wei86]). In few words (inspired also by Section 8 of [@BamN98], in particular Lemmas 8.5 and 8.6, although we work in the simplified case of a finite dimensional phase space), given a generic point of the orbit $\tilde\eta\in{{\mathsf O}}(\tilde\psi_{a,b})$, there exists a neighborhood ${\mathcal W}_0$ of $\tilde\eta$ where a suitable set of coordinates can be introduced. This local representation is based on the decomposition $P_{\tilde\eta} = \nabla H_\Omega(\tilde\eta) \oplus V_{\tilde\eta}$ of the hyperplane $P_{\tilde\eta}$ orthogonal to $X_{H_\Omega}(\tilde\eta)$, for any $\tilde\eta\in{{\mathsf O}}(\tilde\psi_{a,b})$. More precisely, there exists a (tubular) neighborhood ${\mathcal W}_0$ of $\tilde\eta$ such that, for any point $\tilde z\in{\mathcal W}_0$, the hyperplane through $\tilde z$ and orthogonal to ${{\mathsf O}}(\tilde\psi_{a,b})$ is unique. This plane intersects the periodic orbit ${{\mathsf O}}(\tilde\psi_{a,b})$ in a point $\tilde\xi$, which can be obtained as the evolution of $\tilde\eta$ at “time” $\varphi$ along the flow of the periodic orbit. Hence, using the previous notation, such a plane can be decomposed as $P_{\tilde\xi} = \nabla H_\Omega(\tilde\xi) \oplus V_{\tilde\xi}$. This implies that $\tilde z$ can be locally represented by the coordinates $$\label{e.loc.coord} \tilde z\equiv (\varphi,E,v)\in\RR\times\RR\times V_{\tilde\xi}\ ,$$ where $E$ represents the displacement in the $\nabla H_\Omega(\tilde\xi)$ direction and $v$ the displacement in the $V_{\tilde\xi}$ direction(s). Using these local coordinates in order to represent $\tilde z= \tilde\phi(t)\in{\mathcal W}_0$, the orbital distance of $\tilde\phi(t)$ from ${{\mathsf O}}(\tilde\psi_{a,b})$ is controlled in ${\mathcal W}_0$ by $$\label{e.dist.0} d(\tilde\phi(t),{{\mathsf O}}(\tilde\psi_{a,b})) \leq \inf_{w\in {{\mathsf O}}(\tilde\psi_{a,b})\cap {\mathcal W}_0} {\left\|w-\tilde\phi(t)\right\|}\leq c_1 |E(t)| + c_2{\left\|v(t)\right\|}\ ,$$ with $c_{1,2}$ depending on ${\mathcal W}_0$. The first term $|E(t)|$ represents the variation of $|H_\Omega(\tilde\phi(t))-H_\Omega(\tilde\psi_{a,b})| = |H_\Omega(\tilde\phi(t))-H_\Omega(\tilde\xi)|$: indeed, being $E(t)$ the coordinate associated to the direction $\nabla H_\Omega(\tilde\xi)$, with $\tilde\xi\in{{\mathsf O}}(\tilde\psi_{a,b})$, it controls the displacement orthogonal to ${{\mathcal S}}_{\tilde\xi}$. The second term ${\left\|v(t)\right\|}$ is instead related to the variation of $|{\mathcal Z}(\tilde\phi(t)) - {\mathcal Z}(\tilde\psi_{a,b})| = |{\mathcal Z}(\tilde\phi(t)) - {\mathcal Z}(\tilde\xi)|$, which controls the $V_{\tilde\xi}$ directions transverse to the orbit, [*provided ${\left\|v(t)\right\|}$ is small enough*]{}. Here enters the fact that any point $\tilde\xi\in{{\mathsf O}}(\tilde\psi_{a,b})$ is a local extremizer for ${\mathcal Z}$ constrained to[^15] ${\mathcal M}_{\tilde\xi}$. Indeed, if we take a point $\tilde z\in V_{\tilde\xi}$ close enough to $\tilde\xi$ (such that ${\mathcal Z}$ almost coincides with its quadratic part), then a Taylor expansion gives $${\mathcal Z}(\tilde z) - {\mathcal Z}(\tilde\xi) = \frac12 {\mathcal Z}''(\tilde\xi)[\tilde z-\tilde\xi,\tilde z-\tilde\xi]+h.o.t\ ,$$ which provides the bound $$\label{e.Z.control} {\left\|\tilde z-\tilde\xi\right\|}^2 = {\left\|v\right\|}^2\leq \frac3{C_\mu}|{\mathcal Z}(\tilde z) - {\mathcal Z}(\tilde\xi)|\ ,\qquad {\left\|v\right\|}\ll C_\mu\sim \rho^2 \ .$$ Thus there exists a neighborhood ${\mathcal W}_1\subset{\mathcal W}_0$ of $\tilde\eta$ such that if $\tilde\phi(t)\in{\mathcal W}_1$ then becomes $$\label{e.dist.1} d(\tilde\phi(t),{{\mathsf O}}(\tilde\psi_{a,b})) \leq \sqrt{c_3|H_\Omega(\tilde\phi(t))-H_\Omega(\tilde\psi_{a,b})| + \frac{c_4}{C_\mu}|{\mathcal Z}(\tilde\phi(t)) - {\mathcal Z}(\tilde\psi_{a,b})|}\ .$$ with $c_{3,4}$ depending on ${\mathcal W}_1$. Since ${{\mathsf O}}(\tilde\psi_{a,b})$ is compact (being homeomorphic to $S^1$), we can cover a whole neighborhood ${\mathcal U}$ of this orbit with a finite collection (independent of $N$) of local neighborhoods like ${\mathcal W}_1$ and set of coordinates like , such that holds true. Since both $H_\Omega$ and ${\mathcal Z}$ are continuous (analytic, see Lemma \[l.cont.pol\]) constants of motion for $K$, the requirement of staying in ${\mathcal U}$ is translated in a closeness condition for the initial datum $\tilde\phi(0)$: there exists $\delta(\eps)$ such that $$d(\tilde\phi(0),{{\mathsf O}}(\tilde\psi_{a,b}))<\delta \quad\Rightarrow\quad \sqrt{c_3|H_\Omega(\tilde\phi(0))-H_\Omega(\tilde\psi_{a,b})| + \frac{c_4}{C_\mu}|{\mathcal Z}(\tilde\phi(0)) - {\mathcal Z}(\tilde\psi_{a,b})|}<\eps\ .$$ This actually gives $d({{\mathsf O}}(\tilde\phi),{{\mathsf O}}(\tilde\psi_{a,b}))<\eps$, i.e. the orbit we aim to control is contained in the tubular neighborhood of the breather $\tilde\psi_{a,b}$ for the normal form $K$. To conclude the proof we also need the symmetric control, to avoid that our orbit, despite being in ${\mathcal U}$, does not actually follow the whole trajectory of the breather. Indeed, in full generality it could happen that the orbit goes back and forth only in a section of ${\mathcal U}$; or it could happen that such a neighborhood is not homotopic to an $S^1$, e.g. it has an “eight” shape, and in that case the orbit could use the “connection” as a shortcut to follow only a part of the orbit without leaving ${\mathcal U}$. In our case these problems do not arise: indeed the GdNLS breather is given by the action of $e^{i\lambda t}$ on ${{\mathcal S}}$, i.e. it is a maximal circle on a sphere whose radius is of order $\rho$. On the other hand, ${\mathcal U}$ is the cartesian product of the breather and a disc of co-dimension one, whose radius has to be of order $\eps$ which is constrained to be smaller than $\rho^2$. As a first consequence ${\mathcal U}$ is necessarily homotopic to an $S^1$. Moreover the component of the vector field transverse to the disc is a small perturbation of the vector field in the point of the breather orbit which lies in the disc itself. It is thus not possible for any orbit in ${\mathcal U}$ to stop flowing along the tubular neighborhood, and this happen in a time which is a small perturbation of the period of the breather. The above arguments allow us to get also $d({{\mathsf O}}(\tilde\psi_{a,b}),{{\mathsf O}}(\tilde\phi))<\eps$, and this concludes the proof. [[-18pt$\square$]{}]{} #### Acknowledgments: We warmly thank Dario Bambusi for suggesting to separate this application from the normal form result [@PalP14], and for other useful comments. This research is partially supported by MIUR-PRIN program under project 2010 JJ4KPA (“Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite”). i [10]{} , [*A primer of nonlinear analysis*]{}, Cambridge University Press, Cambridge, 1995. 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Pure Appl. Math., 39 (1986), pp. 51–67. height 2pt depth -1.6pt width 23pt, [*Excitation thresholds for nonlinear localized modes on lattices*]{}, Nonlinearity, 12 (1999), pp. 673–691. , [*Stability of discrete breathers in nonlinear [K]{}lein-[G]{}ordon type lattices with pure anharmonic couplings*]{}, J. Math. Phys., 53 (2012), pp. 102701, 20. [^1]: Università degli Studi di Milano, Dipartimento di Matematica, Via Saldini 50, 20133 Milano (Italy) [^2]: [simone.paleari@unimi.it]{} [^3]: [tiziano.penati@unimi.it]{} [^4]: We can’t include the FPU case (which corresponds to $a\neq0$ and $b\neq0$, without the on-site potentials $x_j^2$ and $x_j^4$) since the normal form construction we rely on does not apply for such a model; moreover, in the present context of an application to the stability of Breather solutions, the discussion of such an extension could be pointless since in many FPU-like models, like in [@JamKC13] no fundamental Breathers exist. [^5]: [We recall that, given $E=H(x,y)$ the total energy, the specific energy is the average energy per degree of freedom $E/N$.]{} [^6]: Recently several works appeared on GdNLS models with more than first neighbor interactions, like [@KouKCR13; @ChoCMK11], or with higher nonlinearities, like [@ChoP11; @CarTCM06], where spatially localized periodic orbits, like breathers or multibreathers, are studied. [Please remark that we here use the term “generalized” exactly to indicate [*a generalization*]{}, without any particular reference to other “generalized” dNLS.]{} [^7]: We will follow the following convention to denote profiles, and consequently orbits via the symbol ${{{\mathsf O}}}(\cdot)$: $$\begin{aligned} \phi &&& \text{generic profile for FPU-KG} &&&&&&&& z=(x,y) && \text{original coordinates} \\ \Psi &&& \text{Breather profile for FPU-KG} &&&&&&&& \tilde z=(\tilde x,\tilde y) && \text{transformed coordinates} \\ \psi &&& \text{Breather profile for GdNLS} &&&&&&&& \tilde \psi && \text{objects in transformed coordinates}\end{aligned}$$ See also for the relation between an object in original coordinates and in transformed ones. [^8]: However, in order to get a meaningful result, $\eps$ shouldn’t be taken too small: otherwise the stability time $T_{\eps,r,R}$, which scales as $\eps^2$, could fall shorter than the period of the (true/approximated) Breather. [^9]: We should also remark that different choices for $\sigma_1$ and $\sigma_*$, respectively right before and right after the statement of Theorem \[prop.gen\], could lead to a boundary of the form $cr^\alpha=\text{const}$, with $\alpha>1$. This could further improve the time scale dependence on $c$, at the price of lowering the thresholds of validity for the relevant parameters (see [@PalP14], Section 4.2 for further details); we did not pursue an optimization of this type. [^10]: For an easier notation we drop the Hamiltonian $F$ in the indexes of the components of the vector field. [^11]: An immediate consequence of is that, defining the norm of the vector field as the sum of its components (i.e. a finite $\ell^1$ norm), we would get ${\left\|X_F\right\|}_R = N{\Big\|X_F\Big\|^\oplus}_R$, which in turn justify the definition , and make it consistent with our previous definition . [^12]: As clearly explained for example in [@Wei99], Section 5. [^13]: From this subsection on, we will often use $d(A,B)$, as defined in , in the particular case of the set $A$ given by a single point. [^14]: Unless one decides to take [ $\eps\ll R^2(Rr)^r$]{}, which is not necessary to get a meaningful result of orbital stability. In that case, one would get the stability only of a piece of the periodic orbit. [^15]: One can define ${\mathcal M}_{\tilde\xi}$ as the submanifold tangent to $V_{\tilde\xi}$ as in .

--- address: 'Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA' author: - 'Yuri G. Zarhin' title: Hodge classes on certain hyperelliptic prymians --- Definitions and statements ========================== Throughout this paper $K$ is a field, $\K$ its algebraic closure and $\Gal(K)=\Aut(\K/K)$ the absolute Galois group of $K$. If $X$ is an abelian variety over $\K$ then we write $\End(X)$ for the ring of all its $\K$-endomorphisms ; the notation $1_X$ stands for the identity automorphism of $X$. If $Y$ is an abelian variety over $\K$ then we write $\Hom(X,Y)$ for the corresponding group of all $\K$-homomorphisms. Let $f(x)\in K[x]$ be a polynomial of degree $n\ge 2$ with coefficients in $K$ and without multiple roots, $\RR_f\subset \K_a$ the ($n$-element) set of roots of $f$ and $K(\RR_f)\subset \K_a$ the splitting field of $f$. We write $\Gal(f)=\Gal(f/K)$ for the Galois group $\Gal(K(\RR_f)/K)$ and call it the Galois group of $f(x)$ over $K$; it permutes roots of $f$ and may be viewed as a certain permutation group of $\RR_f$, i.e., as a subgroup of the group $\Perm(\RR_f)\cong\Sn$ of permutation of $\RR_f$. (It is well known that $\Gal(f)$ is transitive if and only if $f$ is irreducible.) Let us put $$g=\left[\frac{n-1}{2}\right].$$ Clearly, $g$ is a nonnegative integer and either $n=2g+1$ or $n=2g+2$. Let us assume that $\fchar(K)\ne 2$. We write $C_{f}$ for the genus $g$ hyperelliptic $K$-curve $y^2=f(x)$ and $J(C_{f})$ for its jacobian. Clearly, $J(C_{f})$ is a $g$-dimensional abelian variety that is defined over $K$. In particular, $J(C_{f})=\{0\}$ if and only if $n=2$. The abelian variety $J(C_{f})$ is an elliptic curve if and only if $n=4$. Let us assume that $K$ is a subfield of the field $\C$ of complex numbers (and $\bar{K}$ is the algebraic closure of $K$ in $\C$). Then one may view $J(C_{f})$ as a complex abelian variety and consider its first rational homology group $\H_1(J(C_{f}),\Q)$ and the Hodge group $\Hdg(J(C_{f}))$ of $J(C_{f})$, which is a certain connected reductive algebraic $\Q$-subgroup of the general linear group $\GL(\H_1(J(C_{f}),\Q))$ [@MumfordSh; @Deligne; @Ribet; @ZarhinIzv; @MZ2]. The canonical principal polarization on $J(C_{f})$ gives rise to the nondegenerate alternating bilinear form $$\H_1(J(C_{f}),\Q) \times \H_1(J(C_{f}),\Q) \to \Q$$ and the corresponding symplectic group $\Sp( \H_1(J(C_{f}),\Q))$ contains $\Hdg(J(C_{f}))$ as a (closed) algebraic $\Q$-subgroup. In addition, $\End(J(C_{f}))$ coincides with the endomorphism ring of the complex abelian variety $J(C_{f})$ and $\End^{0}(J(C_{f}))$ coincides with the centralizer of $\Hdg(J(C_{f}))$ in $\End_{\Q}(\H_1(J(C_{f}),\Q))$ [@MumfordSh; @Ribet; @MZ2]. The following result was obtained by the author in [@ZarhinMRL Th. 2.1], [@ZarhinMMJ Sect. 10]. (See also [@ZarhinPLMS2], [@ZarhinBSMF; @ZarhinL].) \[jacobian\] Suppose that $K\subset \C$, $n \ge 5$ (i.e., $g \ge 2$) and $\Gal(f)=\ST_n$ or the alternating group $\A_n$. Then $\End(J(C_{f}))=\Z$ and $\Hdg(J(C_{f}))=\Sp( \H_1(J(C_{f}),\Q))$. Every Hodge class on each self-product of $J(C_{f})$ can be presented as a linear combination of products of divisor classes. In particular, the Hodge conjecture is valid for each self-product of $J(C_{f})$. The assertion that $\Hdg(J(C_{f}))=\Sp( \H_1(J(C_{f}),\Q))$ was not stated explicitly in [@ZarhinMMJ]. However, it follows immediately from the description of the Lie algebra $\mathrm{mt}$ of the corresponding Mumford-Tate group [@ZarhinMMJ p. 429] as the direct sum of the scalars $\Q \mathrm{Id}$ and the Lie algebra of the symplectic group, because the Lie algebra of the Hodge group coincides with the intersection of Lie algebras of the Mumford-Tate group and the symplectic group. (The same arguments prove the equality $\Hdg(J(C_{f}))=\Sp( \H_1(J(C_{f}),\Q))$ for all $f(x)$ that satisfy the conditions of Theorem 10.1 of [@ZarhinMMJ].) Our next result that was obtained in [@ZarhinSh Th. 1.2 and Theorem 2.5] deals with homomorphisms of hyperelliptic jacobians. \[homo\] Suppose that $\fchar(K)\ne 2$, $n\ge 3$ and $m\ge 3$ are integers and let $f(x)$ and $h(x)$ be irreducible polynomials over $K$ of degree $n$ and $m$ respectively. Suppose that $$\Gal(f)=\ST_n, \ \Gal(h)=\ST_m$$ and the corresponding splitting fields $K(\RR_f)$ and $K(\RR_h)$ are linearly disjoint over $K$. Then either $$\Hom(J(C_f),J(C_h))=\{0\}, \ \Hom(J(C_h),J(C_f))=\{0\}$$ or $\fchar(K)>0$ and both $J(C_f)$ and $J(C_h)$ are supersingular abelian varieties. \[homoC\] If $K\subset \C$ then Theorem \[homo\] implies that (under its assumptions) there are no nonzero homomorphisms between complex abelian varieties $J(C_f)$ and $J(C_h)$. The main results of the present paper are the following statements. \[main\] Suppose that $n=2g+2=\deg(f)\ge 8$. Let $\tilde{C}_f \to C_{f}$ be an unramified double cover of complex smooth projective irreducible curves and let $P$ be the corresponding Prym variety, which is a $(g-1)$-dimensional (principally polarized) complex abelian variety. If $\Gal(f)=\ST_n$ then: - $\End(P)=\Z$ or $\Z\oplus \Z$. - Every Hodge class on each self-product of $P$ can be presented as a linear combination of products of divisor classes. In particular, the Hodge conjecture holds true for each self-product of $P$. \[main1\] Suppose that $n=2g+2\ge 10$, $K\subset\C$ and $f(x)=(x-a)h(x)$ where $a\in K$ and $h(x)\in K[x]$ is an irreducible degree $n-1$ polynomial with $\Gal(h)=\ST_{n-1}$. Let $\tilde{C}_{f} \to C_{f}$ be an unramified double cover of complex smooth projective irreducible curves and let $P$ be the corresponding Prym variety, which is a $(g-1)$-dimensional (principally polarized) complex abelian variety. Then: - $\End(P)=\Z$ or $\Z\oplus \Z$. - every Hodge class on each self-product of $P$ can be presented as a linear combination of products of divisor classes. In particular, the Hodge conjecture holds true for each self-product of $P$. Our proof is based on the explicit description of Prym varieties of hyperelliptic curves [@MumfordP; @Dal] and our results about Hodge groups of hyperelliptic jacobians mentioned above. If $n=2g+2\le 10$ then $\dim(P)=g-1\le 3$. Notice that if $A$ is a complex abelian varietiy of dimension $\le 3$ then it is well known that every Hodge class on each self-product of $A$ can be presented as a linear combination of products of divisor classes [@MZ2 Th. 0.1(iv)]. In particular, the Hodge conjecture holds true for each self-product of $A$. The paper is organized as follows. In Section \[Galois\] we discuss an elementary construction from Galois theory and apply it in Section \[homohyper\] to homomorphisms of hyperelliptic jacobians. Section \[hyperhodge\] deals with Hodge groups of hyperelliptic jacobians. In Section \[hyperprym\] we discuss hyperelliptic prymians and prove the main results. Galois theory {#Galois} ============= Throughout this Section, $K$ is an arbitrary field and $n\ge 3$ is an integer, $f(x)\in K[x]$ is a degree $n$ irreducible polynomial, whose Galois group $$\Gal(f)=\Gal(K(\RR_f)/K)$$ is the full symmetric group $\Perm(\RR_f)=\ST_n$. If $T\subset \RR_f$ is a non-empty subset then we put $$f_T(x)=\prod_{\alpha\in T}(x-\alpha)\in K(\RR_f)[x].$$ By definition, $$\deg(f_T)=\#(T), \ \RR_{f_T}=T.$$ We view $\Perm(T)$ as a subgroup of $\Perm(\RR_f)=\Gal(f)$ that consists of all permutations that leave invariant every element outside $T$. \[oneT\] Let us consider the subfield $E_0=K(\RR_f)^{\Perm(T)}$ of $\Perm(T)$-invariants. Since $\Perm(T)$ leaves invariant $T=\RR_{f_T}$, $$f_T(x)\in E_0[x].$$ Clearly, $$\Gal(K(\RR_f)/E_0)=\Perm(T)\subset \Perm(\RR_f)=\Gal(\RR_f/K).$$ Let us prove that the splitting field $E_0(\RR_{f_T})=E_0(T)$ of $f_T(x)$ over $E_0$ coincides with $K(\RR_f)$. Indeed, $\Gal(K(\RR_f))/E_0(T))$ consists of all elements of $\Perm(T)=\Gal(K(\RR_f)/E_0)$ that leave invariant every element of $T$. Since every element of $\Perm(T)$ leaves invariant every element of $\RR_f\setminus T$, $\Gal(K(\RR_f))/E_0(T))=\{1\}$, i.e., $K(\RR_f)=E_0(T)=E_0(\RR_{f_T})$. This implies that the Galois group $$\Gal(E_0(\RR_{f_T})/E_0)=\Gal(K(\RR_f)/E_0)=\Perm(T).$$ \[key\] Let $T$ and $S$ be two nonempty disjoint subsets of $\RR_f$. Then there exists a field subextension $E/K \subset K(\RR_f)/K$ that enjoys the following properties. - The Galois group $\Gal(K(\RR_f)/E)$ coincides with the subgroup $$\Gal(T)\times \Gal(S)\subset \Perm(\RR_f)=\Gal(K(\RR_f)/K),$$ which consists of all permutations that leave invariant $T,S$ and every element outside $T \sqcup S$. - Both $f_T(x)$ and $f_S(x)$ lie in $E[x]$, i.e., all their coefficients belong to $E$. - Let $E(\RR_T)=E(T)$ and $E(\RR_S)=E(S)$ be the splitting fields over $E$ of $f_T(x)$ and $f_S(x)$ respectively. Then the natural injective homomorphisms $$\Gal(E(T)/E)\hookrightarrow \Perm(T), \ \Gal(E(S)/E)\hookrightarrow \Perm(S)$$ are group isomorphisms, i.e., $$\Gal(E(T)/E)=\Perm(T), \ \Gal(E(S)/E)= \Perm(S).$$ - $E(T)$ and $E(S)$ are linearly disjoint over $E$. - The compositum $E(T) E(S)$ coincides with $K(\RR_f)$. Recall that $$\RR_{f_T}=T, \ \RR_{f_S}=S.$$ We define $E$ as the subfield $K(\RR_f)^{\Perm(T)\times \Perm(S)}$ of $\Perm(T)\times \Perm(S)$-invariants. Now Galois theory gives us (i). The subgroup $\Perm(T)\times \Perm(S)$ leaves invariant both sets $T=\RR_T$ and $S=\RR_S$. This implies that all the coefficients of $f_T(x)$ and $f_S(x)$ are $[\Perm(T)\times \Perm(S)]$-invariant, i.e., lie in $E$. This proves (ii). Clearly, $$[K(\RR_f):E]=\#(\Perm(T)\times \Perm(S)).$$ Clearly, the subgroup of $\Perm(T)\times \Perm(S)$ that consists of all permutations that act identically on $S$ coincides with $\Perm(T)$. Similarly, the subgroup of $\Perm(T)\times \Perm(S)$ that consists of all permutations that act identically on $T$ coincides with $\Perm(S)$. This implies that $$\Gal(E(T)/E)=[\Perm(T)\times \Perm(S)]/\Perm(S)=\Perm(T),$$ $$\Gal(E(S)/E)=[\Perm(T)\times \Perm(S)]/\Perm(T)=\Perm(S),$$ which proves (iii). This implies that $$[E(T):E]=\#(\Perm(T)), \ [E(S):E]=\#(\Perm(S)).$$ Let $L$ be the compositum $E(T) E(S)$. Clearly, $L$ contains $T$, $S$ and $E$. Therefore $\Gal(K(\RR_f)/L)$ consists of elements of $\Perm(T)\times \Perm(S)=\Gal(K(\RR_f)/E)$ that act identically on $T$ and $T$. Since all elements of $\Perm(T)\times \Perm(S)$ act identically on the complement to $T \sqcup S$, we conclude that $\Gal(K(\RR_f)/L)=\{1\}$, i.e., $$K(\RR_f)=L=E(T) E(S).$$ This proves (v). We also obtain that $$[E(T) E(S):E]=[K(\RR_f):E]=\#(\Perm(T)\times \Perm(S))=[E(T):E] [E(S):E],$$ i.e. $$[E(T) E(S):E]=[E(T):E] [E(S):E],$$ which means that $E(T)/E$ and $E(S)/E$ are linearly disjoint. This proves (iv). Homomorphisms of hyperelliptic jacobians {#homohyper} ======================================== We keep the notation and assumptions of Section \[Galois\]. Also we assume that $\fchar(K)\ne 2$. \[homoG\] Let $T$ and $S$ be disjoint nonempty subsets of $\RR_f$ and consider the hyperelliptic curves $$C_{f_T}:y^2=f_T(x), \ C_{f_S}:y^2=f_S(x)$$ and their jacobians $J(C_{f_T})$ and $J(C_{f_S})$. Then either $$\Hom(J(C_{f_T}), J(C_{f_S}))=\{0\}, \ \Hom(J(C_{f_S}), J(C_{f_T}))=\{0\}$$ or $\fchar(K)>0$ and both $J(C_{f_T})$ and $J(C_{f_S})$ are supersingular abelian varieties. If $\#(T)<3$ (resp. $\#(S)<3$) then $C_{f_T}$ (resp. $C_{f_S}$) has genus zero and $J(C_{f_T})=0$ (resp. $J(C_{f_S})=0$), which implies that there are no nonzero homomorphisms between $J(C_{f_T})$ and $J(C_{f_S})$. So, further we assume that $$n_1:=\#(T)\ge 3, \ n_2:=\#(S)\ge 3.$$ By Lemma \[key\], there exists a field $E$ such that both $f_T(x)$ and $f_S(x)$ lie in $E[x]$, their Galois groups are $\Perm(T)\cong\ST_{n_1}$ and $\Perm(S)\cong\ST_{n_2}$ respectively. In addition, their splitting fields are linearly disjoint over $E$. Now the result follows from Theorem \[homo\] applied to $E, f_T(x), f_S(x)$ instead of $K, f(x), h(x)$. \[odd\] Suppose that $m:=\#(S)=2r+1$ is odd and let $b$ be an arbitrary element of $K$. Let us consider the hyperelliptic curve $C^{b}_{f_S}:y^2=(x-b)f_S(x)$. By Remark \[oneT\], there exists a field $E_0\subset K(\RR_f)$ such that $f_S(x)$ lies in $E_0[x]$ and $\Gal(E_0(\RR_{f_S})/E_0)=\Perm(S)=\ST_m$. Then the standard substitution [@ZarhinPLMS2 p. 25] $$x_1=\frac{1}{x-b}, \ y_1:=\frac{y}{(x-b)^{r+1}}$$ gives us a degree $m$ irreducible polynomial $h(x_1)\in E[x_1]$ such that $$E_0(\RR_h)=E_0(\RR_{f_S}), \ \Gal(E_0(\RR_h)/E)=\Gal(E_0(\RR_{f_S})/E)=\ST_m$$ and $C^{b}_{f_S}$ is $E_0$-birationally isomorphic to the hyperelliptic curve $C_{h}:y_1^2=h(x_1)$. (It is assumed in [@ZarhinPLMS2 p. 25] that $m \ge 5$ but the substitution works for any positive odd $m$.) \[homoGt\] Let $T$ and $S$ be disjoint nonempty subsets of $\RR_f$ and assume that $\#(S)$ is odd. Let $b$ be an arbitrary element of $K$. Let us consider the hyperelliptic curves $$C_{f_T}:y^2=f_T(x), \ C^{b}_{f_S}:y^2=(x-b)f_S(x)$$ and their jacobians $J(C_{f_T})$ and $J(C^{b}_{f_S})$. Then either $$\Hom(J(C_{f_T}), J(C^{b}_{f_S}))=\{0\}, \ \Hom(J(C^{b}_{f_S}), J(C_{f_T}))=\{0\}$$ or $\fchar(K)>0$ and both $J(C_{f_T})$ and $J(C^{b}_{f_S})$ are supersingular abelian varieties. We may assume that both $m_1=\#(T)$ and $m_2=\#(S)$ are, at least, $3$. Let $E$ be as in Lemma \[key\]. In particular, the splitting fields $E(T)$ and $E(S)$ are linearly disjoint over $E$ and $$\Gal(E(T)/E)\cong \ST_{m_1}, \ \Gal(E(S)/E) \cong \ST_{m_2}.$$ Using Remark \[homoGt\] over $E$ (instead of $E_0$), we obtain that there is a degree $m$ irreducible polynomial $h(x)\in E[x]$ such that $$E(\RR_h)=E(\RR_{f_S}), \ \Gal(E(\RR_h)/E)=\Gal(E(\RR_{f_S})/E)=\ST_{m_2}$$ and $C^{b}_{f_S}$ is birationally $E$-isomorphic to the hyperelliptic curve $C_{h}:y^2=h(x)$. Clearly, the jacobians $J(C^{b}_{f_S})$ and $J(C_h)$ are isomorphic. Applying Theorem \[homo\] to $E, f_T(x),h(x)$, we conclude that either $$\Hom(J(C_{f_T}), J(C_h))=\{0\}, \ \Hom(J(C_h)), J(C_{f_T}))=\{0\}$$ or $\fchar(K)>0$ and both $J(C_{f_T})$ and $J(C_{f_S})$ are supersingular abelian varieties. Since $J(C^{b}_{f_S})$ and $J(C_h)$ are isomorphic, we are done. \[homoGab\] Let $T$ and $S$ be disjoint nonempty subsets of $\RR_f$ and assume that both $\#(T)$ and $\#(S)$ are odd. Let $a$ and $b$ be arbitrary (not necessarily distinct) elements of $K$. Let us consider the hyperelliptic curves $$C^{a}_{f_T}:y^2=(x-a)f_T(x), \ C^{b}_{f_S}:y^2=(x-b)f_S(x)$$ and their jacobians $J(C^{a}_{f_T})$ and $J(C^{b}_{f_S})$. Then either $$\Hom(J(C^{a}_{f_T}), J(C^{b}_{f_S}))=\{0\}, \ \Hom(J(C^{b}_{f_S}), J(C^{a}_{f_T}))=\{0\}$$ or $\fchar(K)>0$ and both $J(C^{b}_{f_T})$ and $J(C^{a}_{f_S})$ are supersingular abelian varieties. We may assume that both $m_1:=\#(T)$ and $m_2:=\#(S)$ are, at least, $3$. Again, let $E$ be as in Lemma \[key\]. Applying Remark \[homoGt\] two times over $E$ (instead of $E_0$) to the polynomials $(x-a)f_T(x)$ and $(x-b)f_S(x)$, we conclude that there are degree $m$ irreducible polynomials $h_1(x)\in E[x]$ and $h_2 (x)\in E[x]$ such that $$E(\RR_{h_1})=E(T), \ E(\RR_{h_2})=E(S),$$ $$\Gal(E(\RR_{h_1})/E)=\ST_{m_1}, \ \Gal(E(\RR_{h_2})/E)=\ST_{m_2},$$ $C^{a}_{f_T}$ is $E$-birationally isomorphic to $C_{h_1}$ and $C^{b}_{f_T}$ is $E$-birationally isomorphic to $C_{h_2}$. Clearly, $J(C^{a}_{f_T})\cong J(C_{h_1})$ and $J(C^{b}_{f_S})\cong J(C_{h_2})$. Applying Theorem \[homo\] to $E, h_1(x),h_2(x)$, we conclude that either $$\Hom(J(C_{h_1}), J(C_{h_2}))=\{0\}, \ \Hom(J(C_{h_2})), J(C_{h_1}))=\{0\}$$ or $\fchar(K)>0$ and both $J(C_{h_1})$ and $J(C_{h_2})$ are supersingular abelian varieties. The rest is clear. Let $K_2/K$ be the only quadratic subextension of $K(\RR_f)/K$. Clearly, $K_2(\RR_f)=K(\RR_f)$ and the Galois group $\Gal(K_2(\RR_f)/K_2)$ coincides with the alternating group $\A_n$. \[ellipticZ\] Suppose that $\fchar(K)\ne 2$. Let $T \subset \RR_f$ be a $4$-element subset. Let us consider the corresponding elliptic curve $$C_{f_T}:y^2=f_T(x)$$ and its jacobian $J(C_{f_T})$. If $n \ge 8$ then one of the following conditions holds: - $\End(J(C_{f_T}))=\Z$ for all $T$. - $\fchar(K)>0$ and all $J(C_{f_T})$’s are supersingular elliptic curves mutually isomorphic over $\K$. Let $j_T$ be the $j$-invariant of the elliptic curve $J(C_{f_T})$ ([@Tate], [@Knapp Ch. III, Sect. 2]). Clearly, $$j_T \in K(T)\subset K(\RR_f)$$ and $$j_{\sigma T}=\sigma j_T \ \forall \sigma\in \Gal(K(\RR_f)/K)=\Gal(f).$$ Suppose that $J(C_{f_T})$ admits complex multiplication. Then one of the following two conditions holds. - $p=\fchar(K)>0$. Then a classical result of M. Deuring asserts that $j_T$ is [*algebraic*]{}, i.e., lies in a finite field $\F_q$ where $q$ is a power of the prime $p$. (See [@Deuring], [@Oort Sect. 3.2], [@Lang Ch. 13, Sect. 5].) In particular, $K(j_T)/K$ is an abelian field extension. - $\fchar(K)=0$. Then there exists an imaginary quadratic field $k$ such that $\End^0(J(C_{f_T}))=k$. In addition, a classical result of the theory of complex multiplication asserts that $j_T$ is an algebraic number such that the field extension $k(j_T)/k$ is abelian. (See [@Shimura Sect. 5.4], [@Lang Ch. 10, Sect. 3].) Let us consider the overfield $K^{\prime}$ of $K$ that is defined as follows. If $\fchar(K)>0$ then $K^{\prime}=K_2$. If $\fchar(K)=0$ then $K^{\prime}$ is the compositum $K_2 k$ of $K_2$ and the imaginary quadratic field $k$; in particular, $K^{\prime}$ contains $k$. Since $\A_n=\Gal(K^{\prime}(\RR_f)/K^{\prime})$ is simple nonabelian, the field extension $K^{\prime}(\RR_f)/K^{\prime}$ does not contain nontrivial abelian subextensions. However, $j_T \in K^{\prime}(\RR_f)$ and the field (sub)extension $K^{\prime}(j_T)/K^{\prime}$ is abelian. This implies that this subextension is trivial, i.e., $j_T\in K^{\prime}$. This means that for all $\sigma \in \Gal(K^{\prime}(\RR_f)/K^{\prime})=\A_n$ $$j_T=\sigma j_T=j_{\sigma T}.$$ Since $n \ge 8$, the permutatation group $\A_n$ is $4$-transitive and therefore the jacobians $J(C_{f_T})$’s are mutually isomorphic over $\K$ for all $4$-element subsets $T\subset \RR_f$. Let $T_1$ and $T_2$ be two [*disjoint*]{} $4$-element subsets of $\RR_f$. (Since $n\ge 8$, such $T_1$ and $T_2$ do exist.) Applying Theorem \[homoG\] to $T_1$ and $T_2$ (instead of $T$ and $S$) and taking into account that $J(C_{f_{T_1}})$ and $J(C_{f_{T_2}})$ are isomorphic over $\K$ (i.e., $\Hom(J(C_{f_{T_1}}),J(C_{f_{T_2}}))\ne\{0\}$), we conclude that $\fchar(K)>0$ and both $J(C_{f_{T_1}})$ and $J(C_{f_{T_2}}))$ are supersingular elliptic curves. \[ellipticZ1\] Suppose that $\fchar(K)\ne 2$. Let $a$ be an arbitrary element of $K$. Let $T \subset \RR_f$ be a $3$-element subset. Let us consider the corresponding elliptic curve $$C^{a}_{f_T}:y^2=(x-a)f_T(x)$$ and its jacobian $J(C^{a}_{f_T})$. If $n \ge 6$ then one of the following conditions holds: - $\End(J(C^{a}_{f_T}))=\Z$ for all $T$. - $\fchar(K)>0$ and all $J(C^{a}_{f_T})$’s are supersingular elliptic curves mutually isomorphic over $\K$. Let $j_{T,a}$ be the $j$-invariant of the elliptic curve $J(C^{a}_{f_T})$. Clearly, $$j_{T,a} \in K(T)\subset K(\RR_f)$$ and $$j_{\sigma T,a}=\sigma j_{T,a} \ \forall \sigma\in \Gal(K(\RR_f)/K)=\Gal(f).$$ Suppose that $J(C^{a}_{f_T})$ admits complex multiplication. Then, as in the proof of Theorem \[ellipticZ\], there exists an overfield $K^{\prime}\supset K_2$ such that either $K^{\prime}=K_2$ or $K^{\prime}$ is a quadratic extension of $K_2$ and in both cases $K^{\prime}(j_{T,a})\subset K^{\prime}(\RR_f)$ and the field (sub)extension $K^{\prime}(j_{T,a})/K^{\prime}$ is abelian. Again, $\A_n=\Gal(K^{\prime}(\RR_f)/K^{\prime})$ is simple nonabelian and therefore there are no nontrivial abelian subextensions of $K^{\prime}(\RR_f)/K^{\prime}$. This implies that $j_{T,a}\in K^{\prime}$, i.e., for all $\sigma \in \Gal(K^{\prime}(\RR_f)/K^{\prime})=\A_n$ $$j_{T,a}=\sigma j_{T,a}=j_{\sigma T, a}.$$ Since $n \ge 6$, the permutatation group $\A_n$ is $3$-transitive and therefore the jacobians $J(C^{a}_{f_T})$’s are mutually isomorphic over $\K$ for all $3$-element subsets $T\subset \RR_f$. Let $T_1$ and $T_2$ be two [*disjoint*]{} $3$-element subsets of $\RR_f$. (Since $n\ge 6$, such $T_1$ and $T_2$ do exist.) Applying Theorem \[homoGab\] to $T_1,a$ and $T_2,a$ (instead of $T,a$ and $S,b$) and taking into account that $J(C^{a}_{f_{T_1}})$ and $J(C^{a}_{f_{T_2}})$ are isomorphic over $\K$ (i.e., $\Hom(J(C_{f_{T_1}}),J(C_{f_{T_2}}))\ne\{0\}$), we conclude that $\fchar(K)>0$ and both $J(C^{a}_{f_{T_1}})$ and $J(C^{a}_{f_{T_2}}))$ are supersingular elliptic curves. The rest is clear. Hodge groups of hyperelliptic jacobians {#hyperhodge} ======================================= We keep the notation and assumptions of Sections \[Galois\] and \[homohyper\]. Also we assume that $K\subset \C$. We say (as in [@MZ2 Sect. 1.8]) that a complex abelian variety $X$ satisfies property (D) if every Hodge class on each self-product $X^r$ of $X$ can be presented as a linear combination of products of divisor classes. If this condition is satisfied then the Hodge conjecture is true for all $X^r$. Abelian varieties that satisfy (D) are also called [*stably nondegenerate*]{} [@hazamaT]; see also [@murty]. \[DE\] If $Y$ is an elliptic curve over $\C$ with $\End(Y)=\Z$ then it is well known [@MZ2 Th. 0.1(iv)] that $Y$ satisfies (D) and $\Hdg(Y)=\Sp(\H_1(Y,\Q))$. \[nonsimple\] Let $X_1$ and $X_2$ be complex abelian varieties of positive dimension and $X=X_1\times X_2$. Suppose that $$\End(X_1)=\Z, \ \End(X_2)=\Z, \ \Hom(X_1,X_2)=\{0\}.$$ Then: 1. $\End(X_1\times X_2)=\Z\oplus \Z$. 2. If both $X_1$ and $X_2$ satisfy (D) then $\Hdg(X)= \ \Hdg(X_1)\times \Hdg(X_2)$ and $X$ satisfies (D). \(i) is obvious. (ii) follows from [@Hazama Th. 0.1 and Prop. 1.8] (see also Theorem 3.2(i) of [@MZ2]). Since $\End(X_i)=\Z$ and $X_i$ satisfies (D), $$\Hdg(X_i)=\Sp(H_1(X_i,\Q)$$ [@hazamaT; @murty]. (See also [@MZ2 Sect. 1.8].) \[jacprym\] Suppose that $T$ is a subset of $\RR_f$ with $\#(T)\ge 5$. Let us consider the hyperelliptic curve $C_{f_T}:y^2=f_T(x)$ and its jacobian $J(C_{f_T})$. Then $\End(J(C_{f_T}))=\Z$ and $\Hdg(J(C_{f_T}))=\Sp( \H_1(J(C_{f_T}),\Q))$. In addition, $J(C_{f_T})$ satisfies (D). Let us put $m=\#(T)$. We have $m \ge 5$ and $\deg(f_T)=m\ge 5$. By Remark \[oneT\], there exists a (sub)field $$E_0\subset K(\RR_f)\subset\K\subset \C$$ such that $f_T(x)\in E_0(T)$ and the Galois group of $f_T(x)$ over $E_0$ is $\Perm(T)\cong \ST_m$. Now the result follows from Theorem \[jacobian\] applied to $m,E_0, f_T(x)$ instead of $n,K,f(x)$. \[jacfam\] Suppose that $T$ is a subset of $\RR_f$ with $\#(T)\ge 5$. Suppose that $m$ is odd and let $a$ be an arbitrary element of $K$. Let us consider the hyperelliptic curve $C^{a}_{f_T}:y^2=(x-a)f_T(x)$ and its jacobian $J(C^{a}_{f_T})$. Then $\End(J(C^{a}_{f_T}))=\Z$ and $\Hdg(J(C^{a}_{f_T}))=\Sp( \H_1(J(C^{a}_{f_T}),\Q))$. In addition, $J(C^{a}_{f_T})$ satisfies (D). By Remark \[odd\], there exists a field $$E_0\subset K(\RR_f)\subset\K\subset\C$$ and a degree $m$ irreducible polynomial $h(x)\in E_0[x]$ such that $$E(\RR_h)=E(\RR_{f_T}), \ \Gal(E(\RR_h)/E)=\Gal(E(\RR_{f_T})/E)=\ST_m$$ and $C^{a}_{f_S}$ is birationally $E$-isomorphic to to the hyperelliptic curve $C_{h}:y^2=h(x)$. Clearly, the jacobians $J(C^{a}_{f_S})$ and $J(C_h)$ are isomorphic. It follows from Lemma \[jacprym\] applied to $m,E_0,h(x)$ (instead of $n,K,f(x)$) that $\End(J(C_{h}))=\Z$, $\Hdg(J(C_h)=\Sp( \H_1(J(C_h),\Q))$ and $J(C_{h})$ satisfies (D). Since $J(C^{a}_{f_S})$ and $J(C_h)$ are isomorphic, we are done. Prym varieties {#hyperprym} ============== Following [@MumfordP; @Dal], let us give an explicit description of hyperelliptic prymians $P$, assuming that $\fchar(K)\ne 2$. Suppose that $n=2g+2\ge 6$ and $$f(x)\in K[x]\subset \K[x]$$ is a degree $n$ polynomial without multiple roots. Let us split the $n$-element set $\RR_f$ of roots of $f(x)$ into a [*disjoint*]{} union $$\RR_f =\RR_1 \sqcup \RR_2$$ of [*non-empty*]{} sets $\RR_1$ and $\RR_2$ of [*even*]{} cardinalities $n_1$ and $n_2$ respectively. Further we assume that $$n_1 \ge n_2\ge 2.$$ and put $$f_1(x)=\prod_{\alpha\in\RR_1}(x-\alpha), \ f_2(x)=\prod_{\alpha\in\RR_2}(x-\alpha).$$ We have $n_1+n_2=n$ and define nonnegative integers $g_1$ and $g_2$ by $$n_1=2g_1+2, \ n_2=2g_2+2.$$ Clearly, $$g_1+g_2=g-1.$$ Let us consider the hyperelliptic curves $C_{f_1}:y^2=f_1(x)$ and $C_{f_2}:y^2=f_2(x)$ of genus $g_1$ and $g_2$ respectively and corresponding hyperellptic jacobians $J(C_{f_1})$ and $J(C_{f_2})$ of dimension $g_1$ and $g_2$ respectively. Then the prymians $P$ of $C_f: y^2=f(x)$ are just the products $J(C_{f_1})\times J(C_{f_2})$ for all the partitions $\RR_f =\RR_1 \sqcup \RR_2$. Now Theorem \[main\] becomes an immediate corollary of the following statement. \[help\] Suppose that $n=2g+2\ge 8$, $K\subset\C$ and $\Gal(f)=\ST_n$. Let us put $P=J(C_{f_1})\times J(C_{f_2})$. Then: - $\Hom(J(C_{f_1}),J(C_{f_2}))=\{0\}, \ \Hom(J(C_{f_2}),J(C_{f_1}))=\{0\}$. - Suppose that $g_i \ge 1$, i.e., $n_i \ge 4$. Then $$\End(J(C_{f_i}))=\Z, \ \Hdg(J(C_{f_i}))=\Sp( \H_1(J(C_{f_i}),\Q))$$ and $J(C_{f_i})$ satisfies (D). - If $n_2=2$ then $P=J(C_{f_1})$. In particular, $\End(P)=\Z$, $\Hdg(P)=\Sp( \H_1(P,\Q))$ and $P$ satisfies (D). - If $n_2 \ge 4$ then $\End(P)=\Z\oplus\Z$, $$\Hdg(P)=\Hdg(J(C_{f_1}))\times \Hdg(J(C_{f_2}))=\Sp( \H_1(J(C_{f_1}),\Q))\times \Sp( \H_1(J(C_{f_2}),\Q))$$ and $P$ satisfies (D). The assertion (i) follows from Theorem \[homoG\]. If $n_i\ge 6$ then (ii) follows from Lemma \[jacprym\]. Suppose that $n_i=4$. Then $J(C_{f_i})$ is an elliptic curve. It follows from Theorem \[ellipticZ\] that $\End(J(C_{f_i}))=\Z$. Now the assertion about its Hodge group and property (D) follows from Example \[DE\]. This completes the proof of (ii). Let us prove (iii). If $n_2=2$ then $J(C_{f_2})=0$ and therefore $P=J(C_{f_1})$. Now the assertion follows from (ii). Let us prove (iv). We assume that $$n_1 \ge n_2 \ge 4.$$ By already proven (i) and (ii), $$\End(J(C_{f_1}))=\Z, \ \End(J(C_{f_2}))=\Z, \ \Hom(J(C_{f_1}),J(C_{f_2}))=\{0\}.$$ Now (iv) follows from Theorem \[nonsimple\] applied to $X_1=J(C_{f_1})$, $X_2=J(C_{f_2})$ and $X=P$. Theorem \[main1\] is an immediate corollary of the following statement. \[help1\] Suppose that $n=2g+2\ge 10$, $K\subset\C$ and $f(x)=(x-a)h(x)$ where $a\in K$ and $h(x)\in K[x]$ is an irreducible degree $(n-1)$ polynomial with $\Gal(h)=\ST_{n-1}$. Let us put $P=J(C_{f_1})\times J(C_{f_2})$. Then: - $\Hom(J(C_{f_1}),J(C_{f_2}))=\{0\}, \ \Hom(J(C_{f_2}),J(C_{f_1}))=\{0\}$. - Suppose that $g_i \ge 1$, i.e., $n_i \ge 4$. Then $$\End(J(C_{f_i}))=\Z, \ \Hdg(J(C_{f_i}))=\Sp( \H_1(J(C_{f_i}),\Q))$$ and $J(C_{f_i})$ satisfies (D). - If $n_2=2$ then $P=J(C_{f_1})$. In particular, $\End(P)=\Z$, $\Hdg(P)=\Sp( \H_1(P,\Q))$ and $P$ satisfies (D). - If $n_2 \ge 4$ then $\End(P)=\Z\oplus\Z$, $$\Hdg(P)=\Hdg(J(C_{f_1}))\times \Hdg(J(C_{f_2}))=\Sp( \H_1(J(C_{f_1}),\Q))\times \Sp( \H_1(J(C_{f_2}),\Q))$$ and $P$ satisfies (D). Clearly, $\RR_f=\RR_h \sqcup\{a\}$; in particular, $a$ belongs to precisely one of $\RR_1$ and $\RR_2$. Suppose that $a$ lies in $\RR_j$ and does [*not*]{} belong to $\RR_k$ and put $$T=\RR_k\subset\RR_h, \ S=\RR_j\setminus \{a\}\subset\RR_h.$$ Now the assertion (i) follows from Corollary \[homoGt\] applied to $h(x)$ (instead of $f(x)$). If $n_k\ge 6$ then the assertion (ii) for $J(C_{f_k})$ follows from Lemma \[jacprym\]. Suppose that $n_k=4$, i.e., $J(C_{f_k})$ is an elliptic curve. Then it follows from Theorem \[ellipticZ\] applied to $m=n-1\ge 9$ and $h(x)$ (instead of $f(x)$) that $\End(J(C_{f_k}))=\Z$. Now the assertion about its Hodge group and property (D) follows from Example \[DE\]. If $n_j\ge 6$ then the assertion (ii) for $J(C_{f_j})$ follows from Lemma \[jacfam\]. Suppose that $n_j=4$, i.e., $J(C_{f_j})$ is an elliptic curve. Then it follows from Theorem \[ellipticZ1\] applied to $m=n-1$ and $h(x)$ (instead of $f(x)$) that $\End(J(C_{f_j}))=\Z$. Now the assertion about its Hodge group and property (D) follows from Example \[DE\]. This ends the proof of (ii). The proof of the remaining assertions (iii) and (iv) goes literally as the proof of the corresponding assertions of Theorem \[help\]. Let us take $K=\Q$ and $f_n(x)=x^n-x-1$. It is known [@SerreGalois p. 42] that $\Gal(f_n)=\ST_n$. Let $a$ be a rational number. Suppose that $n=2g+2$ and let us consider the hyperelliptic genus $g$ curves $C_{f_n}: y^2=f_n(x)$ and $C^{a}_{f_{n-1}}: y^2=(x-a) f_{n-1}(x)$. Then: - If $n=2g+2\ge 8$ then all $(2^{2g}-1)$ Prym varieties $P$ of $C_{f_{n}}$ satisfy (D). Among them there are exactly $n(n-1)/2$ complex abelian varieties with $\End(P)=\Z$; for all others $\End(P)=\Z\oplus \Z$. - If $n=2g+2\ge 10$ then all $(2^{2g}-1)$ Prym varieties $P$ of $C^{a}_{f_{n-1}}$ satisfy (D). Among them there are exactly $n(n-1)/2$ complex abelian varieties with $\End(P)=\Z$; for all others $\End(P)=\Z\oplus \Z$. Let $z_1, \dots , z_n$ be algebraically independent (transcendental) complex numbers and $L=\Q(z_1, \dots , z_n)\subset \C$ the corresponding subfield of $\C$, which is isomorphic to the field of rational functions in $n$ variables over $\Q$. Let $K\subset L$ be the (sub)field of symmetric rational functions. Then $$f(x)=\prod_{i=1}^n (x-z_i)\in K[x], \ \RR_f=\{z_1, \dots , z_n\}, \ \Gal(f)=\ST_n.$$ Suppose that $n=2g+2$ and let us consider the hyperelliptic genus $g$ curve $C_f:y^2=f(x)$. If $g\ge 3$ (i.e., $n\ge 8$) then all $(2^{2g}-1)$ Prym varieties $P$ of $C_{f}$ satisfy (D). 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--- abstract: 'This article re-examines a classic question in liquid-crystal physics: What are the elastic modes of a nematic liquid crystal? The analysis uses a recent mathematical construction, which breaks the director gradient tensor into four distinct types of mathematical objects, representing splay, twist, bend, and a fourth deformation mode. With this construction, the Oseen-Frank free energy can be written as the sum of squares of the four modes, and saddle-splay can be regarded as bulk rather than surface elasticity. This interpretation leads to an alternative way to think about several previous results in liquid-crystal physics, including: (1) free energy balance between cholesteric and blue phases, (2) director deformations in hybrid-aligned-nematic cells, (3) spontaneous twist of achiral liquid crystals confined in a torus or cylinder, and (4) curvature of smectic layers.' author: - 'Jonathan V. Selinger' bibliography: - 'saddlesplay2.bib' date: 'January 17, 2019' title: 'Interpretation of saddle-splay and the Oseen-Frank free energy in liquid crystals' --- Introduction ============ One of the oldest problems in liquid-crystal research is to characterize how the director field can be distorted away from a uniform state, or in other words, to identify the elastic modes of a nematic liquid crystal. Through the mid-20th century, this issue was investigated in classic work by Oseen [@Oseen1933] and Frank [@Frank1958], and further by Nehring and Saupe [@Nehring1971; @Nehring1972]. This body of research led to the Oseen-Frank free energy density, which is discussed in many textbooks, such as Ref. [@Kleman2003], and which is widely used in liquid-crystal science and technology. This free energy density includes terms representing the cost of three distortion modes—splay, twist, and bend—and also includes a further term called saddle-splay. The saddle-splay contribution to the free energy density is the total divergence of a vector field. As a result, the volume integral of this term can be transformed into a surface integral. For that reason, the saddle-splay contribution is often considered as surface elasticity, in contrast with the splay, twist, and bend contributions, which are bulk elasticity. Over the years, the role of saddle-splay in liquid-crystal physics has been rather subtle. In many cases, the saddle-splay free energy is fixed by boundary conditions, and hence does not affect the behavior of a system. In other cases, the saddle-splay free energy is a variable quantity, and it can induce complex nonuniform structures in the director field [@Crawford1995]. Many theoretical studies have successfully analyzed the role of saddle-splay in specific systems [@Pairam2013; @Koning2014; @Davidson2015; @Kos2016; @Tran2016], but saddle-splay is still often difficult to understand on any intuitive basis. This difficulty arises for several reasons, including: (1) saddle-splay is normally not visualized by itself, (2) it can be regarded as either a bulk or surface free energy, and (3) if it is regarded as a surface contribution, it can accumulate along defects as internal surfaces. The purpose of this article is to discuss an alternative interpretation of liquid-crystal elasticity, which may help to clarify the role of saddle-splay in the Oseen-Frank free energy. This interpretation is based on a mathematical construction that was recently suggested by Machon and Alexander [@Machon2016]. Their paper decomposes the director gradient tensor into four modes: splay, twist, bend, and a fourth mode that they call $\bm{\Delta}$. The $\bm{\Delta}$ mode is related to saddle-splay but is not exactly the same—in the following sections, we discuss the terminology and suggest the name “biaxial splay.” The Machon-Alexander paper uses this mathematical construction to analyze the topological properties of umbilic lines where $\bm{\Delta}=0$. Here, we use the same construction for a different purpose. We re-express the Oseen-Frank free energy in terms of the four modes, and use this new expression to re-analyze several previous problems in liquid-crystal physics where saddle-splay was found to be important. Through this re-analysis, we suggest that the new construction provides a simpler and more intuitive way to understand the role of saddle-splay. We emphasize that our re-analysis does not change any predictions for experiments. It gives exactly the same predictions as previous studies, because the theories are mathematically equivalent. Hence, the significance of our argument is purely a matter of theoretical understanding. The plan of this paper is as follows. In Sec. II, we explain how the director gradient tensor can be decomposed into the four modes splay, twist, bend, and $\bm{\Delta}$. We first present the argument mathematically, and then visualize and discuss each of the modes. In Sec. III, we express the Oseen-Frank free energy in terms of these modes, and show that it takes a simple form as the sum of squares. In particular, we see how the saddle-splay term is related to splay, twist, and $\bm{\Delta}$. This relation leads to a discussion of terminology. In Sec. IV, we discuss the distinction between double splay and single splay, as well as double twist and cholesteric single twist. This analysis implies that single splay should really be understood as a combination of double splay and $\bm{\Delta}$, and cholesteric single twist as a combination of double twist and $\bm{\Delta}$. Based on that argument, we assess the free energy balance between cholesteric and blue phases. In Sec. V, we apply this analysis to several further examples, particularly director deformations in hybrid-aligned-nematic cells, spontaneous twist of achiral liquid crystals confined in a torus or cylinder, and curvature of smectic layers. In all of these cases, the new analysis provides an alternative way to think about previous results about saddle-splay. Finally, in Sec. VI, we discuss some related issues for future research. We also provide two appendices with specific calculations that might be useful for other investigators. Appendix A shows how the four modes can be expressed in terms of the nematic order tensor $\bm{Q}$. These expressions might be used for analyzing simulations of blue phases, for example. Appendix B shows the two-dimensional (2D) version of how to decompose the director gradient tensor. In 2D, the only two normal modes are splay and bend. Director gradient modes ======================= Following Machon and Alexander [@Machon2016], we separate the director gradient tensor into four normal modes. We first explain the decomposition mathematically, and then visualize and discuss each of the modes. Mathematics ----------- A nematic liquid crystal has a director field $\hat{\bm{n}}(\bm{r})$, and hence a tensor of director gradients $\partial_i n_j$. Let us first consider the number of degrees of freedom in this tensor. In three dimensions (3D), the tensor has $3\times3=9$ components. However, because $\hat{\bm{n}}$ is a unit vector, we must have $$(\partial_i n_j)n_j = \frac{1}{2}\partial_i(n_j n_j) = \frac{1}{2}\partial_i(1) = 0.$$ That equation is actually 3 constraints, for $i=1$ to $3$. Hence, the tensor $\partial_i n_j$ should have $9-3=6$ degrees of freedom. In other words, the first leg of $\partial_i n_j$ might have components parallel or penpendicular to $\hat{\bm{n}}$, but the second leg must be perpendicular to $\hat{\bm{n}}$. We can break $\partial_i n_j$ into parts where the first leg is parallel or perpendicular to $\hat{\bm{n}}$, $$\partial_i n_j = -n_i B_j + \alpha_{ij}, \label{decomposition1}$$ where $\bm{B}$ is a vector perpendicular to $\hat{\bm{n}}$ and $\alpha_{ij}$ is a tensor in the plane perpendicular to $\hat{\bm{n}}$. Contracting both sides of Eq. (\[decomposition1\]) with $n_i$ gives $$\bm{B} = -(\hat{\bm{n}}\cdot\bm{\nabla})\hat{\bm{n}} = \hat{\bm{n}}\times(\bm{\nabla}\times\hat{\bm{n}}). \label{benddefinition}$$ Hence, $\bm{B}$ is the standard *bend vector* in liquid-crystal physics. Because $\bm{B}$ is perpendicular to $\hat{\bm{n}}$, it has two degrees of freedom. Now we are left with the tensor $\alpha_{ij}$ in the plane perpendicular to $\hat{\bm{n}}$, which has 4 degrees of freedom. We can break it into an antisymmetric tensor $\beta_{ij}$ and symmetric tensor $\gamma_{ij}$. Because $\beta_{ij}$ is an antisymmetric tensor in the plane perpendicular to $\hat{\bm{n}}$, it can be expressed as $\beta_{ij}=\frac{1}{2}T\epsilon_{ijk}n_k$, for some pseudoscalar $T$. Hence, the director gradient tensor becomes $$\partial_i n_j = -n_i B_j + \frac{1}{2}T\epsilon_{ijk}n_k + \gamma_{ij}. \label{decomposition2}$$ Contracting both sides of Eq. (\[decomposition2\]) with $\epsilon_{ijm}n_m$ gives $$T=\hat{\bm{n}}\cdot(\bm{\nabla}\times\hat{\bm{n}}). \label{twistdefinition}$$ Hence, $T$ is the standard *twist* in liquid-crystal physics. Because $T$ is a pseudoscalar, it has one degree of freedom. At this point, we are left with the symmetric tensor $\gamma_{ij}$ in the plane perpendicular to $\hat{\bm{n}}$, which has 3 degrees of freedom. We can break it into its trace $S$, multiplied by half of the identity tensor in the plane perpendicular to $\hat{\bm{n}}$, which is $\frac{1}{2}(\delta_{ij}-n_i n_j)$, plus a traceless symmetric tensor $\Delta_{ij}$. Hence, the director gradient tensor becomes $$\partial_i n_j = -n_i B_j + \frac{1}{2}T\epsilon_{ijk}n_k + \frac{1}{2}S(\delta_{ij}-n_i n_j) + \Delta_{ij}. \label{decomposition}$$ Taking the trace of Eq. (\[decomposition\]) gives $$S=\bm{\nabla}\cdot\hat{\bm{n}}. \label{splaydefinition}$$ Hence, $S$ is the standard *splay* in liquid-crystal physics. Because $S$ is a scalar, it has one degree of freedom. Finally, we have the traceless symmetric tensor $\Delta_{ij}$ in the plane perpendicular to $\hat{\bm{n}}$. As a traceless symmetric tensor in a plane, it has two degrees of freedom. For an explicit expression for $\Delta_{ij}$, we combine Eq. (\[decomposition\]) with its transpose to obtain $$\begin{aligned} \Delta_{ij} = & \frac{1}{2} [\partial_i n_j + \partial_j n_i + n_i B_j + n_j B_i - S (\delta_{ij} - n_i n_j)] \nonumber\\ =& \frac{1}{2} [\partial_i n_j + \partial_j n_i - n_i n_k \partial_k n_j - n_j n_k \partial_k n_i \nonumber\\ &\quad - \delta_{ij} \partial_k n_k + n_i n_j \partial_k n_k ]. \label{biaxialsplaydefinition}\end{aligned}$$ We note that $\Delta_{ij}=\Delta_{ji}$, $\Delta_{ii}=0$, $n_i \Delta_{ij}=0$, and $\Delta_{ij}n_j =0$. Equation (\[decomposition\]) decomposes the director gradient tensor $\partial_i n_j$ into the four normal modes $\bm{B}$, $T$, $S$, and $\bm{\Delta}$. Together, these modes account for the six degrees of freedom in $\partial_i n_j$. These modes are four distinct types of mathematical objects: $\bm{B}$ is a vector, $T$ a pseudoscalar, $S$ a scalar, and $\bm{\Delta}$ a symmetric traceless tensor. In a more precise mathematical language, Machon and Alexander [@Machon2016] write that these modes are distinct irreducible representations of the rotation group. Visualization and discussion of each mode ----------------------------------------- In this section, we discuss each of the four modes by visualizing director configurations in which all the other modes are zero, at least locally, if not globally. ### Bend $\bm{B}$ --------------------------------- ---------------------------------- ---------------------------------- \(a) Bend $\bm{B}$ \(b) Bend $\bm{B}$ \(c) Twist $T$ {height="4.4cm"} {height="4.4cm"} {height="4.4cm"} \(d) Splay $S$ \(e) Biaxial splay $\Delta_{ij}$ \(f) Biaxial splay $\Delta_{ij}$ {height="4.4cm"} {height="4.4cm"} {height="4.4cm"} --------------------------------- ---------------------------------- ---------------------------------- A bend deformation has the structure shown in Fig. 1(a,b). It has the standard form of bend in liquid-crystal physics, with $\hat{\bm{n}}$ varying along the direction parallel to the local $\hat{\bm{n}}$. It has two components, because the change in $\hat{\bm{n}}$ can be in either of two directions perpendicular to the local $\hat{\bm{n}}$. Hence, it can be represented by a vector in the plane perpendicular to the local $\hat{\bm{n}}$, as shown by the red arrows in the figures. It is possible to fill up 3D Euclidean space with pure bend. For example, consider the director field $$\hat{\bm{n}}(x,y,z)=\frac{(-y,x,0)}{\sqrt{x^2+y^2}}.$$ Explicit calculations give $$\bm{B}=\frac{(x,y,0)}{x^2+y^2},\quad T=0,\quad S=0,\quad \bm{\Delta}=0.$$ Hence, this example has pure bend, which decreases in magnitude as we move away from the $z$-axis. In a nematic liquid crystal, $\hat{\bm{n}}$ and $-\hat{\bm{n}}$ are equivalent ways to describe the same state. The bend vector $\bm{B}$ defined by Eq. (\[benddefinition\]) is invariant under the transformation $\hat{\bm{n}}\to-\hat{\bm{n}}$, and hence it is a physical object that does not depend on this arbitrary choice of sign. ### Twist $T$ A twist deformation has the structure shown in Fig. 1(c). In the liquid-crystal literature, this type of deformation is commonly known as “double twist,” because $\hat{\bm{n}}$ varies in both directions perpendicular to the local $\hat{\bm{n}}$. We emphasize that twist $T$ means double twist. In other words, the deformation with nonzero $T$ but zero $\bm{B}$, $S$, and $\bm{\Delta}$ is double twist, not single twist. This statement must be true because $T$ is a pseudoscalar, and hence it has no direction in the plane perpendicular to $\hat{\bm{n}}$. If the deformation were single twist, with a helical axis perpendicular to $\hat{\bm{n}}$, then it could not be described by a pseudoscalar. We will discuss single twist in Sec. IV. It is impossible to fill up 3D Euclidean space with pure double twist, but we can construct pure double twist locally. For example, consider the director field $$\hat{\bm{n}}(x,y,z)=\frac{(-qy,qx,1)}{\sqrt{1+q^2(x^2+y^2)}}$$ for $q\sqrt{x^2+y^2}\ll1$. Along the $z$-axis, for $x=y=0$, explicit calculations give $$\bm{B}=0,\quad T=2q,\quad S=0,\quad \bm{\Delta}=0.$$ Hence, this example has pure double twist along the $z$-axis. (Farther from the $z$-axis, it has a mixture of double twist and bend.) The twist pseudoscalar $T$ defined by Eq. (\[twistdefinition\]) is invariant under the transformation $\hat{\bm{n}}\to-\hat{\bm{n}}$, and hence it is a physical object that does not depend on this arbitrary choice of sign. ### Splay $S$ A splay deformation has the structure shown in Fig. 1(d). By analogy with double twist, this type of deformation might be called “double splay,” because $\hat{\bm{n}}$ varies in both directions perpendicular to the local $\hat{\bm{n}}$. As in the previous case, we emphasize that splay $S$ means double splay; i.e. the deformation with nonzero $S$ but zero $\bm{B}$, $T$, and $\bm{\Delta}$ is double splay, not single splay. This statement must be true because $S$ is a scalar, and hence has no direction in the plane perpendicular to $\hat{\bm{n}}$. If the deformation were single splay, with variation in only one direction perpendicular to $\hat{\bm{n}}$, then it could not be described by a scalar. We will discuss single splay in Sec. IV. It is possible to fill up 3D Euclidean space with pure double splay. For example, consider the director field for a hedgehog, $$\hat{\bm{n}}(x,y,z)=\frac{(x,y,z)}{\sqrt{x^2+y^2+z^2}}.$$ Explicit calculations give $$\bm{B}=0,\quad T=0,\quad S=\frac{2}{\sqrt{x^2+y^2+z^2}},\quad \bm{\Delta}=0.$$ Hence, this example has pure double splay, which decreases in magnitude as we move away from the origin. The splay scalar $S$ defined by Eq. (\[splaydefinition\]) changes sign under the transformation $\hat{\bm{n}}\to-\hat{\bm{n}}$. If we want a physical object that does not depend on this arbitrary choice of sign, we can construct the splay vector $\bm{S}=S\hat{\bm{n}}={\hat{\bm{n}}(\bm{\nabla}\cdot\hat{\bm{n}})}$. This splay vector is well-known in the theory of flexoelectricity [@Meyer1969]. It is shown by the red arrow in Fig. 1(d). ### “Biaxial splay” $\bm{\Delta}$ The fourth deformation mode $\bm{\Delta}$ has the structure shown in Fig. 1(e,f). In this deformation, $\hat{\bm{n}}$ tips outward along one axis perpendicular to the local $\hat{\bm{n}}$, and tips inward along the other axis perpendicular to the local $\hat{\bm{n}}$. In other words, there is a combination of positive splay along one axis and negative splay along the other axis. The symmetry of this deformation is similar to a biaxial nematic liquid crystal, because of the two distinct axes perpendicular to $\hat{\bm{n}}$. For that reason, we suggest that this deformation might be called “biaxial splay.” We will discuss the terminology further in Sec. III. The biaxial splay deformation has two components, which are rotated with respect to each other by $45^\circ$ in the plane perpendicular to $\hat{\bm{n}}$. In Fig. 1(e), the splay is outward along $\pm(1,0,0)$ and inward along $\pm(0,1,0)$. By comparison, in Fig. 1(f), the splay is outward along $\pm(1/\sqrt{2},1/\sqrt{2},0)$ and inward along $\pm(1/\sqrt{2},-1/\sqrt{2},0)$. A further rotation of $45^\circ$ gives a splay outward along $\pm(0,1,0)$ and inward along $\pm(1,0,0)$, which is just the negative of the first component in Fig. 1(e). It is impossible to fill up 3D Euclidean space with pure biaxial splay, but we can construct pure biaxial splay locally. For example, consider the director field $$\hat{\bm{n}}(x,y,z)=\frac{(qx,-qy,1)}{\sqrt{1+q^2(x^2+y^2)}}$$ for $q\sqrt{x^2+y^2}\ll1$. Along the $z$-axis, for $x=y=0$, explicit calculations give $$\bm{B}=0,\quad T=0,\quad S=0,\quad \bm{\Delta}= \begin{pmatrix} q & 0 & 0\\ 0 & -q & 0\\ 0 & 0 & 0 \end{pmatrix} .$$ Hence, this example has pure biaxial splay along the $z$-axis. (Farther from the $z$-axis, it has a mixture of biaxial splay with bend and double splay.) The biaxial splay tensor $\Delta_{ij}$ defined by Eq. (\[biaxialsplaydefinition\]) changes sign under the transformation $\hat{\bm{n}}\to-\hat{\bm{n}}$. If we want a physical object that does not depend on this arbitrary choice of sign, we can construct the third-rank tensor $\Delta_{ij}n_k$. This third-rank tensor is represented by the red tetrahedra in Fig. 1(e,f). Free energy, saddle-splay, and $K_{24}$ ======================================= The Oseen-Frank free energy gives the elastic free energy associated with deformations in the director field. We consider first the simplified version of the free energy with equal elastic constants, and then the full free energy with unequal elastic constants. In the simple approximation of equal elastic constants, the Oseen-Frank free energy density is $$F=\frac{1}{2}K(\partial_i n_j)(\partial_i n_j). \label{oseenfranksimple}$$ By inserting Eq. (\[decomposition\]) for $\partial_i n_j$ into Eq. (\[oseenfranksimple\]), we obtain $$F=\frac{1}{4}KS^2 + \frac{1}{4}KT^2 + \frac{1}{2}K|\bm{B}|^2 + \frac{1}{2}K\operatorname{Tr}(\bm{\Delta}^2),$$ where $\operatorname{Tr}(\bm{\Delta}^2)=\Delta_{ij}\Delta_{ji}$. This expression shows that all four of the modes cost elastic free energy. There are no cross terms between the modes. Indeed, bilinear cross terms are forbidden by symmetry: There is no bilinear coupling that can generate a scalar for the free energy density. (The coupling $S T$ is a pseudoscalar, which is permitted in a chiral liquid crystal, but not in an achiral nematic phase.) In this approximation, the coefficients of $S^2$, $T^2$, $|\bm{B}|^2$, and $\operatorname{Tr}(\bm{\Delta}^2)$ are all the same, except for the factors of $\frac{1}{4}$ and $\frac{1}{2}$. To understand these factors, we might say that $S$ and $T$ are double deformations, while $\bm{B}$ and $\bm{\Delta}$ are single deformations with two components. In general, the full Oseen-Frank free energy density is conventionally written as [@Kleman2003] $$\begin{aligned} \label{oseenfrankconventional} F=&\frac{1}{2}K_{11}S^2 + \frac{1}{2}K_{22}T^2 + \frac{1}{2}K_{33}|\bm{B}|^2 \\ &-K_{24}\bm{\nabla}\cdot\left[\hat{\bm{n}}(\bm{\nabla}\cdot\hat{\bm{n}})+\hat{\bm{n}}\times(\bm{\nabla}\times\hat{\bm{n}})\right],\nonumber\end{aligned}$$ where the last term is the saddle-splay term. In the literature, there are some variations in the notation for the saddle-splay term. Instead of $K_{24}$, the coefficient is sometimes written as $\frac{1}{2}K_{24}$ or as $(K_{22}+K_{24})$. Those variations are not important for the following argument, which still applies with a minor change of notation. The saddle-splay term can be expressed in terms of the four modes discussed in the previous section. An explicit calculation gives $$\begin{aligned} \label{saddlesplay} &\bm{\nabla}\cdot\left[\hat{\bm{n}}(\bm{\nabla}\cdot\hat{\bm{n}})+\hat{\bm{n}}\times(\bm{\nabla}\times\hat{\bm{n}})\right]\\ &=\bm{\nabla}\cdot\left[\hat{\bm{n}}(\bm{\nabla}\cdot\hat{\bm{n}})-(\hat{\bm{n}}\cdot\bm{\nabla})\hat{\bm{n}})\right] =\partial_j \left[n_j \partial_i n_i - n_i \partial_i n_j \right]\nonumber\\ &=(\partial_i n_i)(\partial_j n_j)-(\partial_i n_j)(\partial_j n_i) =\frac{1}{2}S^2 +\frac{1}{2}T^2 - \operatorname{Tr}(\bm{\Delta}^2)\nonumber\end{aligned}$$ Combining Eqs. (\[oseenfrankconventional\]) and (\[saddlesplay\]), the Oseen-Frank free energy density becomes $$\begin{aligned} \label{oseenfranknew} F=&\frac{1}{2}K_{11}S^2 + \frac{1}{2}K_{22}T^2 + \frac{1}{2}K_{33}|\bm{B}|^2 \nonumber\\ &-K_{24}\left[\frac{1}{2}S^2 +\frac{1}{2}T^2 - \operatorname{Tr}(\bm{\Delta}^2)\right]\nonumber\\ =&\frac{1}{2}(K_{11}-K_{24})S^2 + \frac{1}{2}(K_{22}-K_{24})T^2 \nonumber\\ &+\frac{1}{2}K_{33}|\bm{B}|^2 + K_{24}\operatorname{Tr}(\bm{\Delta}^2) .\end{aligned}$$ This expression shows that all four of the modes cost different amounts of elastic free energy. The elastic constant for (double) splay is $(K_{11}-K_{24})$, the elastic constant for (double) twist is $(K_{22}-K_{24})$, the elastic constant for bend is $K_{33}$, and the elastic constant for the mode $\bm{\Delta}$ is $2K_{24}$. The simple approximation of equal elastic constants then corresponds to $K_{11}=K_{22}=K_{33}=2K_{24}\equiv K$. We can now make a remark about the terminology for the mode $\bm{\Delta}$. Because of the saddle-like shape of the deformation in Figs. 1(e,f), one might be tempted to consider $\bm{\Delta}$ as saddle-splay. However, Eq. (\[saddlesplay\]) shows that the saddle-splay term in the free energy is actually a combination of $S$, $T$, and $\bm{\Delta}$. Because “saddle-splay” is already well-established as the name for that term in the free energy, we cannot use the same name for $\bm{\Delta}$. Hence, the mode $\bm{\Delta}$ needs another name. Machon and Alexander [@Machon2016] refer to this mode as “anisotropic orthogonal gradients of $\hat{\bm{n}}$,” but that phrase is too long for common use. For that reason, we suggest the name “biaxial splay,” to emphasize the similarity with the symmetry of a biaxial nematic phase. Up to now, we have not yet used the fact that the saddle-splay term is a total divergence. Because it is a total divergence, the volume integral of this term can be reduced to a surface integral. Hence, from Eq. (\[saddlesplay\]), the volume integral of $[\frac{1}{2}S^2 +\frac{1}{2}T^2 - \operatorname{Tr}(\bm{\Delta}^2)]$ can be reduced to a surface integral. In many liquid-crystal systems, that surface integral is a constant determined by the boundary conditions. In those systems, liquid-crystal theorists normally use this constraint to eliminate $\bm{\Delta}$ from the theory, and work in terms of the three remaining modes $S$, $T$, and $\bm{B}$. As an alternative, in principle, one might eliminate $S$ or $T$ from the theory, and work in terms of the other three modes. We do not actually recommend either of those alternatives. Rather, we suggest treating all four of the modes as bulk elastic modes, and using the bulk free energy density of Eq. (\[oseenfranknew\]). This expression explicitly shows the free energy associated with each of the four modes, and it is clearly positive-definite as long as the four coefficients are positive. Moreover, it applies even to liquid-crystal systems in which the surface integral is not a constant determined by the boundary conditions, but rather is variable. In the following sections, we discuss several problems that have been previously treated using the concept of saddle-splay as a surface integral, and re-analyze them in terms of the four bulk elastic modes. We argue that this view gives an interesting new perspective on those problems, although it still gives the same predictions for experiments. Single vs. double deformations ============================== Planar (single) splay --------------------- --------------------------------- --------------------------------- --------------------------------- --------------------------------- \(a) Planar (single) splay \(b) Splay Frederiks transition \(c) Cholesteric (single) twist \(d) Twist Frederiks transition {height="4.2cm"} {height="4.2cm"} {height="4.2cm"} {height="4.2cm"} --------------------------------- --------------------------------- --------------------------------- --------------------------------- In Sec. II, we argued that a deformation with nonzero $S$ but zero $\bm{B}$, $T$, and $\bm{\Delta}$ is 3D double splay, as in a hedgehog. For comparison, we might consider a deformation with splay only in the 2D plane, as in slices of pizza. This deformation might be called planar splay or single splay. For example, consider the director field $$\hat{\bm{n}}(x,y,z)=\frac{(x,y,0)}{\sqrt{x^2+y^2}},$$ which is shown in Fig. 2(a). For this director field, explicit calculations give $$\begin{aligned} &\bm{B}=0,\quad T=0,\quad S=\frac{1}{\sqrt{x^2+y^2}},\\ &\bm{\Delta}=\frac{1}{2(x^2+y^2)^{3/2}} \begin{pmatrix} y^2 & -xy & 0\\ -xy & x^2 & 0\\ 0 & 0 & -x^2-y^2 \end{pmatrix} .\nonumber\end{aligned}$$ This result shows that single splay is a linear combination of $S$ and $\bm{\Delta}$. In particular, the tensor $\bm{\Delta}$ contains the directional information that identifies which of the directions perpendicular to the local $\hat{\bm{n}}$ is the splayed direction, and which is the uniform direction. From Eq. (\[oseenfranknew\]), the free energy density has the components $$\begin{aligned} &F_S = \frac{1}{2}(K_{11}-K_{24})S^2 = \frac{K_{11}-K_{24}}{2(x^2+y^2)}, \nonumber\\ &F_\Delta=K_{24}\operatorname{Tr}(\bm{\Delta}^2) = \frac{K_{24}}{2(x^2+y^2)},\end{aligned}$$ and hence the total $$F = \frac{K_{11}}{2(x^2+y^2)}.$$ We see that $K_{24}$ drops out of the free energy for single splay, precisely because single splay is a combination of $S$ and $\bm{\Delta}$. Thus, $K_{11}$ is the relevant elastic constant for single splay. This free energy expression has an important consequence for the splay Frederiks transition. The experimental geometry for this transition is shown in Fig. 2(b). We can see that this geometry has single splay, not double splay, because the director field stays in a plane. Hence, the field-induced distortion is a combination of $S$ and $\bm{\Delta}$, and the relevant elastic constant is $K_{11}$. Hence, the critical field for this transition is determined by $K_{11}$, not by $(K_{11}-K_{24})$. Of course, this result is consistent with how the splay Frederiks transition has been analyzed for many years. Cholesteric (single) twist -------------------------- The same argument for single and double splay also applies to single and double twist. In Sec. II, we argued that a deformation with nonzero $T$ but zero $\bm{B}$, $S$, and $\bm{\Delta}$ is 3D double twist. For comparison, we might consider a deformation with single twist, as in a cholesteric phase. For example, consider the director field $$\hat{\bm{n}}(x,y,z)=(\cos qz,\sin qz,0),$$ which is shown in Fig. 2(c). For this director field, we calculate $$\begin{aligned} &\bm{B}=0,\quad T=-q,\quad S=0,\\ &\bm{\Delta}=\frac{q}{2} \begin{pmatrix} 0 & 0 & -\sin qz\\ 0 & 0 & \cos qz\\ -\sin qz & \cos qz & 0 \end{pmatrix} .\nonumber\end{aligned}$$ This result shows that cholesteric twist is a linear combination of $T$ and $\bm{\Delta}$. The tensor $\bm{\Delta}$ contains the directional information about which of the directions perpendicular to the local $\hat{\bm{n}}$ is the helical axis, and which is the uniform direction. From Eq. (\[oseenfranknew\]), the free energy density has the components $$\begin{aligned} &F_T = \frac{1}{2}(K_{22}-K_{24})T^2 = \frac{1}{2}(K_{22}-K_{24})q^2, \nonumber\\ &F_\Delta=K_{24}\operatorname{Tr}(\bm{\Delta}^2) = \frac{1}{2}K_{24}q^2,\end{aligned}$$ and hence the total $$F = \frac{1}{2}K_{22}q^2.$$ Thus, $K_{24}$ drops out of the free energy for cholesteric twist, because it is a combination of $T$ and $\bm{\Delta}$, and $K_{22}$ is the relevant elastic constant for cholesteric twist. The argument for the splay Frederiks transition also applies to the twist Frederiks transition. As shown in Fig. 2(d), this transition involves single twist, not double twist. Hence, the critical field is determined by $K_{22}$, not by $(K_{22}-K_{24})$. Cholesteric vs. blue phase -------------------------- The distinction between single and double twist provides a simple way to compare the free energies of cholesteric and blue phases in chiral liquid crystals. If a liquid crystal is chiral, the free energy has the achiral nematic terms of Eq. (\[oseenfranknew\]), plus an additional chiral term proportional to the twist $T$. The combination of the achiral and chiral terms favor s a certain optimal value of $T$. The question is: How does the chiral liquid crystal achieve this twist? Does it form a cholesteric or a blue phase? In a cholesteric phase, the director field forms a helix with single twist everywhere. This single twist is a combination of $T$ and $\bm{\Delta}$. The mode $T$ provides a favorable, negative contribution to the free energy, but the mode $\bm{\Delta}$ provides an unfavorable, positive contribution to the free energy. --------------------------------------- --------------------------------------- --------------------------------------- \(a) $\bm{\nabla}\cdot\hat{\bm{c}}=0$ \(b) $\bm{\nabla}\cdot\hat{\bm{c}}<0$ \(c) $\bm{\nabla}\cdot\hat{\bm{c}}>0$ {width="5.8cm"} {width="5.8cm"} {width="5.8cm"} --------------------------------------- --------------------------------------- --------------------------------------- In a blue phase, the director field forms a complex network of double twist tubes separated by disclination lines. Because the tubes have double rather than single twist, they have pure $T$ with no $\bm{\Delta}$. Hence, $T$ provides a favorable contribution to the free energy, while $\bm{\Delta}$ makes no contribution. However, it is impossible to fill Euclidean space with double twist, as discussed below in Sec. VI(B). Rather, geometric considerations require a finite density of disclination lines. These disclinations make an unfavorable contribution to the free energy. Comparing these two possiblities, we can see that the favorable contributions to the free energy are the same, but the unfavorable contributions are different. Thus, the relative stabilities of cholesteric and blue phases depends on which is worse: the $\bm{\Delta}$ mode in a cholesteric phase, or the disclination lines in a blue phase. That issue depends on the magnitude of $K_{24}$ compared with the free energy of the disclination cores, in which the liquid-crystal order is disrupted. If the disclination core energy is high compared with $K_{24}$, then the liquid crystal will form a cholesteric phase. If $K_{24}$ is high compared with the disclination core energy, then the liquid crystal will form a blue phase. This conclusion that large $K_{24}$ is necessary to stabilize a blue phase is certainly not new. It goes back to early work by Meiboom *et al.* [@Meiboom1981], who considered saddle-splay as a surface free energy along the disclination lines as internal surfaces. Related arguments have been made in the context of smectic blue phases [@DiDonna2002; @DiDonna2003] and double-tilt blue phases [@Chakrabarti2006]. We only suggest that this theoretical approach with modes $T$ and $\bm{\Delta}$ provides a particularly simple way to understand why $K_{24}$ is important. Further examples ================ Hybrid aligned nematic liquid crystal ------------------------------------- In the hybrid aligned nematic geometry, a liquid crystal is confined between two isotropic media, so that the director field has homeotropic (perpendicular) anchoring on the bottom and degenerate planar (tangential) anchoring on the top, or vice versa. This type of system has been studied experimentally and theoretically over many years, as in Refs. [@Sparavigna1994; @Lavrentovich1995]. Experimentally, these systems exhibit complex modulated structures in the director field. Theoretically, the modulated structures have been explained by effects of saddle-splay, regarded as surface elasticity. Here, we re-analyze the same system in terms of the four bulk modes discussed in this article. Suppose that a cell extends from $z=0$ to $d$. At the bottom surface $z=0$, there is homeotropic anchoring, so that $\hat{\bm{n}}(x,y,0)=\hat{\bm{z}}$. At the top surface $z=d$, there is degenerate planar anchoring, so that $\hat{\bm{n}}$ must be in the $xy$ plane, and all orientations in the $xy$ plane have the same energy. Hence, we can write $\hat{\bm{n}}(x,y,d)=\hat{\bm{c}}(x,y)$, where $\hat{\bm{c}}(x,y)$ is a unit vector in the $xy$ plane. For the liquid crystal in the interior, an approximate form for the director field is $$\hat{\bm{n}}(x,y,z)=\hat{\bm{c}}(x,y)\sin\frac{\pi z}{2d}+\hat{\bm{z}}\cos\frac{\pi z}{2d}. \label{ninterior}$$ This approximation is shown schematically in Fig. 3. It is reasonable if $\hat{\bm{c}}(x,y)$ is slowly varying and the elastic constants are approximately equal. The question is: In the lowest-free-energy state, does the liquid crystal have a uniform surface orientation $\hat{\bm{c}}(x,y)$? Or can the liquid crystal reduce its free energy with some modulation in $\hat{\bm{c}}(x,y)$? ------------------------------ ------------------------------ ------------------------------ \(a) $T=0$ \(b) $T<0$ \(c) $T>0$ {width="5cm"} {width="5cm"} {width="5cm"} ------------------------------ ------------------------------ ------------------------------ To answer that question, we use Eq. (\[ninterior\]) for the 3D director field to calculate the 3D distortions $S$, $T$, $\bm{B}$, and $\bm{\Delta}$. We put those distortions into Eq. (\[oseenfranknew\]) to calculate the 3D Frank free energy density. We then integrate over $z=0$ to $d$ to calculate the effective 2D Frank free energy density, $F_{2D}(x,y)=\int_0^d dz F(x,y,z)$, in terms of the surface orientation $\hat{\bm{c}}(x,y)$. This calculation gives the four components $$\begin{aligned} &F_{2D}^B=K_{33}\left[\frac{\pi^2}{16d} +\frac{3d}{16}(\bm\nabla\times\hat{\bm{c}})^2\right],\\ &F_{2D}^T=(K_{22} -K_{24})\left[\frac{d}{16}(\bm\nabla\times\hat{\bm{c}})^2\right],\nonumber\\ &F_{2D}^S=(K_{11} -K_{24})\left[\frac{\pi^2}{16d} -\frac{\pi}{4}\bm\nabla\cdot\hat{\bm{c}} +\frac{d}{4}(\bm\nabla\cdot\hat{\bm{c}})^2\right],\nonumber\\ &F_{2D}^\Delta=K_{24}\left[\frac{\pi^2}{16d} +\frac{\pi}{4}\bm\nabla\cdot\hat{\bm{c}} +\frac{d}{4}(\bm\nabla\cdot\hat{\bm{c}})^2 +\frac{d}{16}(\bm\nabla\times\hat{\bm{c}})^2\right],\nonumber\end{aligned}$$ where $\bm\nabla\cdot\hat{\bm{c}}$ and $\bm\nabla\times\hat{\bm{c}}$ are the 2D divergence and curl of the surface orientation, respectively. As a result, the total effective 2D free energy density becomes $$\begin{aligned} F_{2D}=&\frac{\pi^2 (K_{11} +K_{33})}{16d} -\frac{\pi(K_{11} -2K_{24})}{4}\bm\nabla\cdot\hat{\bm{c}}\\ &+\frac{K_{11} d}{4}(\bm\nabla\cdot\hat{\bm{c}})^2 +\frac{(K_{22} +3K_{33})d}{16}(\bm\nabla\times\hat{\bm{c}})^2.\nonumber\end{aligned}$$ To interpret these expressions, the most important feature to notice is the linear dependence on $\bm\nabla\cdot\hat{\bm{c}}$. First consider a state with uniform surface orientation, so that $\bm\nabla\cdot\hat{\bm{c}}=0$, as in Fig. 3(a). This state is loaded with three of the modes, $S$, $\bm{B}$, and $\bm{\Delta}$, which all contribute to the free energy. Now suppose the surface orientation has a small variation with $\bm\nabla\cdot\hat{\bm{c}}<0$, as in Fig. 3(b). This variation increases the $S$ and reduces the $\bm{\Delta}$, compared with the uniform state. Hence, this variation reduces the free energy compared with the uniform state, provided that the splay coefficient $(K_{11}-K_{24})$ is less than the biaxial splay coefficient $K_{24}$. By comparison, suppose the surface orientation has a small variation with $\bm\nabla\cdot\hat{\bm{c}}>0$, as in Fig. 3(c). This variation reduces the $S$ and increases the $\Delta$, compared with the uniform state. Hence, that variation reduces the free energy compared with the uniform state, provided that $(K_{11}-K_{24})>K_{24}$. From this argument, we see that the lowest-free-energy state has uniform surface orientation only in the special case that $(K_{11}-K_{24})=K_{24}$, or $K_{11}=2K_{24}$. Interestingly, that is exactly the case of equal elastic constants, discussed in Sec. III. Although that case is a theoretical possibility, an experimental liquid crystal will normally have elastic constants that are at least slightly different. If $K_{11}<2K_{24}$, then the energetic benefit of reducing $\Delta$ exceeds the energetic cost of increasing $S$, and hence the system can reduce its free energy by going to a modulation with $\bm\nabla\cdot\hat{\bm{c}}<0$. Conversely, if $K_{11}>2K_{24}$, then the energetic benefit of reducing $S$ exceeds the energetic cost of increasing $\Delta$, and hence the system can reduce its free energy by going to a modulation with $\bm\nabla\cdot\hat{\bm{c}}>0$. This substitution of $S$ for $\Delta$, or vice versa, can be regarded as the origin for the complex textures that are observed in hybrid aligned nematic liquid crystals. Of course, there is no contradiction between this interpretation and previous theories based on surface elasticity; they are just two ways of describing the same behavior. Liquid crystal in torus ----------------------- Suppose we have a nematic liquid crystal inside a torus, with degenerate planar anchoring on the surface. Does the director field form a simple, achiral configuration, running along the long axis of the torus, as shown in Fig. 4(a)? Or does it break reflection symmetry and form a chiral configuration, with a double twist from the central axis to the surface of the torus, as shown in Fig. 4(b,c)? This system was investigated theoretically in Ref. [@Koning2014], and further, more mathematically, in Ref. [@Pedrini2018]. Those studies use the perspective of saddle-splay as surface elasticity, and show that it favors alignment of the director field along the highly curved direction on the surface of the torus. If $K_{24}$ is small, then the bulk free energy favoring alignment along the long axis exceeds the surface free energy favoring alignment in the highly curved direction, and the director field forms the simple, achiral configuration. However, if $K_{24}$ is large enough, then the surface free energy exceeds the bulk free energy, and the director field forms a chiral configuration. ------------------------------ ------------------------------ ------------------------------ \(a) $T=0$ \(b) $T<0$ \(c) $T>0$ {width="5cm"} {width="5cm"} {width="5cm"} ------------------------------ ------------------------------ ------------------------------ Here, we consider the same problem in terms of the four bulk deformation modes. For this calculation, we use the same toroidal coordinate system and the same director ansatz as Ref. [@Koning2014]. We assume that their chiral order parameter $\omega$ is small, and expand the free energy in powers of $\omega$. The four terms in the free energy, integrated over the interior of the torus, then become $$\begin{aligned} & F_B = K_{33}\pi^2 R_1\biggl[2\left(1-\frac{(\xi^2 -1)^{1/2}}{\xi}\right)\nonumber\\ &\qquad-\left(6\xi-\frac{6\xi^4 -9\xi^2 +1}{(\xi^2 -1)^{3/2}}\right)\xi\omega^2 +O(\omega^4)\biggr],\nonumber\\ & F_T = (K_{22}-K_{24})\pi^2 R_1 \frac{4\xi^3}{(\xi^2 -1)^{3/2}}\omega^2 +O(\omega^6),\nonumber\\ & F_S = 0, \quad F_\Delta = O(\omega^6), \label{ftorus}\end{aligned}$$ where $\xi=R_1/R_2>1$ is the aspect ratio of the torus. From these expressions, we see that the splay free energy is exactly zero (by construction), and the $\bm{\Delta}$ mode free energy is approximately zero, so the important physics arises from the competition between bend and twist. In the achiral configuration with $\omega=0$, the bend free energy is high, and the twist free energy is zero. As the magnitude of $\omega$ increases, the bend free energy decreases and the twist free energy increases. This trade-off might or might not reduce the total free energy, depending on the total coefficient of $\omega^2$. From Eq. (\[ftorus\]), we see that this coefficient is negative, and hence a chiral configuration is favored, provided that $$\begin{aligned} \frac{K_{22}-K_{24}}{K_{33}}&<\frac{6\xi(\xi^2 -1)^{3/2} -6\xi^4 +9\xi^2 -1}{4\xi^2}\nonumber\\ &\approx\frac{5}{16\xi^2}\text{ if }\xi\gg1. \label{torusinequality}\end{aligned}$$ This result is exactly the same critical threshold found in Ref. [@Koning2014]. Thus, we see that their result can be understood from the competition between the bulk free energies of bend and twist, without reference to any surface alignment. We note that the $\bm{\Delta}$ mode is not involved with this competition. Rather, $K_{24}$ enters into the prediction because the twist is double twist, and the elastic constant for double twist is $(K_{22}-K_{24})$. Chromonic liquid crystal in cylinder ------------------------------------ Let us apply the results from the torus to the simpler case of a liquid crystal in a cylinder. We can ask whether the director field forms a simple uniform configuration parallel to the cylinder axis, as in Fig. 5(a), or whether it breaks reflection symmetry and forms a chiral configuration, as in Fig. 5(b,c). A cylinder can be regarded as a limiting case of a torus, in the limit where $R_1\to\infty$ and $R_2$ remains finite, so that $\xi\to\infty$. In that case, Eq. (\[torusinequality\]) implies that a chiral configuration is favored if $(K_{22}-K_{24})/K_{33}<0$. Hence, we must ask: Is it possible for a liquid crystal to have $K_{22}<K_{24}$? This issue was addressed by Ericksen [@Ericksen1966], who developed several inequalities for liquid-crystal elasticity. Translated into our current notation, Ericksen’s argument is essentially: We assume that a uniform nematic state is stable, so any gradients of the director must cost free energy, and hence all of the elastic coefficients in Eq. (\[oseenfranknew\]) must be positive. Thus, he concludes that $0<K_{24}<K_{11}$, $0<K_{24}<K_{22}$, and $0<K_{33}$. If these inequalities are correct, then the cylinder never has a transition from achiral to chiral; the uniform, achiral state is always stable. However, this reasoning has a flaw: We are concluding that the uniform state is stable based on a theory that *assumes* the uniform state is stable. We cannot draw any conclusions from this circular argument. Surprisingly, there are certain liquid crystal materials that violate the Ericksen inequalities, and actually exhibit $K_{22}<K_{24}$. In particular, several studies have investigated the lyotropic chromonic liquid crystals Sunset Yellow (SSY) and disodium cromoglycate (DSCG) in cylindrical capillaries [@Davidson2015; @Nayani2015; @Fu2017] or cylindrical shells [@Javadi2018]. Experimentally, those studies show that the uniform, achiral state can become unstable to the formation of a chiral state. Theoretically, they model the transition in terms of saddle-splay as surface elasticity, which favors alignment of the director field along the curved direction on the surface of the torus. Here, we see that this symmetry-breaking transition has a simple interpretation in terms of the double twist $T$. When $K_{22}<K_{24}$, the coefficient of $T^2$ in the free energy density (\[oseenfranknew\]) becomes negative. Hence, the state with $T=0$ becomes unstable to the formation of $T\not=0$, which can be either positive or negative. The director field will then form a configuration determined by the combination of the favorable free energy associated with double twist and the unfavorable free energy associated with bend. Based on this argument, there is an analogy between the double twist instability of chromonic liquid crystals and the bend instability of bent-core liquid crystals. Dozov [@Dozov2001] argued that the bend elastic constant $K_{33}$ of bent-core liquid crystals can become negative, leading to the formation of a nonuniform phase, and such phases have been found experimentally; see the discussion in Ref. [@Jakli2018]. In chromonic liquid crystals, the double twist elastic constant $(K_{22}-K_{24})$ can become negative, which also leads to the formation of a nonuniform phase. In both cases, the nonuniform phase has a combination of bend and double twist, and the double twist is randomly right- or left-handed. If the coefficient of $T^2$ in the free energy density becomes negative, then we need some mechanism to stabilize the free energy so that it cannot decrease without limit. The free energy might be stabilized by a higher-order power, such as $T^4$, or a higher derivative of the director field $\hat{\bm{n}}$, as discussed below in Sec. VI(A). Alternatively, the free energy might be stabilized by a compatibility constraint, as discussed in Sec. VI(B). Curvature of smectic layers --------------------------- In the smectic-A liquid crystal phase, the molecules lie in (approximately) equally spaced layers, and the director field is normal to the layers. The layers are curved surfaces, which can be treated through the methods of differential geometry. Hence, the deformations of the director field can be related to the curvature of the layers. In particular, the splay of the director field is twice the mean curvature of the layers, and the saddle-splay is twice the Gaussian curvature [@Kleman2003]. Here, because we are analyzing director deformations in terms of the four modes $\bm{B}$, $T$, $S$, and $\bm{\Delta}$, we should determine how all four of these modes are related to the layer curvature. Suppose the equilibrium smectic layers are in the $xy$ plane, and there are small local displacements from $z$ to $z+u(x,y,z)$. To lowest nontrivial order in $u$, the director field is $$\hat{\bm{n}}=\pm(\hat{\bm{z}}-\bm{\nabla}_\perp u).$$ Hence, the director deformations become $$\begin{aligned} &\bm{B}=\bm{\nabla}_\perp (\partial_z u),\quad T=0,\quad S=\mp\nabla_\perp^2 u,\\ &\bm{\Delta}=\pm\begin{pmatrix} \frac{1}{2}(\partial_y^2 u -\partial_x^2 u) & -\partial_x \partial_y u & 0\\ -\partial_x \partial_y u & \frac{1}{2}(\partial_x^2 u -\partial_y^2 u) & 0\\ 0 & 0 & 0 \end{pmatrix},\nonumber\end{aligned}$$ with the $\pm$ signs depending on the arbitrary choice of $\hat{\bm{n}}$ or $-\hat{\bm{n}}$. The twist is zero, as it must be for a director field that is perpendicular to layers. The bend is the perpendicular gradient of $\partial_z u$, which is the variation in layer spacing. If the layers are equally spaced, then the bend is also zero. Hence, the important two deformations are the splay and biaxial splay. We can relate these properties to the curvature of the smectic layers. To lowest nontrivial order in $u$, the curvature tensor is $$K_\alpha^\beta = \begin{pmatrix} -\partial_x^2 u & -\partial_x \partial_y u \\ -\partial_x \partial_y u & -\partial_y^2 u \\ \end{pmatrix}.$$ If $\kappa_1$ and $\kappa_2$ are the two principal curvatures, then the mean curvature becomes $$\frac{1}{2}(\kappa_1 +\kappa_2)=\frac{1}{2}K_\alpha^\alpha = -\frac{1}{2}\nabla_\perp^2 u = \pm\frac{1}{2}S.$$ Hence, the geometric meaning of the splay $S$ is twice the mean curvature. Likewise, the traceless part of the curvature tensor becomes $$K_\alpha^\beta - \frac{1}{2}K_\gamma^\gamma \delta_\alpha^\beta = \begin{pmatrix} \frac{1}{2}(\partial_y^2 u -\partial_x^2 u) & -\partial_x \partial_y u \\ -\partial_x \partial_y u & \frac{1}{2}(\partial_x^2 u -\partial_y^2 u) \end{pmatrix}.$$ Hence, the geometric meaning of the biaxial splay tensor $\bm{\Delta}$ is the traceless part of the curvature tensor. Its eigenvalues are $\pm\frac{1}{2}(\kappa_1 -\kappa_2)$, and its eigenvectors are the principal curvature directions, along with a third eigenvalue of zero corresponding the eigenvector $\hat{\bm{n}}$. From Eq. (\[saddlesplay\]), the standard saddle-splay becomes $$\begin{aligned} &\bm{\nabla}\cdot\left[\hat{\bm{n}}(\bm{\nabla}\cdot\hat{\bm{n}})+\hat{\bm{n}}\times(\bm{\nabla}\times\hat{\bm{n}})\right] =\frac{1}{2}S^2 +\frac{1}{2}T^2 - \operatorname{Tr}(\bm{\Delta}^2)\nonumber\\ &\quad=\frac{1}{2}(\kappa_1 +\kappa_2)^2 - \frac{1}{2}(\kappa_1 -\kappa_2)^2 = 2\kappa_1 \kappa_2.\end{aligned}$$ This agrees with the textbook result that the saddle-splay is twice the Gaussian curvature. We have verified that these results hold exactly, beyond the approximation of small layer displacements, for a toroidal focal conic structure. Related issues ============== Second-derivative elasticity and $K_{13}$ ----------------------------------------- In the literature on elasticity of liquid crystals, the Oseen-Frank free energy density is sometimes written with two divergence terms [@Nehring1971; @Nehring1972; @Kleman2003], $$\begin{aligned} F=&\frac{1}{2}K_{11}S^2 + \frac{1}{2}K_{22}T^2 + \frac{1}{2}K_{33}|\bm{B}|^2 \\ &-K_{24}\bm{\nabla}\cdot\left[\hat{\bm{n}}(\bm{\nabla}\cdot\hat{\bm{n}})+\hat{\bm{n}}\times(\bm{\nabla}\times\hat{\bm{n}})\right]\nonumber\\ &+K_{13}\bm{\nabla}\cdot\left[\hat{\bm{n}}(\bm{\nabla}\cdot\hat{\bm{n}})\right].\nonumber\end{aligned}$$ The origin and effects of the $K_{13}$ term have been analyzed in detail [@Pergamenshchik1998; @Pergamenshchik1999; @Pergamenshchik2000]. In this article, we have discussed how the $K_{24}$ term can be written in terms of the modes $S$, $T$, and $\bm{\Delta}$. Can the $K_{13}$ term be written in a similar way? To answer that question, we note that the $K_{24}$ and $K_{13}$ terms are actually different types of mathematical objects. Although the $K_{24}$ saddle-splay term appears superficially as if it includes second derivatives of $\hat{\bm{n}}$, the second derivatives of the first piece exactly cancel the second derivatives of the second piece. As a result, this term depends only on first derivatives of $\hat{\bm{n}}$. That is why it can be expressed in terms of $S$, $T$, and $\bm{\Delta}$, which are all combinations of first derivatives of $\hat{\bm{n}}$. Hence, it is quite appropriate to include this term in the Oseen-Frank free energy, along with the $K_{11}$, $K_{22}$, and $K_{33}$ terms, which also involve first derivatives of $\hat{\bm{n}}$. By contrast, in the $K_{13}$ term, the second derivatives of $\hat{\bm{n}}$ do not cancel. Instead, that term becomes $$\begin{aligned} \bm{\nabla}\cdot\left[\hat{\bm{n}}(\bm{\nabla}\cdot\hat{\bm{n}})\right] &=\bm{\nabla}\cdot\left[\hat{\bm{n}}S\right] =S^2 + (\hat{\bm{n}}\cdot\bm{\nabla})S\nonumber\\ &=(\partial_i n_i)(\partial_j n_j) + n_i \partial_i \partial_j n_j.\end{aligned}$$ If we include this term in the free energy, with no other second-derivative terms, then the theory would become unstable, because it would favor arbitrary large second derivatives of $\hat{\bm{n}}$. To avoid that problem, we would need to add other second-derivative terms, such as $(\partial_i \partial_j n_k)(\partial_i \partial_j n_k)$, to stabilize the free energy. We recognize that such terms are formally smaller than other terms in the free energy, because they include more derivatives, but still they are necessary for stability. From this discussion, we can see two reasonable options. First, we might consider only first derivatives of $\hat{\bm{n}}$ in the theory. In that case, the free energy would include the $K_{11}$, $K_{22}$, $K_{33}$, and $K_{24}$ terms, but not $K_{13}$. Alternatively, we might develop a higher-order elasticity theory that includes all second derivatives of $\hat{\bm{n}}$. For this higher-order elasticity, we could begin by decomposing the tensor $\partial_i \partial_j n_k$ into its normal modes (by analogy with the calculation in Sec. II) and expressing the general second-derivative free energy in terms of those modes (by analogy with Sec. III). That analysis is beyond the scope of this article. For almost all liquid crystal physics problems, the first option is sufficient. However, there may be a few unusual problems where second-derivative elasticity is needed. One example might be the chromonic liquid crystals discussed in Sec. V(C), where the double twist elastic constant $(K_{22}-K_{24})$ becomes negative. Another example might be the bend instability of bent-core liquid crystals, where the effective bend elastic constant $K_{33}$ becomes negative [@Dozov2001; @Jakli2018]. In those unusual cases, second-derivative elasticity might be important for stabilizing the free energy. Compatibility ------------- In general, some director deformations can exist everywhere in space, and other deformations cannot. For example, it is possible to fill up space with cholesteric single twist (which is a specific combination of $T$ and $\bm{\Delta}$), but it is impossible to fill up space with double twist (pure $T$). That is the reason why blue phases must have tubes of double twist separated by disclination lines, rather than uniform double twist [@Sethna1983]. Similarly, it is possible to fill up space with pure bend of varying magnitude, as in Sec. II(B1), or pure splay of varying magnitude, as in the hedgehog of Sec. II(B3), but it is impossible to fill up space with pure bend or splay of constant magnitude. This issue can be regarded as a *compatibility* problem. For any director field $\hat{\bm{n}}$, we can calculate derivatives to define the modes $\bm{B}$, $T$, $S$, and $\bm{\Delta}$. However, that procedure does not work in reverse: We cannot begin with any arbitrary set of $\bm{B}$, $T$, $S$, and $\bm{\Delta}$, and calculate the corresponding director field. Only certain combinations of $\bm{B}$, $T$, $S$, and $\bm{\Delta}$ can be constructed from the same $\hat{\bm{n}}$. Those combinations can be called compatible, while other combinations are incompatible. An analogous issue of compatibility occurs in the theory of elastic solids. For any displacement field $\bm{u}$, we can calculate derivatives to define the strain tensor $\bm{\epsilon}$. However, we cannot begin with any arbitrary strain tensor and calculate a corresponding displacement field. In this sense, the displacement field of an elastic solid is analogous to the director field of a liquid crystal, and the strain tensor is analogous to the liquid crystal deformation modes $\bm{B}$, $T$, $S$, and $\bm{\Delta}$. The issue of compatibility has been studied extensively in the theory of elastic solids. It is known that the strain tensor must satisfy certain constraints in order to be compatible with a displacement field. By contrast, this issue has not been studied much in the theory of liquid crystals. To our knowledge, it has only been investigated theoretically by Niv and Efrati [@Niv2018], for the case of 2D liquid crystals. They derived the compatibility constraint, which depends on the Gaussian curvature of the 2D surface in which the liquid crystal exists. For a flat surface, it is possible to have a simple case of a uniform liquid crystal, with zero bend and zero splay everywhere, but it is impossible to have uniform nonzero bend or uniform nonzero splay. For a positively curved surface, such as a sphere, it is impossible to have even uniform zero bend or uniform zero splay. By contrast, for a negatively curved surface, such as a saddle, it is possible to have uniform nonzero bend or uniform nonzero splay. So far, the corresponding compatibility constraint or constraints have not yet been derived for 3D liquid crystals. We anticipate that this calculation will be done in the future, and then it will establish what combinations of $\bm{B}$, $T$, $S$, and $\bm{\Delta}$ are compatible with 3D Euclidean flat space. A compatibility constraint may be important for understanding the stability of the Oseen-Frank free energy in the case where one of the elastic constants becomes negative. The free energy does not need to be positive-definite for all variations of $\bm{B}$, $T$, $S$, and $\bm{\Delta}$. Rather, it only needs to be positive-definite for all *compatible* variations of $\bm{B}$, $T$, $S$, and $\bm{\Delta}$, so that it will be positive-definite for all possible variations of $\hat{\bm{n}}$. A mathematical constraint should help to determine what are the compatible variations. Conclusions =========== In this article, we have presented a theoretical formalism to analyze director deformations in liquid crystals. This formalism is based on a mathematical construction by Machon and Alexander [@Machon2016], which decomposes the director gradient tensor $\partial_i n_j$ into four modes: splay, twist, bend, and $\bm{\Delta}$. We re-express the Oseen-Frank free energy in terms of these modes, and show that it takes a simple form as the sum of squares. Using that expression for the free energy, we re-analyze several previous problems in liquid-crystal physics, and show how they can be understood based on the four modes. The main difference between our current approach and previous work is that we now regard all four modes as bulk elastic modes, while previous work usually considered saddle-splay as surface elasticity. We emphasize that there is no contradiction between these perspectives. Indeed, the theories are mathematically equivalent, and give the same predictions for experiments. However, we suggest that the current approach provides a simpler and more intuitive way to understand the role of saddle-splay, and hence provides a useful tool for future theoretical research. We would like to thank E. Efrati, A. Jákli, O. D. Lavrentovich, R. Mosseri, and J.-F. Sadoc for helpful discussions. This work was supported by National Science Foundation Grant DMR-1409658. Part of this work was performed at the Aspen Center for Physics, which is supported by National Science Foundation Grant PHY-1607761. Calculation of four modes from tensor $\bm{Q}$ ============================================== One application of the formalism discussed in this article may be to analyze simulations of complex liquid-crystal structures, such as skyrmions, half-skyrmions, and blue phases. Such simulations are often done using the nematic order tensor field $\bm{Q}(\bm{r})$, rather than the director field $\hat{\bm{n}}(\bm{r})$. For use in these simulations, we would like to express the four modes $\bm{B}$, $T$, $S$, and $\bm{\Delta}$ in terms of the nematic order tensor. The nematic order tensor field is usually written as the traceless form $$Q_{ij}=s\left(\frac{3}{2}n_i n_j -\frac{1}{2}\delta_{ij}\right),$$ where $s$ is the scalar order parameter, i.e. the magnitude of nematic order, not to be confused with the splay $S$. In any defect-free region where $s$ is approximately constant, we can just work with the tensor $$q_{ij}=n_i n_j.$$ It is then straightforward to convert $Q_{ij}=s(\frac{3}{2}q_{ij}-\frac{1}{2}\delta_{ij})$, or $q_{ij}=\frac{1}{3}(\delta_{ij}+2Q_{ij}/s)$. Because $\hat{\bm{n}}$ is a unit vector, we have $$n_i \partial_j n_k = q_{il} \partial_j q_{kl}.$$ By making appropriate contractions of that equation, we can derive the bend vector $$B_k = -n_i \partial_i n_k = -q_{il} \partial_i q_{kl},$$ and the twist pseudoscalar $$T=\epsilon_{ijk} n_i \partial_j n_k =\epsilon_{ijk} q_{il} \partial_j q_{kl}.$$ It is impossible to define the splay scalar $S=\bm\nabla\cdot\hat{\bm n}$ uniquely in terms of $q_{ij}$, because $S$ is odd in $\hat{\bm{n}}$; i.e., it depends on the arbitrary choice of $\hat{\bm{n}}$ or $-\hat{\bm{n}}$. However, we can uniquely define the splay vector $\bm{S}=S\hat{\bm{n}}$, because it is even in $\hat{\bm{n}}$. The splay vector becomes $$S_i = S n_i = n_i \partial_j n_j = q_{il} \partial_j q_{jl}.$$ Likewise, it is impossible to define the second-rank tensor $\Delta_{ij}$ uniquely in terms of $q_{ij}$, because $\Delta_{ij}$ is also odd in $\hat{\bm{n}}$. Instead, we can define the third-rank tensor $$\begin{aligned} \Delta_{ij} n_k = & \frac{1}{2} [n_k \partial_i n_j + n_k \partial_j n_i + n_k n_i B_j + n_k n_j B_i \nonumber\\ &\quad - S n_k (\delta_{ij} - n_i n_j)] \nonumber\\ = & \frac{1}{2} [q_{kl} \partial_i q_{jl} + q_{kl} \partial_j q_{il} + q_{ki} B_j + q_{kj} B_i \nonumber\\ &\quad - S_k (\delta_{ij} - q_{ij})].\end{aligned}$$ Director gradient modes in 2D ============================= In Sec. II(A), we decompose the director gradient tensor $\partial_i n_j$ into the four modes $\bm{B}$, $T$, $S$, and $\bm{\Delta}$ in 3D. Some researchers also investigate nematic liquid crystals in 2D. In this appendix, we do the analogous decomposition in 2D, for use in such studies. First, consider the number of degrees of freedom. In 2D, the tensor $\partial_i n_j$ has $2 \times 2 = 4$ components. Because $\hat{\bm{n}}$ is a unit vector, we have the constraint $(\partial_i n_j)n_j = 0$. That equation is actually 2 constraints, for $i=1$ and $2$. Hence, the tensor $\partial_i n_j$ should have $4-2=2$ degrees of freedom. The first leg of this tensor might have components parallel or perpendicular to $\hat{\bm{n}}$, but the second leg must be perpendicular to $\hat{\bm{n}}$. We break $\partial_i n_j$ into parts where the first leg is parallel or perpendicular to $\hat{\bm{n}}$, $$\partial_i n_j = -n_i B_j + \alpha_{ij},$$ where $\bm{B}$ is a vector perpendicular to $\hat{\bm{n}}$ and $\alpha_{ij}$ is a tensor perpendicular to $\hat{\bm{n}}$. Contracting both sides of this equation with $n_i$ gives $$\bm{B}=-(\hat{\bm{n}}\cdot\bm{\nabla})\hat{\bm{n}}.$$ Hence, $\bm{B}$ is the 2D version of the standard bend vector. Because it is perpendicular to $\hat{\bm{n}}$ in 2D, it has one degree of freedom. Because it is invariant under the transformation $\hat{\bm{n}}\to-\hat{\bm{n}}$, it is a physical object that does not depend on this arbitrary choice of sign. Now we are left with the tensor $\alpha_{ij}$ perpendicular to $\hat{\bm{n}}$. In 2D, this tensor can be written as $\alpha_{ij}=S(\delta_{ij}-n_i n_j)$ for some scalar $S$. Hence, the director gradient tensor becomes $$\partial_i n_j = -n_i B_j + S(\delta_{ij}-n_i n_j), \label{decomposition2D}$$ Taking the trace of both sides of this equation gives $$S=\bm{\nabla}\cdot\hat{\bm{n}}.$$ Hence, $S$ is the 2D version of the standard splay scalar. Because it is a scalar, it has one degree of freedom. Just as in 3D, $S$ changes sign under the transformation $\hat{\bm{n}}\to-\hat{\bm{n}}$. If we want a physical object that does not depend on the arbitrary sign of $\hat{\bm{n}}$, we can construct the splay vector $\bm{S}=S\bm{\hat{n}}=\bm{\hat{n}}(\bm{\nabla}\cdot\hat{\bm{n}})$. Equation (\[decomposition2D\]) decomposes the director gradient tensor $\partial_i n_j$ into the two normal modes $\bm{B}$ and $S$ in 2D. These two modes account for the two degrees of freedom in $\partial_i n_j$. In 2D, there is no analogue for the twist $T$ or the biaxial splay $\bm{\Delta}$, or for saddle-splay.

--- abstract: | The Shapovalov determinant for a class of pointed Hopf algebras is calculated, including quantized enveloping algebras, Lusztig’s small quantum groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl groupoids, and Lusztig type isomorphisms. We elaborate powerful novel techniques for the algebras at roots of unity, and pass to the general case using a density argument. Key words: Hopf algebra, Nichols algebra, quantum group, representation MSC: 16W30; 17B37, 81R50 address: - 'István Heckenberger, Universität zu Köln, Mathematisches Institut, Weyertal 86–90, D-50931 Köln, Germany' - 'Hiroyuki Yamane, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka 560-0043, Japan' author: - 'I. Heckenberger' - 'H. Yamane' title: ' Drinfel’d doubles and Shapovalov determinants' --- Introduction ============ We study finite-dimensional representations of a large class of Hopf algebras $U(\chi )$, where $\chi $ is a bicharacter on ${\mathbb{Z}}^I$ for some finite index set $I$. These algebras emerged from a program of Andruskiewitsch and Schneider to classify pointed Hopf algebras [@a-AndrSchn98], [@p-Heck07b]. Prominent examples are quantized enveloping algebras of semisimple Lie algebras, where the deformation parameter is not a root of $1$, and Lusztig’s (finite-dimensional) small quantum groups, see Sect. \[sec:Uqg\]. Other relevant examples are quantized enveloping algebras of Lie superalgebras, see [@a-KhorTol91] and [@a-Yam99; @a-Yam99e], and Drinfeld doubles of bosonizations of Nichols algebras of diagonal type classified in [@p-Heck06b]. Our main combinatorial tools towards the study of representations are the root system and the Weyl groupoid associated to $\chi $. For quantized enveloping algebras of semisimple Lie algebras the Weyl groupoid is nothing but the Weyl group of the Lie algebra. The main concern of this paper is the determination of the Shapovalov determinants for all algebras $U(\chi )$ with finite root system. We obtain a natural analog of Shapovalov’s original formula (for complex semisimple Lie algebras) as a product of linear factors. For our approach we need that $\chi (\beta ,\beta )\not=1$ for all positive roots $\beta $. This assumption is fulfilled for the special cases mentioned above. The generality of our setting forces us to understand the representation theory of algebras $U(\chi )$, where many values of $\chi $ are roots of $1$. We turn this bondage into a promising leading principle of our approach. We concentrate first on those bicharacters, which take values in the set of roots of $1$. In this case the positive and negative parts of $U(\chi )$ are finite-dimensional algebras. For these algebras we develop special techniques, which are very different from the usual ones for semisimple Lie algebras, based on reflections in the Weyl groupoid, With these techniques we are able to characterize easily the irreducibility of Verma modules. The characterization leads quickly to a formula for the Shapovalov determinants of Verma modules by a variant of the usual density argument. In a next step we extend our results to more general bicharacters by a new density argument. This is possible because of our good knowledge of the root system of bicharacters. The history of Shapovalov determinants started with Shapovalov’s work [@a-Shapov72], where he defined a bilinear form on Verma modules and determinants on homogeneous subspaces to obtain information on the reducibility of Verma modules. These structures have been generalized by Kac and Kazhdan [@a-KK79] to symmetrizable Kac-Moody algebras and by Kac [@inb-Kac79] [@inp-Kac86] to Lie superalgebras with symmetrizable Cartan matrix. Shapovalov determinants have been calculated for quantized enveloping algebras by de Concini and Kac [@inp-dCK90] and for quantized Kac-Moody algebras by Joseph [@b-Joseph]. For Lusztig’s small quantum groups Kumar and Letzter [@a-KumLetz97] factorize the Shapovalov determinants under the assumption that the deformation parameter $q$ has prime order and the base field is a cyclotomic field. Shapovalov determinants have been considered recently in various contexts, see for example [@a-BrunKles02], [@inp-GorSerg05], [@a-Gorel06], [@a-Alek05], [@a-Hill08]. Our approach yields in particular an entirely new proof of the formula of de Concini and Kac. In Sect. \[sec:Uqg\] we improve Kumar’s and Letzter’s result by allowing arbitrary base fields and arbitrary orders of $q$. The paper is organized as follows. In Sect. \[sec:prelims\] the axioms of Cartan schemes, Weyl groupoids, and root systems are recalled. The Weyl groupoid of a bicharacter fits into this framework. Besides collecting the most important facts we introduce a character $\rho ^\chi $ on ${\mathbb{Z}}^I$ which will play a similar role as the linear form $2\rho $ on the root lattice. In Sect. \[sec:DD\] the definition and properties of the Drinfeld doubles $U(\chi )$ are recalled. Sect. \[sec:Lusztig\] deals with Lusztig type isomorphisms between two (usually different) Drinfeld doubles. With Thm. \[th:PBWtau\] we establish a Lusztig type PBW basis of these algebras. Moreover, in Thm. \[th:EErel\] we develop important properties for $q$-commutators of root vectors. In Sect. \[sec:Verma\] we start to study Verma modules and special maps between them. Prop. \[pr:VTMiso\] gives a criterion for bijectivity of such maps, and Prop. \[pr:M=L\] identifies irreducible Verma modules. In Sect. \[sec:shapdet\] we study Shapovalov determinants following the approach in [@b-Joseph]. Here our main result is Thm \[th:Shapdet\], which gives a formula for the Shapovalov determinant of $U(\chi )$, where all values of $\chi $ are roots of $1$, and the root system of $\chi $ is finite. Then we pass to more general bicharacters: Thm. \[th:Shapdet2\] states a similar result for bicharacters with finite root system. We conclude the paper with the adaptation of our formulas to quantized enveloping algebras and Lusztig’s small quantum groups in Sect. \[sec:Uqg\], and with some commutative algebra in the Appendix. Preliminaries {#sec:prelims} ============= Let ${\Bbbk }$ be a field and ${{\Bbbk }^\times }={\Bbbk }\setminus \{0\}$. For all $n\in {\mathbb{N}}_0$ and $q\in {{\Bbbk }^\times }$ let $${(n)_{q}}=\sum _{j=0}^{n-1}q^j,\qquad {(n)^!_{q}}=\prod _{j=1}^n{(j)_{q}},$$ where ${(0)^!_{q}}=1$. For any finite set $I$ let $\{{\alpha }_i\,|\,i\in I\}$ be the standard basis of the free ${\mathbb{Z}}$-module ${\mathbb{Z}}^I$. Cartan schemes, Weyl groupoids, and root systems {#ssec:CS} ------------------------------------------------ The combinatorics of a Drinfel’d double of a Nichols algebra of diagonal type is controlled to a large extent by its Weyl groupoid. We use the language developed in [@p-CH08]. Substantial part of the theory was obtained first in [@a-HeckYam08]. We recall the most important definitions and facts. Let $I$ be a non-empty finite set. By [@b-Kac90 §1.1] a generalized Cartan matrix $C=(c_{ij})_{i,j\in I}$ is a matrix in ${\mathbb{Z}}^{I\times I}$ such that 1. $c_{ii}=2$ and $c_{jk}\le 0$ for all $i,j,k\in I$ with $j\not=k$, 2. if $i,j\in I$ and $c_{ij}=0$, then $c_{ji}=0$. \[de:CS\] Let $I$ be a non-empty finite set, $A$ a non-empty set, ${r}_i : A \to A$ a map for all $i\in I$, and $C^a=(c^a_{jk})_{j,k \in I}$ a generalized Cartan matrix in ${\mathbb{Z}}^{I \times I}$ for all $a\in A$. The quadruple $${\mathcal{C}}= {\mathcal{C}}(I,A,({r}_i)_{i \in I}, (C^a)_{a \in A})$$ is called a *Cartan scheme* if 1. ${r}_i^2 = {\operatorname{id}}$ for all $i \in I$, 2. $c^a_{ij} = c^{{r}_i(a)}_{ij}$ for all $a\in A$ and $i,j\in I$. Let $A=\{a\}$ be a set with a single element, and let $C$ be a generalized Cartan matrix. Then $r_i={\operatorname{id}}$ for all $i\in I$, and ${\mathcal{C}}$ becomes a Cartan scheme. One says that a Cartan scheme ${\mathcal{C}}$ is *connected*, if the group $\langle {r}_i\,|\,i\in I\rangle \subset {\mathrm{Aut}}(A)$ acts transitively on $A$, that is, if for all $a,b\in A$ with $a\not=b$ there exist $n\in {\mathbb{N}}$ and $i_1,i_2,\ldots ,i_n\in I$ such that $b=r_{i_n}\cdots r_{i_2} r_{i_1}(a)$. Two Cartan schemes ${\mathcal{C}}={\mathcal{C}}(I,A,({r}_i)_{i\in I},(C^a)_{a\in A})$ and ${\mathcal{C}}'={\mathcal{C}}'(I',A',$ $({r}'_i)_{i\in I'},({C'}^a)_{a\in A'})$ are called *equivalent*, if there are bijections $\varphi _0:I\to I'$ and $\varphi _1:A\to A'$ such that $$\begin{aligned} \label{eq:equivCS} \varphi _1({r}_i(a))={r}'_{\varphi _0(i)}(\varphi _1(a)), \qquad c^{\varphi _1(a)}_{\varphi _0(i) \varphi _0(j)}=c^a_{i j} \end{aligned}$$ for all $i,j\in I$ and $a\in A$. Let ${\mathcal{C}}= {\mathcal{C}}(I,A,({r}_i)_{i \in I}, (C^a)_{a \in A})$ be a Cartan scheme. For all $i \in I$ and $a \in A$ define ${\sigma }_i^a \in \Aut({\mathbb{Z}}^I)$ by $$\begin{aligned} {\sigma }_i^a ({\alpha }_j) = {\alpha }_j - c_{ij}^a{\alpha }_i \qquad \text{for all $j \in I$.} \label{eq:sia} \end{aligned}$$ This map is a reflection. The *Weyl groupoid of* ${\mathcal{C}}$ is the category ${\mathcal{W}}({\mathcal{C}})$ such that ${\mathrm{Ob}}({\mathcal{W}}({\mathcal{C}}))=A$ and the morphisms are generated by the maps ${\sigma }_i^a\in {\mathrm{Hom}}(a,{r}_i(a))$ with $i\in I$, $a\in A$. Formally, for $a,b\in A$ the set ${\mathrm{Hom}}(a,b)$ consists of the triples $(b,f,a)$, where $$f={\sigma }_{i_n}^{{r}_{i_{n-1}}\cdots {r}_{i_1}(a)}\cdots {\sigma }_{i_2}^{{r}_{i_1}(a)}{\sigma }_{i_1}^a$$ and $b={r}_{i_n}\cdots {r}_{i_2}{r}_{i_1}(a)$ for some $n\in {\mathbb{N}}_0$ and $i_1,\ldots ,i_n\in I$. The composition is induced by the group structure of ${\mathrm{Aut}}({\mathbb{Z}}^I)$: $$(a_3,f_2,a_2)\circ (a_2,f_1,a_1) = (a_3,f_2f_1, a_1)$$ for all $(a_3,f_2,a_2),(a_2,f_1,a_1)\in {\mathrm{Hom}}({\mathcal{W}}({\mathcal{C}}))$. By abuse of notation we will write $f\in {\mathrm{Hom}}(a,b)$ instead of $(b,f,a)\in {\mathrm{Hom}}(a,b)$. The cardinality of $I$ is termed the *rank of* ${\mathcal{W}}({\mathcal{C}})$. A Cartan scheme is called *connected* if its Weyl groupoid is connected. Recall that a groupoid is a category such that all morphisms are isomorphisms. The Weyl groupoid ${\mathcal{W}}({\mathcal{C}})$ of a Cartan scheme ${\mathcal{C}}$ is a groupoid, see [@p-CH08]. For all $i\in I$ and $a\in A$ the inverse of ${\sigma }_i^a$ is ${\sigma }_i^{r_i(a)}$. If ${\mathcal{C}}$ and ${\mathcal{C}}'$ are equivalent Cartan schemes, then ${\mathcal{W}}({\mathcal{C}})$ and ${\mathcal{W}}({\mathcal{C}}')$ are isomorphic groupoids. A groupoid $G$ is called *connected*, if for each $a,b\in {\mathrm{Ob}}(G)$ the class ${\mathrm{Hom}}(a,b)$ is non-empty. Hence ${\mathcal{W}}({\mathcal{C}})$ is a connected groupoid if and only if ${\mathcal{C}}$ is a connected Cartan scheme. \[de:RSC\] Let ${\mathcal{C}}={\mathcal{C}}(I,A,({r}_i)_{i\in I},(C^a)_{a\in A})$ be a Cartan scheme. For all $a\in A$ let $R^a\subset {\mathbb{Z}}^I$, and define $m_{i,j}^a= |R^a \cap (\ndN_0{\alpha }_i + \ndN_0{\alpha }_j)|$ for all $i,j\in I$ and $a\in A$. We say that $${\mathcal{R}}= {\mathcal{R}}({\mathcal{C}}, (R^a)_{a\in A})$$ is a *root system of type* ${\mathcal{C}}$, if it satisfies the following axioms. 1. $R^a=R^a_+\cup - R^a_+$, where $R^a_+=R^a\cap \ndN_0^I$, for all $a\in A$. 2. $R^a\cap {\mathbb{Z}}{\alpha }_i=\{{\alpha }_i,-{\alpha }_i\}$ for all $i\in I$, $a\in A$. 3. ${\sigma }_i^a(R^a) = R^{{r}_i(a)}$ for all $i\in I$, $a\in A$. 4. If $i,j\in I$ and $a\in A$ such that $i\not=j$ and $m_{i,j}^a$ is finite, then $({r}_i{r}_j)^{m_{i,j}^a}(a)=a$. If ${\mathcal{R}}$ is a root system of type ${\mathcal{C}}$, then ${\mathcal{W}}({\mathcal{R}})={\mathcal{W}}({\mathcal{C}})$ is the *Weyl groupoid of* ${\mathcal{R}}$. Further, ${\mathcal{R}}$ is called *connected*, if ${\mathcal{C}}$ is a connected Cartan scheme. If ${\mathcal{R}}={\mathcal{R}}({\mathcal{C}},(R^a)_{a\in A})$ is a root system of type ${\mathcal{C}}$ and ${\mathcal{R}}'={\mathcal{R}}'({\mathcal{C}}',({R'}^a_{a\in A'}))$ is a root system of type ${\mathcal{C}}'$, then we say that ${\mathcal{R}}$ and ${\mathcal{R}}'$ are *equivalent*, if ${\mathcal{C}}$ and ${\mathcal{C}}'$ are equivalent Cartan schemes given by maps $\varphi _0:I\to I'$, $\varphi _1:A\to A'$ as in Def. \[de:CS\], and if the map $\varphi _0^*:\ndZ^I\to \ndZ^{I'}$ given by $\varphi _0^*({\alpha }_i)={\alpha }_{\varphi _0(i)}$ satisfies $\varphi _0^*(R^a)={R'}^{\varphi _1(a)}$ for all $a\in A$. There exist many interesting examples of root systems of type ${\mathcal{C}}$ related to semisimple Lie algebras, Lie superalgebras and Nichols algebras of diagonal type, respectively. For further details and results we refer to [@a-HeckYam08] and [@p-CH08]. \[con:uind\] In connection with Cartan schemes ${\mathcal{C}}$, upper indices usually refer to elements of $A$. Often, these indices will be omitted if they are uniquely determined by the context. In particular, for any $w,w'\in {\mathrm{Hom}}({\mathcal{W}}({\mathcal{C}}))$ and $a\in A$, the notation $1_aw$ and $w'1_a$ means that $w\in {\mathrm{Hom}}(\underline{\,\,},a)$ and $w'\in {\mathrm{Hom}}(a,\underline{\,\,})$, respectively. A fundamental result about Weyl groupoids is the following theorem. [@a-HeckYam08 Thm.1]\[th:Coxgr\] Let ${\mathcal{C}}={\mathcal{C}}(I,A,({r}_i)_{i\in I},(C^a)_{a\in A})$ be a Cartan scheme and ${\mathcal{R}}={\mathcal{R}}({\mathcal{C}},(R^a)_{a\in A})$ a root system of type ${\mathcal{C}}$. Let ${\mathcal{W}}$ be the abstract groupoid with ${\mathrm{Ob}}({\mathcal{W}})=A$ such that ${\mathrm{Hom}}({\mathcal{W}})$ is generated by abstract morphisms $s_i^a\in {\mathrm{Hom}}(a,{r}_i(a))$, where $i\in I$ and $a\in A$, satisfying the relations $$\begin{aligned} s_i s_i 1_a=1_a,\quad (s_j s_k)^{m_{j,k}^a}1_a=1_a, \qquad a\in A,\,i,j,k\in I,\, j\not=k, \end{aligned}$$ see Conv. \[con:uind\]. Here $1_a$ is the identity of the object $a$, and $(s_j s_k)^\infty 1_a$ is understood to be $1_a$. The functor ${\mathcal{W}}\to {\mathcal{W}}({\mathcal{R}})$, which is the identity on the objects, and on the set of morphisms is given by $s _i^a\mapsto \s_i^a$ for all $i\in I$, $a\in A$, is an isomorphism of groupoids. If ${\mathcal{C}}$ is a Cartan scheme, then the Weyl groupoid ${\mathcal{W}}({\mathcal{C}})$ admits a length function $\ell :{\mathcal{W}}(\cC )\to {\mathbb{N}}_0$ such that $$\begin{aligned} \ell (w)=\min \{k\in {\mathbb{N}}_0\,|\,\exists i_1,\dots ,i_k\in I,a\in A: w={\sigma }_{i_1}\cdots {\sigma }_{i_k}1_a\} \label{eq:ell}\end{aligned}$$ for all $w\in {\mathcal{W}}({\mathcal{C}})$. If there exists a root system of type ${\mathcal{C}}$, then $\ell $ has very similar properties to the well-known length function for Weyl groups, see [@a-HeckYam08]. Let ${\mathcal{C}}$ be a Cartan scheme and ${\mathcal{R}}$ a root system of type ${\mathcal{C}}$. Let $a\in A$. Then $-c^a_{ij}=\max\{m\in {\mathbb{N}}_0\,|\,{\alpha }_j+m{\alpha }_i\in R^a_+\}$ for all $i,j\in I$ with $i\not=j$. \[le:cm\] By (C2) and (R3), ${\sigma }_i^{r_i(a)}({\alpha }_j)={\alpha }_j-c^a_{ij}{\alpha }_i\in R^a_+$. Hence $-c^a_{ij}\le \max\{m\in {\mathbb{N}}_0\,|\,{\alpha }_j+m{\alpha }_i\in R^a_+\}$. On the other hand, if ${\alpha }_j+m{\alpha }_i\in R^a_+$, then ${\sigma }_i^a({\alpha }_j+m\al _i)={\alpha }_j+(-c^a_{ij}-m){\alpha }_i\in R^{r_i(a)}_+$ by (R3) and (R1), and hence $m\le -c^a_{ij}$. This proves the lemma. Let ${\mathcal{C}}$ be a Cartan scheme and ${\mathcal{R}}$ a root system of type ${\mathcal{C}}$. We say that ${\mathcal{R}}$ is *finite*, if $R^a$ is finite for all $a\in A$. The following lemmata are well-known for traditional root systems. [@p-CH08 Lemma 2.11] Let ${\mathcal{C}}$ be a connected Cartan scheme and ${\mathcal{R}}$ a root system of type ${\mathcal{C}}$. The following are equivalent. 1. ${\mathcal{R}}$ is finite. 2. $R^a$ is finite for at least one $a\in A$. 3. ${\mathcal{W}}({\mathcal{R}})$ is finite. \[le:Rfincond\] [@a-HeckYam08 Cor.5] Let ${\mathcal{C}}$ be a connected Cartan scheme and ${\mathcal{R}}$ a finite root system of type ${\mathcal{C}}$. Then for all $a\in A$ there exist unique elements $b\in A$ and $w\in {\mathrm{Hom}}(b,a)$ such that $|R^a_+|=\ell (w)\ge \ell (w')$ for all $w'\in {\mathrm{Hom}}(b',a')$, $a',b'\in A$. \[le:longestw\] The Weyl groupoid of a bicharacter {#ssec:Weylgroupoid} ---------------------------------- Let $I$ be a non-empty finite set. Recall that a bicharacter on ${\mathbb{Z}}^I$ with values in ${{\Bbbk }^\times }$ is a map $\chi :{\mathbb{Z}}^I\times {\mathbb{Z}}^I\to {{\Bbbk }^\times }$ such that $$\begin{aligned} \chi (a+b,c)=&\chi (a,c)\chi (b,c),& \chi (c,a+b)=&\chi (c,a)\chi (c,b) \label{eq:bichar}\end{aligned}$$ for all $a,b,c\in {\mathbb{Z}}^I$. Then $\chi (0,a)=\chi (a,0)=1$ for all $a\in {\mathbb{Z}}^I$. Let ${\mathcal{X}}$ be the set of bicharacters on ${\mathbb{Z}}^I$. If $\chi \in {\mathcal{X}}$, then $$\begin{aligned} \label{eq:chiop} \chi {^\mathrm{op}}: & {\mathbb{Z}}^I\times {\mathbb{Z}}^I\to {{\Bbbk }^\times },& \chi {^\mathrm{op}}(a,b)=&\,\chi (b,a),\\ \label{eq:chiinv} \chi ^{-1} : & {\mathbb{Z}}^I\times {\mathbb{Z}}^I\to {{\Bbbk }^\times },& \chi ^{-1}(a,b)=&\,\chi (a,b)^{-1}, \intertext{and for all $w\in {\mathrm{Aut}}_{\mathbb{Z}}({\mathbb{Z}}^I)$ the map} \label{eq:w*chi} w^*\chi : & {\mathbb{Z}}^I\times {\mathbb{Z}}^I\to {{\Bbbk }^\times },& w^*\chi (a,b)=&\,\chi (w^{-1}(a),w^{-1}(b)),\end{aligned}$$ are bicharacters on ${\mathbb{Z}}^I$. The equation $$\begin{aligned} \label{eq:w*functor} (ww')^*\chi =w^*(w'{}^*\chi )\end{aligned}$$ holds for all $w,w'\in {\mathrm{Aut}}_{{\mathbb{Z}}}({\mathbb{Z}}^I)$ and all $\chi \in {\mathcal{X}}$. \[de:Cartan\] Let $\chi \in {\mathcal{X}}$, $p\in I$, and $q_{ij}=\chi ({\alpha }_i,{\alpha }_j)$ for all $i,j\in I$. We say that $\chi $ is $p$-*finite*, if for all $j\in I$ there exists $m\in {\mathbb{N}}_0$ such that $\qnum{m+1}{q_{pp}}=0$ or $q_{pp}^m q_{pj}q_{jp}=1$. Assume that $\chi $ is $p$-finite. Let $c_{p p}^\chi =2$, and for all $j\in I\setminus \{p\}$ let $$c_{pj}^\chi =-\min \{m\in {\mathbb{N}}_0 \,|\, (m+1)_{q_{pp}}(q_{pp}^m q_{pj} q_{jp}-1)=0\}.$$ If $\chi $ is $i$-finite for all $i\in I$, then the matrix $C^\chi =(c_{ij}^\chi )_{i,j\in I}$ is called the *Cartan matrix* associated to $\chi $. It is a generalized Cartan matrix, see Sect. \[ssec:CS\]. For all $p\in I$ and $\chi \in {\mathcal{X}}$, where $\chi $ is $p$-finite, let ${\sigma }_p^\chi \in {\mathrm{Aut}}_{\mathbb{Z}}({\mathbb{Z}}^I)$, $$\begin{aligned} {\sigma }_p^\chi ({\alpha }_j)={\alpha }_j-c_{pj}^\chi {\alpha }_p \quad \text{for all $j\in I$.}\end{aligned}$$ Towards the definition of the Weyl groupoid of a bicharacter, we define bijections $r_p:{\mathcal{X}}\to {\mathcal{X}}$ for all $p\in I$. Namely, let $$\begin{aligned} r_p: {\mathcal{X}}\to \cX,\quad r_p(\chi )= \begin{cases} ({\sigma }_p^\chi )^*\chi & \text{if $\chi $ is $p$-finite,}\\ \chi & \text{otherwise.} \end{cases}\end{aligned}$$ Let $p\in I$, $\chi \in {\mathcal{X}}$, $q_{ij}=\chi ({\alpha }_i,{\alpha }_j)$ for all $i,j\in I$. If $\chi $ is $p$-finite, then $$\begin{aligned} r_p(\chi )({\alpha }_p,{\alpha }_p)=&q_{p p}, & r_p(\chi )({\alpha }_p,{\alpha }_j)=&q_{p j}^{-1}q_{p p}^{c_{pj}^\chi },\\ r_p(\chi )({\alpha }_i,{\alpha }_p)=&q_{i p}^{-1}q_{p p}^{c_{pi}^\chi },& r_p(\chi )({\alpha }_i,{\alpha }_j)=&q_{i j} q_{i p}^{-c_{p j}^\chi } q_{p j}^{-c_{p i}^\chi } q_{p p}^{c_{pi}^\chi c_{p j}^\chi } \end{aligned} \label{eq:rpchi}$$ for all $i,j\in I\setminus \{p\}$. It is a small exercise to check that then $({\sigma }_p^\chi )^*\chi $ is $p$-finite, and $$\begin{aligned} \label{eq:rp2} c_{pj}^{r_p(\chi )}=c_{pj}^\chi \quad \text{for all $j\in I$}, \qquad r_p^2(\chi )=\chi .\end{aligned}$$ The reflections $r_p$, $p\in I$, generate a subgroup $$\begin{aligned} {\mathcal{G}}=\langle r_p\,|\, p\in I\rangle\end{aligned}$$ of the group of bijections of the set ${\mathcal{X}}$. For all $\chi \in {\mathcal{X}}$ let ${\mathcal{G}}(\chi )$ denote the ${\mathcal{G}}$-orbit of $\chi $ under the action of ${\mathcal{G}}$. Let $\chi \in {\mathcal{X}}$ such that $\chi '$ is $p$-finite for all $\chi '\in {\mathcal{G}}(\chi )$ and $p\in I$. By Eq.  we obtain that $${\mathcal{C}}(\chi )= {\mathcal{C}}(I,{\mathcal{G}}(\chi ),(r_p)_{p\in I}, (C^{\chi '})_{\chi '\in {\mathcal{G}}(\chi )})$$ is a connected Cartan scheme. The Weyl groupoid of $\chi $ is then the Weyl groupoid of the Cartan scheme ${\mathcal{C}}(\chi )$ and is denoted by ${\mathcal{W}}(\chi )$. Clearly, ${\mathcal{C}}(\chi )={\mathcal{C}}(\chi ')$ and ${\mathcal{W}}(\chi )={\mathcal{W}}(\chi ')$ for all $\chi '\in {\mathcal{G}}(\chi )$. \[ex:Cartan\] Let $C=(c_{i j})_{i,j\in I}$ be a generalized Cartan matrix. Let $\chi \in {\mathcal{X}}$, $q_{ij}=\chi ({\alpha }_i,{\alpha }_j)$ for all $i,j\in I$, and assume that $q_{ii}^{c_{ij}}=q_{ij}q_{ji}$ for all $i,j\in I$, and that $\qnum{m+1}{q_{ii}}\not=0$ for all $i\in I$ and $m\in {\mathbb{N}}_0$ with $m<\max \{-c_{ij}\,|\,j\in I\setminus \{i\}\}$. (The latter is not an essential assumption, since if it fails, then one can replace $C$ by another generalized Cartan matrix $\tilde{C}$, such that $\chi $ has this property with respect to $\tilde{C}$.) One says that $\chi $ is of *Cartan type*. Then $\chi $ is $i$-finite for all $i\in I$, and $c_{ij}^\chi =c_{ij}$ for all $i,j\in I$. Eq.  gives that $$\begin{aligned} r_p(\chi )({\alpha }_i,{\alpha }_i)=&q_{ii}=\chi ({\alpha }_i,{\alpha }_i),\\ r_p(\chi )({\alpha }_i,{\alpha }_j)\,r_p(\chi )({\alpha }_j,{\alpha }_i)=&q_{ij}q_{ji}= r_p(\chi )({\alpha }_i,{\alpha }_i)^{c_{pi}} \end{aligned}$$ for all $p,i,j\in I$. Hence $r_p(\chi )$ is again of Cartan type with the same Cartan matrix $C$. Thus $\chi '$ is $i$-finite for all $\chi '\in {\mathcal{G}}(\chi )$ and $i\in I$. Let $C=(c_{ij})_{i,j\in I}$ be a symmetrizable generalized Cartan matrix, and for all $i\in I$ let $d_i\in {\mathbb{N}}$ such that $d_ic_{ij}=d_jc_{ji}$ for all $i,j\in I$. Let $q\in {{\Bbbk }^\times }$ such that $\qnum{m+1}{q^{2d_i}}\not=0$ for all $m\in {\mathbb{N}}_0$ with $m<-c_{ij}$ for some $j\in I$. Define $\chi \in {\mathcal{X}}$ by $\chi ({\alpha }_i,{\alpha }_j)=q^{d_ic_{ij}}$. Then $\chi $ is of Cartan type, hence $\chi $ is $p$-finite for all $p\in I$. Eq.  implies that $r_p(\chi )=\chi $ for all $p\in I$, and hence ${\mathcal{G}}(\chi )$ consists of precisely one element. In this case the Weyl groupoid ${\mathcal{W}}(\chi )$ is a group, which is precisely the Weyl group associated to the generalized Cartan matrix $C$. We will study this example in Sect. \[sec:Uqg\] under the assumption that $C$ is of finite type. Roots {#ssec:roots} ----- Let $\chi \in {\mathcal{X}}$. There exists a canonical root system of type ${\mathcal{C}}(\chi)$ which we describe in this subsection. It is based on the construction of a restricted PBW basis of Nichols algebras of diagonal type. Nichols algebras are braided Hopf algebras defined by a universal property. More details can be found in [@inp-AndrSchn02] on braided Hopf algebras and Nichols algebras, in [@a-Khar99] on the PBW basis, and in [@a-Heck06a] on the root system. Let $V\in { {}_{{\Bbbk }{\mathbb{Z}}^I}^{{\Bbbk }{\mathbb{Z}}^I}\mathcal{YD}}$ be a $|I|$-dimensional module of diagonal type. Let ${\delta }:V\to {\Bbbk }{\mathbb{Z}}^I{\otimes }V$ and ${\boldsymbol{\cdot}}:{\Bbbk }{\mathbb{Z}}^I{\otimes }V\to V$ denote the left coaction and the left action of ${\Bbbk }{\mathbb{Z}}^I$ on $V$, respectively. Fix a basis $\{x_i\,|\,i\in I\}$ of $V$, elements $g_i$, where $i\in I$, and a matrix $(q_{ij})_{i,j\in I}\in (\fienz)^{I\times I}$, such that $${\delta }(x_i)=g_i{\otimes }x_i,\quad g_i{\boldsymbol{\cdot}}x_j=q_{ij}x_j \quad \text{for all $i,j\in I$.}$$ Assume that $\chi ({\alpha }_i,{\alpha }_j)=q_{ij}$ for all $i,j\in I$. For all ${\alpha }\in {\mathbb{Z}}^I$ define the “bound function” $$\begin{aligned} {b^{\chi}} ({\alpha })=& \begin{cases} \min \{ m\in {\mathbb{N}}\,|\, \qnum{m}{\chi ({\alpha },{\alpha })}=0\} & \text{if $\qnum{m}{\chi ({\alpha },{\alpha })}=0$}\\ & \text{for some $m\in {\mathbb{N}}$,}\\ \infty & \text{otherwise}. \end{cases} \label{eq:height}\end{aligned}$$ If $p\in I$ such that $\chi $ is $p$-finite, then $$\begin{aligned} \bfun{r_p(\chi )}({\sigma }_p^\chi ({\alpha }))={b^{\chi}} ({\alpha })\quad \text{for all ${\alpha }\in {\mathbb{Z}}^I$} \label{eq:hghtrpchi}\end{aligned}$$ by Eq. . The tensor algebra $T(V)$ admits a universal braided Hopf algebra quotient ${\mathfrak{B}(V)}$, called the *Nichols algebra of* $V$. As an algebra, ${\mathfrak{B}(V)}$ has a unique ${\mathbb{Z}}^I$-grading $$\begin{aligned} {\mathfrak{B}(V)}=\oplus _{{\alpha }\in {\mathbb{Z}}^I}{\mathfrak{B}(V)} _{\alpha }\label{eq:NAVgrading}\end{aligned}$$ such that $\deg x_i={\alpha }_i$ for all $i\in I$. This is also a coalgebra grading. There exists a totally ordered index set $(L,\le )$ and a family $(y_l)_{l\in L}$ of ${\mathbb{Z}}^I$-homogeneous elements $y_l\in {\mathfrak{B}(V)}$ such that the set $$\begin{aligned} \{ y_{l_1}^{m_1}y_{l_2}^{m_2}\cdots y_{l_k}^{m_k}\,|\, &k\ge 0,\,l_1,\dots ,l_k\in L,\,l_1>l_2>\cdots >l_k,\\ &m_i\in {\mathbb{N}},\,m_i<{b^{\chi}} (\deg y_{l_i}) \quad \text{for all $i\in I$}\} \end{aligned} \label{eq:PBWbasis}$$ forms a vector space basis of ${\mathfrak{B}(V)}$. The set $$\begin{aligned} \label{eq:roots} R^\chi _+=\{\deg y_l\,|\,l\in L\}\subset {\mathbb{Z}}^I\end{aligned}$$ depends on the matrix $(q_{ij})_{i,j\in I}$, but not on the choice of the basis $\{x_i\,|\,i\in I\}$, the set $L$, and the elements $g_i$, $i\in I$, and $y_l$, $l\in L$. Let $$R^\chi =R^\chi _+\cup -R^\chi _+.$$ [@p-Heck07b Thm.3.13] \[th:rschi\] Let $\chi \in {\mathcal{X}}$ such that $\chi '$ is $p$-finite for all $p\in I$. $\chi '\in {\mathcal{G}}(\chi )$. Then ${\mathcal{R}}(\chi )={\mathcal{R}}({\mathcal{C}}(\chi ), (R^{\chi '})_{\chi '\in {\mathcal{G}}(\chi )})$ is a root system of type ${\mathcal{C}}(\chi )$. Roots with finite bounds often play a distinguished role. For all $\chi \in {\mathcal{X}}$ let $$\begin{aligned} R^\chi _{+{\mathrm{fin}}}=\{\beta \in R^\chi _+\,|\, {b^{\chi}} (\beta )<\infty \},\quad R^\chi _{+\infty }=R^\chi _+\setminus R^\chi _{+{\mathrm{fin}}}. \label{eq:R+fin}\end{aligned}$$ We will use several finiteness properties of bicharacters. $$\begin{aligned} \label{eq:X1} {\mathcal{X}}_1=\{\chi \in {\mathcal{X}}\,|\,& \text{$\chi $ is $p$-finite for all $p\in I$}\},\\ {\mathcal{X}}_2=\{\chi \in {\mathcal{X}}\,|\,& \text{$\chi '$ is $p$-finite for all $\chi '\in {\mathcal{G}}(\chi )$, $p\in I$}\},\\ {\mathcal{X}}_3=\{\chi \in {\mathcal{X}}\,|\,& \text{$R^\chi $ is finite}\},\\ {\mathcal{X}}_4=\{\chi \in {\mathcal{X}}\,|\,& \text{$R^\chi $ is finite, $R^\chi _+=R^\chi _{+{\mathrm{fin}}}$}\},\\ \label{eq:X5} {\mathcal{X}}_5=\{\chi \in {\mathcal{X}}_4\,|\,& \text{$\chi ({\alpha },{\alpha })\not=1$ for all ${\alpha }\in R^\chi _+$}\}.\end{aligned}$$ Clearly, ${\mathcal{X}}_i\supset {\mathcal{X}}_j$ for $1\le i<j\le 5$. By Eq. , $\chi \in {\mathcal{X}}_5$ if and only if $R^\chi $ is finite and $\chi ({\alpha },{\alpha })$ is a root of $1$ different from $1$ for all ${\alpha }\in R^\chi _+$. \[le:equalrs\] Let $\chi ,\chi '\in {\mathcal{X}}_2$. \(i) If $R^\chi _+=R^{\chi '}_+$, then $C^{w^*\chi }=C^{w^*\chi '}$ for all $w\in {\mathrm{Hom}}(\chi ,\underline{\,\,}) \subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))$. \(ii) Assume that $\chi ,\chi '\in {\mathcal{X}}_3$. If $C^{w^*\chi }=C^{w^*\chi '}$ for all $w\in {\mathrm{Hom}}(\chi ,\underline{\,\,}) \subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))$, then $R^\chi _+=R^{\chi '}_+$. By Thm. \[th:rschi\], ${\mathcal{R}}(\chi )$ is a root system of type $\cC (\chi )$. \(i) Assume that $R^\chi _+=R^{\chi '}_+$. Then $C^{\chi }=C^{\chi '}$ by Lemma \[le:cm\]. Therefore ${\sigma }_i^\chi ={\sigma }_i^{\chi '}$ in ${\mathrm{Aut}}({\mathbb{Z}}^I)$. Since $\chi ,\chi '\in {\mathcal{X}}_2$, by induction it follows that $\s_{i_1}\cdots {\sigma }_{i_k}^\chi ={\sigma }_{i_1}\cdots {\sigma }_{i_k}^{\chi '}$ in ${\mathrm{Aut}}({\mathbb{Z}}^I)$ and $C^{(\s_{i_1}\cdots {\sigma }_{i_k}^\chi )^*\chi }= C^{(\s_{i_1}\cdots {\sigma }_{i_k}^{\chi '})^*\chi '}$ for all $k\in {\mathbb{N}}_0$ and $i_1,\dots ,i_k\in I$. Hence $C^{w^*\chi }=C^{w^*\chi '}$ for all $w\in {\mathrm{Hom}}(\chi ,\underline{\,\,}) \subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))$. \(ii) Since $\chi \in {\mathcal{X}}_3$, $R^\chi =\{w^{-1}({\alpha }_i)\,|\,w\in \Hom (\chi ,\underline{\,\,})\subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))\}$ by [@p-CH08 Prop.2.12]. By assumption on the Cartan matrices, $\s_{i_1}\cdots {\sigma }_{i_k}^\chi ={\sigma }_{i_1}\cdots {\sigma }_{i_k}^{\chi '}$ in ${\mathrm{Aut}}({\mathbb{Z}}^I)$ for all $k\in {\mathbb{N}}_0$ and $i_1,\dots,i_k\in I$. Hence $R^\chi =R^{\chi '}$, and the lemma holds by (R1). For our study of Drinfel’d doubles we will use an analog of the sum of fundamental weights, commonly known as $\rho $. More precisely, we define a character version of the linear form $(2\rho ,\cdot )$, where $(\cdot ,\cdot )$ is the usual bilinear form on the weight lattice. Let ${\widehat{{\mathbb{Z}}^I}}={\mathrm{Hom}}({\mathbb{Z}}^I,{{\Bbbk }^\times })$ denote the group of characters of ${\mathbb{Z}}^I$ with values in ${{\Bbbk }^\times }$. \[de:rhomap\] Let $\chi \in {\mathcal{X}}$. Let ${\rho ^{\chi}} \in \widehat{{\mathbb{Z}}^I}$ such that $${\rho ^{\chi}} ({\alpha }_i)=\chi ({\alpha }_i,{\alpha }_i) \quad \text{for all $i\in I$}.$$ Let $\chi \in {\mathcal{X}}$, $p\in I$, and ${b}={b^{\chi}} ({\alpha }_p)$. Assume that ${b}<\infty $. Then $\chi $ is $p$-finite and $$\chi ({\alpha }_p,\beta )^{{b}-1}\chi (\beta ,{\alpha }_p)^{{b}-1}= \frac{\rhomap{r_p(\chi )}({\sigma }_p^\chi (\beta ))}{\rhomap\chi (\beta )}$$ for all $\beta \in {\mathbb{Z}}^I$. \[le:rho\] Define $\xi _1,\xi _2:{\mathbb{Z}}^I\to {{\Bbbk }^\times }$ by $$\xi _1(\beta )=\chi ({\alpha }_p,\beta )^{{b}-1}\chi (\beta ,\al _p)^{{b}-1}, \qquad \xi _2(\beta )=\frac{\rhomap{r_p(\chi )}({\sigma }_p^\chi (\beta ))}{ \rhomap\chi (\beta )}.$$ Then $\xi _1,\xi _2\in \ZIdual$. Thus it suffices to prove that $\xi _1({\alpha }_j)=\xi _2({\alpha }_j)$ for all $j\in I$. Let $q_{jk}=\chi ({\alpha }_j,{\alpha }_k)$ for all $j,k\in I$. Then $q_{pp}^{b}=1$ since $\qnum{{b}}{q_{pp}}=0$. Moreover, $$\begin{aligned} \rhomap{r_p(\chi )}({\alpha }_j)=&r_p(\chi )({\alpha }_j,{\alpha }_j)\\ =&\chi ({\alpha }_j-c_{pj}^\chi {\alpha }_p,{\alpha }_j-c_{pj}^\chi {\alpha }_p) =q_{jj}(q_{pj}q_{jp})^{-c_{pj}^\chi }q_{pp}^{c_{pj}^\chi c_{pj}^\chi } \end{aligned}$$ for all $j\in I$. By assumption, $\chi $ is $p$-finite, and hence for all $j\in I\setminus \{p\}$ we have $q_{pp}^{c_{pj}^\chi }=q_{pj}q_{jp}$ or $q_{pp}^{1-c_{pj}^\chi }=1$. Let $j\in I$. If $q_{pp}^{c_{pj}^\chi }=q_{pj}q_{jp}$, then $$\begin{gathered} \rhomap{r_p(\chi )}({\alpha }_j)=q_{jj},\\ \xi _2({\alpha }_j)=q_{jj}^{-1}\rhomap{r_p(\chi )}({\sigma }_p^\chi ({\alpha }_j))= q_{jj}^{-1}\rhomap{r_p(\chi )}({\alpha }_j-c_{pj}^\chi {\alpha }_p) =q_{pp}^{-c_{pj}^\chi},\\ \xi _1({\alpha }_j)=q_{pj}^{{b}-1} q_{jp}^{{b}-1}=q_{pp}^{({b}-1)c_{pj}^\chi } =q_{pp}^{-c_{pj}^\chi }, \end{gathered}$$ and hence $\xi _1({\alpha }_j)=\xi _2({\alpha }_j)$. Otherwise ${b}=1-c_{pj}^\chi $, $q_{pp}^{c_{pj}^\chi }=q_{pp}$, and then $$\begin{gathered} \rhomap{r_p(\chi )}({\alpha }_p)=q_{pp},\quad \rhomap{r_p(\chi )}({\alpha }_j)=q_{jj}(q_{pj}q_{jp})^{-c_{pj}^\chi }q_{pp},\\ \xi _2({\alpha }_j)=q_{jj}^{-1}\rhomap{r_p(\chi )}({\sigma }_p^\chi ({\alpha }_j))= q_{jj}^{-1}\rhomap{r_p(\chi )}({\alpha }_j-c_{pj}^\chi {\alpha }_p) =(q_{pj}q_{jp})^{-c_{pj}^\chi },\\ \xi _1({\alpha }_j)=q_{pj}^{{b}-1} q_{jp}^{{b}-1} =(q_{pj}q_{jp})^{-c_{pj}^\chi }. \end{gathered}$$ Hence $\xi _1({\alpha }_j)=\xi _2({\alpha }_j)$ also in this case. This proves the lemma. Multiparameter Drinfel’d doubles {#sec:DD} ================================ In this paper we study Verma modules for a class of Hopf algebras introduced in [@p-Heck07b]. This class contains multiparameter quantizations of semisimple Lie algebras and basic classical Lie superalgebras. The precise definition is given in Eq. . It uses the Drinfel’d double construction and the theory of Nichols algebras. The Drinfel’d double [@b-Joseph Sect.3.2] can be defined via a skew-Hopf pairing of two Hopf algebras or as the quotient of a free associative algebra by a certain ideal, see also Rem. \[re:ideal\]. The first approach is more technical, but also more powerful. We present here the second definition. For proofs see [@p-Heck07b]. Let $I$ be a non-empty finite set, $\chi $ a bicharacter on ${\mathbb{Z}}^I$ with values in ${{\Bbbk }^\times }$, and $q_{i j}=\chi ({\alpha }_i,{\alpha }_j)$ for all $i,j\in I$. Let ${\mathcal{U}}(\chi )$ be the unital associative ${\Bbbk }$-algebra with generators $K_i$, $K_i^{-1}$, $L_i$, $L_i^{-1}$, $E_i$, and $F_i$, where $i\in I$, and defining relations $$\begin{aligned} XY= YX \quad & \makebox[0pt][l]{for all $X,Y\in \{K_i,K_i^{-1}, L_i,L_i^{-1}\,|\,i\in I\}$,} \label{eq:KLrel}\\ K_iK_i^{-1}=&\,1, & L_iL_i^{-1}=&\,1, \label{eq:KKrel} \\ K_iE_jK_i^{-1}=&\,q_{ij}E_j, & L_iE_jL_i^{-1}=&\,q_{ji}^{-1}E_j, \label{eq:KErel} \\ K_iF_jK_i^{-1}=&\,q_{ij}^{-1}F_j, & L_iF_jL_i^{-1}=&\,q_{ji}F_j, \label{eq:KFrel}\\ E_iF_j&\makebox[0pt][l]{$-F_jE_i=\delta _{i,j}(K_i-L_i)$,} \label{eq:EFrel}\end{aligned}$$ where $i,j\in I$, and $\delta _{i,j}$ denotes Kronecker’s $\delta $. The algebra ${\mathcal{U}}(\chi )$ can be given a Hopf algebra structure in many different ways. We will use the unique Hopf algebra structure determined by $$\begin{gathered} \left\{ \begin{aligned} {\varepsilon }(K_i)=&\,1,\quad {\varepsilon }(E_i)=0, & {\varepsilon }(L_i)=&\,1,\quad {\varepsilon }(F_i)=0, \\ {\varDelta }(K_i)=&\,K_i\otimes K_i,& {\varDelta }(L_i)=&\,L_i\otimes L_i,\\ {\varDelta }(K_i^{-1})=&\,K_i^{-1}\otimes K_i^{-1},& {\varDelta }(L_i^{-1})=&\,L_i^{-1}\otimes L_i^{-1},\\ {\varDelta }(E_i)=&\,E_i\otimes 1+K_i\otimes E_i,& {\varDelta }(F_i)=&\,1\otimes F_i+F_i\otimes L_i \end{aligned} \right. \label{eq:coprcU}\end{gathered}$$ for all $i\in I$. Let ${\mathcal{U}}^{+0},{\mathcal{U}}^{-0}$, and ${{\mathcal{U}}^0}$ denote the commutative cocommutative Hopf subalgebras of ${\mathcal{U}}(\chi )$ generated by $\{K_i,K_i^{-1}\,|\,i\in I\}$, $\{L_i,L_i^{-1}\,|\,i\in I\}$, and $\{K_i,K_i^{-1},L_i,L_i^{-1}\,|\,i\in I\}$, respectively. They are isomorphic to the ring of Laurent polynomials in $|I|$, $|I|$, and $2|I|$ variables, respectively, in the natural way. For any ${\alpha }=\sum _{i\in I}m_i{\alpha }_i\in {\mathbb{Z}}^I$ let $K_{\alpha }=\prod _{i\in I} K_i^{m_i}$ and $L_{\alpha }=\prod _{i\in I}L_i^{m_i}$. Let ${\mathcal{U}}^+(\chi )$, ${\mathcal{V}}^+(\chi )$, ${\mathcal{U}}^-(\chi )$, and ${\mathcal{V}}^-(\chi )$ denote the subalgebras of ${\mathcal{U}}(\chi )$ generated by $\{E_i\,|\,i\in I\}$, $\{E_i,K_i,K_i^{-1}\,|\,i\in I\}$, $\{F_i\,|\,i\in I\}$, and $\{F_i,L_i,L_i^{-1}\,|\,i\in I\}$, respectively. Then ${\mathcal{V}}^+(\chi )$ and ${\mathcal{V}}^-(\chi )$ are Hopf subalgebras of ${\mathcal{U}}(\chi )$. The algebra ${\mathcal{U}}(\chi )$ admits a unique ${\mathbb{Z}}^I$-grading $$\begin{gathered} {\mathcal{U}}(\chi )=\oplus _{\beta \in {\mathbb{Z}}^I}{\mathcal{U}}(\chi )_\beta ,\\ 1\in {\mathcal{U}}(\chi )_0,\quad {\mathcal{U}}(\chi )_\beta {\mathcal{U}}(\chi )_\gamma \subset {\mathcal{U}}(\chi )_{\beta +\gamma } \quad \text{for all $\beta ,\gamma \in {\mathbb{Z}}^I$,} \end{gathered} \label{eq:Zngrading}$$ such that $K_i,K_i^{-1},L_i,L_i^{-1}\in {\mathcal{U}}(\chi )_0$, $E_i\in {\mathcal{U}}(\chi )_{{\alpha }_i}$, and $F_i\in {\mathcal{U}}(\chi )_{-{\alpha }_i}$ for all $i\in I$. Let $${\mathbb{N}}_0^I=\Big\{\sum _{i\in I}a_i{\alpha }_i\,|\,a_i\in {\mathbb{N}}_0\Big\}\subset {\mathbb{Z}}^I,$$ and for any subspace ${\mathcal{U}}'\subset {\mathcal{U}}(\chi )$ and any $\beta \in {\mathbb{Z}}^I$ let ${\mathcal{U}}'_\beta ={\mathcal{U}}'\cap {\mathcal{U}}(\chi )_\beta $. Then $$\begin{aligned} {\mathcal{U}}^+(\chi )=&\oplus _{\beta \in {\mathbb{N}}_0^I}{\mathcal{U}}^+(\chi )_\beta ,& {\mathcal{U}}^-(\chi )=&\oplus _{\beta \in {\mathbb{N}}_0^I}{\mathcal{U}}^-(\chi )_{-\beta }.\end{aligned}$$ For all $\beta \in {\mathbb{Z}}^I$ let $$\begin{aligned} |\beta |=\sum _{i\in I} a_i\in {\mathbb{Z}},\quad \text{ where } \beta =\sum _{i\in I}a_i{\alpha }_i. \label{eq:abs}\end{aligned}$$ The decomposition $${\mathcal{U}}(\chi )=\oplus _{m\in {\mathbb{Z}}}{\mathcal{U}}(\chi )_m,\quad \text{where}\quad {\mathcal{U}}(\chi )_m=\oplus _{\beta :|\beta |=m}{\mathcal{U}}(\chi )_\beta , \label{eq:Zgrading}$$ gives a ${\mathbb{Z}}$-grading of ${\mathcal{U}}(\chi )$ called the *standard grading*. \[pr:algiso\] Let $\chi \in {\mathcal{X}}$. \(1) Let ${\underline{a}}=(a_i\,|\,i\in I)\in ({{\Bbbk }^\times })^I$. Then there exists a unique algebra automorphism $\varphi _{{\underline{a}}}$ of ${\mathcal{U}}(\chi )$ such that $$\varphi _{{\underline{a}}}(K_i)=K_i,\,\, \varphi _{{\underline{a}}}(L_i)=L_i,\,\, \varphi _{{\underline{a}}}(E_i)=a_iE_i,\,\, \varphi _{{\underline{a}}}(F_i)=a_i^{-1}F_i. \label{eq:cUauto1}$$ \(2) There is a unique algebra antiautomorphism ${\Omega }$ of ${\mathcal{U}}(\chi )$ such that $$\begin{aligned} {\Omega }(K_i)=&K_i,& {\Omega }(L_i)=&L_i,& {\Omega }(E_i)=&F_i,& {\Omega }(F_i)=&E_i. \label{eq:cUantiauto}\end{aligned}$$ It satisfies the relation ${\Omega }^2={\operatorname{id}}$. \[le:commEFi\] For all $i\in I$ there exist unique linear maps ${\partial ^K}_i,{\partial ^L}_i\in {\mathrm{End}}_{\Bbbk }({\mathcal{U}}^+(\chi ))$ such that $$\begin{aligned} [E,F_i]={\partial ^K}_i(E)K_i-L_i{\partial ^L}_i(E)\quad \text{for all $E\in {\mathcal{U}}^+(\chi )$.} \end{aligned}$$ The maps ${\partial ^K}_i,{\partial ^L}_i\in {\mathrm{End}}_{\Bbbk }({\mathcal{U}}^+(\chi ))$ are skew-derivations. More precisely, $$\begin{gathered} {\partial ^K}_i(1)={\partial ^L}_i(1)=0,\quad {\partial ^K}_i(E_j)={\partial ^L}_i(E_j)=\delta _{i,j},\label{eq:derKL1}\\ \begin{aligned} {\partial ^K}_i(EE')=&{\partial ^K}_i(E)(K_i{\boldsymbol{.}}E')+E{\partial ^K}_i(E'),\\ {\partial ^L}_i(EE')=&{\partial ^L}_i(E)E'+(L_i^{-1}{\boldsymbol{.}}E){\partial ^L}_i(E') \end{aligned} \label{eq:derKL2} \end{gathered}$$ for all $i,j\in I$ and $E,E'\in {\mathcal{U}}^+(\chi )$. Let ${\mathcal{I}}^+(\chi )$ be the unique maximal ideal of ${\mathcal{U}}^+(\chi )$ such that ${\mathcal{I}}^+(\chi )\subset \ker \varepsilon $ and ${\partial ^K}_i({\mathcal{I}}^+(\chi ))\subset {\mathcal{I}}^+(\chi )$ for all $i\in I$. Equivalently, ${\mathcal{I}}^+(\chi )$ is the unique maximal ideal of ${\mathcal{U}}^+(\chi )$ such that ${\mathcal{I}}^+(\chi )\subset \varepsilon $ and ${\partial ^L}_i({\mathcal{I}}^+(\chi ))\subset \cI ^+(\chi )$ for all $i\in I$, see [@p-Heck07b Prop.5.4]. Let ${\mathcal{I}}^-(\chi )={\Omega }({\mathcal{I}}^+(\chi ))$. Let $$\begin{aligned} U^+(\chi )=&{\mathcal{U}}^+(\chi )/{\mathcal{I}}^+(\chi ), & U^-(\chi )=&{\mathcal{U}}^-(\chi )/{\mathcal{I}}^-(\chi ),\\ V^+(\chi )=&{\mathcal{V}}^+(\chi )/{\mathcal{I}}^+(\chi ){\mathcal{U}}^{+0},& V^-(\chi )=&{\mathcal{V}}^-(\chi )/{\mathcal{I}}^-(\chi ){\mathcal{U}}^{-0},\end{aligned}$$ and $$\begin{aligned} U(\chi )={\mathcal{U}}(\chi )/({\mathcal{I}}^+(\chi ),{\mathcal{I}}^-(\chi )). \label{eq:Uchi}\end{aligned}$$ The canonical inclusions ${\mathcal{U}}^\pm (\chi )\subset {\mathcal{U}}(\chi )$, ${{\mathcal{U}}^0}\subset {\mathcal{U}}(\chi )$ induce maps $$\iota _+:U^+(\chi )\to U(\chi ),\quad \iota _0:{{\mathcal{U}}^0}\to U(\chi ),\quad \iota _-:U^-(\chi )\to U(\chi ).$$ \[re:ideal\] (i) The vector space $V=\oplus _{i\in I}{\Bbbk }E_i$ is a module over the group algebra ${\Bbbk }{\mathbb{Z}}^I\simeq {\Bbbk }[K_i,K_i^{-1}\,|\,i\in I]\subset {{\mathcal{U}}^0}$, where the left action ${\boldsymbol{\cdot}}:{\Bbbk }{\mathbb{Z}}^I\otimes V\to V$ and the left coaction ${\delta }:V\to {\Bbbk }{\mathbb{Z}}^I{\otimes }V$ are defined by $$\begin{aligned} K_i {\boldsymbol{\cdot}}E_j=q_{ij}E_j,\qquad {\delta }(E_i)=K_i{\otimes }E_i \end{aligned}$$ for all $i,j\in I$. The algebra $U^+(\chi )$ is commonly known as the *Nichols algebra* of the module $V$. \(ii) There are various descriptions of the ideal ${\mathcal{I}}^+(\chi )$, see *e.g.* [@inp-AndrSchn02]. In case of quantized enveloping algebras, see Sect. \[sec:Uqg\], Serre relations generate the ideal ${\mathcal{I}}^+(\chi )$. A more general case is studied by Angiono [@p-Angi08]. For quantized Lie superalgebras the defining relations are determined in [@a-Yam99; @a-Yam99e]. It is in general an open problem to give a nice set of generators of ${\mathcal{I}}^+(\chi )$, see [@inp-Andr02 Question5.9]. (Triangular decomposition) The map $$\mathrm{m}(\iota _-{\otimes }\iota _0{\otimes }\iota _+): U^-(\chi ){\otimes }{{\mathcal{U}}^0}{\otimes }U^+(\chi )\to U(\chi )$$ is an isomorphism of ${\mathbb{Z}}^I$-graded vector spaces, where $\mathrm{m}$ denotes the multiplication map. \[pr:tridec\] Following the convention in [@b-Joseph Sect.3.2.1], a skew-Hopf pairing ${\eta }:A\times B\to {\Bbbk }$, $(x,y)\mapsto {\eta }(x,y)$, of two Hopf algebras $A$, $B$ is a bilinear map satisfying the equations $$\begin{aligned} \label{eq:sHp1} {\eta }(1,y)=&\,{\varepsilon }(y),& {\eta }(x,1)=&\,{\varepsilon }(x),\\ \label{eq:sHp2} {\eta }(xx',y)=&\,{\eta }(x',y_{(1)}){\eta }(x,y_{(2)}),& {\eta }(x,yy')=&\,{\eta }(x_{(1)},y){\eta }(x_{(2)},y'),\\ \label{eq:sHp3} &\qquad \makebox[0pt][l]{${\eta }({S}(x),y)={\eta }(x,{S}^{-1}(y))$}\end{aligned}$$ for all $x,x'\in A$ and $y,y'\in B$. \[pr:sHpdef\] (i) There exists a unique skew-Hopf pairing ${\eta }$ of ${\mathcal{V}}^+(\chi )$ and ${\mathcal{V}}^-(\chi )$ such that for all $i,j\in I$ one has $$\begin{aligned} {\eta }(E_i,F_j)=-\delta _{i,j},\quad {\eta }(E_i,L_j)=0,\quad {\eta }(K_i,F_j)=0,\quad {\eta }(K_i,L_j)=q_{ij}. \end{aligned}$$ \(ii) The skew-Hopf pairing ${\eta }$ satisfies the equations $$\begin{aligned} {\eta }(EK,FL)={\eta }(E,F){\eta }(K,L) \end{aligned}$$ for all $E\in {\mathcal{U}}^+(\chi )$, $F\in {\mathcal{U}}^-(\chi )$, $K\in {\mathcal{U}}^{+0}$, and $L\in {\mathcal{U}}^{-0}$. \(iii) If $\beta ,\gamma \in {\mathbb{N}}_0^I$ with $\beta \not=\gamma $, $E\in {\mathcal{U}}^+(\chi )_\beta $, $F\in {\mathcal{U}}^-(\chi )_{-\gamma }$, then ${\eta }(E,F)=0$. \(iv) The restriction of ${\eta }$ to ${\mathcal{U}}^+(\chi )\times {\mathcal{U}}^-(\chi )$ induces a non-degenerate pairing ${\eta }:U ^+(\chi )\times U^-(\chi )\to {\Bbbk }$. \(i) and (ii) are [@p-Heck07b Prop.4.3]. (iii) follows from the definition of ${\eta }$ and since ${\varDelta }$ is a ${\mathbb{Z}}^I$-homogeneous map. (iv) was proven in [@p-Heck07b Thm.5.8]. By the general theory, see [@b-Joseph 3.2.2], the pairing ${\eta }$ in Prop. \[pr:sHpdef\] can be used to describe commutation rules in ${\mathcal{U}}(\chi )$ and $U(\chi )$. Namely, $$\begin{aligned} \label{eq:Ucomm1} yx=&{\eta }(x{_{(1)}},S(y{_{(1)}}))x{_{(2)}}y{_{(2)}}{\eta }(x{_{(3)}},y{_{(3)}}),\\ \label{eq:Ucomm2} xy=&{\eta }(x{_{(1)}},y{_{(1)}})y{_{(2)}}x{_{(2)}}{\eta }(x{_{(3)}},S(y{_{(3)}}))\end{aligned}$$ for all $x\in {\mathcal{V}}^+(\chi )$ and $y\in {\mathcal{V}}^-(\chi )$. Note that the second formula follows from the first one and Eqs. –. Later we will also need some other general facts about $U(\chi )$. Some of them are collected here. Let $$\begin{aligned} \label{eq:kerderK} U^+_{i,K}(\chi )=&\ker {\partial ^K}_i\subset U^+(\chi ),& U^+_{i,L}(\chi )=&\ker {\partial ^L}_i\subset U^+(\chi ),\\ \label{eq:kerderL} U^-_{i,K}(\chi )=&{\Omega }(U^+_{i,K}),& U^-_{i,L}(\chi )=&{\Omega }(U^+_{i,L}).\end{aligned}$$ Recall the definition of ${b^{\chi}} $ in Eq. . Let $i\in I$. \(i) Let $m\in {\mathbb{N}}$. The following are equivalent. - $E_i^m=0$ in $U(\chi )$, - $F_i^m=0$ in $U(\chi )$, - $m\ge {b^{\chi}} ({\alpha }_i)$. \(ii) Let ${\Bbbk }[E_i]$ and ${\Bbbk }[F_i]$ be the subalgebras of $U(\chi )$ generated by $E_i$ and $F_i$, respectively. The multiplication maps $$\begin{aligned} U^+_{i,K}(\chi ) {\otimes }{\Bbbk }[E_i] \to &\,U^+(\chi ),& {\Bbbk }[E_i] {\otimes }U^+_{i,K}(\chi ) \to &\,U^+(\chi ),\\ U^+_{i,L}(\chi ) {\otimes }{\Bbbk }[E_i] \to &\,U^+(\chi ),& {\Bbbk }[E_i] {\otimes }U^+_{i,L}(\chi ) \to &\,U^+(\chi ),\\ U^-_{i,K}(\chi ) {\otimes }{\Bbbk }[F_i] \to &\,U^-(\chi ),& {\Bbbk }[F_i] {\otimes }U^-_{i,K}(\chi ) \to &\,U^-(\chi ),\\ U^-_{i,L}(\chi ) {\otimes }{\Bbbk }[F_i] \to &\,U^-(\chi ),& {\Bbbk }[F_i] {\otimes }U^-_{i,L}(\chi ) \to &\,U^-(\chi ), \end{aligned}$$ are isomorphisms of ${\mathbb{Z}}^I$-graded algebras. \[le:Eheight\] \(i) is standard in the theory of Nichols algebras. It follows from Eqs. , and the definitions of ${\mathcal{I}}^+(\chi )$ and ${\mathcal{I}}^-(\chi )$. The proof of (ii) for $U^+(\chi )$ can be performed as in [@a-Heck06a]. The formulas with $U^-(\chi )$ follow from those with $U^+(\chi )$ and Eqs. , . \[le:EmFn\] Let $m,n\in {\mathbb{N}}_0$ and $p\in I$. Then $$\begin{aligned} E_p^m F_p^n=\sum _{i=0}^{\makebox[0pt]{\scriptsize $\min \{m,n\}$}} {\textstyle \frac{\qfact{m}{q_{p p}}\qfact{n}{q_{p p}}}{ \qfact{i}{q_{p p}}\qfact{m-i}{q_{p p}}\qfact{n-i}{q_{p p}}} } F_p^{n-i}\prod _{j=1}^i (q_{p p}^{i+j-m-n}K_p-L_p)E_p^{m-i}. \end{aligned}$$ For $n=0$ the claim is trivial. By [@p-Heck07b Cor. 5.4], $$E_p^m F_p-F_p E_p^m= \qnum{m}{q_{p p}}(q_{p p}^{1-m}K_p-L_p)E_p^{m-1}.$$ Hence the lemma holds for $n=1$. It suffices to check the claim for $m\ge n$, since then it also holds for $m<n$ using the algebra antiisomorphism ${\Omega }$. The proof of the lemma for $m\ge n$ is a standard calculation by induction on $n$. An analogue of Lusztig’s PBW basis {#sec:Lusztig} ================================== Let $\chi \in {\mathcal{X}}$ and $p\in I$. Assume that $\chi $ is $p$-finite. Let $q_{i j}=\chi ({\alpha }_i,{\alpha }_j)$ and $c_{p i}=c_{p i}^\chi $ for all $i,j\in I$. For all $m\in {\mathbb{N}}_0$ and $i\in I\setminus \{p\}$ define recursively $E^\pm _{i,m}\in U^+_{{\alpha }_i+m{\alpha }_p}$, $F^\pm _{i,m}\in U^-_{{\alpha }_i+m{\alpha }_p}$ by $$\begin{aligned} E^+_{i,0}=&E_i, & E^+_{i,m+1}=&\,E_p E^+_{i,m}-(K_p{\boldsymbol{.}}E^+_{i,m})E_p,\\ E^-_{i,0}=&E_i, & E^-_{i,m+1}=&\,E_p E^-_{i,m}-(L_p{\boldsymbol{.}}E^-_{i,m})E_p,\\ F^+_{i,0}=&F_i, & F^+_{i,m+1}=&\,F_p F^+_{i,m}-(L_p{\boldsymbol{.}}F^+_{i,m})F_p,\\ F^-_{i,0}=&F_i, & F^-_{i,m+1}=&\,F_p F^-_{i,m}-(K_p{\boldsymbol{.}}F^-_{i,m})F_p.\end{aligned}$$ We also define $E^+_{i,-1}=E^-_{i,-1}=F^+_{i,-1}=F^-_{i,-1}=0$. The above definitions depend essentially on $p$. If we want to emphasize this, we will write $E^\pm _{i,m(p)}$ and $F^\pm _{i,m(p)}$ instead of $E^\pm _{i,m}$ and $F^\pm _{i,m}$, respectively. For all $i\in I\setminus \{p\}$ define $$\lambda _i^\chi =\qfact{-c _{p i}}{q_{p p}} \prod _{j=0}^{-c _{p i}-1}(q_{p p}^j q_{p i}q_{i p}-1).$$ Then $\lambda _i^\chi \not=0$ by definition of $c_{p i}=c^\chi _{p i}$. The next theorem was proven in [@p-Heck07b Thm.6.11]. \[th:Liso\] Let $\chi \in {\mathcal{X}}$ and $p\in I$. Assume that $\chi $ is $p$-finite. Let $c_{pi}=c_{pi}^\chi $ for all $i\in I$. \(i) There exist unique algebra isomorphisms ${T}_p, {T}_p^-: U (\chi )\to U (r_p(\chi ))$ such that $$\begin{aligned} {T}_p(K_p)=&{T}_p^-(K_p)=K _p^{-1},& {T}_p(K_i)=&{T}_p^-(K_i)=K _iK _p^{-c_{p i}},\\ {T}_p(L_p)=&{T}_p^-(L_p)=L _p^{-1},& {T}_p(L_i)=&{T}_p^-(L_i)=L _iL _p^{-c_{p i}},\\ {T}_p(E_p)=&F _p L _p^{-1},& {T}_p(E_i)=&E ^+_{i,-c_{p i}},\\ {T}_p(F_p)=&K _p^{-1}E _p,& {T}_p(F_i)=&\lambda _i(r_p(\chi ))^{-1}F ^+_{i,-c_{p i}},\\ {T}_p^-(E_p)=&K _p^{-1}F _p,& {T}_p^-(E_i)=&\lambda _i(r_p(\chi ^{-1}))^{-1} E ^-_{i,-c_{pi}},\\ {T}_p^-(F_p)=&E _p L _p^{-1},& {T}_p^-(F_i)=&(-1)^{c_{p i}} F ^-_{i,-c_{p i}}. \end{aligned}$$ \(ii) The maps ${T}_p$, ${T}_p^-$ satisfy ${T}_p {T}_p^-={T}_p^-{T}_p={\operatorname{id}}_{U(\chi )}$. \(iii) There exists a unique ${\underline{a}}\in ({{\Bbbk }^\times })^I$ such that ${T}_p {\Omega }={\Omega }{T}^-_p \varphi _{{\underline{a}}}$ in ${\mathrm{Hom}}(U(\chi ),U(r_p(\chi )))$. Note that ${T}_p{T}_p^-$ is an automorphism of $U(\chi )$ if one regards ${T}_p^-$ as a map from $U(\chi )$ to $U(r_p(\chi ))$ and ${T}_p$ as a map from $U(r_p(\chi ))$ to $U(r_p r_p(\chi ))=U(\chi )$. \[pr:LTdeg\] Let $\chi \in {\mathcal{X}}$ and $p\in I$. Assume that $\chi $ is $p$-finite. Then $${T}_p(U(\chi )_{\alpha })=U(r_p(\chi ))_{{\sigma }_p^\chi ({\alpha })} \quad \text{for all ${\alpha }\in {\mathbb{Z}}^I$.}$$ The maps ${T}_p:U(\chi )\to U(r_p(\chi ))$ and ${T}_p^-:U(r_p(\chi ))\to U(\chi )$ are mutually inverse algebra isomorphisms, and send generators of degree ${\alpha }$ into the homogeneous component of degree ${\sigma }_p({\alpha })$. \[le:TpU+U+\] Let $\chi \in {\mathcal{X}}$ and $p\in I$. Assume that $\chi $ is $p$-finite. Then $$\begin{aligned} {T}_p(U^+_{p,L}(\chi ))=&\,U^+_{p,K}(r_p(\chi )), & {T}_p(U^-_{p,K}(\chi ))=&\,U^-_{p,L}(r_p(\chi )),\\ {T}^-_p(U^+_{p,K}(\chi ))=&\,U^+_{p,L}(r_p(\chi )), & {T}^-_p(U^-_{p,L}(\chi ))=&\,U^-_{p,K}(r_p(\chi )). \end{aligned}$$ Since $\chi $ and $r_p(\chi )$ are $p$-finite, [@p-Heck07b Prop.5.10] and [@p-Heck07b Prop.6.7(d)] give that $${T}_p(U^+_{p,L}(\chi ))\subset U^+_{p,K}(r_p(\chi )),\qquad {T}_p^-(U^+_{p,K}(r_p(\chi )))\subset U^+_{p,L}(\chi ).$$ Thus ${T}_p(U^+_{p,L}(\chi ))=U^+_{p,K}(r_p(\chi ))$ by Thm. \[th:Liso\](ii). Similar arguments yield that ${T}^-_p(U^+_{p,K}(\chi ))=U^+_{p,L}(r_p(\chi ))$. The remaining two equations can be obtained from these and Thm. \[th:Liso\](iii). In the rest of the section assume that $\chi \in {\mathcal{X}}_3$. Let $n=|R_+^\chi |\in {\mathbb{N}}$. The following construction generalizes the Poincaré-Birkhoff-Witt basis of quantized enveloping algebras given by Lusztig. Let $i_1,i_2,\dots ,i_n\in I$ such that $\ell (1_{\chi }{\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_n})=n$. For all $\nu \in \{1,2,\dots,n\}$ let $$\begin{gathered} \label{eq:betak} \beta _\nu ^\chi =1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\dots {\sigma }_{i_{\mu -1}}({\alpha }_{i_\nu }).\end{gathered}$$ Then the elements $\beta _\nu ^\chi $, $1\le \nu \le n$, are pairwise different and $$\begin{aligned} R^\chi _+=\{\beta _\nu ^\chi \,|\,1\le \nu \le n\} \label{eq:proots}\end{aligned}$$ by [@p-CH08 Prop.2.12]. For all $\nu \in \{1,2,\dots,n\}$ let $$\begin{gathered} \label{eq:Ebetak} E_{\beta _\nu }= E_{\beta _\nu }^\chi ={T}_{i_1}\dots {T}_{i_{\nu -1}}(E_{i_\nu }),\quad F_{\beta _\nu }= F_{\beta _\nu }^\chi ={T}_{i_1}\dots {T}_{i_{\nu -1}}(F_{i_\nu }),\\ \label{eq:Ebarbetak} {\bar{E}}_{\beta _\nu }= {\bar{E}}_{\beta _\nu }^\chi ={T}^-_{i_1}\dots {T}^-_{i_{\nu -1}}(E_{i_\nu }),\quad {\bar{F}}_{\beta _\nu }= {\bar{F}}_{\beta _\nu }^\chi ={T}^-_{i_1}\dots {T}^-_{i_{\nu -1}}(F_{i_\nu }),\end{gathered}$$ where $E_{i_\nu },F_{i_\nu }\in U(r_{i_{\nu -1}}\dots r_{i_2}r_{i_1}(\chi ))$. Then $$\begin{aligned} E_{\beta _\nu },{\bar{E}}_{\beta _\nu }\in U^+(\chi )_{\beta _\nu },\qquad F_{\beta _\nu },{\bar{F}}_{\beta _\nu }\in U^-(\chi )_{-\beta _\nu } \label{eq:EbetainU+}\end{aligned}$$ for all $\nu \in \{1,\dots ,n\}$ by [@p-Heck07b Thm.6.19], Thm. \[th:Liso\](iii) and Prop. \[pr:LTdeg\]. \[le:rvrel\] Assume that $\chi \in {\mathcal{X}}_3$. Let $\nu \in \{1,2,\dots ,n\}$, and assume that ${b^{\chi}} (\beta _\nu )<\infty $. Then $E_{\beta _\nu }^{{b^{\chi}} (\beta _\nu )}= F_{\beta _\nu }^{{b^{\chi}} (\beta _\nu )}=0$ in $U(\chi )$. By Eq.  and since $T_i$ is an isomorphism for each $i\in I$, it suffices to prove that $E_i^{\bfun{\chi '}({\alpha }_i)}=F_i^{\bfun{\chi '}({\alpha }_i)}=0$ in $U(\chi ')$ for all $\chi '\in {\mathcal{X}}$ and $i\in I$ with $\bfun{\chi '}({\alpha }_i)<\infty $. This follows from Lemma \[le:Eheight\](i). \[th:PBW\] Assume that $\chi \in {\mathcal{X}}_3$. Let $n=|R_+^\chi |\in {\mathbb{N}}$. Both sets $$\begin{aligned} \label{eq:LusztigPBW+} \big\{ E_{\beta _1}^{m_1} E_{\beta _2}^{m_2}\cdots E_{\beta _n}^{m_n}\,&|\, 0\le m_\nu <{b^{\chi}} (\beta _\nu ) \text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\},\\ \label{eq:LusztigPBW-} \big\{ {\bar{E}}_{\beta _1}^{m_1} {\bar{E}}_{\beta _2}^{m_2}\cdots {\bar{E}}_{\beta _n}^{m_n}\,&|\, 0\le m_\nu <{b^{\chi}} (\beta _\nu ) \text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\} \end{aligned}$$ form vector space bases of $U ^+(\chi )$. We prove the claim for the basis in Eq. . For the other set the proof is analogous. By Eqs.  and , $$\begin{aligned} \dim U^+(\chi )_{\alpha }=\Big|\big\{(m_1,m_2,&\dots ,m_n)\in {\mathbb{N}}_0^n\,\big|\, \sum _{\nu =1}^n m_\nu \beta _\nu ={\alpha },\\ &m_\nu <{b^{\chi}} (\beta _\nu ) \text{ for all $\nu \in \{1,\dots ,n\}$} \big\}\Big| \end{aligned}$$ for all ${\alpha }\in {\mathbb{N}}_0^I$. Since $\deg E_{\beta _\nu }=\beta _\nu $ for all $\nu \in \{1,2,\dots ,n\}$, it suffices to show that for all $\mu \in \{1,2,\dots ,n+1\}$ the elements of the set $$\begin{aligned} \big\{ E_{\beta _\mu }^{m_\mu } E_{\beta _{\mu +1}}^{m_{\mu +1}}\cdots E_{\beta _n}^{m_n}\,|\, 0\le m_\nu <{b^{\chi}} (\beta _\nu ) \text{ for all $\nu \in \{\mu ,\mu +1 ,\dots ,n\}$} \big\} \end{aligned}$$ are linearly independent. We proceed by induction on $n+1-\mu $. If $\mu =n+1$, then the above set is empty, and hence its elements are linearly independent. Let now $\mu \in \{1,2,\dots ,n\}$. For all $m_\mu ,\dots ,m_n\in {\mathbb{N}}_0$ with $m_\nu <{b^{\chi}} (\beta _\nu )$ for all $\nu \in \{\mu ,\mu +1,\dots ,n\}$ let $a_{m_\mu ,\dots ,m_n}\in {\Bbbk }$. Assume that $$\begin{aligned} \sum _{m_\mu ,\dots ,m_n}a _{m_\mu ,\dots ,m_n} E_{\beta _\mu }^{m_\mu } E_{\beta _{\mu +1}}^{m_{\mu +1}}\cdots E_{\beta _n}^{m_n}=0 \label{eq:lindep} \end{aligned}$$ in $U^+(\chi )$. Let ${T}^-={T}^-_{i_\mu }\cdots {T}^-_{i_2}{T}^-_{i_1}$. Since ${T}^-(E_{\beta _\mu })={T}^-_{i_\mu }(E_{i_\mu }) =K_{i_\mu }^{-1}F_{i_\mu }$, we obtain that $$\sum _{m_\mu ,\dots ,m_n} a _{m_\mu ,\dots ,m_n} (K_{i_\mu }^{-1}F_{i_\mu })^{m_\mu }{T}^-(E_{\beta _{\mu +1}})^{m_{\mu +1}} \cdots {T}^-(E_{\beta _n})^{m_n}=0.$$ Since ${T}^-(E_{\beta _\nu })\in U^+(r_{i_\mu }\cdots r_{i_2}r_{i_1}(\chi ))$ for all $\nu \in \{\mu +1,\mu +2,\dots ,n\}$, Prop. \[pr:tridec\] implies that $$\sum _{m_{\mu +1},\dots ,m_n}a _{m_\mu ,m_{\mu +1},\dots ,m_n} {T}^-(E_{\beta _{\mu +1}})^{m_{\mu +1}}\cdots {T}^-(E_{\beta _n})^{m_n} =0$$ for all $m_\mu \in {\mathbb{N}}_0$, $m_\mu <{b^{\chi}} (\beta _\mu )$. Therefore $$\sum _{m_{\mu +1},\dots ,m_n}a _{m_\mu ,m_{\mu +1},\dots ,m_n} E_{\beta _{\mu +1}}^{m_{\mu +1}}\cdots E_{\beta _n}^{m_n}=0$$ for all $m_\mu \in {\mathbb{N}}_0$, $m_\mu <{b^{\chi}} (\beta _\mu )$. Then $a_{m_\mu ,m_{\mu +1},\dots ,m_n}=0$ for all $(m_\mu ,\dots ,m_n)$ by induction hypothesis, which proves the induction step. Thus the theorem holds. Assume that $\chi \in {\mathcal{X}}_3$. Then $\ker ({\partial ^K}_{i_1}:U^+(\chi )\to U^+(\chi ))$ coincides with the subalgebra of $U^+(\chi )$ generated by the elements $E_{\beta _\nu }$, $\nu \in \{2,3,\dots ,n\}$. The set $$\begin{aligned} \big\{ E_{\beta _2}^{m_2} E_{\beta _3}^{m_3}\cdots E_{\beta _n}^{m_n}\,&|\, 0\le m_\nu <{b^{\chi}} (\beta _\nu ) \text{ for all $\nu \in \{2,3,\dots ,n\}$} \big\} \end{aligned}$$ forms a vector space basis of $\ker {\partial ^K}_{i_1}$. \[le:kerderK\] Let $\nu \in \{2,3,\dots ,n\}$. By [@p-Heck07b Lemma5.10] and Lemma \[le:rvrel\] there exist $m\in {\mathbb{N}}_0$ and $x_0,x_1,\dots ,x_m\in \ker {\partial ^K}_{i_1}$ such that $m<{b^{\chi}} ({\alpha }_{i_1})$ and $E_{\beta _\nu }=\sum _{\mu =0}^m x_\mu E_{i_1}^\mu $. Then $${T}_{i_1}^-(E_{\beta _\nu })= \sum _{\mu =0}^m{T}^-_{i_1}(x_\mu ) (K_{i_1}^{-1}F_{i_1})^\mu .$$ Moreover, ${T}^-_{i_1}(x_\mu )\in U^+(r_{i_1}(\chi ))$ for all $\mu \in \{0,1,\dots ,m\}$ by [@p-Heck07b Prop.5.19,Lemma6.7(d)]. Since ${T}^-_{i_1}(E_{\beta _\nu })\in U^+(r_{i_1}(\chi ))$, triangular decomposition of $U(r_{i_1}(\chi ))$ implies that $x_\mu =0$ for all $\mu >0$. Hence $E_{\beta _\nu }=x_0\in \ker {\partial ^K}_{i_1}$. Then the claim of the lemma follows from the inclusions $$\langle E_{\beta _\kappa }\,|\,\kappa \in \{2,3,\dots ,n\}\rangle \subset \ker {\partial ^K}_{i_1} \subset \mathop{\oplus }_{ {(m_2,m_3,\dots ,m_n) \atop 0\le m_\kappa <{b^{\chi}} (\beta _\kappa )\,\text{for all $\kappa $}}} {\Bbbk }E_{\beta _2}^{m_2}\cdots E_{\beta _n}^{m_n},$$ where the second inclusion is obtained from Thm. \[th:PBW\] and the formula $${\partial ^K}_{i_1}( E_{\beta _1}^{m_1}E_{\beta _2}^{m_2}\cdots E_{\beta _n}^{m_n})= \qnum{m_1}{q_{i_1i_1}} E_{\beta _1}^{m_1-1}K_{i_1}{\boldsymbol{\cdot}}(E_{\beta _2}^{m_2}\cdots E_{\beta _n}^{m_n}).$$ The analogous version of Lemma \[le:kerderK\] for $\ker {\partial ^L}_{i_1}$ is obtained by replacing $E_{\beta _\nu }$ by ${\bar{E}}_{\beta _\nu }$ for all $\nu \in \{2,3,\dots ,n\}$. \[th:EErel\] Assume that $\chi \in {\mathcal{X}}_3$. Let $n=|R_+^\chi |\in {\mathbb{N}}$. Then $$\begin{aligned} E_{\beta _\mu }E_{\beta _\nu }-\chi (\beta _\mu ,\beta _\nu ) E_{\beta _\nu }E_{\beta _\mu } \in \, & \langle E_{\beta _\kappa }\,|\,\mu <\kappa <\nu \rangle \subset U^+(\chi ),\\ {\bar{E}}_{\beta _\mu }{\bar{E}}_{\beta _\nu } -\chi ^{-1}(\beta _\nu ,\beta _\mu ){\bar{E}}_{\beta _\nu }{\bar{E}}_{\beta _\mu } \in \, & \langle {\bar{E}}_{\beta _\kappa } \,|\,\mu <\kappa <\nu \rangle \subset U^+(\chi ),\\ F_{\beta _\mu }F_{\beta _\nu }-\chi (\beta _\nu ,\beta _\mu ) F_{\beta _\nu }F_{\beta _\mu } \in \, & \langle F_{\beta _\kappa }\,|\,\mu <\kappa <\nu \rangle \subset U^-(\chi ),\\ {\bar{F}}_{\beta _\mu }{\bar{F}}_{\beta _\nu }-\chi ^{-1}(\beta _\mu ,\beta _\nu ) {\bar{F}}_{\beta _\nu }{\bar{F}}_{\beta _\mu } \in \, & \langle {\bar{F}}_{\beta _\kappa }\,|\,\mu <\kappa <\nu \rangle \subset U^-(\chi ) \end{aligned}$$ for all $\mu ,\nu \in \{1,2,\dots ,n\}$ with $\mu <\nu $. We prove the first relation for $\mu =1$ and all $\nu \in \{2,3,\dots ,n\}$. Then the first relation for $\mu >1$ follows from $$E_{\beta _\mu }E_{\beta _\nu }-\chi (\beta _\mu ,\beta _\nu ) E_{\beta _\nu }E_{\beta _\mu } ={T}_{i_1}\cdots {T}_{i_{\mu -1}}( E_{i_\mu }E'_\nu -\chi (\beta _\mu ,\beta _\nu )E'_\nu E_{i_\mu }),$$ where $E'_\nu ={T}_{i_\mu }\cdots {T}_{i_{\nu -1}}(E_{i_\nu })$, by using the case $\mu =1$, Eq.  and the first relation in . The proof of the second relation of the theorem is similar. The third and fourth relations can be obtained from the first two by applying ${\Omega }$ and using the formulas $$\begin{aligned} {\Omega }(E_{\beta _\kappa })\in {{\Bbbk }^\times }{\bar{F}}_{\beta _\kappa },\quad {\Omega }({\bar{E}}_{\beta _\kappa })\in {{\Bbbk }^\times }F _{\beta _\kappa }, \qquad \kappa \in \{1,2,\dots ,n\}, \label{eq:aaaU(E)} \end{aligned}$$ which follow from Thm. \[th:Liso\](iii). Let $\nu \in \{2,3,\dots ,n\}$. For all $(m_1,m_2,\dots ,m_n)\in {\mathbb{N}}_0^I$ with $m_\kappa <{b^{\chi}} (\beta _\kappa )$ for all $\kappa \in \{1,2,\dots ,n\}$ let $a_{m_1,\dots ,m_n}\in {\Bbbk }$ such that $$\begin{aligned} E_{i_1}E_{\beta _\nu }-\chi ({\alpha }_{i_1},\beta _\nu )E_{\beta _\nu }E_{i_1} =\sum _{m_1,\dots ,m_n}a_{m_1,\dots ,m_n}E_{\beta _1}^{m_1} \cdots E_{\beta _n}^{m_n}. \label{eq:EE-EE} \end{aligned}$$ The numbers $a_{m_1,\dots ,m_n}\in {\Bbbk }$ exist and are unique by Thm. \[th:PBW\]. Let $\chi _\nu =r_{i_\nu }\cdots r_{i_2}r_{i_1}(\chi )$. Apply to Eq.  the isomorphism ${T}^-={T}^-_{i_\nu }\cdots {T}^-_{i_2}{T}^-_{i_1}\in {\mathrm{Hom}}(U(\chi ), U(\chi _\nu ))$. For all $\kappa \in \{1,2,\dots ,\nu \}$, $$\begin{aligned} {T}^-_{i_\nu }\cdots {T}^-_{i_2}{T}^-_{i_1}(E_{\beta _\kappa }) =&{T}^-_{i_\nu }\cdots {T}^-_{i_{\kappa +1}} {T}^-_{i_\kappa }(E_{i_\kappa })\\ =&{T}^-_{i_\nu }\cdots {T}^-_{i_{\kappa +1}} (K_{i_\kappa }^{-1}F_{i_\kappa }) \in U^-(\chi _\nu ){{\mathcal{U}}^0}& \end{aligned}$$ by Eq. . Hence $$\begin{aligned} &\sum _{m_1,\dots ,m_n}a_{m_1,\dots ,m_n} {T}^-(E_{\beta _1}^{m_1}\cdots E_{\beta _\nu }^{m_\nu }) {T}^-(E_{\beta _{\nu +1}}^{m_{\nu +1}}\cdots E_{\beta _n}^{m_n})\\ &\quad ={T}^-(E_{i_1}E_{\beta _\nu } -\chi ({\alpha }_{i_1},\beta _\nu )E_{\beta _\nu }E_{i_1})\in U^-(\chi _\nu ){{\mathcal{U}}^0}. \end{aligned}$$ By triangular decomposition of $U(\chi )$ it follows that $a_{m_1,\dots ,m_n}=0$ for all $(m_1,\dots ,m_n)$ with $m_\kappa >0$ for some $\kappa \in \{\nu +1,\nu +2,\dots ,n\}$. By Lemma \[le:kerderK\], $E_{\beta _\nu }\in \ker {\partial ^K}_{i_1}$. Hence $E_{i_1}E_{\beta _\nu }-\chi ({\alpha }_{i_1},\beta _\nu )E_{\beta _\nu }E_{i_1} \in \ker {\partial ^K}_{i_1}$ by Lemma \[le:commEFi\]. Thus Lemma \[le:kerderK\] implies that $a_{m_1,\dots ,m_n}=0$ whenever $m_1>0$. Suppose that there exists $(m_1,\dots ,m_n)$ with $m_\nu >0$ and $a_{m_1,\dots ,m_n}\not=0$. Since $E_{i_1}E_{\beta _\nu }-\chi ({\alpha }_{i_1},\beta _\nu )E_{\beta _\nu }E_{i_1}$ is ${\mathbb{Z}}^I$-homogeneous of degree ${\alpha }_{i_1}+\beta _\nu $, the only possibility is that $m_1=1$, $m_\nu =1$, and $m_\kappa =0$ for all $\kappa \notin \{1,\nu \}$. Since $a_{m_1,\dots ,m_n}\not=0$, this is a contradiction to the previous paragraph. Thus the theorem is proven. Next we prove a generalization of Thm. \[th:PBW\]. \[th:PBWtau\] Assume that $\chi \in {\mathcal{X}}_3$. Let $n=|R^\chi _+|$ and let $\tau $ be a permutation of the set $\{1,2,\dots ,n\}$. Then the sets $$\begin{aligned} \big\{ E_{\beta _{\tau (1)}}^{m_{\tau (1)}} E_{\beta _{\tau (2)}}^{m_{\tau (2)}}\cdots E_{\beta _{\tau (n)}}^{m_{\tau (n)}}\,&|\, 0\le m_\nu <{b^{\chi}} (\beta _\nu ) \text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\},\\ \big\{ {\bar{E}}_{\beta _{\tau (1)}}^{m_{\tau (1)}} {\bar{E}}_{\beta _{\tau (2)}}^{m_{\tau (2)}}\cdots {\bar{E}}_{\beta _{\tau (n)}}^{m_{\tau (n)}}\,&|\, 0\le m_\nu <{b^{\chi}} (\beta _\nu ) \text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\} \end{aligned}$$ form vector space bases of $U ^+(\chi )$, and the sets $$\begin{aligned} \big\{ F_{\beta _{\tau (1)}}^{m_{\tau (1)}} F_{\beta _{\tau (2)}}^{m_{\tau (2)}}\cdots F_{\beta _{\tau (n)}}^{m_{\tau (n)}}\,&|\, 0\le m_\nu <{b^{\chi}} (\beta _\nu ) \text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\},\\ \big\{ {\bar{F}}_{\beta _{\tau (1)}}^{m_{\tau (1)}} {\bar{F}}_{\beta _{\tau (2)}}^{m_{\tau (2)}}\cdots {\bar{F}}_{\beta _{\tau (n)}}^{m_{\tau (n)}}\,&|\, 0\le m_\nu <{b^{\chi}} (\beta _\nu ) \text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\} \end{aligned}$$ form vector space bases of $U ^-(\chi )$. It suffices to prove that the first set is a basis of $U^+(\chi )$. Indeed, the proof for the second set can be obtained by using the maps ${T}^-_i$, where $i\in I$, instead of ${T}_i$. The second part of the claim follows from the first part by applying the algebra antiautomorphism ${\Omega }$ and using Eq. . For any ${\underline{m}}=(m_1,\ldots ,m_n)\in {\mathbb{N}}_0^n$ let $|{\underline{m}}|=\sum _{\mu =1}^n m_\mu |\beta _\mu |$, where $|{\alpha }|=\sum _{j\in I}a_j$ for all ${\alpha }=\sum _{j\in I}a_j{\alpha }_j \in {\mathbb{N}}_0^I$. Let $N$ be the (additive) monoid ${\mathbb{N}}_0^n$ equipped with the following ordering: $${\underline{m}}'<{\underline{m}}\quad \Leftrightarrow \quad |{\underline{m}}'|<|{\underline{m}}| \quad \text{or} \quad |{\underline{m}}'|=|{\underline{m}}|,\,{\underline{m}}'<_{\mathrm{lex}}{\underline{m}},$$ where $<_{\mathrm{lex}}$ means lexicographical ordering. We use the convention ${\underline{m}}<_{\mathrm{lex}}{\underline{m}}$. The ordering $<$ is a total ordering. For all ${\underline{m}}\in {\mathbb{N}}_0^n$ define $${\mathcal{F}}^{{\underline{m}}}U^+(\chi )=\mathop{\oplus }_{{\underline{m}}'\in N,{\underline{m}}'<\ulm} {\Bbbk }E_{\beta _1}^{m'_1} E_{\beta _2}^{m'_2}\cdots E_{\beta _n}^{m'_n} \subset U^+(\chi ).$$ The vector spaces ${\mathcal{F}}^{{\underline{m}}}U^+(\chi )$, where ${\underline{m}}\in N$, are finite-dimensional, since the degrees of their elements are bounded. Moreover, $${\mathcal{F}}^0 U^+(\chi )={\Bbbk }1,\qquad {\mathcal{F}}^{{\underline{m}}}U^+(\chi ) {\mathcal{F}}^{{\underline{m}}'}U^+(\chi )\subset {\mathcal{F}}^{{\underline{m}}+{\underline{m}}'}U^+(\chi )$$ for all ${\underline{m}},{\underline{m}}'\in N$ by Thm. \[th:EErel\] and since $U^+(\chi )$ is ${\mathbb{Z}}^I$-graded. Thus ${\mathcal{F}}$ defines a filtration of $U^+(\chi )$ by the monoid $N$, and the corresponding graded algebra $$\mathop{\oplus }_{{\underline{m}}\in N} \Big({\mathcal{F}}^{{\underline{m}}}U^+(\chi ) / \sum _{{\underline{m}}'<{\underline{m}},\ulm '\not={\underline{m}}} {\mathcal{F}}^{{\underline{m}}'}U^+(\chi )\Big)$$ is a skew-polynomial ring in $n$ variables by Thm. \[th:EErel\]. By a standard conclusion we obtain that the first set in the claim of the theorem is indeed a basis of $U^+(\chi )$. For later purpose we define additional elements in $U(\chi )$. For each $\nu \in \{n+1,n+2, \dots ,2n\}$ there exists a unique element $i_\nu \in I$ such that $$\begin{aligned} \ell (1_{r_{i_{\nu -n}}\cdots r_{i_2}r_{i_1}(\chi )} {\sigma }_{i_{\nu +1-n}}{\sigma }_{i_{\nu +2-n}}\cdots {\sigma }_{i_\nu })=n. \label{eq:ellsss} \end{aligned}$$ For this $i_\nu $ we get $1_{r_{i_{\nu -n}}\cdots r_{i_2}r_{i_1}(\chi )} {\sigma }_{i_{\nu +1-n}}{\sigma }_{i_{\nu +2-n}}\cdots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu }) ={\alpha }_{i_{\nu -n}}$. \[le:longlongw\] We proceed by induction on $\nu $. First, $$\ell (1_{r_{i_{\nu -n}}\cdots r_{i_2}r_{i_1}(\chi )} {\sigma }_{i_{\nu +1-n}}{\sigma }_{i_{\nu +2-n}}\cdots {\sigma }_{i_{\nu -1}})=n-1$$ by induction hypothesis (if $\nu >n+1$) or by the choice of $i_1,\dots ,i_n$ (for $\nu =n+1$). Thus, by [@a-HeckYam08 Lemma8(iii)] there exists a unique positive root ${\alpha }$ such that $1_{r_{i_{\nu -n}}\cdots r_{i_2}r_{i_1}(\chi )} {\sigma }_{i_{\nu +1-n}}{\sigma }_{i_{\nu +2-n}}\cdots {\sigma }_{i_{\nu -1}}({\alpha })>0$. Clearly, this has to be a simple root. Let $i_\nu \in I$ such that ${\alpha }=\al _{i_\nu }$. Then the first claim follows from [@a-HeckYam08 Cor.3]. By Eq.  for $\nu -1$ and by [@a-HeckYam08 Cor.3] we get $1_{r_{i_{\nu -n-1}}\cdots r_{i_2}r_{i_1}(\chi )} {\sigma }_{i_{\nu -n}}{\sigma }_{i_{\nu +1-n}}\cdots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu })<0$. Together with the first claim of the lemma this implies the second claim. For each $\nu \in \{1,2,\dots,2n\}$ let $$\begin{aligned} \label{eq:beta2n} \beta _\nu =\beta _\nu ^\chi =&1_\chi {\sigma }_{i_1}\dots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu }),& F_{\beta _\nu }=& F_{\beta _\nu }^\chi ={T}_{i_1}\dots {T}_{i_{\nu -1}}(F_{i_\nu }),\end{aligned}$$ where $F_{i_\nu }\in U^-(r_{i_{\nu -1}}\cdots r_{i_2}r_{i_1}(\chi ))$ and $F_{\beta _\nu }\in U(\chi )$. Since $$\ell (1_{r_{i_n}\cdots r_{i_2}r_{i_1}(\chi )}{\sigma }_{i_{n+1}}{\sigma }_{i_{n+2}} \cdots {\sigma }_{i_{2n}})=n,$$ Lemma \[le:longestw\] implies that $$\begin{aligned} 1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\dots {\sigma }_{i_n}=& {\sigma }_{i_1}{\sigma }_{i_2}\dots {\sigma }_{i_n}1_{r_{i_n}\cdots r_{i_2}r_{i_1}(\chi )}\\ =& (1_{r_{i_n}\cdots r_{i_2}r_{i_1}(\chi )} {\sigma }_{i_{n+1}}{\sigma }_{i_{n+2}}\dots \s_{i_{2n}})^{-1}.\end{aligned}$$ Moreover, $$\begin{aligned} \beta _\nu ^\chi =-\beta _{\nu -n}^\chi \in - R^\chi _+ \label{eq:betak>n}\end{aligned}$$ for all $\nu \in \{n+1,n+2,\dots ,2n\}$ by Lemma \[le:longlongw\]. In $U(\chi )$ we also get $$\begin{aligned} F_{\beta ^\chi _\nu }=T_{i_1}\cdots T_{i_{\nu -n}}(F_{i_{\nu -n}})\in \fienz K^{-1}_{\beta _{\nu -n}^\chi }E_{\beta ^\chi _{\nu -n}} \label{eq:Fbetak}\end{aligned}$$ for all $\nu \in \{n+1,\dots ,2n\}$ by [@p-Heck07b Thm.6.19,Prop.6.8(ii)]. Verma modules and morphisms {#sec:Verma} =========================== We consider Verma modules for the algebras $U(\chi )$, $\chi \in {\mathcal{X}}$. We observe that the fundaments of the theory of Verma modules for quantized enveloping algebras can be carried over to a great extent to $U(\chi )$. New phenomena appear if some generators of $U(\chi )$ are nilpotent. Let ${\mathbb{K}}$ be a field extension of ${\Bbbk }$. Let ${{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ denote the set of ${{\mathbb{K}}^\times }$-valued characters (algebra maps from ${{\mathcal{U}}^0}$ to ${{\mathbb{K}}^\times }$) of the group algebra ${{\mathcal{U}}^0}$. For all $\chi \in {\mathcal{X}}$ there is a natural group homomorphism $$\begin{aligned} {\zeta ^{\chi}} : {\mathbb{Z}}^I\to {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })},\quad {\zeta ^{\chi}} ({\alpha })(K_\beta L_{\beta '})= \chi (\beta ,{\alpha })\chi (\al,\beta ')^{-1} \label{eq:ZIch}\end{aligned}$$ for all ${\alpha },\beta ,\beta ' \in {\mathbb{Z}}^I$. If $\chi \in {\mathcal{X}}$ and $p\in I$ such that $\chi $ is $p$-finite, then $$\begin{aligned} \ZIch{r_p(\chi )}({\sigma }_p^\chi ({\alpha })) (K_{{\sigma }_p^\chi (\beta )}L_{{\sigma }_p^\chi (\beta ')})= {\zeta ^{\chi}} ({\alpha })(K_\beta L_{\beta '}) \label{eq:ZIchrefl}\end{aligned}$$ by Eq. . Let $\chi \in {\mathcal{X}}$. Given a ${{\mathbb{K}}^\times }$-valued character $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$, one can regard ${\mathbb{K}}$ as a one-dimensional $U ^+(\chi ){{\mathcal{U}}^0}$-module with generator $1_\Lambda =1$ via $$\begin{aligned} uE 1_\Lambda ={\varepsilon }(E)\Lambda (u)1_\Lambda \qquad \text{for all $E\in U ^+(\chi )$, $u\in {{\mathcal{U}}^0}$.} \label{eq:fieLambda}\end{aligned}$$ We write ${\mathbb{K}}_\Lambda $ for this module. A *Verma module* of $U (\chi )$ is a $U (\chi )$-module of the form $$M^\chi (\Lambda )=U (\chi ){\otimes }_{U ^+(\chi ){{\mathcal{U}}^0}}{\mathbb{K}}_\Lambda ,\quad \text{where $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$.}$$ We write $v^\chi _\Lambda $ or just $v_\Lambda $ for $1{\otimes }1_\Lambda \in M^\chi (\Lambda )$. Any Verma module is also a ${\mathbb{K}}$-module via the ${\mathbb{K}}$-module structure of the second tensor factor. Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Triangular decomposition of $U (\chi )$ gives the following standard fact. The map $U ^-(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}\to M^\chi (\Lambda )$, $u{\otimes }x\mapsto uxv_\Lambda =u{\otimes }x1_\Lambda $, is an isomorphism of vector spaces over ${\mathbb{K}}$. \[le:MLiso\] The isomorphism in Lemma \[le:MLiso\] and the ${\mathbb{Z}}^I$-grading of $U^-(\chi )$ induce a unique ${\mathbb{Z}}^I$-grading on $M^\chi (\Lambda )$ such that $$\begin{aligned} M^\chi (\Lambda )_{\alpha }=U^-(\chi )_{\alpha }{\otimes }_{\Bbbk }{\mathbb{K}}_\Lambda \quad \text{for all ${\alpha }\in {\mathbb{Z}}^I$.} \label{eq:MLgrading}\end{aligned}$$ Then $$\begin{aligned} M^\chi (\Lambda )_0={\mathbb{K}}v_\Lambda ,\,\, U(\chi )_{\alpha }M^\chi (\Lambda )_\beta \subset M^\chi (\Lambda )_{{\alpha }+\beta } \,\,\text{for all ${\alpha },\beta \in {\mathbb{Z}}^I$.} \label{eq:MLgrading2}\end{aligned}$$ Moreover, $M^\chi (\Lambda )_{\alpha }\not=0$ implies that $-{\alpha }\in {\mathbb{N}}_0^I$. The group algebra ${{\mathcal{U}}^0}$ acts on $M^\chi (\Lambda )$ via left multiplication. This action is given by characters: $$\begin{aligned} uv=(\Lambda +{\zeta ^{\chi}} ({\alpha }))(u)v \quad \text{for all $u\in {{\mathcal{U}}^0}$, ${\alpha }\in {\mathbb{Z}}^I$, $v\in M^\chi (\Lambda )_{\alpha }$,} \label{eq:U0M}\end{aligned}$$ see Eqs. , , , and . Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. The family of those $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-submodules of $M^\chi (\Lambda )$, which are contained in $\oplus _{{\alpha }\not=0}M^\chi (\Lambda )_{\alpha }$, have a unique maximal element $I^\chi (\Lambda )$. Let $$\begin{aligned} L^\chi (\Lambda )=M^\chi (\Lambda )/I^\chi (\Lambda ) \label{eq:LLambda}\end{aligned}$$ be the quotient $U(\chi )$-module. The maximality of $I^\chi (\Lambda )$ implies that $$I^\chi (\Lambda )=\big(I^\chi (\Lambda )\cap (U^-(\chi ){\otimes }1)\big)\fiee$$ and that $L^\chi (\Lambda )$ is ${\mathbb{Z}}^I$-graded. For all ${\alpha }\in {\mathbb{Z}}^I$ let $$I^\chi (\Lambda )_{\alpha }=M^\chi (\Lambda )_{\alpha }\cap I^\chi (\Lambda ), \quad L^\chi (\Lambda )_{\alpha }=M^\chi (\Lambda )_{\alpha }/I^\chi (\Lambda )_{\alpha }.$$ Since $M^\chi (\Lambda )_0={\mathbb{K}}v_\Lambda $, and any ${\mathbb{Z}}^I$-graded quotient of $M^\chi (\Lambda )$ by a $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-submodule containing $v_\Lambda $ is zero, $L^\chi (\Lambda )$ is the unique simple ${\mathbb{Z}}^I$-graded $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-module quotient of $M^\chi (\Lambda )$. \[de:fchar\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ and $V$ a ${\mathbb{Z}}^I$-graded subquotient of $M^\chi (\Lambda )$. The *(formal) character of* $V$ is the sum $$\fch{V}=\sum _{{\alpha }\in {\mathbb{N}}_0^I} (\dim V_{-{\alpha }}) e^{-{\alpha }},$$ where $e$ is a formal variable. Eq.  implies that $$\begin{aligned} \fch{M^\chi (\Lambda )}=\sum _{{\alpha }\in {\mathbb{N}}_0^I}\dim U^-(\chi )_{-{\alpha }} e^{-{\alpha }} \label{eq:chML}\end{aligned}$$ for all $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. \[re:fch\] For all ${\alpha },\beta \in {\mathbb{Z}}^I$ we let $e^{\alpha }e^\beta =e^{{\alpha }+\beta }$. Thus we can consider formal characters as elements of the ring $\cup _{{\alpha }\in {\mathbb{N}}_0^I}e^{\alpha }{\mathbb{Z}}[[e^{-{\alpha }_i}\,|\,i\in I]]$, where $e^{\alpha }{\mathbb{Z}}[[e^{-{\alpha }_i}\,|\,i\in I]]\subset e^{{\alpha }+\beta }{\mathbb{Z}}[[e^{-{\alpha }_i}\,|\,i\in I]]$ for all ${\alpha },\beta \in {\mathbb{N}}_0^I$ in the natural way. [*From now on let $\chi \in {\mathcal{X}}$, $p\in I$, and ${b}={b^{\chi}} ({\alpha }_p)={b^{r_p(\chi )}}({\alpha }_p)$. Assume that ${b}<\infty $.*]{} Then $\chi $ and $r_p(\chi )$ are $p$-finite. We deduce some phenomena which arise from the finiteness assumption on ${b}$. For all $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ define ${t}_p^\chi (\Lambda )\in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ by $$\begin{aligned} {t}_p^\chi (\Lambda )(K_{\alpha }L_\beta )=\Lambda (K_{{\sigma }_p^{r_p(\chi )}({\alpha })} L_{{\sigma }_p^{r_p(\chi )}(\beta )}) \frac{r_p(\chi ) ({\alpha },{\alpha }_p)^{{b}-1}}{r_p(\chi )({\alpha }_p,\beta )^{{b}-1}} \label{eq:tpLambda}\end{aligned}$$ for all ${\alpha },\beta \in {\mathbb{Z}}^I$. By Eq.  this is equivalent to $$\begin{aligned} {t}_p^\chi (\Lambda )(K_{{\sigma }^\chi _p({\alpha })}L_{{\sigma }_p^\chi (\beta )}) =\Lambda (K_{\alpha }L_\beta ) \frac {\chi ({\alpha }_p,\beta )^{{b}-1}} {\chi ({\alpha },{\alpha }_p)^{{b}-1}} \label{eq:tpLambda1}\end{aligned}$$ for all ${\alpha },\beta \in {\mathbb{Z}}^I$. Recall the definition of ${\rho ^{\chi}} $ from Def. \[de:rhomap\]. Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Then $$\begin{aligned} {\rho ^{r_p(\chi )}}({\sigma }_p^\chi ({\alpha }))\, {t}_p^\chi (\Lambda ) (K_{{\sigma }_p^\chi ({\alpha })} L_{{\sigma }_p^\chi ({\alpha })}^{-1})= {\rho ^{\chi }}({\alpha }) \, \Lambda (K_{\alpha }L_{{\alpha }}^{-1}) \end{aligned}$$ for all ${\alpha }\in {\mathbb{Z}}^I$ and $p\in I$. \[le:VTinv\] Insert Eq.  and use Lemma \[le:rho\]. Let $C$ be a symmetrizable Cartan matrix and $q\in {{\Bbbk }^\times }$, $\chi \in {\mathcal{X}}$ as in the second part of Ex. \[ex:Cartan\]. In particular, $\chi ({\alpha }_i,{\alpha }_j)=q^{d_i c_{i j}}$. Let $p\in I$. Assume that ${b}={b^{\chi}} ({\alpha }_p)<\infty $. Then $q^{2d_p \bnd }=1$, and hence $q^{2{b}({\alpha },{\alpha }_p)}=1$ for all ${\alpha }\in \ndZ ^I$. Further, $r_p(\chi )=\chi $. Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Assume that $\Lambda (K_{\alpha }L_\al ^{-1})=q^{2({\alpha },\lambda )}$ for some $\lambda $ in the weight lattice. Then $$\begin{aligned} {t}_p^\chi (\Lambda )(K_{\alpha }L_{\alpha }^{-1}) =&\Lambda (K_{{\sigma }_p^\chi ({\alpha })} L_{{\sigma }_p^\chi ({\alpha })}^{-1}) q^{2(b-1)(\al ,{\alpha }_p)}\\ =&q^{2({\sigma }_p^\chi ({\alpha }),\lambda )}q^{-2({\alpha },{\alpha }_p)} =q^{2({\alpha },{\sigma }_p^\chi (\lambda )-{\alpha }_p)}, \end{aligned}$$ which recovers the dot action of the Weyl group on the weight lattice. If we consider a composition ${t}_i^{\chi '}{t}_j^{\chi ''}$, where $i,j\in I$, $\chi ',\chi ''\in {\mathcal{X}}$, then we will always assume that $$\chi '=r_j(\chi '').$$ For simplicity, we will omit the upper index $\chi '$ if it is uniquely determined by another bicharacter in the expression. Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Then ${t}_p {t}_p^\chi (\Lambda ) =\Lambda $. \[le:VTMrel1\] By Eqs.  and , and since $r_p^2(\chi )=\chi $, $$\begin{aligned} &{t}_p^{r_p(\chi )}{t}_p^\chi (\Lambda )(K_{\alpha }L_\beta )= {t}_p^\chi (\Lambda )(K_{{\sigma }_p^\chi ({\alpha })}L_{{\sigma }_p^\chi (\beta )}) \frac{\chi ({\alpha },{\alpha }_p)^{{b}-1}}{\chi ({\alpha }_p,\beta )^{{b}-1}} =\Lambda (K_{\alpha }L_\beta ) \end{aligned}$$ for all ${\alpha },\beta \in {\mathbb{Z}}^I$. This proves the lemma. \[le:MLmap\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. There exist unique ${\mathbb{K}}$-linear maps $$\begin{gathered} {\hat{T}}_p={\hat{T}}^\chi _{p,\Lambda },\, {\hat{T}}^-_p={\hat{T}}^{\chi ,-}_{p,\Lambda }: \,\, M^{r_p(\chi )}({t}_p^\chi (\Lambda ))\to M^{\chi }(\Lambda ),\\ \intertext{such that for all $u\in U(\chi )$,} {\hat{T}}_p(uv_{{t}_p^\chi (\Lambda )})= {T}_p(u)F_p^{{b}-1}v_{\Lambda },\quad {\hat{T}}^-_p(uv_{{t}_p^\chi (\Lambda )})= {T}^-_p(u)F_p^{{b}-1}v_{\Lambda }. \end{gathered}$$ If $V\subset M^{r_p(\chi )}({t}_p^\chi (\Lambda ))$ is a $U(r_p(\chi ))$-submodule, then ${\hat{T}}_p(V), {\hat{T}}^-_p(V)$ are $U(\chi )$-submodules of $M^{\chi }(\Lambda )$. The uniqueness of the maps ${\hat{T}}_p$, ${\hat{T}}^-_p$ is clear. We prove that ${\hat{T}}_p$ is well-defined. The proof for ${\hat{T}}^-_p$ is analogous. Let $\chi '=r_p(\chi )$ and $\Lambda '={t}_p^\chi (\Lambda )$. By Lemma \[le:MLiso\] and Thm. \[th:PBWtau\], $$M^{\chi }(\Lambda )_{{\alpha }_j+a{\alpha }_p}=0= M^{\chi }(\Lambda )_{-{b}{\alpha }_p} \quad \text{for all $a\in {\mathbb{Z}}$.}$$ Thus, since $\deg {T}_p(E_j)={\sigma }_p^{\chi '}({\alpha }_j)$ for $E_j\in U(\chi ')$, $$\begin{aligned} {T}_p(E_j)F_p^{{b}-1}v_\Lambda \in M^{\chi }(\Lambda )_{{\alpha }_j+(1-{b}-c^{\chi '}_{pj}){\alpha }_p}=0 \quad \text{for all $j\in I$.} \end{aligned}$$ Moreover, Eqs. , give that $$\begin{aligned} K_{{\alpha }}L_{\beta }F_p^{{b}-1} v_{\Lambda } =\chi ({\alpha },{\alpha }_p)^{1-{b}}\, \chi ({\alpha }_p,\beta )^{{b}-1}\, \Lambda (K_{\alpha }L_\beta ) F_p^{{b}-1}v_{\Lambda }\qquad &\\ =\Lambda '(K_{{\sigma }_p^\chi ({\alpha })} L_{{\sigma }_p^\chi (\beta )}) F_p^{{b}-1}v_\Lambda & \end{aligned}$$ for all ${\alpha },\beta \in {\mathbb{Z}}^I$. Hence ${T}_p(u)F_p^{{b}-1}v_{\Lambda }={\hat{T}}_p(\Lambda '(u)v_{\Lambda '})$ for all $u\in {{\mathcal{U}}^0}$. Therefore ${\hat{T}}_p$ is well-defined. The last claim of the lemma follows from the equations $$\begin{aligned} {\hat{T}}_p(uv)={T}_p(u){\hat{T}}_p(v),\quad {\hat{T}}^-_p(uv)={T}^-_p(u){\hat{T}}^-_p(v), \label{eq:Tuv} \end{aligned}$$ where $u\in U(r_p(\chi ))$ and $v\in M^{r_p(\chi )}({t}_p^\chi (\Lambda ))$, and from the surjectivity of the maps ${T}_p,{T}^-_p$. \[le:VTMrel2\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ and $j,k\in I$ such that $j\not=k$. Assume that $m_{j,k}^\chi =|({\mathbb{N}}_0{\alpha }_j+{\mathbb{N}}_0{\alpha }_k)\cap R^\chi _+|<\infty $ and that ${b^{\chi}} ({\alpha })<\infty $ for all ${\alpha }\in ({\mathbb{N}}_0{\alpha }_j+{\mathbb{N}}_0{\alpha }_k)\cap R^\chi _+$. Then $$\begin{aligned} \label{eq:VTCox} ({t}_j{t}_k)^{m _{j,k}^\chi -1}{t}_j{t}_k^\chi (\Lambda )=\Lambda . \end{aligned}$$ Let $m=m^\chi _{j,k}$, and let $i_0,i_1,\dots ,i_{m_{j,k}^\chi }\in \{j,k\}$ such that $i_\nu =j$ if $\nu $ is even and $i_\nu =k$ if $\nu $ is odd. Let $\chi '=r_{i_0}r_{i_1}\cdots r_{i_{m-1}}(\chi )= r_{i_1}r_{i_2}\cdots r_{i_m}(\chi )$ (by (R4)), and $$\Lambda '={t}_{i_0}{t}_{i_1}\cdots {t}_{i_{m-1}}^\chi (\Lambda ),\quad \Lambda ''={t}_{i_1}{t}_{i_2}\cdots {t}_{i_m}^\chi (\Lambda ).$$ These definitions make sense, since ${b^{\chi}} ({\alpha })<\infty $ for all ${\alpha }\in ({\mathbb{N}}_0{\alpha }_j+{\mathbb{N}}_0{\alpha }_k)\cap R^\chi _+$. Lemma \[le:VTMrel1\] implies that the claim of the lemma is equivalent to $\Lambda '=\Lambda ''$. Let ${\hat{T}}'={\hat{T}}_{i_0}{\hat{T}}_{i_1}\cdots {\hat{T}}_{i_{m-1}}: M^{\chi '}(\Lambda ')\to M^\chi (\Lambda )$ and ${\hat{T}}''={\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_m}: M^{\chi '}(\Lambda '')\to M^\chi (\Lambda )$. For all $\nu \in \{1,2,\dots ,m\}$ let $$\beta '_\nu =1_\chi {\sigma }_{i_0}{\sigma }_{i_1}\cdots {\sigma }_{i_{\nu -2}} ({\alpha }_{i_{\nu -1}}), \quad \beta ''_\nu =1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_{\nu -1}} ({\alpha }_{i_\nu }).$$ By definition of ${\hat{T}}_j$ and ${\hat{T}}_k$, $$\begin{aligned} {\hat{T}}''(v_{\Lambda ''})=&{\hat{T}}_{i_1}\cdots {\hat{T}}_{i_{m-1}} (F_{i_m}^{{b^{\chi '}}({\alpha }_{i_m})-1}v_{{t}_{i_m}^{\chi '}(\Lambda '')}) =\cdots \\ =&F_{\beta ''_m}^{{b^{\chi}} (\beta ''_m)-1}\cdots F_{\beta ''_2}^{{b^{\chi}} (\beta ''_2)-1} F_{\beta ''_1}^{{b^{\chi}} (\beta ''_1)-1}v_\Lambda ,\\ {\hat{T}}'(v_{\Lambda '})=&F_{\beta '_m}^{{b^{\chi}} (\beta '_m)-1}\cdots F_{\beta '_2}^{{b^{\chi}} (\beta '_2)-1} F_{\beta '_1}^{{b^{\chi}} (\beta '_1)-1}v_\Lambda . \end{aligned}$$ Both expressions are nonzero by Thm. \[th:PBWtau\]. Since $$\{\beta '_\nu \,|\,1\le \nu \le m\}=\{\beta ''_\nu \,|\,1\le \nu \le m\} =R^\chi _+\cap ({\mathbb{N}}_0{\alpha }_j+{\mathbb{N}}_0{\alpha }_k),$$ we obtain that - ${\hat{T}}'({\mathbb{K}}v_{\Lambda '})$ and ${\hat{T}}''({\mathbb{K}}v_{\Lambda ''})$ are isomorphic ${{\mathcal{U}}^0}$-modules. By Thm. \[th:Coxgr\], $${T}_{i_0}{T}_{i_1}\cdots {T}_{i_{m-1}}(u_0)= {T}_{i_1}{T}_{i_2}\cdots {T}_{i_m}(u_0)\quad \text{for all $u_0\in {{\mathcal{U}}^0}$.}$$ Hence Lemma \[le:MLmap\] yields that $$\begin{aligned} \Lambda '(u_0){\hat{T}}'(v_{\Lambda '})={\hat{T}}'(u_0v_{\Lambda '}) =&{T}_{i_0}{T}_{i_1}\cdots {T}_{i_{m-1}}(u_0) {\hat{T}}'(v_{\Lambda '}),\\ \Lambda ''(u_0){\hat{T}}''(v_{\Lambda ''})={\hat{T}}''(u_0v_{\Lambda ''}) =&{T}_{i_0}{T}_{i_1}\cdots {T}_{i_{m-1}}(u_0) {\hat{T}}''(v_{\Lambda ''}). \end{aligned}$$ Thus $\Lambda '=\Lambda ''$ by ($*$). This proves the lemma. In view of Thm. \[th:Coxgr\] and Lemmata \[le:VTMrel1\], \[le:VTMrel2\] we can say that Eq.  defines an action of the groupoid ${\mathcal{W}}(\chi )$ on ${{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Then Lemma \[le:VTinv\] says that the numbers ${\rho ^{\chi}} ({\alpha })\Lambda (K_{\alpha }L_{\alpha }^{-1})$, where ${\alpha }\in {\mathbb{Z}}^I$, are invariants of this action. In general, the maps ${\hat{T}}_p$ and ${\hat{T}}^-_p$ are not isomorphisms. Assume that $\Lambda (K_pL_p^{-1})\not=\chi ({\alpha }_p,{\alpha }_p)^{t-1}$ for all $t\in \{1,2,\dots ,{b}-1\}$. Then ${\hat{T}}_p,{\hat{T}}^-_p: \,M^{r_p(\chi )}({t}_p^\chi (\Lambda ))\to M^\chi (\Lambda )$ are isomorphisms of vector spaces over ${\mathbb{K}}$. \[pr:VTMiso\] Let $q=\chi ({\alpha }_p,{\alpha }_p)$, $\chi '=r_p(\chi )$, $\Lambda '={t}_p^\chi (\Lambda )$, and ${\hat{T}}'={\hat{T}}^\chi _{p,\Lambda } {\hat{T}}^{\chi ',-}_{p,\Lambda '}$. By Lemma \[le:VTMrel1\] and since $r_p^2(\chi )=\chi $, ${\hat{T}}'$ is a $U(\chi )$-module endomorphism of $M^\chi (\Lambda )$. We calculate ${\hat{T}}'(v_\Lambda )$. $$\begin{aligned} {\hat{T}}'(v_\Lambda ) =&{\hat{T}}_p(F_p^{{b}-1}v_{\Lambda '}) ={T}_p(F_p^{{b}-1})F_p^{{b}-1}v_\Lambda \\ =&(K_p^{-1}E_p)^{{b}-1}F_p^{{b}-1}v_\Lambda =q^{({b}-2)({b}-1)/2}\Lambda (K_p^{1-{b}})E_p^{{b}-1}F_p^{{b}-1}v_\Lambda \\ =&q^{({b}-2)({b}-1)/2}\Lambda (K_p^{1-{b}})\qfact{{b}-1}{q} \prod _{t=1}^{{b}-1}(q^{t+1-{b}}\Lambda (K_p)-\Lambda (L_p))v_\Lambda \end{aligned}$$ by Lemma \[le:EmFn\]. By assumption, ${\hat{T}}'(v_\Lambda )\not=0$, and hence ${\hat{T}}'$ is a nonzero multiple of ${\operatorname{id}}_{M^\chi (\Lambda )}$. Therefore ${\hat{T}}_p$ is an isomorphism. The proof for ${\hat{T}}_p^-$ is analogous. \[le:hwvector\] Let $t\in \{1,2,\dots ,{b}-1\}$. Let $q=\chi ({\alpha }_p,{\alpha }_p)$. Assume that $\Lambda (K_pL_p^{-1})=q^{t-1}$. Then in $M^\chi (\Lambda )$ $$\begin{aligned} E_pF_p^m v_\Lambda =\qnum{m}{q}\Lambda (L_p)(q^{t-m}-1)F_p^{m-1} v_\Lambda \quad \text{for all $m\in {\mathbb{N}}_0$.} \label{eq:EpFpmv} \end{aligned}$$ In particular, if $q\not=1$, then $EF_p^mv_\Lambda ={\varepsilon }(E)F_p^mv_\Lambda $ for all $E\in U^+(\chi )$ if and only if $m=0$, $m=t$ or $m\ge {b}$. If $q=1$, then $EF_p^mv_\Lambda ={\varepsilon }(E)F_p^mv_\Lambda $ for all $E\in U^+(\chi )$, $m\in {\mathbb{N}}_0$. Eq.  follows from Lemma \[le:EmFn\]. By definition of ${b}={b^{\chi}} ({\alpha }_p)$, either $q\not=1$ and $q$ is a primitive ${b}$-th root of $1$, or $q=1$ and ${b}=\mathrm{char}\,{\Bbbk }$. Therefore, if $q\not=1$ and $m\in \{0,1,\dots ,{b}-1\}$, then $q^{t-m}=1$ if and only if $t=m$. If $E=E_i$ with $i\not=p$, then $EF_p^m=0$ by Eqs. , . The rest is a consequence of Lemma \[le:Eheight\](i). \[pr:VTMker\] Assume that $\Lambda (K_pL_p^{-1})=\chi ({\alpha }_p,{\alpha }_p)^{t-1}$ for some $t\in \{1,2,\dots ,{b}-1\}$. \(i) If $\chi ({\alpha }_p,{\alpha }_p)\not=1$, then $t$ is unique, and $$\begin{aligned} \ker {\hat{T}}^{\chi }_{p,\Lambda }=\ker {\hat{T}}^{\chi ,-}_{p,\Lambda } = & \, U^-(\chi )F_p^{{b}-t}{\otimes }{\mathbb{K}}_{{t}_p^\chi (\Lambda )},\\ {\operatorname{Im}}{\hat{T}}^{\chi }_{p,\Lambda }={\operatorname{Im}}{\hat{T}}^{\chi ,-}_{p,\Lambda } = & \, U^-(\chi )F_p^t {\otimes }{\mathbb{K}}_\Lambda . \end{aligned}$$ \(ii) If $\chi ({\alpha }_p,{\alpha }_p)=1$, then $\mathrm{char}\,{\Bbbk }={b}>0$ and $$\begin{aligned} \ker {\hat{T}}^{\chi }_{p,\Lambda }=\ker {\hat{T}}^{\chi ,-}_{p,\Lambda } = & \, U^-(\chi )F_p{\otimes }{\mathbb{K}}_{{t}_p^\chi (\Lambda )},\\ {\operatorname{Im}}{\hat{T}}^{\chi }_{p,\Lambda }={\operatorname{Im}}{\hat{T}}^{\chi ,-}_{p,\Lambda } = & \, U^-(\chi )F_p^{{b}-1} {\otimes }{\mathbb{K}}_\Lambda . \end{aligned}$$ Let $q=\chi ({\alpha }_p,{\alpha }_p)$, $\chi '=r_p(\chi )$, and $\Lambda '={t}_p^\chi (\Lambda )$. We prove the claims about ${\hat{T}}^\chi _{p,\Lambda }$ in (i). The rest is analogous. By Lemma \[le:EmFn\], for all $m\in \{0,1,\dots ,{b}-1\}$, $$\begin{aligned} {\hat{T}}^\chi _{p,\Lambda }(F_p^mv_{\Lambda '}) =&{T}_p(F_p^m)F_p^{{b}-1}v_\Lambda =(K_p^{-1}E_p)^mF_p^{{b}-1}v_\Lambda \\ =&aE_p^mF_p^{{b}-1}v_\Lambda =a'F_p^{{b}-1-m} \prod _{j=1}^m (q^{j+1-{b}}\Lambda (K_pL_p^{-1})-1) v_\Lambda \end{aligned}$$ for some $a,a'\in {{\Bbbk }^\times }$. Thus, ${\hat{T}}^\chi _{p,\Lambda }(F_p^mv_{\Lambda '})=0$ if and only if $q^j=q^{{b}-1}\Lambda (K_pL_p^{-1})^{-1}$ for some $j\in \{1,2,\dots ,m\}$. By the assumption on $\Lambda (K_pL_p^{-1})$, this is equivalent to $j={b}-t$ for some $j\in \{1,2,\dots ,m\}$. Therefore ${\hat{T}}^\chi _{p,\Lambda }(F_p^mv_{\Lambda '})=0$ if and only if $m\ge {b}-t$. Let $F\in U^-(\chi ')$. By Lemma \[le:Eheight\](ii), there exist unique $F'_m\in U^-_{p,K}(\chi ')$, where $m\in \{0,1,\dots ,{b}-1\}$, such that $F=\sum _{m=0}^{{b}-1}F'_mF_p^m$. By the previous paragraph, and since $\Lambda (K_pL_p^{-1})\in {\Bbbk }$, for each $m\in \{0,1,\dots ,{b}-1-t\}$ there is a unique $a_m\in {{\Bbbk }^\times }$ such that $${\hat{T}}^\chi _{p,\Lambda }(F v_{\Lambda '})=\sum _{m=0}^{{b}-1-t} a_m{T}_p (F'_m)F_p^{{b}-1-m}v_\Lambda \in U^-(\chi ){\otimes }1_\Lambda .$$ By Lemma \[le:Eheight\](ii), the latter expression is zero if and only if ${T}_p(F'_m)=0$ for all $m\in \{0,1,\dots ,{b}-1-t\}$. Therefore, Lemma \[le:TpU+U+\] and relations $F'_m\in U^-_{i,K}(\chi ')$ imply that ${\hat{T}}^\chi _{p,\Lambda }(F v_{\Lambda '})=0$ if and only if $F'_m=0$ for all $m\in \{0,1,\dots ,{b}-1-t\}$. Hence $\ker {\hat{T}}^{\chi }_{p,\Lambda }=U^-(\chi )F_p^{{b}-t}{\otimes }{\mathbb{K}}_{\Lambda '}$ and ${\operatorname{Im}}{\hat{T}}^{\chi }_{p,\Lambda } = U^-(\chi )F_p^t {\otimes }{\mathbb{K}}_\Lambda $. For all $w\in {\mathrm{Aut}}({\mathbb{Z}}^I)$ and ${\alpha }\in {\mathbb{Z}}^I$ let $w(e^{\alpha })=e^{w({\alpha })}$, and extend this definition linearly on formal characters. We investigate the effect of the maps ${\hat{T}}_p,{\hat{T}}^-_p$ on formal characters. For all $\chi '\in {\mathcal{G}}(\chi )$ and $i\in I$ with $\bfun{\chi '}({\alpha }_i)<\infty $ let ${\dot{\sigma }}^{\chi '}_i:{\mathbb{Z}}^I\to {\mathbb{Z}}^I$ be the affine transformation $$\begin{aligned} {\dot{\sigma }}_i^{\chi '}({\alpha })={\sigma }_i^{\chi '}({\alpha }) +(1-\bfun{\chi '}({\alpha }_i)){\alpha }_i. \label{eq:sdot}\end{aligned}$$ Note that then ${\dot{\sigma }}_i^{r_i(\chi ')}{\dot{\sigma }}_i^{\chi '}({\alpha })={\alpha }$ for all ${\alpha }\in {\mathbb{Z}}^I$. \[le:Tpfch\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ and ${\alpha }\in {\mathbb{Z}}^I$. Then $$\begin{aligned} \label{eq:Tpweight+} {\hat{T}}_p(M^{r_p(\chi )}({t}^\chi _p(\Lambda ))_{\alpha }) \subset &M^\chi (\Lambda )_{{\dot{\sigma }}_p^{r_p(\chi )}({\alpha })},\\ \label{eq:Tpweight-} {\hat{T}}^-_p(M^{r_p(\chi )}({t}^\chi _p(\Lambda ))_{\alpha }) \subset &M^\chi (\Lambda )_{{\dot{\sigma }}_p^{r_p(\chi )}({\alpha })}. \end{aligned}$$ In particular, $$\begin{aligned} \fch{M^\chi (\Lambda )} = {\dot{\sigma }}_p^{r_p(\chi )} (\fch{M^{r_p(\chi )}({t}_p^\chi (\Lambda ))}). \label{eq:fchM} \end{aligned}$$ Let $\Lambda '={t}^\chi _p(\Lambda )$ and $u\in U(r_p(\chi ))_{\alpha }$. Then $${\hat{T}}_p(uv_{\Lambda '})={T}_p(u)F_p^{{b}-1}v_\Lambda \in U(\chi )_{{\sigma }_p^{r_p(\chi )}({\alpha })}U(\chi )_{(1-{b}){\alpha }_p}v_\Lambda$$ by Prop. \[pr:LTdeg\]. This proves Eq. , since $v_\Lambda \in M^\chi (\Lambda )_0$, The proof of Eq.  is similar. By Eq. , ${\mathrm{ch}\,M}^\chi (\Lambda )$ does not depend on $\Lambda $. Hence by Prop. \[pr:VTMiso\] we may assume that ${\hat{T}}_p$ is an isomorphism. Then Eq.  follows from Eq. . \[le:chsub\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ and $t\in \{1,2,\dots ,{b}-1\}$. Assume that $\Lambda (K_pL_p^{-1})=\chi ({\alpha }_p,{\alpha }_p)^{t-1}$. Then $V=U^-(\chi )F_p^t {\otimes }{\mathbb{K}}_\Lambda $ is a $U(\chi ){\otimes }{\mathbb{K}}$-submodule of $M^\chi (\Lambda )$ with $$\fch{V}=\fch{M^\chi }(\Lambda ) \frac{e^{-t{\alpha }_p}-e^{-{b}{\alpha }_p}}{1-e^{-{b}{\alpha }_p}}.$$ The formal character of the subspace $\oplus _{m=t}^{{b}-1}F_p^m$ of $M^\chi (\Lambda )$ is $$e^{-t{\alpha }_p}+e^{-(t+1){\alpha }_p}+\cdots +e^{-({b}-1){\alpha }_p}= \frac{e^{-t{\alpha }_p}-e^{-{b}{\alpha }_p}}{1-e^{-{\alpha }_p}}.$$ Thus the lemma is a consequence of Lemmata \[le:hwvector\], \[le:MLiso\] and Eqs. , . *In the rest of this section assume that $\chi \in {\mathcal{X}}_4$.* Let $n=|R^\chi _+|$ and $i_1,\dots ,i_n\in I$ with $\ell (1_\chi {\sigma }_{i_1}\cdots \s_{i_n})=n$. Recall the definitions of $\beta _\nu $ and $F_{\beta _\nu }$, where $1\le \nu \le n$, from Eq. . Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. We characterize irreducible Verma modules (see also Lemma \[le:subfch\]). \[le:Fhweight\] Let $m\in \{0,1,\dots ,n\}$, $\chi '=r_{i_m}\cdots r_{i_2}r_{i_1}(\chi )$ and $w={\sigma }_{i_m}\cdots {\sigma }_{i_2}{\sigma }_{i_1}^\chi $. Then $$\begin{aligned} K_{\alpha }&L_{\alpha }^{-1} F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1} F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda \\ &=\frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha }))} \Lambda (K_{{\alpha }}L_{{\alpha }}^{-1}) F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1} F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda . \end{aligned}$$ for all ${\alpha }\in {\mathbb{Z}}^I$. By Eq. , the claim is equivalent to the equation $$\begin{aligned} {\zeta ^{\chi}} \Big(-\sum _{k=1}^{m-1}({b^{\chi}} (\beta _k)-1)\beta _k\Big ) (K_{\alpha }L_{\alpha }^{-1})= \frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha }))}. \label{eq:Fhweight} \end{aligned}$$ For each $k\in \{1,\dots ,n\}$ let $\chi _k=r_{i_{k-1}}\cdots r_{i_2}r_{i_1}(\chi )$, $w_k={\sigma }_{i_{k-1}}\cdots {\sigma }_{i_2}{\sigma }_{i_1}^\chi $. Since $$\begin{aligned} \frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha }))}= \prod _{k=1}^{m-1}\frac{{\rho ^{\chi _k}}(w_k({\alpha }))}{ {\rho ^{\chi _{k+1}}}(w_{k+1}({\alpha }))}, \label{eq:rhochiprod} \end{aligned}$$ Lemma \[le:rho\] gives that $$\begin{aligned} \frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha }))}= \prod _{k=1}^{m-1} \chi _k({\alpha }_{i_k},w_k({\alpha })) ^{1-\bfun{\chi _k}({\alpha }_{i_k})} \prod _{k=1}^{m-1} \chi _k (w_k({\alpha }),{\alpha }_{i_k}) ^{1-\bfun{\chi _k}({\alpha }_{i_k})}. \end{aligned}$$ Using Eqs.  and , this implies that $$\begin{aligned} \frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha })))}= \prod _{k=1}^{m-1} \chi (\beta _k,{\alpha })^{1-{b^{\chi}} (\beta _k)} \chi ({\alpha },\beta _k)^{1-{b^{\chi}} (\beta _k)}. \end{aligned}$$ Thus Eq.  holds, and the lemma is proven. Let ${\alpha }\in {\mathbb{Z}}^I$ and $\Lambda '={t}_{i_m}\cdots {t}_{i_2}\VT _{i_1}^\chi (\Lambda )$. Then $$\begin{aligned} &{\rho ^{\chi '}}(w({\alpha }))K_{\alpha }L_{\alpha }^{-1} F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1} F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda \\ &={\hat{T}}_{i_1}\big( {\rho ^{\chi '}}(w({\alpha }))K_{{\sigma }_{i_1}^{\chi }({\alpha })} L_{{\sigma }_{i_1}^{\chi }({\alpha })}^{-1} {T}_{i_1}^-(F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1})v_{{t}_{i_1}^\chi (\Lambda )} \big)\\ &={\hat{T}}_{i_1}{\hat{T}}_{i_2}\big( {\rho ^{\chi '}}(w({\alpha }))K_{{\sigma }_{i_2}{\sigma }_{i_1}^{\chi }({\alpha })} L_{{\sigma }_{i_2}{\sigma }_{i_1}^{\chi }({\alpha })}^{-1} \times \\ &\qquad \qquad {T}_{i_2}^-{T}_{i_1}^-(F_{\beta _m}^{{b^{\chi}} (\beta _m)-1} \cdots F_{\beta _3}^{{b^{\chi}} (\beta _3)-1}) v_{{t}_{i_2}{t}_{i_1}^\chi (\Lambda )} \big) = \dots\\ &={\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_m}\big( {\rho ^{\chi '}}(w({\alpha }))K_{w({\alpha })}L_{w({\alpha })}^{-1} v_{\Lambda '} \big)\\ &={\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_m}\big( {\rho ^{\chi '}}(w({\alpha })) \, \Lambda '(K_{w({\alpha })}L_{w({\alpha })}^{-1}) v_{\Lambda '} \big). \end{aligned}$$ Similarly, one obtains that $$\begin{aligned} {\rho ^{\chi }}({\alpha })\,\Lambda (K_{\alpha }L_{\alpha }^{-1}) F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1} F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda \qquad &\\ = {\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_m}\big( {\rho ^{\chi }}({\alpha }) \, \Lambda (K_{{\alpha }}L_{{\alpha }}^{-1}) v_{\Lambda '} \big).& \end{aligned}$$ Now the claim of the lemma follows from Lemma \[le:VTinv\]. \[le:TEFF\] Let $\nu \in \{0,1,\dots ,n\}$. For all $j\in I$, $$\begin{aligned} T_{i_1}\cdots T_{i_\nu }(E_j^{r_{i_\nu }\cdots r_{i_2}r_{i_1}(\chi )}) F_{\beta _\nu }^{{b^{\chi}} (\beta _\nu )-1}\cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1} F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda =0 \label{eq:TEFF} \end{aligned}$$ in $M^\chi (\Lambda )$. If $0\le m\le n-1$, then in $M^\chi (\Lambda )$ $$\begin{aligned} F_{\beta _{m+n+1}} F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1} F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda =0. \end{aligned}$$ We proceed by induction on $m$. For $m=0$ the claim is that $E_jv_\Lambda =0$ for all $j\in I$, which is clear from the definition of $M^\chi (\Lambda )$. Assume now that $m\ge 1$ and that the claim holds for all smaller values of $m$. Let $j\in I$. Suppose first that $j=i_m$. Then $$T_{i_1}\cdots T_{i_m}(E_j)= T_{i_1}\cdots T_{i_{m-1}}(F_{i_m}L_{i_m}^{-1}) =F_{\beta _m}L_{\beta _m}^{-1}.$$ Hence Eq.  follows from Lemma \[le:rvrel\]. Suppose now that $j\not=i_m$. Let $\chi '=r_{i_{m-1}}\cdots r_{i_2}r_{i_1}(\chi )$. Then $$\begin{aligned} \label{eq:TEFF1} T_{i_1}\cdots T_{i_m}(E_j) F_{\beta _m}^{{b^{\chi}} (\beta _m)-1} =T_{i_1}\cdots T_{i_{m-1}}(E^+_{j,a(i_m)} F_{i_m}^{\bfun{\chi '}({\alpha }_{i_m})-1}) \end{aligned}$$ where $a=-c^{\chi '}_{i_m j}$. Let $Z=\oplus _{k=0}^a\fie E^+_{j,k(i_m)}\subset U^+(\chi ')$. By [@p-Heck07b Cor.5.4], $$E^+_{j,k(i_m)}F_{i_m}-F_{i_m}E^+_{j,k(i_m)}\in \fie L_{i_m}E^+_{j,k-1}\subset Z \quad \text{for all $k\in {\mathbb{N}}_0$.}$$ Hence $ZF_{i_m}^{\bfun{\chi '}({\alpha }_{i_m})-1}\subset U(\chi ')Z$. Therefore $$\begin{aligned} T_{i_1}\cdots T_{i_m}(E_j) F_{\beta _m}^{{b^{\chi}} (\beta _m)-1} &\in T_{i_1}\cdots T_{i_{m-1}}(U(\chi ')Z)\\ &\subset \sum _{j'\in I}U(\chi )T_{i_1}\cdots T_{i_{m-1}}(E_{j'}). \end{aligned}$$ by Eq.  and since $Z\subset \sum _{j'\in I}U(\chi ')E_{j'}$. This and induction hypothesis imply Eq. . The last claim of the lemma follows from inserting $j=i_{m+1}$ in Eq.  and using Eq. . \[le:MLbasis\] Let $m\in \{0,1,\dots ,n\}$ and $\tau $ a permutation of the set $\{1,2,\dots ,n\}$. For all $k\in \{1,2,\dots ,m\}$ let $\chi _k=r_{i_{k-1}}\cdots r_{i_2}r_{i_1}(\chi )$. Assume that $$\begin{aligned} \label{eq:MLass} \prod _{k=1}^m \prod _{t=1}^{{b^{\chi}} (\beta _k)-1} \big( {\rho ^{\chi}} (\beta _k)\Lambda (K_{\beta _k}L_{\beta _k}^{-1}) -{\rho ^{\chi _k}}({\alpha }_{i_k})^t\big)\not=0. \end{aligned}$$ Then the set $$\label{eq:MLbasis} \begin{aligned} \{ F_{\beta _{m+\tau (1)}}^{l_{m+\tau (1)}} F_{\beta _{m+\tau (2)}}^{l_{m+\tau (2)}}\cdots F_{\beta _{m+\tau (n)}}^{l_{m+\tau (n)}} F_{\beta _m}^{{b^{\chi}} (\beta _m)-1} \cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1} F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda \,|&\\ 0\le l_k<{b^{\chi}} (\beta _k)\quad \text{for all $k\in \{m+1,m+2,\dots ,m+n\}$}& \end{aligned}$$ forms a vector space basis of $M^\chi (\Lambda )$. For all $k\in \{1,2,\dots ,m\}$ let $\Lambda _k={t}_{i_{k-1}}\cdots {t}_{i_2}{t}_{i_1}^\chi (\Lambda )$. By Lemma \[le:VTinv\] and Eq. , Eq.  is equivalent to $${\rho ^{\chi _k}} ({\alpha }_{i_k})\Lambda _k(K_{i_k}L_{i_k}^{-1})\not= \rhomap{\chi _k}({\alpha }_{i_k},{\alpha }_{i_k})^t$$ for all $k\in \{1,2,\dots ,m\}$, $t\in \{1,2,\dots ,{b^{\chi _k}}({\alpha }_{i_k})-1\}$. Hence, by Prop. \[pr:VTMiso\], the map $${\hat{T}}_{i_m}\cdots {\hat{T}}_{i_2}{\hat{T}}_{i_1}: M^{\chi _m}(\Lambda _m)\to M^\chi (\Lambda )$$ is an isomorphism. Thus the claim of the lemma holds by Lemma \[le:MLiso\] for $M^{\chi _m}(\Lambda _m)$ and by Thm. \[th:PBWtau\]. \[pr:M=L\] Assume that $$\begin{aligned} \label{eq:MLass} \prod _{\nu =1}^n \prod _{t=1}^{{b^{\chi}} (\beta _\nu )-1} \big( {\rho ^{\chi}} (\beta _\nu )\Lambda (K_{\beta _\nu } L_{\beta _\nu }^{-1})-\chi (\beta _\nu ,\beta _\nu )^t\big)\not=0. \end{aligned}$$ Then $I^\chi (\Lambda )=0$. For all $\nu \in \{1,2,\dots ,n\}$ let $\chi _\nu =r_{i_{\nu -1}}\cdots r_{i_2}r_{i_1}(\chi )$ and $\Lambda _\nu ={t}_{i_{\nu -1}}\cdots {t}_{i_2} {t}_{i_1}^\chi (\Lambda )$. By Lemma \[le:VTinv\] and Eq. , Eq.  is equivalent to $${\rho ^{\chi _\nu }} ({\alpha }_{i_\nu })\Lambda _\nu (K_{i_\nu }L_{i_\nu }^{-1})\not= \chi _\nu ({\alpha }_{i_\nu },{\alpha }_{i_\nu })^t$$ for all $\nu \in \{1,2,\dots ,n\}$, $t\in \{1,2,\dots ,{b^{\chi _\nu }}({\alpha }_{i_\nu })-1\}$. Hence, by Prop. \[pr:VTMiso\], the map $${\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_n}: M^{r_{i_n}(\chi _n)} ({t}_{i_n}(\Lambda _n))\to M^\chi (\Lambda )$$ is an isomorphism. Thus $v=F_{\beta _n}^{{b^{\chi}} (\beta _n)-1} \cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1} F_{\beta _1}^{{b^{\chi}} (\beta _1)-1} v_\Lambda \not=0$ and $(U^+(\chi ){\otimes }_{\Bbbk }{\mathbb{K}})v=M^\chi (\Lambda )$. Since $v$ is contained in any nonzero $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-submodule of $M^\chi (\Lambda )$ by Thms. \[th:EErel\], \[th:PBWtau\], it follows that $I^\chi (\Lambda )=0$. \[le:FFv\] Let $t\in \{1,2,\dots ,{b}-1\}$. Assume that $\Lambda (K_{i_1}L_{i_1}^{-1})=\chi ({\alpha }_{i_1},{\alpha }_{i_1})^{t-1}$. Then in $M^\chi (\Lambda )$ $$\begin{aligned} F_{\beta _{n+\nu -\mu }} F_{\beta _{n+1-\mu }}^{{b^{\chi}} (\beta _{n+1-\mu })-1} F_{\beta _{n+2-\mu }}^{{b^{\chi}} (\beta _{n+2-\mu })-1}\cdots F_{\beta _n}^{{b^{\chi}} (\beta _n)-1} F_{i_1}^t v_\Lambda =0 \end{aligned}$$ for all $\mu \in \{0,1,\dots ,n-1\}$ and $\nu \in \{\mu +1,\mu +2,\dots ,n\}$. By Eq.  and Lemma \[le:hwvector\] it suffices to prove that $$F_{\beta _{n+\nu -\mu }} F_{\beta _{n+1-\mu }}^{{b^{\chi}} (\beta _{n+1-\mu })-1} F_{\beta _{n+2-\mu }}^{{b^{\chi}} (\beta _{n+2-\mu })-1}\cdots F_{\beta _n}^{{b^{\chi}} (\beta _n)-1} \in \sum _{\kappa =1}^{\nu -\mu } U(\chi )F_{\beta _{n+\kappa }}$$ for all $\mu \in \{0,1,\dots ,n-1\}$ and $\nu \in \{\mu +1,\mu +2,\dots ,n\}$. By replacing $\chi $ with an appropriate element in ${\mathcal{G}}(\chi )$ and applying isomorphisms ${T}_i^-$, it suffices to show that $$F_{\beta _\nu } F_{\beta _1}^{{b^{\chi}} (\beta _1)-1} F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}\cdots F_{\beta _\mu }^{{b^{\chi}} (\beta _\mu )-1} \in \sum _{\kappa =1}^{\nu -\mu } U^-(\chi )F_{\beta _{\mu +\kappa }}$$ for all $\mu \in \{0,1,\dots ,n-1\}$ and $\nu \in \{\mu +1,\mu +2,\dots ,n\}$. The latter follows from Thms. \[th:EErel\] and \[th:PBW\] by considering ${\mathbb{Z}}^I$-degrees. \[th:hwv\] Assume that $\chi \in {\mathcal{X}}_4$. Let $\mu \in \{1,2,\dots ,n\}$, $t\in \{1,2, \dots ,{b^{\chi}} (\beta _\mu )-1\}$, and $\chi _\mu =r_{i_{\mu -1}}\cdots r_{i_2}r_{i_1}(\chi )$. Assume that $$\begin{aligned} {\rho ^{\chi}} (\beta _\mu )\Lambda (K_{\beta _\mu }L_{\beta _\mu }^{-1}) ={\rho ^{\chi _\mu }}({\alpha }_{i_\mu })^t. \label{eq:maid1} \end{aligned}$$ Then $$\begin{aligned} E_i F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1} F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} F_{\beta _\mu }^t \times \qquad & \\ F_{\beta _{\mu -1}}^{{b^{\chi}} (\beta _{\mu -1})-1}\cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1} F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda =0& \end{aligned}$$ for all $i\in I$. By Eq.  it suffices to prove that $$\begin{aligned} F_{\beta _{n+\nu }} F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1} F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} F_{\beta _\mu }^t \times \qquad & \\ F_{\beta _{\mu -1}}^{{b^{\chi}} (\beta _{\mu -1})-1}\cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1} F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda =0& \end{aligned} \label{eq:hwv}$$ for all $\nu \in \{1,2,\dots ,n\}$. Let first $\nu \in \{1,2,\dots ,\mu -1\}$. Then $$F_{\beta _{n+\nu }} F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1} F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} =0$$ by Thms. \[th:PBW\] and \[th:EErel\] by considering the ${\mathbb{Z}}^I$-degree. Thus Eq.  holds. Let now $\nu \in \{\mu ,\mu +1,\dots ,n\}$, $\Lambda _\mu ={t}_{i_{\mu -1}}\cdots {t}_{i_2}{t}_{i_1}^\chi (\Lambda )$, and $\beta '_\kappa =1_{\chi _\mu }{\sigma }_{i_\mu }{\sigma }_{i_{\mu +1}}\cdots {\sigma }_{i_{\mu +\kappa -2}}({\alpha }_{i_{\mu +\kappa -1}})$ for all $\kappa \in \{1,2,\dots ,2n\}$. Then by Lemma \[le:VTinv\], Eq.  is equivalent to $${\rho ^{\chi _\mu }}({\alpha }_{i_\mu })\Lambda _\mu (K_{i_\mu }L_{i_\mu }^{-1}) ={\rho ^{\chi _\mu }}({\alpha }_{i_\mu })^t,$$ and ${\alpha }_{i_\mu }=\beta '_1$. Thus we can apply Lemma \[le:FFv\] (with $\Lambda _\mu $ instead of $\Lambda $, $\chi _\mu $ instead of $\chi $, $\beta '_\kappa $ instead of $\beta _\kappa $, and $\mu -1$ instead of $\mu $). We obtain that $$\begin{aligned} F_{\beta '_{n+\nu +1-\mu }} F_{\beta '_{n+2-\mu }}^{{b^{\chi}} (\beta '_{n+2-\mu })-1} F_{\beta '_{n+3-\mu }}^{{b^{\chi}} (\beta '_{n+3-\mu })-1}\cdots F_{\beta '_n}^{{b^{\chi}} (\beta '_n)-1} F_{i_\mu }^t v_{\Lambda _\mu }=0 \end{aligned}$$ for all $\mu \in \{1,2,\dots ,n\}$ and $\nu \in \{\mu ,\mu +1,\dots ,n\}$. Apply on both sides of this equation the map ${\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_{\mu -1}}: M^{\chi _\mu }(\Lambda _\mu )\to M^\chi (\Lambda )$ to obtain Eq. . The Shapovalov form {#sec:shapdet} =================== We discuss the analog of the Shapovalov form for the algebras $U(\chi )$ following the construction in [@b-Joseph 3.4.10]. Let $\chi \in {\mathcal{X}}$. By Prop. \[pr:tridec\], there exists a decomposition $${\mathcal{U}}(\chi )=\Big(\sum _{i\in I}F_i{\mathcal{U}}(\chi )+\sum _{i\in I}{\mathcal{U}}(\chi )E_i \Big) \oplus \Uz$$ and hence a unique projection $${\theta ^{\chi}} :{\mathcal{U}}(\chi )\to \Uz$$ with kernel $\sum _{i\in I}F_i{\mathcal{U}}(\chi )+\sum _{i\in I}{\mathcal{U}}(\chi )E_i$. This map is commonly known as the *Harish-Chandra map*. By definition, ${\theta ^{\chi}} $ satisfies the property $$\begin{aligned} {\theta ^{\chi}} (u_-uu_+)={\varepsilon }(u_-){\theta ^{\chi}} (u){\varepsilon }(u_+) \label{eq:HCprop}\end{aligned}$$ for all $u_-\in U^-(\chi )$, $u\in U(\chi )$, $u_+\in U^+(\chi )$. Since ${\Omega }(u)=u$ for all $u\in {{\mathcal{U}}^0}$, $$\begin{aligned} {\theta ^{\chi}} ({\Omega }(u))={\theta ^{\chi}} (u)\qquad \text{for all $u\in {\mathcal{U}}(\chi )$.} \label{eq:HCprop2}\end{aligned}$$ The bilinear map $$\begin{aligned} {\mathrm{Sh}}: {\mathcal{U}}(\chi )\times {\mathcal{U}}(\chi )\to {{\mathcal{U}}^0},\qquad {\mathrm{Sh}}(u,v)={\theta ^{\chi}} ({\Omega }(u)v), \label{eq:Shf}\end{aligned}$$ is called the *Shapovalov form*. By Eq.  and since ${\Omega }^2={\operatorname{id}}$, $$\begin{aligned} {\mathrm{Sh}}(u,v)={\mathrm{Sh}}(v,u) \quad \text{for all $u,v\in {\mathcal{U}}(\chi )$.} \label{eq:Shfprop}\end{aligned}$$ Moreover, by definition of ${\mathrm{Sh}}$ and ${\theta ^{\chi}} $, $$\begin{aligned} {\mathrm{Sh}}(u,v)=0 \quad \text{if $u\in \sum _{i\in I}{\mathcal{U}}(\chi )E_i$ or $v\in \sum _{i\in I}{\mathcal{U}}(\chi )E_i$.} \label{eq:Shfprop2}\end{aligned}$$ Recall the definitions of $U(\chi )$, ${\mathcal{I}}^+(\chi )$ and ${\mathcal{I}}^-(\chi )$ from Sect. \[sec:DD\]. Since ${\mathcal{U}}(\chi ){\mathcal{I}}^+(\chi ){\mathcal{U}}(\chi )+ {\mathcal{U}}(\chi ){\mathcal{I}}^-(\chi ){\mathcal{U}}(\chi )\subset \ker {\theta ^{\chi}} $, ${\theta ^{\chi}} $ and ${\mathrm{Sh}}$ induce maps $${\theta ^{\chi}} : U(\chi )\to {{\mathcal{U}}^0},\qquad {\mathrm{Sh}}: U(\chi )\times U(\chi )\to {{\mathcal{U}}^0}.$$ The map ${\theta ^{\chi}} $ is ${\mathbb{Z}}^I$-homogeneous, that is, ${\theta ^{\chi}} (u)=0$ for all $u\in U(\chi )_{\alpha }$, where ${\alpha }\in {\mathbb{Z}}^I\setminus \{0\}$. The map ${\Omega }$ reverses degrees, that is, ${\Omega }(u)\in U(\chi)_{-{\alpha }}$ for all $u\in U(\chi )_{\alpha }$, where ${\alpha }\in {\mathbb{Z}}^I$. Therefore, for all ${\alpha },\beta \in {\mathbb{Z}}^I$, where $\al \not=\beta $, we get $$\begin{aligned} {\mathrm{Sh}}(u,v)=0 \quad \text{for all $u\in U(\chi )_{\alpha }$, $v\in U(\chi )_\beta $ \quad $({\alpha }\not=\beta )$.} \label{eq:Shfprop3}\end{aligned}$$ \[de:Shapdet\] The family of determinants $$\det \nolimits ^\chi _{\alpha }= \det {\mathrm{Sh}}(F'_i,F'_j)_{i,j\in \{1,\dots ,k\}}\in {{\mathcal{U}}^0}/{{\Bbbk }^\times },$$ where ${\alpha }\in {\mathbb{N}}_0^I$, $k=\dim U^-(\chi )_{-{\alpha }}$, and $\{F'_1,F'_2,\dots ,F'_k\}$ is a basis of $U^-(\chi )_{-{\alpha }}$, is called the *Shapovalov determinant* of $U(\chi )$. \[re:Shdet\] Let ${\alpha }\in {\mathbb{N}}_0^I$ and $k=\dim U^-(\chi )_{-{\alpha }}$. By the above considerations, ${\mathrm{Sh}}:U^-(\chi )_{-{\alpha }}\times U^-(\chi )_{-{\alpha }}\to {{\mathcal{U}}^0}$ is a symmetric bilinear form for all ${\alpha }\in {\mathbb{N}}_0^I$. Let $F'=\{F'_1,F'_2,\dots ,F'_k\}$ be a basis of $U^-(\chi )_{-{\alpha }}$, and let $d(F')=\det {\mathrm{Sh}}(F'_i,F'_j)_{i,j\in \{1,\dots ,k\}} \in {{\mathcal{U}}^0}$. Then $d(A'F')=(\det A')^2d(F')$ for all $A'\in \mathrm{GL}(k,{\Bbbk })$, and hence $\det ^\chi _{\alpha }=d(F')/{{\Bbbk }^\times }$ does not depend on the choice of the basis $F'$ of $U^-(\chi )_{-{\alpha }}$. \[le:Uzideal\] Let $\chi \in {\mathcal{X}}$. Let $J$ be an ideal of ${{\mathcal{U}}^0}$. Assume that $J$ is contained in the center of $U(\chi )$. Let $J_U$ be the ideal of $U(\chi )$ generated by $J$. Then $\Shf :U(\chi )\times U(\chi )\to {{\mathcal{U}}^0}$ induces a map ${\mathrm{Sh}}:U(\chi )/J_U\times U(\chi )/J_U\to {{\mathcal{U}}^0}/J$. Since $J$ is contained in the center of $U(\chi )$, triangular decomposition of $U(\chi )$ yields that $J_U=U^-(\chi )JU^+(\chi )$. This and ${\Omega }(J)=J$ imply the claim of the lemma. Let $\chi \in {\mathcal{X}}$, ${\alpha }\in {\mathbb{Z}}^I$, $E\in U^+(\chi )_{\alpha }$, and $F\in U^-(\chi )_{-{\alpha }}$. Then $${\mathrm{Sh}}(\Omega (E),F)\in \sum _{\beta ,\gamma \in {\mathbb{N}}_0^I,\, \beta +\gamma ={\alpha }} {\Bbbk }K_\beta L_\gamma$$ Moreover, the coefficients of $K_{\alpha }$ and $L_{\alpha }$ of ${\mathrm{Sh}}({\Omega }(E),F)$ are ${\eta }(E,S(F))$ and ${\eta }(E,F)$, respectively. \[le:Shfcoeffs\] If ${\alpha }\notin {\mathbb{N}}_0^I$ or ${\alpha }=0$, then $U ^+(\chi )_{\alpha }=0$ or $U ^+(\chi )_{\alpha }={\Bbbk }$. In this case the claim of the lemma holds by definition of ${\mathrm{Sh}}$. Assume now that ${\alpha }\in {\mathbb{N}}_0^I\setminus \{0\}$. Using Eqs. –, by induction on $\beta $ and $\beta '$ one can show that $$E'F'\in \sum _{\gamma _1,\gamma _2,\gamma _3,\gamma _4\in {\mathbb{N}}_0^I, \gamma _4-\gamma _1=\beta -\beta ',\gamma _2+\gamma _3+\gamma _4=\beta } U ^-(\chi )_{-\gamma _1}K_{\gamma _2}L_{\gamma _3}U ^+(\chi )_{\gamma _4}$$ for all $\beta ,\beta '\in {\mathbb{N}}_0^I$ and $E'\in U ^+(\chi )_\beta $, $F'\in U ^-(\chi )_{-\beta '}$. This implies the first claim of the lemma by letting $\beta =\beta '={\alpha }$ and $E'=E$, $F'=F$. The second claim follows from $$\begin{aligned} {\varDelta }(E)-K_{\alpha }{\otimes }E-E{\otimes }1\in &\mathop{\oplus } _{\beta ,\gamma \in {\mathbb{N}}_0^I, \beta +\gamma ={\alpha },\,\beta ,\gamma \not=0} U ^+(\chi )_\beta K_\gamma {\otimes }U ^+(\chi )_\gamma ,\\ {\varDelta }(F)-1 {\otimes }F-F{\otimes }L_{\alpha }\in &\mathop{\oplus } _{\beta ,\gamma \in {\mathbb{N}}_0^I, \beta +\gamma ={\alpha },\beta ,\gamma \not=0} U ^-(\chi )_{-\beta }{\otimes }U ^-(\chi )_{-\gamma }L_\beta , \end{aligned}$$ and from Eq.  (with $x=E$, $y=F$) and Prop. \[pr:sHpdef\](iii). Let ${\mathbb{K}}$ be a field extension of ${\Bbbk }$. The importance of the Shapovalov form arises from the fact that it induces a form on the Verma modules $M^\chi (\Lambda )$ and on their simple quotients $L^\chi (\Lambda )$, where $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Define $$\begin{aligned} \Lambda {\mathrm{Sh}}:U(\chi )\times U(\chi )\to {\mathbb{K}},\quad (u,v)\mapsto \Lambda ({\mathrm{Sh}}(u,v)). \label{eq:LSh}\end{aligned}$$ By Eq. , $$\begin{aligned} \Lambda {\mathrm{Sh}}(u_-u_0u_+,v_-v_0v_+)=& {\varepsilon }(u_+){\varepsilon }(v_+)\Lambda (u_0{\mathrm{Sh}}(u_-,v_-)v_0)\\ =&\Lambda (u_0)\Lambda (v_0){\varepsilon }(u_+){\varepsilon }(v_+) \Lambda {\mathrm{Sh}}(u_-,v_-).\end{aligned}$$ Thus, by Eq. , $\Lambda {\mathrm{Sh}}$ induces a ${\mathbb{K}}$-bilinear form on $M^\chi (\Lambda )$ by letting $$\Lambda {\mathrm{Sh}}:M(\Lambda )\times M(\Lambda )\to {\mathbb{K}},\quad (u{\otimes }1_\Lambda , v{\otimes }1_\Lambda )\mapsto \Lambda {\mathrm{Sh}}(u,v)$$ for all $u,v\in U(\chi )$. Moreover, Eq.  gives that $$\begin{aligned} \label{eq:LShf} \Lambda {\mathrm{Sh}}( u{\otimes }1_\Lambda , v) =\Lambda {\mathrm{Sh}}(1{\otimes }1_\Lambda ,{\Omega }(u)v) =0\end{aligned}$$ for all $u\in U(\chi )$ and $v\in I^\chi (\Lambda )$, since ${\Omega }(u)v \in I^\chi (\Lambda )\subset \oplus _{{\alpha }\not=0}M^\chi (\Lambda )_{\alpha }$. Thus by Eq. , $\Lambda {\mathrm{Sh}}$ induces a symmetric bilinear form on $L^\chi (\Lambda )$, also denoted by $\Lambda {\mathrm{Sh}}$. The radical of this form is a ${\mathbb{Z}}^I$-graded $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-submodule of $L^\chi (\Lambda )$, but does not contain $1{\otimes }1_\Lambda $, and hence it is zero. Thus $\Lambda {\mathrm{Sh}}$ is a nondegenerate symmetric bilinear form on $L^\chi (\Lambda )$. [*For the rest of this section let $\chi \in {\mathcal{X}}_4$, $n=|R^\chi _+|$, and $i_1,\dots ,i_n\in I$ with $\ell (1_\chi {\sigma }_{i_1}\cdots \s_{i_n})=n$.*]{} For all $\nu \in \{1,2,\dots ,n\}$ let $$\beta _\nu =1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu }),\qquad \chi _\nu =r_{i_{\nu -1}}\cdots r_{i_2}r_{i_1}(\chi ).$$ For all $\nu \in \{1,2,\dots ,n\}$, ${\alpha }\in {\mathbb{N}}_0^I$, and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$ let $$\begin{aligned} {P}^\chi ({\alpha },\beta _\nu ;t)=\Big|\Big\{(m_1,\dots ,m_n)\in {\mathbb{N}}_0^n\,\big|\, \sum _{\mu =1}^n m_\mu \beta _\mu ={\alpha },\,m_\nu \ge t,\quad &\\ m_\mu <{b^{\chi}} (\beta _\mu )\quad \text{for all $\mu \in \{1,2,\dots ,n\}$} \Big\}\Big|.& \label{eq:PF} \end{aligned}$$ We will use two important facts on the function ${P}^\chi $. \[le:P1\] For all $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu)-1\}$, $$\sum _{{\alpha }\in {\mathbb{N}}_0^I}{P}^\chi ({\alpha },\beta _\nu ;t)e^{-{\alpha }}= \frac {e^{-t\beta _\nu}-e^{-{b^{\chi}} (\beta _\nu )\beta _\nu }} {1-e^{-\beta _\nu }} \prod _{\mu \in \{1,\dots ,n\},\, \mu \not=\nu } \frac {1-e^{-{b^{\chi}} (\beta _\mu )\beta _\mu }} {1-e^{-\beta _\mu }}.$$ By Thm. \[th:PBWtau\], the two sides of the equation are two different expressions for the formal character of the subspace of $U^-(\chi ){\otimes }{\mathbb{K}}$ spanned by the elements $$\prod _{ {m_1,\dots ,m_n \atop m_\nu \ge t,\, 0\le m_\mu <{b^{\chi}} (\beta _\mu ) \,\text{for all $\mu $}}} F_{\beta _1}^{m_1}F_{\beta _2}^{m_2}\cdots F_{\beta _n}^{m_n}.$$ \[le:P2\] For all ${\alpha }\in {\mathbb{N}}_0^I$, $$\begin{aligned} {\alpha }\dim U^-(\chi )_{-{\alpha }}= \sum _{\nu =1}^n \sum _{t=1}^{{b^{\chi}} (\beta _\nu )-1} {P}^\chi ({\alpha },\beta _\nu ;t) \beta _\nu . \end{aligned}$$ By Thm. \[th:PBWtau\], for each ${\alpha }\in {\mathbb{N}}_0^I$ there is a basis of $U^-(\chi )_{-{\alpha }}$ parametrized by the set $$\Big\{(m_1,\dots ,m_n)\in {\mathbb{N}}_0^n\,\big|\, \sum _{\mu =1}^n m_\mu \beta _\mu ={\alpha },\,\, m_\mu <{b^{\chi}} (\beta _\mu )\,\, \text{for all $\mu $}\Big\}. \label{eq:dimU-}$$ Each $(m_1,\dots ,m_n)$ in this set contributes to ${P}^\chi ({\alpha },\beta _\nu ;t)$ with a summand $1$, for all $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,m_\nu \}$. Thus the claim of the lemma follows from the decomposition of the ${P}^\chi ({\alpha },\beta _\nu ;t)$ into $1+1+\cdots +1$ by reordering the summands. \[le:subfch\] Let $\nu \in \{1,2,\dots ,n\}$, $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu)-1\}$, and $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Assume that ${\rho ^{\chi}} (\beta _\nu )\Lambda (K_{\beta _\nu } L_{\beta _\nu }^{-1}) =\chi (\beta _\nu ,\beta _\nu )^t$ and $$\prod _{\mu =1}^{\nu -1} \prod _{m=1}^{{b^{\chi}} (\beta _\mu )-1} ({\rho ^{\chi}} (\beta _\mu ) \Lambda (K_{\beta _\mu }L_{\beta _\mu }^{-1})-\chi (\beta _\mu , \beta _\mu )^m)\not=0.$$ Then $M^\chi (\Lambda )$ contains a $U(\chi ){\otimes }{\mathbb{K}}$-submodule $V$ with $$\begin{aligned} {\mathrm{ch}\,V}=\sum _{{\alpha }\in {\mathbb{N}}_0^I}{P}^\chi ({\alpha },\beta _\nu ;t)e^{-{\alpha }}. \label{eq:fchV} \end{aligned}$$ In particular, $0\not=V\subset I^\chi (\Lambda )$. We proceed by induction on $\nu $. Let first $\nu =1$. By Lemma \[le:chsub\], $V=U^-(\chi )F_{i_1}^t{\otimes }{\mathbb{K}}_\Lambda $ is a $U(\chi ){\otimes }{\mathbb{K}}$-submodule of $M^\chi (\Lambda )$. Then Eq.  follows from Thm. \[th:PBWtau\]. Assume now that $\nu \in \{2,3,\dots ,n\}$ and that the lemma holds for $\nu -1$. Let $\chi _\mu =r_{i_{\mu -1}}\cdots r_{i_2}r_{i_1}(\chi )$ and $\Lambda _\mu ={t}_{i_{\mu -1}}\cdots {t}_{i_2} {t}_{i_1}^\chi (\Lambda )$ for all $\mu \in \{1,2,\dots ,\nu \}$. By Lemma \[le:VTinv\], the assumptions on $\Lambda $ are equivalent to the relations $$\prod _{\mu =1}^{\nu -1} \prod _{m=1}^{{b^{\chi _\mu }}({\alpha }_{i_\mu })-1} (\Lambda _\mu (K_{i_\mu }L_{i_\mu }^{-1}) -{\rho ^{\chi _\mu }}({\alpha }_{i_\mu })^{m-1}) \not=0$$ and $\Lambda _\nu (K_{i_\nu }L_{i_\nu }^{-1})= {\rho ^{\chi _\nu }}({\alpha }_{i_\nu })^{t-1}$. Let $$\beta '_\nu ={\sigma }_{i_1}^\chi (\beta _\nu )=1_{\chi _2}{\sigma }_{i_2}\s _{i_3}\cdots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu }).$$ By induction hypothesis there exists a $U(\chi _2)$-submodule $V'$ of $M^{\chi _2}(\Lambda _2)$ with $$\begin{aligned} \fch{V'}=\sum _{{\alpha }\in {\mathbb{N}}_0^I}{P}^{\chi _2}({\alpha },\beta '_\nu ;t)e^{-{\alpha }}. \end{aligned}$$ Moreover, $\Lambda (K_{i_1}L_{i_1}^{-1})\not=\rho ^\chi ({\alpha }_{i_1})^{m-1}$ for all $m\in \{1,2,\dots ,{b^{\chi}} ({\alpha }_{i_1})-1\}$, and hence ${\hat{T}}_{i_1}:M^{\chi _2}(\Lambda _2)\to M^\chi (\Lambda )$ is an isomorphism. Let $V={\hat{T}}_{i_1}(V')$. By Lemmata \[le:MLmap\] and \[le:Tpfch\], $V$ is a $U(\chi )$-submodule of $M^\chi (\Lambda )$ and $$\begin{aligned} {\mathrm{ch}\,V}={\dot{\sigma }}_{i_1}^{\chi _2}(\fch{V'})=\sum _{{\alpha }\in {\mathbb{N}}_0^I} {\dot{\sigma }}_{i_1}^{\chi _2}({P}^{\chi _2}({\alpha },\beta '_\nu ;t)e^{-{\alpha }}). \end{aligned}$$ Thus, by Lemma \[le:P1\], $$\begin{aligned} {\mathrm{ch}\,V}=&e^{(1-{b^{\chi}} ({\alpha }_{i_1})){\alpha }_{i_1}}{\sigma }_{i_1}^{\chi _2}\Big( \frac {e^{-t\beta '_\nu}-e^{-\bfun{\chi _2}(\beta '_\nu )\beta '_\nu }} {1-e^{-\beta '_\nu }} \prod _{\beta \in R_+^{\chi _2}\setminus \{\beta '_\nu \}} \frac {1-e^{-\bfun{\chi _2}(\beta )\beta }} {1-e^{-\beta }}\Big). \end{aligned}$$ Recall that $\beta '_\nu \not={\alpha }_{i_1}$, since $\nu >1$. Moreover, $$\begin{aligned} &e^{(1-{b^{\chi}} ({\alpha }_{i_1})){\alpha }_{i_1}}{\sigma }_{i_1}^{\chi _2}\Big( \frac {1-e^{-\bfun{\chi _2}({\alpha }_{i_1}){\alpha }_{i_1}}} {1-e^{-{\alpha }_{i_1}}}\Big)\\ &\quad =e^{(1-{b^{\chi}} ({\alpha }_{i_1})){\alpha }_{i_1}} \frac {1-e^{\bfun\chi ({\alpha }_{i_1}){\alpha }_{i_1}}} {1-e^{{\alpha }_{i_1}}} =\frac {1-e^{-\bfun\chi ({\alpha }_{i_1}){\alpha }_{i_1}}} {1-e^{-{\alpha }_{i_1}}}. \end{aligned}$$ Therefore $$\begin{aligned} {\mathrm{ch}\,V}=&\frac {e^{-t\beta _\nu}-e^{-\bfun{\chi }(\beta _\nu )\beta _\nu }} {1-e^{-\beta _\nu }} \prod _{\beta \in R_+^\chi \setminus \{\beta _\nu \}} \frac {1-e^{-\bfun{\chi }(\beta )\beta }} {1-e^{-\beta }}=\sum _{{\alpha }\in {\mathbb{N}}_0^I} {P}^\chi ({\alpha },\beta _\nu ;t) e^{-{\alpha }} \end{aligned}$$ by Lemma \[le:P1\]. This proves Eq. . Since $t>0$, ${\mathrm{ch}\,V}\not={\mathrm{ch}\,M}^\chi (\Lambda )$. By assumption on $t$, $V_{t\beta _\nu }\not=0$, and hence $V\not=0$. Since $V$ is a ${\mathbb{Z}}^I$-graded $U(\chi )$-submodule of $M^\chi (\Lambda )$, the lemma is proven. \[th:Shapdet\] Let $\chi \in {\mathcal{X}}_5$. For all ${\alpha }\in {\mathbb{N}}_0^I$, the Shapovalov determinant of $U(\chi )$ is the family $(\det ^\chi _\al )_{{\alpha }\in {\mathbb{N}}_0^I}$, where $$\begin{aligned} \label{eq:det} \det \nolimits ^\chi _{\alpha }= \prod _{\beta \in R^\chi _+} \prod _{t=1}^{{b^{\chi}} (\beta _\nu )-1} ({\rho ^{\chi}} (\beta )K_{\beta } -\chi (\beta ,\beta )^t L_{\beta }) ^{{P}^\chi ({\alpha },\beta ;t)}. \end{aligned}$$ Let ${\alpha }\in {\mathbb{N}}_0^I$, $k=\dim U^-(\chi )_{-{\alpha }}$, and let $\{F'_1,F'_2,\dots ,F'_k\}$ be a basis of $U^-(\chi )_{-{\alpha }}$. Then ${\mathrm{Sh}}(F'_i,F'_j)\in \sum _{\beta ,\gamma \in {\mathbb{N}}_0^I,\beta +\gamma ={\alpha }}{\Bbbk }K_\beta L_\gamma $ by Lemma \[le:Shfcoeffs\], and hence $$\begin{aligned} \label{eq:detsummands} \det \nolimits ^\chi _{\alpha }\in \sum _{\beta ,\gamma \in {\mathbb{N}}_0^I,\beta +\gamma =k{\alpha }} {\Bbbk }K_\beta L_\gamma . \end{aligned}$$ The polynomials $${\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu } -\chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu }={T}_{i_1}\cdots {T}_{i_{\nu -1}}({\rho ^{\chi}} (\beta _\nu )K_{i_\nu } -\chi (\beta _\nu ,\beta _\nu )^t L_{i_\nu })$$ are irreducible and pairwise distinct for all $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$. Thus by Lemma \[le:P2\] it suffices to prove that $\det ^\chi _{\alpha }\not=0$ and that the polynomials $({\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu } -\chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu }) ^{{P}^\chi ({\alpha },\beta _\nu ;t)}$ are factors of $\det ^\chi _{\alpha }$. Let ${\bar{{\Bbbk }}}$ be the algebraic closure of ${\Bbbk }$ and ${\mathbb{T}}={\mathrm{maxspec}\,}{{\mathcal{U}}^0}{\otimes }_{\Bbbk }{\bar{{\Bbbk }}}$ the algebraic torus. The points of ${\mathbb{T}}$ are just the ${{\bar{{\Bbbk }}}^\times }$-valued characters of ${{\mathcal{U}}^0}$. The equation $\det ^\chi _{\alpha }=0$ defines a closed affine subvariety ${\mathbb{T}}'_{\alpha }\subset {\mathbb{T}}$. Let $\Lambda \in {\mathbb{T}}$. By definition, $\Lambda \in {\mathbb{T}}'_{\alpha }$ if and only if $\Lambda {\mathrm{Sh}}: U^-(\chi )_{-{\alpha }}\times U^-(\chi )_{-{\alpha }}\to {\bar{{\Bbbk }}}$ is a degenerate symmetric bilinear form, that is, if $I^\chi (\Lambda )_{-{\alpha }}\not=0$. Thus, by Prop. \[pr:M=L\], ${\mathbb{T}}'_{\alpha }$ is a subset of the finite union of irreducible varieties $${\mathbb{T}}_{{\alpha },\nu ,t}={\mathrm{maxspec}\,}({{\mathcal{U}}^0}{\otimes }_{\Bbbk }{\bar{{\Bbbk }}})/ ({\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }- \chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu }),$$ where $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$. By Eq.  $$\det \nolimits ^\chi _{\alpha }= f\prod _{\nu =1}^n\prod _{t=1}^{{b^{\chi}} (\beta _\nu )-1} ({\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }- \chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu })^{N_{\nu ,t}}$$ for some $N_{\nu ,t}\in {\mathbb{N}}_0$ and an element $f\in {\Bbbk }[K_i,L_i\,|\,i\in I]$ which is invertible on ${\mathbb{T}}$. In particular, $\det ^\chi _{\alpha }\not=0$. We finish the proof of the theorem by showing that $N_{\nu ,t}\ge {P}^\chi ({\alpha },\beta _\nu ;t)$ for all $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$. The essential ingredients will be Lemmata \[le:subfch\] and \[le:detXfactor\]. Let $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$. Let $w=1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_{\nu -1}}$. Then $${{\mathcal{U}}^0}={\Bbbk }[K_{w({\alpha }_j)},K_{w({\alpha }_j)}^{-1}, L_{w({\alpha }_j)},L_{w({\alpha }_j)}^{-1}\,|\,j\in I]$$ and $w({\alpha }_{i_\nu })=\beta _\nu $. Let $$B ={\Bbbk }[L_{\beta _\nu },L_{\beta _\nu }^{-1}, K_{w({\alpha }_j)},K_{w({\alpha }_j)}^{-1}, L_{w({\alpha }_j)},L_{w({\alpha }_j)}^{-1}\,|\,j\in I \setminus \{i_\nu \}]$$ and $x={\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }- \chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu }$. Then $${{\mathcal{U}}^0}\simeq B[x,(x+ \chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu })^{-1}].$$ Let $X'=(x'_{ij})_{i,j\in \{1,2,\dots ,k\}}\in ({{\mathcal{U}}^0})^{k\times k}$ with $$x'_{ij}={\mathrm{Sh}}(F'_i,F'_j) \quad \text{for all $i,j\in \{1,2,\dots ,k\}$.}$$ Let $l\in {\mathbb{Z}}$ such that $K_{\beta _\nu }^l X'\in B[x]^{k\times k}$, and let $X=K_{\beta _\nu }^l X'$. By Lemma \[le:subfch\] and Eq.  there is a non-empty open subset of the variety of $B\simeq {{\mathcal{U}}^0}/(x)$ such that $\mathrm{rk}\,X(0)_p\le k-{P}^\chi ({\alpha },\beta _\nu ;t)$ for all $p$ in this set. By Lemma \[le:detXfactor\], $\det X=x^{{P}^\chi ({\alpha },\beta _\nu ;t)}b'$ for some $b'\in B[x]$. In particular, $x^{{P}^\chi ({\alpha },\beta _\nu ;t)}$ is a factor of $\det ^\chi _{\alpha }$, and the proof of the theorem is complete. Shapovalov determinants for bicharacters with finite root systems {#sec:shapdetgen} ================================================================= In Sect. \[sec:shapdet\] we mainly considered bicharacters $\chi \in {\mathcal{X}}_5$. Here we extend our results to all $\chi \in {\mathcal{X}}_3$ with $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi _+$. In what follows let ${\overline{{\mathcal{X}}}}$ denote the set of ${{\bar{{\Bbbk }}}^\times }$-valued bicharacters on ${\mathbb{Z}}^I$. Identify ${\overline{{\mathcal{X}}}}$ with $({{\bar{{\Bbbk }}}^\times })^{I\times I}$ via $\chi \mapsto (\chi ({\alpha }_i,{\alpha }_j) )_{i,j\in I}$ for all $\chi \in {\overline{{\mathcal{X}}}}$. For all $i\in \{1,2,3,4,5\}$ define ${\overline{{\mathcal{X}}}}_i\subset {\overline{{\mathcal{X}}}}$ in analogy to Eqs. –. Note that ${\mathcal{X}}_i={\mathcal{X}}\cap {\overline{{\mathcal{X}}}}_i$ for all $i\in \{1,2,3,4,5\}$. For all $\beta ,\beta '\in {\mathbb{Z}}^I$ let $f_{\beta ,\beta '}$ be the rational function on the affine variety ${\overline{{\mathcal{X}}}}=({{\bar{{\Bbbk }}}^\times })^{I\times I}$ such that $$f_{\beta ,\beta '}(\chi )=\chi (\beta ,\beta ') \quad \text{for all $\chi \in {\overline{{\mathcal{X}}}}$.}$$ Clearly, the functions $f_{\beta ,\beta '}$ with $\beta ,\beta '\in \{\al _i,-{\alpha }_i\,|\,i\in I\}$ generate the algebra ${\bar{{\Bbbk }}}[{\overline{{\mathcal{X}}}}]$. Recall that a subset of ${\overline{{\mathcal{X}}}}$ is locally closed, if it is the intersection of an open and a closed subset of ${\overline{{\mathcal{X}}}}$. \[pr:Vchi\] Let $\chi \in {\overline{{\mathcal{X}}}}_3$. Assume that $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi _+$. Let $\underline{n}=(n_\beta )_{\beta \in R^\chi _{+\infty }}$ with $n_\beta \in {\mathbb{N}}$ for all $\beta \in R^\chi _{+\infty }$. Then there exists an ideal $J\subsetneq {\bar{{\Bbbk }}}[{\overline{{\mathcal{X}}}}]$ generated by products of polynomials of the form $$q-\prod _{i,j\in I}f_{{\alpha }_i,{\alpha }_j}^{m_{ij}},\quad \text{$q$ is a root of $1$, $m_{ij}\in {\mathbb{Z}}$ for all $i,j\in I$,}$$ such that the set $$\begin{aligned} V^\chi _{\underline{n}}=\{\chi '\in {\overline{{\mathcal{X}}}}\,|\,& R^{\chi '}_+ =R^\chi _+ ,\, {b^{\chi '}}(\beta )={b^{\chi}} (\beta ) \text{ for all } \beta \in R^\chi _{+{\mathrm{fin}}},\\ &\chi '(\beta ,\beta )^n\not=1 \text{ for all $\beta \in R^\chi _{+\infty }$, $1\le n\le n_\beta $} \} \end{aligned} \label{eq:Vchin}$$ is an open subset of ${\mathrm{maxspec}\,}{\bar{{\Bbbk }}}[{\overline{{\mathcal{X}}}}]/J$. We use Lemma \[le:equalrs\] and Def. \[de:Cartan\] to reformulate the equation $R^\chi _+=R^{\chi '}_+$. Let $\chi '\in {\mathcal{G}}(\chi )$. Since $\chi \in {\overline{{\mathcal{X}}}}_3$, $\chi '$ is $p$-finite for all $p\in I$. Further, $\chi '({\alpha }_p,{\alpha }_p)\not=1$ since $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi _+$, see Eq. . Thus $$(\chi '({\alpha }_p,{\alpha }_p)^{-c_{pj}^{\chi '}}\chi '({\alpha }_p,{\alpha }_j)\chi '(\al _j,{\alpha }_p)-1)(\chi '({\alpha }_p,{\alpha }_p)^{1-c_{pj}^{\chi '}}-1)=0$$ for all $p,j\in I$ with $p\not=j$. Let $w\in {\mathrm{Hom}}(\chi ,\chi ')\subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))$. Identify $w$ with the corresponding element in ${\mathrm{Aut}}({\mathbb{Z}}^I)$ in the usual way. Then $\chi '=w^*\chi $ and hence $$\begin{aligned} \label{eq:chirel} (\chi (\gamma _p,\gamma _p)^{-c_{pj}^{\chi '}}\chi (\gamma _p,\gamma _j) \chi (\gamma _j,\gamma _p)-1)(\chi (\gamma _p,\gamma _p) ^{1-c_{pj}^{\chi '}}-1)=0 \end{aligned}$$ for all $p,j\in I$ with $p\not=j$, where $\gamma _p=w^{-1}({\alpha }_p)$ and $\gamma _j=w^{-1}({\alpha }_j)$. Let $$\begin{aligned} &\begin{aligned} &J'=\big( (f_{\gamma _p,\gamma _p}^{-c_{pj}^{w^*\chi }} f_{\gamma _p,\gamma _j} f_{\gamma _j,\gamma _p}-1) (f_{\gamma _p,\gamma _p}^{1-c_{pj}^{w^*\chi }}-1)\,|\\ &\quad j,p\in I,j\not=p,\, w\in {\mathrm{Hom}}(\chi ,\underline{\,\,}),\,\gamma _p =w^{-1}({\alpha }_p),\,\gamma _j=w^{-1}({\alpha }_j)\big) \end{aligned} \label{eq:J'} \end{aligned}$$ and $$\begin{aligned} J=J'+\big( f_{\beta ,\beta }^{{b^{\chi}} (\beta )}-1\,|\, \beta \in R^\chi _{+{\mathrm{fin}}} \big). \label{eq:J} \end{aligned}$$ Then, by Lemma \[le:equalrs\], Def. \[de:Cartan\], and Eq. , $V^\chi _{\underline{n}}$ is the set of points $\chi ''\in \maxspec {\bar{{\Bbbk }}}[{\overline{{\mathcal{X}}}}]/J$ such that - $f_{\beta ,\beta }^n(\chi '')\not=1$ for all $\beta \in R^\chi _{+\infty }$, $1\le n\le n_\beta $ and - $(f_{\gamma _p,\gamma _p}^m f_{\gamma _p,\gamma _j} f_{\gamma _j,\gamma _p}-1)(\chi '')\, (f_{\gamma _p,\gamma _p}^{m+1} -1)(\chi '')\not=0$ for all $j,p\in I$, $w\in {\mathrm{Hom}}(\chi ,\underline{\,\,})$, and $m\in \{0,1,\dots ,-c^{w^*\chi }_{pj}-1\}$, where $j\not=p$ and $\gamma _p=w^{-1}({\alpha }_p)$, $\gamma _j=w^{-1}({\alpha }_j)$. This is clearly an open subset, which proves the proposition. \[pr:X5dense\] Let $\chi \in {\overline{{\mathcal{X}}}}_3$. Assume that $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi _+$. Let $\underline{n}=(n_\beta )_{\beta \in R^\chi _{+\infty }}$, where $n_\beta \in {\mathbb{N}}$ for all $\beta \in R^\chi _{+\infty }$. Let $V^\chi _{\underline{n}}$ be as in Prop. \[pr:Vchi\]. Then ${\overline{{\mathcal{X}}}}_5\cap V^\chi _{\underline{n}}$ is Zariski dense in $V^\chi _{\underline{n}}$. Prop. \[pr:Vchi\] gives that $V^\chi _{\underline{n}}\subset {\overline{{\mathcal{X}}}}_3$ satisfies the conditions on $V$ in Lemma \[le:gendensity\], where $k=|I|^2$ and $\{x_i\,|\,i=1,2,\dots,k\}=\{f_{{\alpha }_i,{\alpha }_j}\,|\,i,j\in I\}$. Since ${\overline{{\mathcal{X}}}}_4$ contains all finite sets $V_{n_1,\dots ,n_k}$ in Lemma \[le:gendensity\], and ${\overline{{\mathcal{X}}}}_5\cap V^\chi _{\underline{n}}= {\overline{{\mathcal{X}}}}_4\cap V^\chi _{\underline{n}}$ by definition of $V^\chi _{\underline{n}}$, the proof is completed. Similarly to Eq.  define ${P}^\chi ({\alpha },\beta _\nu ;t)$ for all $\chi \in {\overline{{\mathcal{X}}}}_3$, ${\alpha }\in {\mathbb{N}}_0^I$, $\beta _\nu \in R^\chi _+$, and $t\in {\mathbb{N}}$ with $t<{b^{\chi}} (\beta _\nu )$ by $$\begin{aligned} {P}^\chi ({\alpha },\beta _\nu ;t)=\Big|\Big\{(m_1,\dots ,m_n)\in {\mathbb{N}}_0^n\,\big|\, \sum _{\mu =1}^n m_\mu \beta _\mu ={\alpha },\,m_\nu \ge t,\quad &\\ m_\mu <{b^{\chi}} (\beta _\mu )\quad \text{for all $\mu \in \{1,2,\dots ,n\}$} \Big\}\Big|.& \label{eq:PF2} \end{aligned}$$ \[th:Shapdet2\] Let $\chi \in {\mathcal{X}}_3$. Assume that $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi _+$. The Shapovalov determinant of $U(\chi )$ is the family $(\det ^\chi _{\alpha })_{{\alpha }\in {\mathbb{N}}_0^I}$, where $$\begin{aligned} \label{eq:det2} \det \nolimits ^\chi _{\alpha }= \prod _{\beta \in R^\chi _+} \prod _{t=1}^{{b^{\chi}} (\beta )-1} ({\rho ^{\chi}} (\beta )K_{\beta } -\chi (\beta ,\beta )^t L_{\beta }) ^{{P}^\chi ({\alpha },\beta ;t)}. \end{aligned}$$ Let ${\alpha }\in {\mathbb{N}}_0^I$. Choose a basis $\{F'_1,\dots ,F'_k\}$ of $U^-(\chi )_{-{\alpha }}$ consisting of monomials $F_{i_1}F_{i_2}\cdots F_{i_l}$, where $k,l\in {\mathbb{N}}_0$ and $i_1,\dots ,i_l\in I$. Identify $\oplus _{\beta ,\gamma \in {\mathbb{N}}_0^I,\,\beta +\gamma ={\alpha }}\Fie K_\beta L_\gamma $ with ${\bar{{\Bbbk }}}^N$ for an appropriate $N\in {\mathbb{N}}$. By the commutation relations – and the definition of ${\mathrm{Sh}}$, the map $$d:{\overline{{\mathcal{X}}}}\to {\bar{{\Bbbk }}}^N, \quad \chi '\mapsto \det ({\mathrm{Sh}}(F'_i,F'_j))_{i,j\in \{1,2,\dots ,k\}}$$ is a morphism of affine varieties. Further, $d(\chi )\not=0$ by Lemma \[le:Shfcoeffs\], the choice of $\{F'_1,\dots ,F'_k\}$, and the nondegeneracy of the pairing $\eta $, see Prop. \[pr:sHpdef\](iv). Recall the definition of $|\beta |$, $\beta \in {\mathbb{Z}}^I$, from Eq. . Restrict $d$ to the set $V^\chi _{\underline{n}}$ defined in Prop. \[pr:Vchi\], with $n_\beta =|{\alpha }|/|\beta |$ for all $\beta \in R^\chi _{+\infty }$. The set $$V'=\{\chi '\in V^\chi _{\underline{n}}\,|\,d(\chi ')\not=0 \}$$ is open in $V^\chi _{\underline{n}}$ and contains $\chi $. Thus by Prop. \[pr:X5dense\] the set $$V''=\{\chi '\in {\overline{{\mathcal{X}}}}_5\cap V^\chi _{\underline{n}}\,| \,d(\chi ')\not=0\}$$ is Zariski dense in all irreducible components of $V^\chi _{\underline{n}}$ containing $\chi $. The definition of $V^\chi _{\underline{n}}$ and the choice of $\underline{n}$ yield that $R^{\chi '}_+=R^\chi _+$ and ${b^{\chi '}}(\beta )\le {b^{\chi}} (\beta )$ for all $\chi '\in V^\chi _{\underline{n}}$ and $\beta \in R^\chi _+$. Thus $\dim U(\chi ')_{-{\alpha }}\le \dim U(\chi )_{-{\alpha }}$ for all $\chi ' \in V^\chi _{\underline{n}}$ by Eqs. , . Hence $d(\chi ')$ is a multiple of $\det ^{\chi '}_{\alpha }$ for all $\chi '\in V^\chi _{\underline{n}}$. By Thm. \[th:Shapdet\], $$\begin{aligned} \label{eq:d} d(\chi ')=a(\chi ') \prod _{\beta \in R^\chi _+} \prod _{t=1}^{{b^{\chi}} (\beta _\nu )-1} ({\rho ^{\chi}} (\beta )K_{\beta } -\chi (\beta ,\beta )^t L_{\beta }) ^{{P}^\chi ({\alpha },\beta ;t)} \end{aligned}$$ for all $\chi '\in V''$, where $a(\cdot )$ is some regular function on ${\overline{{\mathcal{X}}}}$ which does not vanish on $V''$. By the density of $V''$, Eq.  holds for all $\chi '\in V'$ in the irreducible components of $V^\chi _{\underline{n}}$ containing $\chi $, and $a(\chi ')\not=0$ for all $\chi '\in V'$ by definition of $V'$. In particular, Eq.  holds for $\chi '=\chi $. Thus the theorem is proven. Quantized enveloping algebras {#sec:Uqg} ============================= We adapt our main result to quantized enveloping algebras. Let $I$ be a finite set and let $C=(c_{ij})_{i,j\in I}$ be a symmetrizable Cartan matrix of finite type. Let ${\mathfrak{g}}$ be the associated semisimple Lie algebra and $R_+$ the set of positive roots. For all $i\in I$ let $d_i\in {\mathbb{N}}$ such that $d_ic_{ij}=d_jc_{ji}$ for all $i,j\in I$. Assume that the numbers $d_i$, where $i\in I$, are relatively prime. Identify ${\mathbb{Z}}^I$ with the root lattice by considering $\{{\alpha }_i\,|\,i\in I\}$ as the set of simple roots. Let $(\cdot ,\cdot ):{\mathbb{Z}}^I\to {\mathbb{Z}}^I$ be the (positive definite) symmetric bilinear form defined by $({\alpha }_i,{\alpha }_j)=d_ic_{ij}$. Let $\rho :{\mathbb{Z}}^I\to {\mathbb{Z}}$ be the linear form defined by $\rho ({\alpha }_i)=d_i$ for all $i\in I$. Let ${\Bbbk }$ be a field, and let $q\in {{\Bbbk }^\times }$. Assume that $q^{2m}\not=1$ for all $m\in {\mathbb{N}}$ with $m\le \max \{d_i\,|\,i\in I\}$. The quantized enveloping algebra of ${\mathfrak{g}}$ is the associative algebra $U_q({\mathfrak{g}})$ generated by the elements $E_i$, $F_i$, $K_i$, and $K_i^{-1}$, where $i\in I$, and defined by the relations $$\begin{gathered} K_iK_i^{-1}=K_i^{-1}K_i=1,\quad K_iK_j=K_jK_i,\\ K_iE_jK_i^{-1}=q^{d_ic_{ij}}E_j,\quad K_iF_jK_i^{-1}=q^{-d_ic_{ij}}F_j,\\ E_iF_j-F_jE_i=\delta_{ij}(K_i-K_i^{-1}),\\ ({\mathrm{ad}}E_i)^{1-c_{ij}}(E_j)=0,\quad ({\mathrm{ad}}F_i)^{1-c_{ij}}(F_j)=0 \quad (i\not=j)\end{gathered}$$ for all $i,j\in I$. Here ${\mathrm{ad}}$ denotes adjoint action: $$\begin{aligned} ({\mathrm{ad}}E_i)(x)=E_ix-K_ixK_i^{-1}E_i,\quad ({\mathrm{ad}}F_i)(x)=xF_i-F_iK_i^{-1}xK_i\end{aligned}$$ for all $i\in I$ and $x\in \langle E_j,F_j,K_j,K_j^{-1}\,|\,j\in I\rangle $. Traditionally, in the third line of the defining relations of $U_q({\mathfrak{g}})$ one inserts a denominator $q^{d_i}-q^{-d_i}$ on the right hand side, but this denominator can be eliminated by rescaling *e.g.* the variables $E_i$, $i\in I$. Assume first that $q$ is not a root of $1$. Then, by [@b-Lusztig93 Ch.1] and [@inp-AndrSchn02 Prop.2.10], $U_q({\mathfrak{g}})\simeq U(\chi )/(K_iL_i-1\,|\,i\in I)$, where $\chi \in {\mathcal{X}}$ with $\chi ({\alpha }_i,{\alpha }_j)=q^{d_ic_{ij}}$ for all $i,j\in I$. Assume now that $q$ is a root of $1$. Let again $\chi \in {\mathcal{X}}$ with $\chi ({\alpha }_i,{\alpha }_j)=q^{d_ic_{ij}}$ for all $i,j\in I$. Then $$U(\chi )/(K_iL_i-1,K_\beta ^{{b^{}}(\beta )}-1\,|\,i\in I, \beta \in R^\chi _+),$$ where ${b^{}}(\beta )$ is the order of $q^{(\beta ,\beta )}$ for all $\beta \in R_+$, is isomorphic to Lusztig’s small quantum group $u_q({\mathfrak{g}})$. This was observed *e.g.* in [@inp-AndrSchn02 Thm.4.3] by referring to results of Lusztig, de Concini, Procesi, Rosso, and Müller. Similarly to Eq.  and Def. \[de:Shapdet\] one defines the Shapovalov form and the Shapovalov determinant $(\det _{\alpha })_{{\alpha }\in {\mathbb{N}}_0^I}$ of $U_q({\mathfrak{g}})$ and $u_q({\mathfrak{g}})$, respectively. Alternatively, since $K_iL_i$ for $i\in I$ and $K_\beta ^{\bfun{}(\beta )}$ for $\beta \in R^\chi _+$ (the latter only if $q$ is a root of $1$) are central elements in $U(\chi )$ for all $i\in I$, the Shapovalov form can also be obtained from the definition in Sect. \[sec:shapdet\] via Lemma \[le:Uzideal\]. \[th:ShapdetUqg\] Let $I$, $C$, $(d_i)_{i\in I}$, and ${\mathfrak{g}}$ as above. Let $q\in {{\Bbbk }^\times }$. Assume that $q^{2m}\not=1$ for all $m\in {\mathbb{N}}$ with $m\le \max \{d_i\,|\,i\in I\}$. \(i) [@inp-dCK90] If $q$ is not a root of $1$, then the Shapovalov determinant of $U_q({\mathfrak{g}})$ is the family $(\det _{\alpha })_{{\alpha }\in {\mathbb{N}}_0^I}$, where $$\begin{aligned} \label{eq:detUqg} \det \nolimits _{\alpha }= \prod _{\beta \in R_+} \prod _{t=1}^\infty (q^{2\rho (\beta )}K_{\beta } -q^{t(\beta ,\beta )} K_{\beta }^{-1}) ^{{P}({\alpha },\beta ;t)}. \end{aligned}$$ \(ii) Assume that $q$ is a root of $1$. Then the Shapovalov determinant of $u_q({\mathfrak{g}})$ is the family $(\det _{\alpha })_{{\alpha }\in {\mathbb{N}}_0^I}$, where $$\begin{aligned} \label{eq:detuqg} \det \nolimits _{\alpha }= \prod _{\beta \in R_+} \prod _{t=1}^{\bfun{}(\beta )-1} (q^{2\rho (\beta )}K_{\beta } -q^{t(\beta ,\beta )} K_{\beta }^{-1}) ^{{P}({\alpha },\beta ;t)}. \end{aligned}$$ Let $\chi \in {\mathcal{X}}$ with $\chi ({\alpha }_i,{\alpha }_j)=q^{d_ic_{ij}}$ for all $i,j\in I$. Choose the ideal $J$ in Lemma \[le:Uzideal\] as explained above. Then one gets the Shapovalov determinants of $U_q({\mathfrak{g}})$ and $u_q({\mathfrak{g}})$ from the one of $U(\chi )$ in Thm. \[th:Shapdet2\]. The second part of Thm. \[th:ShapdetUqg\] was proved in [@a-KumLetz97] in the case when the order of $q$ is prime and ${\Bbbk }$ is the cyclotomic field $\mathbb{Q}[q]$. Appendix ======== For the proofs of Thms. \[th:Shapdet\] and \[th:Shapdet2\] we need some commutative algebra which is considered here. Let ${\bar{{\Bbbk }}}$ be an algebraically closed field. \[le:rankX\] Let $B$ be an integral domain, $x$ an indeterminate, $k\in {\mathbb{N}}$, and $X\in B[x]^{k\times k}$. Then there exist $s\in \{0,1,\dots ,k\}$, $D_1,D_2\in B^{k\times k}$, $D_0\in B[x]^{k\times k}$ and $b\in B\setminus \{0\}$ such that $\det D_1,\det D_2\not=0$, $$\begin{aligned} D_1XD_2=xD_0+b\,\mathrm{diag} (\underbrace{1,\dots ,1}_s,0,\dots ,0). \label{eq:rankX} \end{aligned}$$ Let ${\mathrm{Frac}}(B)$ be the field of fractions of $B$. Then there exist $s\in \{0,1,\dots ,k\}$ and $D'_1,D'_2\in {\mathrm{Frac}}(B)^{k\times k}$ such that $\det D'_1,\det D'_2\not=0$ and $$D'_1X(0)D'_2=\mathrm{diag} (\underbrace{1,\dots ,1}_s,0,\dots ,0).$$ Let $b_1,b_2\in B\setminus \{0\}$ such that $b_1D'_1,b_2D'_2\in B[x]^{k\times k}$. Let $b=b_1b_2$, $D_1=b_1D'_1$, and $D_2=b_2D'_2$. Then $$D_1X(0)D_2=b\,\mathrm{diag} (\underbrace{1,\dots ,1}_s,0,\dots ,0),$$ and hence the lemma holds for $D_0=D_1X'D_2$, where $X'\in B[x]^{k\times k}$ such that $X=X(0)+xX'$. \[le:detXfactor\] Let $B$ be a finitely generated integral domain over ${\bar{{\Bbbk }}}$, $x$ an indeterminate, $k\in {\mathbb{N}}$, $r\in \{0,1,\dots ,k\}$, and $X\in B[x]^{k\times k}$. Assume that $\mathrm{rk}\,X(0)_p\le r$ for all points $p$ in a non-empty Zariski open subset of the affine variety of $B$. Then $\det X=x^{k-r}b$ for some $b\in B[x]$. By Lemma \[le:rankX\] there exist $s\in \{0,1,\dots ,k\}$, $b\in B\setminus \{0\}$, $D_1,D_2\in B^{k\times k}$, and $D_0\in B[x]^{k\times k}$ such that $\det D_1,\det D_2\not=0$ and Eq.  holds. Let $V$ be a non-empty Zariski open subset of the affine variety of $B\simeq B[x]/(x)$ such that $(\det D_1)_p,(\det D_2)_p, b_p\not=0$ and $\mathrm{rk}\,X(0)_p\le r$ for all $p\in V$. This exists by the assumption on $r$ and since the variety of $B$ is irreducible. Then $s\le r$ by Eq.  in the points $(p,0)$ of the variety of $B[x]$, where $p\in V$. Therefore $$\det D_1\, \det X \det D_2=x^{k-r}b'$$ for some $b'\in B[x]$. Since $\det D_1,\det D_2\in B$ and $B$ is an integral domain, we conclude that $\det X\in x^{k-r}B[x]$. For all $k\in {\mathbb{N}}$ let ${\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$ denote the ring of Laurent polynomials in $k$ variables. For all $M=(m_{ij})_{i,j\in \{1,2,\dots ,k\}}\in \mathrm{GL}(k,{\mathbb{Z}})$ let $$X^{(M)} _i=\prod _{j=1}^kx_j^{m_{ij}},\quad 1\le i\le k.$$ Then the ring endomorphism of ${\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$ given by $x_i\mapsto X^{(M)}_i$ for all $i\in \{1,2,\dots ,k\}$ is an isomorphism with inverse map given by $x_i\mapsto X^{(M^{-1})}_i$ for all $i\in \{1,2,\dots ,k\}$. Let $k\in {\mathbb{N}}$. Let $J\subsetneq {\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$ be an ideal generated by elements of the form $q-\prod _{i=1}^k x_i^{m_i}$, where $m_1,\dots ,m_k\in {\mathbb{Z}}$ and $q\in {{\bar{{\Bbbk }}}^\times }$ is a root of $1$. Then $J$ is a finite intersection of ideals of the form $$\begin{aligned} ( X_1^{(M)}-q_1, X_2^{(M)}-q_2,\dots ,X_l^{(M)}-q_l), \label{eq:primeid} \end{aligned}$$ where $l\in \{0,1,\dots ,k\}$, $q_1,\dots ,q_l\in {{\bar{{\Bbbk }}}^\times }$ are roots of $1$, and $M\in \mathrm{GL}(k,{\mathbb{Z}})$. \[le:torusideal\] Proceed by induction on $k$. If $J$ is empty, then the claim is true. Assume now that $q-\prod _{i=1}^kx_i^{m_i}$ is one of the generators of $J$, where $q$ is a root of $1$ and $(m_1,\dots ,m_k)\in {\mathbb{Z}}^k\setminus \{0\}$. Let $m_0=\mathrm{gcd}(m_1,\dots ,m_k)$. Let $M'\in \mathrm{GL}(k,{\mathbb{Z}})$ such that $m'_{1i}=m_i/m_0$ for all $i\in \{1,2,\dots ,k\}$. Then $(X_1^{(M')})^{m_0}-q\in J$, and hence $J$ is the intersection of the (finite number of) ideals $J+(X^{(M')}_1-q')$, where $q'\in {\bar{{\Bbbk }}}$, ${q'}^{m_0}=q$. By assumption, $J+(X^{(M')}_1-q')$ is generated by $X^{(M')}_1-q'$ and by elements of the form $q''-\prod _{i=2}^k (X^{(M')}_i)^{m'_i}$, where $m'_2,\dots ,m'_k\in {\mathbb{Z}}$ and $q''\in {{\bar{{\Bbbk }}}^\times }$ is a root of $1$. Then the claim follows by the induction hypothesis. The ideals in Eq.  are prime ideals of ${\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$, since the quotient ring ${\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]/J \simeq {\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k-l]$ is an integral domain. Let $k\in {\mathbb{N}}$. Let $J\subsetneq {\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$ be an ideal generated by polynomials of the form $$\big(q_1-\prod _{i=1}^k x_i^{m_{1i}}\big) \big(q_2-\prod _{i=1}^k x_i^{m_{2i}}\big)\cdots \big(q_l-\prod _{i=1}^k x_i^{m_{li}}\big),$$ where $l\in {\mathbb{N}}$, $m_{j1},\dots ,m_{jk}\in {\mathbb{Z}}$ and $q_j\in {{\bar{{\Bbbk }}}^\times }$ is a root of $1$ for all $j\in \{1,2,\dots ,l\}$. Let $V\subset {\mathrm{maxspec}\,}{\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]/J$ be an open subset. Then the union of the subsets $$V_{n_1,\dots ,n_k}=\{p\in V\,|\,p_1^{n_1}=1,\dots ,p_k^{n_k}=1\}, \quad n_1,\dots ,n_k\in {\mathbb{N}},$$ is dense in $V$ with respect to the Zariski topology. \[le:gendensity\] We can assume that $V={\mathrm{maxspec}\,}{\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]/J$. Moreover, it suffices to prove the lemma for the irreducible components of $V$. Thus, as a first reduction, $J$ can be assumed to be as in the assumptions of Lemma \[le:torusideal\]. 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--- abstract: 'The 21cm line brightness temperature brings rich information about Epoch of Reionizaton (EoR) and high-$z$ universe (Cosmic Dawn and Dark Age). While the power spectrum is a useful tool to investigate the EoR signal statistically, higher-order statistics such as bispectrum are also valuable because the EoR signal is expected to be highly non-Gaussian. In this paper, we develop a formalism to calculate the bispectrum contributed from the thermal noise taking array configularion of telescopes into account, by extending a formalism for the power spectrum. We apply our formalism to the ongoing and future telescopes such as expanded Murchison Widefield Array (MWA), LOw Frequency ARray (LOFAR) , Hydrogen Epoch of Reionization Array (HERA) and Square Kilometre Array (SKA). We find that expanded MWA does not have enough sensitivity to detect the bispectrum signal. On the other hand, LOFAR has better sensitivity and will be able to detect the peaks of the bispectrum as a function of redshift at large scales with comoving wavenumber $k \lesssim 0.03~{\rm Mpc}^{-1}$. The SKA has enough sensitivity to detect the bispectrum for much smaller scales $k \lesssim 0.3~{\rm Mpc}^{-1}$ and redshift $z \lesssim 20$' author: - 'Shintaro Yoshiura$^1$, Hayato Shimabukuro$^{1,2}$, Keitaro Takahashi$^1$, Rieko Momose$^3$, Hiroyuki Nakanishi$^4$, and Hiroshi Imai$^4$' title: Sensitivity to 21cm Bispectrum from Epoch of Reionization --- Introduction ============ The redshifted 21cm line emission from neutral hydrogens is a promising way to probe Epoch of Reionization (EoR), Cosmic Dawn and Dark Age [@2006PhR...433..181F; @2012RPPh...75h6901P] because it reflects the physical state of intergalactic gas. Actually, the brightness temperature depends on quantities crucial for the understanding of these epochs, such the neutral hydrogen fraction, spin temperature and baryon density. However, the observation of the redshifted 21cm signal is very challenging due to the presence Galactic and extragalactic foreground emissions, Low-frequency radio telescopes such as Murchison Widefield Array (MWA) [@2013MNRAS.429L...5B], LOw Frequency ARray (LOFAR) [@2013MNRAS.435..460J] and PAPER [@2014ApJ...788..106P] have started their observations and set upper bounds on the brightness temperature. The upper bounds will improve further as our understanding of the foreground proceeds and the subtraction techniques become more sophisticated. Ultimately, the Square Kilometre Array (SKA) [@SKA; @2013ExA....36..235M] and Hydrogen Epoch of Reionization Array (HERA) [@2014ApJ...782...66P] will perform precise observations and will reveal the physical process of EoR and Cosmic Dawn. One of the useful tools to extract information from observed data is to take the power spectrum of fluctuations in brightness temperature at a fixed redshift (frequency). This is effective even for relatively low S/N data, which could be obtained by ongoing telescopes, while making a map of brightness temperature through imaging requires much higher sensitivity the SKA is expected to have. Actually, the power spectrum of brightness temperature has been studied by many authors . When fluctuations follow Gaussian probability distribution, they can be well characterized by the power spectrum and higher-order statistics such as bispectrum and trispectrum have no further independent information. However, since reionization is a highly non-Gaussian process which involves non-linear density fluctuations, star formation and expansion of HII bubbles, the brightness temperature fluctuations are also expected to be strongly non-Gaussian [@shi-P]. In this case, the power spectrum does not have sufficient information to describe the fluctuations and higher-order statistics have independent and complimentary information [@2005MNRAS.363.1049C; @2007ApJ...662....1P]. In this paper, we develop a formalism to calculate the errors in bispectrum measurement contributed from thermal noise. Noise estimation has been studied by many authors in case of power spectrum [@2004ApJ...615....7M; @2005ApJ...619..678M; @2006ApJ...653..815M], and we extend the formalism given in [@2006ApJ...653..815M]. Starting from the error in visibility obtained by a single baseline, we consider its summation over the baseline distribution in $uv$ plane. A striking feature of thermal-noise bispectrum is that its ensemble average vanishes because thermal noise is Gaussian. Nevertheless, thermal noise contributes to the bispectrum error through its variance. Considering the variance of thermal noise error is the main extention to the previous formalism. The structure of this paper is the following. In section 2, we define the brightness temperature, it’s power spectrum and bispectrum. In section 3, we review the formalism of calculation of thermal-noise power spectrum given by [@2006ApJ...653..815M]. Then, we develop a formalism for bispectrum and estimate thermal-noise bispectrum for several specific configuration of the wave number in section 4. The summary and discussion will be given in section 5. Throughout this paper, we assume $\Lambda$CDM cosmology with $(\Omega_{\rm m},\Omega_\Lambda,\Omega_{\rm b},H_0) = (0.27,0.73,0.046,70~{\rm km/s/Mpc})$ [@2011ApJS..192...18K]. 21cm line signal ================ In this section, we define basic quantities concerning the 21cm signal. The brightness temperature $\delta T_b$ is defined by spin temperature offsetting from CMB temperature, $$\begin{aligned} \delta T_b(z) &=& \frac{T_s - T_\gamma}{1+z}(1 - e^{-\tau_{\nu_{0}}}) \nonumber\\ &\approx& 27 x_{\rm HI} (1 + \delta_m) \bigg(\frac{H}{dv_r/dr + H} \bigg) \bigg(1 - \frac{T_\gamma}{T_{\rm S}}\bigg) \bigg(\frac{1+z}{10} \frac{0.15}{\Omega_m h^2}\bigg)^{1/2} \bigg(\frac{\Omega_b h^2}{0.023} \bigg) ~ [{\rm mK}], \label{eq:brightness}\end{aligned}$$ where $x_{HI}$ is the neutral fraction of hydrogen, $\delta_m$ is the matter over density, $H$ is the Hubble parameter and $dv_r/dr$ is the velocity gradient along the line of sight. Then we introduce fluctuation of $\delta T_b({\bf x})$, $$\delta_{21}({\bf x}) = \delta T_b({\bf x}) - \delta \bar{T_b}, \label{delta_T_b}$$ where $\delta \bar{T_b}$ is the average value of brightness temperature, ${\bf{x}}$ is spatial position. The power spectrum of brightness temperature is defined from its Fourier transform, $\delta_{21}({{\boldsymbol{k}}})$, as, $$\langle \delta_{21}({\bold k_1}) \delta_{21}({\bold k_2}) \rangle = \delta({\bold k_1} + {\bold k_2}) P_{21}(k_1), \label{eq:ps_def}$$ where $\langle \rangle$ represents the ensemble average, ${\bf{k}}$ is position in Fourier space. The bispectrum $B_{21}$ can be defined in a similar way: $$\langle \delta_{21}({\bold k_1}) \delta_{21}({\bold k_2}) \delta_{21}({\bold k_3}) \rangle = \delta({\bold k_1}+{\bold k_2}+{\bold k_3}) B_{21}({\bold k_1},{\bold k_2}). \label{eq:bs_def}$$ Here the delta function forces the three wave vectors to make a triangle and $B_{21}$ is dependent on only two of the three vectors (chosen $\bold k_1$ and $\bold k_2$ here) due to this triangle condition. power spectrum sensitivity ========================== In this section, we summarize a formalism to estimate the thermal noise for power spectrum, following [@2006ApJ...653..815M]. First, we define visibility $V(u,v,\nu)$ for a pair of antennae as, $$V(u,v,\nu) = \int{d{{{\hat{\boldsymbol{n}}}}} \, \, T_N({{\hat{\boldsymbol{n}}}},\nu) \, W({{\hat{\boldsymbol{n}}}},\nu) e^{2 \pi {\it i}{u \choose v} \cdot {{\hat{\boldsymbol{n}}}}}}, \label{viseqn}$$ where $T_N$ is the thermal-noise temperature, ${{\hat{\boldsymbol{n}}}}$ is the direction of primely beam, $\nu$ is observed frequency and $W({{\hat{\boldsymbol{n}}}},\nu)$ is a product of the window functions concering the field of view and bandwidth. The rms thermal-noise fluctuation per visibility is given by, $$\begin{aligned} V_N = \frac{\lambda^2 T_{\rm sys}}{A_e \sqrt{\Delta \nu t_0}} ~ [\rm K], \label{visnoise}\end{aligned}$$ where $\lambda$ is the observed wavelength, $T_{\rm sys}$ is the total system temperature, $A_e$ is the effective area of antenna, $\Delta \nu$ is the width of the frequency channel and $t_0$ is total observing time. By Fourier transforming the visibility in the frequency direction, we obtain, $$\begin{aligned} \tilde{I}(u,v,\eta) &=& \int d\nu V_N(u,v,\nu) \exp(2\pi i \nu \eta) \nonumber\\ &=& \sum^{B/\Delta \nu}_{i=1} V_N(u,v,\nu_i) \exp(2 \pi i \nu_i \eta) \Delta\nu ~ [\rm K \cdot Hz],\end{aligned}$$ where $B (\gg \Delta \nu)$ is the bandwidth, $\nu_i$ is the $i$-th frequency channel and we define ${{\boldsymbol{u}}}= (u,v,\eta)$. The covariance matrix of detector noise for a single baseline is given by, $$\begin{aligned} C_N({{\boldsymbol{u}}}_i,{{\boldsymbol{u}}}_j) &=& \langle \tilde{I}_N({{\boldsymbol{u}}}_i) \tilde{I}^{*}_N({{\boldsymbol{u}}}_j) \rangle \nonumber \\ &=& \int d{{\boldsymbol{u}}}' \int d{{\boldsymbol{u}}}'' \langle \tilde{T}_N({{\boldsymbol{u}}}') \tilde{T}_N({{\boldsymbol{u}}}'') \rangle \tilde{W}({{\boldsymbol{u}}}_i - {{\boldsymbol{u}}}') \tilde{W}({{\boldsymbol{u}}}_j - {{\boldsymbol{u}}}'') \nonumber\\ &=& \int d{{\boldsymbol{u}}}' \int d{{\boldsymbol{u}}}'' P_N({{\boldsymbol{u}}}') \delta^3_D({{\boldsymbol{u}}}' - {{\boldsymbol{u}}}'') \tilde{W}({{\boldsymbol{u}}}_i - {{\boldsymbol{u}}}') \tilde{W}({{\boldsymbol{u}}}_j - {{\boldsymbol{u}}}'') \nonumber\\ &=& \int d{{\boldsymbol{u}}}' \int d{{\boldsymbol{u}}}'' P_N({{\boldsymbol{u}}}') \tilde{W}({{\boldsymbol{u}}}_i - {{\boldsymbol{u}}}') \tilde{W}({{\boldsymbol{u}}}_j - {{\boldsymbol{u}}}') \nonumber\\ &\approx& \delta_{ij} P_N({{\boldsymbol{u}}}_i) \int d^3{{\boldsymbol{u}}}' \, |\tilde{W}({{\boldsymbol{u}}}_i - {{\boldsymbol{u}}}')|^2,\end{aligned}$$ where, we used the definition of power spectrum for noise temperature (Eq. \[eq:ps\_def\]) in third equality and assumed that the covariance vanishes when ${{\boldsymbol{u}}}_i \neq {{\boldsymbol{u}}}_j$ in the last equality. Further, we assumed that the power spectrum is constant for a range where the window function have non-zero value and we pulled $P_N$ out of the integration. Then the integration of window functions can be evaluated as follows: $$\begin{aligned} \int d^3{{\boldsymbol{u}}}' \, |\tilde{W}({{\boldsymbol{u}}}- {{\boldsymbol{u}}}')|^2 &=& \int d^3{{\boldsymbol{u}}}' \int d^3{\bf r} \int d^3{\bf r'} \, |{W}({\bf r})| |{W}({\bf r'})| e^{2\pi i({{\boldsymbol{u}}}- {{\boldsymbol{u}}}') \cdot (\bf{r} + \bf{r'})}\nonumber\\ &=& \int d^3{\bf{r_0}} \int d^3 {\bf r' } \delta_{D}({\bf r_0}) \, |{W}({\bf{r_0 - r'}})| |{W}({\bf{r'}})| e^{2 \pi i {{\boldsymbol{u}}}\cdot ({\bf r}_0)} \nonumber\\ &=& \int d^3{\bf r'} \, |{W}({\bf r'})| |{W}({\bf -r'})| \approx \Omega B \approx \frac{\lambda^2 B}{A_e}, \label{window}\end{aligned}$$ where $\Omega$ is the field of view. Thus we obtain, $$C_N({{\boldsymbol{u}}}_i,{{\boldsymbol{u}}}_j) \approx \frac{\lambda^2 B}{A_e} P_N({{\boldsymbol{u}}}_i) \delta_{ij}. \label{eq:cn}$$ On the other hand, the covariance matrix for a single baseline can be evaluated from Eq. (\[visnoise\]), $$\begin{aligned} C_{N,1b}({{\boldsymbol{u}}}_i,{{\boldsymbol{u}}}_j) &=& \langle \tilde{I}({{\boldsymbol{u}}}_i) \tilde{I}^*({{\boldsymbol{u}}}_j) \rangle_{1b} = \sum_l^{B/\Delta \nu} \sum_m^{B/\Delta\nu} |V_N(u_i,v_i,\nu_l) \Delta \nu |^2 \delta_{ij} \delta_{lm} \nonumber\\ &=& \sum_l^{B/\Delta\nu} |V_N(u_i,v_i,\nu_l)|^2 \Delta \nu ~ \delta_{ij} = \frac{B}{\Delta \nu} (\Delta \nu)^2 ({V_N}(u_i,v_i,\nu))^2 \delta_{ij} \nonumber\\ &=& \Bigl(\frac{\lambda^2 B T_{\rm sys}}{A_e}\Bigr)^2 \frac{\delta_{ij}}{B t_0}. \label{eq2}\end{aligned}$$ Again, we assumed that there is no correlation between the thermal noise with different $u, v$ and $\nu$. If multiple baselines contribute to the same pixel, the observing time is effectively increased. Here we assume that the number density of the baselines in $uv$-plane is constant under rotation with respect to $\eta$-axis, that is, depends only on $|{{\boldsymbol{u}}}_\perp| = |{{\boldsymbol{u}}}| \sin{\theta}$ where $\theta$ is the angle between ${{\boldsymbol{u}}}$ and $\eta$-axis. Therefore, the effective observing time $t_{{{\boldsymbol{u}}}}$ can be written as, $$t_{{\boldsymbol{u}}}\approx \frac{A_e}{\lambda^2} n(|{{\boldsymbol{u}}}| \sin{\theta}) t_0. \label{eq:tu}$$ Here $A_e/\lambda^2$ represents area per pixel on $uv$-plane which reflects the resolution on $uv$-plane and $n(|{{\boldsymbol{u}}}| \sin{\theta})$ is the number density of baselines on $uv$-plane. Thus, we obtain the covariance matrix for a pixel in $uv\eta$-space, replacing $t_0$ with $t_k$, as, $$\begin{aligned} C_N({{\boldsymbol{u}}}_i,{{\boldsymbol{u}}}_j) = \Bigl(\frac{\lambda^2 B T_{\rm sys}}{A_e}\Bigr)^2 \frac{\delta_{ij}}{B t_{{\boldsymbol{u}}}}. \label{eq:noise_pix}\end{aligned}$$ Thus, comparing with Eq. (\[eq:cn\]) and substituting Eq. (\[eq:tu\]), we obtain, $$P_N({{\boldsymbol{u}}}) = \frac{\lambda^4 T_{\rm sys}^2}{A_e^2 n(|{{\boldsymbol{u}}}| \sin{\theta}) t_0}.$$ Now we convert the noise power spectrum of ${{\boldsymbol{u}}}$ space to the one of cosmological Fourier space ${{\boldsymbol{k}}}$. Using the following relations $$\begin{aligned} && {{\boldsymbol{u}}}_{\perp} = \frac{D_M(z)}{2 \pi} {{\boldsymbol{k}}}_{\perp} \equiv \frac{x}{2 \pi} {{\boldsymbol{k}}}_{\perp}, \\ && \eta \approx \frac{c (1+z)^2}{2 \pi H_0 f_{21}E(z)} k_z \equiv \frac{y}{2 \pi B} k_z,\end{aligned}$$ where, $H_0$ is the Hubble constant, $f_{21}$ is the frequency of 21cm radiation and $$\begin{aligned} && D_M(z) = \frac{c}{H_0} \int^z_0 \frac{dz'}{E(z')} \\ && E(z) = \sqrt{\Omega_{\rm M} (1+z)^3 + 1 - \Omega_{\rm M}}\end{aligned}$$ where $\Omega_{\rm M}$ is the density parameter of matter and we assumed the flat universe. Thus, we obtain, $$P_N({{\boldsymbol{k}}}) = \frac{x^2 y}{B} P_N({{\boldsymbol{u}}}) = \frac{x^2 y \lambda^4 T_{\rm sys}^2}{B A_e^2 n(|{{\boldsymbol{u}}}| \sin{\theta}) t_0}. \label{eq:Ppix}$$ Because the power spectrum of 21cm signal is dependent only on the length of the wave vector, we take a sum of the above noise power spectrum over a spherical shell which corresponds to the same $k$. First, we consider an annulus with radial width $\Delta k$ and angular width $\Delta \theta$. Noting that the baseline distribution is assumed to be uniform in an annulus, the number of pixels in the annulus is, $$\begin{aligned} N_a = 2 \pi k^2 \sin{\theta} ~ \Delta \theta ~ \Delta k \frac{V}{(2\pi)^3}, \label{pixnum}\end{aligned}$$ where $V = \lambda^2 x^2 y/A_e$ is the observed volume in real space, $(2\pi)^3 / V$ is the resolution in Fourier space and the other factor, $2\pi k^2 \sin{\theta} \Delta \theta \Delta k$, is the annulus volume in Fourier space. Then the noise power spectrum ruduces by a factor of $1/\sqrt{N_a}$. Next, we consider a sum over $\theta$. Taking $\Delta k = \epsilon k$, where $\epsilon$ is a constant factor which we set equal to 0.5, the spherically averaged sensitivity is given by, $$\begin{aligned} \delta P_N(k) &=& \left[ \sum_{\theta} \left(\frac{1}{P_N(k, \theta)/\sqrt{N_a}} \right)^2 \right]^{-1/2} \approx \left[ k^3 \int_{\arccos[\min(\frac{y k}{2 \pi},1)]}^{\arcsin[\min(\frac{k_*}{k}, 1)]} ~ d \theta \,\sin{\theta} \frac{\epsilon (n(k \sin{\theta}))^2 A_e^3 B^2 t_0^2}{(2 \pi)^2 x^2 y \lambda^6 T_{\rm sys}^4} \right]^{-1/2}, \label{eq:dPk}\end{aligned}$$ where $k_*$ is the longest transverse wave vector, which corresponds to the maximum baseline length. The lower limit of the integral corresponds to the pixel size. bispectrum sensitivity ====================== In this section, we estimate the bispectrum from the thermal noise in a similar way in the previous section. However, we should notice that, because the thermal noise is Gaussian, its bispectrum is actually zero. Nontheless, its statistical fluctuation, that is, its variance is non zero and contributes to the noise to the bispectrum signal. Thus, the calculation in this case is more subtle than that of the power spectrum, although we can use similar techniques as we see below. In [@2006MNRAS.366..213S], an order estimation for the thermal noise bispectrum has been done without considering this fact and also the baseline distribution. covariance of bispectrum ------------------------ Remembering the definition of the bispectrum in Eq. (\[eq:bs\_def\]), the covariance of the bispectrum can be defined by, $$\begin{aligned} && {\rm Cov}(B_N({{\boldsymbol{u}}}_1,{{\boldsymbol{u}}}_2,{{\boldsymbol{u}}}_3) B_N({{\boldsymbol{u}}}_4,{{\boldsymbol{u}}}_5,{{\boldsymbol{u}}}_6)) D \nonumber \\ && = \langle \bigl(\tilde{T}_N({{\boldsymbol{u}}}_1) \tilde{T}_N({{\boldsymbol{u}}}_2) \tilde{T}_N({{\boldsymbol{u}}}_3) - \langle \tilde{T}_N({{\boldsymbol{u}}}_1) \tilde{T}_N({{\boldsymbol{u}}}_2) \tilde{T}_N({{\boldsymbol{u}}}_3) \rangle \bigr) \bigl(\tilde{T}_N({{\boldsymbol{u}}}_4) \tilde{T}_N({{\boldsymbol{u}}}_5) \tilde{T}_N({{\boldsymbol{u}}}_6) - \langle \tilde{T}_N({{\boldsymbol{u}}}_4) \tilde{T}_N({{\boldsymbol{u}}}_5) \tilde{T}_N({{\boldsymbol{u}}}_6) \rangle \bigr) \rangle \nonumber\\ && = \langle \tilde{T}_N({{\boldsymbol{u}}}_1) \tilde{T}_N({{\boldsymbol{u}}}_2) \tilde{T}_N({{\boldsymbol{u}}}_3) \tilde{T}_N({{\boldsymbol{u}}}_4) \tilde{T}_N({{\boldsymbol{u}}}_5) \tilde{T}_N({{\boldsymbol{u}}}_6) \rangle \label{eq:CovBB}\end{aligned}$$ where each bispectrum satisfies the triangular condition (${{\boldsymbol{u}}}_1 + {{\boldsymbol{u}}}_2 + {{\boldsymbol{u}}}_3 = 0$ and ${{\boldsymbol{u}}}_4 + {{\boldsymbol{u}}}_5 + {{\boldsymbol{u}}}_6 = 0$) and $$\begin{aligned} D &=& \delta({{\boldsymbol{u}}}_1-{{\boldsymbol{u}}}_4) \delta({{\boldsymbol{u}}}_2-{{\boldsymbol{u}}}_5) + \delta({{\boldsymbol{u}}}_1-{{\boldsymbol{u}}}_4) \delta({{\boldsymbol{u}}}_2-{{\boldsymbol{u}}}_6) + \delta({{\boldsymbol{u}}}_1-{{\boldsymbol{u}}}_5) \delta({{\boldsymbol{u}}}_2-{{\boldsymbol{u}}}_4) \nonumber \\ & & + \delta({{\boldsymbol{u}}}_1-{{\boldsymbol{u}}}_5) \delta({{\boldsymbol{u}}}_2-{{\boldsymbol{u}}}_6) + \delta({{\boldsymbol{u}}}_1-{{\boldsymbol{u}}}_6) \delta({{\boldsymbol{u}}}_2-{{\boldsymbol{u}}}_4) + \delta({{\boldsymbol{u}}}_1-{{\boldsymbol{u}}}_6) \delta({{\boldsymbol{u}}}_2-{{\boldsymbol{u}}}_5), \label{eq:D}\end{aligned}$$ comes from the fact that there is no correlation unless the two triangles, (${{\boldsymbol{u}}}_1,{{\boldsymbol{u}}}_2,{{\boldsymbol{u}}}_3$) and (${{\boldsymbol{u}}}_4,{{\boldsymbol{u}}}_5,{{\boldsymbol{u}}}_6$), coincide. Next, we consider ensemble average of the product of six noise intensities, denoted as $\rm C_B$, $$\begin{aligned} C_B({{{{\boldsymbol{u}}}_1 ,{{\boldsymbol{u}}}_2 ,{{\boldsymbol{u}}}_3 ,{{\boldsymbol{u}}}_4 ,{{\boldsymbol{u}}}_5 ,{{\boldsymbol{u}}}_6}}) &=& \langle \tilde{I}({{\boldsymbol{u}}}_1) \tilde{I}({{\boldsymbol{u}}}_2) \tilde{I}({{\boldsymbol{u}}}_3) \tilde{I}({{\boldsymbol{u}}}_4) \tilde{I}({{\boldsymbol{u}}}_5) \tilde{I}({{\boldsymbol{u}}}_6) \rangle\nonumber\\ &=& \int d{{\boldsymbol{u}}}_1' \int d{{\boldsymbol{u}}}_2' \int d{{\boldsymbol{u}}}_3' \int d{{\boldsymbol{u}}}_4' \int d{{\boldsymbol{u}}}_5' \int d{{\boldsymbol{u}}}_6' \langle \tilde{T}({{\boldsymbol{u}}}_1') \tilde{T}({{\boldsymbol{u}}}_2') \tilde{T}({{\boldsymbol{u}}}_3') \tilde{T}({{\boldsymbol{u}}}_4') \tilde{T}({{\boldsymbol{u}}}_5') \tilde{T}({{\boldsymbol{u}}}_6') \rangle \nonumber\\ & & \times \tilde W({{\boldsymbol{u}}}_1-{{\boldsymbol{u}}}_1') \tilde W({{\boldsymbol{u}}}_2-{{\boldsymbol{u}}}_2') \tilde W({{\boldsymbol{u}}}_3-{{\boldsymbol{u}}}_3') \tilde W({{\boldsymbol{u}}}_4-{{\boldsymbol{u}}}_4') \tilde W({{\boldsymbol{u}}}_5-{{\boldsymbol{u}}}_5') \tilde W({{\boldsymbol{u}}}_6-{{\boldsymbol{u}}}_6'). \label{eq:C_b}\end{aligned}$$ To proceed further, we substitute Eq. (\[eq:CovBB\]) and consider the first term in Eq. (\[eq:D\]). $$\begin{aligned} C_B({{{{\boldsymbol{u}}}_1 ,{{\boldsymbol{u}}}_2 ,{{\boldsymbol{u}}}_3 ,{{\boldsymbol{u}}}_4 ,{{\boldsymbol{u}}}_5 ,{{\boldsymbol{u}}}_6}}) &=& \int d{{\boldsymbol{u}}}_1' \int d{{\boldsymbol{u}}}_2' \int d{{\boldsymbol{u}}}_3' \int d{{\boldsymbol{u}}}_4' \int d{{\boldsymbol{u}}}_5' \int d{{\boldsymbol{u}}}_6' \nonumber\\ & & \times {\rm Cov}(B_N({{\boldsymbol{u}}}_1',{{\boldsymbol{u}}}_2',{{\boldsymbol{u}}}_3') B_N({{\boldsymbol{u}}}_4',{{\boldsymbol{u}}}_5',{{\boldsymbol{u}}}_6')) \delta({{\boldsymbol{u}}}'_1-{{\boldsymbol{u}}}'_4) \delta({{\boldsymbol{u}}}'_2-{{\boldsymbol{u}}}'_5) \nonumber\\ & & \times W({{\boldsymbol{u}}}_1-{{\boldsymbol{u}}}_1') W({{\boldsymbol{u}}}_2-{{\boldsymbol{u}}}_2') W({{\boldsymbol{u}}}_3-{{\boldsymbol{u}}}_3') W({{\boldsymbol{u}}}_4-{{\boldsymbol{u}}}_4') W({{\boldsymbol{u}}}_5-{{\boldsymbol{u}}}_5') W({{\boldsymbol{u}}}_6-{{\boldsymbol{u}}}_6') \nonumber \\ &=& \int d{{\boldsymbol{u}}}_1' \int d{{\boldsymbol{u}}}_2' \int d{{\boldsymbol{u}}}_3' \int d{{\boldsymbol{u}}}_6' {\rm Cov}(B_N({{\boldsymbol{u}}}_1',{{\boldsymbol{u}}}_2',{{\boldsymbol{u}}}_3') B_N({{\boldsymbol{u}}}_1',{{\boldsymbol{u}}}_2',{{\boldsymbol{u}}}_6')) \nonumber\\ & & \times W({{\boldsymbol{u}}}_1-{{\boldsymbol{u}}}_1') W({{\boldsymbol{u}}}_2-{{\boldsymbol{u}}}_2') W({{\boldsymbol{u}}}_3-{{\boldsymbol{u}}}_3') W({{\boldsymbol{u}}}_4-{{\boldsymbol{u}}}_1') W({{\boldsymbol{u}}}_5-{{\boldsymbol{u}}}_2') W({{\boldsymbol{u}}}_6-{{\boldsymbol{u}}}_6')\end{aligned}$$ This is non-zero only when ${{\boldsymbol{u}}}_1 \approx {{\boldsymbol{u}}}_4$ and ${{\boldsymbol{u}}}_2 \approx {{\boldsymbol{u}}}_5$ (and then ${{\boldsymbol{u}}}_3 \approx {{\boldsymbol{u}}}_6$ from the triangular conditions). If these conditions are satisfied, $$\begin{aligned} C_B({{{{\boldsymbol{u}}}_1 ,{{\boldsymbol{u}}}_2 ,{{\boldsymbol{u}}}_3 ,{{\boldsymbol{u}}}_4 ,{{\boldsymbol{u}}}_5 ,{{\boldsymbol{u}}}_6}}) &\approx& {\rm Cov}(B_N({{\boldsymbol{u}}}_1,{{\boldsymbol{u}}}_2,{{\boldsymbol{u}}}_3) B_N({{\boldsymbol{u}}}_1,{{\boldsymbol{u}}}_2,{{\boldsymbol{u}}}_3)) \int d{{\boldsymbol{u}}}_1' \int d{{\boldsymbol{u}}}_2' \int d{{\boldsymbol{u}}}_3' \int d{{\boldsymbol{u}}}_6' \nonumber\\ & & \times (W({{\boldsymbol{u}}}_1-{{\boldsymbol{u}}}_1'))^2 (W({{\boldsymbol{u}}}_2-{{\boldsymbol{u}}}_2'))^2 W({{\boldsymbol{u}}}_3-{{\boldsymbol{u}}}_3') W({{\boldsymbol{u}}}_6-{{\boldsymbol{u}}}_6') \nonumber \\ &=& \bigg(\frac{\lambda^2 B}{A_e}\bigg)^2 {\rm Cov}(B_N({{\boldsymbol{u}}}_1,{{\boldsymbol{u}}}_2,{{\boldsymbol{u}}}_3) B_N({{\boldsymbol{u}}}_1,{{\boldsymbol{u}}}_2,{{\boldsymbol{u}}}_3)) \label{eq:CovB}\end{aligned}$$ where we used Eq. (\[window\]) and $$\int d{{\boldsymbol{u}}}' W({{\boldsymbol{u}}}-{{\boldsymbol{u}}}') = 1,$$ and assumed ${\rm Cov}(B_N({{\boldsymbol{u}}}_1,{{\boldsymbol{u}}}_2,{{\boldsymbol{u}}}_3) B_N({{\boldsymbol{u}}}_4,{{\boldsymbol{u}}}_5,{{\boldsymbol{u}}}_6))$ is approximately constant within the window function. Thus, taking other terms in Eq. (\[eq:D\]) into account, we have, $$C_B({{{{\boldsymbol{u}}}_1 ,{{\boldsymbol{u}}}_2 ,{{\boldsymbol{u}}}_3 ,{{\boldsymbol{u}}}_4 ,{{\boldsymbol{u}}}_5 ,{{\boldsymbol{u}}}_6}}) = D \bigg(\frac{\lambda^2 B}{A_e}\bigg)^2 {\rm Cov}(B_N({{\boldsymbol{u}}}_1,{{\boldsymbol{u}}}_2,{{\boldsymbol{u}}}_3) B_N({{\boldsymbol{u}}}_1,{{\boldsymbol{u}}}_2,{{\boldsymbol{u}}}_3)). \label{eq:C_B1}$$ On the other hand, the product of six noise intensities can also be calculated as follows. $$\begin{aligned} C_B({{{{\boldsymbol{u}}}_1 ,{{\boldsymbol{u}}}_2 ,{{\boldsymbol{u}}}_3 ,{{\boldsymbol{u}}}_4 ,{{\boldsymbol{u}}}_5 ,{{\boldsymbol{u}}}_6}}) &=& \langle \tilde{I}({{\boldsymbol{u}}}_1) \tilde{I}({{\boldsymbol{u}}}_2) \tilde{I}({{\boldsymbol{u}}}_3) \tilde{I}({{\boldsymbol{u}}}_4) \tilde{I}({{\boldsymbol{u}}}_5) \tilde{I}({{\boldsymbol{u}}}_6) \rangle \nonumber \\ &=& \langle \tilde{I}({{\boldsymbol{u}}}_1) \tilde{I}({{\boldsymbol{u}}}_4) \rangle \langle \tilde{I}({{\boldsymbol{u}}}_2) \tilde{I}({{\boldsymbol{u}}}_5) \rangle \langle \tilde{I}({{\boldsymbol{u}}}_3) \tilde{I}({{\boldsymbol{u}}}_6) \rangle + ({\rm 5~permutations }) \nonumber\\ &=& D \langle \tilde{I}({{\boldsymbol{u}}}_1) \tilde{I}({{\boldsymbol{u}}}_1) \rangle \langle \tilde{I}({{\boldsymbol{u}}}_2) \tilde{I}({{\boldsymbol{u}}}_2) \rangle \langle \tilde{I}({{\boldsymbol{u}}}_3) \tilde{I}({{\boldsymbol{u}}}_3) \rangle \nonumber\\ &=& D \Bigl(\frac{\lambda^{2} B T_{\rm sys}}{A_{e}}\Bigr)^6 \frac{1}{B^3 t_{{{\boldsymbol{u}}}_1} t_{{{\boldsymbol{u}}}_2} t_{{{\boldsymbol{u}}}_3}}, \label{eq:C_B2}\end{aligned}$$ where we used Wick theorem in the second equality and Eq. (\[eq:noise\_pix\]) in the last equality. Thus, from Eqs. (\[eq:C\_B1\]) and (\[eq:C\_B2\]), we obtain, $${\rm Cov}(B_N({{\boldsymbol{u}}}_1,{{\boldsymbol{u}}}_2,{{\boldsymbol{u}}}_3) B_N({{\boldsymbol{u}}}_1,{{\boldsymbol{u}}}_2,{{\boldsymbol{u}}}_3)) = \bigg(\frac{A_e}{\lambda^2 B}\bigg)^2 \Bigl(\frac{\lambda^{2} B T_{\rm sys}}{A_e}\Bigr)^6 \frac{1}{B^3 t_{{{\boldsymbol{u}}}_1}t_{{{\boldsymbol{u}}}_2}t_{{{\boldsymbol{u}}}_3}}.$$ Converting the argument from ${{\boldsymbol{u}}}$ to ${{\boldsymbol{k}}}$, we finally obtain, $$\begin{aligned} {\rm Cov}(B_N({{\boldsymbol{k}}}_1,{{\boldsymbol{k}}}_2,{{\boldsymbol{k}}}_3) B_N({{\boldsymbol{k}}}_1,{{\boldsymbol{k}}}_2,{{\boldsymbol{k}}}_3)) &=& \bigg(\frac{x^2 y}{B}\bigg)^4 {\rm Cov}(B_N({{\boldsymbol{u}}}_1,{{\boldsymbol{u}}}_2,{{\boldsymbol{u}}}_3) B_N({{\boldsymbol{u}}}_4,{{\boldsymbol{u}}}_5,{{\boldsymbol{u}}}_6)) \nonumber \\ &=& \bigg(\frac{x^2 y \lambda^2}{A_e}\bigg)^4 \frac{T^6_{\rm sys}}{B^3 t_{{{\boldsymbol{k}}}_1} t_{{{\boldsymbol{k}}}_2} t_{{{\boldsymbol{k}}}_3}}. \label{eq:CovBB}\end{aligned}$$ This equation corresponds to Eq. (\[eq:noise\_pix\]) for the power spectrum, if we substitute Eq. (\[eq:tu\]). spherical average ----------------- In this subsection, we take a sum of the noise bispectrum over spherical shell as we did for the power spectrum in the previous section. However, the situation is much more complicated in the case of bispectrum, because $|{{\boldsymbol{k}}}_1|, |{{\boldsymbol{k}}}_2|$ and $|{{\boldsymbol{k}}}_3|$ can be all different with each other in general so that we must consider two spherical shells with the radius $|{{\boldsymbol{k}}}_1|$ and $|{{\boldsymbol{k}}}_2|$, while $|{{\boldsymbol{k}}}_3|$ is determined by the triangular condition, ${{\boldsymbol{k}}}_1 + {{\boldsymbol{k}}}_2 + {{\boldsymbol{k}}}_3 = 0$. In this paper, we calculate the noise bispectrum for equilateral type ($|{{\boldsymbol{k}}}_1| = |{{\boldsymbol{k}}}_2| = |{{\boldsymbol{k}}}_3|$) and isosceles type ($|{{\boldsymbol{k}}}_2| = |{{\boldsymbol{k}}}_3|$) and define $K \equiv |{{\boldsymbol{k}}}_1|$ and $k \equiv |{{\boldsymbol{k}}}_2| = |{{\boldsymbol{k}}}_3|$. First, as in the case of the power spectrum, ${{\boldsymbol{k}}}_1$ can run over a spherical shell with radius $k$ which can be parametrized by two of the spherical coordinate of ${{\boldsymbol{k}}}_1$, $(\theta_1,\phi_1)$. Further, for a fixed ${{\boldsymbol{k}}}_1$, there is a rotational degree of freedom for ${{\boldsymbol{k}}}_2$ with respect to ${{\boldsymbol{k}}}_1$, which is denoted by an angle $\alpha$ with $0 \leq \alpha < 2 \pi$. Thus, we need to integrate the covariance matrix in Eq. (\[eq:CovBB\]) with respect to $\theta_1$, $\phi_1$ and $\alpha$. Noting that the covariance matrix does not depend on $\phi_1$, the weight of the integration, which corresponds to Eq. (\[pixnum\]), is given by $$\begin{aligned} N_a = \left[ 2 \pi \sin{\theta_1} K^2 \Delta K \Delta \theta_1 \frac{V}{(2\pi)^3} \right] \times \left[ k^2 \sin{\theta_2} \sin{\gamma} ~ \Delta k \Delta \theta_2 \Delta \alpha \frac{V}{(2\pi)^3} \right].\end{aligned}$$ where the first factor comes from the sum for ${{\boldsymbol{k}}}_1$ over the spherical shell and the second factor takes the rotational degree of freedom of ${{\boldsymbol{k}}}_2$ for each ${{\boldsymbol{k}}}_1$ into account. Here $\theta_2$ is the polar angle of ${{\boldsymbol{k}}}_2$ and $\gamma$ is the angle $\partial {{\boldsymbol{k}}}_2/\partial \alpha$ and $\partial {{\boldsymbol{k}}}_2/\partial \theta_2$. $\Delta \theta_2$ is the width of the annulas of ${{\boldsymbol{k}}}_2$ when ${{\boldsymbol{k}}}_1$ is fixed, which we set equal to the resolution in Fourier space, $2 \pi/V^{1/3}$. It is convenient to express $\theta_2$ by $\theta_1$, $\alpha$ and the angle between ${{\boldsymbol{k}}}_1$ and ${{\boldsymbol{k}}}_2$ denoted as $\beta$. Noting ${{\boldsymbol{k}}}_2$ can be express as $${{\boldsymbol{k}}}_2 = k (\cos{\theta_1} \cos{\alpha} \sin{\beta} + \sin{\theta_1} \cos{\beta}, \sin{\alpha} \sin{\beta}, - \sin{\theta_1} \cos{\alpha} \sin{\beta} + \cos{\theta_1} \cos{\beta}),$$ we obtain, $$\cos{\theta_2} = - \sin{\theta_1} \cos{\alpha} \sin{\beta} + \cos{\theta_1} \cos{\beta}$$ Then, setting $\Delta k = \epsilon k$ and $\Delta K = \epsilon K$, the bispectrum variance due to the thermal noise is written by an integrate with respect to $\theta_1$ and $\alpha$, $$\begin{aligned} \delta B_N(k,K,\beta) &=& \left[ \sum_{\theta} \sum_{\alpha} \left(\frac{1}{\sqrt{N_a}} \sqrt{{\rm Cov}(B_1B_2)(k, K, \theta_1, \alpha)} \right)^{-2} \right]^{-1/2} \nonumber \\ &=& \frac{(2\pi)^{\frac{5}{2}}}{\sqrt{\Delta \theta_2} k K^{3/2} \epsilon} \bigg(\frac{x^2y \lambda^2}{A_e}\bigg) \bigg(\frac{T^2_{\rm sys} \lambda^2}{A_e B t_0}\bigg)^{\frac{3}{2}} \nonumber \\ & & \times \left[\int {d\theta_1} \int {d\alpha} \sin{\theta_1} \sin{\theta_2} \sin{\gamma(\theta_1,\alpha)} ~ n({{\boldsymbol{k}}}_1) n({{\boldsymbol{k}}}_2) n({{\boldsymbol{k}}}_3) \right]^{-\frac{1}{2}}.\end{aligned}$$ This is a general expression for isosceles-type bispectrum. For equilateral type, we just set $K = k$ and $\beta = 2 \pi/3$. estimation of noise bispectrum ------------------------------ To calculate the bispectrum sensitivity, we need the number density of baselines on uv-plane. In this paper, we consider expanded MWA, LOFAR and SKA. The expanded MWA will have 500 antennae within a radius of 750 m with $r^{-2}$ distribution [@2006ApJ...638...20B]. LOFAR has 24 antennae within a radius of 2000 m with $r^{-2}$ distribution . HERA has 547 antennae within 200 m with constant distribution [@2014ApJ...782...66P]. SKA will have 466 antennae within 600 m with $r^{-2}$ distribution, 670 antennae within 1000 m, 866 antennae within 3000 m [@SKA]. For simplicity, we assume that the antennae density is constant between 600 m to 1000 m and 1000 m to 3000 m, respectively. We list parameters in table \[table:parameter\]. Further, we assume $t_0 = 1000~{\rm hour}$ for the total observing time and $6~{\rm MHz}$ bandwidth. redshift 8 10 12 17 $N_{\rm station}$ --------------------- ----- ----- ------ ------ ------------------- frequency \[MHz\] 158 129 109 79 $T_{\rm sys}~[K]$ 440 600 1000 1900 $A_e~[m^2]$ (MWA) 14 18 18 18 500 $A_e~[m^2]$ (LOFAR) 512 600 900 900 24 $A_e~[m^2]$ (HERA) 68 106 154 154 547 $A_e~[m^2]$ (SKA) 462 728 962 962 866 : Parameters for telescopes: $T_{\rm sys}$ is system temperature, $A_e$ is effective area of a station and $N_{\rm station}$ is the number of stations.[]{data-label="table:parameter"} For comparison, we show the bispectrum of 21cm signal from the epoch of reionization, using a public code, 21cmFAST [@2011MNRAS.411..955M]. This is based on a semi-analytic model of reionization and we can obtain 3D brightness temperature maps at arbitrary redshifts. We set the simulation box to $(200~{\rm Mpc})^3$ with $300^3$ grids and take a fiducial set of model parameters as $(\zeta, \zeta_X, T_{\rm vir}, R_{\rm mfp}) = (31.5, 10^{56}/M_{\odot}, 10^4~{\rm K}, 30~{\rm Mpc}$). Here, $\zeta$ is the ionizing efficiency, $\zeta_X$ is the number of X-ray photons per solar mass, $T_{\rm vir}$ is the minimum virial temperature of halos which host stars and $R_{\rm mfp}$ is the mean free path of ionizing photons. [We also estimate the sample variance of the bispectrum by calculating the average and variance from 19 brightness-temperature maps with different realizations of the initial condition.]{} In Fig. \[eqi1\], we compare the equilateral-type bispectrum signal with thermal noise at $z = 8,10,12$ and $17$. [Here the sample variance for the fiducial model and the average signal for a variant model with $\zeta = 26.5$ are also shown for reference. The ionization is less effective for this variant model so that the reionization proceeds slowly compared with the fiducial model.]{} Generally, the noise increases toward smaller scales which reflects the deficiency of longer baselines. On the other hand, the sensitivity for larger scales are limited by the survey volume. We see the signals are larger than SKA noise for $k \lesssim 0.3~{\rm Mpc}^{-1}$ at all redshifts. However, the thermal noise dominates over the signal for the expanded MWA at almost all scales and redshifts, while the bispectrum may be observable for large scales $k \lesssim 0.05~{\rm Mpc}^{-1}$ at $z = 10$. LOFAR has better sensitivity and the signal will be observable at scales with $k \lesssim 0.1~{\rm Mpc}^{-1}$ at $z = 10$ and $17$. Here it should be noted that the bispectrum signal has several peaks as a function of redshift and they are at $z = 10$ and $17$ [@shi-P]. The peak redshifts depend on the specific values of the model parameters and observation of them will give us information on the process of reionization. Thus, it is expected that LOFAR is enough sensitive to detect the bispectrum at the peak redshifts for large scales. [On the other hand, due to the short baselines of HERA, equilateral triangle does not exist in the $(u,v,\eta)$ space of HERA for large $k$, while it has a sufficient sensitivity for small $k$.]{} The isosceles-type bispectrum with $K = 0.06~{\rm Mpc}^{-1}$ bispectra are plotted in Fig. \[iso1\]. The behavior and relative amplitudes of the signal and noise are very similar to the case of the equilateral type but SKA is more sensitive at smaller scales. [Finally, we calculate the total signal-to-noise ratio for equilateral and isosceles types, considering a $k$ range from $5.0 \times 10^{-2}~{\rm Mpc^{-1}}$ to $1.0~{\rm Mpc^{-1}}$ in Figs. \[eqi1\] and \[iso1\], respectively. Here the total signal-to-noise ratio is calculated by summing the square of signal-to-noise ratio over $k$ bins and taking its square root. The result is shown in the Table \[table:signal\_to\_noise\].]{} ![Comparison of equilateral-type bispectrum of 21cm signal [of the fiducial model with sample variance (thin dashed line and shaded area), signal of a variant model with $\zeta = 26.5$ (thin dot dashed line)]{} and thermal noise of MWA (thick dashed), LOFAR (thick dotted), HERA(thick dot dashed) and SKA (thick solid) at $z = 8,10,12$ and $17$, [Here $x_{\rm HI}$ is the neutral fraction of the fiducial model at each redshift]{}.[]{data-label="eqi1"}](Noise_eqi.eps){width="11.5cm"} ![Comparison of isosceles-type bispectrum of 21cm signal (thin dashed line) and thermal noise of MWA (thick dashed), LOFAR (thick dotted), HERA(dot dashed) and SKA (thick solid) at $z = 8,10,12$ and $17$. Here $x_{\rm HI}$ is the neutral fraction of standard model at each redshift. Here one of the wavenumber $K$ is fixed to $0.06~{\rm Mpc}^{-1}$.[]{data-label="iso1"}](Noise_iso.eps){width="11.5cm"} redshift    8        10        12        17     ------------ -------------------- ------------------- -------------------- -------------------- -- MWA(equ) $5.4\times10^{-2}$ $2.2\times10^{1}$ $3.1\times10^{-2}$ $8.0\times10^{-1}$ LOFAR(equ) $3.5\times10^{-1}$ $1.9\times10^{2}$ $1.3$ $6.0\times10^{1}$ HERA(equ) $2.6\times10^{1}$ $2.4\times10^{4}$ $1.2\times10^{2}$ $3.7\times10^{3}$ SKA(equ) $1.7\times10^{3}$ $1.9\times10^{6}$ $1.1\times10^{4}$ $5.7\times10^{5}$ MWA(iso) $4.3\times10^{-3}$ $6.4$ $8.9\times10^{-3}$ $2.0\times10^{-1}$ LOFAR(iso) $4.8\times10^{-2}$ $1.0\times10^{2}$ $5.5\times10^{-1}$ $1.8\times10^{1}$ HERA(iso) $3.0$ $2.4\times10^{4}$ $4.0\times10^{1}$ $0.0$ SKA(iso) $1.1\times10^{2}$ $4.3\times10^{5}$ $2.3\times10^{3}$ $1.0\times10^{5}$ : [Total signal-to-noise ratios of bispectra of equilateral (equ) and isosceles (iso) types for each array and redshift.]{}[]{data-label="table:signal_to_noise"} summary and discussion ====================== In this paper, we esimated the bispectrum of thermal noise for redshifted 21cm signal observation for Epock of Reionization by extending the formalism of the noise power spectrum estimation given by [@2006ApJ...653..815M]. Because thermal noise was assumed to be Gaussian, the ensamble average of the bispectrum vanishes and its variance contributes to the noise to the bispectrum signal. We developed a formalism to calculate the noise bispectrum for an arbitrary triangle, taking the array configuration into account. We applied it to the cases with equilateral and isosceles triangles and estimated the noise bispectrum for expanded MWA, LOFAR and the SKA. Consequently, it was found that the SKA has enough sensitivity for $k \lesssim 0.3~{\rm Mpc}^{-1}$ for both types of triangles. On the other hand, LOFAR will have sensitivity for the peaks of the bispectrum as a function of redshift. The expanded MWA has even less sensitivity but it will be possible to put a meaningful constraints on model parameters which induce larger signals than those with the parameters used in this paper. Not only the themal noise but signal of bispectrum depend on the configuration of the triangle of three wave numbers. It is possible that the signal bispectrum has a large amplitude for a specific configuration of the triangle and observation may become easier in that case. An investigation of the details of the bispectrum signal and comparison with noise bispectrum will be presented elsewhere [@shi-B]. Actually, thermal noise is just one of many obstacles for the observation of 21cm signal. Other serious sources of noise are Galactic and extragalactic foreground and sample variance, and the foreground emission has not been well understood even for power spectrum. [It may be helpful to consider the “EoR window” for bispectrum as in the case of power spectrum [@2014ApJ...782...66P]. It would turn out that small-k modes should be discarded to avoid foreground and, if this is the case, the total signal-to-noise ratios in Table \[table:signal\_to\_noise\] are overestimation. Further, for a practical application of bispectrum analysis to survey data, we need to take survey geometry into account [@2000ApJ...544..597S; @2001ApJ...546..652S; @2014arXiv1407.5668G].]{} Nevertheless, the observation of the bispectrum is very important because 21cm signal from Epoch of Reionization is highly non-Gaussian so that the bispectrum will give us enormous information complementary to the power spectrum. Acknowledgement {#acknowledgement .unnumbered} =============== This work is supported by Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, Nos. 24340048 and 26610048(K.T), No. 25-3015(H.S.), No. 26800104(H.N.) and No. 25610043(H.I.). 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--- bibliography: - 'auto\_generated.bib' title: 'Search for heavy resonances decaying to a top quark and a bottom quark in the lepton+jets final state in proton-proton collisions at 13' --- =1 $Revision: 439399 $ $HeadURL: svn+ssh://svn.cern.ch/reps/tdr2/papers/B2G-17-010/trunk/B2G-17-010.tex $ $Id: B2G-17-010.tex 439399 2017-12-18 19:13:49Z drankin $ \[1\] Introduction ============ Despite the broad success of the standard model (SM), the absence of answers to the hierarchy problem, among other shortcomings, has led to the development of many theories for new physics that lies beyond the SM. A common prediction of many of these theories is the existence of new heavy gauge bosons [@doi:10.1146/annurev.nucl.55.090704.151502; @PhysRevD.64.035002; @PhysRevD.64.065007; @PhysRevD.53.5258; @PhysRevD.11.566]. These particles typically arise from additional symmetries in the theories, and it is common to generically refer to charged instances of these resonances as $\PWpr$ bosons. In scenarios where the $\PWpr$ boson is sufficiently heavy, the decay $\PWpr\to\PQt\PQb$ has several features that make it an appealing search channel. Searches in this channel directly probe the $\PWpr$ boson coupling to third generation quarks, which, in some models [@Muller1996345; @Malkawi1996304], can be enhanced with respect to the coupling to lighter quarks. Additionally, the large continuum multijet background has less impact on searches for $\PWpr\to\PQt\PQb$ decay than on searches for the decay to light quarks ($\PWpr\to \PQq\PQq'$). The $\PWpr\to\PQt\PQb$ search is complementary to searches for $\PWpr\to{\ell}\PGn$ and $\PWpr\to\PW\Z$, where $\ell$ denotes a charged lepton and $\PGn$ denotes a neutrino. Unlike searches for $\PWpr\to\ell\PGn$, the search for $\PWpr\to\PQt\PQb\to\PQb\PQb\ell\PGn$ decay allows the $\PWpr$ boson mass to be fully reconstructed, up to a quadratic ambiguity. Searches for $\PWpr$ bosons in the top and bottom quark (tb) decay channel have been performed at the Fermilab Tevatron [@D0:2010; @D0Wprime; @PhysRevLett.115.061801] and at the CERN LHC by both CMS [@Chatrchyan:2014koa; @Chatrchyan:2016had; @Sirunyan:2017ukk] and ATLAS [@Aad:2014xra; @Aad:2014xea] Collaborations. The most stringent limits to date on the production of $\PWpr$ bosons come from the CMS search performed at $\sqrt{s}=13\TeV$ [@Sirunyan:2017ukk], using 2.2of data collected in 2015. This Letter presents a search for $\PWpr$ bosons decaying via the tb channel using proton-proton collision data at $\sqrt{s}=13\TeV$, collected by the CMS experiment in 2016. The analyzed data correspond to an integrated luminosity of 35.9. Events with exactly one electron or muon, significant missing transverse momentum, and multiple jets in the final state are selected. This search focuses on $\PWpr$ bosons with widths that are narrow compared to their masses. In addition to searching for $\PWpr$ bosons with purely right- or left-handed couplings, we also search for $\PWpr$ bosons with varying combinations of these couplings. This analysis is sensitive to $\PWpr$ bosons with masses between 1 and 4. The CMS detector\[sec:detector\] ================================ The central feature of the CMS apparatus [@Chatrchyan:2008zzk] is a superconducting solenoid of 6 internal diameter, providing a magnetic field of 3.8. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. Forward calorimeters extend the pseudorapidity ($\eta$) coverage provided by the barrel and endcap detectors. Muons are measured in gas-ionization detectors embedded in the steel flux-return yoke outside the solenoid. The particle-flow (PF) algorithm [@Sirunyan:2017ulk] reconstructs and identifies individual particle candidates with an optimized combination of information from relevant elements of the CMS detector. The energy of photons is measured using the ECAL and corrected for zero-suppression effects. The energy of electrons is determined from a combination of the electron momentum at the primary interaction vertex as determined by the tracker, the energy of the corresponding ECAL cluster, and the energy sum of all bremsstrahlung photons spatially compatible with originating from the electron track. The primary interaction vertex is defined as the vertex with the largest sum of $\pt^2$ of associated tracks. The energies of muons are obtained from the curvature of the corresponding tracks. The energy of charged hadrons is determined from a combination of their momentum measured in the tracker and the matching ECAL and HCAL energy deposits. This measurement is then corrected for zero-suppression effects and for the response function of the calorimeters to hadronic showers. Finally, the energy of neutral hadrons is obtained from the corresponding corrected ECAL and HCAL energy. In the barrel section of the ECAL, an energy resolution of about 1% is achieved for unconverted or late-converting photons in the tens of energy range. The resolution for photons not belonging to this category is about 1.3% up to $\abs{\eta} = 1$, rising to about 2.5% at $\abs{\eta} = 1.4$. In the endcaps, the resolution of unconverted or late-converting photons is about 2.5%, while the remaining photons have a resolution between 3 and 4% [@CMS:EGM-14-001]. The momentum resolution for electrons with transverse momentum $\pt\approx45$from $\Z \to \Pe \Pe$ decays ranges from 1.7% for nonshowering electrons in the barrel region to 4.5% for showering electrons in the endcaps [@Khachatryan:2015hwa]. When combining information from the entire detector, the jet energy resolution amounts typically to 15% at 10, 8% at 100, and 4% at 1, to be compared to about 40, 12, and 5% obtained when the ECAL and HCAL calorimeters alone are used [@Chatrchyan:2011ds]. Muons are measured in the range $\abs{\eta} < 2.4$, with detection planes made using three technologies: drift tubes, cathode strip chambers, and resistive-plate chambers. Matching muons to tracks measured in the silicon tracker results in a relative transverse momentum resolution for muons with $20 <\pt < 100\GeV$ of 1.3–2.0% in the barrel, and better than 6% in the endcaps. The resolution in the barrel is better than 10% for muons with up to 1 [@Chatrchyan:2012xi]. The missing transverse momentum vector is defined as the projection on the plane perpendicular to the beams of the negative vector sum of the momenta of all reconstructed particles in an event. A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [@Chatrchyan:2008zzk]. Signal and background modeling\[sec:model\] =========================================== Signal modeling --------------- Simulated signal samples are generated at leading order and their cross sections are scaled to next-to-leading order with a K-factor of 1.25 [@Sullivan:2002jt; @Duffty:2012rf] appropriate for our signal mass range of interest. All signal samples are generated using the [@comphep] 4.5.2 package according to the following lowest-order effective Lagrangian [@Sullivan:2002jt]: $$\mathcal{L}=\frac{V_{f_if_j}}{2\sqrt{2}}g_{\PW}\bar{f}_i\gamma_\mu \left[a_\mathrm{R}(1+\gamma^5)+a_\mathrm{L}(1-\gamma^5)\right]\mathrm{W}'^\mu f_j+\text{h.c.},\label{eq:lag}$$ where $V_{f_if_j}$ is the Cabibbo–Kobayashi–Maskawa matrix if $f$ is a quark and $V_{f_if_j}=\delta_{ij}$ if $f$ is a lepton, $g_{\PW}$ is the SM weak coupling constant, and $a_\mathrm{R}$ and $a_\mathrm{L}$ are the coupling strengths of the to right- and left-handed fermions, respectively. We consider values of $a_\mathrm{L}$ and $a_\mathrm{R}$ that range from 0 to 1, and any signal with $a_\mathrm{L}>0$ takes into account interference with the SM W boson. The signal simulation includes decays involving a $\tau$ lepton, and no distinction is made in the analysis selection or strategy between an electron or muon produced directly from the W boson decay, and an electron or muon from a subsequent $\tau$ lepton decay. We use boson width values computed in for each mass point, and use a narrow-width approximation for the generation of bosons that have both left- and right-handed couplings. The typical width is approximately 3% of the signal resonance mass. The widths of all generated samples are significantly smaller than the detector and reconstruction resolutions, and therefore the precise values of the width do not affect our results. For $\PWpr_\mathrm{R}$ bosons we consider two scenarios for the mass of the hypothetical right-handed neutrinos. If the right-handed neutrinos are lighter than the $\PWpr_\mathrm{R}$ boson ($M_{\nu_\mathrm{R}}<M_{\PWpr_\mathrm{R}}$), then both $\PWpr_\mathrm{R}\to\ell\nu_\mathrm{R}$ and $\PWpr_\mathrm{R}\to\PQq\PQq'$ decays are allowed. However, if the right-handed neutrinos are heavier than the $\PWpr_\mathrm{R}$ boson ($M_{\nu_\mathrm{R}}>M_{\PWpr_\mathrm{R}}$), then the $\PWpr_\mathrm{R}\to\ell\nu_\mathrm{R}$ decay is forbidden, resulting in an enhancement of the branching fraction for $\PWpr\to\PQt\PQb$. This branching fraction varies slightly with mass and ranges from 0.32 to 0.33 if $M_{\nu_\mathrm{R}}>M_{\PWpr_\mathrm{R}}$ and from 0.24 to 0.25 if $M_{\nu_\mathrm{R}}<M_{\PWpr_\mathrm{R}}$ for $\PWpr_\mathrm{R}$ boson masses between 1 and 4. For the purposes of signal generation all neutrinos are assumed to be massless. When calculating the number of expected signal events (in Table \[table:lep\_yields\]), showing expected signal distributions (in Figs. \[fig:lep\_mtb\_A\] and \[fig:lep\_mtb\_B\]), or presenting results for arbitrary left- and right-handed couplings (in Fig. \[fig:2dlimit\]), it is always assumed that the masses of hypothetical right-handed neutrinos are much lighter than that of the $\PWpr_\mathrm{R}$ boson. Both scenarios are considered when presenting results for $\PWpr_\mathrm{R}$ (in Figs. \[fig:sepxseclim\_sep\] and \[fig:sepxseclim\]). Background modeling ------------------- The most significant contributions to the background come from W+jets and production. Smaller contributions, from $s$- and $t$-channel single top quark production, associated production of a top quark and a W boson, $\Z/\gamma^*$+jets, and diboson production ($\mathrm{VV}$), are also included in the total background estimate. Predictions for all background processes are taken from simulation with corrections applied in cases where initial modeling is found to be inaccurate. Further details on the background modeling can be found in Section \[sec:anstrat\]. The contribution to the total background from the multijet background is found to be negligible after the full selection and is therefore not included. Simulated samples for $\Z/\gamma^*$+jets, $s$- and $t$-channel single-top quark, and W+jets events are produced using  [@Alwall:2014hca; @Frederix:2012ps; @Alwall:2007fs] v2.2.2, and associated production of a top quark and a W boson are produced using v2  [@Nason:2004rx; @powheg; @Alioli:2010xd; @Frixione:2007nw; @Re:2010bp], and all other background processes are produced using 8.212  [@Sjostrand:2014zea]. The process contribution is then assigned a correction based on the top quark , which is known to be improperly modeled [@Khachatryan:2016mnb]. A correction for the relative fraction of W+light quark/gluon jets and W+charm/bottom jets in W+jets events is derived and then checked in a control region. More details on the background estimation methods can be found in Section \[sec:anstrat\]. All simulated signal and background samples are processed through for parton fragmentation and hadronization. The simulation of the CMS detector is performed by 4 [@Agostinelli:2002hh; @Allison:2006ve]. The NNPDF 3.0 parton distribution function (PDF) set is used for sample generation [@nnpdf]. All simulated samples include additional proton-proton interactions (pileup) and are weighted such that the distribution of the number of interactions in each event agrees with that in the data. Event selection\[sec:select\] ============================= All leptons, jets, and used in this search are reconstructed using the particle-flow algorithm. Jets are clustered using the anti-algorithm [@Cacciari:2008gp; @Cacciari:2011ma] with a size parameter of 0.4(AK4), and dedicated jet energy corrections [@Chatrchyan:2011ds; @Khachatryan:2016kdb] are then applied. Any charged hadrons that are not associated with the leading vertex are removed from the event, using the charged hadron subtraction method [@Krohn:2013lba]. The leading vertex is defined as the primary vertex with the largest squared sum of the transverse momenta of its associated tracks. The neutral-hadron contribution to jets from pileup is also subtracted, using the jet area method [@areasubtract]. Charged hadron subtraction is applied before any jet clustering, while area-based subtractions are applied after clustering but before the final level of jet energy corrections. Jet momentum is determined as the vectorial sum of all particle momenta in the jet, and is found from simulation to be within 5 to 10% of the true momentum over the whole spectrum and detector acceptance [@Chatrchyan:2008zzk]. An offset correction is applied to jet energies to take into account the contribution from pileup. Jet energy corrections are derived from simulation, and are confirmed with in situ measurements of the energy balance in dijet, multijet, photon+jet, and leptonically decaying Z+jets events. Additional selection criteria are applied to each event to remove spurious jet-like features originating from isolated noise patterns in certain HCAL regions. The combined secondary vertex version 2 algorithm [@btagging; @CMS:2016kkf] is used to identify jets that have originated from a b quark. The algorithm combines secondary vertex and track based lifetime information to discriminate b jets from light quark and gluon jets. The operating point used has a b jet identification (b tagging) efficiency of 80% and a light-flavor jet misidentification (mistag) probability of 10%. Our signal selection requires at least one of the two leading jets to be b-tagged. This requirement is critical in reducing the contributions from some SM background processes like W+jets. Scale factors to account for observed differences between data and simulation are applied as a function of . The event selection, which is optimized separately for the electron and muon channels, results in different requirements for the two channels. Most notably, the multijet background, through misidentification of showers, is significantly larger in the electron channel than in the muon channel. For electron events we therefore require higher $\abs{\ptvecmiss}$ and correspondingly lower leading jet than for muon events, in order to keep acceptance high for signal events. Events are required to have at least two jets with $\pt>30$and $\abs{\eta}<2.4$, and the leading jet must have $\pt>350\,(450)$in the electron (muon) channel. One lepton in each event is required to have fired a single-lepton trigger that has no isolation requirement, be within the detector acceptance ($\abs{\eta} <2.5$ for electrons, excluding the barrel endcap transition region, $1.444<\abs{\eta}<1.566$, and $\abs{\eta} < 2.4$ for muons) and be associated with a reconstructed primary vertex. For heavy resonance masses, the top quark from the decay is highly boosted, causing the b-jet and lepton to be close to each other. For this reason, leptons are not required to be isolated. Electrons and muons are required to have $\pt > 180$and to fulfill several identification criteria. Electron candidates are selected using a boosted decision tree based on the shower shape information, the quality of the track, the match between the track and electromagnetic cluster, the fraction of total cluster energy in the hadronic calorimeter, the amount of activity in the surrounding regions of the tracker and calorimeters, and the probability of the electron originating from a converted photon. The track associated with a muon candidate is required to have hits in the pixel and muon detectors, a good-quality fit, and be consistent with originating from the primary vertex. To reduce the multijet background, the candidate lepton is required to satisfy either $\Delta R$(lepton, nearest jet) $> 0.4$ or $\pt^{\text{rel}}$(lepton, nearest jet) ${>}60\,(50)$for electrons (muons), where $\Delta R=\sqrt{\smash[b]{(\Delta\eta)^2+(\Delta\phi)^2}}$ and $\pt^{\text{rel}}$ is defined as the magnitude of the lepton momentum orthogonal to the jet axis. Events with additional charged leptons with $\pt > 35$and $\abs{\eta} < 2.5$ for electrons and $\abs{\eta} < 2.4$ for muons are vetoed. The four-vectors of identified lepton candidate particles are subtracted from those of jets containing them. This procedure helps to ensure the reconstructed jets are not contaminated by nearby high-energy leptons as is common in the characteristic boosted signal topology. Scale factors resulting from small differences between lepton identification and trigger efficiencies in data and simulation are derived in a $\Z\to\ell\ell$ sample as a function of $\abs{\eta}$ and $\pt$ and applied as a correction to simulated events. Events are required to have at least $\abs{\ptvecmiss}>120\,(50)$in the electron (muon) channel. Additionally, events in the electron channel must have $\abs{\Delta\phi(\Pe,\ptvecmiss)}<2$ radians. These requirements are responsible for differences between the two channels in yields from some background processes. This selection, along with the other requirements, also helps reject nearly all multijet background events. Mass reconstruction ------------------- The tb invariant mass is reconstructed from the momenta of the charged lepton and two jets in the event, together with the . The transverse components of the neutrino momentum are set to the and the longitudinal component $p^\nu_z$ is calculated by constraining the invariant mass of the lepton and neutrino to the W boson mass. This method leads to a quadratic equation in $p^\nu_z$. In the case that the two solutions are real numbers, both solutions are used to reconstruct W boson candidates. If both solutions contain imaginary parts, then $p^\nu_z$ is set to the real part of the solutions, and then recompute $\pt^\nu$, which yields another quadratic ambiguity. In this case, we use only the solution with the mass closest to 80.4. Once all the components of the neutrino momentum have been assigned, the viable solutions for the neutrino are combined with the charged lepton to define W boson candidate(s). The top quark candidate is then reconstructed by combining the four-momenta of each W boson candidate with each jet with $\pt>25$and $\abs{\eta}<2.4$. The jet that yields a top quark mass closest to the nominal top quark mass is used to reconstruct the top quark candidate. In the case of two W candidates, only the candidate that yields the best top quark mass is used. Finally, the top quark candidate is combined with the highest jet remaining in the event, yielding the reconstructed $\PWpr$ candidate. The mass of the $\PWpr$ candidate is referred to as [$M_{\PQt\PQb}$]{}. Additional requirements that improve the rejection of background events are placed on the combinations of objects involved in the mass reconstruction. The top quark candidate is required to have $\pt^{\PQt}>250$and $100<m_{\PQt}<250$, and $\pt^\mathrm{j_1+j_2}>350$, where $\pt^\mathrm{j_1+j_2}$ is the of the four-vector sum of the two leading jets. Two event categories based on $\pt^{{\PQt}}$ and $\pt^\mathrm{j_1+j_2}$ are used when setting cross section limits. All events satisfying the above criteria are classified as Type A except for those with $\pt^{{\PQt}}>650$and $\pt^\mathrm{j_1+j_2}>700$, which are labeled Type B events. This categorization improves the sensitivity to high signal masses without sacrificing the performance for lower masses. Finally, events are also separated into two categories based on whether both (2 b tags) or only one (1 b tag) of the two leading jets is b-tagged. Event yields in all these categories after the event selection are shown in Table \[table:lep\_yields\]. Backgrounds\[sec:anstrat\] ========================== The W+jets background --------------------- For the W+jets background, the relative fractions of the heavy and light flavor components in simulation are known to differ from those in data [@Chatrchyan:2011yy]. The validity of the modeling of the flavor content is tested and two scale factors are derived for W+jets heavy and light flavor events using two samples that differ from the signal selection only in b tagging. The *pre tag* sample does not have any b tagging requirements, while the events in the *0 tag* sample must not have any b-tagged jets. In these two regions the relative fractions of the W+jets heavy and light flavor events are distinctly different. The yields from data and simulation in these two regions are used to solve a system of equations for the relative fractions of W+jets heavy and light flavor components, while requiring that the overall W+jets yield remains unchanged. Uncertainties are determined from repeating the calculation after varying the b tagging efficiencies and mistag rates within their uncertainties. The scale factors are found to be $2.10\pm^{0.21}_{0.18}$ and $0.49\pm^{0.08}_{0.10}$ for W+jets heavy and light flavor events, respectively. The corresponding scale factor is then applied to all simulated W+jets events. The top quark pair production background ---------------------------------------- For the background, we verify normalization as well as the modeling of the top quark . This check is performed in two signal-depleted -enriched regions: one that requires $450<{\ensuremath{M_{\PQt\PQb}}\xspace}<750$and at least two b tags, and another that removes the second-lepton veto and instead requires an additional electron or muon with a of at least 35. These comparisons motivate a reweighting of the background using a correction factor obtained from measurements of the differential top quark distribution. This correction factor is applied to the simulation, as a function of the generator-level top quark . The simulation without the correction factor applied is used as an estimate of the systematic uncertainty in the reweighting procedure. Systematic uncertainties\[sec:syst\] ==================================== The systematic uncertainties in this analysis can be grouped into two categories: uncertainties in the overall normalization and in the shape of the [$M_{\PQt\PQb}$]{}distribution. The normalization uncertainties include the uncertainty in the integrated luminosity (2.5%) [@CMS-PAS-LUM-17-001], the and W+jets cross sections (8 and 10%, respectively), the lepton identification (2%), and the trigger efficiencies (2%). The uncertainty due to variations in the renormalization and factorization scales ($\mu_\mathrm{R}$ and $\mu_\mathrm{F}$, respectively) is evaluated at the matrix element level using event weights from varying the scales by 0.5 and 2 while restricting to $0.5\le\mu_\mathrm{R}/\mu_\mathrm{F}\le2$ [@Cacciari:2003fi; @Catani:2003zt]. Uncertainties resulting from $\pm1$ standard deviation (s.d.) variations in the b tagging efficiency and mistagging rate scale factors, jet energy scale, and jet energy resolution are also included. A correction is applied to all simulated samples to better match the distribution of pileup interactions observed in data. This procedure uses a total inelastic cross section of 69.2, and an uncertainty is calculated by varying the cross section by $\pm$5% [@Aaboud:2016mmw]. To estimate the uncertainty arising from the choice of PDF, we evaluate the root-mean-square of the distribution of 100 NNPDF 3.0 replicas as the $\pm$1 s.d. uncertainties according to the guidelines in Ref. [@pdf4lhc]. When considering signal samples only the shape component of the uncertainty due to PDFs is included. The uncertainty in the W+jets heavy and light flavor scale factors is included as a variation in the W+jets background. The background with an uncorrected top quark spectrum is included as a one-sided $+1$ s.d. variation. All uncertainties are listed in Table \[table:systematics\]. The uncertainties with the largest effect on the overall background normalization are those associated with the top quark reweighting, $\mu_\mathrm{R}$ and $\mu_\mathrm{F}$ scales, and PDFs, which have effects of approximately 15, 15, and 6%, respectively. \[table:systematics\] Source Rate uncertainty Signal ---------------------------------------------- ------------------ -------- Integrated luminosity 2.5% $\ttbar$ cross section 8% — W+jets cross section 10% — Trigger eff. ($\Pe/\mu$) 2%/2% Lepton id. eff. ($\Pe/\mu$) 2%/2% \[2ex\] Jet energy scale 3% Jet energy resolution 1% b/c tagging 2% Light quark mistagging 2% Pileup 1% PDF 6% Top quark reweighting 15% — W+jets heavy/light flavor 1% — $\mu_\mathrm{R}$ and $\mu_\mathrm{F}$ scales 15% — Results\[sec:results\] ====================== Distributions of [$M_{\PQt\PQb}$]{}are shown in Figs. \[fig:lep\_mtb\_A\] and \[fig:lep\_mtb\_B\]. The binning is chosen to reduce uncertainties due to the size of the simulated event samples and is one bin from 0 to 500, eight bins of 200width from 500 to 2100, one bin from 2100 to 2400, one bin from 2400 to 3000, and one bin above 3000. Having observed that data agree with the predicted SM background processes, we set 95% confidence level (CL) upper limits on the $\PWpr$ boson production cross section for masses between 1 and 4. The analysis separates events into eight independent categories in order to improve the signal sensitivity. Categories are created according to lepton type (electron or muon), the number of b-tagged jets among the first two leading jets (1 or 2), and $\pt^{\PQt}$ and $\pt^\mathrm{j_1+j_2}$ (Type A or B). Categorization according to the number of b tags allows the analysis to maintain acceptance for signal events where one of the jets is not correctly b tagged, and categorization according to the $\pt^{\PQt}$ and $\pt^\mathrm{j_1+j_2}$ allows the analysis to perform well over a large range of possible signal masses. {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} Limits on the cross section of $\PWpr$ bosons are calculated using a Bayesian method with a prior uniform in the signal cross section, as implemented with the <span style="font-variant:small-caps;">theta</span> package [@theta-stat]. The Bayesian approach uses a binned likelihood in order to calculate the 95% CL upper limits on the product of the signal production and the branching fraction $\sigma(\Pp\Pp\to{\PWpr})\,\mathcal{B}(\PWpr\to\PQt\PQb)$. Statistical uncertainties related to the background prediction are treated using the “Barlow–Beeston lite" method [@barlowbeeston]. All uncertainties given in Section \[sec:syst\] are included as nuisance parameters. Uncertainties in the shape of the [$M_{\PQt\PQb}$]{}distribution are treated using template interpolation and all rate uncertainties are included with log-normal priors. Results for right-handed bosons are shown in Figs. \[fig:sepxseclim\_sep\] and \[fig:sepxseclim\]. $\PWpr_\mathrm{R}$ bosons with masses below 3.4are excluded at 95% CL. ![Upper limit at 95% CL on the $\PWpr_\mathrm{R}$ boson production cross section separately in the electron () and muon () channels. Signal masses for which the theoretical cross section (in red and blue for $M_{\nu_\mathrm{R}}\ll M_{\PWpr_\mathrm{R}}$ and $M_{\nu_\mathrm{R}}>M_{\PWpr_\mathrm{R}}$, respectively) exceeds the observed upper limit (in solid black) are excluded at 95% CL. The green and yellow bands represent the $\pm$1 and 2 s.d. uncertainties in the expected limit, respectively.\[fig:sepxseclim\_sep\]](Figure_003-a.pdf "fig:"){width="48.00000%"} ![Upper limit at 95% CL on the $\PWpr_\mathrm{R}$ boson production cross section separately in the electron () and muon () channels. Signal masses for which the theoretical cross section (in red and blue for $M_{\nu_\mathrm{R}}\ll M_{\PWpr_\mathrm{R}}$ and $M_{\nu_\mathrm{R}}>M_{\PWpr_\mathrm{R}}$, respectively) exceeds the observed upper limit (in solid black) are excluded at 95% CL. The green and yellow bands represent the $\pm$1 and 2 s.d. uncertainties in the expected limit, respectively.\[fig:sepxseclim\_sep\]](Figure_003-b.pdf "fig:"){width="48.00000%"} ![Upper limit at 95% CL on the $\PWpr_\mathrm{R}$ boson production cross section for the combined electron and muon channels. Signal masses for which the theoretical cross section (in red and blue for $M_{\nu_\mathrm{R}}\ll M_{\PWpr_\mathrm{R}}$ and $M_{\nu_\mathrm{R}}>M_{\PWpr_\mathrm{R}}$, respectively) exceeds the observed upper limit (in solid black) are excluded at 95% CL. The green and yellow bands represent the $\pm$1 and 2 s.d. uncertainties in the expected limit, respectively.\[fig:sepxseclim\]](Figure_004.pdf){width="48.00000%"} Although models with a boson that couples exclusively to right-handed fermions are simpler because of the lack of interference, the effective Lagrangian in Eq. (\[eq:lag\]) allows us to analyze models with arbitrary combinations of left- and right-handed couplings. In order to accomplish this the interference between the SM $s$-channel tb production and the tb production via an intermediate left-handed W’ boson must be accounted for since these processes initial and final states are identical. The cross section for single top quark production given a boson can be written for any set of $a_\mathrm{L}$ and $a_\mathrm{R}$ coupling values in terms of the cross sections of four simulated signal samples. It is assumed that the couplings to fermions are independent of generation, such that each signal can be described by a single value of $a_\mathrm{L}$ and a single value of $a_\mathrm{R}$. The four simulated signals are then $\sigma_\mathrm{L}$ for purely left-handed couplings $(a_\mathrm{L},a_\mathrm{R})=(1,0)$, $\sigma_\mathrm{R}$ for purely right-handed couplings $(a_\mathrm{L},a_\mathrm{R})=(0,1)$, $\sigma_\mathrm{LR}$ for mixed couplings $(a_\mathrm{L},a_\mathrm{R})=(1/\sqrt{2},1/\sqrt{2})$, and $\sigma_\mathrm{SM}$ for SM couplings $(a_\mathrm{L},a_\mathrm{R})=(0,0)$, and the cross section for single top quark production is By combining four signal samples according to this equation we are able to produce invariant mass distributions for a boson with arbitrary $a_\mathrm{L}$ and $a_\mathrm{R}$ couplings. A notable adjustment for this paper with respect to previous CMS publications is in the definition of the mixed coupling sample, which was previously defined as $(a_\mathrm{L},a_\mathrm{R})=(1,1)$. This change results in slightly different expressions for the total cross section, and is chosen to ensure that the widths of all three simulated signal samples are identical. It should be noted that in the case that the boson couples exclusively to right-handed fermions, this equation reduces to the sum of SM $s$-channel tb production and $\PWpr_\mathrm{R}$ production, as expected. For pure $\PWpr_\mathrm{L}$ or $\PWpr_\mathrm{LR}$ boson production, the equation reduces to the cross section of the respective sample, which is generated already including SM $s$-channel tb production and interference with $\PWpr$ production. A scan is performed over the $a_\mathrm{L}$ and $a_\mathrm{R}$ plane in 0.1 steps from 0 to 1 to produce cross section limits for arbitrary combinations of $a_\mathrm{L}$ and $a_\mathrm{R}$. For each point in the scan the expected and observed 95% CL upper limits on the cross section are calculated using the same method described above. Figure \[fig:2dlimit\] shows the excluded boson mass for each ($a_\mathrm{L}$, $a_\mathrm{R}$) point, in addition to an interpolation between points to create smooth contours of equivalent signal mass limits. ![Expected () and observed () limits on the boson mass as function of the left-handed ($a_\mathrm{L}$) and right-handed ($a_\mathrm{R}$) couplings. Black lines represent contours of equal boson mass separated by 200. \[fig:2dlimit\]](Figure_005-a.pdf "fig:"){width="48.00000%"} ![Expected () and observed () limits on the boson mass as function of the left-handed ($a_\mathrm{L}$) and right-handed ($a_\mathrm{R}$) couplings. Black lines represent contours of equal boson mass separated by 200. \[fig:2dlimit\]](Figure_005-b.pdf "fig:"){width="48.00000%"} Summary\[sec:summary\] ====================== A search for a narrow heavy $\PWpr$ boson resonance decaying to a top quark and a bottom quark has been performed in $\mbox{lepton+jets}$ final states using data collected at $\sqrt{s}=13$by the CMS detector in 2016, corresponding to an integrated luminosity of 35.9. No evidence is observed for the production of a $\PWpr$ boson, and 95% CL upper limits on the product of the right-handed $\PWpr$ ($\PWpr_\mathrm{R}$) boson production cross section and its branching fraction to a top and a bottom quark are calculated as a function of the $\PWpr_\mathrm{R}$ boson mass. The observed (expected) 95% CL upper limit is 3.4 (3.3)if $M_{\PWpr_\mathrm{R}}\gg M_{\nu_\mathrm{R}}$ and 3.6 (3.5)if $M_{\PWpr_\mathrm{R}}<M_{\nu_\mathrm{R}}$, where $M_{\nu_\mathrm{R}}$ is the mass of the right-handed neutrino. Exclusion limits are also presented for $\PWpr$ bosons with varied left- and right-handed couplings to fermions, for the first time at $\sqrt{s}=13$. These results are the most stringent limits to date on the production of bosons that decay to a top and a bottom quark. We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMWFW and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF (Cyprus); SENESCYT (Ecuador); MoER, ERC IUT, and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NIH (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of Korea); LAS (Lithuania); MOE and UM (Malaysia); BUAP, CINVESTAV, CONACYT, LNS, SEP, and UASLP-FAI (Mexico); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS, RFBR and RAEP (Russia); MESTD (Serbia); SEIDI, CPAN, PCTI and FEDER (Spain); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR, and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU and SFFR (Ukraine); STFC (United Kingdom); DOE and NSF (USA). Individuals have received support from the Marie-Curie programme and the European Research Council and Horizon 2020 Grant, contract No. 675440 (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Council of Science and Industrial Research, India; the HOMING PLUS programme of the Foundation for Polish Science, cofinanced from European Union, Regional Development Fund, the Mobility Plus programme of the Ministry of Science and Higher Education, the National Science Center (Poland), contracts Harmonia 2014/14/M/ST2/00428, Opus 2014/13/B/ST2/02543, 2014/15/B/ST2/03998, and 2015/19/B/ST2/02861, Sonata-bis 2012/07/E/ST2/01406; the National Priorities Research Program by Qatar National Research Fund; the Programa Clarín-COFUND del Principado de Asturias; the Thalis and Aristeia programmes cofinanced by EU-ESF and the Greek NSRF; the Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University and the Chulalongkorn Academic into Its 2nd Century Project Advancement Project (Thailand); and the Welch Foundation, contract C-1845. The CMS Collaboration \[app:collab\] ==================================== =5000=500=5000

--- abstract: 'Existence check of non-trivial, stationary axisymmetric black hole solutions in Brans-Dicke theory of gravity in different direction from those of Penrose, Thorne and Dykla, and Hawking is performed. Namely, working directly with the known explicit spacetime solutions in Brans-Dicke theory, it is found that non-trivial Kerr-Newman-type black hole solutions [*different*]{} from general relativistic solutions could occur for the generic Brans-Dicke parameter values $-5/2\leq \omega <-3/2$. Finally, issues like whether these new black holes carry scalar hair and can really arise in nature and if they can, what the associated physical implications would be are discussed carefully.' address: | Department of Astronomy and Atmospheric Sciences\ Kyungpook National University, Taegu, 702-701, KOREA author: - 'Hongsu Kim[^1]' date: 'October, 1998' title: 'New Black Hole Solutions in Brans-Dicke Theory of Gravity' --- [**I. Introduction**]{}\ Of all the alternative theories of classical gravity to Einstein’s general relativity, perhaps the Brans-Dicke (BD) theory \[1\] is the most studied and hence the best-known. This theory can be thought of as a minimal extension of general relativity designed to properly accomodate both Mach’s principle \[2\] and Dirac’s large number hypothesis \[2\]. Namely, the theory employs the viewpoint in which the Newton’s constant $G$ is allowed to vary with space and time and can be written in terms of a scalar (“BD scalar”) field as $G = 1/ \Phi $. In this work, we are interested in the existence of exact solutions to the BD field equations that can describe rotating, charged black hole spacetimes and their detailed structure. And if there are, we would like to know whether they are non-trivial ones different from general relativistic black hole solutions. As is well-known, even in Einstein’s general relativity, to find the exact solutions to the highly non-linear Einstein field equations is a formidable task. For this reason, algorithms generating exact, new solutions from the known solutions of simpler situations have been actively looked for and actually quite a few were found. In BD theory of gravity, the field equations are even more complex and thus it is natural to seek similar algorithms generating exact solutions from the already known simpler solutions either of the BD theory or of the conventional Einstein gravity. To the best of our knowledge, methods thus far discovered along this line includes those of Janis et al., Buchdahl, McIntosh, Tupper, Tiwari and Nayak, and Singh and Rai \[3\]. In particular, Tiwari and Nayak \[3\] proposed an algorithm that allows us to generate stationary, axisymmetric solutions in vacuum BD theory from the known Kerr solution \[6\] in vacuum Einstein theory and later on Singh and Rai \[3\] generalized this method to the one that generates stationary, axisymmetric, charged solutions in BD-Maxwell theory from the known Kerr-Newman (KN) solution \[6\] in Einstein-Maxwell theory. Thus in the present work, we shall take, as the Kerr-Newman-type solutions in BD-Maxwell theory (henceforth “BDKN” solutions), the ones constructed by Singh and Rai to explore if it can descibe non-trivial black hole spacetimes [*different*]{} from those described by the standard KN solution in Einstein-Maxwell theory.\ [**II. Non-trivial BDKN black hole solutions**]{}\ We begin by briefly reviewing the algorithm proposed first by Tiwari and Nayak and generalized later by Singh and Rai. Consider the BD-Maxwell theory described by the action $$\begin{aligned} S = \int d^4x \sqrt{g}\left[{1\over 16\pi}\left(\Phi R - \omega {{\nabla_{\alpha}\Phi \nabla^{\alpha}\Phi }\over \Phi}\right) - {1\over 4}F_{\alpha \beta}F^{\alpha \beta}\right]\end{aligned}$$ where $\Phi $ is the BD scalar field and $\omega $ is the generic parameter of the theory. Extremizing this action then with respect to the metric $g_{\mu \nu}$, the BD scalar field $\Phi $, and the Maxwell gauge field $A_{\mu}$ (with the field strength $F_{\mu \nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$) yields the classical field equations given respectively by $$\begin{aligned} G_{\mu \nu} &=& R_{\mu \nu} - {1\over 2}g_{\mu \nu}R = {8\pi \over \Phi}T^{M}_{\mu \nu} + 8\pi T^{BD}_{\mu \nu}, \nonumber \\ {\rm where} \\ T^{M}_{\mu \nu} &=& F_{\mu \alpha}F_{\nu}^{\alpha} - {1\over 4}g_{\mu \nu}F_{\alpha \beta} F^{\alpha \beta}, \nonumber \\ T^{BD}_{\mu \nu} &=& {1\over 8\pi}\left[{\omega \over \Phi^2}(\nabla_{\mu}\Phi \nabla_{\nu}\Phi - {1\over 2}g_{\mu \nu}\nabla_{\alpha}\Phi \nabla^{\alpha}\Phi) + {1\over \Phi}(\nabla_{\mu} \nabla_{\nu}\Phi - g_{\mu \nu}\nabla_{\alpha}\nabla^{\alpha}\Phi)\right] \nonumber \\ {\rm and} \nonumber \\ \nabla_{\alpha}\nabla^{\alpha}\Phi &=& {8\pi \over (2\omega + 3)}T^{M\lambda}_{\lambda} = 0, ~~~~\nabla_{\mu}F^{\mu \nu} = 0, ~~~~\nabla_{\mu}\tilde{F}^{\mu \nu} = 0 \nonumber \end{aligned}$$ with the last equation being the Bianchi identity and $\tilde{F}_{\mu \nu}={1\over 2} \epsilon_{\mu \nu}^{\alpha \beta}F_{\alpha \beta}$. And the Einstein-Maxwell theory is the $\omega \rightarrow \infty$ limit of this BD-Maxwell theory. Note that in the action and hence in the classical field equations, there are no [*direct*]{} interactions between the BD scalar field $\Phi$ and the ordinary matter, i.e., the Maxwell gauge field $A_{\mu}$. Indeed this is the essential feature of the BD scalar field $\Phi$ that distinguishes it from “dilaton” fields in other scalar-tensor theories such as Kaluza-Klein theories or low-energy effective string theories where the dilaton-matter couplings generically occur as a result of dimensional reduction. (Here we would like to stress that we shall work in the context of original BD theory format not some conformal transformation of it.) As a matter of fact, it is the original spirit \[1\] of BD theory of gravity in which the BD scalar field $\Phi$ is prescribed to remain strictly massless by forbidding its direct interaction with matter fields. Now the algorithm of Tiwari and Nayak, and Singh and Rai goes as follows. Let the metric for a stationary, axisymmetric, charged solution to Einstein-Maxwell field equations take the form $$\begin{aligned} ds^2 = - e^{2U_{E}}(dt + W_{E}d\phi)^2 + e^{2(k_{E}-U_{E})}[(dx^1)^2 + (dx^2)^2] + h^2_{E}e^{-2U_{E}}d\phi^2 \end{aligned}$$ while the metric for a stationary, axisymmetric, charged solution to BD-Maxwell field equations be $$\begin{aligned} ds^2 = - e^{2U_{BD}}(dt + W_{BD}d\phi)^2 + e^{2(k_{BD}-U_{BD})}[(dx^1)^2 + (dx^2)^2] + h^2_{BD}e^{-2U_{BD}}d\phi^2\end{aligned}$$ where $U$, $W$, $k$ and $h$ are functions of $x^1$ and $x^2$ only. The significance of the choice of the metric in this form has been thoroughly discussed by Matzner and Misner \[4\] and Misra and Pandey \[5\]. Tiwari and Nayak, and Singh and Rai first wrote down the Einstein-Maxwell and BD-Maxwell field equations for the choice of metrics in eq.(3) and (4) respectively. Comparing the two sets of field equations closely, they realized that stationary, axisymmetric solutions of the BD-Maxwell field equations are obtainable from those of Einstein-Maxwell field equations provided certain relations between metric functions hold.\ That is, if $(W_{E}, ~k_{E}, ~U_{E}, ~h_{E}, ~A^{E}_{\mu})$ form a stationary, axisymmetric solution to the Einstein-Maxwell field equations for the metric in eq.(3), then a corresponding stationary, axisymmetric solution to the BD-Maxwell field equations for the metric in eq.(4) is given by $(W_{BD}, ~k_{BD}, ~U_{BD}, ~h_{BD}, ~A^{BD}_{\mu})$ where $$\begin{aligned} W_{BD} &=& W_{E}, ~~~k_{BD} = k_{E}, ~~~U_{BD} = U_{E} - {1\over 2}\log \Phi, \\ h_{BD} &=& [h_{E}]^{(2\omega - 1)/(2\omega + 3)}, ~~~\Phi = [h_{E}]^{4/(2\omega + 3)}, ~~~A^{BD}_{\mu} = A^{E}_{\mu}. \nonumber\end{aligned}$$ Now what remains is to apply this method to obtain the Kerr-Newman-type solution in BD-Maxwell theory (BDKN solution) from the known Kerr-Newman (KN) solution \[6\] in Einstein-Maxwell theory. And to do so, one needs some preparation which involves casting the KN solution given in Boyer-Lindquist coordinates \[7\] $(t, r, \theta, \phi)$ in the metric form in eq.(3) by performing a coordinate transformation (of $r$ alone) suggested by Misra and Pandey \[5\]. Namely, we start with the KN solution written in Boyer-Lindquist coordinates $$\begin{aligned} ds^2 &=& -dt^2 + \Sigma (d\theta^2 + {dr^2\over \Delta}) + (r^2 + a^2)\sin^2 \theta d\phi^2 + {(2Mr - e^2)\over \Sigma}[dt - a\sin^2 \theta d\phi]^2, \nonumber \\ A_{\mu} &=& -{er\over \Sigma}[\delta^{t}_{\mu} - a\sin^2 \theta \delta^{\phi}_{\mu}] \end{aligned}$$ where $\Sigma = r^2+a^2 \cos^2 \theta $ and $\Delta = r^2 -2Mr + a^2 + e^2$ with $M$, $a$, and $e$ denoting the ADM mass, angular momentum per unit mass, and the electric charge respectively. Consider now the transformation of the radial coordinate introduced by Misra and Pandey \[5\] $$\begin{aligned} r = e^{R} + M + {(M^2-a^2-e^2)\over 4}e^{-R}\end{aligned}$$ which gives $dr^2/ \Delta = dR^2$. Then the KN solution can now be cast in the form in eq.(3), i.e., $$\begin{aligned} ds^2 = -[{\Delta - a^2\sin^2 \theta \over \Sigma}](dt + W d\phi)^2 + \Sigma [d\theta^2 + dR^2 + {\Delta \sin^2 \theta \over {\Delta - a^2\sin^2 \theta}}d\phi^2]\end{aligned}$$ with now $\Sigma = L^2+a^2 \cos^2 \theta $ and $\Delta = L^2 -2ML + a^2 + e^2$ where we set, as a short-hand notation, $L \equiv e^{R} + M + {(M^2-a^2-e^2)\over 4}e^{-R}$. Now we can read off the metric components as $$\begin{aligned} e^{2U_{E}} &=& [{\Delta - a^2\sin^2 \theta \over \Sigma}], ~~~W_{E} = {{a\sin^2 \theta (L^2 + a^2 - \Delta)}\over {\Delta - a^2\sin^2 \theta}}, \nonumber \\ e^{2k_{E}} &=& (\Delta - a^2\sin^2 \theta), ~~~h^2_{E} = \Delta \sin^2 \theta. \end{aligned}$$ Then using the rule in eq.(5) in the algorithm by Tiwari and Nayak, and Singh and Rai, we can now construct BDKN solution in BD-Maxwell theory as $$\begin{aligned} ds^2 &=& - \left({{L^2 + a^2\cos^2 \theta - 2ML + e^2}\over {L^2 + a^2\cos^2 \theta}}\right) (L^2 + a^2 - 2ML + e^2)^{-2/(2\omega+3)}\sin^{-4/(2\omega+3)}\theta \nonumber \\ &\times &\left[dt + {{a\sin^2 \theta (2ML-e^2)}\over {L^2 + a^2\cos^2 \theta - 2ML + e^2}}d\phi \right]^2 \nonumber \\ &+& (L^2 + a^2 - 2ML + e^2)^{2/(2\omega+3)}\sin^{4/(2\omega+3)}\theta (L^2 + a^2\cos^2 \theta) [d\theta^2 + dR^2] \nonumber \\ &+& (L^2 + a^2 - 2ML + e^2)^{(2\omega+1)/(2\omega+3)}\sin^{2(2\omega+1)/(2\omega+3)}\theta \left({{L^2 + a^2\cos^2 \theta}\over {L^2 + a^2\cos^2 \theta - 2ML + e^2}}\right)d\phi^2, \nonumber \\ \Phi (R, \theta) &=& (L^2 + a^2 - 2ML + e^2)^{2/(2\omega+3)}\sin^{4/(2\omega+3)}\theta, \\ A_{\mu} &=& -{eL \over {L^2 + a^2\cos^2 \theta}}[\delta^{t}_{\mu} - a\sin^2 \theta \delta^{\phi}_{\mu}]. \nonumber\end{aligned}$$ Then by transforming back to the standard Boyer-Lindquist coordinates using eq.(7), we finally arrive at the BDKN solution in Boyer-Lindquist coordinates given by $$\begin{aligned} ds^2 &=& \Delta^{-2/(2\omega+3)}\sin^{-4/(2\omega+3)}\theta \left[ - \left({\Delta - a^2\sin^2 \theta \over \Sigma}\right)dt^2 - {2a\sin^2 \theta (r^2+a^2-\Delta)\over \Sigma}dt d\phi \right. \nonumber \\ &+&\left. \left({(r^2+a^2)^2 - \Delta a^2\sin^2 \theta \over \Sigma }\right)\sin^2 \theta d\phi^2 \right] + \Delta^{2/(2\omega+3)}\sin^{4/(2\omega+3)}\theta \left[ {\Sigma \over \Delta}dr^2 + \Sigma d\theta^2 \right], \nonumber \\ \Phi (r, \theta) &=& \Delta^{2/(2\omega+3)}\sin^{4/(2\omega+3)}\theta, ~~~A_{\mu} = -{er\over \Sigma}[\delta^{t}_{\mu} - a\sin^2 \theta \delta^{\phi}_{\mu}]. \end{aligned}$$ Note that as $\omega \rightarrow \infty$, this BDKN solution goes over to the standard KN solution as it should since the $\omega \rightarrow \infty$ limit of BD theory is the Einstein gravity. Now that we have an exact, electrovac, stationary axisymmetric solution to the BD-Maxwell theory. Then it is natural to ask if this BDKN solution can describe a black hole spacetime resulted from a gravitational collapse. And if it can, furthermore one might be curious whether this BDKN solution or its three special cases (i.e., BD-Schwarzschild, BD-Reissner-Nordstrom or BD-Kerr solutions) could describe non-trivial black hole spacetimes which are [*different*]{} from those described by their general relativistic counterparts. Indeed, questions of this sort had been raised long ago, and actually Penrose \[8\] [*conjectured*]{} that even in BD theory of gravity, the relativistic gravitational collapse in three spatial dimensions would produce black holes identical to those in general relativity. And this conjecture received some support from the work of Thorne and Dykla \[8\] in which they presented four pieces of evidence in favor of the conjecture by employing mainly the “large-$\omega $” expansion scheme (recall that in the limit $\omega \rightarrow \infty$, the BD theory goes over to the general relativity). As Thorne and Dykla mentioned in their work, however, the conjecture of Penrose was not fully proved since detailed analysis of the collapse with arbitrary, finite values of the generic BD parameter $\omega$ is needed. In this regard, we now seem to be in a better shape toward the serious investigation on the validity of the conjecture since we have an exact, stationary axisymmetric solution possessing arbitrary $\omega $ values which was not available at the time. Therefore we begin with the bottomline qualification for the BDKN solution in BD-Maxwell theory to describe a rotating, charged black hole spacetime, namely the possible occurrence of non-singular event horizon. Then along this line, perhaps the most natural first step is to ask under what circumstances the Killing horizons develop. Just like the KN solution in Einstein-Maxwell theory, this BDKN solution is stationary and axisymmetric and hence possesses the time translational Killing field $\xi^{\mu}=(\partial /\partial t)^{\mu}$ and the rotational Killing field $\psi^{\mu}=(\partial /\partial \phi)^{\mu}$ correspondingly and it is their linear combination, $\chi^{\mu} = \xi^{\mu} + \Omega_{H}\psi^{\mu}$ which is normal to the Killing horizons, if any (here $\Omega_{H}$ denotes the angular velocity of the Killing horizon). And if Killing horizons are present, they occur at points where $\chi^{\mu}$ becomes null which turn out to be zeroes of $\Delta^{(2\omega+1)/(2\omega+3)}=0$. Thus first for $\omega = -1/2$, obviously no horizon occurs. Next for $\left({2\omega +1 \over 2\omega +3}\right) > 0$, i.e., for $\omega < -3/2$ or $\omega > -1/2$, two Killing horizons occur at $r_{\pm} = M \pm (M^2-a^2-e^2)^{1/2}$ (provided $M^2\geq a^2+e^2$) which are precisely the same locations as those of Killing horizons of KN black holes in Einstein-Maxwell theory. Now since the formation of horizons appears to be possible, next we investigate their nature. And to this end, we examine behaviors of the invariant curvature polynomials such as $R$, $R_{\mu\nu}R^{\mu\nu}$ and $R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$, the surface gravity $\kappa$ and the energy density of the BD scalar field $T^{BD}_{\mu \nu}\xi^{\mu}\xi^{\nu}$ on these candidates for Killing horizons. And as we mentioned above, since the bottomline qualification for the black hole interpretation of BDKN solution is the regularity of the horizon candidate, we begin with the examination of behavior of invariant curvature polynomials on the horizon candidate at which $\Delta = 0$. First, the curvature scalar is calculated to be $$\begin{aligned} R &=& {\omega \over \Phi^2}g^{\alpha \beta}\nabla_{\alpha}\Phi \nabla_{\beta}\Phi + {3\over \Phi}\nabla_{\alpha}\nabla^{\alpha}\Phi \\ &=& \omega \left({4\over 2\omega+3}\right)^2 {1\over \Sigma}\sin^{-4/(2\omega+3)}\theta [(r-M)^2\Delta^{-(2\omega+5)/(2\omega+3)} + \cot^2 \theta \Delta^{-2/(2\omega+3)}]. \nonumber\end{aligned}$$ As was the case with KN black hole solutions, it also blows up at $\Sigma = 0$ (i.e., $r=0$, $\theta = \pi/2$) indicating that the BDKN black hole solution also has the curvature singularity with the same “ring” structure. The direct computation of the other two curvature polynomials, i.e., the Ricci square $R_{\mu\nu}R^{\mu\nu}$ and the Kretschmann curvature invariant $R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$ for this BDKN solution is a formidable job. But a close inspection reveals that indeed we can save considerable amount of labor. Namely, consider now the Brans-Dicke-Schwarzschild (BDS) spacetime solution that can be obtained by setting $a = e = 0$ in the BDKN solution in eq.(11) $$\begin{aligned} ds^2 &=& \Delta^{-2/(2\omega+3)}\sin^{-4/(2\omega+3)}\theta \left[-\left(1 - {2M\over r}\right)dt^2 + r^2 \sin^2 \theta d\phi^2 \right] \nonumber \\ &+& \Delta^{2/(2\omega+3)}\sin^{4/(2\omega+3)}\theta \left[\left(1 - {2M\over r}\right)^{-1}dr^2 + r^2 d\theta^2 \right], \\ &\Phi& (r, \theta) = \Delta^{2/(2\omega+3)}\sin^{4/(2\omega+3)}\theta \nonumber\end{aligned}$$ where now $\Delta = r(r - 2M)$. A remarkable feature of this BDS solution is the fact that, unlike the Schwarzschild solution in general relativity, the spacetime it describes is static (i.e., non-rotating) but [*not*]{} spherically-symmetric. Thus computing $R_{\mu\nu}R^{\mu\nu}$ and $R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$ for this BDS solution and examining their behaviors on the horizon candidate at which $\Delta = 0$ would be sufficient to envisage the possibility of regular event horizon for both BDS and BDKN solutions. In addition, another noticeable characteristic of both BDKN and BDS spacetime solutions is that they have possible coordinate singularities not only at the outer event horizon where $\Delta = 0$ but also along the symmetry axis $\theta = 0, ~\pi$. Thus in order to explore the nature of this metric singularity along the symmetry axis, the computation of invariant curvature polynomials looks necessary. The result of the computation of the Ricci square and the Kretschmann curvature invariant for this BDS solution is given in the appendix. And it is a straightforward matter to realize that the two invariant curvature polynomials $R_{\mu\nu}R^{\mu\nu}$ and $R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$ given in the appendix and the curvature scalar $R$ given above in eq.(12) become finite both on the horizon candidate at which $\Delta = 0$ and along the symmetry axis $\theta = 0, ~\pi$ provided the generic BD $\omega$-parameter takes values in the range $-5/2 \leq \omega < -3/2$. Next, we turn to the computation of surface gravity at the Killing horizons, $\kappa_{\pm}$. Generally, the surface gravity $\kappa$ is defined in a gravity theory- independent manner as follows. Since the horizon is a null surface, there we have $\chi^{\mu}\chi_{\mu}=0$ where $\chi^{\mu}$ is the Killing field normal to the horizon we defined above. This implies that $\nabla^{\mu}(\chi^{\nu}\chi_{\nu})$ is also normal to the horizon. Thus on the horizon, there exists a function $\kappa$ such that $$\begin{aligned} \nabla^{\mu}(\chi^{\nu}\chi_{\nu}) = - 2\kappa \chi^{\mu} ~~~~{\rm or} ~~~~\chi^{\nu}\nabla_{\nu}\chi_{\mu} = \kappa \chi_{\mu} \nonumber\end{aligned}$$ from which it can be derived that $$\begin{aligned} \kappa^2 = -{1\over 2}(\nabla^{\mu}\chi^{\nu})(\nabla_{\mu}\chi_{\nu})\end{aligned}$$ where the evaluation on the horizon is understood. Now, for the non-trivial BDKN black hole solution at hand, a straightforward albeit somewhat tedious calculation yields $$\begin{aligned} \kappa_{\pm} = \Delta^{-2/(2\omega+3)}(r_{\pm})\sin^{-4/(2\omega+3)}\theta \left({2\omega +1 \over 2\omega +3}\right) {(r_{\pm} - r_{\mp})\over 2(r^2_{\pm}+a^2)}.\end{aligned}$$ Lastly, the energy density of the BD scalar field is computed using eq.(2) as $$\begin{aligned} &T^{BD}_{\mu \nu}&\xi^{\mu}\xi^{\nu} = T^{BD}_{tt} \\ &=& {1\over 8\pi}\Delta^{-4/(2\omega+3)}\sin^{-8/(2\omega+3)}\theta \left[{\omega \over 2} \left({4\over 2\omega +3}\right)^2 \left({(r-M)^2\over \Delta} + \cot^2 \theta \right) \left({\Delta - a^2\sin^2 \theta \over \Sigma^2}\right) \right. \nonumber \\ &+& \left({4\over 2\omega+3}\right){1\over \Sigma^2}\left\{\cot^2 \theta \left(\left({2\omega+1 \over 2\omega +3}\right)a^2\sin^2 \theta + \left({2\over 2\omega +3}\right)\Delta - a^2\sin^2 \theta \left({\Delta - a^2\sin^2 \theta \over \Sigma}\right)\right) \right. \nonumber \\ &-& \left. \left. \left({r-M \over \Delta}\right)\left(\left({2\omega+1 \over 2\omega +3}\right)(r-M)\Delta + \left({2\over 2\omega +3}\right)(r-M)a^2\sin^2 \theta - r\Delta \left({\Delta - a^2\sin^2 \theta \over \Sigma}\right)\right)\right\} \right]. \nonumber\end{aligned}$$ It is interesting to note that this energy density of the BD scalar field also blows up at the curvature singularity $\Sigma = 0$.\ Now (i) for $\omega \rightarrow \infty$, $R=0$, $\kappa_{\pm}=(r_{\pm} - r_{\mp})/2(r^2_{\pm}+a^2)$ and $T^{BD}_{\mu \nu}\xi^{\mu}\xi^{\nu}=0$ on the surfaces $r=r_{\pm}$. This is an anticipated result since this is the correct KN black hole limit in Einstein-Maxwell theory. (ii) Next for $\infty > \omega > -1/2$, on the surfaces $r=r_{\pm}$, $(R, ~R_{\mu\nu}R^{\mu\nu}, ~R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}) \rightarrow \infty$, $\kappa_{\pm} \rightarrow \infty$ and $T^{BD}_{\mu \nu}\xi^{\mu}\xi^{\nu}\rightarrow \infty$ with $\Phi (r_{\pm}, \theta)=0$. This indicates that the surfaces $r=r_{\pm}$ are singular and fail to act as horizons and hence the corresponding metric cannot describe a black hole spacetime. (iii) Finally for $\omega < -3/2$, or more precisely for $-5/2 \leq \omega <-3/2$, on the surfaces $r=r_{\pm}$, $(R, ~R_{\mu\nu}R^{\mu\nu}, ~R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}) = 0$ (or $const.$ particularly for $\omega = -5/2$), $\kappa_{\pm}=0$ and $T^{BD}_{\mu \nu}\xi^{\mu}\xi^{\nu}=0$ with $\Phi (r_{\pm}, \theta)\rightarrow \infty$. Namely the curvature invariants are finite, surface gravity is zero and the BD scalar field satisfies the weak energy condition although its value diverges there. (Here, infinite value of $\Phi$ indicates that the effective Newton’s constant tends to zero.) Thus in this range of the $\omega $-values, the surfaces $r=r_{\pm}$ may act as regular Killing horizons and hence the corresponding BDKN metric solution appears to describe non-trivial black hole spacetimes [*different*]{} from those in Einstein-Maxwell theory. In particular for $\omega = -5/2$, the corresponding non-trivial BDKN black hole solution singles out with a relatively simple form given by $$\begin{aligned} ds^2 &=& - \left[{\Delta - a^2\sin^2 \theta \over \Sigma}\right]\Delta \sin^2 \theta dt^2 - {2a\sin^4 \theta (r^2+a^2-\Delta)\Delta \over \Sigma}dt d\phi \nonumber \\ &+& \left[{(r^2+a^2)^2 - \Delta a^2\sin^2 \theta \over \Sigma }\right]\Delta \sin^4 \theta d\phi^2 + {\Sigma \over \Delta^2 \sin^2 \theta }dr^2 + {\Sigma \over \Delta \sin^2 \theta } d\theta^2, \nonumber \\ \Phi (r, \theta) &=& {1\over \Delta \sin^2 \theta}.\end{aligned}$$ Lastly, for the rest of the $\omega $-values, i.e., for $\omega <-5/2$, the surfaces $r=r_{\pm}$ are singular horizons on which $(R,~R_{\mu\nu}R^{\mu\nu}, ~R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}) \rightarrow \infty$, $\kappa_{+}=0$ and $T^{BD}_{\mu \nu}\xi^{\mu}\xi^{\nu}=0$ and for $-3/2<\omega \leq -1/2$, no Killing horizon develops and thus the curvature singularity at $\Sigma =0$ is naked. Therefore it now appears that for the values of the generic BD $\omega $-parameter in the limited range $-5/2 \leq \omega < -3/2$, the BDKN solution in BD-Maxwell theory may describe non-trivial black hole spacetimes.\ [**III. Nature of BDKN black holes**]{}\ Now that we have non-trivial BDKN black hole solutions. It seems then natural to explore its thermodynamics and causal structure in some more detail. Firstly, these BDKN black hole solutions have vanishing surface gravity at the event horizon, $\kappa_{+}=0$ and hence [*zero*]{} Hawking temperature, $T_{H}=\kappa_{+}/2\pi=0$. In other words, they do not radiate and hence are completely “dark and cold”. Certainly, this is a very bizzare feature in sharp contrast to evaporating black holes in general relativity. Next, we turn to their causal structure. As noted earlier, the two Killing horizons, i.e., the outer event horizon and the inner Cauchy horizon turn out to occur precisely at the same locations (i.e., same coordinate distances) as those of KN black hole solutions in Einstein-Maxwell theory, i.e., at $r_{\pm} = M \pm (M^2-a^2-e^2)^{1/2}$. Also it is interesting to note that the proper area of the event horizon at $r = r_{+}$, $$\begin{aligned} A = \int_{r_{+}} d\theta d\phi (g_{\theta \theta}g_{\phi \phi})^{1/2} = 4\pi (r^2_{+}+a^2)\end{aligned}$$ is again exactly the same as that of standard KN black hole spacetime. In addition, its angular velocity at the event horizon coincides with that of standard KN solution as well $$\begin{aligned} -W^{-1}_{BD}(r_{+}) = {a\over {r^2_{+}+a^2}} = -W^{-1}_{E}(r_{+}).\end{aligned}$$ Next, observe that the norm of the time translational Killing field $$\begin{aligned} \xi^{\mu}\xi_{\mu} = g_{tt} = -\Delta^{-2/(2\omega+3)}\sin^{-4/(2\omega+3)}\theta \left[{\Delta - a^2\sin^2 \theta \over \Sigma}\right]\end{aligned}$$ goes like negative $(r_{-}<r<r_{+})$ $\rightarrow$ positive $(r_{+}<r<r_{s})$ $\rightarrow$ negative $(r>r_{s})$ with $r_{s} = M + (M^2-a^2\cos^2 \theta - e^2)^{1/2} > r_{+}$ being the larger root of $\xi^{\mu}\xi_{\mu}$, indicating that $\xi^{\mu}$ behaves as timelike $\rightarrow$ spacelike $\rightarrow$ timelike correspondingly. And particularly the region in which $\xi^{\mu}$ stays spacelike extends outside hole’s event horizon. This region is the so-called “ergoregion” and its outer boundary on which $\xi^{\mu}$ becomes null, i.e., $r=r_{s}$ is called “static limit” since inside of which no observer can possibly remain static. Thus if we recall the location of the static limit in standard KN black hole solution, we can realize that even the locations of ergoregions in two black hole spacetimes, KN and BDKN, are the same as well. Namely in two theories, i.e., the BD-Maxwell theory and the Einstein-Maxwell theory, rotating, charged black hole solutions turn out to possess [*identical*]{} causal structure (i.e., the locations of ring singularities, two Killing horizons and static limits are the same) and hence exhibit the same global topology. Thus actually what distinguishes the BDKN black hole spacetime from its general relativity’s counterpart, i.e., the KN black hole is the local geometry alone such as the curvature characterized by the specific $\omega $-values, $-5/2 \leq \omega < -3/2$. At this point, perhaps it is relevant to mention the behavior of the BD scalar field which plays unique role only in BD theory of gravity. Independently of Penrose \[8\] and of Thorne and Dykla \[8\], Hawking \[9\] also explored the possible existence of black hole solutions in BD theory and put forward a theorem which states that stationary black holes in BD theory are identical to those in general relativity. To be a little more concrete, Hawking extended some of his theorems for general relativistic black holes to BD theory and showed that any object collapsing to a black hole in BD theory must settle into final equilibrium state which is either Schwarzschild or Kerr spacetime. And in doing so, he “assumed” that the BD scalar field $\Phi$ satisfies the weak energy condition and is constant outside the black hole. Therefore now one may be puzzled as we realized in the present work that non-trivial BDKN black hole solutions [*different*]{} from general relativistic KN solution could exist in seemingly contradiction to Hawking’s theorem. There is, however, no contradiction. Hawking deduced the theorem by manipulating the BD field equations and most crucially “assuming” the strict conditions on the BD scalar field stated above but not by working with an explicit spacetime solution which was not available at the time. In the present work, however, we investigated closely the known, explicit stationary axisymmetric solution in BD theory. And in particular, when the BDKN solutions can describe non-trivial black hole spacetimes for the BD $\omega $-parametr values $-5/2 \leq \omega < -3/2$, the accompanying BD scalar field solution $\Phi (r, \theta) = \Delta^{2/(2\omega+3)}\sin^{4/(2\omega+3)} \theta$ turns out [*not*]{} to be a constant field outside the event horizon at $r=r_{+}$. Besides, the energy density of this BD scalar field, $T^{BD}_{\mu \nu}\xi^{\mu}\xi^{\nu}$ whose explicit form was given earlier in eq.(16) does not strictly obey the weak energy condition for all $r$. Namely the value of $T^{BD}_{\mu \nu}\xi^{\mu}\xi^{\nu}$ does not remain non-negative for all $r$. Rather, its value and hence the signature changes from point to point. In short, the Hawking’s theorem simply cannot be applied to the present situation and hence the results of the present study needs not be restricted by Hawking’s theorem. At this point, it seems appropriate to ask whether the non-trivial BDKN black hole solution studied in the present work can be viewed as a counterexample to the no-hair theorem of black holes. In the loose sense, one may think of the non-trivial behavior of the BD scalar field outside the event horizon as indicating the appearance of “scalar hair”. Here, however, we need to be more precise with the nature of no-hair theorem. Following Bizon \[11\], for instance, a certain theory is said to allow a hairy black hole solution if there is a need to specify quantities other than conserved charges defined at asymptotic infinity such as the mass, angular momentum and the electric charge in order to characterize comletely a stationary black hole solution within that theory. Thus in this stricter sense, the non-trivial BDKN black hole solution studied in the present work does not constitute a hairy black hole solution since both the metric and BD scalar field solutions in eq.(11) are specified completely by the ADM mass $M$, angular momentum per unit mass $a$, and the electric charge $e$ only and no other quantities.\ [**IV. Discussions**]{}\ Before we address the physical implication of the non-trivial BDKN black hole solution found in the present work, we would like to comment on a technical issue, i.e., the divergent behavior of the BD scalar field solution on the horizon. And in relation to this, it is interesting to note that our BDKN black hole solution shares two peculiar features, i.e., the divergent behavior of the scalar field on the horizon and the null Hawking radiation, with the well-known Bekenstein black hole solution in Einstein-conformal scalar field theory \[12\]. Namely, using a suitable solution generation technique, long ago, Bekenstein constructed a static, spherically-symmetric black hole solution in which the metric part corresponds to that of extreme Reissner-Nordstrom (RN) black hole and thus represents non-radiating black hole spacetime and the conformal scalar field solution diverges on the horizon. Therefore for direct and parallel comparison between the two solutions, it seems appropriate to take the BDS solution analyzed in detail in the appendix. The two theories, of course, have completely different nature and motivations. The Einstein-conformal scalar field theory has been devised guided mainly by a particular (Weyl) symmetry and the scalar field there is supposed to describe a matter. The BD theory, on the other hand, is an alternative theory to Einstein gravity and the BD scalar field here represents a spacetime-varying effective Newton’s constant, not a matter. Thus the divergent behavior of the scalar field in Einstein-conformal scalar field theory could be disastrous but that of the BD scalar field in BD theory essentially represents the vanishing effective Newton’s constant in a certain region of spacetime. Besides, since the energy density of the BD scalar field $T^{BD}_{\mu\nu}\xi^{\mu}\xi^{\nu}$ given in eq.(16) vanishes and hence satisfies the weak energy condition on the horizon at which $\Delta = 0$ (of course for $-5/2 \leq \omega <-3/2$), we do not worry too much about the divergent behavior of the BD scalar field there. Another interesting contrast is that the metric solution of Bekenstein black hole spacetime there corresponds to a familiar extreme RN metric which is static and spherically-symmetric whereas the BDS metric solution here exhibits a remarkable feature that it is static (i.e., non-rotating) but not spherically-symmetric as we pointed out earlier. Besides, the Bekenstein solution represents a “hairy” black hole \[13\] whereas our BDS black hole solution carries no hair as we mentioned above. Now, the point of central interest we would like to make is about the issue raised recently by Sudarsky and Zannias \[13\]. To be a little more concrete, they showed that the divergent behavior of the conformal scalar field solution on the horizon essentially leads the Bekenstein black hole solution to fail to satisfy Einstein field equations particularly on the horizon. They, thus, concluded that the Bekenstein solution cannot be considered as a genuine black hole solution and therefore the black hole no-hair theorem is saved. And as a manifest evidence for their argument against the black hole interpretation, Sudarsky and Zannias demonstrated that by working in Eddington-Finkelstein null coordinates, parts of the Einstein field equations can be shown not to hold in a rigorous sense as the left-hand side of the equation, say $R_{\mu\nu}$ vanishes on the horizon (since the metric is that of extreme RN) while the right-hand side, namely the energy-momentum tensor of the conformal scalar field, $8\pi [T_{\mu\nu}-g_{\mu\nu}T^{\lambda}_{\lambda}/2]$ is ill-defined on the horizon as the conformal scalar field diverges there. Thus with this in mind, now we consider the validity of our BDS solution on the horizon. As one can see in the appendix, the 5 non-vanishing components of BD field equations, i.e., $tt$, $rr$, $r\theta$, $\theta \theta$ and $\phi \phi$ parts appear to hold perfectly. In particular, on the horizon and for $-5/2 \leq \omega < -3/2$, $tt$ and $\phi \phi$ parts hold as “$0 = 0$”. The other 3 parts, however, hold as “$\infty = \infty$”. Namely, in these 3 equations, not only the energy-momentum tensor of the BD scalar field on the right-hand side of BD field equations but also the Ricci tensor components on the left-hand side diverges in exactly the same manner on the horizon. Indeed precisely these 3 equations are the ones we need to be careful with. Here, however, we must say that a naive attempt toward the validity check of the BDS solution on the horizon in exactly the same way as Sudarsky and Zannias did for Bekenstein solution seems to be obscured. In fact, the demonstration of Sudarsky and Zannias was successful largely because the simple transformation from spherical to Eddington-Finkelstein null coordinates for the Bekenstein solution was available. Indeed, the virtue of null coordinates is that in terms of which $g_{rr}=0$ and hence $l^{\mu} = (\partial/\partial r)^{\mu}$ becomes a smooth null vector field such that the quantity $R_{\mu\nu}l^{\mu}l^{\nu}$ can be shown to be finite (zero) on the horizon while the right-hand side of Einstein equation, $8\pi [T_{\mu\nu}-g_{\mu\nu}T^{\lambda}_{\lambda}/2] l^{\mu}l^{\nu}$ is ill-defined. In contrast, however, taking a Eddington-Finkelstein-type null coordinates for the BDS solution is not so obvious as in the case of static, spherically-symmetric black hole solutions in Einstein theory and is indeed practically awkward. It is essentially due to the peculiar feature of BDS metric solution which is static but not spherically-symmetric as has been stressed earlier. Thus in the present work, we do not pursue validity check of the BDS solution on the horizon in this direction and leave it as an issue for future investigation. As a result, for the 3 parts of the BD field equations yielding “$\infty = \infty$”, no definite statement can be made yet concerning whether or not they are valid, i.e., these 3 equations are really satisfied on the horizon. Nevertheless, one interesting point we can make is that if the BDS solution does represent a genuine black hole spacetime, then the black hole no-hair theorem appears to survive even in the BD theory of gravity since the BDS (and BDKN as well) spacetime is not a hairy black hole solution as we discussed earlier.\ It seems that now the most relevant question to ask is ; if they are genuine, would these non-trivial BDKN black hole spacetimes in BD theory of gravity exhibiting bizzare features such as null radiation really arise in nature? Of course this question needs to be answered very carefully and honestly and the answer for now does not seem to be in the affirmative. Firstly, from field theory’s viewpoint, the generic BD theory $\omega $-parameter has to be “positive” in order for the BD scalar field $\Phi$ to have canonical (positive-definite) kinetic energy as can be seen in the BD gravity theory action given in eq.(1). Secondly, it is well-known that the BD gravity theory is in reasonable accord with all available observations and experiments thus far provided $|\omega | \gtrsim 500$ \[10\]. Since both these constraints on the values of the $\omega $-parameter seem to rule out the range $-5/2 \leq \omega < -3/2$ in which the BDKN solution could describe non-trivial black hole spacetimes, for now it seems fair to say that these non-trivial BDKN black hole spacetimes different from their general relativistic counterparts are unlikely to arise in nature. This, however, may not be the end of the story. As we have seen in this work, the energy density of the explicit BD scalar field solution (which essentially consists of its kinetic energy) turns out not to satisfy the weak energy condition irrespective of the $\omega $-value. Perhaps this implies that we may abandon the “canonical kinetic energy” condition on the BD scalar field and allow negative-$\omega $ values. Moreover, the lower bound $|\omega | \gtrsim 500$ may be relaxed considerably with the advances in technology associated with astronomical observations and astrophysical experiments. Thus perhaps it might be wise to keep the possibility of non-trivial BDKN black holes alive. As a matter of fact, there is another type of possibility of greater physical significance and relevance. Note that the generic BD $\omega $-parameter is a kind of coupling constant appearing in the BD gravity action. Thus in principle, it should be considered as a “running” coupling constant as a result of renormalization in the quantum gravity context. And its scale-dependent behavior can be envisaged as follows. In the BD gravity action given in eq.(1), the term $\sim \omega (\nabla_{\alpha}\Phi \nabla^{\alpha}\Phi /\Phi)$, like other terms in the action, should be finite. Thus large-$\omega$ indicates the regime where the BD scalar field $\Phi$ remains nearly constant which corresponds to the large-scale present universe limit (in which the BD theory goes over to the general relativity). On the other hand, small-$\omega$ indicates the regime where the BD scalar field varies sizably with space and time which would presumably correspond to the small-scale early universe limit. Thus if we are willing to accept the BD theory as a “better” effective theory of quantum gravity than general relativity to describe the entire stages (scales) of the universe evolution, then at early times when the value of $\omega$ was small such as $-5/2 \leq \omega < -3/2$, the non-trivial BDKN black holes like the ones studied in this work would have had a chance to form. These “primordial” black holes, unlike their general relativistic counterparts, however, do not evaporate as we discussed earlier. Thus it can be speculated that they might still hide somewhere in the dark side of the space today as a possible constituent of the cold dark matter. After all, new discoveries can be made when we keep our minds as well as eyes open. [**Acknowledgements**]{} This work was supported by grant of Post-doc. Program at Kyungpook National University (1998). [**Appendix : Computation of invariant curvature polynomials**]{} In this appendix, we shall explicitly write down the BD field equations satisfied by the BDS solution given in eq.(13) and then provide the result of the computation of its invariant curvature polynomials such as the Ricci square $R_{\mu\nu}R^{\mu\nu}$ and the Kretschmann curvature invariant $R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$.\ First we start with the BD field equations. The tensor part of the vacuum BD field equations that this BDS solution satisfies, $$\begin{aligned} R_{\mu\nu} = {\omega \over \Phi^2}\nabla_{\mu}\Phi \nabla_{\nu}\Phi + {1\over \Phi}\nabla_{\mu}\nabla_{\nu}\Phi\end{aligned}$$ has the following 5 non-vanishing components. Here we show that actually they are all satisfied by the BDS solution in the sense that the explicit computation of the left-hand side, i.e., $R_{\mu\nu}$ and the right-hand side, i.e., ${\omega \over \Phi^2}\nabla_{\mu}\Phi \nabla_{\nu}\Phi + {1\over \Phi}\nabla_{\mu}\nabla_{\nu}\Phi $ performed with the BDS solution given in eq.(13) precisely agree. $$\begin{aligned} R_{tt} &=& {\omega \over \Phi^2}\nabla_{t}\Phi \nabla_{t}\Phi + {1\over \Phi}\nabla_{t}\nabla_{t}\Phi \nonumber \\ &=& {1\over r^4}\Delta^{-4/(2\omega+3)}\sin^{-8/(2\omega+3)}\theta {4\over (2\omega+3)}\left[{2\over (2\omega+3)}(r-M)^2 - M(r-M) + {2\over (2\omega+3)} \cot^2 \theta \Delta \right], \nonumber \\ R_{rr} &=& {\omega \over \Phi^2}\nabla_{r}\Phi \nabla_{r}\Phi + {1\over \Phi}\nabla_{r}\nabla_{r}\Phi \nonumber \\ &=& {1\over \Delta^2} {4\over (2\omega+3)}\left[{-4\over (2\omega+3)}(r-M)^2 + M(r-M) + \left\{1 + {2\over (2\omega+3)}\cot^2 \theta \right\}\Delta \right], \nonumber \\ R_{r\theta} &=& {\omega \over \Phi^2}\nabla_{r}\Phi \nabla_{\theta}\Phi + {1\over \Phi}\nabla_{r}\nabla_{\theta}\Phi \nonumber \\ &=& {1\over \Delta}\cot \theta {4\over (2\omega+3)}\left[{4\omega \over (2\omega+3)}(r-M) - (r-2M)\right], \\ R_{\theta \theta} &=& {\omega \over \Phi^2}\nabla_{\theta}\Phi \nabla_{\theta}\Phi + {1\over \Phi}\nabla_{\theta}\nabla_{\theta}\Phi \nonumber \\ &=& {1\over \Delta}{4\over (2\omega+3)}\left[{2\over (2\omega+3)}(r-M)^2 + (r-M)(r-2M) - \left\{1 - \left({2\omega-1\over 2\omega+3}\right)\cot^2 \theta \right\}\Delta \right], \nonumber \\ R_{\phi \phi} &=& {\omega \over \Phi^2}\nabla_{\phi}\Phi \nabla_{\phi}\Phi + {1\over \Phi}\nabla_{\phi}\nabla_{\phi}\Phi \nonumber \\ &=& {-1\over \Delta}\Delta^{-4/(2\omega+3)}\sin^{-8/(2\omega+3)}\theta \times \nonumber \\ &&{4\over (2\omega+3)}\left[\left\{{2\over (2\omega+3)}(r-M)^2 - (r-M)(r-2M)\right\}\sin^2 \theta - \left({2\omega+1\over 2\omega+3}\right)\cos^2 \theta \Delta \right]. \nonumber\end{aligned}$$ And the scalar part of the vacuum BD field equations can easily be seen to hold.\ Next we turn to the expressions for the invariant curvature polynomials. The Ricci square is calculated to be $$\begin{aligned} &&R_{\mu\nu}R^{\mu\nu} = g^{\mu\alpha}g^{\nu\beta}R_{\mu\nu}R_{\alpha\beta} \\ &=& {1\over r^4}\Delta^{-2(2\omega+5)/(2\omega+3)}\sin^{-8/(2\omega+3)}\theta \left[\left\{{4\over (2\omega+3)}\left[{2\over (2\omega+3)}(r-M)^2 - M(r-M) \right. \right. \right. \nonumber \\ &+& \left. \left. {2\over (2\omega+3)}\cot^2 \theta \Delta \right]\right\}^2 \nonumber \\ &+& \left\{{4\over (2\omega+3)}\left[{-4\over (2\omega+3)}(r-M)^2 + M(r-M) + \left(1 + {2\over (2\omega+3)} \cot^2 \theta \right)\Delta \right]\right\}^2 \nonumber \\ &+& \left\{{4\over (2\omega+3)}\left[{2\over (2\omega+3)}(r-M)^2 + (r-M)(r-2M) - \left(1 - \left({2\omega-1\over 2\omega+3}\right)\cot^2 \theta \right)\Delta \right]\right\}^2 \nonumber \\ &+& \left. \left\{{4\over (2\omega+3)}\left[{2\over (2\omega+3)}(r-M)^2 - (r-M)(r-2M) - \left({2\omega+1\over 2\omega+3}\right)\cot^2 \theta \Delta \right]\right\}^2\right] \nonumber \\ &+& {2\over r^4}\Delta^{-(2\omega+7)/(2\omega+3)}\sin^{-2(2\omega+7)/(2\omega+3)}\theta \cos^2 \theta \left[\left\{{4\over (2\omega+3)}\left[{4\omega \over (2\omega+3)}(r-M) - (r-2M)\right] \right\}^2\right]. \nonumber\end{aligned}$$ It is worth noting that this Ricci square vanishes in the limit $\omega \rightarrow \infty$ as it should since in which the BDS solution goes over to the Schwarzschild solution. And the Kretschmann curvature invariant is computed to be $$\begin{aligned} &&R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta} = g^{\mu\sigma}g^{\nu\rho}g^{\alpha\lambda}g^{\beta\delta} R^{\mu}_{\nu\alpha\beta}R^{\sigma}_{\rho\lambda\delta} \\ &=& {2\over r^4}\Delta^{-2(2\omega+5)/(2\omega+3)}\sin^{-8/(2\omega+3)}\theta \left[4\left\{2{(2\omega+5)\over (2\omega+3)^2}(r-M)^2 - \left({2\omega+7\over 2\omega+5}\right)M(r-M) + M^2 \right. \right. \nonumber \\ &-& \left. {1\over (2\omega+3)}\left[1 + {2\over (2\omega+3)}\cot^2 \theta\right]\Delta \right\}^2 \nonumber \\ &+& 4\left\{\left[{2\over (2\omega+3)}(r-M) + (r-2M)\right]\left[{2\over (2\omega+3)}(r-M) - M \right] - {1\over (2\omega+3)}\left[1 + {2\over (2\omega+3)}\cot^2 \theta\right]\Delta \right\}^2 \nonumber \\ &+& \left\{\left[{2\over (2\omega+3)}(r-M) - (r-2M)\right]\left[{2\over (2\omega+3)}(r-M) - M \right] - 2{(2\omega+1)\over (2\omega+3)^2}\cot^2 \theta \Delta \right\}^2 \nonumber \\ &+& \left. \left\{\left({2\omega+7 \over 2\omega+3}\right)(r-M)\left[{2\over (2\omega+3)}(r-M) - (r-2M)\right] + \left({2\omega+1 \over 2\omega+3}\right)\left[1 + {2\over (2\omega+3)}\cot^2 \theta\right]\Delta \right\}^2 \right] \nonumber \\ &+& {2\over r^4}\Delta^{-2(2\omega+5)/(2\omega+3)}\sin^{-4(2\omega+5)/(2\omega+3)}\theta \left[\left\{\left[{2\over (2\omega+3)}(r-M) + (r-2M)\right] \times \right. \right. \nonumber \\ &&\left. \left[{2\over (2\omega+3)}(r-M) - M \right] \sin^2 \theta - {2\over (2\omega+3)}\left[1 + {2\over (2\omega+3)}\cos^2 \theta\right]\Delta \right\}^2 \nonumber \\ &+& \left\{\left[{2\over (2\omega+3)}(r-M) - (r-2M)\right]\left[{2\over (2\omega+3)}(r-M) - M \right] \sin^2 \theta - 2{(2\omega+1)\over (2\omega+3)^2}\cos^2 \theta \Delta \right\}^2 \nonumber \\ &+& 2\left\{(r-M)\left[{2\over (2\omega+3)}(r-M) + (r-2M)\right]\sin^2 \theta + \left[{2\over (2\omega+3)} - \left({2\omega+5 \over 2\omega+3}\right)\sin^2 \theta \right]\Delta \right\}^2 \nonumber \\ &+& \left\{\left({2\omega+7 \over 2\omega+3}\right)(r-M)\left[{2\over (2\omega+3)}(r-M) - (r-2M)\right] \sin^2 \theta + \left({2\omega+1 \over 2\omega+3}\right)\left[1 - \left({2\omega+5 \over 2\omega+3}\right) \cos^2 \theta\right]\Delta \right\}^2 \nonumber \\ &+& \left\{\left[{2\over (2\omega+3)}(r-M) + (r-2M)\right]\left[{2\over (2\omega+3)}(r-M) - M \right] \sin^2 \theta \right. \nonumber \\ &-& \left. {2\over (2\omega+3)}\left[1 + {4\over (2\omega+3)}\cos^2 \theta\right]\Delta \right\}^2 \\ &+& \left. 2\left\{\left[{4\over (2\omega+3)^2}(r-M)^2 - (r-2M)^2\right]\sin^2 \theta + \left({2\omega+1 \over 2\omega+3}\right)\left[1 - \left({2\omega-1 \over 2\omega+3}\right) \cos^2 \theta\right]\Delta \right\}^2 \right] \nonumber \\ &+& {32\over r^4}\Delta^{-(2\omega+7)/(2\omega+3)}\sin^{-2(2\omega+7)/(2\omega+3)}\theta \cos^2 \theta \left[\left\{{1\over (2\omega+3)}\left[{6\over (2\omega+3)}(r-M) + (r-4M)\right]\right\}^2 \right. \nonumber \\ &+& \left. \left\{{1\over (2\omega+3)}\left[{4\omega\over (2\omega+3)}(r-M) + (r-2M)\right]\right\}^2 \right]. \nonumber\end{aligned}$$ Again, it is straightforward to check that this Kretschmann curvature invariant reduces to that of Schwarzschild solution, $48M^2/r^6$ in the Einstein gravity limit (i.e., as $\omega \rightarrow \infty$) as it should. In conclusion, a close inspection reveals that $R_{\mu\nu}R^{\mu\nu}$ and $R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$ are finite (more precisely vanishes) both on the horizon candidate at which $\Delta = 0$ and along the symmetry axis $\theta = 0, ~\pi$ provided the generic BD $\omega $-parameter takes values in the range $-5/2 \leq \omega <-3/2$. [**References**]{} [\[1\]]{} C. Brans and C. H. Dicke, Phys. Rev. [**124**]{}, 925 (1961). [\[2\]]{} S. Weinberg, [*Gravitation and Cosmology*]{} (Wiley, New York, 1972). [\[3\]]{} A. I. Janis, D. C. Robinson, and J. Winicour, Phys. Rev. [**186**]{}, 1729 (1969) ; H. A. Buchdahl, Int. J. Theor. Phys. [**6**]{}, 407 (1972) ; C. B. G. McIntosh, Commun. Math. Phys. [**37**]{}, 335 (1974) ; B. O. J. Tupper, Nuovo Cimento [**19B**]{}, 135 (1974) ; R. N. Tiwari and B. K. Nayak, Phys. Rev. [**D14**]{}, 2502 (1976) ; J. Math. Phys. [**18**]{}, 289 (1977) ; T. Singh and L. N. Rai, Gen. Rel. Grav. [**11**]{}, 37 (1979). [\[4\]]{} R. A. Matzner and C. 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Sudarsky and T. Zannias, Phys. Rev. [**D58**]{}, 087502 (1998) ; we thank the referee for drawing our attention to this reference. [^1]: e-mail : hongsu@vega.kyungpook.ac.kr

[**The spin of the $\mu$ mesons**]{} Introduction ============ We have shown earlier \[1\] that the spin of the “stable" mesons and baryons can be explained with the standing wave model \[2\]. In \[3\] we have determined the rest mass of the $\mu$ mesons with the concepts used in the standing wave model and explained why m($\mu^\pm$) is $\cong$ 3/4 $\cdot\, \mathrm{m}(\pi^\pm)$. According to \[3\] the $\mu^\pm$ mesons consist of a lattice of $\nu_\mu$ (respectively $\bar{\nu}_\mu$), $\nu_e$ and $\bar{\nu}_e$ neutrinos which remain from the cubic neutrino lattice of the $\pi^\pm$ mesons after their decay, plus an electric charge. It has been argued that our explanation of the mass of the $\mu$ mesons cannot be correct because in our model the $\mu$ meson lattice has a diameter of 0.88$\cdot$10$^{-13}$ cm, whereas the $\mu$ mesons are commonly said to be point particles. However, since in our model the $\mu$ mesons consist of neutrinos, but for the electric charge, and since neutrinos do not, in a first approximation, interact with electric charge or with mass, it will not be possible to establish the size of the $\mu$ meson lattice, i.e. of the $\mu$ mesons, through conventional scattering experiments. Therefore the $\mu$ mesons will *appear* to be point particles. We will now show that the neutrino lattice does not only determine the mass of the $\mu^\pm$ mesons but also provides an explanation of the spin of the $\mu$ mesons which has, so far, not been explained. The spin of the $\mu^\pm$ mesons ================================ The spin s = 1/2 or the intrinsic angular momentum $\hbar$/2 of the $\mu^\pm$ mesons can, theoretically, be the sum of the angular momentum vectors of all neutrino lattice oscillations of frequency $\nu_i$ in the $\mu$ mesons, plus the sum of all spin vectors of the $n_i$ neutrinos in the lattice, plus the spin vector of the electric charge which each $\mu$ meson carries. In a formula $$j(\mu^\pm) = \sum_{i} \,j(\nu_i) + \sum_{i}j(n_i) + j(e^\pm) \quad 0\,\leq\, i\, \leq\,N_\mu ,$$ where $N_\mu$ is the number of all neutrinos in the $\mu$ meson lattice, $N_\mu = 2.14\cdot 10^9$. This procedure is completely analogous to the way how the spin of the $\pi^\pm$ mesons is determined, only that then a cubic neutrino lattice is considered consisting of $\nu_\mu, \bar{\nu}_\mu, \nu_e$ and $\bar{\nu}_e$ neutrinos with N = 2.854$\cdot 10^9$ neutrinos. The lattice oscillations in the $\mu$ mesons are longitudinal and hence do not have an angular momentum because for each oscillation $\vec{r} \times \vec{p} = 0$. So the lattice oscillations do not contribute to the intrinsic angular momentum of the $\mu$ mesons, or $\sum_{i} j(\nu_i) = 0$. Each of the O($10^9$) neutrinos in the $\mu$ meson lattice has, however, an angular momentum $\hbar$/2. The sum of the spin vectors of all neutrinos in the lattice of the $\mu$ mesons must be zero; otherwise the sum of the spin vectors of all neutrinos in the lattice plus the spin vector of the electric charge of a $\mu^\pm$ meson could not be $\hbar$/2, as it must be.  > Fig.1. The $\mu^+$ meson lattice. The dotted rectangle marks the front-side of the $\pi^\pm$ meson lattice in Fig.5 of \[2\]. In order to show that the angular momentum vectors around a central axis of the lattice caused by the spin vectors of all neutrinos in the $\mu$ meson lattice is zero we have to consider this lattice in detail, Fig.1. As we have learned in \[3\] the, say, $\mu^+$ meson lattice is obtained from the cubic neutrino lattice of the $\pi^+$ meson through the removal of all $\nu_\mu$ neutrinos. That means that in the $\mu^+$ meson lattice are then N/4 = (2.854$\cdot 10^9$)/4 vacancies at the location where originally $\nu_\mu$ neutrinos were. Each vacancy is surrounded in the xy plane by combinations of four electron neutrinos. In the z-direction there is on top as well as below each vacancy another electron neutrino. The same applies for the electron neutrinos around each antimuon neutrino, which are the centers of the cells of a $\mu^+$ meson lattice. The cells of the $\mu$ meson lattice are octahedrons, two pyramids joint at their square base. As can be seen on Fig.1 each lattice point in the upper right quadrant has at the opposite position in the lower left quadrant the *same* type of neutrino. The same applies in the upper left and lower right quadrant. As is well-known it appears that only left-handed neutrinos and right-handed antineutrinos exist, at least as long as the neutrinos are massless. Suppose this also holds in the case of neutrinos with a rest mass. The angular momentum vectors originating from the spin of the neutrinos would then not cancel on a diagonal through the center of the lattice. The spin vectors on either side of the same length on the diagonal are then from the same type of neutrino (Fig.1) and therefore point in the same direction and do not cancel. However, the polarization vector of the spin of a neutrino depends on the direction of the velocity of the neutrino, because the helicity H is given by $$H = \vec{P}\cdot\vec{v}/Pv ,$$ where $\vec{P}$ is the polarization vector. If only left-handed neutrinos (H = $\mathrm{-}$$\beta$) and right-handed antineutrinos (H = $+\beta$) exist, the direction of $\vec{P}$ must be reversed if the direction of motion of the neutrinos during their oscillation in the $\mu$ meson lattice is, in the lower left quadrant of Fig.1, opposite to the direction of motion in the upper right quadrant. This change of the direction of motion follows from the equation of motion for the displacements $u_n$ of the lattice points in Eq.(7) of \[2\] $$u_n = Ae^{i(\omega t\,\, +\,\, n\phi)} ,$$ from which follows that $\dot{u}_n = v_n = i\omega u_n$. The frequencies are given by $$\nu_n = \nu_0 \phi_n ,$$ as in Eq.(19) of \[2\]. Since n$\phi$ = kx, $\phi_n$ is proportional to x with x = n*a*, where *a* is the lattice constant. It follows that the direction of motion of the neutrinos in the upper right quadrant ($\phi> 0$) is opposite to the direction of motion of the neutrinos in the lower left ($\phi < 0$) quadrant. Consequently the angular momentum vectors around the center of the lattice caused by the spin of the neutrinos in the $\mu$ meson lattice are opposite and of the same magnitude, they cancel. The only point without an opposite is the point at the center of the lattice, at which there is no neutrino, so the center does not contribute a spin vector. The sum of the angular momentum vectors caused by the spin of all neutrinos in the $\mu$ meson lattice is zero, $\sum_{i} j(n_i)$ = 0. Together with $\sum_{i} j(\nu_i)$ = 0, as shown above, we arrive from Eq.(1) at $$j(\mu^\pm) = j(e^\pm) .$$ The spin s = 1/2 of the $\mu^\pm$ mesons is caused exclusively by the spin of the electric charge that a $\mu$ meson carries. Crucial for this point is the absence of a neutrino at the center of the $\mu$ meson lattice. The cubic lattice of the $\pi^\pm$ mesons is easily recovered from the $\mu$ meson lattice by filling all vacancies with either $\nu_\mu$ (or $\bar{\nu}_\mu$) neutrinos. The $\pi^\pm$ mesons do not have spin. As in the $\mu$ meson lattice all neutrino spin vectors around the center of the cubic $\pi^\pm$ lattice cancel, but for the angular momentum $\hbar$/2 of the center neutrino. This angular momentum is canceled by the spin of the electric charge which the $\pi^\pm$ mesons carry, so s($\pi^\pm$) = 0, as it must be. This explanation supercedes the explanation given in \[1\] for the spin of the particles of the neutrino branch which applies only for a static lattice. Conclusions =========== If the $\mu^\pm$ mesons consist of a lattice of $\bar{\nu}_\mu$ (respectively $\nu_\mu$), $\nu_e$ and $\bar{\nu}_e$ neutrinos, as is suggested by the decay of the $\pi^\pm$ mesons, then it can be shown \[3\] that the theoretical mass of the $\mu^\pm$ mesons is, within 1%, equal to the measured mass of the $\mu^\pm$ mesons. Using the octahedronal lattice structure of the $\mu^\pm$ mesons suggested by the determination of the mass of the $\mu^\pm$ mesons it follows that the angular momentum vectors of all longitudinal lattice oscillations are zero and that the sum of the angular momentum vectors caused by the spin of the neutrinos of the lattice, taken around the center of the lattice, is also zero. The only contribution to the intrinsic angular momentum of the $\mu$ mesons comes from the spin of the electric charge which the $\mu^\pm$ mesons carry. Consequently the spin of the $\mu$ mesons is s($\mu^\pm$) = 1/2, as it must be. We note that the spin of the $\mu$ mesons can be explained, without any additional assumption, from the structure of the $\mu$ mesons which we have used for the explanation of the mass of the $\mu$ mesons. Both the $\pi^\pm$ mesons and the $\mu^\pm$ mesons carry an electric charge. The $\pi^\pm$ mesons do not have spin, whereas the $\mu^\pm$ mesons have spin 1/2. The presence or absence of a neutrino at the center of the lattice makes the difference. The spin of the electric charge in $\pi^\pm$ is canceled by the spin of the central neutrino, whereas the spin of the electric charge in $\mu^\pm$ remains because there is no central neutrino to cancel the spin of the electric charge. **REFERENCES** E.L.Koschmieder, arXiv: physics/0301060 (2003), Hadr.J. (to appear). E.L.Koschmieder, arXiv: physics/0211100 (2002), Chaos, Solitons and\ Fractals [**18**]{},1129 (2003). E.L.Koschmieder, arXiv: physics/0110005 (2001).

--- abstract: 'We have performed magnetic susceptibility, heat capacity, muon spin relaxation, and neutron scattering measurements on three members of the family Ba$_3M$Ru$_2$O$_9$, where $M$ $=$ In, Y and Lu. These systems consist of mixed-valence Ru dimers on a triangular lattice with antiferromagnetic interdimer exchange. Although previous work has argued that charge order within the dimers or intradimer double exchange plays an important role in determining the magnetic properties, our results suggest that the dimers are better described as molecular units due to significant orbital hybridization, resulting in one spin-1/2 moment distributed equally over the two Ru sites. These molecular building blocks form a frustrated, quasi-two-dimensional triangular lattice. Our zero and longitudinal field $\mu$SR results indicate that the molecular moments develop a collective, static magnetic ground state, with oscillations of the zero field muon spin polarization indicative of long-range magnetic order in the Lu sample. The static magnetism is much more disordered in the Y and In samples, but they do not appear to be conventional spin glasses.' author: - 'D. Ziat' - 'A. A. Aczel' - 'R. Sinclair' - 'Q. Chen' - 'H. D. Zhou' - 'T. J. Williams' - 'M. B. Stone' - 'A. Verrier' - 'J. A. Quilliam' title: 'Frustrated spin-1/2 molecular magnetism in the mixed-valence antiferromagnets Ba$_3M$Ru$_2$O$_9$ ($M$ $=$ In, Y, Lu)' --- I. Introduction =============== The 6H-perovskites, with formula Ba$_3MA_2$O$_9$, have provided fertile ground for recent discoveries in frustrated quantum magnetism. Materials in this family with magnetic $M$-sites have been shown to exhibit quantum spin liquid behavior, in particular 6HB-Ba$_3$NiSb$_2$O$_9$ [@Cheng2011; @Quilliam2016; @Fak2016] and Ba$_3$IrTi$_2$O$_9$ [@Dey2012BITO] while others, Ba$_3$CuSb$_2$O$_9$ [@Zhou2011; @Nakatsuji2012; @Quilliam2012bcso] and Ba$_3$ZnIr$_2$O$_9$ [@Nag2016], exhibit possible quantum spin-orbital liquids. Furthermore, Ba$_3$CoSb$_2$O$_9$ has allowed for some of the first studies on the magnetization process of a truly triangular spin-1/2 antiferromagnet [@Shirata2012; @Zhou2012Co; @Susuki2013; @Koutroulakis2015; @Quirion2015]. The flexibility of this crystal structure means that we are also at liberty to include magnetic 4$d$/5$d$ transition metal $A$-site ions and thereby study spin dimers distributed on a triangular lattice with significant spin-orbit coupling and orbital hybridization. In the case of the ruthenates Ba$_3M$Ru$_2$O$_9$, where $M^{3+}$ is non-magnetic, one obtains a triangular lattice of magnetic, mixed-valence Ru dimers. A total of seven electrons occupy each dimer and this leads to the possibility of charge, orbital and spin degrees of freedom. For analogous 3$d$ transition metal-based dimer systems with more than two electrons per dimer, Hund’s coupling is usually dominant and therefore needs to be treated before turning to intersite effects such as electron hopping and the interdimer Coulomb interaction. However, recent theoretical [@Streltsov2014; @Streltsov2016] and experimental [@Kimber2012] work has shown that this approach can break down in some 4$d$ and 5$d$ transition metal-based dimer systems, where Hund’s coupling is expected to be significantly weaker due to the spatially-extended $d$-orbitals. This more complicated regime may be realized in the Ba$_3M$Ru$_2$O$_9$ family, as any simple picture based on dominant Hund’s coupling cannot describe all of the known magnetic properties of the Ru dimers. [ll]{} {width="3in"} & {width="2.5in"}\ \ (a) & (b)\ \ More specifically, two different magnetic ground states for the Ru dimers in Ba$_3M$Ru$_2$O$_9$ have been proposed previously that are consistent with dominant Hund’s coupling. Doi *et al.* [@Doi2002] first assumed that all seven electrons were localized at particular Ru sites, which leads to Ru$^{4+/5+}$ charge order within the dimers and antiferromagnetic intradimer exchange. They argued that the latter should produce dimers with a magnetic ground state of total spin $S$ $=$ 1/2, which could explain the loss of effective magnetic moment with decreasing temperature in their $M$ $=$ In, Y and Lu magnetic susceptibility measurements and an entropy release of $R \ln(2)$ in their specific heat data at the low temperature magnetic phase transitions ($T_m$ $=$ 4.5, 4.5, and 9.5 K for In, Y, and Lu). However, their model fails to explain the very different, monotonic susceptibility in the $M =$ La sample, as the intradimer exchange interaction would have to change dramatically, from strongly antiferromagnetic to strongly ferromagnetic with only a tiny modification of the crystal structure. Even if that were possible, the model would imply $S=5/2$ dimers in the La compound which would lead to much larger values of susceptibility than are measured. Furthermore, subsequent neutron diffraction measurements of the Y system found no evidence for the required charge ordering within the dimers down to 2 K [@Senn2013]. For these reasons, the magnetic ground state of the Ru dimers has also been discussed more recently in the context of molecular double exchange [@Senn2013], but this simple model cannot explain the non-monotonic $T$-dependence of the magnetic susceptibility, the low-$T$ entropy release in the specific heat data, and the small ordered moment sizes for the Y and La systems found in neutron diffraction. This means that there is currently no comprehensive understanding of the magnetic ground states for single Ru dimers in the Ba$_3M$Ru$_2$O$_9$ family. The collective magnetic ground states of these materials may also be interesting in their own right, as the interdimer interactions are likely frustrated due to the triangular lattice geometry of the dimers. For these reasons, we have used magnetic susceptibility, heat capacity, muon spin relaxation ($\mu$SR), and neutron scattering to investigate both the single dimer and collective magnetic properties of the $M$ $=$ In, Y and Lu systems. II. Experimental details ======================== The polycrystalline samples of Ba$_3M$Ru$_2$O$_9$ ($M$ $=$ In, Y and Lu) studied here were prepared by the standard solid state reaction method. Appropriate amounts of BaCO$_3$, In$_2$O$_3$/Y$_2$O$_3$/Lu$_2$O$_3$ (Y$_2$O$_3$ and Lu$_2$O$_3$ were pre-dried at 980 $^\circ$C overnight), and RuO$_2$ were mixed in agate mortars, compressed into pellets, and annealed for 20 hours in air at temperatures of $900^\circ$C, $1200^\circ$C and $1300^\circ$C, respectively. Magnetic susceptibility and specific heat measurements were performed using *Quantum Design MPMS* and *PPMS* systems. The DC magnetic susceptibility was measured with a magnetic field of 1 kG. AC susceptibility measurements were also performed at various frequencies (from 333 Hz to 9999 Hz) to look for evidence of spin freezing. Neutron powder diffraction (NPD) was performed with polycrystalline Ba$_3M$Ru$_2$O$_9$ ($M$ $=$ In, Y and Lu) using the HB-2A powder diffractometer of the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory (ORNL). The Lu sample was loaded in a vanadium can, and the data was collected at $T$ $=$ 1.5 K with a neutron wavelength of 1.54 Å and a collimation of 12$'$-open-12$'$. The In and Y samples were loaded in aluminum cans, and the data was collected at $T$ $=$ 3.5 K with a neutron wavelength of 1.54 Å and a collimation of 12$'$-21$'$-6$'$. Complementary elastic neutron scattering measurements were performed on the fixed-incident-energy triple-axis spectrometer HB-1A of HFIR at ORNL, using the same polycrystalline samples. A series of two pyrolytic graphite (PG) crystal monochromators provided the fixed incident energy $E_i$ of 14.6 meV and two-highly oriented PG filters were placed in the incident beam to remove higher order wavelength contamination. A PG analyzer crystal was located before the single He-3 detector for energy discrimination. A collimation of 40$'$-40$'$-40$'$-80$'$ resulted in an energy resolution at the elastic line of $\approx$ 1 meV. The elastic scattering was measured at 1.5 K for all three samples, with higher temperature background data collected at 20 K for the Lu system and 10 K for the In and Y systems. Inelastic neutron scattering (INS) measurements were collected on the direct-geometry time-of-flight chopper spectrometer SEQUOIA of the Spallation Neutron Source (SNS) at ORNL, using the same polycrystalline samples loaded in aluminum cans. Spectra were collected at a variety of temperatures by operating in high flux mode (elastic resolution of $\sim$ 4 % $E_i$) with $E_i$ $=$ 50 and 100 meV. The monochromatic incident beam was obtained by using a Fermi chopper rotating at a frequency of either 180 or 240 Hz for $E_i$ $=$ 50 and 100 meV respectively. The background from the prompt pulse was removed with a $T_0$ chopper operating at 90 Hz. An empty aluminum can was measured in identical experimental conditions for a similar counting time. The resulting spectra were subtracted from the corresponding sample spectra after normalization with a vanadium standard to account for variations of the detector response and the solid angle coverage. This procedure ensured that temperature-independent scattering was removed from the spectra before applying the appropriate Bose corrections to calculate $f(Q)^2 \chi''(Q, \omega)$, where $\chi''(Q, \omega)$ is the imaginary part of the dynamic magnetic susceptibility and $f(Q)$ is the magnetic form factor. Muon spin relaxation measurements were performed at TRIUMF, Canada on the M20 beam line with the LAMPF spectrometer and a He-flow cryostat. Samples were encapsulated in Ag-coated mylar adhesive and suspended between copper supports in the path of the muon beam where they were cooled by helium vapour to as low as $\sim 2$ K. This style of sample mount and a veto counter behind the sample allow us to almost completely eliminate any background asymmetry. Measurements were taken in zero-field (ZF), longitudinal field (LF), and weak transverse field (TF) geometries using forward and backward positron counters to determine the asymmetry, $a(t) = (n_B - \alpha n_F)/(n_B + \alpha n_F)$. $\alpha$ is determined with weak transverse field measurements in the paramagnetic phase and $a(t)$ is divided by the initial asymmetry to obtain the muon polarization, $P(t)$. $B'$ In (3.5 K) Y (3.5 K) Lu (1.5 K) La (11 K) [@Senn2013] ----------------- ------------ ------------ ------------ ----------------------- $a$ 5.7947(1) 5.8565(1) 5.8436(1) 5.9492 $c$ 14.2738(2) 14.4589(1) 14.3978(2) 14.9981 Ba$_2$ $z$ 0.9116(2) 0.9075(1) 0.9084(2) 0.8909 Ru $z$ 0.1611(1) 0.1632(1) 0.1620(1) 0.16556 O$_1$ $x$ 0.4874(5) 0.4879(4) 0.4887(5) 0.4873 O$_2$ $x$ 0.1712(4) 0.1758(2) 0.1741(3) 0.17889 O$_2$ $z$ 0.4150(1) 0.4124(1) 0.4138(1) 0.40471 R$_\mathrm{wp}$ 8.82 % 6.27 % 6.18 % 6.66 % Ru-O$_1$ 2.001(3) 2.009(2) 2.019(2) 2.030 Ru-O$_2$ 1.956(2) 1.936(1) 1.947(2) 1.909 Ru-Ru 2.538(3) 2.511(2) 2.533(3) 2.533 Ru-O$_1$-Ru 78.8(1) 77.4(1) 77.7(1) 77.2 : Structural parameters for Ba$_3M$Ru$_2$O$_9$ ($M$ $=$ In, Y, and Lu) extracted from the refinements of the $\lambda$ $=$ 1.54 Å neutron powder diffraction data. The lattice constants and bond distances are in Å, and the bond angles are in degrees. III. Search for static charge order =================================== It is important to understand the magnetic ground state of a single Ru dimer before moving on to a discussion of these materials’ collective magnetic properties. As shown in Fig. \[Structure\](a), each Ru site is in an octahedral oxygen environment, and the Ru dimers form via face-sharing octahedra. It is well-known that all three materials crystallize in the space group $P 6_3/m m c$ at room temperature, which ensures that both Ru sites forming a dimer are crystallographically-equivalent due to the crystal symmetry. However, static charge order is a distinct possibility for these materials upon cooling due to the mixed Ru$^{4+/5+}$ nominal valence, which has been found in isostructural systems with a mixed Ru$^{5+/6+}$ nominal valence such as Ba$_3$NaRu$_2$O$_9$ [@Kimber2012]. Neutron powder diffraction (NPD) is a sensitive probe to look for this effect, as one can investigate the $T$-dependence of the charge distribution in the dimers indirectly via Ru-O bond lengths. Figure \[Diffraction\] shows NPD data collected using $\lambda$ $=$ 1.54 Å for Ba$_3M$Ru$_2$O$_9$, with $T$ $=$ 1.5 K for the Lu system and $T$ $=$ 3.5 K for the In and Y analogs. Rietveld refinements were performed using FullProf [@Rodriguez1993]. In all cases, we find that the data is best refined in the room temperature $P 6_3/m m c$ space group with only one unique Ru site and no Ru/$M$ site mixing, and therefore we find no evidence for static charge ordering down to these temperatures. We also do not detect any magnetic Bragg peaks, which would be indicative of long-range magnetic order, in this data. Table I shows lattice constants, atomic fractional coordinates, and selected bond distances and angles extracted from the refinements. We note that our O$_2$ $z$ parameter for the Y system is significantly different from the value reported in Ref. [@Senn2013]. Upon careful inspection their value appears to be somewhat unphysical [@Footnote] ![\[Diffraction\] Neutron powder diffraction measurements with a wavelength of 1.54 Å for (a) Ba$_3$InRu$_2$O$_9$ (3.5 K), (b) Ba$_3$YRu$_2$O$_9$ (3.5 K), and (c) Ba$_3$LuRu$_2$O$_9$ (1.5 K). The corresponding structural refinements (black lines) are superimposed on the data points. The extra peaks in the In and Y patterns arise from the Al sample can.](Diffraction.pdf){width="3.35in"} IV. Molecular magnetism ======================= Since there is no evidence for static charge order of the Ru dimers in Ba$_3M$Ru$_2$O$_9$ ($M$ $=$ In, Y and Lu), we now consider other possibilities for the single dimer ground states that are consistent with the known magnetic properties. We first revisit the DC magnetic susceptibility of these materials, as a satisfactory explanation for the complex $T$-dependence is still lacking. Our own results, shown in Fig. \[Orbitals\](c), are very similar to previous work by Doi *et al.* [@Doi2002]. Between $\sim 100$ K and 300 K, $\chi$ is an increasing function of temperature ($d\chi/dT > 0$), suggestive of gapped spin excitations. Below $\sim 100$ K, however, $\chi$ becomes a decreasing function of temperature ($d\chi/dT<0$), *i.e.* begins to resemble a Curie-Weiss law. A logical explanation for this non-monotonic behavior is a change in spin number with temperature. For example, the ground state of each Ru dimer may be a $S=1/2$ doublet with a relatively low-lying excited $S=3/2$ manifold (with energy $\Delta_1$). As $T>100$ K, we begin to populate the $S$ $=$ 3/2 manifold, which naturally has a larger susceptibility. If we assume that there is also a $S$ $=$ 5/2 manifold with higher energy $\Delta_2$, a minimal functional form for the susceptibility [@Doi2002] can be written as $$\chi(T) = \frac{\mathcal{C}}{T + \Theta_W}\cdot \frac{1 + 10e^{-\Delta_1/k_BT} + 35e^{-\Delta_2/k_BT}}{1 + 2e^{-\Delta_1/Tk_B} + 3e^{-\Delta_2/k_BT}} \label{ChiModel}$$ This equation accounts for interactions between dimers via the $\Theta_W$ term. Fits of this form provide an adequate description of the susceptibility data over a broad temperature range. Without fixing any parameters, these fits yield $\Delta_1 = 38.9(4)$ meV (In), 28.6(3) meV (Y) and 34.1(4) (Lu). However, fits of this form are somewhat over parametrized and a more direct method for exploring the excitation spectrum is desirable. ![(a) Energy-level and spin occupation diagram for a hybridized Ru$^{4.5+}$-Ru$^{4.5+}$ dimer with large bonding energy, which is likely to apply to the In, Y and Lu samples [@Streltsov2016]. (b) Energy level diagram with lower bonding energy as expected to apply to the La sample. (c) Magnetic susceptibility of the In, Y and Lu samples, with the fits using Eq. \[ChiModel\].\[Orbitals\] superimposed on the data.](EnergyLevels.pdf "fig:"){width="3.5in"}\ ![(a) Energy-level and spin occupation diagram for a hybridized Ru$^{4.5+}$-Ru$^{4.5+}$ dimer with large bonding energy, which is likely to apply to the In, Y and Lu samples [@Streltsov2016]. (b) Energy level diagram with lower bonding energy as expected to apply to the La sample. (c) Magnetic susceptibility of the In, Y and Lu samples, with the fits using Eq. \[ChiModel\].\[Orbitals\] superimposed on the data.](Susceptibility.pdf "fig:"){width="3.5in"}\ To this end, we have employed inelastic neutron scattering measurements, carried out on the SEQUOIA spectrometer with an incident energy of $E_i$ $=$ 100 meV. We plot $f(Q)^2 \chi''(Q, \omega$) for Ba$_3M$Ru$_2$O$_9$ in Fig. \[Neutrons\](a)-(c) as a temperature difference $f(Q)^2 \Delta \chi'' = f(Q)^2[\chi''(5\text{ K})-\chi''(225\text{ K})]$ to isolate the low-temperature magnetic scattering. Two dispersive magnetic modes are visible in the spectra of each system. The lower modes are located just above the elastic line and appear more clearly in the complementary $E_i$ $=$ 50 meV datasets shown in Fig. \[LowIncidentEnergy\]. The finite dispersion of these modes likely arises from significant interdimer interactions. Constant-$Q$ cuts taken from the same datasets with an integration range of \[2-2.5\] Å$^{-1}$ are depicted in Fig. \[Neutrons\](d). These cuts indicate that the higher energy mode is centered about 34(1) meV, 31.5(1.5) meV, and 34(1) meV for the In, Y, and Lu systems respectively. These excitation energies correspond reasonably well to the values of $\Delta_1$ obtained from freely fitting the DC susceptibility. Ultimately, we have fitted the susceptibility data by fixing the values of $\Delta_1$ to those measured with our INS measurements, resulting in only 3-parameter fits and eliminating the over-parametrization problem. The Curie constants $\mathcal{C}$ obtained from this fitting give effective moment sizes, $\mu_\mathrm{eff}$, in the ground state manifold of $1.40(3)\mu_B$, $1.65(3)\mu_B$, and $1.53(3)\mu_B$ per dimer for the In, Y and Lu samples, respectively. These values are only slightly under the value of $1.73\mu_B$ expected for a free spin-1/2, and therefore this result is consistent with our proposal that a single dimer has a total spin $S$ $=$ 1/2 ground state. The Weiss constants, $\Theta_W$, are found to be 43(3) K, 110(10) K and 113(2) K for the In, Y and Lu systems respectively, which are indicative of significant antiferromagnetic interdimer exchange. The $S=5/2$ state is found at $\Delta_2 = $ 81(1) meV (In), 72(1) meV (Y) and 80(1) meV (Lu). Despite the high energy of $\Delta_2$, these states cannot be ignored in the susceptibility fitting. This model includes a number of simplifications, most importantly that the Weiss constant, $\Theta_W$, is the same in all manifolds of total spin. This is counterintuitive since one would expect a higher total spin to yield a larger Weiss constant, all things being equal, since $\Theta_W = 2zJS(S+1)/3k_B$ where $z$ is the number of nearest neighbors. The success of this simplistic model, in which $\Theta_W$ is constant, therefore implies that the interaction strength, $J$, between dimers is smaller when they are excited into their $S$ $=$ 3/2 or $S$ $=$ 5/2 manifolds, compensating for the increase in spin number. This single dimer picture supported by our susceptibility and INS measurements can be better understood by drawing on the work of Streltsov and Khomskii who have investigated the possibility of covalent bonds forming between 4$d$/5$d$ ions in various cases [@Streltsov2014; @Streltsov2016]. For the current Ba$_3 M$Ru$_2$O$_9$ structure, one should consider the transition metal Ru ions in the strong crystal field regime. Since these ions are in an octahedral oxygen environment, this assumption leads to the usual low energy $t_{2g}$ orbitals and higher energy $e_g$ orbitals. A trigonal distortion, inherent to this family of materials crystallizing in the $P6_3/m m c$ space group, then splits the $t_{2g}$ orbitals into an $a_{1g}$ singlet and an $e_{g}^\pi$ doublet [@Kugel2015]. The unique face-sharing octahedral geometry of two neighboring Ru sites is argued to produce strong orbital hybridization, with the $a_{1g}$ orbitals experiencing the largest bonding energy as shown in Fig. \[Orbitals\](a) and (b). If the two Ru sites are close enough, then the $e_{g}^\pi$ orbitals can also participate in molecular bonding. The choice of magnetic ground state for a single dimer in a particular system depends critically on the ratio of the molecular bonding energy to Hund’s coupling, as illustrated in Fig. \[Orbitals\](a) and (b). In the present materials Ba$_3M$Ru$_2$O$_9$ with $M$ $=$ Y, In and Lu, the molecular bonding energy appears to be higher than Hund’s coupling, and therefore the electrons prefer to occupy the $e_g^\pi$ bonding orbitals rather than the $e_g^{\pi\ast}$ anti-bonding orbitals. In other words, three covalent bonds form and one uncompensated electron is left over. This situation is illustrated in Fig. \[Orbitals\](a). This model suggests that the higher-energy dispersive modes observed in the INS spectra, shown in Fig. \[Neutrons\](a)-(c) and highlighted in the cuts of Fig. \[Neutrons\](d), can be assigned to electron transitions from bonding to antibonding molecular orbitals, which would cause the total spin of a dimer to change from $S$ $=$ 1/2 to 3/2. The origin of the lower energy INS modes can also be understood in the context of the molecular magnet model, as they may simply represent electron transitions between the antibonding orbitals shown in Fig. \[Orbitals\](a). Any origin associated with collective magnetic ordering or spin freezing for these low energy modes can be ruled out as there was no significant change observed in their temperature-dependence between 1.5 and 20 K in complementary $E_i$ $=$ 50 meV datasets, which is illustrated for Ba$_3$YRu$_2$O$_9$ in Fig. \[LowIncidentEnergy\]. On the other hand, the magnetic susceptibility of Ba$_3$LaRu$_2$O$_9$ [@Doi2002] is consistent with a total spin $S$ $=$ 3/2 dimer ground state, and a $S=1/2$ excited state, which implies that the molecular bonding energy is not as large and therefore only two covalent bonds form in this case, as illustrated in Fig. \[Orbitals\](b). This also explains the much larger magnetic moment observed in neutron diffraction experiments [@Senn2013]. It is natural to ask what structural parameter gives rise to this dramatic difference between the La sample and the In, Y, and Lu analogs studied here. Although there is no discernible correlation with Ru-Ru distance as shown in Table I, the La sample does have a larger Ru-O(1) distance and smaller Ru-O(1)-Ru bond angle than the other materials. These parameters may play an important role in determining the molecular bonding energy of the $e_g^\pi$ orbitals, especially since the O$_1$ ions form the common octahedral face of the Ru$_2$O$_9$ units. As can be seen from Fig. \[Orbitals\](b), a smaller bonding energy leads to the $S=3/2$ configuration expected for the La sample. ![$f(Q)^2 \Delta \chi'' = f(Q)^2 \chi''$(5 K) - $f(Q)^2 \chi''$(225 K) (arbitrary units) obtained with inelastic neutron scattering for (a) Ba$_3$LuRu$_2$O$_9$, (b) Ba$_3$InRu$_2$O$_9$ and (c) Ba$_3$YRu$_2$O$_9$ as a function of wave vector and energy transfer. (d) Cuts of $f(Q)^2\Delta \chi''$ integrated between $Q = 2$ and 2.5 Å$^{-1}$. \[Neutrons\]](Neutrons_4panel.pdf){width="3.35in"} ![$f(Q)^2 \Delta \chi'' = f(Q)^2 \chi''$($T$) - $f(Q)^2 \chi''$(225 K) (arbitrary units) obtained with inelastic neutron scattering as a function of wave vector and energy transfer, using a lower incident energy of $E_i = 50$ meV for (a) Ba$_3$LuRu$_2$O$_9$ at $T=1.5$ K (b) Ba$_3$InRu$_2$O$_9$ at $T=1.5$ K and Ba$_3$YRu$_2$O$_9$ at (c) $T=1.5$ K and (d) $T = 20$ K. \[LowIncidentEnergy\]](LowIncidentEnergy.pdf){width="3.5in"} V. Collective magnetic ground states ==================================== Specific heat, presented in Fig. \[C\](a), shows peaks at 3.0 K, 5.2 K and 10.5 K for the In, Y and Lu samples, respectively, presumably indicating the onset of long range order (LRO) or spin freezing. First, it is quite clear that these materials are highly frustrated as the values of $\Theta_W$ we have determined are much higher than $T_m$, with the frustration likely arising from the triangular lattice geometry of the Ru dimers and the strong antiferromagnetic interactions between them. More specifically, we find frustration parameters, $f = \Theta_W/T_m$, of 13 (In), 21 (Y) and 11 (Lu). While our results are qualitatively consistent with previous work [@Doi2002], there is some variability in transition temperatures between our samples and those of Doi *et al.* [@Doi2002]. Whereas our $M$ $=$ Lu sample has a peak in the specific heat $C(T)$ at 10.5 K, their sample seems to have a 9.5 K ordering transition. The low-$T$ specific heat anomaly of our $M$ $=$ Y sample is also somewhat elevated when compared to Doi *et al.* [@Doi2002]. Meanwhile, our $M$ $=$ In sample has a peak in $C(T)$ that is broader and somewhat lower in temperature. Evidently there is some sample-dependence of the magnetic properties of these materials. Since there are possible indications of magnetic order or spin freezing in the $C(T)$ measurements, we performed elastic neutron scattering on the Ba$_3M$Ru$_2$O$_9$ ($M$ $=$ In, Y and Lu) samples using the HB-1A fixed incident energy triple axis spectrometer at HFIR of ORNL. The HB-1A experiment was designed to maximize the possibility of observing a magnetic signal, so this data is complementary to the HB-2A measurements described above where magnetic Bragg peaks were not observed. Specific advantages for the HB-1A experiment, as compared to the HB-2A measurements, are as follows: (1) The low-$T$ datasets were all measured at $T$ $=$ 1.5 K to ensure that we were well below the $C(T)$ anomalies in each case. (2) The signal-to-noise at HB-1A relative to HB-2A is enhanced due to a double-bounce monochromator and the use of an analyzer for energy discrimination. Despite these improvements in the experimental set-up, the HB-1A measurements show no evidence of a magnetic signal below the $C(T)$ anomalies in each case, as illustrated in Fig. \[MagDiff\]. Although the HB-1A result for the Y sample appears to be inconsistent with previous work by Senn *et al.* [@Senn2013] using the WISH diffractometer at ISIS, it is important to note that the magnetic Bragg peaks observed in the WISH experiment were extremely weak. In fact, the ordered moment for the Y system reported in Ref. [@Senn2013] is only 0.5(6)$\mu_B$ per Ru site, so there is a great deal of uncertainty in this value. The apparent discrepancy with the HB-1A data may simply arise due to a slightly different signal-to-noise ratio on WISH as compared to HB-1A, or there may be an extreme sensitivity of the Y magnetic ground state to some form of disorder. ![(a) Specific heat ($C$) of the samples measured here with dashed lines identifying the low-$T$ anomalies as $T_m$. (b) The paramagnetic fraction of the samples as a function of temperature, obtained by applying a transverse field and assessing the amplitude of the $\mu^+$ precession generated. (c) Fast relaxation rates, $\lambda_F$, for all three samples and the highest oscillation frequency in the Lu sample, $\omega_1 = 2\pi f_1$, as functions of temperature. (d) Slow relaxation rate, or $1/T_1$, vs. temperature for all three samples.\[C\]](C_f_T1_w_longer.pdf){width="3.3in"} Ref. [@Senn2013] also reported the observation of significantly stronger magnetic Bragg peaks for Ba$_3$LaRu$_2$O$_9$. An ordered moment of $1.4(2)\mu_B$ per Ru site was determined from the subsequent magnetic refinement, which is consistent with a total spin of $S=3/2$ per dimer. Similar magnetic reflections were observed for both the La and Y samples, and this finding led the authors to conclude that these two materials host the same magnetic structure. Specifically, they find a (0 1/2 0) propagation vector which they attribute to a magnetic structure with ferromagnetic dimers. We note that their assumption of a ferromagnetic intradimer interaction is not consistent with our interpretation of the single dimer ground state for these materials, as discussed above. However, this discrepancy is resolved simply by replacing the single ion spins in their work with a single spin-1/2 moment distributed over each dimer in the case of the Y sample or a spin-3/2 moment in the case of La. The revised magnetic structure is then simply a collinear stripe phase, which has been predicted to arise for the quasi-2D triangular lattice when the NN and NNN in-plane exchange interactions are antiferromagnetic and comparable in magnitude [@Seabra2011]. These materials could therefore be considered to be the molecular magnet equivalents of isostructural compounds (for instance Ba$_3$CoSb$_2$O$_9$ [@Shirata2012; @Zhou2012Co; @Susuki2013; @Koutroulakis2015; @Quirion2015]) where the $M$-site is magnetic and forms a quasi-2d triangular lattice, albeit with a more important NNN interaction strength. ![Elastic neutron diffraction data from HB-1A for all three samples. The open circles are data taken at 1.5 K. Red points were taken above $T_m$, at 10 K (for Y and In) and 20 K for Lu. The Lu data has been shifted downwards by 200 counts / minute for ease of view. No evidence of magnetic order is observed, possibly because the ordered moment is extremely small or spread out over an entire dimer.\[MagDiff\]](MagneticDiffraction.pdf){width="3.3in"} Since our neutron scattering measurements found no evidence for magnetic Bragg peaks in the In, Y, and Lu samples, possibly due to the small ordered moment sizes, it is natural to study these materials with $\mu$SR, which is one of the most sensitive probes of weak magnetism. As can be seen in Fig. \[PandFT\], the ZF-$\mu$SR data of all three samples show indications of a magnetic phase transition with greatly increased relaxation at low $T$. The In and Y samples do not show any oscillations of the muon spin polarization thus the fast relaxation may arise from static disordered magnetism or slow spin fluctuations. As shown in Fig. \[PandFT\](a) and (b), the muon spin polarization is well-described by the following two-component relaxation function $$\label{DblExp} P(t) = (1-x)e^{-\lambda_f t} + x e^{-t/T_1}$$ where $x$ is close to 1/3 at low $T$ and $1/T_1$ is the spin-lattice relaxation rate [@Colman2011]. Assuming that we are in the quasi-static limit, $\lambda_f$ results largely from inhomogeneities (disorder) in the static internal fields at the muon site(s), whereas $1/T_1$ is caused by residual spin fluctuations. ![Zero-field $\mu$SR asymmetry at various temperatures for (a) In, (b) Y and (c) Lu, with the corresponding fits superimposed on the 2 K data only. Fourier transforms of the 2 K data, with the fits superimposed, are shown in (d) In, (e) Y, and (f) Lu . The Lu sample data shows long-lived oscillations, whereas the data of the In and Y samples shows only fast exponential relaxation (along with a slowly relaxing 1/3 tail). The narrow zero-frequency peak in the Fourier transforms comes from the $T_1$ time of the 1/3 tail, and is a measure of spin fluctuations rather than static internal fields.\[PandFT\]](PandFT.pdf){width="3.5in"} As shown in Fig. \[PandFT\](c), clear oscillations of the polarization are observed in the low $T$ regime for the Lu sample. The Fourier transform of this data, illustrated in Fig. \[PandFT\](f), shows two distinct frequencies corresponding to rather small internal fields of 6.1(2) and 14.1(1) mT. The two frequencies are indicative of two distinct muon stopping sites, which can likely be associated with the two crystallographically-inequivalent oxygen atoms in the crystal structure. It is also possible that one crystallographic muon stopping site could give rise to two distinct frequencies as a result of a complex magnetic structure. However, the magnetic structure reported by Senn *et al.* [@Senn2013] should only lead to one frequency per crystallographic site, so in that particular case our spectrum would arise from two crystallographically distinct muon stopping sites. The Fourier transform also shows that these two peaks are superimposed on a broad feature, which is consistent with the fast exponential relaxation observed in the time domain. Hence, the Lu data can be fit with the following equation : $$\label{PLu} P(t) = (1-x) \sum_{n=0}^2 a_n \cos(2\pi f_n t) e^{-\lambda_n t} + x e^{-t/T_1}$$ where $f_0$ $=$ 0 and $\lambda_0 = \lambda_f = 10.9(7)$ $\mu$s$^{-1}$ is the fast relaxing exponential component. Despite the fact that the oscillations in the muon spin polarization are very well-resolved, our fits reveal that they come from a relatively small portion of the sample, 15%, with the remainder of the sample behaving more similarly to the In and Y analogs. The fitting parameters obtained in ZF at the lowest temperatures are presented in Table \[ParamTable\]. TF-$\mu$SR measurements (in a field of $\sim$50 G) were used to rapidly map out the transitions. The data was fit with the following equation: $$\label{PTF} P(t) = f_\mathrm{PM}\cos(\gamma B_\mathrm{TF} t + \phi) e^{-\lambda t} + (1-f_\mathrm{PM})$$ where $f_\mathrm{PM}$, shown as a function of temperature in Fig. \[C\](b), is the fraction of the sample that remains paramagnetic (and therefore has oscillations of the muon spin polarization induced by the applied magnetic field). The other fraction of the sample hosts either static magnetism or strong spin dynamics that dwarf the small applied transverse field. It is interesting to compare the temperature evolution of the paramagnetic fraction to the specific heat, the maximum of which can be taken as the transition temperature, $T_m$. For the Lu sample, $f_\mathrm{PM}$ begins to drop below 100% precisely at $T_m$. On the other hand, the paramagnetic volume fraction deviates from 100% well above $T_m$ for the In and Y samples, which suggests that there is a broad temperature regime of short-range magnetic order. [lllll]{} Technique & Parameter & In & Y & Lu\ $C$ & $T_M$ (K) & 3.0(3) & 5.2(1) & 10.5(2)\ \ INS & $\Delta_1$ (meV) & 34.0(1.0) & 31.5(1.5) & 34.0(1.0)\ \ $\chi$ & $\Delta_2$ (meV) & 81(1) & 72(1) & 80(1)\ &$\Theta_W$ (K) & 43(3) & 110(10) & 113(2)\ &$\mu_\mathrm{eff}/\mu_B$ & 1.40(3) & 1.65(3) & 1.53(3)\ \ $\mu$SR & $\lambda_f$ ($\mu$s$^{-1}$) & 9.9(3) & 15.7(6) & 10.9(7)\ \ & $f_1$ (MHz) & – & – & 0.83(3)\ & $\lambda_1$ ($\mu$s$^{-1}$) & – & – & 0.7(4)\ \ &$f_2$ (MHz) & – & – & 1.91(2)\ &$\lambda_2$ ($\mu$s$^{-1}$) & – & – & 0.74(16)\ \[default\] Performing ZF-$\mu$SR measurements as a function of $T$ has allowed us to extract the temperature dependence of $1/T_1$, as well as the fast relaxation rate, $\lambda_f$. In the case of the Lu system, we can also track one of the precession frequencies, $f_1$, as a function of temperature, whereas the lower frequency, $f_2$, is only quantifiable at the lowest temperatures. These results are shown in panels (c) and (d) of Fig. \[C\]. $f_1(T)$ develops rather sharply at the Lu transition temperature and the $T$-dependence resembles a standard order-parameter plot. On the other hand, $\lambda_f$ evolves very gradually for all three samples with no sharp change at $T_m$. $1/T_1$ shows a peak near 4 K in the data for the Y and In samples, which is typical of critical spin dynamics. In the case of the Lu sample, there is a much weaker and broader feature in $1/T_1$. ![Longitudinal field $\mu$SR scans for the Y and In samples.\[LF\]](LFfigure2.pdf){width="3.3in"} The two-component exponential relaxation observed in the Y and In samples can be interpreted in two ways. First, in a quasi-static picture, the slow relaxation arises from a so-called 1/3 tail with a weak $1/T_1$ relaxation rate coming from residual spin fluctuations, and the fast relaxation is the 2/3 component coming from random internal fields. Alternatively, one could suspect a dynamic, but inhomogeneous, material with two different $T_1$ times. Longitudinal field scans at the lowest temperature, shown in Fig. \[LF\], confirm that the fast relaxation is a result of static inhomogeneities as it is decoupled fairly quickly. More precisely, in the Y sample, the fast relaxation is $\lambda_f = 15.7(6)$ $\mu$s$^{-1}$, implying an internal field distribution of width $\Delta B \simeq \lambda_f/\gamma_\mu = 184(7)$ G. Thus, the application of a longitudinal field equal to $B_\mathrm{LF} = 10\Delta B$, should entirely decouple the muon spins from the internal field and eliminate the fast-relaxing 2/3 component of $P(t)$ [@YaouancBook]. As shown in Fig. \[LF\](a), this appears to be the case for the LF $=$ 2000 G spectrum. Furthermore, as seen in Fig. \[LF\](b), the ZF fast relaxation for the In sample ($\lambda_f = 9.9(3)$ $\mu$s$^{-1}$) is somewhat more easily decoupled via application of a longitudinal field, as expected. It is thus clear that these materials host static magnetic ground states from the perspective of $\mu$SR. It is tempting to attribute the lack of oscillations in the ZF muon spin polarization of the In and Y samples to spin glass physics, especially since a zero-field-cooled/field-cooled divergence has been previously observed at $T_m$ in the DC susceptibility of the former system [@Shylk2007]. Furthermore, many geometrically-frustrated magnetic materials show a strong sensitivity to tiny amounts of quenched crystalline disorder which can lead to a spin glass transition [@Gingras1997; @Schiffer1995; @QuilliamGarnets; @Bisson2008]. However, we have also measured the AC susceptibility of these materials at several different frequencies (ranging from 333 Hz to 9999 Hz) and found no evidence of spin glassiness. More specifically, as shown in Fig. \[acChi\], the position, $T_\mathrm{max}$, of the real part of the ac susceptibility, $\chi'(T)$, is independent of frequency, in the frequency range studied. A conventional spin glass will show a maximum in $\chi'$ at the freezing temperature, $T_f$, which then depends strongly on the frequency of measurement, with an extrapolation to zero-frequency allowing for a determination of the true glass temperature, $T_g$ [@Paulsen1987]. Whereas the In and Y samples have a single peak in $\chi'$, the Lu sample has a somewhat more complicated susceptibility, with a relatively sharp peak at $\sim 11$ K, corresponding to the peak in specific heat and the onset of oscillations in $\mu$SR and a lower-temperature peak, similar to that of the Y sample, which likely corresponds to the gradual onset of fast relaxation ($\lambda_f$) in the $\mu$SR spectra. In other words, the broad, lower temperature peak, is associated with the disordered portion of the sample. Nonetheless, this peak does not seem to show an appreciable dependence on frequency, but simply a very slight increase in magnitude at 9999 Hz. These two features end up forming a rather broad critical temperature regime which is consistent with the broad $1/T_1$ feature observed in our $\mu$SR experiments on the Lu sample. Given our AC susceptibility results and the fact that magnetic Bragg peaks were observed in neutron diffraction measurements on a different Y sample [@Senn2013], it appears that these materials are not conventional spin glasses and likely have long-range ordered ground states. There are several possible origins for strongly-damped oscillations in the $\mu$SR data. We will concentrate on static origins only, since the well-defined 1/3 tail in our data indicates that the spins are mostly static, or fluctuating so slowly that they are essentially static from the point of view of $\mu$SR. The two possible static origins of the strong damping are (1) a large number of inequivalent muon stopping sites and (2) a modulation of the internal fields by disorder. The first scenario is highly unlikely given the two well-defined oscillations in the Lu data, which imply that there are two preferred crystallographic sites for the muons. On the other hand, the second scenario appears to be compatible with our $\mu$SR and neutron scattering results. An antiferromagnetic, symmetry-breaking long-range order can coexist with a large random modulation of the moments. This large amount of disorder can lead to the loss of oscillations in the ZF muon spin polarization and a reduced magnetic signal in neutron scattering that is not observed in our measurements. Given the discrepancy between our results and earlier neutron diffraction work on the Y system [@Senn2013], it is logical to suspect the influence of sample-dependent disorder on the magnetic ground state. It is also valuable to consider the implications of the observed $\mu$SR signals for the molecular magnet model proposed above, notably through the size of the measured internal fields. Dipolar coupling to point-like dipoles of $0.5\mu_B$ per site ($S=1/2$ per dimer) should give rise to an oscillation frequency of $\sim 7$ MHz for a $\mu^+$ stopping $\sim1\,\mathrm{\AA}$ away from the O$_1$ site. Hence, the fact that we observe $f_1 = 1.91(2)$ MHz in the Lu sample implies a magnetic moment of only $0.14\mu_B$ ($0.28\mu_B$ per dimer). Evidently a model of point-like dipoles on the Ru sites is highly simplistic. Even so, our results indicate that the spins are probably very much extended over an entire Ru$_2$O$_9$ “molecule” which is consistent with the orbital hybridization picture discussed above. Indeed, the slow oscillations seen here resemble those observed in molecular magnets where each spin is distributed over an entire molecular unit [@Le1993; @Blundell2004]. The ordered magnetic moments for the Y and In samples appear to be similarly weak, which is likely why no magnetic signal was detected in our elastic neutron diffraction measurements. ![AC susceptibility ($\chi'$) of all three samples at two different frequencies, 333 Hz and 9999 Hz. The high frequency data has been normalized so that the peak susceptibilities are equal, since the frequency response of the *PPMS* system is not perfectly flat. Otherwise, the temperature dependence of the susceptibilities are very similar. In particular, the peak positions are independent of frequency. \[acChi\]](acChi.pdf){width="3.3in"} Finally, we can speculate as to why the Y and In samples show such a high level of disordered static magnetism. As can be seen in Fig. \[Orbitals\](a) and (b), the $e_g^\pi$ and $e_g^{\pi\ast}$ orbitals remain degenerate in the Ba$_3M$Ru$_2$O$_9$ structure. For the Lu, Y and In samples, only one electron occupies the anti-bonding $e_g^{\pi\ast}$ orbitals and therefore they are Jahn-Teller (JT) active. Importantly, this degeneracy is not lifted by the spin-orbit coupling [@Kugel2015]. This may leave these materials vulnerable to local structural distortions that relieve the degeneracy, but lead to disorder in the interdimer exchange or the crystalline electric field, both of which can modulate the size of the ordered moments. An important parallel can be found in the sister compound Ba$_3$CuSb$_2$O$_9$, which is also based on $S=1/2$ moments and Jahn-Teller active [@Nakatsuji2012]. In Ba$_3$CuSb$_2$O$_9$, two distinct behaviors are observed, depending on the precise stoichiometry of the samples [@Nakatsuji2012; @Katayama2015]. In some off-stoichiometric samples, the orbital degeneracy is relieved by an orthorhombic distortion (a collective JT transition) near 200 K. Ultimately, these orthorhombic samples order magnetically at low temperatures. More stoichiometric samples manage to preserve their room-temperature hexagonal symmetry down to much lower temperatures either through a dynamic JT effect [@Ishiguro2013] or else local distortions that nonetheless preserve the global symmetry of the structure and give rise to a random-singlet magnetic ground state [@Quilliam2012bcso]. The most recent experimental results, thermal conductivity measurements, on nearly stoichiometric single-crystal Ba$_3$CuSb$_2$O$_9$ point toward the local-distortion picture [@Sugii2016] which is consistent with the random singlet magnetic ground state and excitation gap [@Quilliam2012bcso]. Since a hexagonal to orthorhombic collective JT transition can be ruled out by the neutron diffraction results on the materials studied here, similar random distortions might then apply, and they may be extremely important for understanding the collective magnetic ground states and possible sample dependence of the magnetic properties. Future work should search for these local distortions, possibly via x-ray absorption fine structure measurements. VI. Conclusions =============== We have used a wide array of experimental techniques to characterize both the single dimer and collective magnetic properties of the mixed valence Ru dimer systems Ba$_3M$Ru$_2$O$_9$ ($M$ $=$ In, Y and Lu). Our combined neutron powder diffraction, DC magnetic susceptibility, and inelastic neutron scattering results indicate that the Ru dimers are best described as molecular units with one spin-1/2 moment distributed equally over the two Ru sites. Two dispersive magnetic excitations are observed in the inelastic neutron scattering spectrum of each system. We attribute the lower energy mode to electron transitions between antibonding orbitals, while the upper mode is argued to arise from electron transitions between bonding and antibonding orbitals. The dimers form a quasi-2D triangular lattice, which is strongly frustrated due to significant antiferromagnetic interdimer exchange. Our heat capacity and muon spin relaxation results reveal that the molecular moments develop a static magnetic ground state in each case, with clear evidence of long-range magnetic order for the Lu sample. The size of the static internal fields observed in $\mu$SR at low temperatures are consistent with $S=1/2$ moments distributed over an entire Ru$_2$O$_9$ dimer, similar to molecular magnets. Although the static magnetism is much more disordered for the Y and In samples, they do not appear to be conventional spin glasses, for example. Overall, the current work demonstrates that the 6H-perovskites Ba$_3MA_2$O$_9$ are excellent model systems for detailed investigations of frustrated quantum magnetism arising from spin-1/2 molecular building blocks on a triangular lattice. Given the strong theoretical interest in $S=1/2$ triangular-lattice antiferromagnets and the rarity of representative materials, these systems should be attractive for future studies of the magnetic ground state and magnetization process, albeit with the added complexity of orbital degrees of freedom. Finally, we note that our results can likely be directly applied to understanding the magnetic properties of the related Ir-dimer system, Ba$_3$InIr$_2$O$_9$, which also seem to be consistent with spin-1/2 dimers at low temperature, and moreover appear to indicate a gapless quantum spin liquid ground state [@Dey2017]. We are grateful to the staff of the Centre for Molecular and Materials Science at TRIUMF for extensive technical support, in particular G. Morris, B. Hitti, D. Arseneau, and I. MacKenzie. We acknowledge useful conversations with S. Johnston and G.E. Granroth and are particularly grateful to S. Streltsov for helping us to understand the intradimer physics of these systems. A portion of this research used resources at the High Flux Isotope Reactor and Spallation Neutron Source, which are DOE Office of Science User Facilities operated by Oak Ridge National Laboratory. J. Q. acknowledges research funding obtained from NSERC and the FRQNT. R.S. and H. Z. acknlowedge the support of NSF-DMR-1350002. C. Q. acknowledges the CEM, and NSF MRSEC, under Grant No. DMR-1420451. [33]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , , , , , ****, (). , , , , , , , , , ****, (). , , , , , , , , , , (). , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , ****, (). , , , , , , , ****, (). , , , , , , ****, (). , ****, (). , ****, (). , , , , , , , , , ****, (). , , , ****, (). , , , , , ****, (). , ****, (). The value reported by Senn *et al.* is very far from the values obtained in our work and by Doi *et al.* [@Doi2002] and moreover nearly places the Ru ions outside of the surrounding oxygen octahedra. , , , , ****, (). , , ****, (). , , , , , , , ****, (). , ** (, ). , , , , , , ****, (). , , , , , ****, (). , , , , , , , ****, (). , , , , , , , , , , ****, (). , ****, (). , , , ****, (). , , , , , , , , , , , ****, (). , ****, (). , , , , , , , , , , , ****, (). , , , , , , , ****, (). , , , , , , , , , (). , , , , , , , , , , ().

--- abstract: | While the individuals chosen for a genome-wide association study (GWAS) may not be closely related to each other, there can be distant (cryptic) relationships that confound the evidence of disease association. These cryptic relationships violate the GWAS assumption regarding the independence of the subjects’ genomes, and failure to account for these relationships results in both false positives and false negatives. This paper presents a method to correct for these cryptic relationships. We accurately detect distant relationships using an expectation maximization (EM) algorithm for finding the identity coefficients from genotype data with know prior knowledge of relationships. From the identity coefficients we compute the kinship coefficients and employ a kinship-corrected association test. We assess the accuracy of our EM kinship estimation algorithm. We show that on genomes simulated from a Wright-Fisher pedigree, our method converges quickly and requires only a relatively small number of sites to be accurate. We also demonstrate that our kinship coefficient estimates outperform state-of-the-art covariance-based approaches and PLINK’s kinship estimate. To assess the kinship-corrected association test, we simulated individuals from deep pedigrees and drew one site to recessively determine the disease status. We applied our EM algorithm to estimate the kinship coefficients and ran a kinship-adjusted association test. Our approach compares favorably with the state-of-the-art and far out-performs a naïve association test. We advocate use of our method to detect cryptic relationships and for correcting association tests. Not only is our model easy to interpret due to the use of identity states as latent variables, but also inference provides state-of-the-art accuracy. cryptic relatedness, identity states, kinship coefficients, expectation maximization author: - 'Bonnie Kirkpatrick and Alexandre Bouchard-Côté' bibliography: - 'pedbib.bib' title: 'Correcting for Cryptic Relatedness in Genome-Wide Association Studies' --- Introduction ============ Accuracy in disease association studies is heavily influenced by cryptic relatedness and population substructure, [@Astle2009]. Both false positives and false negatives result from these influences, because of false assumptions of independence between individuals that are actually related. Disease associations are primarily detected using the genome-wide association study (GWAS) which is typically a case-control or cohort association study implemented using a test for correlation between the disease and the genotypes, [@Risch1996]. While GWAS typically assumes independence between the individuals, a growing number of methods are designed to detect relationships in the form of statistical dependence between the genomes of individuals and correct the correlation calculation for these dependencies. This paper presents a novel method for detecting relationships and correcting the association analysis. Methods exist that correct association test for relationships that are either known or unknown. These methods often use a two-step approach. First, pair-wise relationships in the form of identity states or kinship coefficients must be inferred either from a known pedigree structure, [@thompson1985], or from data, [@Purcell2007; @Browning2011; @Milligan2003; @Sun2014]. Second, the inferred relationships are used to correct the test for association, [@Yu2005; @Thornton2007; @Rakovski2009; @Kang2010; @Cordell2014]. In this work, we focus on inferring pair-wise relationships, for which existing methods can be split into four categories. First, there are exact methods that compute the kinship coefficients from pedigree structures in quadratic time in the number of individuals in the pedigree, [@Kirkpatrick2016a; @Abney2009; @thompson1985; @Karigl81]. Second, there are ways to estimate the probabilities of the condensed identity coefficients, [@Jacquard1972], from data using a maximum likelihood estimator, [@Milligan2003]. Third, there are ways to estimate the probabilities of the outbred condensed identity coefficients, [@Purcell2007; @Browning2011][^1], but such methods work only for outbred pedigrees and fail to capture more complicated relationships, as we see later in this paper. Forth, there has been work on inferring kinship coefficients from admixed individuals, [@Thornton2012]. Our EM method has a running time, for a pair of individuals, that is linear in the number of sites. To estimate a kinship coefficient matrix for $n$ individuals, our method is quadratic in $n$, due to the number of pairs of individual. Our method avoids the assumption of having a known pedigree, taking only the genotypes as an input. We introduce one latent variable for each pair of individuals and each site, which encodes our uncertainty on the identity by descent states. The distribution over these states for each pair of individuals is learned using a simple expectation maximization (EM) algorithm and provides an informative, yet concise and learnable, summary of the relationship of each pair of individuals. We show that this EM algorithm quickly converges to an accurate estimate. We assess the accuracy in two ways. First, we measure the accuracy of the reconstructed latent variables using simulated pedigree data (the true value of the latent variable can be computed efficiently from the held-out pedigree structure). We asses the kinship estimates compared with PLINK and with a covariance-based estimate. Second, we demonstrate that our method can be used to correct GWAS for cryptic relatedness without assuming the knowledge of the pedigree structure. Background {#sec:background} ========== A pedigree is an annotated directed acyclic graph where the nodes are diploid individuals and the edges are directed from parent to child (Figure \[fig:ped-ibd\]A). The graph is acyclic, because no individual can be their own ancestor. The pedigree graph is typically annotated with the gender (male or female), affection status, and the genotype status of the individuals. The gender is used to enforce on the graph the marriage constraint that each individual with parents in the graph has at most one parent of each gender. The *founders* are the individuals who do not have parents in the pedigree while *non-founders* are all the individuals having parent(s) in the pedigree graph. The *affection status* indicates whether an individual has a disease or phenotype. The genotype status indicates whether the individual has available genotypes. The pedigree graph depicts familial relationships between individuals. Each diploid individual inherits one copy of each chromosome from each parent, and at different sites (loci) in the genome, an individual may inherit from different copies of the parent’s chromosome. In representing inheritance, we consider a single site in the genome and the binary inheritance decision made along each parent-child edge in the pedigree graph. We represent this with an *inheritance path* (Figure \[fig:ped-ibd\]B). For a particular genomic site, the nodes of the inheritance path represent the alleles in all the individuals in the pedigree graph (i.e. for individual $a$ the allele nodes are $a_1$ and $a_2$). The edges of the inheritance path graph proceed from a parent allele to a child allele and indicate that the parent allele is copied with fidelity (i.e. without mutation) to the child allele. For each pedigree graph with $n$ edges there are $2^n$ possible inheritance path graphs. In an inheritance path, if two alleles are in the same connected component, we say those alleles are *identical by descent (IBD)*. For example, the paternal alleles from two siblings are IBD if and only if the two children inherit from the same allele of the father’s two alleles. The two alleles from the same individual whose parents are related to each other may appear in the same connected component due to inbreeding. When we focus on two individuals, $a$ and $b$, who are genotyped, a natural question is: which subsets of the four alleles of those two individuals are IBD? For a given inheritance path, the *identity state* captures the answer to this question (Figure \[fig:ped-ibd\]C). The identity state is a graph with four nodes, one for each of the four alleles $\{a_1,a_2,b_1,b_2\}$, and with an edge between two alleles if and only if these two alleles are IBD. Therefore, there is an edge between two alleles in the identity state if and only if they appear in the same connected component of the inheritance path graph. The identity state is related to the genotype data in the following way. Each of these two individuals may be either heterozygous (two different alleles, i.e. $a_1 \ne a_2$) or homozygous (two copies of the same allele, $a_1 = a_2$) at each site assayed. Certainly if alleles $x,y$ appear together in a connected component of the identity state, they are IBD and must be identical alleles. However, if alleles $x,y$ appear in different connected components of the identity state, then they are not IBD and they may or may not be identical alleles. For any given pedigree, there are 15 possible identity states, [@Jacquard1972]. For a pedigree with $n$ edges and a pair of individuals $a$ and $b$, the identity state is a random variable, $S$, having a corresponding distribution called the *identity coefficients*, $\bbP[S=s]$, which is the fraction of the $2^n$ inheritance paths for the pedigree that exhibit identity state $s$. ![[**Example pedigree, inheritance path, and identity state.**]{} [**A)**]{} An example of a pedigree dag is shown at the left with the edges directed implicitly downwards, the circles representing females, and the boxes representing males. The two individuals with known genotypes are labeled $a$ and $b$. [**B)**]{} An example of an inheritance path graph for this pedigree is shown in the center with the dots being nodes representing alleles. Although the allele nodes are unlabeled, here, for each individual the inheritance path edges are confined to the options permitted by the pedigree graph. Notice that while there are cycles in graph (A), the inheritance path graph, which has alleles (dots) as nodes, has no cycles (such a graph is called a forest, as it is not connected in general). There are three trees in this inheritance path forest. [**C)**]{} An example of an identity state is shown on the right. This identity state is the one produced from the connected components of the inheritance path graph in the center (B).[]{data-label="fig:ped-ibd"}](ped-path-is.eps){width="5in"} While the identity state describes the possible IBD between two genotyped individuals, the *condensed identity state*, [@Jacquard1972], more directly relates the IBD to the genotype data. In the identity state the two alleles of one individual are distinguishable, however in the genotype, due to our inability to measure which allele is on which chromosome, the alleles are exchangeable. The condensed identity states are a grouping or partition of the identity states such that two identity states appearing in the same group are isomorphic when the labels on the nodes belonging to individual $a$ are permuted and the labels on the nodes belonging to the individual $b$ are permuted (any valid permutation is permitted, including the permutation that does not change the node labels of the individual). According to this definition, there are nine condensed identity states. Let $c_i$ be the number of condensed identity states which contain $i$ distinct identity states. Then the 9 condensed identity states group the 15 identity states into groups $c_1=5, c_2=3, c_3=0, c_4=1$ (see Figure \[fig:identstates\] for drawings of all the identity state graphs grouped into the condensed identity states). Similar to the identity coefficient, each of these condensed identity states has a corresponding *condensed identity coefficient* which is the fraction of the $2^n$ inheritance paths for the pedigree with $n$ edges that exhibit the condensed identity state. ![[**Identity States.**]{} The 15 identity states are grouped so that each row corresponds to one of the 9 condensed identity states. The number of founder alleles for each identity state is listed, along with whether the identity state is outbred. []{data-label="fig:identstates"}](identitystates.eps){width="3in"} We need one last concept, still defined with respect to a pair of individuals in a pedigree. The *kinship coefficient* for that pair of individuals is defined as the probability of IBD when choosing one allele from each individual uniformly at random. The exact kinship coefficients for every pair of individuals are typically computed from known relationships–a pedigree graph–using a recursive computation, [@Kirkpatrick2016a; @Abney2009; @Karigl81; @thompson1985]. Methods {#sec:methods} ======= \[em\] We focus on estimating the identity coefficients. [^2] We do not directly estimate the kinship coefficients from the genotypes, because we have no model by which the kinship coefficients can generate the genotypes. On the other hand, if we know the identity states, their coefficients, and the allele frequencies in the founders, there is a generative model for producing the genotype. Since the kinship coefficients can be expressed as an expectation with respect to the identity coefficients, they contain a degenerate version of the information contained in the identity coefficients and do not provide a convenient generative model for the genotype. The estimated identity coefficients are then transformed into estimates of the kinship using equations developed by [@Kirkpatrick2011xxxx; @Kirkpatrick2016a], which we repeat for convenience. For an identity state, let $t \in \{aa, ab, bb\}$ denote an *edge type*, for example, $ab$ indicates any edge between the alleles in two different individuals $a$ and $b$ (i.e., and edge between one of the nodes $\{a_1, a_2\}$ and one of the nodes $\{b_1,b_2\}$). Let ${{\mathcal S}}$ be the set of all the identity states for a pedigree and pair of individuals. Let $e(s,t)$ for $s \in {{\mathcal S}}$ be an indicator function that is one if and only if the identity state has an edge of type $t$. Equations (\[eq:phi\_ab\]) and (\[eq:phi\_aa\]) give the kinship in terms of the identity state distribution. $$\label{eq:phi_ab} \Phi_{a,b} = \sum_{s\in{{\mathcal S}}} \frac{e(s,ab)}{4} \; \bbP[S=s].$$ where $s$ is the identity state. Rather than the kinship coefficient for the diagonal, we consider the inbreeding coefficient (see [@Kirkpatrick2016a] for details of how these quantities are related): $$\label{eq:phi_aa} \Phi_{a,a} = \sum_{s\in{{\mathcal S}}} e(s,aa) \; \bbP[S=s],$$ where $e(s,aa)$ indicates whether the single possible edge between nodes $a_1$ and $a_2$ exists. In this section, we first introduce a model with the identity states as latent variables. Second, we discuss our EM algorithm for doing inference with this model. Accuracy discussions appear in the Results section. Model ----- Let the identity states for the $m$ sites be represented as $S = (S_1, \dots,S_m)$. Let the pair genotype data be represented as $G = (G_1, \dots,G_m)$ where each $G_j$ is a tuple of two genotypes $G_j=(G^1_j,G^2_j)$, one genotype for each individual of interest. For each genotype, the values $G^i_j$ takes values in $\{0,1,2\}$, the count of the minor allele. The vector $p = (p_1, \dots, p_m)$ contains the founder allele frequencies for each site. In our model, the likelihood is a function of the allele frequencies and is defined as a marginal probability where the identity states are marginalized out $$\bbP[G=g|p] = \sum_{s\in{{\mathcal S}}} \bbP[G=g,S=s|p].$$ The joint probability of the data and the identity states is defined as the product of independent sites:[^3] $$\begin{aligned} \bbP[G=g,S=s|p] &=& \prod_j \bbP[G_j=g_j,S_j=s_j|p_j] \\ &=& \prod_j \bbP[G_j=g_j|S_j=s_j,p_j]\; \bbP[S_j=s_j].\end{aligned}$$ Since all of the inheritance paths are drawn from the same pedigree, the distribution on the identity states is the same for all sites: $\bbP[S_j=s] = \bbP[S_{j'}=s]$ for all $s$. We denote the parameters of this shared categorical distribution by $(d_s : s\in{{\mathcal S}})$, i.e. $d_s = \bbP[S_j=s]$. A key component of our model is the conditional probability $\bbP[G_j=g|S_j=s,p_j]$. To write an expression for this conditional probability, we introduce the *allele assignment* which must be consistent with both the genotype and the identity state. Recall that an identity state $s$ has vertices $V = \{a_1,a_2,b_1,b_2\}$. Let an allele assignment be given by a map $a:V \to \{0,1\}$. We say that an allele assignment $a$ with $1$ as the minor allele is consistent with the genotypes for two individuals $g=(g^1,g^2)$ if and only if $a(a_1)+a(a_2) = g^1$ and $a(b_1)+a(b_2) = g^2$. Indicate this genotype consistency by the indicator function $C$ (i.e., $C(a,g) = 1$, otherwise $C(a,g) = 0$). Further let $CC(s)$ be the partition of $V$ into connected components extracted from the identity state graph of $s$. Then we say that an allele assignment $a$ is legal with respect to the identity state if and only if the function $a$ is constant on each connected component (i.e., for each connected component $c \in CC(s)$, $a(x) = a(y)$ whenever $x,y \in c$). We represent a legal allele assignment with the indicator $L(a,s)=1$ and $L(a,s)=0$ otherwise. Now, let $$A(s,g) = \{a | C(a,g)=1 \textrm{ and } L(a,s)=1\}$$ be the set of legal and consistent assignments of the genotype alleles to the allele nodes of the identity state. We can now write an expression for the conditional probability $\bbP[G_j=g|S_j=s,p_j]$ which is: $$\bbP[G_j=g|S_j=s,p_j] \propto \sum_{a \in A(s,g)} p_j^{n_0(a)} (1-p_j)^{N_{cc}-n_0(a)},$$ where $A(s,g)$ is the set of legal assignments of the genotype alleles to the allele nodes of the identity state, $N_{cc}$ is the number of connected components in the identity state, $n_0(a)$ is the number of connected components of the identity state that are labeled with the minor allele by assignment $a$. Inference --------- For inference, we use the EM algorithm below. This algorithm takes as input an estimate of the allele frequencies $\hat{p}$ together with the genotypes for a pair of individuals and predicts the identity state coefficients for those individuals. First, we will describe a method to estimate $\hat{p}$ from many genotypes, and second, we will give the details of the EM estimator. For simplicity and computational efficiency, we estimate the allele frequencies $\hat{p}$ using the Laplace estimator based on the genotypes of independent individuals.[^4] Since our simulations produce pairs of related individuals, we consider the genotypes of one individual per simulated pair, but pool the data from many pairs. Later, in our simulations, this same allele frequency estimate $\hat{p}$ is then given to all the methods that infer kinship, ours and others. To estimate the identity state coefficient, we use an EM algorithm that iteratively produces successive estimates $d_s^{(t)}$ of the probability distribution $\bbP[S_j=s]$ from above. These estimates are obtained using simple and efficient update rules. For the [**E-step**]{} we update our estimate of $N^{(t)}_s$ which is the expected number of times that identity state $s$ occurred $$\begin{aligned} \label{eq:ml-eq} N^{(t)}_s = \sum_{j} \frac{d_s^{(t-1)} \times \bbP[G_j=g_j|S_j=s,\hat{p}]}{\sum_{s'} d_{s'}^{(t-1)} \times \bbP[G_j=g_j|S_j=s',\hat{p}]}\end{aligned}$$ while the [**M-step**]{} consists of updating our estimate of the identity coefficients $$d_s^{(t)} = \frac{N^{(t)}_s}{m}.$$ Iterative application of the E-step and M-step yields a sequence of estimates: $$(N^{(t)}_s,d_s^{(t)})~~\textrm{for } t=1,2,\dots~.$$ As we show in the Supplement, this algorithm is efficient, requiring a small number of iterations before convergence. Inferred relationships can be used to correct association tests for cryptic relatedness thereby reducing spurious, or false-positive, associations. This combined algorithm of our estimates of kinship coefficients informing MQLS, we call *pedigree-free MQLS (PFMQLS)*. Results {#sec:results} ======= There are two categories of results, estimate accuracy and corrections for spurious associations, and simulations that go along with each category. We simulated pedigree replicates from the Wright-Fisher (WF) model with parameters: $N$, the number of male and the number of female individuals per generation (meaning there are $2N$ individuals per generation), and $G$, the number of generations. The data was simulated by holding the pedigree fixed, and drawing an inheritance path for each site from the uniform distribution over inheritance paths. For each site and inheritance path, the allelic data was drawn using the parameter, $p$—the vector of minor allele frequencies, one for each site. The founder alleles for site $j$, with possible values in $\{0,1\}$, were each drawn from the Bernoulli distribution with parameter $p_j$. Once the founder alleles were selected, they were copied along the inheritance path without mutation, so that a descendant inherited a copy of a founding allele from their founding ancestor if there was an undirected path in the inheritance path graph between the descendant and the founder allele. This simulated data consisted of haplotypes, since it is known which allele was inherited from each parent. We discuss accuracy in three ways. Each accuracy measure uses variations on how the genotyped individuals are sampled from the simulation. Improved Estimation of Kinship Coefficients ------------------------------------------- To asses the accuracy of the kinship estimates, we simulated pedigree replicates. For each pedigree, we sampled two extant individuals of interest for which to simulate allelic data. Recall that the data simulation provides haplotypes. To obtain the genotypes, the record of each allele’s parent was discarded. The genotype data of the two extant individuals, and not the pedigree, was given to the estimation algorithms. To estimate the allele frequencies, since they vary from site to site, our method requires a set of individuals all simulated using the same allele frequencies. Therefore, our full simulation produces a number of pairs of individuals, where each pair is simulated from a single pedigree, and all the pedigrees share the same parameter $p$. Recall that we obtain $p$ using the Laplace estimator applied to independent individuals, one individual from each pedigree. To assess estimation accuracy, we use a gold standard estimates that are computed during pedigree and inheritance path simulation. Once the pedigree has been drawn, we apply the algorithm for computing the exact kinship coefficients and the inbreeding coefficients, [@Kirkpatrick2016a]. Once the inheritance paths have been drawn, we can compute the empirical identity state distribution represented in the data[^5]. These quantities 1) the kinship and inbreeding coefficients and 2) the identity state distribution, are used as the gold-standard for estimates. Table \[table:kinship\] shows the results of two estimation methods: the EM algorithm introduced in the Methods section and the covariance-based kinship coefficient estimator introduced by [@Astle2009]. We found that REAP, which infers kinship from admixed individuals, [@Thornton2012], does not apply to our setting even when the parameters are set for no admixture, as it found unreasonable kinship and inbreeding coefficients. Comparison with PLINK --------------------- Essentially the worst case of population structure would be if a sample of individuals was from a family, yet they were thought to be unrelated individuals. Suppose also that the founders of the family are potentially inbred. Even if the pedigree were known, the pedigree relationships would under-represent the inbreeding since some of the inbreeding occurred chronologically before the known relationships. In order to compare our method to PLINK, [@Purcell2007], which estimates the kinship coefficients, we simulate just such a scenario. We use the pedigree Wright-Fisher simulation to produce the founder haplotypes from an inbred population with $N=8$ individuals per generation and $2,4,6,8,...,40$ generations all with $m=500$ sites. From the most recent generation of the Wright-Fisher pedigree, we draw $4$ founders for an outbred $12$-individual pedigree, see Figure \[fig:pedigree\]. We simulated the recent pedigree genotypes with recombination and considered $6$ of the individuals to have observed genotypes from which we estimated the kinship coefficients. To compute the accuracy of the kinship estimates, we found the actual kinship coefficients of the $12$ person pedigree. This required using a new method of computing kinship coefficients from known founder kinship coefficients, due to [@Kirkpatrick2016a]. Both methods, ours and PLINK provide estimates of the kinship coefficients. We computed the sum of the absolute value of the differences between the matrix entries of the estimates and the actual kinship coefficients. This sum is the $L_1$ accuracy. In all cases, our method has accuracy far superior to that of PLINK, see Figure \[fig:plink\]. ![ [**Outbred Pedigree.**]{} This outbred pedigree was used to simulate genotypes from inbred founder haplotypes. The shaded individuals had genotypes that were typed and used to estimate kinship coefficients with PLINK and the EM method in this paper. []{data-label="fig:pedigree"}](pedigree.pdf){width="3in"} ![ [**Accuracy of Kinship Estimates.**]{} Comparing kinship estimates of PLINK and our EM method using the $L_1$ accuracy demonstrates that our method has superior accuracy to that of PLINK. Our method’s accuracy margin improves as the amount of inbreeding in the founders increases. []{data-label="fig:plink"}](cmpsim-paper.pdf){width="3in"} Mathematically, we compare the outbred inference method used by PLINK, [@Purcell2007], to our identity state approach which considers both outbred and inbred identity states. Notice that PLINK’s approach is limited to considering the 7 outbred identity states which have a “Yes” in the first column of Figure \[fig:identstates\]. Our approach considers both inbred and outbred identity states—all states shown in Figure \[fig:identstates\]—in a structured learning setting where the identity state for each site is selected along with the frequency for that state. From the bar plot in Figure \[fig:overfitting\] we can compute the average number of parameters for the outbreeding condensed identity states ($9/3 = 3.00$) and for both inbreeding and outbreeding condensed identity states ($13/9 \simeq 1.44$). This shows that on average, PLINK’s outbred model will over-fit as compared with our method which selects the best model from both inbred and outbred identity states. ![[**Over-fitting.**]{} The number of founder alleles are on the x-axis, and the y-axis shows the count of the number of condensed identity states with the given number of founder alleles. The left, blue bars count only the outbred condensed identity states, while the right, red bars count all the condensed identity states (outbred and inbred). Thus, the red bars are at least the height of the blue bars. From this bar plot we can compute the average number of parameters for the outbreeding condensed identity states ($9/3 = 3.00$) and for both inbreeding and outbreeding condensed identity states ($13/9 \simeq 1.44$).[]{data-label="fig:overfitting"}](overfitting.pdf){width="3in"} This over-fitting by PLINK is what we see, when PLINK’s $\hat{\pi}$ estimate of kinship nearly recapitulates the kinship computed from an outbred pedigree. This happens because PLINK explains away the excess homozygosity in the data by outbreeding which introduces many independent founder alleles—the founder alleles all appear in distinct connected components of the identity states. On the other hand our approach discovers that outbreeding explains the excess homozygosity less well than inbreeding, because the outbreeding explanation has a lower likelihood than an explanation involving inbreeding. Therefore our approach estimates, from the data, kinship coefficients that deviate from that predicted by the (outbred) pedigree structure, precisely because inbreeding among the founders provides a better model. Indeed, under the setting with a lot of inbreeding among the founders of a pedigree, an outbred-only model like that used by PLINK might have a significantly lower likelihood than a model that allows inbreeding as an explanation for the observed excess homozygosity. Fewer Spurious Associations --------------------------- Similar to the simulations in the previous section, we simulated pedigree replicates from the Wright-Fisher (WF) model. Unlike the previous simulations, we sampled $k$ extant individuals and discard the pedigree graph. The genotypes at site $j$ were simulated as before by drawing the founder alleles uniformly at random from the population distribution which is Bernoulli with parameter $p_j$, and then inheriting those alleles along the edges indicated by the inheritance path.The case-control simulation then involves two pedigree replicates which share the same founder allele frequencies. The presence of the two sets of family relationships confounds most association tests and results in very low power. Each site was independently taken to be the disease site, resulting in $m$ true positive tests per simulated pair of pedigrees. The affection status of each individual was computed from the genotype assuming an almost recessive trait and the minor allele to be the disease allele. Assuming the minor allele is $0$, the penetrance probabilities for the disease given the genotype of person $i$ were $\bbP(D|G^i=0)=0.95$ and $\bbP(D|G^i=1)=\bbP(D|G^i=2)=0.05$. Several association tests were applied to the simulated genotypes of the two pedigrees: the Cochran-Armitage trend test, the ROADTRIPS RM test, [@Thornton2010], and the PFMQLS test which is the MQLS test given the kinship coefficients estimated by the EM algorithm. The simulations where conducted with the following parameters: $N=50$ number of male/females per generation, $G=25$ number of generations, $k=10$ individuals sampled, and $m=400$ sites. We suggest using the Bonferroni threshold.[^6] Overall, we find that the Bonferroni threshold can favor the PFMQLS test, see Supplement. We summarized the data in a receiver operating characteristic (ROC) plot. The $(x,y)$ points for the ROC plot are the false positive rate (FPR) and true positive rate (TPR) for a particular p-value threshold for the test. By considering multiple thresholds, we can look across all the simulations to find the FPR for all the non-disease site and to find the TPR for all the disease sites, see Figure \[fig:roc\]. While there is a slight difference between the performance of the PFMQLS and ROADTRIPS, the number of simulations suggests that this difference may not be significant. The performance of the Cochran-Armitage trend test (CATT) is very poor due to many false positives. Both PFMQLS and ROADTRIPS avoid the spurious false positives. We conclude that PFMQLS is as good as the state-of-the-art represented by ROADTRIPS while providing an intuitive and interpretable model of relatedness. ![[**ROC plot.**]{} The x-axis is the false positive rate and the y-axis is the true positive rate. There are several tests shown: the Cochran-Armitage trend test (CATT), the pedigree-free MQLS test (PFMQLS), and the ROADTRIPS RM test. The performances of RM and PFMQLS are almost indistinguishable and far superior to that of CATT. []{data-label="fig:roc"}](roc-105.eps){width="3in"} Conclusion ========== We present a method for inferring the kinship coefficients that relies on identity states—a more detailed description of the pedigree than used by most kinship inference methods. Our method is an EM algorithm that infers the identity state distribution without assuming a known pedigree. The accuracy of our method depends on the number of sites and is reliable with as few as 64 independent sites as input[^7]. Our results show that our kinship estimates out-perform the covariance kinship method and other recent methods for kinship estimation by a large margin. Our EM kinship estimates can also correct an association test to produce state-of-the-art accuracy. Constructing the kinship matrix with our method requires a pair-wise comparison of individuals’ genomes. Such an approach can be computationally intensive, and future work includes a method avoiding this quadratic cost. A potential drawback of using an EM algorithm is that it finds local optima: our current results each use a single run of EM, and so random restarts could potentially improve the results. Other areas of future work involve simulating complex diseases which produce the disease trait by interaction of multiple sites in the genome. The results presented here were for a simple nearly recessive disease which probably accounts for the high accuracy of PFMQLS and ROADTRIPS as seen in the area under the curve for the ROC plot. In addition to simulating complex diseases, future work involves tailoring a test to the setting of using kinship corrections to detect epistasis. Author Disclosure Statement. {#author-disclosure-statement. .unnumbered} ============================ BK is the owner of Intrepid Net Computing. Supplement: Correcting for Cryptic Relatedness in Genome-Wide Association Studies {#supplement-correcting-for-cryptic-relatedness-in-genome-wide-association-studies .unnumbered} ================================================================================= Reducing Spurious Associations ------------------------------ Inferred relationships can be used to correct association tests for cryptic relatedness thereby reducing spurious, or false-positive, associations. Notably, the MQLS, [@Thornton2007] test relies on kinship coefficients calculated from a known pedigree to correct for the dependencies caused by relatedness that would confound tests that assume independence, such as the $\chi^2$ test. We propose to reduce spurious associations in data sets having an unknown pedigree by using our EM algorithm for estimating the kinship coefficients for every pair of individuals. Recall that for every pair of individuals, we obtain estimates of the inbreeding coefficients of each individual and the kinship coefficient between them. For each of the $N$ individuals, we will have $N-1$ estimates of the inbreeding coefficient which we average to obtain a single estimate. This leaves us with a matrix of estimates with the off-diagonals being estimates of the kinship coefficients and the diagonal being the estimates of the inbreeding coefficients. This combined algorithm of our estimates of kinship coefficients informing MQLS, we call *pedigree-free MQLS (PFMQLS)*. While it is possible to tailor-design a test based on these EM kinship coefficients, this approach of running MQLS on our EM results allows us to judge whether the kinship coefficients estimated by the EM algorithm can successfully reduce spurious associations. Results: Improved Estimation of Kinship Coefficients ---------------------------------------------------- Figure \[fig:sites\] shows the effect of the number of sites on the accuracy of the estimates, both in terms of the kinship coefficient estimates and of the identity state distribution estimate. The simulations were performed exactly as they were for the table (see paper) with the actual allele frequencies generated uniformly and the estimated allele frequencies being obtained empirically from independent individuals in the simulation. The estimated allele frequencies were then used to estimate the identity state distribution and the kinship coefficients. We show that it is possible to use only a few EM iterations to obtain a stable solution, Figure \[fig:l1\]. ![[**The number of sites and the accuracy of the estimates.**]{} As the number of sites increase, log-scale on the x-axis, the accuracy, on the y-axis of both the kinship and the identity state estimates, improves. The maximum number of sites shown here is $2^6 = 64$.[]{data-label="fig:sites"}](mse-kinship.eps "fig:"){width="3in"} ![[**The number of sites and the accuracy of the estimates.**]{} As the number of sites increase, log-scale on the x-axis, the accuracy, on the y-axis of both the kinship and the identity state estimates, improves. The maximum number of sites shown here is $2^6 = 64$.[]{data-label="fig:sites"}](mse-is-dist.eps "fig:"){width="3in"} ![[**$L_1$ distances between the EM solutions of successive iterations.**]{} For a single pair of individuals, the figure records the $L_1$ distance between every pair of solutions for successive iterations. As the iterations, x-axis, increase the $L_1$ distances, y-axis, decrease rapidly towards zero.[]{data-label="fig:l1"}](em_convergence.eps){width="3in"} Results: Fewer Spurious Associations ------------------------------------ We ran a simulation as described in the paper with $m=400$ sites. Figure \[fig:tppval\] shows the true positive p-values while Figure \[fig:fppval\] shows the false positive p-values.[^8] Overall, these figures illustrate that the Bonferroni threshold can favor the PFMQLS test. ![[**True positives.**]{} On the x-axis are the sites, and the y-axis the negative log of the p-value. We show here the negative log p-value for the pedigree-free MQLS test (PFMQLS), the Cochran-Armitage trend test (CATT), and the ROADTRIPS RM test. The horizontal line in the figure gives the Bonferroni-corrected threshold for a site-specific significance of $0.05$. Any spikes that protrude above the line indicate sites that are significant and true positive.[]{data-label="fig:tppval"}](plot-tp-105.pdf){width="5in"} ![[**False positives.**]{} On the x-axis are the sites, and the y-axis the negative log of the p-value. We show here the negative log p-value for the pedigree-free MQLS test (PFMQLS), the Cochran-Armitage trend test (CATT), and the ROADTRIPS RM test. The horizontal line in the figure gives the Bonferroni-corrected threshold for a site-specific significance of $0.05$. Any spikes that protrude above the line indicate sites that are significant and false positive. []{data-label="fig:fppval"}](plot-fp-105.pdf){width="5in"} [^1]: The probabilities of having zero, one, or two alleles identical-by-descent (IBD) are considered. [^2]: One could re-express our method in terms of the condensed identity coefficients (since the data do not provide information to distinguish between identity states that fall in the same condensed identity states), but the presentation is simpler in the non-condensed setting. [^3]: In practice, sites are clearly dependent because of recombination, but we assume sites have been sufficiently subsampled to approximate independence. We show in the next section that our method requires relatively few sites, so this is a reasonable approximation in practice. [^4]: Smoothing of $\hat{p}$ is required to avoid degeneracies where our method fails due to divisions by zero in Equation (\[eq:ml-eq\]). [^5]: Unlike the kinship coefficients, the exact algorithm for obtaining the identity state distribution is exponential, so we use a Monte Carlo approximation similar to [@Sun2014ugrad]. [^6]: Our association results are demonstrated with p-values, but our method is equally amenable to the use of false discovery rates and the computation of q-values. [^7]: From the whole genome of correlated sites, it is feasible to extract many more than 64 independent sites for input to our method. [^8]: Our association results are demonstrated with p-values, but our method is equally amenable to the use of false discovery rates and the computation of q-values.

--- abstract: | We report the discovery of a very cool d/sdL7+T7.5p common proper motion binary system, SDSS J1416+13AB, found by cross-matching the UKIDSS Large Area Survey Data Release 5 against the Sloan Digital Sky Survey Data Release 7. The d/sdL7 is blue in J-H and H-K and has other features suggestive of low-metallicity and/or high gravity. The T7.5p displays spectral peculiarity seen before in earlier type dwarfs discovered in UKIDSS LAS DR4, and referred to as CH$_4$-J-early peculiarity, where the CH$_4$-J index, based on the absorption to the red side of the $J$-band peak, suggests an earlier spectral type than the H$_2$O-J index, based on the blue side of the $J$-band peak, by $\sim 2$ subtypes. We suggest that CH$_4$-J-early peculiarity arises from low-metallicity and/or high-gravity, and speculate as to its use for classifying T dwarfs. UKIDSS and follow-up UKIRT/WFCAM photometry shows the T dwarf to have the bluest near-infrared colours yet seen for such an object with $H-K = -1.31 \pm 0.17$. Warm [*Spitzer*]{} IRAC photometry shows the T dwarf to have extremely red $H - [4.5]~=~4.86 \pm 0.04$, which is the reddest yet seen for a substellar object. The lack of parallax measurement for the pair limits our ability to estimate parameters for the system. However, applying a conservative distance estimate of 5–15 pc suggests a projected separation in range 45–135 AU. By comparing $H - K:H - [4.5]$ colours of the T dwarf to spectral models we estimate that $T_{\rm eff} = 500$ K and \[M/H\]$\sim -0.30$, with $\log g \sim 5.0$. This suggests a mass of $\sim$30 M$_{Jupiter}$ for the T dwarf and an age of $\sim$10 Gyr for the system. The primary would then be a 75 M$_{Jupiter}$ object with $\log~g \sim 5.5$ and a relatively dust-free $T_{\rm eff} \sim 1500$K atmosphere. Given the unusual properties of the system we caution that these estimates are uncertain. We eagerly await parallax measurements and high-resolution imaging which will constrain the parameters further. author: - | Ben Burningham$^{1}$[^1], S. K. Leggett$^{2}$, P.W. Lucas$^{1}$, D.J. Pinfield$^{1}$, R.L. Smart$^{3}$, A.C. Day-Jones$^{4}$, H.R.A. Jones$^1$, D.Murray$^1$ E. Nickson$^{5,1}$, M. Tamura$^{6}$, Z. Zhang$^{1}$, N. Lodieu$^{7}$, C.G. Tinney$^{8}$, M. R. Zapatero Osorio$^{9}$\ $^{1}$ Centre for Astrophysics Research, Science and Technology Research Institute, University of Hertfordshire, Hatfield AL10 9AB\ $^{2}$ Gemini Observatory, 670 N. A’ohoku Place, Hilo, HI 96720, USA\ $^{3}$ Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Torino, Strada Osservatrio 20, 10025 Pino Torinese, Italy\ $^{4}$ Universidad de Chile,Camino el Observatorio \# 1515, Santiago, Chile, Casilla 36-D\ $^{5}$ University of Southampton, Southampton, UK\ $^{6}$ National Astronomical Observatory, Mitaka, Tokyo 181-8588\ $^{7}$ Instituto de Astrofísica de Canarias, 38200 La Laguna, Spain\ $^{8}$ School of Physics, University of New South Wales, 2052. Australia\ $^{9}$ Centro de Astrobiología (CSIC-INTA), E-28850 Torrejón de Ardoz, Madrid, Spain\ bibliography: - 'refs.bib' title: The discovery of a very cool binary system --- surveys - stars: low-mass, brown dwarfs Introduction {#sec:intro} ============ The current generation of wide-field surveys [e.g. UKIRT Infrared Deep Sky Survey, UKIDSS; Canada-France Brown Dwarf Survey, CFBDS; @ukidss; @cfbds] is significantly expanding the sample of late type T dwarfs [e.g. @delorme08; @lod07; @pinfield08; @ben10]. Recent discoveries of extremely cool T dwarfs probe new low-temperature extremes, with $T_{\rm eff}$ as low as 500K [@ben08; @delorme08; @ben09; @sandy09]. In addition to probing new $T_{\rm eff}$ regimes, we can expect the expanded sample to populate other hitherto unexplored regions of T dwarf parameter space. Of particular interest is the growing diversity seen in metallicity and gravity for late-T dwarfs [e.g. @sandy10], and the potential for extending the low-metallicity subdwarf sequence to very low temperatures. To date, the sample of ultracool subdwarfs (UCSDs) consists of just one proposed T subdwarf, 2MASS J09373487+2931409 [@burgasser02; @burgasser06], along with a small number of L subdwarfs [e.g. 2MASS J1626+3925 - sdL4; SDSS J1256–0224 - sdL4; 2MASS J0616–6407 - sdL5; ULAS J1350+0815 - sdL5; 2MASS J0532+8246 - sdL7; @burgasser04a; @sivarani09; @cushing09; @lodieu2009; @burgasser03 respectively]. Recent parallax determinations and model comparisons by @schilbach09 suggest that of these, only the earliest type objects (2MASS J1626+39 and SDSS J1256-02) have metallicities consistent with subdwarf classification on the scheme that @gizis97 defined for M subdwarfs. Based on this @schilbach09 suggest that an intermediate d/sd classification should be applied to the two coolest objects (2MASS J0532+82 and 2MASS J0937+29). It is important to remember, however, that the subdwarf classification scheme is empirically based, and metallicities are associated with specific subdwarf classes only by model comparisons. That the model comparisons for the latest type UCSDs suggest higher metallicities than seen for earlier type objects should not be a sole basis for reclassification. The higher metallicity inferred from the colours of the coldest objects may actually highlight problems with the models to which they are compared. The spectral classification of subdwarfs should be based on observed spectral features that distinguish these objects from “normal” ultracool dwarfs (UCDs). As such, in this paper we adopt the position that the sdL objects described above are subdwarfs, since their spectra are clearly distinct from those of the bulk population of L dwarfs in a manner broadly consistent with subdwarf status. The more limited sample of T dwarfs, however, precludes such classification at this time, and we adopt the “peculiar” description for possible subdwarfs of this type [e.g. @burgasser06; @ben10]. However, in both cases the limited sample of “subdwarf” objects means that the current classification system may require significant revision as the true diversity of the spectra of low-metallicity UCDs becomes apparent in the era of larger, deeper surveys such as VISTA and WISE. We report here the discovery of a nearby d/sdL7+T7.5p common proper motion binary. The rest of the paper is laid out as follows. In Section \[sec:ident\] we describe the identification, photometric follow-up, spectral classification and proper motion determination for the two objects. In Section \[sec:binary\] we demonstrate their association as a common proper motion binary pair, and we provide initial estimates for some of their properties in Section \[sec:properties\]. Our results and conclusions are summarised in Section \[sec:summ\]. Two new ultracool dwarfs {#sec:ident} ======================== Our searches of the UKIDSS Large Area Survey [LAS; see @ukidss] have been successful at identifying late-type T dwarfs [e.g. @lod07; @warren07; @pinfield08; @ben08; @ben09; @ben10]. Using the same search methodology as previously described in detail in @pinfield08 and @ben10, we identified ULAS J141623.94+134836.30 (hereafter ULAS J1416+13) as a candidate late-T dwarf in Data Release 5 of the LAS with unusually blue $H-K =-1.35$. The subsequent photometric and spectroscopic follow-up, which resulted in its classification as a T7.5p dwarf, are described in the following sub-sections. Inspection of the surrounding field in SDSS, required to establish the red nature of ULAS J1416+13, revealed the presence of a nearby, very red object at a separation of 9. Interrogation of SDSS DR7 revealed this object, SDSS J141624.08+134826.7 (hereafter SDSS J1416+13), to have an SDSS spectrum with L dwarf spectral morphology (see also Table \[tab:optmags\] for SDSS photometry of this object). Since our initial identification of this L dwarf, its discovery has been published by @schmidt10 and @bowler10, who have classified it as a blue L5 and L6pec $\pm 2$ dwarf respectively. In the following sub-sections, we also describe our follow-up photometry of this target, and describe our analysis of this source that was carried out independently prior to the @schmidt10 and @bowler10 publications. Figure \[fig:finder\] shows a UKIDSS $J$ band finding chart for both the L and T dwarf. ![A 1’$\times$1’ $J$ band finding chart for ULAS J1416+13 and SDSS J1416+13 taken from the UKIDSS database. []{data-label="fig:finder"}](Jfinder_11.ps){height="200pt"} Near-infrared photometry {#subsec:photo} ------------------------ Near-infrared follow-up photometry was obtained using the Wide Field CAMera [WFCAM; @wfcam] on UKIRT on the night of 17$^{th}$ June 2009, and the data were processed using the WFCAM science pipeline by the Cambridge Astronomical Surveys Unit (CASU) [@irwin04], and archived at the WFCAM Science Archive [WSA; @wsa]. Observations consisted of a three point jitter pattern in the $Y$ and $J$ bands, and five point jitter patterns in the $H$ and $K$ bands repeated twice, all with 2x2 microstepping and individual exposures of 10 seconds resulting in total integration times of 120 seconds in $Y$ and $J$ and 400 seconds in $H$ and $K$. The resulting photometry for both our targets is given in Table \[tab:nirmags\]. The WFCAM filters are on the Mauna Kea Observatories (MKO) photometric system [@mko] Object $u'$ $g'$ $r'$ $i'$ $z'$ $g'-r'$ $r'-i'$ $i'-z'$ --------------- ------------------ ------------------ ------------------ ------------------ ------------------ ----------------- ----------------- ----------------- SDSS J1416+13 $23.55 \pm 0.57$ $23.08 \pm 0.18$ $20.69 \pm 0.04$ $18.38 \pm 0.01$ $15.92 \pm 0.01$ $2.39 \pm 0.19$ $2.31 \pm 0.04$ $2.46 \pm 0.01$ Object $Y$ $J$ $H$ $K$ $Y-J$ $J-H$ $H-K$ --------------- ------------------ ------------------ ------------------ ------------------ ----------------- ------------------ ------------------ -- -- -- -- ULAS J1416+13 $18.16 \pm 0.02$ $17.26 \pm 0.02$ $17.58 \pm 0.03$ $18.93 \pm 0.24$ $0.90 \pm 0.03$ $-0.32 \pm 0.03$ $-1.35 \pm 0.25$ $18.13 \pm 0.02$ $17.35 \pm 0.02$ $17.62 \pm 0.02$ $18.93 \pm 0.17$ $0.78 \pm 0.03$ $-0.27 \pm 0.03$ $-1.31 \pm 0.17$ SDSS J1416+13 $14.25 \pm 0.01$ $12.99 \pm 0.01$ $12.47 \pm 0.01$ $12.05 \pm 0.01$ $1.26 \pm 0.01$ $0.52 \pm 0.01$ $0.42 \pm 0.01$ $14.28 \pm 0.01$ $13.04 \pm 0.01$ $12.49 \pm 0.01$ $12.08 \pm 0.01$ $1.24 \pm 0.01$ $0.55 \pm 0.01$ $0.41 \pm 0.01$ Warm-Spitzer IRAC photometry {#subsec:irac} ---------------------------- The [*Spitzer*]{} General Observer program 60093 allowed us to obtain IRAC photometry of apparently very late-type T dwarfs discovered in the UKIDSS data. This Cycle 6 warm mission program provides only photometry at the shortest two wavelengths, \[3.6\] and \[4.5\]. Note that \[3.6\] and \[4.5\] are nominal filter wavelengths and, as the photometry is not colour-corrected for the dwarfs’ spectral shapes, the results cannot be translated to a flux at the nominal wavelength [e.g. @cushing08; @reach05]. Data were obtained for SDSS J1416+13 and ULAS J1416+13 on 23$^{rd}$ August 2009. The telescope was pointed mid-way between the L and T dwarf; with a separation of 9  both dwarfs were near the centre of the 5.2 arcminute field of view. Individual frame times were 30 seconds, repeated three times, with a 16 position spiral dither pattern, for a total integration time of 24 minutes in each band. The post-basic-calibrated-data (pbcd) mosaics generated by the [ *Spitzer*]{} pipeline were used to obtain aperture photometry. The photometry was derived using a 0.6-arcsecond pixel aperture radius, with separate (i.e. not annular) skies chosen to avoid the flaring due to the bright primary. The aperture correction was taken from the IRAC handbook[^2]. The error is estimated by the variation with sky aperture, which is larger than that implied by the uncertainty images (noise pixel maps) provided by the [*Spitzer*]{} pipeline, and is much less than 1% for the A component in both bands, and 4% and 0.7% for the B component at \[3.6\] and \[4.5\] respectively. The description of the primary issues with early release warm IRAC data[^3] indicates that the only significant concern is the uncertainty in the linearity correction for SDSS J1416+13; the total uncertainty due to this correction is estimated to be 5–7% at \[3.6\] and 4% at \[4.5\] for bright sources. Otherwise the photometry for both sources is uncertain by the usual 3% due to uncertainties in the absolute calibration and pipeline processing. Table \[tab:irac\] gives the photometry and the total uncertainties for both dwarfs. Near-infrared spectroscopy {#subsec:spectra} -------------------------- We used $JH$ and $HK$ grisms in the InfraRed Camera and Spectrograph [IRCS; @IRCS2000] on the Subaru telescope on Mauna Kea to obtain a R$\sim 100$ $JH$ and $HK$ spectra for ULAS J1416+13 on 7$^{th}$ May 2009 and 31$^{st}$ December 2009 respectively. The observations were made up of a set of eight 300s sub-exposures for the $JH$ spectrum and eighteen 200s sub-exposures in an ABBA jitter pattern to facilitate effective background subtraction, with a slit width of 1 arcsec. The length of the A-B jitter was 10 arcsecs. The spectrum was extracted using standard IRAF packages. The AB pairs were subtracted using generic IRAF tools, and median stacked. The data were found to be sufficiently uniform in the spatial axis for flat-fielding to be neglected. We used a comparison argon arc frame to obtain the dispersion solution, which was then applied to the pixel coordinates in the dispersion direction on the images. The resulting wavelength-calibrated subtracted pairs had a low-level of residual sky emission removed by fitting and subtracting this emission with a set of polynomial functions fit to each pixel row perpendicular to the dispersion direction, and considering pixel data on either side of the target spectrum only. The spectra were then extracted using a linear aperture, and cosmic rays and bad pixels removed using a sigma-clipping algorithm. Telluric correction was achieved by dividing each extracted target spectrum by that of the F4V star HIP 72303, which was observed just after the target and at a similar airmass. Prior to division, hydrogen lines were removed from the standard star spectrum by interpolating the stellar continuum. Relative flux calibration was then achieved by multiplying through by a blackbody spectrum of the appropriate $T_{\rm eff}$. The spectra were then normalised using the measured near-infrared photometry to place the spectra on an absolute flux scale, and rebinned by a factor of three to increase the signal-to-noise, whilst avoiding under-sampling of the spectral resolution. Spectral types {#subsec:sptypes} -------------- As noted in the Section \[sec:intro\], the discovery SDSS J1416+13 has recently been published by @schmidt10 and @bowler10. @schmidt10 find an optical spectral type of L5 and an infrared type of L5–6 [using the @geballe02 indices]. @bowler10 similarly find an optical type of L6$\pm$0.5 and an infrared type of L7–7.5. The template fits carried out in both papers show some discrepancies beyond 9000Å however, and here we use the SDSS spectrum of the source to produce an alternative classification as follows. The top two panels of Figure \[fig:sdssspec\] show the SDSS DR7 spectrum of SDSS J1416+13 along with the optical spectra of the L6 and L7 spectral templates 2MASS J0103+19 and DENIS J0205–11. Whilst the SDSS J1416+13 is good match over much of the range to the L6 template, they disagree significantly beyond 9000Å. On the other hand, the slope of the pseudo-continuum is very similar to that of an L7 across the entire 6000–9200Å range, although the prominent TiO, FeH and CrH features are considerably stronger in the spectrum of SDSS J1416+13. This behaviour is more typical of low-metallicity objects, where it has been speculated that that the low-metallicity atmosphere inhibits the formation of the condensate dust clouds, allowing the opacity due to alkali and hydride species to become more apparent [e.g. @burgasser03; @reiners06]. Hence, we do not classify this object following the system for L dwarfs defined by @kirkpatrick99, and instead rely on comparison to other metal-poor L dwarfs. The lower panel of Figure \[fig:sdssspec\] shows the close similarity between the spectrum of SDSS J1416+13 and that of the metal-poor L dwarf 2MASS J0532+8246. @burgasser03 demonstrated that this object not only displays features characteristic of a low-metallicity atmosphere, but also has kinematics consistent with halo membership, and classify it as sdL7. Whilst the general agreement between the spectrum of SDSS J1416+13 and the sdL7 spectrum is good across the entire range considered, there are specific areas of disagreement that should be noted. In particular the Cs[I]{} and Na[I]{} absorption features are somewhat deeper than in the sdL7 template, and more suggestive of dwarf classification than that of a subdwarf. This suggests that SDSS J1416+13 may be less metal poor than 2MASS J0532+82. Given the apparent intermediate nature of SDSS J1416+13 between the L7 and sdL7 spectra, we classify it as d/sdL7 (optical). We note that @bowler10 suggest that SDSS J1416+13 is unlikely to have significantly reduced metallicity based on the optical TiO and CaH features. @burgasser08b and @stephens09 discuss various mechanisms which may lead to unusually blue L dwarfs including low metallicity, high gravity and thin condensate cloud decks. We explore the physical properties of the L dwarf further in Section \[sec:properties\]. The IRCS spectrum of the T dwarf, ULAS J1416+13, is shown in Figure \[fig:JHKspec\], along with spectra of the T7 and T8 spectral standards [@burgasser06]. With the exception of the poor match to both templates on the red side of the J-band peak and the heavily suppressed $K$ band peak, the spectrum appears intermediate between the two. This is reflected in the spectral typing ratios (see Table \[tab:indices\]), and we classify this object as T7.5p. The early type suggested by the CH$_{4}$-K index clearly reflects the small amount of flux in the $K$ band peak. The type of peculiarity seen here in the red side of the $J$ band peak, and reflected in the spectral typing ratios, has been described for at least three other T dwarfs in @ben10, and has been suggested as a possible tracer of low-metallicity and/or high-gravity. The significance of this feature is discussed in more detail in Section \[sec:properties\]. Name \[3.6\] \[4.5\] --------------- ------------------ ------------------ SDSS J1416+13 $10.99 \pm 0.07$ $10.98 \pm 0.05$ ULAS J1416+13 $14.69 \pm 0.05$ $12.76 \pm 0.03$ : [*Spitzer*]{} IRAC photometry for the d/sdL7 and T7.5p dwarfs presented here. \[tab:irac\] {height="300pt"} {height="500pt"} [**Index**]{} [**Ratio**]{} [**Value**]{} [**Type**]{} ------------------- ----------------------------------------------------------------------------------------- ----------------- -------------- H$_2$O-J $\frac{\int^{1.165}_{1.14} f(\lambda)d\lambda}{\int^{1.285}_{1.26}f(\lambda)d\lambda }$ $0.07 \pm 0.01$ T7/8 \[+1mm\] CH$_4$-J $\frac{\int^{1.34}_{1.315} f(\lambda)d\lambda}{\int^{1.285}_{1.26}f(\lambda)d\lambda }$ $0.34 \pm 0.01$ T6 $W_J$ $\frac{\int^{1.23}_{1.18} f(\lambda)d\lambda}{2\int^{1.285}_{1.26}f(\lambda)d\lambda }$ $0.34 \pm 0.01$ T7/8 H$_2$O-$H$ $\frac{\int^{1.52}_{1.48} f(\lambda)d\lambda}{\int^{1.60}_{1.56}f(\lambda)d\lambda }$ $0.20 \pm 0.01$ T7/8 CH$_4$-$H$ $\frac{\int^{1.675}_{1.635} f(\lambda)d\lambda}{\int^{1.60}_{1.56}f(\lambda)d\lambda }$ $0.20 \pm 0.01$ T7 NH$_3$-$H$ $\frac{\int^{1.56}_{1.53} f(\lambda)d\lambda}{\int^{1.60}_{1.57}f(\lambda)d\lambda }$ $0.61 \pm 0.01$ ... CH$_4$-K $\frac{\int^{2.255}_{2.215} f(\lambda)d\lambda}{\int^{2.12}_{2.08}f(\lambda)d\lambda }$ $0.29 \pm 0.02$ T4 : The spectral flux ratios for ULAS J1416+13. Those used for spectral typing are indicated on Figure \[fig:JHKspec\].The NH$_3$ index is not used for assigning a type [see @ben08 and Burningham et al 2010 for a discussion of this], but is included for completeness and to permit future comparison with other late T dwarfs. []{data-label="tab:indices"} Proper motions {#subsec:propermotion} -------------- The photometric follow-up observations that were carried out provided a second epoch of imaging data, showing the position of the two sources of interest 1.1 years after the LAS image was measured. We used the [IRAF]{} task [GEOMAP]{} to derive spatial transformations from the WFCAM follow-up $J$-band image into the original UKIDSS LAS $J$-band image based on the positions of 18 reference stars. The transform allowed for linear shifts and rotation, although the rotation that was required was negligible. We then transformed the WFCAM follow-up pixel coordinates of the targets into the LAS images using [GEOXYTRAN]{}, and calculated their change in position (relative to the reference stars) between the two epochs. The uncertainties associated with our proper motion measurement primarily come from the spatial transformations, and the accuracy with which we have been able to measure the position of the targets (by centroiding) in the image data. Centroiding uncertainties for the targets should be small, since the seeing and signal-to-noise of the sources was good in both epochs, so this latter source of uncertainty will be neglected. For the LAS image the seeing was $\sim$0.9 in the $J$-band, whilst for the WFCAM image it was $\sim 1.1$. The root-mean-square (rms) scatter in the difference between the transformed positions of the reference stars and their actual measured positions was $\pm$0.24 pixels in declination and $\pm$0.18 pixels in right ascension, corresponding to 0.048 and 0.036  in the $J$-band LAS image. We thus estimate proper motion uncertainties of $\pm$45 mas/yr and $\pm$33 mas/yr in declination and right ascension respectively. The final, relative, proper motion measurements are $\mu_{\alpha cos\delta}=248 \pm 33$mas/yr, $\mu_{\delta}= 100 \pm 45$mas/yr for SDSS J1416+13 and $\mu_{\alpha cos\delta}=221 \pm 33$mas/yr, $\mu_{\delta}= 115 \pm 45$mas/yr for ULAS J1416+13. It should be noted that the relative proper motions calculated here disagree with the absolute values found for the primary by @schmidt10 and @bowler10 at the 4$\sigma$ level. This discrepancy likely arises as a result of two factors. Firstly, in the first epoch images both targets lie within 30 of the detector edge. As a result, the distribution of reference stars is not even about the targets. Since there is likely to considerable geometric distortion across the field of view, this poor distribution of reference stars will likely result in an unreliable absolute fit to the coordinates. Secondly, they do not take into account the parallax of the targets. The first and second epoch data were taken on 12$^{th}$ May 2008 and 17$^{th}$ June 2009 respectively, which would suggest the influence of parallax should be small. However, given that the distance for both objects may be as low as 5 pc (see Section \[sec:properties\]), we do not rule this out as a significant effect. These concerns should not effect the reliability of these proper motions as relative values, but we caution that they include systematic effects that prevent their use in any absolute manner. A wide low-mass binary {#sec:binary} ====================== The close agreement of the proper-motions for these two objects, and their 9  proximity on the sky suggests that they represent a common proper motion binary pair. To estimate the probability that the proper motions are aligned by chance, rather than because of a bona-fide association, we have considered the proper motions of objects in the SuperCosmos Sky Survey [@supercos1] in the direction of our targets. Since we do not have a parallax for either object, we instead estimate a liberal range of distances based on their spectral types and apparent magnitudes for the purposes of placing broad limits on their shared volume. In Section \[sec:properties\] we refine this distance estimate based on subsequent analysis of these objects. If we apply the $M_J$ vs. spectral type relations of @liu06 we find that an L7 and a T7.5 dwarf with the apparent magnitudes of our objects can be expected to lie at distances ranging from 5 pc to 25 pc. Of the $\sim 50$ SuperCosmos objects with apparent distances (based on colour-magnitude relation for field stars) similar to those of our targets, none shared a common motion to within $2 \sigma$. We thus conclude that the likelihood of a common proper motion occurring by chance in this direction is less than $1/50$. Since the statistics for the properties of the ultracool subdwarf population are not currently known, we will use the space density of “normal” L dwarfs to estimate a conservative probability that this pair are unrelated, and are found in close proximity by chance. Using our liberal distance range of 5–25 pc, and given the separation of 9, we can thus estimate that two objects likely share a volume of $\leq$0.01 pc$^3$. The space density for field L dwarfs was determined by @cruz07 to be 0.0038 pc$^{-3}$. The probability of finding an L dwarf within the same 0.01 pc$^3$ as our T7.5 dwarf is thus $3.8 \times 10^{-5}$. It is reasonable to surmise that the probability of finding two ultracool subdwarfs within this volume would be considerably smaller. These combined arguments suggest the probability of a chance alignment in space and motion for these two objects is less than $10^{-6}$. If we apply these arguments to the total UKIDSS LAS T dwarf sample up to DR4 [@ben10] we find that we would need a sample of approximately 1000 times larger before we would expect to identify one chance alignment such as this. It is worth stressing that our estimate for this probability is somewhat conservative. Given the apparently unusual nature of the objects discussed here, it is likely that true probability for chance alignment is considerably lower. We thus conclude that SDSS J1416+13 and ULAS J1416+13 represent a binary pair, which we shall henceforth refer to as SDSS J1416+13AB. The properties of SDSS J1416+13AB {#sec:properties} ================================= The optical spectral classification of SDSS J1416+13A as a dwarf/subdwarf implies that we could reasonably classify the secondary as a dwarf/subdwarf also, given that most binary systems are expected to be coevally formed in the same cloud core. Figure \[fig:colplot\] shows near-infrared colours as a function of spectral type for L and T dwarfs, with SDSS J1416+13AB indicated with red asterisks. Blue $H-K$ near-infrared colours for mid-to late T dwarfs have typically been interpreted as indicative of low-metallicity and/or high-gravity [e.g. @burgasser02; @knapp04; @liu07], caused by $K$ band suppression by pressure sensitive collisionally induced absorption by hydrogen [CIA H$_2$; @saumon94]. Blue $J-H$ colours in metal poor L dwarfs have also been interpreted in terms of $H$ band suppression by CIA H$_2$ [e.g. @burgasser03]. The blue $J-H$ colour of SDSS J1416+13A, and the blue $H-K$ colour of SDSS J1416+13B, therefore, support the interpretation that both objects have low-metallicity and/or high-gravity, and we interpret the peculiar spectral shape of SDSS J1416+13B in this context. ![$J-H$ and $H-K$ colour as a function of spectral type for L and T dwarfs. Data for L and T  dwarfs on the MKO system are taken from @knapp04 with T spectral types updated to the @burgasser06 system. Additional data for late-T dwarfs taken from @ben10. Known binary systems are shown as green dots, whilst known metal poor objects discussed in the text are shown as red dots, and labelled in the lower plot. The only other known T dwarf with $K$ band photometry that displays CH$_4$-J-early peculiarity is shown as an orange dot, whilst SDSS J1416+13AB are shown as red asterisks. With the exception of SDSS J1416+13A and 2MASS J0532+82, all spectral types are near-infrared types. 2MASS photometry for 2MASS0532+82 has been converted to the MKO system using the @stephens04 relationships, which give consistent results with synthetic colours calculated from the object’s near-infrared spectrum. []{data-label="fig:colplot"}](colplot.ps){height="300pt"} The spectral morphology in the $J$ band peak of SDSS J1416+13B is reminiscent of a number of T dwarfs recently discovered that have been classified as peculiar [@ben10]. These also show a $J$ band peak that appears earlier in type on the red side (as indicated by the CH$_4$-J index) compared to the blue side (as indicated by the H$_2$0-J and $W_J$ indices). This morphology was referred to by @ben10 as CH$_4$-J-early peculiarity, and we continue this convention here. Only one of the objects already found with CH$_4$-J-early peculiarity, ULAS J1233+1219, currently has $K$ band photometry. It also appears very blue, with $H-K = -0.75$ (indicated by an orange filled circle in Figure \[fig:colplot\]), and is as notable an outlier in $H-K$ for its type as SDSS J1416+13AB. It thus seems plausible that CH$_4$-J-early peculiarity is indicative of low-metallicity and/or high gravity. There is some theoretical basis for preferring a low-metallicity interpretation of CH$_4$-J-early peculiarity. Figure \[fig:metmod\] shows comparisons of @bsh2006 model spectra for $\log~g = 5.0$, $T_{\rm eff} = 700$K T dwarfs with solar and \[Fe/H\]=-0.5 metallicity, and also for solar metallicity with $\log g = 5.0$ and $\log g = 5.5$. Enhancement of the red side of the $J$ band peak is apparent in both the low-metallicity and high-gravity cases, but is most pronounced in the former. We speculate that CH$_4$-J-early peculiarity may represent a useful tracer of low-metallicity atmospheres, although its presence in a system with fiducial metallicity and age constraints will need to be observed before a robust interpretation will be possible. It is interesting to note that the spectral shape of SDSS J1416+13B also deviates from that of the spectral templates blueward of 1.1$\mu$m, in a manner similar to that seen in Figure \[fig:metmod\] for the low-metallicity case. The same behaviour is not predicted for the high-gravity case. A spectrum with better coverage in the $Y$ band may provide a useful means of breaking the gravity-metallicity degeneracy. The need for a more complex spectral classification scheme to take account of spectral variations that result from changes in metallicity and gravity in addition to $T_{\rm eff}$ has been highlighted by @kirkpatrick05. As more objects that exhibit CH$_4$-J-early spectral peculiarity are identified, its behaviour may provide a convenient method for more detailed classification of T dwarf spectra. ![@bsh2006 models for $\log~g = 5.0$ 700K T dwarfs with solar and \[Fe/H\]0=-0.5 metallicity, and for solar metallicity combined with $\log g = 5.0$ and $\log g = 5.5$. []{data-label="fig:metmod"}](modcomp.ps){height="250pt"} The lack of a known parallax for this binary pair precludes a detailed assessment of their properties, since spectroscopic distances are not well constrained for ultracool T dwarfs. In the case of the one previously identified sdL7, 2MASS J0532+82, the determined absolute magnitude ($M_J$ = 13.00) is 1-2 mags brighter than might otherwise be expected for a field dwarf of type L7 [@burgasser08]. If we assume that SDSS J1416+13A has $M_J = 13.0$ as was the case for for 2MASS J0532+82 we arrive at a distance estimate of 10 pc. However, SDSS J1416+13A is considerably less blue in $J-H$ than 2MASS J0532+82 ($J-H = 0.55$ vs $J-H = 0.08$ respectively, see Figure \[fig:colplot\]) and, as previously discussed, may have rather different properties. Assuming spectral types L7 and T7.5, however, suggests distances of 5 pc and 20 pc for the objects respectively by applying the $M_J$ vs spectral type relations of @liu06. In the case of the metal poor T dwarf 2MASS J0937+29 this method would overestimate the distance by $\sim 30$%. Using this as a correction suggests a distance for SDSS J1416+13AB of 14 pc. This would represent a significant discrepancy in the distances of the primary and secondary members of SDSS J1416+13AB, implying that the primary could be an unresolved binary. However, the $K$ band suppression in SDSS J1416+13B is greater than in the case of 2MASS J0937+29 and, as discussed below, it appears to be considerably cooler. It thus seems likely that SDSS J1416+13B is fainter still, and a distance as close as 10 pc seems plausible. A distance of $\sim 10$pc is also in broad agreement with that estimated by @schmidt10 and @bowler10 for the primary. We thus conservatively estimate the distance to SDSS J1416+13AB to lie in the 5-15 pc range. The implied projected separation of the binary pair at this range of distances is 45 – 135 AU. It is thus possible that this pair also represents a rare very low-mass wide binary system [e.g. Figure 9 in @lafreniere08]. The longer baseline provided by our [*Spitzer*]{} IRAC photometry offers the opportunity to estimate parameters of the system through comparison to predictions of model spectra. The IRAC colours of SDSS J1416+13A are normal for a late-type L dwarf, although the colours of these objects show significant scatter (see Figure \[fig:mircol\]). All of the low-metallicity late-T dwarfs plotted in Figure \[fig:mircol\] display $H - [4.5]$ that is at least 0.5 magnitudes redder than would otherwise be expected for a “normal” T dwarf of their subtype. However, the $H_{MKO}$ - \[4.5\] colour of SDSS J1416+13B is the reddest yet measured. In addition to apparently indicating low-metallicity atmospheres, this colour is a good indicator of $T_{\rm eff}$ [e.g. @warren07; @stephens09; @sandy10] and so SDSS J1416+13B appears to be very cool. ![Spitzer IRAC colours as a function of spectral type for L and T dwarfs. Data for L and T  dwarfs on the MKO system taken from @knapp04 with T spectral types updated to the @burgasser06 system. Additional data for late-T dwarfs taken from @sandy10. Known binary systems are shown as green dots, whilst known metal poor objects discussed in the text are shown as red dots, and labelled in the upper plot. SDSS J1416+13AB are shown as red asterisks. []{data-label="fig:mircol"}](mircol.ps){height="300pt"} Figure \[fig:h449\] reproduces Figure 11 of @sandy10 with the location of SDSS J1416+13B indicated. It can be seen that this T dwarf forms a sequence with the other known metal-poor (\[m/H\]$\sim$ -0.3) high-gravity ($\log g \sim 5.0 - 5.3$) dwarfs: 2MASS J0937347+293142, 2MASS J12373919+6526148, 2MASS J11145133–2618235, 2MASS J09393548–2448279. The dwarfs have $T_{\rm eff} \sim 950, 825, 750$ and 600 K respectively [@geballe09; @lb07; @sandy07; @sandy09; @burgasser08a]. Extrapolating these values using Figure \[fig:h449\] and the models [@marley02; @sm08] shown in the figure, implies that SDSS J1416+13B has \[m/H\]$\sim$–0.3, ${\rm log}~g \sim$ 5.0 to 5.3 and $T_{\rm eff} \sim 500$ K. This indicates a mass of 30–40 $M_{Jupiter}$ for SDSS J1416+13B and an age around 10 Gyr or older for the system using the evolutionary models of @sm08. The near-infrared indices of @geballe02 for SDSS 1416+13A suggest a near-infrared spectral type of L7-7.5 [@bowler10]. The near-infrared spectral type-$T_{\rm eff}$ relations of @stephens09 suggest that blue L7 dwarfs have $T_{\rm eff} \sim$1500 K. If the system is aged at $\sim 10$Gyr as implied by the secondary then the evolutionary models of @sm08 suggest that the primary is a $\sim$75M$_{Jupiter}$ dwarf with $\log~g \sim 5.5$. Hence the L dwarf is at the stellar/substellar boundary as also suggested by @bowler10. @stephens09 have shown that the atmospheres of blue L dwarfs are less dusty than the bulk population, deriving a high value of $f_{sed} \sim 3$ using the @marley02 models. These authors also find that there is an indication that dust clearing may occur at higher temperatures for higher gravity systems. The blue colours and almost dust-free atmosphere of this relatively warm $\sim$1500 K L dwarf is therefore consistent with a high gravity and relatively old age for the system. Both @schmidt10 and @bowler10 have estimated $(U,V,W)_{\rm LSR}$ for SDSS J1416+13A, finding $(-7.9 \pm 2.1, 10.2 \pm 1.2, -31.4 \pm 4.7)$ kms$^{-1}$ and $(-6 \pm 4, 10.2 \pm 1.4, -27 \pm 9)$ kms$^{-1}$ respectively[^4], and interpret its kinematics as indicative of thin disk membership. This is consistent with the age of  10 Gyr and slight metal-paucity that we find here [@robin03; @haywood97]. It is intriguing that SDSS J1416+13B appears to be $\sim$250 K and $\sim$100 K cooler than 2MASS J1114–26 2MASS J0939–24 respectively, which are of similar spectral type (T7.5 and T8). It is plausible that the near-infrared spectral type vs. $T_{\rm eff}$ relation for very late T dwarfs shows significant dependence on metallicity and gravity, with lower-metallicity dwarfs of a given subtype having lower $T_{\rm eff}$ than similar type objects of higher metallicity. The cool nature of 2MASS J0939–24 compared to other T8 dwarfs spectral type would tend to support this assertion, although interpretation of this object is complicated by its probable binarity [@burgasser08a]. If SDSS J1416+13B had significantly lower metallicity and/or higher gravity than 2MASS J1114–26 2MASS J0939–24, such an effect might account for their similar types but diverse $T_{\rm eff}$s. However, the same model predictions seen in Figure \[fig:h449\] that suggest such different $T_{\rm eff}$ also suggest fairly similar metallicities and gravities for the three objects. Finally, it should be noted that we cannot currently rule out the presence of a cooler unresolved companion to SDSS J1416+13B, which might explain its extremely red $H-[4.5]$ colour coupled with is T7.5p near-infrared morphology. Unfortunately, the lack of parallax and high-resolution imaging for this target prevent us from adequately exploring this issue here. {height="400pt"} Summary {#sec:summ} ======= We have identified what appears to be the coolest binary system yet found. The association of the T7.5p component with the d/sdL7 primary allows us to now extend the high-gravity and low-metallicity sequence to the lowest observed temperatures, and we suggest that CH$_4$-J-early peculiarity [@ben10] may in future prove to be a useful discriminator for this type of object. The likely close proximity of the system to the Sun should facilitate the determination of the trigonometric parallax in the near-future, which will allow a more robust determination of the properties for this exciting system. Acknowledgements {#acknowledgements .unnumbered} ================ SKL is supported by the Gemini Observatory, which is operated by AURA, on behalf of the international Gemini partnership of Argentina, Australia, Brazil, Canada, Chile, the United Kingdom, and the United States of America. EN received support from a Royal Astronomical Society small grant. NL was funded by the Ramón y Cajal fellowship number 08-303-01-02. CGT is supported by ARC grant DP0774000. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and has benefited from the SpeX Prism Spectral Libraries, maintained by Adam Burgasser at http://www.browndwarfs.org/spexprism. [^1]: E-mail: B.Burningham@herts.ac.uk [^2]: http://ssc.spitzer.caltech.edu/irac/dh/ [^3]: http://ssc.spitzer.caltech.edu/irac/documents/iracwarmdatamemo.txt [^4]: $U$ positive towards the Galactic centre.

--- abstract: 'Large discrepancies between quasi-free neutron-neutron (nn) cross section data from neutron-deuteron (nd) breakup and theoretical predictions based on standard nucleon-nucleon (NN) and three-nucleon (3N) forces are pointed out. The nn $^1S_0$ interaction is shown to be dominant in that configuration and has to be increased to bring theory and data into agreement. Using the next-to-leading order (NLO) $^1S_0$ interaction of chiral perturbation theory ($\chi$PT) we demonstrate that the nn QFS cross section only slightly depends on changes of the nn scattering length but is very sensitive to variations of the effective range parameter. In order to account for the reported discrepancies one must decrease the nn effective range parameter by about $\approx 12 \% $ from its value implied by charge symmetry and charge independence of nuclear forces.' author: - 'H. Wita[ł]{}a' - 'W. Glöckle' title: 'The nn quasi-free nd breakup cross section: discrepancies to theory and implications on the $^1S_0$ nn force' --- Introduction {#one} ============ The knowledge of the NN interaction is fundamental for interpreting nuclear phenomena. The proton-proton (pp) experiments provide a solid data basis [@nijm90; @nijm93], which restricts theoretical assumptions about the strong part of the pp force. In case of the neutron-proton (np) system this is only true to a smaller extent. The partial wave analysis of the np data [@nijm93] relies on the assumption that the isospin $ t=1 $ piece can be taken over from the pp system and only the $ t=0 $ part is free in the adjustment to the data. The lack of a free neutron target forbids neutron-neutron (nn) experiments, therefore the information on the nn interaction can be deduced only in an indirect way. To that aim the best tool seems to be the study of the three-nucleon (3N) system composed from two neutrons and the proton. It is simple enough to allow a rigorous theoretical treatment, e.g. in the framework of Faddeev equations [@physrep96]. The neutron-deuteron elastic scattering together with the neutron induced deuteron breakup, supplemented with the triton properties, offer a data basis which can be used to test properties of the nn force. Especially the nd breakup process with its rich set of configurations for free three outgoing nucleons seems to be a powerfull tool to test the nuclear Hamiltonian. By comparing theoretical predicitons to the nd breakup data in different configurations not only can the present day models of two-nucleon (2N) interactions be tested, but also effects of three-nucleon forces (3NF’s) can be studied. The nn quasi-free scattering (QFS) refers to a situation where the outgoing proton is at rest in the laboratory system. In the nd breakup also np QFS is possible. Here one of the neutrons is at rest while the second neutron together with the proton form a quasi-freely scattered pair. The reported nn QFS cross sections taken at $E_n^{lab}=26$ MeV [@siepe2] and at $E_n^{lab}=25$ MeV [@ruan1] overestimate the nd theory by $\approx 18 \%$. Surprisingly, when instead of the nn pair the np pair is quasi-freely scattered, the theory follows nicely the np QFS cross section data taken in the $E_n^{lab}=26$ MeV nd breakup measurement [@siepe2]. That good description of the np QFS cross section contrasts with the drastic discrepancy between the theory and the nn QFS cross section data taken in the same experiment [@siepe2]. We do not expect surprises in case of the pp QFS data [@raup1; @przyb1; @zejma1], since the information of the rich set of pp data has been incorporated into the pp forces. In fact a recent analysis [@wit_coul] including the Coulomb force to pp QFS data lead to a nice agreement, while in previous analysis [@raup1; @przyb1; @zejma1] the Coulomb force was not yet included. Additional theoretical efforts to include all effects of the Coulomb force beyond the ones in [@wit_coul] are underway. In section \[two\] we exemplify the stability of the QFS cross sections against changes of modern nuclear forces. We also demonstrate that below $\approx 30$ MeV the $^1S_0$ and $^3S_1$-$^3D_1$ NN force components dominate the QFS cross sections. In section \[three\] we analyse the np as well as the nn QFS data from [@siepe2] in terms of rigorous solutions of the 3N Faddeev equation and discuss necessary changes in the $^1S_0$ nn force component to remove the discrepancies in the nn QFS cross section. Thereby a detailed study is performed using the next-to-leading order (NLO) chiral NN force, composed of contact interactions and the one-pion exchange potential. It reveals that the effective range parameter is decisive to reconcile theory and data. The outcome is discussed in section \[four\] and further experimental insights on the nn force are proposed. Finally we summarize in section \[five\]. Stability and sensitivity studies {#two} ================================== It is known that nd scattering theory provides QFS cross sections which are highly independent from the realistic NN potential used in the calculations and that they practically do not change when any of the present day 3NF’s is included [@physrep96; @kur2002; @wit2010]. We exemplify it in Fig. \[fig\_qfs\_nn\_np\_indep\_NN\_3NF\] for the nn and np QFS geometries of ref. [@siepe2]. There results of 3N Faddeev calculations [@physrep96] based on different high precision NN forces (CD Bonn [@cdbonn], Nijm I and Nijm II [@nijm]) alone or combined with the TM99 3NF [@TM; @tm99] are shown. The sensitivity study performed in [@wit2010] revealed that at energies below $\approx 30$ MeV the $^1S_0$ and $^3S_1$-$^3D_1$ NN force components provide the most dominant contribution to the QFS cross sections with much less contributions of higher partial waves. Specifically, in the np QFS geometries the $^3S_1$-$^3D_1$ is the dominant force component while for nn QFS it is the $^1S_0$ force which contributes decisively. Again we exemplify it for nn and np QFS geometries of ref. [@siepe2] in Fig. \[fig\_sensiv\_pw\_qfsnnnp\]. Such a dominance for the QFS peak is understandable since the QFS cross sections are practically insensitive to the action of the presently available 3NF. Then at low energy the largest contribution should be provided by the S-wave components of the NN potential. In case of free np and nn scattering these are the $^1S_0$(np)+$^3S_1-^3D_1$ and $^1S_0$(nn) contributions, respectively. In the simple minded spirit that under QFS condition one of the three nucleons (at rest in the lab system) is just a spectator such a dominance of a two-nucleon encounter is to be expected. In reality, however, the projectile nucleon also interacts with that “ spectator” particle and the three nucleons at low energies undergo higher order rescatterings [@physrep96; @hwit89]. Thus the scattering to the final nn (np) QFS configuration also receives contributions from the np $ ^3S_1-^3D_1 $ (nn $ ^1S_0 $) interaction. Despite of all that the numerical results clearly reveal that for the np QFS configuration the $ ^3S_1-^3D_1 $ force is the most dominant contribution and for the nn QFS it is the $ ^1S_0 $ force (for free nn scattering there is no $ ^3S_1-^3D_1 $ interaction possible). This implies that the nn QFS is a powerful tool to study the $ ^1S_0 $ nn force component. That extreme sensitivity of the nn QFS cross section to the $^1S_0 $ nn force component is demonstrated in Fig. \[fig\_1s0nn\_changes\_qfs\] for the QFS geometries of ref. [@siepe2]. To that aim we multiplied the $ ^1S_0 $ nn matrix element of the CD Bonn potential by a factor $ \lambda$. The result is, that the nn QFS cross section undergoes significant variations while the np QFS cross section is practically unchanged. The displayed $\lambda$-parameters include also the value $ \lambda=1.08 $ which is necessary to get agreement with the nn QFS data of ref.[@siepe2]. While both, $^1S_0$ and $^3S_1-^3D_1$, np forces are well determined by np scattering data (with the restrictions mentioned above) and by the deuteron properties, the $^1S_0$ nn force is determined up to now only indirectly due to lack of free nn data. The disagreement between data and theory in the nn QFS peak points to the possibility of a flaw in the nn $^1S_0$ force. It was shown in [@wit2010] that in order to remove the $\approx 18 \%$ discrepancy found in [@siepe2] for the nn QFS cross section required an increased strength of the $^1S_0$ nn interaction which when given in terms of a factor $\lambda$ amounts to $\lambda \approx 1.08$. In Fig. \[ann\_reff\_fac\] we show the effect of the $ \lambda$-modification for the nn scattering length $a_{nn}$ and for the effective range parameter $r_{eff}$, and in Fig. \[ebind1s0\_fac\] for the binding energy of two neutrons in the $^1S_0$ state. It is seen that taking $\lambda =1.08$ leads to a nearly bound state of two neutrons. Implications on the $^1S_0$ nn effective range parameter {#three} ========================================================= Since the multiplication of the $^1S_0$ potential matrix element by a factor $\lambda$ induces changes in the effective range as well as in the scattering length the question arises, which from both effects is more important for the nn QFS cross section variations ? To answer that question we performed 3N Faddeev calculations based on the next-to-leading (NLO) order $\chi$PT potential [@epel2000; @epel] including all np and nn forces up to the total angular momentum $j_{max}=3$ in the two-nucleon subsystem. The $^1S_0$ component of that interaction is composed of the one-pion exchange potential and contact interactions parametrized by two parameters $\tilde C_{^1S_0}$ and $C_{^1S_0}$ $$V(^1S_0) = \tilde C_{^1S_0} + C_{^1S_0} (p^2 + p'^2) ~.$$ Standard values are $\tilde C_{^1S_0} =-0.1557374 * 10000$ GeV$^{-2}$ and $C_{^1S_0} = 1.5075220 * 10000$ GeV$^{-4}$ for cut-off combinations $\{\Lambda, \tilde {\Lambda} \} = \{450$ MeV , $500$ MeV} [@epel]. Multiplying $\tilde C_{^1S_0}$ by a factor $C_2(^1S_0)$ and $C_{^1S_0}$ by a factor $C_1(^1S_0)$ one can induce changes of the nn $^1S_0$ interaction. Requiring either the value of the scattering length $a_{nn}$ or the value of the effective range parameter $r_{eff}$ to be constant correlates the $C_1(^1S_0)$ and $C_2(^1S_0)$ factors. Changing $C_1({^1S_0})$ and $C_2(^1S_0)$ in such a way that the scattering length is kept constant and equal $a_{nn}=-17.6$ fm leads to changes of the effective range $r_{eff}$ shown in Fig. \[reff\_from\_c1\_c2\_new\]. The resulting changes of the nn and np QFS cross sections for geometries of ref. [@siepe2] are shown in Fig. \[e26p0\_qfs\_NLO\_reff\] for five sets of $C_1(^1S_0)$ and $C_2(^1S_0)$ factors with different nn $^1S_0$ effective range parameters ranging from $r_{eff}=2.03$ fm to $r_{eff}=3.07$ fm; one of them corresponding to the value required by the data. Similarily, changing $C_1(^1S_0)$ and $C_2(^1S_0)$ while keeping the effective range constant to $r_{eff}=2.75$ fm, leads to changes of the nn $^1S_0$ scattering length $a_{nn}$ shown in Fig. \[ann\_from\_c1\_c2\_new\]. The resulting changes of the nn and np QFS cross sections are presented in Fig. \[e26p0\_qfs\_NLO\_ann\] for four values of the nn $^1S_0$ scattering length ranging from $a_{nn}=-10.9$ fm to $a_{nn}=-75.9$ fm. It is clearly seen that the nn QFS cross sections depend only slightly on a change of the scattering length. The variations of the QFS cross section maximum stays below $\approx \pm 4 \%$. On the other side much stronger variations of the nn QFS cross sections result from changes of the effective range (see Fig. \[e26p0\_qfs\_NLO\_reff\]). Thus we can conclude that the $\lambda$-enhancement mechanism for the $^1S_0$ nn force studied in [@wit2010] acts mainly through the change of the effective range parameter. Thus in order to remove the discrepancies found in [@siepe2] and [@ruan1] for the nn QFS cross section a change of the nn $^1S_0$ effective range parameter is required. Its value taken under the assumption of charge symmetry and charge independence of nuclear forces is $r_{eff}=2.75$ fm and it has to be changed to $r_{eff} \approx 2.41$ fm. That implies a large charge symmetry and charge independence breaking effect of about $\approx 12 \%$ for that parameter. We would like to add that the discussed changes of $r_{eff}$ did not affect the elastic nd cross section nor vector or tensor analysing powers to a measurable extent. Only more complicated spin observables in elastic nd scattering are affected but the present day experimental errors are much larger than those changes. Discussion and further experimental information {#four} ================================================ Is such a large isospin breaking effect at all possible in view of the present understanding of nuclear forces? First of all it seems unprobable that only the effective range would reveal large isospin breaking and the scattering length will be left unaffected. In $\chi$PT the leading isospin breaking contribution is provided by isospin breaking contact interaction without derivatives [@epel_privat]. It turns out that the effective range parameter is quite insensitive to that isospin breaking contact force and typical isospin breaking effects for $r_{eff}$ are small and under $\approx 1 \%$ [@epel_privat]. The reported discrepancies for nn QFS require however a much larger effect for $r_{eff}$ of the order $\approx 12 \%$. Only when the contact terms in next orders would be unnaturally large one could expect larger isospin breaking effects for $r_{eff}$. Assuming naturalness it seems rather unprobable. Since it seems unlikely that isospin breaking effects will show up, if at all, in the effective range parameter alone without affecting simultaneously the nn scattering length, the question of a possible existence of a bound state of two neutrons reappears. Present day NN interactions allow only one bound state of two nucleons, namely the deuteron, where the neutron and the proton are interacting in a state with angular momenta $l=0$ or $2$, total spin $s=1$, and total angular momentum $j=1$. When the neutron and proton are interacting with the $^1S_0$ force no bound state exists and only a virtual resonant state occurs as documented by the negative scattering length $a_{np}=-21.73$ fm. Also the data for the proton-proton system exclude a $^1S_0$ pp bound state; however in this case the nuclear force is overpowered by the strong pp Coulomb repulsion. Assuming charge-independence and charge-symmetry of strong interactions also the two neutrons should not bind in the $^1S_0$ state. It also seems that modern nuclear forces do not allow for the 3n and 4n systems to be bound [@pieper]. However, in view of the strong discrepancies between theory and data found in the nd breakup measurements for the nn QFS geometry, which cannot be explained by present day nuclear forces, it appears reasonable to check experimentally the possibility of two neutrons being bound. There are reactions which provide conditions advantegous for a hypothetical di-neutron bound state. Such conditions can be found e.g. when two neutrons are moving with equal momenta and with relative energy close to zero. That occurs in the so called final-state-interaction (FSI) geometry of the nd breakup. Incomplete nd breakup measurements have been performed in the past to study properties of the $^1S_0$ nn force [@tornow96]. Even a dedicated experiment was performed in order to look for a hypothetical $^1S_0$ nn bound state [@vwitsch2006] in which the spectrum of the proton going in forward direction has been measured with the aim of a precise determination of its high energy region. The negative result of [@vwitsch2006] showed that the nd reaction is not suitable for such a study. It seems that much more appropriate would be reactions in which from the begining two neutrons occupy a configuration advantegous for their binding. It is known [@blank; @friar] that $^3$He is predominantely a spatially symmetric S state with its two protons mainly in opposite spin states. This component amounts for $\approx 90 \%$ of the $^3$He wave function. Similarily, the two neutrons in $^3$H are restricted to be in a spin-singlet state. That makes the triton target a very suitable tool to look for a nn bound state in $\gamma$ induced breakup of $^3$H. The idea is to measure the spectra of the outgoing protons in such a reaction. The two-neutron bound state, if existant, should reveal itself as a peak above the highest available proton energy from the 3-body decay of $^3$H. We show in Figs. \[fig1\] to \[fig2\] the outgoing proton spectra from the $\gamma(^3H,p)nn$ reaction for a number of $\gamma$ energies and angles of the outgoing protons. The big advantage of that reaction is that $\gamma$ interacts predominantely with the proton. Also other reactions, such as e.g. $^3$H(n,d)nn and $^3$H(d,$^3$He)nn, provide conditions advantegous for two neutrons to bind. They are complimentary and independent from the $^3$H($\gamma$,p)nn reaction and the data from all three processes should provide an answer to the question whether two neutrons can form a bound state. The reaction $^3$H(d,$^3$He)nn cannot presently be treated in a theoretically rigorous manner, however with the rapid increase in computer power such a treatment based on Fadeev-Yakubovsky equations can be expected in the near future. Summary {#five} ======= The strong discrepancy in the nn QFS nd break up configuration found in [@siepe2; @ruan1] is reconsidered. It is documented again that at low energies (below $\approx 30$ MeV) the nn (np) QFS cross section depends dominantly only on the $^1S_0$ ($^3S_1-^3D_1$) NN force component and higher partial wave contributions are quite small. Furthermore the theoretical results are quite stable under exchange of the standard nuclear forces. Also the present day available 3N forces have negligible effect on the QFS configurations. Since no direct measurement of the nn force is available there is the possibility that the properties of the nn force are still unsettled. Thus simply multiplying the nn $ ^1S_0$ force matrix element by a factor $ \lambda=1.08$ one can perfectly well reconcile theory and data. In addition we performed a more detailed study using the NLO chiral potential, which is composed of the one-pion exchange and contact interactions depending on two parameters. That dependence allowed us to study separately variations in the scattering length $ a_{nn}$ leaving the effective range parameter $ r_{eff}$ constant and vice versa. Thereby it turned out that the nn QFS peak height is very sensitive to $ r_{eff}$ and hardly sensitive to $ a_{nn}$. The outcome for an agreement with the data is the requirement that $ r_{eff}$ decreases from the value $r_{eff} =2.75$ fm to a significantly smaller one, $r_{eff} = 2.41$ fm. That strongly breaks charge symmetry and charge independence and is not supported by present day chiral potential theory. So, what might be a solution to remove the discrepancy? If the data are taken for granted there remains the possibility that a di-neutron exists. We propose additional experimental investigations, like the $ ^3H( \gamma,p)nn$ process and evaluated the proton spectra at various emission angles emphasizing its high energy region. The direct inclusion of $\Delta$-degrees of freedom into $\chi$PT allows for a rich set of additional NN and 3N force diagrams which are presently under investigation [@epel_privat1]. This might reconcile theory and data also for the space-star discrepancy [@physrep96] in the nd breakup process. Right now the situation is unsettled. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the Polish 2008-2011 science funds as the research project No. N N202 077435. It was also partially supported by the Helmholtz Association through funds provided to the virtual institute “Spin and strong QCD”(VH-VI-231) and by the European Community-Research Infrastructure Integrating Activity “Study of Strongly Interacting Matter” (acronym HadronPhysics2, Grant Agreement n. 227431) under the Seventh Framework Programme of EU. 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Woloshyn, Phys. Rev. C [**29**]{}, 538 (1984). J.L. Friar, B.F. Gibson, G.L. Payne, A.M. Bernstein, and T.E. Chupp, Phys. Rev. C [**42**]{}, 2310 (1990). R. B. Wiringa, V. G. J. Stoks, R. Schiavilla, Phys. Rev. C[**51**]{}, 38 (1995). J. Golak, R. Skibiński, H. Wita[ł]{}a, W. Glöckle, A. Nogga, H. Kamada, Phys. Rep. [**[415]{}**]{}, 89 (2005). E. Epelbaum, private communication. ![(color online) The cross section $d^5\sigma/d\Omega_1d\Omega_2dS$ for the $E_n^{lab}=26$ MeV nd breakup reaction $d(n,nn)p$ (upper panel) and $d(n,np)n$ (lower panel) as a function of the S-curve length for two complete configurations of Ref. [@siepe2]. QFS nn refers to the angles of the two neutrons: $\theta_1=\theta_2=42^o$ and QFS np refers to the angle $\theta_1=39^o$ of the detected neutron and $\theta_2=42^o$ for the proton. In both cases $\phi_{12}=180^o$. The (practically overlapping) lines correspond to different underlying dynamics: CD Bonn [@cdbonn] - dashed (blue), Nijm I - dotted (black), Nijm II [@nijm] - dashed-dotted (green), CD Bonn+TM99 - solid (red), Nijm I +TM99 [@TM; @tm99] - dashed-double-dotted (orange), Nijm II + TM99 - double-dashed-dotted (maroon). All partial waves with 2N total angular momenta up to $j_{max}=5$ have been included. []{data-label="fig_qfs_nn_np_indep_NN_3NF"}](e26p0_qfs_S_diff_pot_fig2.eps) ![(color online) The cross section $d^5\sigma/d\Omega_1d\Omega_2dS$ for the $E_n^{lab}=26$ MeV nd breakup reaction $d(n,nn)p$ (upper panel) and $d(n,np)n$ (lower panel) as a function of the S-curve length for two complete configurations of Ref. [@siepe2] specified in Fig. \[fig\_qfs\_nn\_np\_indep\_NN\_3NF\]. The different lines show contributions from different NN force components. The solid (red) line is the full result based on the CD Bonn potential [@cdbonn] and all partial waves with 2N total angular momenta up to $j_{max}=5$ included. The dotted (black), dashed-dotted (green), and dashed (blue) lines result when only contributions from $^1S_0$, $^3S_1-^3D_1$, and $^1S_0+^3S_1-^3D_1$ are kept calculating the cross sections. The dashed-double-dotted (brown) line presents the contribution of all partial waves with the exception of $^1S_0$ and $^3S_1-^3D_1$. []{data-label="fig_sensiv_pw_qfsnnnp"}](e26p0_qfs_S_alfa_fig4.eps) ![(color online) The cross section $d^5\sigma/d\Omega_1d\Omega_2dS$ for the $E_n^{lab}=26$ MeV nd breakup reaction $d(n,nn)p$ (upper panel) and $d(n,np)n$ (lower panel) as a function of the S-curve length for two complete configurations of Ref. [@siepe2] specified in Fig. \[fig\_qfs\_nn\_np\_indep\_NN\_3NF\]. The lines show sensitivity of the QFS cross sections to the changes of the nn $^1S_0$ force component. Those changes were induced by multiplying the $^1S_0$ nn matrix element of the CD Bonn potential by a factor $\lambda$. The solid (red) line is the full result based on the original CD Bonn potential [@cdbonn] ($a_{nn}=-18.8$ fm, $r_{eff}=2.79$ fm) and all partial waves with 2N total angular momenta up to $j_{max}=5$ included. The dashed (blue), dotted (black), and dashed-dotted (green) lines correspond to $\lambda=0.9$ ($a_{nn}=-8.3$ fm, $r_{eff}=3.12$ fm), $0.95$ ($a_{nn}=-11.7$ fm, $r_{eff}=2.96$ fm), and $1.05$ ($a_{nn}=-42.0$ fm, $r_{eff}=2.66$ fm), respectively. The double-dashed-dotted (violet) line shows cross sections obtained with $\lambda=1.08$ ($a_{nn}=-134.7$ fm, $r_{eff}=2.61$ fm), which factor is required to get agreement with nn QFS data of ref. [@siepe2]. []{data-label="fig_1s0nn_changes_qfs"}](e26p0_qfs_S_fac_fig6.eps) ![(color online) The changes of the nn scattering length $a_{nn}$ and the effective range parameter $r_{eff}$ with factor $\lambda$ by which the $^1S_0$ nn matrix element of the CD Bonn potential is multiplied: $V_{nn}(^1S_0)=\lambda * V_{CD~Bonn}(^1S_0)$. []{data-label="ann_reff_fac"}](ann_reff_fac.eps) ![(color online) The range of $\lambda$ values by which the $^1S_0$ nn matrix element of the CD Bonn potential is multiplied ($V_{nn}(^1S_0)=\lambda * V_{CD~Bonn}(^1S_0)$), for which the two neutrons form a bound state with the binding energy $E_b$. []{data-label="ebind1s0_fac"}](ebind1s0_fac.eps) ![(color online) Changes of the effective range parameter $r_{eff}$ in the $^1S_0$ partial wave caused by a correlated change of the factors $C_1(^1S_0)$ and $C_2(^1S_0)$ as shown in bottom part of this figure. This correlation between the factors $C_1(^1S_0)$ and $C_2(^1S_0)$ corresponds to a constant value of the scattering length $a_{nn}=-17.6$ fm. []{data-label="reff_from_c1_c2_new"}](reff_from_c1_c2_new.eps) ![(color online) Changes of QFS cross sections for configurations specified in Fig. \[fig\_qfs\_nn\_np\_indep\_NN\_3NF\] caused by correlated change of factors $C_1(^1S_0)$ and $C_2(^1S_0)$ shown in Fig. \[reff\_from\_c1\_c2\_new\]. All lines show results of Faddeev calculations based on LO $\chi$PT potential and all partial waves with 2N total angular momenta up to $j_{max}=3$ included. They differ in the nn $^1S_0$ force which was obtained keeping constant scattering length $a_{nn}=-17.6$ fm and changing constants $C_1(^1S_0)$ and $C_2(^1S_0)$ to get different effective ranges which are: solid (red line) - $C_1(^1S_0)=1.0$, $C_2(^1S_0)=1.0$, $r_{eff}=2.75$ fm, dashed (blue line) - $C_1(^1S_0)=1.5$, $C_2(^1S_0)=1.0615$, $r_{eff}=3.07$ fm, dotted (black line) - $C_1(^1S_0)=0.8$, $C_2(^1S_0)=0.9275$, $r_{eff}=2.54$ fm, dashed-dotted (green line) - $C_1(^1S_0)=0.5$, $C_2(^1S_0)=0.7675$, $r_{eff}=2.03$ fm. The double-dashed-dotted (violet) line shows cross sections obtained with $C_1(^1S_0)=0.7064$, $C_2(^1S_0)=0.8842$, $r_{eff}=2.41$ fm, which are required to get agreement with nn QFS data of ref.[@siepe2]. []{data-label="e26p0_qfs_NLO_reff"}](e26p0_qfs_NLO_reff.eps) ![(color online) Changes of the nn scattering length $a_{nn}$ in the $^1S_0$ partial wave caused by a correlated change of the factors $C_1(^1S_0)$ and $C_2(^1S_0)$ as shown in the bottom part of this figure. This correlation between the factors $C_1(^1S_0)$ and $C_2(^1S_0)$ corresponds to a constant value of the effective range parameter $r_{eff}=2.75$ fm. []{data-label="ann_from_c1_c2_new"}](ann_from_c1_c2_new.eps) ![(color online) Changes of QFS cross sections for configurations specified in Fig. \[fig\_qfs\_nn\_np\_indep\_NN\_3NF\] caused by a correlated change of the factors $C_1(^1S_0)$ and $C_2(^1S_0)$ shown in Fig. \[ann\_from\_c1\_c2\_new\]. All lines show results of Faddeev calculations based on the NLO $\chi$PT potential and all partial waves with 2N total angular momenta up to $j_{max}=3$ included. They differ in the nn $^1S_0$ force which was obtained keeping the effective range parameter $r_{eff}=2.75$ fm constant and changing the constants $C_1(^1S_0)$ and $C_2(^1S_0)$ to get different scattering lengths which are: solid (red line) - $C_1(^1S_0)=1.0$, $C_2(^1S_0)=1.0$, $a_{nn}=-17.6$ fm, dashed (blue line) - $C_1(^1S_0)=0.8$, $C_2(^1S_0)=0.8953$, $a_{nn}=-10.9$ fm, dotted (black line) - $C_1(^1S_0)=1.3$, $C_2(^1S_0)=1.1139$, $a_{nn}=-45.3$ fm, dashed-dotted (green line) - $C_1(^1S_0)=1.4$, $C_2(^1S_0)=1.1410$, $a_{nn}=-76.0$ fm. []{data-label="e26p0_qfs_NLO_ann"}](e26p0_qfs_NLO_ann.eps) ![(color online) The spectra of the outgoing proton from the reaction $^3H(\gamma,p)nn$ with $E_{\gamma}=10$ MeV at different lab. angles of the proton. They have been calculated using the AV18 [@av18] NN interaction and the current composed of single nucleon and meson exchange currents [@physrep2005]. []{data-label="fig1"}](3Hincl_w10_th_0_to_75.eps) ![(color online) The spectra of the outgoing proton from the reaction $^3H(\gamma,p)nn$ with $E_{\gamma}=10$ MeV at different lab. angles of the proton. They have been calculated using the AV18 [@av18] NN interaction and the current composed of single nucleon and meson exchange currents [@physrep2005]. []{data-label="fig2"}](3Hincl_w10_th_90_to_180.eps)

--- author: - 'Xin-zhou Li,' - 'Xiang-hua Zhai,' - Ping Li title: Generalized Birkhoff theorem and its applications in mimetic gravity --- Introduction ============ The mimetic gravity is one of the particularly interesting theories of modified gravity which has emerged in the past few years. It is possible to describe the dark matter and dark energy of the universe as a purely geometrical effect, without the need of introducing additional dark components. The nature of dark matter is a real puzzle, which persistently evades any kind of detection outside the realm of gravitational interactions at galactic and cosmological scales. Chamseddine and Mukhanov [@Chamseddine1] introduced for the first time the concept of mimetic field. In Ref. [@Chamseddine1], they showed a conformal extension of the general relativity (GR), in which the physical metric is defined in terms of an auxiliary metric and the first derivatives of a scalar field $\phi$ (mimetic field). An equivalent formulation of the mimetic dark matter theory was given in Ref. [@Golovnev1], where the action employed a Lagrange multiplier as a constraint. Furthermore, this theory has been extended to a generalized version with the addition of an arbitrary potential [@Chamseddine2] and as a consequence, one can induce nearly any gravitational properties of the known substance including quintessence and phantom. The ghost-free models and cosmological perturbation of mimetic gravity were discussed in Refs. [@Barvinsky] and [@Chamseddine3] , respectively. An interesting model has been proposed which does not only lead to mimetic dark matter but also provides a new approach to resolve singularities in GR [@Chamseddine4]. There exist several routes to mimetic gravity including the disformal transformation, Lagrange multiplier and singular Brans-Dicke theory [@Bekenstein; @Deruelle]. With the use of these methods, various models have been proposed, for example, mimetic Horava gravity [@Cognola], mimetic Horndeski gravity [@Achour] and mimetic $F(R)$ [@Odintsov]. Golovnev [@Golovnev2] has extended de Rham-Gabadadze-Tolley (dRGT) theory by a disformal transformation of the metric. Recently, Chamseddine and Mukhonov showed a ghost free mimetic massive gravity where the mass of graviton can be generated by using a Brout-Englert-Higgs mechanism with four scalar fields [@Chamseddine5]. Furthermore, they complete an explicit analysis using the methods of cosmological perturbation theory and consider quantum fluctuations of the massive graviton and mimetic field [@Chamseddine6]. In GR, the static spherically symmetric (SSS) solution to the Einstein equation is a benchmark, and its massive deformation also plays a crucial role in massive gravity [@Li1; @Lip]. Especially, we have found that there are seven black hole solutions in dRGT theory [@Li2]. In Refs. [@Myrzakulov1; @Myrzakulov2], the authors demonstrated how to reconstruct the potential for some interesting cases, including a correction to the Schwarzschild metric, a traversable wormhole, and so on. However, it is impossible to obtain an explicit expression for the potential $V(\phi)$ and exact metric solution using a so-called reconstructed method. In this paper, we prove that there is a zero point for one SSS metric function and a pole of first order for another. There is always a modified black hole solution for any given $V(\phi)$ within mimetic gravity, which can pass the test of solar system. Using this zero point theorem, we provide a dynamical mechanism for the maximum size of galaxies. Furthermore, we find a universal functional expression of SSS solution for any potential $V(\phi)$ in mimetic gravity, and we reduce primal high nonlinearity of the Einstein equation to a frequent Riccati’s form of first differential equation. Using this formula, we find some solutions that can be considered as candidates of black hole, or can explain the flat rotation curves of spiral galaxies within the mimetic gravity. The zero point theorem occupied a key position in this paper, which can be regarded as a generalization of Birkhoff theorem in GR. Especially, this theorem predicts that the product of two metric functions is not 1 as in GR but it is a regular function. Functional expression of SSS solution ===================================== The equations of motion under SSS ansatz ---------------------------------------- In a cosmological context, the mimetic field plays the role of a “clock”. Thus one can fancy making the mimetic field dynamical by adding a potential $V(\phi)$. The action of mimetic gravity is $$\label{action} S=\int d^4 x\sqrt{-g}[R+\lambda(1-g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi)+V(\phi)+\mathcal{L}_m],$$ where $R$ is the Ricci scalar, $\lambda$ is a Lagrange multiplier, $\mathcal{L}_m$ is the Lagrangian of usual matter and we set $16\pi G=1$. Variation of action (\[action\]) with respect to the metric gives the following equations $$\label{einsteintensor} G_{\mu\nu}=R_{\mu\nu}-\frac 1 2g_{\mu\nu}=\tilde{T}_{\mu\nu}+T_{\mu\nu},$$ where $T_{\mu\nu}$ is the energy-momentum tensor of the usual matter and $$\tilde{T}_{\mu\nu}=\lambda\partial_\mu \phi\partial_\nu\phi+\frac {\lambda}2(1-g^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi)g_{\mu\nu}+\frac V 2 g_{\mu\nu},$$ which describes the extra contribution to Einstein equations due to the $\phi$-dependent terms. Variation with respect to $\phi$ gives the motion equation of mimetic scalar field as $$\label{scalarequation} \partial_\nu(\sqrt{-g}\lambda g^{\mu\nu}\partial_\mu \phi)+\frac 1 2 \sqrt{-g}V_\phi=0.$$ Obviously, the mimetic scalar field $\phi$ satisfies the constraint $$\label{constraint} g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi=1.$$ Taking the trace of (\[einsteintensor\]), we obtain $$\lambda=G-2V-T,$$ and (\[einsteintensor\]) can be rewritten as $$\label{einsteinequation} G_{\mu\nu}-(G-2V-T)\partial_\mu\phi\partial_\nu\phi-\frac V 2 g_{\mu\nu}=T_{\mu\nu}.$$ By using the constraint (\[constraint\]), $\nabla^\mu G_{\mu\nu}=0$ and $\nabla^\mu T_{\mu\nu}=0$, we can rederive the mimetic scalar equation (\[scalarequation\]). Next, we consider the SSS ansatz as follows $$\label{ansatz} ds^2=-\beta(r)dt^2+\alpha(r)dr^2+r^2d\Omega^2,$$ and $\phi=\phi(r)$. In the SSS case, the constraint (\[constraint\]) can be reduced to $$\label{reducedconstraint} \phi'^2=\alpha.$$ Thus, we obtain the equations of motion under aforesaid ansatz, $$\label{system1} \frac 1 {r^2}\left(1-\left(\frac r \alpha\right)'\right)+\frac 1 2 V=0,$$ $$\label{system2} \frac 1 {r^2}\left(1-\frac {(\beta r)'} {\alpha\beta}\right)-\lambda+\frac 1 2 V=0,$$ $$\label{system3} (\sqrt{\beta}r^2\lambda)'+\frac 1 2\sqrt{\alpha\beta}r^2V_\phi=0,$$ where the primes denote derivatives with respect to $r$. We would show that presetting the potential function $V(r)$ is equivalent to setup of $V(\phi)$. In reality, we can find $\alpha$ from (\[system1\]) and then obtain $\phi(r)$ by the integral of $\sqrt{\alpha(r)}$ for any potential function $V(r)$ and $T^0\hspace{0.1cm}_0(r)$. Combining $V(r)$ with $\phi(r)$, we obtain the potential $V(\phi)$ easily. Furthermore, we have $V_\phi=\sqrt{\alpha}V'$ from (\[reducedconstraint\]). Thus, (\[system1\])-(\[system3\]) are the system containing three differential equations if and only if we take a fixed function $V(r)$. From this point of view, the system contains three equations for three unknowns, $\alpha, \beta$ and $\lambda$. It is solvable even though it is a highly nonlinear coupling system. Reduction of nonlinearity theorem --------------------------------- Meanwhile, we can regard (\[system1\])-(\[system3\]) as functional differential equations on the potential $V(\phi)$. We adopt a strategy as follows: The first step is to find the relation between $\alpha(r), \phi(r)$ and $V(\phi)$. Let $\alpha^{-1}=1-\frac{r_s}r+\Delta(r)$ and using (\[reducedconstraint\]), we have $$\label{Delta} \Delta(r)=\frac 1{2r}\int^r V\rho^2d\rho,$$ and $$\label{potential0} V(r)=\frac{2\Delta'(r)}r+\frac{2\Delta(r)}{r^2}.$$ From (\[reducedconstraint\]), we obtain $$\phi(r)=\int^r \frac{d\rho}{\sqrt{1-\frac{r_s}\rho+\Delta(\rho)}}\equiv\mathbb{I}(r),$$ where $\mathbb{I}$ is an integral operator and $r=\mathbb{I}^{-1}(\phi)$. Thus, $$V(\phi)=\frac{2\Delta'(\mathbb{I}^{-1}(\phi))}{\mathbb{I}^{-1}(\phi)}+\frac{2\Delta(\mathbb{I}^{-1}(\phi))}{\left[\mathbb{I}^{-1}(\phi)\right]^2}.$$ Second, using (\[system1\]) and (\[system2\]) we have $$\label{lambda} \lambda(r)=\mp\frac 1 {\sqrt{\beta}r^2}\int^r\frac{V'}{\sqrt{\alpha}}d\rho.$$ Third, we derive $\beta(r)$ equation from (\[system3\]) and (\[lambda\]) as follows $$\label{betaequation} \left(\sqrt{\beta}\right)''+\left(\frac 1 r-\frac 1 2\frac{\alpha'}\alpha\right)\left(\sqrt{\beta}\right)'+\left[\frac 1 {r^2}(\alpha-1)+\frac{\alpha'}{2r\alpha}\right]\sqrt{\beta}=0.$$ Though (\[reducedconstraint\])-(\[system3\]) compose a system with high nonlinearity at first glance, we can reduce it by certain arithmetic. In the case of constant potential, (\[betaequation\]) will becomes a linear equation of first order $$\left(\sqrt{\beta}\right)'+\frac 1 2\left(\mathrm{ln} \alpha\right)'\sqrt{\beta}=\frac{\mu\alpha}r,$$ and its exact solution is $$\label{solutionB} \beta=\frac 1 \alpha\left(\mu\int^r\frac{\alpha^{\frac 3 2}d\rho}\rho+\nu\right)^2,$$ where $\mu$ and $\nu$ are integral constants. In the case of nonconstant potential, let $v=\beta'/2\beta$, we can transfer (\[betaequation\]) into a Riccati equation $$\label{Riccati} v'+v^2+(\frac 1 r-\frac 1 2 \frac{\alpha'}{\alpha})v=\frac 1 {r^2}(1-\alpha)-\frac 1 {2r}\frac{\alpha'}{\alpha},$$ and the metric function $$\ \beta=\frac{|\alpha|}{\alpha}\left(\mu\mathrm{exp}\left[\int^r v(\rho)d\rho\right]\right)^2, \label{beta}$$ where $\mu$ is an integral constant. There is not a general method for solving Riccati equation. Nevertheless, when a special solution $v_0(r)$ is known by guess or observation, then one can write the general solution in the form $$v=v_0+\frac 1 u,$$ where $$u(r)=\exp\left[\int^r p(\rho)\rho\right]\left(\int^r \exp\left[-\int^{\xi}p(\rho)d\rho\right]d\xi\right),$$ and $$p(r)=\frac 1 r+2v_0-\frac 1 2 \frac{\alpha'}{\alpha}.$$ In summary, (\[Delta\]), (\[lambda\]), (\[solutionB\]) and (\[beta\]) make up the functional expressions of SSS solution, and the primal highly nonlinear problem has been reduced to Riccati’s form. Thus, we have the following theorem **Theorem 1 (Reduction of nonlinearity theorem).** *In the case of constant potential, the high nonlinearity of equations of motion will fade away so that it is exact solvable; In the case of nonconstant potential, this nonlinearity can be reduced to Riccati’s form so that the problem of exact solution becomes the resolution of a certain Riccati equation.* A pedagogical example --------------------- Using theorem 1, we can find a series of SSS solutions for various potential. We take the mimetic bouncing potential as a pedagogical instance of using the formulae above. The bouncing potential is $$\label{potential} V(\phi)=\frac{2s^2}{\left[(\phi-\phi_0)^2+s^2\right]}.$$ Taking $\Delta(r)=-\frac{s^2}{r^2}$, we have $V(r)=\frac{2s^2}{r^4}$ from (\[potential0\]). In the two asymptotic cases, we can obtain the explicit expression of $V(\phi)$. For the case of $r\ll \frac{s^2}{r_s}$, we have $\mathbb{I}(r)=\sqrt{r^2-s^2}+\phi_0$ and bouncing potential (\[potential\]) is sure to recur. For the case of $r\gg \frac{s^2}{r_s}$, *i.e.*, $\Delta(r)\approx0$, then $V(\phi)\approx0$. Therefore, we have $$V(\phi)=\frac{2s^2}{\left[\mathbb{I}^{-1}(\phi)\right]^4}=\begin{cases}\frac{2s^2}{\left[(\phi-\phi_0)^2+s^2\right]^2},\\\\0,\end{cases} \text{for} \begin{array}{c}r\ll \frac{s^2}{r_s},\\\\r\gg \frac{s^2}{r_s}.\end{array}$$ Since $\Delta(r)=-\frac{s^2}{r^2}$, $\alpha(r)$ contains a pole of first order $$r_h=\frac 1 2 (r_s+\sqrt{r_s^2+4s^2}).$$ Thus, we obtain the solution of (\[betaequation\]) as follows $$\label{metricfunction} \beta(r)=\begin{cases}-\mu^2(\frac 2 \pi \sin \frac r s-1)^2,\\\\\mu^2(\frac 2 \pi \mathrm{arccosh}\frac r s)^2,\\\\1-\frac{r_s}r,\end{cases} \text{for} \begin{array}{c}r<r_h,\\\\r_h<r\ll\frac{s^2}{r_s},\\\\r\gg\frac{s^2}{r_s},\end{array}$$ From (\[lambda\]), we have $$\lambda(r)=\frac{\mu}{\sqrt{\beta}r^2}\begin{cases}\int^r \frac{8s^2(\rho^2-s^2)^{\frac 1 2}}{\rho^6}d\rho,\\\\0,\end{cases} \text{for} \begin{array}{c}r\ll\frac{S^2}{r_s},\\\\r\gg\frac{S^2}{r_s}.\end{array}$$ Obviously, this solution will pass the test of solar system if we take suitable parameter $s$. The metric functions (\[metricfunction\]) give surely the description of a black hole. It is known to all that the scalar $I=R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}=12r_s^2/r^6$, so $r_s$ is only a coordinate singularity and $r=0$ is really a geometric singularity for the Schwarzschild metric. In the case of metric (\[metricfunction\]), we have $$I=\frac{12s^4}{r^6}+\begin{cases}\frac{12(s^2-r^2)}{r^6\left(\frac \pi 2 -\mathrm{arcsin} (\frac r s)\right)^2},\\\\\frac{12(r^2-s^2)}{r^6\left(\mathrm{arccosh} (\frac r s)\right)^2},\end{cases} \text{for} \begin{array}{c}r<r_h,\\\\r_h<r\ll\frac{s^2}{r_s},\end{array}$$ which is similar to Schwarzschild one. Because all worldlines within the future light cone of objects terminate on $r=0$, the crash with the singularity cannot be averted. Using the same procedure as in subsection 2.3 via choosing different potential $V(\phi)$, we can find new analytical solutions that are divided into two types according to the number of zero points for $\beta(r)$. Type I solutions have one zero point and the others are called Type II. The solution (\[metricfunction\]) is an outstanding example of Type I. Type I may be considered as candidates of black holes in mimetic gravity. Type II will provide a good fit to the rotation curves of spiral galaxies within the mimetic gravity. Some universal properties of SSS solution ========================================= The zero point theorem ---------------------- Next, we show a universal property of SSS solution within mimetic gravity which may be regard as generalization of Birkhoff theorem in GR. We will prove that $\alpha\beta$ is an analytic function in mimetic gravity whereas $\alpha\beta=1$ in GR. $\alpha(r)$ is bound to a pole of first order since $\alpha^{-1}=1-\frac{r_s}r+\Delta(r)$, so we have $$\alpha(r)=h(r)(r-r_h)^{-1},$$ where $h(r)$ is a positive-definite and analytic function. According to the general theory of differential equation, $r=r_h$ is a regular singularity of (\[betaequation\]) so that the local solution of (\[betaequation\]) can be written as $$\beta(r)=(r-r_h)^{2\rho}\left[\sum_{n=0}^{\infty}C_n(r-r_h)^n\right]^2,$$ where $C_0\neq 0$ and $\rho$ is determined by its index equation. Let $\alpha_{-1}$ be the residual of $\alpha(r)$, we have $$\lim_{r\rightarrow r_h}(r-r_h)\alpha = \alpha_{-1},\lim_{r\rightarrow r_h}\left(\frac 1 \alpha\right)' = \frac 1 {\alpha_{-1}}.$$ Thus, the index equation is $\rho(\rho-1)+\frac 1 2\rho=0$ and $\rho=\frac 1 2$, and $C_n$ satisfy the recursion formula as follows $$C_n=-\frac{\sum_{k=1}^n[(n+\frac 1 2-k)a_k+b_k]C_{n-k}}{n(n+\frac 1 2)},$$ where $a_k$ and $b_k$ are defined by $$\begin{aligned} \sum a_k(r-r_h)^k &=& \frac{(r-r_h)}r\left[1+\frac 1 2 \alpha r(\frac 1 \alpha)'\right],\nonumber\\ \sum b_k(r-r_h)^k &=& \frac{(r-r_h)^2}{r^2}\left[\alpha-1-\frac r 2 \alpha(\frac 1 \alpha)'\right].\end{aligned}$$ Clearly, $r=r_h$ is a zero point of first order of $\beta(r)$. We now have proven theorem 2 that is stated as follows: **Theorem 2 (Zero point theorem).** *In mimetic gravity, there is a metric function solution $\alpha(r)$ with a pole of first order at $r=r_h$ for a corresponding potential $V(\phi)$, but $\beta(r)$ will possess a zero point of first order at $r=r_h$. Therefore, $\alpha\beta$ will be regular.* Existence theorem of black holes -------------------------------- As applications of the zero point theorem, we will obtain some physical corollaries in the small (star) scale and intermediate (galaxy) scale. In the small scale, the sign of $\beta(r)$ will change from positive to negative as $r$ decreases and crosses $r_h$ since $\beta(r)$ has a zero point at $r=r_h$. Let us consider a light signal propagating in the radial direction. The velocity of this signal is $$\frac{dr}{dt}=\pm (r-r_h)\left[\frac{\sum_{n=0}^\infty C_n(r-r_h)^n}{h(r)}\right]^{\frac 1 2}.$$ In the exterior, the axis of the light cones is parallel to the $t$-axis. In the interior, the axis is parallel to the $r$-axis and $r$ becomes a timelike coordinate. When $r$ approaches $r_h$ from outside, the lightcones become very narrow. When $r$ just crosses $r_h$ to the interior, the lightcones suddenly become very broad again, and their timelike regions come to be horizontal, so that the only possible directions of radial motion are towards the singularity $r=0$. Thus, we have the following theorem: **Theorem 3.** *On every account, there are solutions with event horizon for any given potential function $V(\phi)$ in mimetic gravity, which can be considered as candidates of black holes.* The zero point theorem applied to galaxies ========================================== A dynamical mechanism for the maximum size of galaxies ------------------------------------------------------ If $\alpha$ has two poles of first order, $r_{in}$ and $r_{out}$, there are zero points of first order for $\beta(r)$ from the zero point theorem. That is to say, $\beta(r_{in})=\beta(r_{out})=0$ and $\beta(r)>0, r\in(r_{in}, r_{out})$ so that there exists a maximum $r_{max}$ between $r_{in}$ and $r_{out}$. And $r_{max}$ is the turning point of the Newtonian potential $\Phi(r)=\frac 1 2 (\beta-1)$. In other words, $\Phi(r)$ changes from attractive to repulsive at $r_{max}$ which provides a dynamical mechanism for the maximum size of galaxies. Thus, $r_{max}$ is just the maximum size of galaxies since the Newtonian potential $\Phi$ becomes repulsive at $r>r_{max}$. As a corollary of the zero point theorem, we have the following theorem: **Theorem 4.** *The Newtonian potential will change from attractive to repulsive between two poles if $\alpha(r)$ has two poles of first order. This mathematical result provides a dynamical mechanism for the maximum size of galaxies.* This theorem requires that $\beta(r)$ has two zero points at least, so such solutions should belong to Type II. On the other hand, it is common knowledge that the rotational velocity of spiral galaxies does not fall off as expected in GR with only the luminous matter as source. Mannheim *et. al.* [@Mannheim1; @Mannheim2] exhibited a model which provides an explanation for the inferred flat rotation curves of spiral galaxies within the conformal gravity framework [@Mannheim3; @Zhang]. The authors of Ref.[@Cognola] showed the similar pattern in mimetic gravity framework. In this paper, we shall find two solutions which could apply to explain the inferred flat rotation curves of spiral galaxies. Type I solution for the rotation curves of spiral galaxies ---------------------------------------------------------- We consider the potential $V(\phi)$ with an intermediate scale parameter $\gamma_0$ and its asymptotic form is $$V(\phi)=\begin{cases}0,\\\\\frac{16}{(\phi-\phi_0)^2-4\gamma_0^2},\end{cases}\quad \text{for} \quad \begin{array}{c}r\ll \gamma_0,\\\\r\gg r_s,\end{array}$$ where $r_s$ is the event horizon of solution. Thus, we have $\phi=\gamma_0(1+r/\gamma_0)^{\frac 1 2}+\phi_0$ and $V(r)=4/\gamma_0 r$. Furthermore, the metric functions are $$\alpha=\begin{cases}(1-\frac{r_s}r)^{-1},\\\\(1+\frac r{\gamma_0})^{-1},\end{cases}\quad \text{for} \quad \begin{array}{c} r \ll\gamma_0,\\\\ r \gg r_s,\end{array}$$ $$\beta=\begin{cases}1-\frac{r_s}r,\\\\(1+\frac 3 2 \frac r{\gamma_0})^2,\end{cases}\quad \text{for} \quad \begin{array}{c} r \ll\gamma_0,\\\\r \gg r_s.\end{array}$$ On sufficiently large scales, the Newtonian potential $\Phi\simeq\frac{c^2 r}{2\gamma_0}$ where the speed of light $c$ has been resumed. It is easy to see that the rotation velocity profile increases slightly as $\sqrt{r}$. This solution is fascinatingly in accordance with astrophysical data for small and medium sized low surface brightness (LSB) galaxies. However, the high surface brightness (HSB) galaxies are quite different from LSB, where the Newtonian contribution might be sufficient to complete with the rising linear term. When the rotation velocity departs enough far from the center of HSB, its rising behavior is arrested. For this, We have to turn our attention to Type II solution. Type II solution for the rotation curves of spiral galaxies ----------------------------------------------------------- We consider the potential $V(\phi)$ with two intermediate scale parameters $\gamma_0$ and $\lambda_0$, and its asymptotic form is $$V(\phi)=\begin{cases}0,\\\\-\frac 6{\gamma_0}+\frac 8{\sqrt{\mu}\sin\sqrt{\frac 1{\lambda_0}}(\phi-\phi_0)+\frac 1 {\gamma_0}},\end{cases} \text{for}\begin{array}{c}r\ll \gamma_0,\lambda_0,\\\\r\gg r_s,\end{array}$$ where $\mu=4\gamma_0^2\lambda_0+\lambda_0^2$, and $r_s$ is the event horizon of the solution. Thus, we have $$\phi(r)=\sqrt{\lambda_0}\arcsin\left(\frac{2\gamma_0 r-\lambda_0}{\sqrt{\mu}}\right)$$ and $V(r)=-\frac 6 {\gamma_0}+\frac 4{\lambda_0 r}$. Furthermore, the metric functions are $$\label{alpha} \alpha=\begin{cases}(1-\frac{r_s}r)^{-1},\\\\(1+\frac r{\gamma_0}-\frac{r^2}{\lambda_0})^{-1},\end{cases}\quad \text{for} \quad \begin{array}{c} r \ll\gamma_0,\lambda_0\\\\ r \gg r_s,\end{array}$$ $$\beta=\begin{cases}1-\frac{r_s}r,\\\\\sum_{k=0}^\infty(\sum_{i+j=k}a_ia_j)r^k,\end{cases}\quad \text{for} \quad \begin{array}{c} r \ll\gamma_0,\lambda_0\\\\r \gg r_s.\end{array}$$ where $a_0=1, a_1=\frac 3 {2\gamma_0}, a_2=-\frac 1 {2\lambda_0}$ and the recursion formula is $$\label{recursion} a_{k+1}=\frac{(k-2)(k+1)\lambda_0a_{k-1}-(k-1)(k+\frac 3 2)\gamma_0a_k}{\gamma_0\lambda_0(k+1)^2}.$$ From (\[alpha\]-\[recursion\]) and $r\gg r_s$, we have $$\begin{aligned} r_{out}&=&\frac 1 2\left(\frac{\lambda_0}{\gamma_0}+\sqrt{\left(\frac{\lambda_0}{\gamma_0}\right)^2+4\lambda_0}\right),\nonumber\\ r_{max}&=&\frac 3 2\left(\frac {\gamma_0}{\lambda_0}-\frac 9 {\gamma_0}\right)^{-1}.\end{aligned}$$ At the first blush, we should fit the values of two parameters $\gamma_0$ and $\lambda_0$ using the data from rotation curves. In reality, it is possible that one can adopt the results of Refs.[@Mannheim2; @OBrien] where the same pattern were fitted to rotation curves as the higher order terms are neglected by using the recursion relation (\[recursion\]). The total sample is composed of 138 galaxies where 25 galaxies are dwarf galaxies and 21 galaxies have data points that are enough far from the optical disk region. We obtain the following results with the aid of Ref.[@Mannheim2], $$\gamma_0=9.80\times 10^{19}\textmd{cm}, \quad \lambda_0=1.05\times 10^{53}\textmd{cm}^2,$$ therefore, it is reasonable that the higher order terms are neglected. Furthermore, we have $$r_{out}=3.23\times 10^{26}\textmd{cm}, \quad r_{max}=4.81\times 10^{23} \textmd{cm}.$$ Thus, this Type II solution shows that our dynamical mechanism(Theorem 4) is indeed effective for the inferred flat rotation curves of spiral galaxies within the mimetic gravity framework, without the need for particle dark matter. Summary ======= It is a universal property that there is a zero point of first order for the SSS metric function $\beta(r)$ if another metric function $\alpha(r)$ possesses a pole of first order within mimetic gravity. The zero point theorem occupied a key position in this work, which can be regarded as a generalization of Birkhoff theorem in GR. As its corollary, we show that there is a modified black hole solution for any given $V(\phi)$, which can pass the test of solar system. Furthermore, the zero point theorem provides a dynamical mechanism for the maximum size of galaxies. There are two analytic solutions which give good fits to the rotation curves of spiral galaxies without the demand for particle dark matter. Finally, we give a brief discussion for the cosmological horizon. The visible universe is a spherical region centered on us, from within which signal of gravitational wave has had time to reach us since the universe began. 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--- abstract: 'In this paper we study the influence of hard supersymmetry breaking terms in a $N=1$, $d=4$ supersymmetric model, in $S^1\times R^3$ spacetime topology. It is found that for some interaction terms and for certain values of the couplings, supersymmetry is unbroken for small lengths of the compact radius, and breaks dynamically as the radius increases. Also for another class of interaction terms, when the radius is large supersymmetry is unbroken and breaks dynamically as the radius decreases. It is pointed out that the two phenomena have similarities with the theory of metastable vacua at finite temperature and with the inverse symmetry breaking of continuous symmetries at finite temperature (where the role of the temperature is played by the compact dimension’s radius).' author: - | V.K.Oikonomou[^1]\ Dept. of Theoretical Physics Aristotle University of Thessaloniki,\ Thessaloniki 541 24 Greece\ and\ T.E.I. Serres title: 'Inverse and Dynamical Supersymmetry Breaking in $S^1\times R^3$' --- Introduction {#introduction .unnumbered} ============ Supersymmetry serves as the most promising extension of the Standard Model. In the short future supersymmetry will be verified experimentally through the LHC experiments. The elegance of the theory and the simplifications that introduces is of great importance however supersymmetry must be broken in our four dimensional world. There exist various mechanisms of supersymmetry breaking. In this article we are interested in dynamical breaking of supersymmetry induced from a toroidal compact dimension. Theories with one toroidal compact dimension resemble (mathematically) field theories at finite temperature. It is known that supersymmetry is spontaneously broken at finite temperature [@fuji], a fact that is closely related to the boundary conditions that fermions and bosons obey in the ”thermal” compact dimension. Specifically supersymmetry is spontaneously broken due to the periodicity of bosonic degrees of freedom and anti-periodicity of the fermionic degrees of freedom. Field theories at finite temperature are conceptually related to field theories with compact dimensions. Thus it is easily understood that the boundary conditions of the fields in the compact dimensions control the breaking of supersymmetry. In general,when studying supersymmetric theories in flat spacetime, the background metric is ordinary Minkowski. Spacetime topology affect the boundary conditions of the fields that are integrated in the path integral. Given a class of metrics, several spacetime topologies are allowed. Here we shall focus on a model that has ${S}^{1}{\times R}^{3}$ topology underlying the spacetime, $S^{1}$ refers to a spatial dimension. The specific topology is a homogeneous topology of the flat Clifford–Klein type [@ellis].  Non trivial topology, implies non trivial field configurations, that enter dynamically in the action. This non triviality enters in the action through the boundary conditions and as is well known the boundary conditions are controlled from the topology. The effective potential is a strong order parameter indicating when supersymmetry is broken. The appearance in the effective potential of vacuum terms which have different coefficients for fermions and bosons lead to the fact that the effective potential of the theory has no longer its minimum at zero and thus, supersymmetry is spontaneously broken. This quite general phenomenon can only be avoided in field theories with compact spatial dimensions, if in some way these vacuum terms are cancelled [@love]. Indeed the combination of the allowed boundary conditions, as we show later, can save spontaneous supersymmetry breaking at finite volume. Particularly this is due to the fact that we can have periodic fermions and anti-periodic bosons. This cannot be avoided at finite temperature due to the restricted boundary conditions as we already mentioned, except in cases where fermions have a complex chemical potential [@bartolome] as in pure $SU(N)$ Yang Mills theories with adjoint fermions at finite temperature [@kogan], namely, $$\label{adjointfermions} \psi (\beta )=-e^{\frac{i2\pi k}{N}} \psi (0).$$ Normally a question arises while following the above considerations. Why should someone care in avoiding spontaneous supersymmetry breaking at finite volume? This is because when supersymmetry spontaneously breaks (like at finite temperature) then supersymmetry ceases to be a controllable symmetry of the theory, since it is always broken and not dynamically, but by definition. One would like to have control on the way that supersymmetry breaks (especially in our case that the model spacetime we work is 4 dimensional and supersymmetry must be a symmetry and be broken dynamically. This can be avoided in theories with higher dimensions where supersymmetry can hold in the higher dimensional space and break in our world). In this paper we shall study a 4 dimensional N=1 supersymmetric model at one loop, in topology ${S}^{1}{\times R}^{3}$. We shall find the allowed field configurations that are determined from the topology and construct in a correct way the Lagrangian. The calculation of the effective potential follows, through which we shall find when supersymmetry breaks and when does not. After that we shall add in the Lagrangian non holomorphic and hard supersymmetry breaking terms. This terms break supersymmetry hardly. However for some values of the couplings and of the masses, interesting phenomena occur. Particularly it is found that in some cases when the volume of the compact space is small supersymmetry is not broken and when the compact radius exceeds a critical value, supersymmetry breaks dynamically. We shall give a cosmological implication of this case which resembles first order cosmological phase transitions in the early universe. Also other terms have a curious but worthy of mentioning effect. For these terms another strange phenomenon occurs for specific values of the couplings and masses. In detail for large values of the circumference of the compact dimension, supersymmetry is unbroken in contrast to the previous case and breaks after a critical length, as the compact dimension magnitude decreases. Although this is not so interesting from a phenomenological point of view (in four spacetime dimensions), it worths mentioning (and may have application in the physics of extra dimensions). There exist similar works in the literature but with the difference that the symmetry under study is not supersymmetry but a global symmetry (continuous or discrete). At finite temperature in some cases broken $O(N_f)\times O(N_{\psi})$ symmetries become restored at high temperatures. Also unbroken symmetries at small temperatures may break at high temperatures (a phenomenon known as inverse symmetry breaking). The last may have cosmological implications. In our case the same occurs but with supersymmetry in place of the symmetry and for a compact dimension playing the temperature’s role (also note that roughly the high temperature limit is closely related to the small length limit through the transformation $T\rightarrow \frac{1}{L}$. Actually through the last transformation we can relate the two limits where this is possible). One interesting feature of this resemblance is the similarity of the terms in the lagrangian that trigger these phenomena in both cases. Also these terms appear in the new inflationary models and help in the procedure of reheating the universe after inflation. We shall present and describe everything in detail in the forthcoming sections. Also let us mention that our calculations will be in 1-loop level and within the perturbative limits with $mL\leq 1$, where $m$ is the largest mass scale in the theory and $L$ is the circumference of the compact dimension. Also within the four dimensional setup we use, renormalizability of masses and couplings is ensured when $mL\leq 1$. In section 1 we review the mathematical setup needed for field theories with non trivial topology. The resemblance with extra dimensional theories is pointed out. In section 2 we describe the N=1 supersymmetric model we shall use and calculate the effective potential in the case of the compact dimension having infinite length and after that at finite volume. In section 3 we add several supersymmetry breaking terms and study in detail their effect on the vacuum energy of the model. In section 4 we review the continuous symmetry restoration, symmetry non-restoration and inverse symmetry breaking at finite temperature and point out the resemblance of these with our case (a resemblance that stems from the interactions of the scalar sector). In section 5 we present a cosmological application of one of our results and in section 6 a short discussion with the conclusions follow. Non Trivial Topology, Twisted Fields and Supersymmetry ====================================================== The existence of non trivial field configurations (in terms of boundary conditions) due to non trivial topology (twisted fields), was first pointed out by Isham [@isham] and then adopted by other authors [@gongcharov; @ford; @spalluci]. In the spacetime of our case, the topological properties of $S^{1}{\times R}^{3}$ are classified by the first Stieffel class $H^{1}(S^{1}{\times R}^{3},Z_{\widetilde{2}})$ which is isomorphic to the singular (simplicial) cohomology group ${H}_{1}({S} ^{1}{\times R}^{3}{,Z}_{2})$ because of the triviality of the ${Z}_{\widetilde{2}}$ sheaf. It is known that $H^{1}{(S}^{1}{\times R}^{3}{,Z}_{\widetilde{2}}{)=Z}_{2}$ classifies the twisting of a bundle. Specifically, it describes and classifies the orientability of a bundle globally. In our case, the classification group is ${Z}_{2}$ and, we have two locally equivalent bundles, which are however different globally [*[i.e.]{}*]{} cylinder-like and moebius strip-like. The mathematical lying behind, is to find the sections that correspond to these two bundles, classified completely by $Z_{2}$ [@isham]. The sections we shall consider are real scalar fields and Majorana spinor fields which carry a topological number called moebiosity (twist), which distinguishes between twisted and untwisted fields. The twisted fields obey anti-periodic boundary conditions, while untwisted fields periodic in the compactified dimension. Usually (inspired by field theory at finite temperature) one takes scalar fields to obey periodic and fermion fields anti-periodic boundary conditions, disregarding all other configurations that may arise from non trivial topology. We shall consider all these configurations. Let $\varphi _{u}$, $\varphi_{t}$ and $\psi _{t}$, $\psi _{u}$ denote the untwisted and twisted scalar and twisted and untwisted spinor fields respectively. The boundary conditions in the ${S}^{1}$ dimension are, $$\label{bc1} \varphi_{u}(x,0)=\varphi _{u}(x,L),$$ and $$\label{bc2} \varphi _{t}(x,0)=-\varphi _{t}(x,L),$$ for scalar fields and $$\label{bc1} \psi _{u}(x,0)=\psi _{u}(x,L),$$ and $$\label{bc2} \psi_{t}(x,0)=-\psi _{t}(x,L),$$ for fermion fields, where $x$ stands for the remaining two spatial and one time dimension which are not affected by the boundary conditions. Spinors (both Dirac and Majorana), still remain Grassmann quantities. We assign the untwisted fields twist ${h}_{0}$ (the trivial element of ${Z}_{2})$ and the twisted fields twist $h_{1}$ (the non trivial element of ${Z}_{2}$). Recall that $h_{0}+h_{0}=h_{0}$ ($0+0=0$), $% h_{1}+h_{1}=h_{0}$ ($1+1=0$), $h_{1}+h_{0}=h_{1}$ ($1+0=1$). We require the Lagrangian to be scalar under $Z_{2}$ thus to have ${h}_{0}$ moebiosity. The topological charges flowing at the interaction vertices must sum to ${h}_{0}$ under ${H}^{1}{(S}^{1}{\times R}^{3}{,Z% }_{\widetilde{2}}{)}$. For supersymmetric models, supersymmetry transformations impose some restrictions on the twist assignments of the superfield component fields [@gongcharov]. Now which fields can acquire vacuum expectation value? Grassmann fields cannot acquire vacuum expectation value (vev) since we require the vacuum value to be a scalar representation of the Lorentz group. Thus, the question is focused on the two scalars. The twisted scalar cannot acquire non zero vev [@ford], consequently, only untwisted scalars are allowed to develop vev’s. In the literature, twisted fields have frequently been used, for example in the Scherk-Schwarz mechanism [@scherk], where the harmonic expansion of the fields is of the form: $$\phi(x,y)=e^{imy}\sum_{n=-\infty }^{\infty }{\phi}_{n}(x)e^{\frac{i2{\pi}ny}{L}},$$ The ”$m$” parameter incorporates the twist mentioned above. This treatment is closely related to automorphic field theory [@Dowker] in more than 4 dimensions (which is an alternative to the one used by us). The Scherk-Schwarz mechanism is a well known mechanism that generates supersymmetry breaking to our 4 dimensional world after dimensional reduction and is frequently used for compactifications in extra dimensional models. Concerning the automorphic field theory, let us quote here a different approach to the above. Due to the compact dimension we can use generic boundary conditions for bosons and fermions in the compact dimension. These are,$$\begin{aligned} \varphi _{i}(x_{2},x_{3},\tau ,x_{1}) &=&e^{i\pi n_{1}\alpha }\varphi _{i}(x_{2},x_{3},\tau ,x_{1}+L) \\ ~\Psi (x_{2},x_{3},\tau ,x_{1}) &=&e^{i\pi n_{1}\delta }\Psi (x_{2},x_{3},\tau ,x_{1}+L), \notag\end{aligned}$$with, $0<\alpha ,\delta <1$, $i=1,2$, $n_{1}=1,2,3...$. The values $\alpha= 0,1$ correspond to periodic and antiperiodic bosons respectively while $\delta =0,1$ corresponds to periodic and anti-periodic fermions (for details see [@Dowker]). Description of the Supersymmetric Model ======================================= The model we shall present is described by the global ${N=1}$, ${d=4}$ supersymmetric Lagrangian, $$\mathcal{L}=[\Phi _{1}^{+}\Phi _{1}]_{D}+[\Phi ^{+}\Phi ]_{D}+[\frac{m_{1}}{2}\Phi ^{2}+\frac{g_{1}}{6}\Phi ^{3}+\frac{m}{2}\Phi _{1}^{2}+g\Phi \Phi _{1}^{2}]_{F}+{\rm{H.c}}, \label{lagra}$$where $\Phi _{1}$, $\Phi $ are chiral superfields and the superpotential from which the interaction part of the lagrangian arises is $[\frac{m_{1}}{2}\Phi ^{2}+\frac{g_{1}}{6}\Phi ^{3}+\frac{m}{2}\Phi _{1}^{2}+g\Phi \Phi _{1}^{2}]_{F}$. In the above, $$\begin{aligned} \Phi &=&\varphi _{u}(x)+\sqrt{2}\theta \psi _{u}(x)+\theta \theta F_{\varphi _{u}}+i\partial _{\mu }\varphi _{u}(x)\theta \sigma ^{\mu }\bar{\theta } \\ &&-\frac{i}{\sqrt{2}}\theta \theta \partial _{\mu }\psi _{u}(x)\sigma ^{\mu }% \bar{\theta }-\frac{1}{4}\partial _{\mu }\partial ^{\mu }\varphi _{u}^{+}(x)\theta \theta \bar{\theta }\bar{\theta }, \notag\end{aligned}$$is a left chiral superfield. It contains the untwisted scalar field components and the untwisted Weyl fermion. Although the untwisted scalar is complex, we shall use the real components which will be the representatives of the sections of the trivial bundle classified by ${H}^{1}{(S}^{1}{\times R}^{3}{% ,Z}_{\widetilde{2}}{)}$. Moreover,$$\begin{aligned} \Phi _{1} &=&\varphi _{t}(x)+\sqrt{2}\theta \psi _{t}(x)+\theta \theta F_{\varphi _{t}}+i\partial _{\mu }\varphi _{t}(x)\theta \sigma ^{\mu }\bar{\theta } \\ &&-\frac{i}{\sqrt{2}}\theta \theta \partial _{\mu }\psi _{t}(x)\sigma ^{\mu }% \bar{\theta }-\frac{1}{4}\partial _{\mu }\partial ^{\mu }\varphi _{t}^{+}(x)\theta \theta \bar{\theta }\bar{\theta }, \notag\end{aligned}$$is another left chiral superfield containing the twisted scalar field and the twisted Weyl fermion. Writing down (\[lagra\]) in component form, we get (writing Weyl fermions in the Majorana representation):$$\begin{aligned} \mathcal{L}& =\partial _{\mu }\varphi _{u}^{+}\partial ^{\mu }\varphi _{u}-\left \vert m_{1}\varphi _{u}+\frac{g_{1}}{2}\varphi _{u}\varphi _{u}+g\varphi _{t}^{2}\right \vert ^{2}+i\overline{\Psi }_{t}\gamma ^{\mu }\partial _{\mu }\Psi _{t}-\frac{1}{2}m\overline{\Psi }_{t}\Psi _{t} \label{test}\\ & -\frac{g_{1}}{4}(\overline{\Psi }_{u}\Psi _{u}-\overline{\Psi }_{u}\gamma _{5}\Psi _{u})\varphi _{u}-\frac{g_{1}}{4}(\overline{\Psi }_{u}\Psi _{u}+% \overline{\Psi }_{u}\gamma _{5}\Psi _{u})\varphi _{u}^{+}+\partial _{\mu }\varphi _{t}^{+}\partial ^{\mu }\varphi _{t}- \notag \\ & \left \vert m\varphi _{t}+2g\varphi _{t}\varphi _{u}\right \vert ^{2}+i% \overline{\Psi }_{u}\gamma ^{\mu }\partial _{\mu }\Psi _{u}-\frac{1}{2}m_{1}% \overline{\Psi }_{u}\Psi _{u}- \notag \\ & \frac{g}{4}(\overline{\Psi }_{t}\Psi _{t}-\overline{\Psi }_{t}\gamma _{5}\Psi _{t})\varphi _{u}-\frac{g}{4}(\overline{\Psi }_{t}\Psi _{t}+% \overline{\Psi }_{t}\gamma _{5}\Psi _{t})\varphi _{u}^{+}. \notag\end{aligned}$$Notice that moebiosity is conserved at all interaction vertices [*[i.e.]{}*]{} equals ${h}_{0}$. The moebiosity of $\varphi _{u}$ and $\Psi _{u}$ is ${h}_{0}$ while for $\varphi _{t}$ and $\Psi _{t}$ is ${h}_{1}$. Using the ${Z}_{2}$ cyclic group properties we see that the Lagrangian (\[test\]) has moebiosity ${h}_{0}$.The complex field $\varphi _{u}$ can be written in terms of real components as ${\varphi }_{u}{=\chi +i\varphi }_{u_{2}}{/}\sqrt{2}$, where $\chi ={v+(\varphi }_{u_{1}}{)/}\sqrt{2}$ (${v}$ is the classical value). Thus, $\varphi _{u_{1}}$ and $\varphi _{u_{2}}$ are real untwisted field configurations belonging to the trivial element of ${% H}^{1}{(S}^{1}{\times R}^{3}{,{Z}_{\widetilde{2}})}$ and satisfying periodic boundary conditions in the compactified dimension. The twisted scalar field can be written as ${\varphi }_{t}{=(\varphi }_{t_{1}}{+i\varphi }_{t_{2}}{)/}% \sqrt{2}$, since, this field, being a member of the non trivial element of ${H}^{1}{(S}^{1}{\times R}^{3}{,Z}_{\widetilde{2}}{)}$ cannot acquire a vev. Notice we gave a vev for an untwisted boson, This is useful in order to find the minimum of the effective potential minimizing it in terms of $v$. The masses of the two Majorana fermion fields and the four bosonic fields at tree order are calculated to be:$$\begin{aligned} m_{b_{1}}^{2} &=&m_{1}^{2}+3g_{1}m_{1}v+3g_{1}^{2}v^{2}/2 \label{mass}\\ \ m_{b_{2}}^{2} &=&m_{1}^{2}+g_{1}m_{1}v+g_{1}^{2}v^{2}/2 \notag \\ m_{t_{1}}^{2}\bigskip &=&m^{2}+4gmv+4g^{2}v^{2}+g^{2}m_{1}v/\sqrt{2}% +g^{2}g_{1}v^{2}/4\ \notag \\ \ m_{t_{2}}^{2} &=&m^{2}+4gmv-g^{2}m_{1}v/\sqrt{2}-g^{2}g_{1}v^{2}/4 \notag \\ m_{f_{1}} &=&m_{1}+g_{1}v,\ m_{f_{2}}=m+2gv. \notag\end{aligned}$$In (\[mass\]) $m_{b_{1}}$, $m_{b_{2}}$ are the masses of the untwisted bosons (${\varphi }_{u_{1}}$ and ${\varphi }_{u_{2}}$ respectively), $m_{t_{1}}$, $m_{t_{2}}$ are the masses of the twisted bosons (${\varphi }_{t_{1}}$ and ${\varphi }_{t_{2}}$) and, finally, $m_{f_{1}}$, $m_{f_{2}}$ are the untwisted Majorana fermion and twisted Majorana fermion masses respectively ($\Psi _{u}$ and $\Psi _{t}$). The general tree level result for theories with rigid supersymmetry in terms of chiral superfields is satisfied (see [@martin]) [*[i.e.]{}*]{} : $$STr(M^{2})=\sum \limits_{j}(-1)^{2j}(2j+1)m_{j}^{2}=0. \label{str}$$Also, the following relations hold true:$$m_{b_{1}}^{2}+m_{b_{2}}^{2}=2m_{f_{1}}^{2},\ m_{t_{1}}^{2}+m_{t_{2}}^{2}=2m_{f_{2}}^{2}.\label{sp}$$Since twisted scalars cannot acquire vacuum expectation value, supersymmetry is not spontaneously broken at tree level, like in the O’ Raifeartaigh models. Indeed the auxiliary field equations,$$\begin{aligned} F_{\varphi _{u}}^{+} &=&m_{1}\varphi _{u}+\frac{g_{1}}{2}\varphi _{u}^{2}+g\varphi _{t}^{2}=0 \\ F_{\varphi _{t}}^{+} &=&m\varphi _{t}+2g\varphi _{u}\varphi _{t}=0, \notag\end{aligned}$$imply that $\varphi _{u}=0$ and $\varphi _{t}=0$ and consequently $v=0$, thus, at tree level, no spontaneous supersymmetry breaking occurs. Supersymmetric Effective Potential in $S^{1}\times R^{3}$ --------------------------------------------------------- We now proceed by assuming that the topology is changed to ${S}^{1}{\times R}^{3}$, while the local geometry remains Minkowski. The metric reads: $$\mathrm{d}s^{2}=\mathrm{d}t^{2}-\mathrm{d}x_{1}^{2}-\mathrm{d}x_{2}^{2}-\mathrm{d}x_{3}^{2} \label{flat},$$ with $-\infty <x_{2},x_{3},t<\infty $ and $0<x_{1}<L$ with the points${\ x}% _{1}=0~$and ${x}_{1}=L$ periodically identified. The boundary conditions for the fields are:$$\begin{aligned} ~\varphi _{u}(x_{1},x_{2},x_{3},t) &=~~~~~\varphi _{u}(x_{1}+L,x_{2},x_{3},t) \\ ~\varphi _{t}(x_{1},x_{2},x_{3},t) &=~{}-\varphi _{t}(x_{1}+L,x_{2},x_{3},t) \notag \\ ~\Psi _{u}(x_{1},x_{2},x_{3},t) &=~~~~~\Psi _{u}(x_{1}+L,x_{2},x_{3},t) \notag \\ ~\Psi _{t}(x_{1},x_{2},x_{3},t) &=~~-\Psi _{t}(x_{1}+L,x_{2},x_{3},t). \notag\end{aligned}$$We Wick rotate the time direction ${t}\rightarrow {it}$ thus giving the background metric the Euclidean signature [@spalluci]. The twisted fermions and twisted bosons will be summed over odd Matsubara frequencies, while the untwisted fermions and untwisted scalars will be summed over even Matsubara frequencies [@dolan; @coleman; @weinberg]. Adopting the $\overline{DR}^{\prime }$ renormalization scheme [@martin] the Euclidean effective potential at one loop level reads:$$\begin{aligned} V& =V_{0}+\frac{1}{64\pi ^{2}L}\sum \limits_{n=-\infty }^{\infty }\int \frac{% \mathrm{d}^{3}k}{(2\pi )^{3}}{\bigg (}\ln [k^{2}+\frac{4\pi^{2} n^{2}}{L^{2}}% +m_{b_{1}}^{2}] \\ & -2\ln [k^{2}+\frac{4\pi^{2} n^{2}}{L^{2}}+m_{f_{1}}^{2}]+\ln [k^{2}+\frac{% 4\pi^{2} n^{2}}{L^{2}}+m_{b_{2}}^{2}] \notag \\ & -2\ln [k^{2}+\frac{\pi^{2} (2n+1)^2}{L^{2}}+m_{f_{2}}^{2}]+\ln [k^{2}+\frac{% \pi^{2} (2n+1)^2}{L^{2}}+m_{t_{1}}^{2}] \notag \\ & +\ln [k^{2}+\frac{\pi^{2} (2n+1)^2}{L^{2}}+m_{t_{2}}^{2}]{\bigg ).} \notag\end{aligned}$$${V}_{0}~$includes the tree and the one loop corrections for infinite length,$$\begin{aligned} V_{0}& =m_{1}^{2}v^{2}+g_{1}^{2}m_{1}v^{3}+\frac{g_{1}^{2}v^{4}}{4}+\frac{1}{% 64\pi ^{2}}{\bigg (}m_{b_{1}}^{4}(\ln [\frac{m_{b_{1}}^{2}}{\mu ^{2}}]-% \frac{3}{2}) \\ & +m_{b_{2}}^{4}(\ln [\frac{m_{b_{2}}^{2}}{\mu ^{2}}]-\frac{3}{2}% )+m_{t_{1}}^{4}(\ln [\frac{m_{t_{1}}^{2}}{\mu ^{2}}]-\frac{3}{2}% )+m_{t_{2}}^{4}(\ln [\frac{m_{t_{2}}^{2}}{\mu ^{2}}]-\frac{3}{2}) \notag \\ & -2m_{f_{1}}^{4}(\ln [\frac{m_{f_{1}}^{2}}{\mu ^{2}}]-\frac{3}{2}% )-2m_{f_{2}}^{4}(\ln [\frac{m_{f_{2}}^{2}}{\mu ^{2}}]-\frac{3}{2}){\bigg )}, \notag\end{aligned}$$and $\mu $ is the renormalization scale, being of the order of the largest mass [@tamvakis]. Furthermore, we shall assume that $m{L\simeq 1}$ which is required for the validity of perturbation theory [@guth; @s; @weinberg]. How Can Supersymmetry be Broken Spontaneously in $S^1\times R^3$ ---------------------------------------------------------------- It is well known that when one considers only twisted fermions and untwisted bosons in ${S}^{1}{\times R}^{3}$ (like in thermal field theories), vacuum contributions $\sim $${L}^{-4}$ do not cancel and supersymmetry is spontaneously broken. The non-cancellation occurs because bosons and fermions satisfy different boundary conditions. In our model the field content is such that cancellation of vacuum contributions is being enforced, after having included all topologically inequivalent allowed field configurations. This situation is similar to finite temperature calculations. The question if supersymmetry is broken or not requires to check the zero modes of the vacuum state [@witten]. It is easy to see why in conventional finite temperature field theories and their conceptional analogues ${S}^{1}{\times R}^{3}$ topological field theories supersymmetry is broken. The vacuum state, in the Wess–Zumino in ${S}^{1}{\times R}^{3}$, does contain one bosonic zero mode. In our case this does not occur because we have equal vacuum zero modes (twisted spinors, do not have a zero mode). Consequently, in our model, we expect that supersymmetry will not be spontaneously broken [@love] (for a detailed discussion we recommend the paper of Fujikawa [@fuji]). Small $L$ Expansion of the effective potential ---------------------------------------------- The leading order contribution to the one loop effective potential for small $L$ values is given by [@dolan; @elizalde; @kirsten]: $$\begin{aligned} &V=m_{1}^{2}v^{2}+g_{1}^{2}m_{1}v^{3}+\frac{g_{1}^{2}v^{4}}{4} \label{pot}\\ &{-\frac{3(2m_{f_{1}}^{4}-m_{b_{1}}^{4}-m_{b_{2}}^{4})}{4096\pi ^{4}}-\frac{% 3(2m_{f_{2}}^{4}-m_{t_{1}}^{4}-m_{t_{2}}^{4})}{256\pi ^{4}}}\notag\\ &{+\frac{% 3(2m_{f_{1}}^{4}-m_{b_{1}}^{4}-m_{b_{2}}^{4}+2m_{f_{2}}^{4}-m_{t_{1}}^{4}-m_{t_{2}}^{4})% }{128\pi ^{2}}}\notag \\ &{+\frac{(\gamma -\ln [4\pi ])(2m_{f_{1}}^{4}-m_{b_{1}}^{4}-m_{b_{2}}^{4})}{% 1024\pi ^{4}}+\frac{(\gamma +\ln [\frac{2}{\pi }% ])(2m_{f_{2}}^{4}-m_{t_{1}}^{4}-m_{t_{2}}^{4})}{64\pi ^{4}}}\notag \\ &{+\frac{(2m_{f_{1}}^{3}-m_{b_{1}}^{3}-m_{b_{2}}^{3})}{384L\pi ^{3}}-\frac{% (2m_{f_{1}}^{2}-m_{b_{1}}^{2}-m_{b_{2}}^{2})}{768\pi ^{2}L^{2}}+\frac{% (2m_{f_{2}}^{2}-m_{t_{1}}^{2}-m_{t_{2}}^{2})}{384\pi ^{2}L^{2}}}\notag\\ &{+\frac{2m_{f_{1}}^{4}\ln [Lm_{f_{1}}]-m_{b_{2}}^{4}\ln [Lm_{b_{2}}]-m_{b_{1}}^{4}\ln [Lm_{b_{1}}]}{1024\pi ^{4}}}\notag \\ &{+\frac{2m_{f_{2}}^{4}\ln [Lm_{f_{2}}]-m_{t_{2}}^{4}\ln [Lm_{t_{2}}]-m_{t_{1}}^{4}\ln [Lm_{t_{1}}]}{64\pi ^{4}}}\notag \\ &{-\frac{(2m_{f_{1}}^{4}\ln [\frac{m_{f_{1}}^{2}}{\mu ^{2}}]-m_{b_{2}}^{4}\ln [% \frac{m_{b_{2}}^{2}}{\mu ^{2}}]-m_{b_{1}}^{4}\ln [\frac{m_{b_{1}}^{2}}{\mu ^{2}}])}{64\pi ^{2}}}\notag\\ &{-\frac{(2m_{f_{2}}^{4}\ln [\frac{m_{f_{2}}^{2}}{\mu ^{2}}]-m_{t_{2}}^{4}\ln [% \frac{m_{t_{2}}^{2}}{\mu ^{2}}]-m_{t_{1}}^{4}\ln [\frac{m_{t_{1}}^{2}}{\mu ^{2}}])}{64\pi ^{2}}.\notag}\end{aligned}$$ ![The supersymmetric effective potential[]{data-label="susyzero"}](graph1.eps) Since relation (\[sp\]) holds, the terms proportional to $\frac{1}{L^{2}}$ cancel [@love]. Also, the minimum of the potential vanishes at ${v=}$0 and supersymmetry is preserved. Indeed, expanding (\[pot\]) for small ${v}$ we get: $$V\simeq m_{1}^{2}v^{2}+O(v^{3}).$$ In figure \[susyzero\] we plot the effective potential for the upper perturbative limit $mL=1$. The other numerical values are chosen to be: ${m}_{1}{=}$[200, ]{}${m=}$[7000, ]{}${g}_{1}{% =}$[0.001, ]{}${g=}$[0.09, ]{}${\mu =}$[7000]{}. Addition of Explicit Supersymmetry Breaking Terms ================================================= Let us now introduce into the Lagrangian (\[test\]) various hard supersymmetry breaking terms of the form $g_{3}\chi ^{2}\varphi _{i}^{2}$ and $g_2\overline{\psi}\psi \chi$, where ${g}_{3}$ and $g_2$ are dimensionless couplings (and recall $\chi =v+\varphi _{u_{1}}/\sqrt{2}$). This terms, being non holomorphic and hard, break supersymmetry explicitly (also these terms have moebiosity zero). Indeed the addition of such terms re-introduces quadratic divergences in the theory, namely, $$\Delta m_{scalar}=\frac{1}{8\pi^2}(l_s-l_f^2)\Lambda^2_{UV},$$ with $\Lambda^2_{UV}$ a relevant upper cut-off of the theory and $l_s$, $l_f$ boson and fermion couplings. Since $\chi $ develops a vev, the fields coupled to it will acquire an additional mass of the form $g_{3}v^{2}$ and $g_2v$. We can add various combinations of the allowed terms. There exist two class of phenomena occurring, depending on the supersymmetry breaking terms we use. The first type and, from a phenomenological point of view more interesting, resembles the first order phase transition picture in thermal field theory. Actually, regardless that there exist hard supersymmetry breaking terms, supersymmetry is unbroken for small values of the length of the compact dimension $L$. The minimum of the potential is zero at $v=0$, $V(0)=0$. As the length of the compact dimension increases, a second non supersymmetric minimum is created after a critical length is reached. Then phenomena may occur that can be described by the theory of metastable vacua (the reader may find useful the papers [@alles; @kirsten] where similar issues are discussed in $S^1\times R^3$ but for a non supersymmetric $\phi^4$ theory. Again non trivial topology induces similar behavior of the effective potential). The second type of phenomena describes a theory that supersymmetry is unbroken when the length of the compact dimension is large and as the radius decreases, supersymmetry breaks spontaneously after a critical length. Thus supersymmetry is broken only for small lengths. Although this is rather curious for a four dimensional model, we shall present it because it may have application to extra dimensional models. A term of the form $g_{3}\chi ^{2}\varphi _{u_{2}}^{2}$ ------------------------------------------------------- We introduce into the Lagrangian (\[test\]) an interaction term between the two untwisted scalar fields, of the form $-g_{3}\chi ^{2}\varphi _{u_{2}}^{2}$. Since $\chi =v+\varphi _{u_{1}}/\sqrt{2}$, the scalar field $\varphi _{u_{2}}$ will acquire an additional mass term of the form $g_{3}v^{2}$. This way, the masses of the fields now become: $$\begin{aligned} m_{b_{1}}^{2} &=&m_{1}^{2}+3g_{1}m_{1}v+3g_{1}^{2}v^{2}/2 \\ \ m_{b_{2}}^{2} &=&m_{1}^{2}+g_{1}m_{1}v+g_{1}^{2}v^{2}/2+g_{3}v^{2} \notag \\ m_{t_{1}}^{2}\bigskip &=&m^{2}+4gmv+4g^{2}v^{2}+g^{2}m_{1}v/\sqrt{2}% +g^{2}g_{1}v^{2}/4 \notag \\ \ m_{t_{2}}^{2} &=&m^{2}+4gmv-g^{2}m_{1}v/\sqrt{2}-g^{2}g_{1}v^{2}/4 \notag \\ m_{f_{1}} &=&m_{1}+g_{1}v,\ m_{f_{2}}=m+2gv. \notag\end{aligned}$$ As expected, supersymmetry is now broken and relation (\[str\]) becomes, $$2m_{f_{1}}^{2}-m_{b_{1}}^{2}-m_{b_{2}}^{2}=g_{3}v^{2},\ m_{t_{1}}^{2}+m_{t_{2}}^{2}=2m_{f_{2}}^{2}.$$One can see that the supersymmetric minimum at $v=0$ is still preserved. Indeed, $V$ can be written as:$$V\simeq (m_{1}^{2}+\frac{g_{3}}{768\pi ^{2}L^{2}})v^{2}+O(v^{3}).$$We can see that in the continuum limit (infinite $L$), the supersymmetric vacuum becomes metastable and a second non supersymmetric vacuum appears. Including finite size corrections, we see that for small ${L}$ the effective potential has a unique supersymmetric minimum at $v=0$. As ${L}$ increases, a second minimum develops, which becomes supersymmetric at the critical value ${L}_{c}{=}\frac{1}{21571}$. When ${L}\!>\!{L}_{c}$ the second minimum is non supersymmetric and becomes energetically more preferable than the supersymmetric one [@linde; @zeldovich]. This said behavior of the potential is valid whenever $g_{3}\gg g_{1}$ and for $% \frac{m_{1}}{m}\ll g_{3}$. Using the same numerical values as before, we plot the effective potential for ${g}_{3}{=}$[0.5]{}, first in the continuum limit (figure \[continuumnonsusy\]), and then including ${L}$ dependent corrections (figure \[nonsusy\]). ![The continuum effective potential[]{data-label="continuumnonsusy"}](graph3.eps) ![The effective potential including finite size corrections[]{data-label="nonsusy"}](graph2.eps) Let us discuss the above results. $g_{3}$, $g_{1}$ are couplings among the untwisted superfields, $g_{3}$ corresponding to the supersymmetry breaking term. If the $g_{3}$ interaction is stronger than $g_{1}$ and if the mass (${m}$) of the twisted superfield is larger than the untwisted one (${m_{1}}$), then the following phenomenon occurs. For small length ${L}$ of the compact dimension, supersymmetry is not broken (figure \[nonsusy\]). As ${L}$ grows larger, a second minimum appears which is not supersymmetric (${L}\!>\!{L}_{c}$). There exists a small barrier separating the two minima (figure \[nonsusy\]), and there exists the possibility of quantum barrier penetration between them. This resembles the first order phase transition picture of thermal field theories. Upon closer examination, we can see that in the continuum limit, the supersymmetric vacuum becomes metastable and a second non supersymmetric vacuum appears. Including finite size corrections, we see that for small ${L}$ the effective potential has a unique supersymmetric minimum at $v=$0. As ${L}$ increases, a second minimum develops, which becomes supersymmetric at the critical value ${L}_{c}{=}\frac{1}{21571}$. When ${L}\!>\!{L}_{c}$ the second minimum breaks supersymmetry and becomes energetically more preferable than the supersymmetric one [@linde; @zeldovich]. This said behavior of the potential is always valid whenever $g_{3}\gg g_{1}$ and for $% \frac{m_{1}}{m}\ll g_{3}$. Using the same numerical values as before, we plot the effective potential for ${g}_{3}{=}$[0.5]{}, first in the continuum limit (figure \[continuumnonsusy\]), and then including ${L}$ dependent corrections (figure \[nonsusy\]). A term of the form $g_{3}\chi ^{2}\varphi _{t_{1}}^{2}$ ------------------------------------------------------- Let us try something different now. We add an interaction among a twisted boson and the untwisted boson that acquires vev, namely $-g_{3}\chi ^{2}\varphi _{t_{1}}^{2}$. Since $\chi =v+\varphi _{u_{1}}/\sqrt{2}$ the twisted boson $\varphi _{t_{1}}$ will have additional contribution to it’s tree order mass. The masses now read, $$\begin{aligned} m_{b_{1}}^{2} &=&m_{1}^{2}+3g_{1}m_{1}v+3g_{1}^{2}v^{2}/2 \\ \ m_{b_{2}}^{2} &=&m_{1}^{2}+g_{1}m_{1}v+g_{1}^{2}v^{2}/2 \notag \\ m_{t_{1}}^{2}\bigskip &=&m^{2}+4gmv+4g^{2}v^{2}+g^{2}m_{1}v/\sqrt{2}% +g^{2}g_{1}v^{2}/4+g_3v^2 \notag \\ \ m_{t_{2}}^{2} &=&m^{2}+4gmv-g^{2}m_{1}v/\sqrt{2}-g^{2}g_{1}v^{2}/4 \notag \\ m_{f_{1}} &=&m_{1}+g_{1}v,\ m_{f_{2}}=m+2gv. \notag\end{aligned}$$ As expected $m_{t_{1}}^{2}+m_{t_{2}}^{2}-2m_{f_{2}}^{2}\neq 0$, since supersymmetry is hard broken. An interesting phenomenon occurs for this term and for a class of other terms as we shall see. In detail, when the length of the compact dimension is small, supersymmetry is broken and becomes restored when the radius increases (and overcomes a critical length $L_c$). This is strange and rather counterintuitive to what would be expected from a phenomenologically correct four dimensional theory. However we describe it since it might be useful to extra dimensional physics. Also, as we shall see in the next section, the whole behavior resembles the inverse symmetry breaking of continuous symmetries at finite temperature. Let us call it ”inverse supersymmetry breaking” for brevity. This said behavior can appear when $g_3$ is of the order of $\frac{m_1}{m}$ or for values smaller, that is $g_3\leq \frac{m_1}{m}$ when only the term $g_{3}\chi ^{2}\varphi _{t_{1}}^{2}$ appears in the Lagrangian (recall that in the previous subsection, the metastable vacua phenomena of the previous subsection occurred when $\frac{m_{1}}{m}\ll g_{3}$). This whole phenomenon is well seen in figure \[inverse1\]. We used the following numerical values, ${m}_{1}{=}$[200, ]{}${m=}$[7000, ]{}${g}_{1}{% =}$[0.001, ]{}${g=}$[0.09, ]{}${\mu =}$[7000]{} and $g_3=0.05$. ![Inverse Supersymmetry Breaking[]{data-label="inverse1"}](inverse1.eps) As can be seen in figure \[inverse1\] the phenomenon looks like a second order phase transition with the length of the compact dimension playing the role of the temperature. No barrier appears between the vacua at $v=0$ and at $v\neq 0$. The study was limited to perturbation preserving values of $L$. As we see for large $L$ ($L=1/7000$) supersymmetry is unbroken and start’s to break at $L_c=1/50830$. As the length decreases, the breaking is more profound. The two non supersymmetric vacua are not equivalent. \[table1\] $g_3$ $L_c^{-1}$ --------- ------------ 0.1 32319 0.07 41352 0.05 50839 0.03 67994 0.01 121950 0.007 145900 0.005 173083 0.003 223990 0.001 388990 0.0005 557000 0.0001 1232000 0.00005 1740000 0.00001 3942000 \ We tried to find how $L_c$ changes under a change of $g_3$. In the table we present the values of $g_3$ and the corresponding values of $L_c$, and in figure \[plotl3lc\] we plot the dependence. In figure \[fitl3lc\] we fit the curve with a continuous function. The dots are the values that appear in the table, while the continuous line corresponds to the function $0.000091\sqrt{x}$. Thus the dependence of $g_3$ as a function of $L_c$ is roughly, $$g_3\sim 0.000091\sqrt{L_c}.$$ ![Plot of $g_3$ and $L_c$[]{data-label="plotl3lc"}](l3depe.eps) ![Fit of the curve $g_3-L_c$[]{data-label="fitl3lc"}](prosomiosi.eps) Other Hard Supersymmetry Breaking terms and Discussion ------------------------------------------------------ We shall not present here the detailed study of all the allowed terms (this will be done in [@oikonomouunderpreparation]). We shall discuss only the main results. A term of the form $g_2\overline{\psi_i}\psi_i\chi$ always breaks supersymmetry and non of the previous phenomena occurs. As we shall see in the next section this is similar to symmetry non restoration where a continuous symmetry is broken and never get’s restored. The combined addition of terms $-\frac{1}{2}g_2\overline{\psi_t}\psi_t\chi-g_3\chi ^2\varphi _{t_1} ^2$ (that is interaction of the untwisted scalar $\chi$ with a twisted fermion and twisted boson) causes ”inverse supersymmetry breaking” as previous. The conditions that must hold in order this occurs are the same as before ($g_3\leq \frac{m_1}{m}$) and in addition $g_2 \ll g_3$. For this condition the behavior of supersymmetry breaking is well described from figure \[inverse1\]. Also for $g_2=0.0001$ the $g_3-L_c$ dependence is similar to that of figure \[fitl3lc\]. Particularly the $g_3-L_c$ dependence in this case, is described by, $$g_3\sim 0.000091\sqrt{L_c},$$ which is the same as before. This dependence is the same for all the cases we studied. Thus this motivates us to think that there is a universal behavior of $g_3$ as a function of $L_c$. Of course all effects we described in this section appear in one loop level and within perturbative limits. Thus there exist the danger that all these effects are an artifact of perturbation theory. However the theory is at some level supersymmetric and one loop corrections may be adequate enough [@martin]. A detailed study should include higher loop corrections. In the next section similar problems-considerations are encountered and discussed. Continuous and Discrete Symmetry Inverse Symmetry Breaking, Restoration and Non-restoration at Finite Temperature ================================================================================================================= In this section we review some conceptually similar phenomena to the above. The difference is that symmetries are studied at finite temperature and the symmetry is not supersymmetry but a continuous global $O(N_1)\times O(N_2)$ or a discrete $Z_2\times Z_2$. As we shall see symmetry non-restoration and inverse symmetry breaking phenomena occur naturally when similar terms to $\frac{1}{2}g_2\overline{\psi_t}\psi_t\chi$, $g_3\chi ^2\varphi _{t_1} ^2$ appear in the Lagrangian. Symmetry non-restoration means that a symmetry broken at $T=0$ never gets restored at high temperatures. Inverse symmetry breaking means that an unbroken symmetry at $T=0$ may be spontaneously broken at high temperature. These phenomena occur in field theories when cross interactions are included among the scalar fields similar to the bosonic hard supersymmetry breaking term $g_{3}\chi ^{2}\varphi _{u_{2}}^{2}$. Similar to this term scalar interactions and also Yukawa terms like the ones of the previous sections are frequently used in the theory of reheating after inflation. Actually this similarities motivated us to use such terms in order to see what their effect would be on supersymmetry breaking. We shall discuss these in the end of this section. First let us describe the inverse symmetry breaking phenomenon. Consider a theory with real scalar fields $\phi_1$ and $\phi_2$ described by the $O(N_1)\times O(N_2)$ globally symmetric Lagrangian, $$\begin{aligned} \label{onlagrang} &L=\frac{1}{2}\partial _{\mu }\phi_1\partial _{\mu }\phi_1 +\frac{1}{2}\partial _{\mu }\phi_2\partial _{\mu }\phi_2 +\frac{1}{2}m_1^2\phi_1^2+\frac{1}{2}m_2^2\phi_2^2+\frac{1}{4!}\lambda_1\phi_1^4 +\frac{1}{4!}\lambda_1\phi_2^4+\frac{1}{4!}\lambda \phi_1^2 \phi_2^4,\end{aligned}$$ where $\phi_1$ and $\phi_2$ be real scalars with $N_1$ and $N_2$ components. In the above Lagrangian one of the global $O(N_i)$ symmetries may break at high temperature if the $\lambda$ coupling takes negative values. Thus one of the two scalar fields $\phi_1$ or $\phi_2$ may acquire a non zero vacuum expectation value. Thus at high temperature and for certain values of the parameters, the initial $O(N_1)\times O(N_2)$ is broken to $O(N_1)$. This was called inverse symmetry breaking and was first point out by Weinberg [@weinbergantirestor] and extensively studied by many authors [@bimonte1; @bimonte2; @pinto1; @pinto2; @dvali; @pinto3; @barut]. In the case $N_1=1$ and $N_2=1$ the initial $O(N_1)\times O(N_2)$ symmetry reduces to a $Z_2\times Z_2$ symmetry and similar arguments hold, in connection to the breaking of one of the two discrete symmetries (for details see [@pinto1]) Symmetry non-restoration was used in [@dvali] to solve the monopole problem in the $SU(5)$ GUT. Monopoles are usually produced during phase transitions at temperatures of the order $\sim10^{14}$GeV. As was proposed in [@dvali], the symmetric phase of $SU(5)$ was never realized, no matter how high the temperature becomes. In that paper the interaction term $\alpha \vert \chi_{45}\vert ^2 \vert H_{24}\vert$ appearing in the scalar Kibble-Higgs sector is responsible for the non-restoration of the $SU(5)$ symmetry. Actually the scalar interaction of $H_{24}$ and $\chi_{45}$ gives negative contributions to the thermal masses and one of those becomes negative. Once this happens the corresponding Higgs field maintains a vacuum expectation value for high temperatures and the symmetry is never restored. This phenomenon occurs for certain values of the parameters (see [@dvali; @bimonte2]). However these results are very sensitive and are altered when someone includes two loop calculations [@bimonte2]. In the same spirit there are arguments based on large N calculations which can show that symmetry non-restoration is an artifact of perturbation theory. For an interesting discussion on this see [@bimonte2] and references therein. A cross term of the form $g\chi^2\phi^2$ is used in non relativistic models in condensed matter physics, for example in the coupled two field Bose gases. However the effect of this does not break any of the initial symmetry patterns [@pinto3]. In conclusion the intuitive approach to all phenomena at finite temperature consists of the statement that symmetries broken at small temperatures become restored at high temperature (in the same class belong finite volume theories). Many field theory models exist that belong to this class. Counter intuitive phenomena occur in field theories with rich scalar sector. Especially if the multi-scalar sectors interact weakly with negative couplings then the phenomena known as inverse symmetry breaking or symmetry non-restoration occur. This happens at high temperature and refers to the spontaneous breaking of a symmetry at high temperature. Usually the symmetry is a global $O(N_1)\times O(N_2)$ or for the case of symmetry non-restoration a continuous, like $SU(5)$. Although inverse symmetry breaking is counterintuitive, nature has provided us with examples that systems are more symmetric at low temperatures than at high temperature. For example the Rochelle salt which at low temperatures is orthorhombic and after a critical temperature becomes monoclinic. Similar phenomena occur in liquid crystals (for more examples see [@barut] and references therein). Reheating After Inflation and Thermal Inflation ----------------------------------------------- The process of reheating after inflation is one of the most important features of the new inflationary scenario [@lindebook]. The process of reheating is necessary in order that the inflating vacuum like state of the universe transforms to a hot Friedmann universe state. During the reheating process a massive scalar field gives it’s vacuum energy to lighter fermions and bosons. The Lagrangian governing this process is, $$\begin{aligned} \label{reheatinglagra} &L=\frac{1}{2}(\partial_{\mu}\varphi)^2-\frac{m_{\varphi}}{2}\varphi^2+\frac{1}{2}(\partial_{\mu}\chi)^2-\frac{m_{\chi}}{2}\chi^2 +\overline{\psi }(i\gamma ^{\mu }\partial _{\mu}-m_{\psi})\psi-\lambda \varphi \overline{\psi}\psi-\frac{1}{2}g^2\varphi ^2\chi ^2.\end{aligned}$$ The role of the inflaton fields is played from the scalar field $\varphi$. The inflaton field decays to the particles $\psi$ and $\chi$ due to the interaction terms $\lambda \varphi \overline{\psi}\psi$ and $\frac{1}{2}g^2\varphi ^2\chi ^2$. Note that we used similar terms in order to break supersymmetry hard and all the effects we seen in the previous section are due to these interaction terms. Also, in order reheating takes place, the condition $m_{\varphi}\gg m_{\psi}, m_{\varphi}$ must hold (remember that similar conditions hold in the supersymmetric model we studied previously. There $m_1$ was the tree order mass of the untwisted scalar field and $m$ the tree order mass of the twisted scalar. One of the conditions we used is that the untwisted sector has greater mass than the twisted sector, namely $m_1\gg m$. Also the untwisted fermion has mass m). So with the interactions $\lambda \varphi \overline{\psi}\psi$ and $\frac{1}{2}g^2\varphi ^2\chi ^2$ the initially concentrated energy (during inflation) to the field $\varphi$ is transferred to particle creation through the processes $\varphi \rightarrow \psi \psi $ and $\varphi \rightarrow \chi \chi$, and the universe thermalizes [@reheatingpapers]. The cross interaction terms of the form $\frac{1}{2}g^2\varphi ^2\chi ^2$ are used in some versions of hybrid inflation. Let us briefly mention another application of cross interactions between scalar fields. These terms are used to the thermal inflation scenario. This is a modified version of the new inflation and old inflation scenario [@lindebook; @thermalinflation]. In the thermal inflation scenario, a Lagrangian that contains a term $\frac{1}{2}g^2\varphi ^2\chi ^2$ is used. This term is necessary in order a phase transition takes place. Actually there is bump in the effective potential that is solely created from this interaction term and the phase transition is strongly first order. The universe supercools in the false vacuum and after a critical temperature tunnels to the true vacuum through bubble nucleation. At this point thermal inflation ends (for details see [@thermalinflation]). Before closing this section we conclude that cross terms between scalar fields and Yukawa interactions are frequently used in particle models and these terms make the effective potential locally unstable thus triggering first order phase transitions, reheating and other processes. Also at high temperatures and for specific values of the parameters, these terms may cause inverse symmetry breaking or symmetry non restoration. Our study involved these terms but the phenomena studied where related to supersymmetry breaking and inverse supersymmetry breaking when the space has a compact dimension. So the resemblance between the two setups is quite clear. We shall discuss on this later on. A Toy Cosmological Application ============================== In this section a brief qualitative application (although fictitious) of one of the above results is presented. Consider a toy universe that has just come out of it’s strong gravity period and it’s particle content (matter) is described by (\[test\]) with the addition of the hard supersymmetry breaking term $g_{3}\chi^{2}\varphi _{u_{2}}^{2}$. The back-reaction of gravity on field theory is considered small ([*[i.e.]{}*]{}field theory calculations made in the previous part considering flat background, are consistent and thus the metric fluctuations are negligible). This toy universe’s expansion is described by: $$\mathrm{d}s^{2}=\mathrm{d}t^{2}-a^{2}(t)\mathrm{d}x_{1}^{2}-b^{2}(t)\mathrm{d}x_{2}^{2}-c^{2}(t)\mathrm{d}x_{3}^{2} \label{metric},$$ a homogeneous Clifford-Klein metric (known as Bianchi I cosmological model), with ${x_1}$, ${x_2}$, ${x_3}$, as in (\[flat\]). In (\[metric\]), $a(t)$, $b(t)$, $c(t)$, describe the scale factors of the two infinite and of the compact dimension respectively. Also we assume, $$a(t)=b(t)=k{\,}c(t) \label{cond},$$ with $k\gg{1}$. Figure \[nonsusy\] motivates us to think as follows: At small lengths of the compact dimension the toy universe’s ground state is the supersymmetric vacuum although we had broken supersymmetry using a hard term, something usually unexpected. As the circumference of the compact dimension grows, the toy universe ”acknowledges” the presence of the other true vacuum (in terms of it’s quantum one loop effective potential) and at some point (bubbles of the new vacuum create within the false vacuum) quantum penetrates to the other vacuum, the non supersymmetric one. Therefore, at small lengths of the compact dimension, supersymmetry was unbroken and as the radius grows, supersymmetry breaks. It seems that using a compact dimension in the present model, supersymmetry breaks dynamically after some critical radius of the compact dimension, although supersymmetry is expected to be broken for all lengths (this would exactly be the effect of a hard supersymmetry breaking term). Let us now do some toy cosmology on this toy universe. $V(0)$ is the minimum of the effective potential at the origin (note $V(0)=0$), and $V(v_1)$ the minimum after quantum barrier penetration (the non supersymmetric vacuum). We assume this toy universe has a cosmological constant which is chosen to be $(8{\pi}G)^{-1}\Lambda=-\Delta{V}=-(V(v_1)-V(0))$ (a choice which shall be explained below). Note that $\Lambda>0$. The Friedmann equation describing it’s evolution is: $${\Big{(}}\frac{\dot{a}}{a}{\Big{)}}^2 =\frac{8{\pi}G}{3}{\Big{(}}\rho+\frac{\Lambda}{8{\pi}G}{\Big{)}},$$ referred to the $x_1$ dimension (we omit the analysis on the other dimension and to the compact one since it is similar (\[cond\]). For details see [@gongcharov]). In the early post quantum gravity period, this toy universe is at the $V(0)$ vacuum state, the energy density is $\rho={V(0)}=0$. The Friedmann equation reads: $${\Big{(}}\frac{\dot{a}}{a}{\Big{)}}^2 =\frac{8{\pi}G}{3}{\Big{(}}\frac{\Lambda}{8{\pi}G}{\Big{)}},$$ and without getting into much detail (see [@gongcharov]), an inflationary solution (corresponding to a flat universe) follows in all space dimensions, being of the form, $$a(t)\sim e^{\sqrt{\Lambda}t},$$ with $a(t)=b(t)=k{\,}c(t)$. Note that the rate of the expansion is the same for all dimensions. During the inflation period of this toy universe, its quantum vacuum state is the supersymmetric vacuum (false vacuum), until for some length quantum tunnelling occurs (due to one loop quantum effects), and the new vacuum state is $V(v_1)$, the new minimum of the effective action. During the quantum vacuum penetration, the energy release (something like latent heat) [@vilenkin] is of the order $L^{-4}_p$ which thermalizes the matter content at a temperature $T_p$, with $$L^{-4}_p\sim T^{4}_p,$$ $L_p$ and $T_p$ characterizing the ”phase transition” point. After thermalization, the energy density is $\rho \sim T^{4}_p+V(v_1)$ and the Friedmann equation reads: $${\Big{(}}\frac{\dot{a}}{a}{\Big{)}}^2 \sim \frac{8{\pi}G}{3}{\Big{(}}T^{4}_p{\Big{)}},$$ (we fixed $\Lambda$ in order to cancel the value of $V(v_1)$). So after vacuum penetration the toy universe follows a radiation dominated expansion (note that the maximum temperature ever reached was the thermalization temperature $T_p$ [@vilenkin]). Note that the above picture has many similarities with the strongly supercooled first order phase transitions of the early universe (old inflationary scenario). Let us point out its main features. Start with a toy universe filled with fermions and bosons interacting in a non supersymmetric way (due to explicit hard breaking). The toy universe is at a supersymmetric vacuum (unexpectedly) when it’s magnitude (specifically the compact dimension magnitude) is small, but as it evolves spatially (inflation in our setup) quantum penetrates to a non supersymmetric vacuum, which is energetically preferable. So at the early toy universe’s epoch, supersymmetry was not broken (at least the vacuum quantum state did not realize broken susy), although the matter content of it, interacts in a non supersymmetric way, but supersymmetry dynamically breaks (through quantum tunnelling) [@linde; @zeldovich] when the toy universe evolves at larger sizes. Let us note here that in order this toy universe is realistic, one must deal with defects (monopoles, domain walls) and with the cosmological experimental observations that do not suggest non trivial topology in the spatial dimensions. Maybe domain walls may be avoided in first order phase transitions but if we want to include GUTs in this universe we can not avoid defects (only if the temperature after the quantum penetration is smaller compared to the temperature that the defects are created, then defects maybe be avoided). Even if one deals with defects, the non trivial topology problem remains, so the magnitude of the compact dimension must be larger than the particle horizon (which can be achieved during inflation). Conclusions =========== In this paper we studied a simple supersymmetric model in $S^1\times R^3$ spacetime topology. We discussed how topology can affect the boundary conditions of the fields and we seen that in $S^1\times R^3$ bosons and fermions can have periodic and anti-periodic boundary conditions along the compact dimension. Also we discussed how supersymmetry is broken spontaneously and how this can be avoided in terms of the boundary conditions that the fields obey. We confirmed these by calculating the effective potential of the theory. Next we introduced in the Lagrangian interaction terms among scalars and fermions. These terms break supersymmetry hard in $R^4$ topology. However two class of phenomena occur in $S^1\times R^3$: - When an interaction among the two untwisted scalars is added, a term of the form $g_{3}\chi ^{2}\varphi _{u_{2}}^{2}$, supersymmetry remains unbroken for small values of the radius of the compact dimension, and as the length increases, breaks after a critical length. This occurs when $\frac{m_{1}}{m}\ll g_{3}$. This phenomenon resembles first order phase transitions of finite temperature field theories. Also we applied this to a toy cosmological model, which described the evolution of a universe with an initial cosmological constant and filled with the aforementioned fields. - The addition of an interaction among the scalar of the untwisted sector that develops a vev and the twisted scalars or the twisted fermions results in a very peculiar phenomenon. Particularly when certain conditions hold (similar to the aforementioned) supersymmetry is broken for small lengths and as the radius increases, becomes restored after a critical length. This resembles conceptually second order phase transitions. The last is similar to inverse symmetry breaking phenomena at finite temperature, that appear in field theories with rich scalar sector. In that case a symmetry unbroken at low temperatures may break at high temperature. In our case at small lengths supersymmetry breaks while remains unbroken for large radius values. We called this inverse supersymmetry breaking for brevity. The terms in the Lagrangian that trigger inverse symmetry breaking are the same that trigger inverse supersymmetry breaking. Also these terms appear in the theory of reheating and in the thermal inflation. In the study we realized that there is a universality in the $g_3-L_c$ dependence. We shall present these in detail in [@oikonomouunderpreparation]. Finally let us discuss the physical significance of ”inverse supersymmetry breaking”. This phenomenon is not so appealing to a four dimensional theory. What would be expected is that supersymmetry should be unbroken for small values of the radius of the compact dimension and breaks dynamically at large distances. What happens here is the converse. For large values of the radius, supersymmetry is unbroken and breaks dynamically for small values of the radius. However this would be interesting for a five dimensional model. 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--- abstract: 'Fowler introduced the notion of a product system: a collection of Hilbert bimodules $\mathbf{X}=\left\{\mathbf{X}_p:p\in P\right\}$ indexed by a semigroup $P$, endowed with a multiplication implementing isomorphisms $\mathbf{X}_p\otimes_A \mathbf{X}_q\cong \mathbf{X}_{pq}$. When $P$ is quasi-lattice ordered, Fowler showed how to associate a $C^*$-algebra $\mathcal{NT}_\mathbf{X}$ to $\mathbf{X}$, generated by a universal representation satisfying some covariance condition. In this article we prove a uniqueness theorem for these so called Nica–Toeplitz algebras.' address: | School of Mathematics and Statistics\ Victoria University of Wellington\ Wellington 6140, New Zealand author: - James Fletcher bibliography: - 'uniquenessNTalgebra.bib' title: 'A uniqueness theorem for the Nica–Toeplitz algebra of a compactly aligned product system' --- [^1] Introduction ============ Suppose $A$ is a $C^*$-algebra and $X$ is a right Hilbert $A$-module. When $X$ comes equipped with a left action of $A$ by adjointable operators, we call $X$ a Hilbert $A$-bimodule. When this left action of $A$ is faithful, Pimsner showed how to associate a $C^*$-algebra $\mathcal{T}_X$ to $X$, called the Toeplitz algebra of $X$, generated by the raising and lowering operators on the Fock space of $X$ [@MR1426840]. In [@MR1722197], Fowler and Raeburn generalised the situation to arbitrary left actions by adjointable operators, and showed that $\mathcal{T}_X$ may be realised as the universal $C^*$-algebra generated by so called Toeplitz representations. In the spirit of Coburn’s Theorem for the classical Toeplitz algebra [@MR0213906], Fowler and Raeburn also proved a uniqueness theorem for $\mathcal{T}_X$ [@MR1722197 Theorem 2.1] that provides a sufficient condition for a representation of $\mathcal{T}_X$ to be faithful. Loosely speaking, their result states that a representation of $\mathcal{T}_X$ on a Hilbert space $\mathcal{H}$ will be faithful, provided the representation leaves enough room in $\mathcal{H}$ for the coefficient algebra $A$ to act faithfully. Subsequently Fowler introduced the notion of a product system of Hilbert bimodules [@MR1907896], generalising the continuous product systems of Hilbert spaces studied by Arveson [@MR987590] and the discrete product systems studied by Dinh [@MR1138835]. Loosely speaking, a product system of Hilbert $A$-bimodules over a unital semigroup $P$ consists of a semigroup $\mathbf{X}=\bigsqcup_{p\in P} \mathbf{X}_p$, such that each $\mathbf{X}_p$ is a Hilbert $A$-bimodule, and the map $x\otimes_A y\mapsto xy$ extends to an isomorphism from $\mathbf{X}_p\otimes_A \mathbf{X}_q$ to $\mathbf{X}_{pq}$ for each $p,q\in P\setminus \{e\}$. Motivated by the work of Nica [@MR1241114] and Laca and Raeburn on Toeplitz algebras associated to non-abelian groups [@MR1402771], Fowler studied representations of compactly aligned product systems over quasi-lattice ordered groups — semigroups sitting inside groups possessing a semi-lattice like structure, satisfying an additional constraint called Nica covariance. Generalising the Toeplitz algebra associated to a single Hilbert bimodule, Fowler showed how to associate a $C^*$-algebra $\mathcal{NT}_\mathbf{X}$, generated by a universal Nica covariant representation, to each compactly aligned product system $\mathbf{X}$. We call this $C^*$-algebra the Nica–Toeplitz algebra of $\mathbf{X}$. Furthermore, Fowler associates a twisted semigroup crossed product algebra to each compactly aligned product system, and characterises their faithful representations [@MR1907896 Theorem 7.2]. Restricting to the subalgebra $\mathcal{NT}_\mathbf{X}$ then gives a uniqueness theorem for Nica–Toeplitz algebras, generalising both Laca and Raeburn’s uniqueness theorem for Toeplitz algebras of quasi-lattice ordered groups [@MR1402771 Theorem 3.7] and Fowler and Raeburn’s uniqueness theorem for Toeplitz algebras of Hilbert bimodules [@MR1722197 Theorem 2.1] — at least in the case where the bimodule is ‘essential’. In this article we prove a slightly more general version of Fowler’s uniqueness theorem for Nica–Toeplitz algebras associated to compactly aligned product systems over quasi-lattice ordered groups. The result gives a sufficient condition for the induced representation $\psi_*$ of a Nica covariant representation $\psi$ (on a Hilbert space $\mathcal{H}$) to be faithful. This condition basically says that the ranges of all the operators $\left\{\psi_p(x): x\in \mathbf{X}_p, \ p\in P\setminus\{e\}\right\}$ should leave enough room in $\mathcal{H}$ for $A$ to act faithfully. When $A$ acts by compacts on each fibre of $\mathbf{X}$, we will see that this condition is also necessary. Unlike in Fowler’s result, we do not insist that each fibre of the product system is essential (i.e. we do not require that $\mathbf{X}_p=\overline{A\cdot\mathbf{X}_p}$ for each $p\in P$). Whilst there do not seem to many ‘natural’ examples of Hilbert bimodules with nondegenerate left actions, this level of generality was made use of in our article [@2017arXiv170608626F]. Moreover, in contrast to Fowler’s proof, we do not view the Nica–Toeplitz algebra as a subalgebra of a twisted semigroup crossed product, working with $\mathcal{NT}_\mathbf{X}$ directly. Furthermore, we note that Fowler’s result as stated in [@MR1907896 Theorem 7.2] is technically false when applied to any product system over the trivial semigroup $\{e\}$, and correct this error in our result. The article is set out as follows. In Section \[background\] we fix notation and recap the necessary background material for Hilbert bimodules, product systems of Hilbert bimodules, and their associated Nica–Toeplitz algebras. In Section \[proof of the uniqueness theorem\] we present the proof of our uniqueness theorem for Nica-Toeplitz algebras. After completing this article, it was brought to our attention that Kwaśniewski and Larsen had already proved a far-reaching generalisation of our (along with Fowler’s) uniqueness theorem [@2016arXiv161108525K Corollary 10.14] for full Nica–Toeplitz algebras associated to well-aligned ideals of right tensor $C^*$-precategories. Subsequent to the initial version of this article appearing on the arXiv, Kwaśniewski and Larsen showed how their more general result can be applied to product systems over right LCM semigroups (themselves a generalisation of quasi-lattice ordered groups) [@2017arXiv170604951K Theorem 2.19]. Despite this, we still feel that the results in this article will find use amongst those studying product systems over quasi-lattice ordered groups. We provide a direct proof of the uniqueness theorem in the quasi-lattice ordered case, and as such avoid the various technical conditions present in [@2017arXiv170604951K Theorem 2.19]. We also note that in Corollary \[representations in C\*-algebras\], we show how to extend the uniqueneness theorem to representations in arbitrary $C^*$-algebras (rather than just on Hilbert spaces), provided the action on each fibre is compact. Preliminaries {#background} ============= Hilbert bimodules ----------------- We attempt to summarise only those aspects of Hilbert bimodules that we will need. Readers unfamiliar with this material, or looking for more detail, are directed to [@lance]. Let $A$ be a $C^*$-algebra. A (right) inner-product $A$-module is a complex vector space $X$ equipped with a map $\langle \cdot, \cdot \rangle_A:X\times X\rightarrow A$, linear in its second argument, and a right action of $A$, such that for any $x,y\in X$ and $a\in A$, we have 1. $\langle x,y\rangle_A=\langle y,x\rangle_A^*$; 2. $\langle x,y\cdot a \rangle_A=\langle x,y\rangle_Aa$; 3. $\langle x,x\rangle_A\geq 0$ in $A$; and 4. $\langle x,x\rangle_A=0$ if and only if $x=0$. It follows from [@lance Proposition 1.1] that the formula ${\left\|x\right\|}_X:={\left\|\langle x,x\rangle_A\right\|}_A^{1/2}$ defines a norm on $X$. If $X$ is complete with respect to this norm, we say that $X$ is a (right) Hilbert $A$-module. Let $X$ be a (right) Hilbert $A$-module. We say that a map $T:X\rightarrow X$ is adjointable if there exists a map $T^*:X\rightarrow X$ such that $\langle Tx, y\rangle_A=\langle x, T^*y\rangle_A$ for each $x,y\in X$. Every adjointable operator $T$ is automatically linear and continuous, and the adjoint $T^*$ is unique. The collection of adjointable operators on $X$, denoted by $\mathcal{L}_A(X)$, equipped with the operator norm is a $C^*$-algebra. For each $x,y\in X$ there is an adjointable operator $\Theta_{x,y}\in \mathcal{L}_A(X)$ defined by $\Theta_{x,y}(z)=x\cdot \langle y, z\rangle_A$. We call operators of this form (generalised) rank-one operators. The closed subspace $\mathcal{K}_A(X):={\overline{\mathrm{span}}}\{\Theta_{x,y}:x,y\in X\}$ is an essential ideal of $\mathcal{L}_A(X)$, whose elements we refer to as (generalised) compact operators. A (right) Hilbert $B$–$A$-bimodule is a (right) Hilbert $A$-module $X$ together with a $*$-homomorphism $\phi:B\rightarrow \mathcal{L}_A(X)$. When $A=B$, we say that $X$ is a Hilbert $A$-bimodule. We think of $\phi$ as implementing a left action of $B$ on $X$, and frequently write $b\cdot x$ for $\phi(b)(x)$. Since each $\phi(b)\in \mathcal{L}_A(X)$ is $A$-linear, we have that $b\cdot (x\cdot a)=(b\cdot x)\cdot a$ for each $a\in A$, $b\in B$, and $x\in X$. An important example is the Hilbert $A$-bimodule ${}_A A_A$, which is just the set $A$ equipped with the inner product given by $\langle a,b\rangle_A=a^*b$ and left and right actions of $A$ given by multiplication. Then $\mathcal{K}_A({}_A A_A)$ is isomorphic to $A$ via the map $\Theta_{a,b}\mapsto ab^*$, whilst $\mathcal{L}_A({}_A A_A)$ is isomorphic to the multiplier algebra of $A$. The Hewitt–Cohen–Blanchard factorisation theorem [@MR1634408 Proposition 2.31] says that if $x$ is an element of a Hilbert $A$-module $X$, then there exists a unique $x'\in X$ such that $x=x'\cdot \langle x', x'\rangle_A$. Hence, $X$ is a right nondegenerate $A$-module in the sense that $X={\overline{\mathrm{span}}}\{x\cdot a:x\in X,a\in A\}$. It is not necessarily true that every Hilbert $A$-bimodule is left nondegenerate in the sense that $X={\overline{\mathrm{span}}}\{a\cdot x:x\in X,a\in A\}$ (in [@MR1907896], Fowler calls such bimodules essential). The balanced tensor product $X\otimes_A Y$ of two Hilbert $A$-bimodules $X$ and $Y$ is formed as follows. Let $X \odot Y$ be the algebraic tensor product of $X$ and $Y$ as complex vector spaces, and let $X\odot_A Y$ be the quotient of $X \odot Y$ by the subspace spanned by elements of the form $x\cdot a \odot y- x\odot a\cdot y$ where $x\in X$, $y\in Y$, and $a\in A$ (we write $x\odot_A y$ for the coset containing $x\odot y$). Then the formula $ \langle x\odot_A y, w\odot_A z\rangle_A:=\langle y, \langle x,w\rangle_A \cdot z\rangle_A, $ determines a bounded $A$-valued sesquilinear form on $X\odot_A Y$. Let $N$ be the subspace $\mathrm{span}\{n\in X\odot_A Y:\langle n,n\rangle_A=0\}$. The formula ${\left\|z+N\right\|}:=\inf_{n\in N} {\left\|\langle z+n,z+n\rangle_A\right\|}_A^{1/2}$ defines a norm on $(X\odot_A Y)/N$, and we define $X\otimes_A Y$ to be the completion of $(X\odot_A Y)/N$ with respect to this norm. The balanced tensor product $X\otimes_A Y$ carries a left and right action of $A$, such that $a\cdot (x\otimes_A y)\cdot b=(a\cdot x)\otimes_A (y\cdot b)$ for each $x\in X$, $y\in Y$, and $a,b\in A$. Given Hilbert $A$-bimodules $X$ and $Y$ and an adjointable operator $S\in \mathcal{L}_A(X)$, there exists an adjointable operator $S\otimes_A \mathrm{id}_Y\in \mathcal{L}_A(X\otimes_A Y)$ (with adjoint $S^*\otimes_A \mathrm{id}_Y$) determined by the formula $(S\otimes_A \mathrm{id}_Y)(x\otimes_A y)= (Sx)\otimes_A y$ for each $x\in X$ and $y\in Y$. We will also make use of the theory of induced representations. Given a Hilbert $B$–$A$-bimodule $X$ and a nondegenerate representation $\pi:A\rightarrow \mathcal{B}(\mathcal{H})$ of $A$ on a Hilbert space $\mathcal{H}$, [@MR1634408 Proposition 2.66] gives a representation $X\text{-}\mathrm{Ind}_A^B\pi:B\rightarrow \mathcal{B}\left(X\otimes_A \mathcal{H}\right)$ such that $\left(X\text{-}\mathrm{Ind}_A^B\pi\right)(b)(x\otimes_A h)=(b\cdot x)\otimes_A h$ for each $b\in B$, $x\in X$, and $h\in \mathcal{H}$. Product systems of Hilbert bimodules and quasi-lattice ordered groups --------------------------------------------------------------------- Let $A$ be a $C^*$-algebra and $P$ a semigroup with identity $e$. A product system over $P$ with coefficient algebra $A$ is a semigroup $\mathbf{X}=\bigsqcup_{p\in P}\mathbf{X}_p$ such that 1. $\mathbf{X}_p\subseteq \mathbf{X}$ is a Hilbert $A$-bimodule for each $p\in P$; 2. $\mathbf{X}_e$ is equal to the Hilbert $A$-bimodule ${}_A A_A$; 3. For each $p,q\in P\setminus \{e\}$, there exists a Hilbert $A$-bimodule isomorphism $M_{p,q}:\mathbf{X}_p\otimes_A \mathbf{X}_q\rightarrow \mathbf{X}_{pq}$ satisfying $M_{p,q}(x\otimes_A y)=xy$ for each $x\in \mathbf{X}_p$ and $y\in \mathbf{X}_q$; 4. Multiplication in $\mathbf{X}$ by elements of $\mathbf{X}_e=A$ implements the left and right actions of $A$ on each $\mathbf{X}_p$, i.e. $xa=x\cdot a$ and $ax=a\cdot x$ for each $a\in A$, $x\in \mathbf{X}_p$, and $p\in P$. For each $p\in P$, we write $\phi_p:A\rightarrow \mathcal{L}_A(\mathbf{X}_p)$ for the $*$-homomorphism that implements the left action of $A$ on $\mathbf{X}_p$, i.e. $\phi_p(a)(x)=a\cdot x=ax$ for each $a\in A$ and $x\in \mathbf{X}_p$. Since $\mathbf{X}$ is a semigroup, multiplication in $\mathbf{X}$ is associative. In particular, $\phi_{pq}(a)(xy)=(\phi_p(a)x)y$ for all $p,q\in P$, $a\in A$, $x\in \mathbf{X}_p$, and $y\in \mathbf{X}_q$. Also, for each $p\in P$, we write $\langle \cdot,\cdot\rangle_A^p$ for the $A$-valued inner-product on $\mathbf{X}_p$. By (ii) and (iv), for each $p\in P$ there exist $A$-linear inner-product preserving maps $M_{p,e}:\mathbf{X}_p\otimes_A \mathbf{X}_e \rightarrow \mathbf{X}_p$ and $M_{e,p}:\mathbf{X}_e\otimes_A \mathbf{X}_p\rightarrow \mathbf{X}_p$ such that $M_{p,e}(x\otimes_A a)=xa=x\cdot a$ and $M_{e,p}(a\otimes_A x)=ax=a\cdot x$ for each $a\in \mathbf{X}_e=A$ and $x\in \mathbf{X}_p$. By the Hewitt–Cohen–Blanchard factorisation theorem, each $M_{p,e}$ is automatically an $A$-bimodule isomorphism. On the other hand, the maps $M_{e,p}$ need not be isomorphisms, since we do not require that each $\mathbf{X}_p$ is (left) nondegenerate (i.e. $M_{e,p}$ need not be surjective). Given $p\in P\setminus \{e\}$ and $q\in P$, the $A$-bimodule isomorphism $M_{p,q}:\mathbf{X}_p\otimes_A \mathbf{X}_q\rightarrow \mathbf{X}_{pq}$ enables us to define a $*$-homomorphism $\iota_p^{pq}:\mathcal{L}_A\left(\mathbf{X}_p\right)\rightarrow \mathcal{L}_A\left(\mathbf{X}_{pq}\right)$ by $$\iota_p^{pq}(S):=M_{p,q}\circ \left(S\otimes_A \mathrm{id}_{\mathbf{X}_q}\right)\circ M_{p,q}^{-1}$$ for each $S\in \mathcal{L}_A\left(\mathbf{X}_p\right)$. Equivalently, the $*$-homomorphism $\iota_p^{pq}$ is characterised by the formula $\iota_p^{pq}(S)(xy)=(Sx)y$ for each $S\in \mathcal{L}_A\left(\mathbf{X}_p\right)$, $x\in \mathbf{X}_p$, and $y\in \mathbf{X}_q$. Since $\mathbf{X}_e \otimes_A \mathbf{X}_q$ need not in general be isomorphic to $\mathbf{X}_q$, we cannot always define a map from $\mathcal{L}_A\left(\mathbf{X}_e\right)$ to $\mathcal{L}_A\left(\mathbf{X}_q\right)$ using the above procedure. However, as $\mathcal{K}_A\left(\mathbf{X}_e\right)=\mathcal{K}_A\left({}_A A_A\right)\cong A$, we can define $\iota_e^q:\mathcal{K}_A\left(\mathbf{X}_e\right)\rightarrow \mathcal{L}_A\left(\mathbf{X}_q\right)$ by $\iota_e^q(a):=\phi_q(a)$. For notational purposes, we define $\iota_p^r: \mathcal{L}_A\left(\mathbf{X}_p\right)\rightarrow \mathcal{L}_A\left(\mathbf{X}_r\right)$ to be the zero map whenever $p,r\in P$ and $r\neq pq$ for all $q\in P$. We are primarily interested in situations where the underlying semigroup possesses some additional order structure. In particular we focus on the quasi-lattice ordered groups introduced by Nica [@MR1241114]. A quasi-lattice ordered group $(G,P)$ consists of a group $G$ and a subsemigroup $P$ of $G$ such that $P\cap P^{-1}=\{e\}$, and with respect to the partial order on $G$ induced by $p\leq q \Leftrightarrow p^{-1}q\in P$, any two elements $p,q\in G$ which have a common upper bound in $P$ have a least common upper bound in $P$. It is straightforward to show that if two elements in $G$ have a least common upper bound in $P$, then this least common upper bound is unique. We write $p\vee q$ for the least common upper bound of $p,q\in G$ if it exists. For $p,q\in G$, we write $p\vee q=\infty$ if $p$ and $q$ have no common upper bound in $P$, and $p\vee q<\infty$ otherwise. We can also extend the notion of least upper bounds in $(G,P)$ from pairs of elements in $P$ to finite subsets of $P$. We define $\bigvee\emptyset:=e$, $\bigvee \{p\}:=p$ for any $p\in P$, and for any $n \geq 2$ and $C:=\{p_1,\ldots p_n\}\subseteq P$ we define $\bigvee C:=p_1\vee \cdots \vee p_n$ (since $P\cap P^{-1}=\{e\}$, the relation $\leq$ is antisymmetric, and so this is well-defined). Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a product system over $P$. We say that $\mathbf{X}$ is compactly aligned if, whenever $S\in \mathcal{K}_A(\mathbf{X}_p)$ and $T\in \mathcal{K}_A(\mathbf{X}_p)$ for some $p,q\in P$ with $p\vee q<\infty$, we have $\iota_p^{p\vee q}(S)\iota_q^{p\vee q}(T)\in \mathcal{K}_A(\mathbf{X}_{p\vee q})$. Note that this condition does not imply that either $\iota_p^{p\vee q}(S)$ or $\iota_q^{p\vee q}(T)$ is compact. Representations of compactly aligned product systems, Nica covariance, and the Nica–Toeplitz algebra ---------------------------------------------------------------------------------------------------- Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. A representation of $\mathbf{X}$ in a $C^*$-algebra $B$ is a map $\psi:\mathbf{X}\rightarrow B$ such that: 1. each $\psi_p:=\psi|_{\mathbf{X}_p}$ is a linear map, and $\psi_e$ is a $C^*$-homomorphism; 2. $\psi_p(x)\psi_q(y)=\psi_{pq}(xy)$ for all $p,q\in P$ and $x\in \mathbf{X}_p$, $y\in \mathbf{X}_q$; 3. $\psi_p(x)^*\psi_p(y)=\psi_e(\langle x,y\rangle_A^p)$ for all $p\in P$ and $x,y\in \mathbf{X}_p$. It follows from (T1) and (T3) that a representation $\psi$ is always norm-decreasing, and isometric if and only $\psi_e$ is injective. Proposition 8.11 of [@MR2135030] shows that for each $p\in P$, there exists a $*$-homomorphism $\psi^{(p)}:\mathcal{K}_A\left(\mathbf{X}_p\right)\rightarrow B$ such that $\psi^{(p)}\left(\Theta_{x,y}\right)=\psi_p(x)\psi_p(y)^*$ for all $x,y\in \mathbf{X}_p$. We say that a representation $\psi:\mathbf{X}\rightarrow B$ is Nica covariant if, for any $p,q\in P$ and $S\in \mathcal{K}_A(\mathbf{X}_p)$, $T\in \mathcal{K}_A(\mathbf{X}_q)$, we have $$\begin{aligned} \psi^{(p)}(S)\psi^{(q)}(T)= \begin{cases} \psi^{(p\vee q)}\left(\iota_p^{p\vee q}(S)\iota_q^{p\vee q}(T)\right) & \text{if $p\vee q<\infty$}\\ 0 & \text{otherwise.} \end{cases}\end{aligned}$$ It follows from an application of the Hewitt–Cohen–Blanchard factorisation theorem that for any $p,q\in P$, we have $$\psi_p(\mathbf{X}_p)^*\psi_q(\mathbf{X}_q)\in \begin{cases} {\overline{\mathrm{span}}}\{\psi_{p^{-1}(p\vee q)}(\mathbf{X}_{p^{-1}(p\vee q)})\psi_{q^{-1}(p\vee q)}(\mathbf{X}_{q^{-1}(p\vee q)})^*\} & \text{if $p\vee q<\infty$}\\ \{0\} & \text{if $p\vee q=\infty$.} \end{cases}$$ Associated to each product system there exists a canonical Nica covariant representation called the Fock representation. We let $\mathcal{F}_\mathbf{X}:=\bigoplus_{p\in P} \mathbf{X}_p$ denote the space of sequences $(x_p)_{p\in P}$ such that $x_p\in \mathbf{X}_p$ for each $p\in P$ and $\sum_{p\in P}\langle x_p,x_p\rangle_A^p$ converges in $A$. By [@lance Proposition 1.1] there exists a well defined $A$-valued inner product on $\mathcal{F}_\mathbf{X}$ such that $\left\langle (x_p)_{p\in P}, (y_p)_{p\in P}\right\rangle_A=\sum_{p\in P}\langle x_p, y_p\rangle_A^p$, and that $\mathcal{F}_\mathbf{X}$ is complete with respect to the induced norm. Letting $A$ act pointwise from the left and right gives $\mathcal{F}_\mathbf{X}$ the structure of a Hilbert $A$-bimodule, which we call the Fock space of $\mathbf{X}$. Lemma 5.3 of [@MR1907896] shows that there exists an isometric Nica covariant representation $l:\mathbf{X}\rightarrow \mathcal{L}_A(\mathcal{F}_\mathbf{X})$ such that $l_p(x)(y_q)_{q\in P}=\left(xy_q\right)_{q\in P}$ for each $p\in P$, $x\in \mathbf{X}_p$, and $(y_q)_{q\in P}\in \mathcal{F}_\mathbf{X}$. We call $l$ the Fock representation of $\mathbf{X}$. Using [@MR2679392 Theorem 2.10] it can be shown that there exists a $C^*$-algebra $\mathcal{NT}_\mathbf{X}$, which we call the Nica–Toeplitz algebra of $\mathbf{X}$, and a Nica covariant representation $i_\mathbf{X}:\mathbf{X}\rightarrow \mathcal{NT}_\mathbf{X}$, that are universal in the following sense: 1. $\mathcal{NT}_\mathbf{X}$ is generated by the image of $i_\mathbf{X}$; 2. if $\psi:\mathbf{X}\rightarrow B$ is any other Nica covariant representation of $\mathbf{X}$, then there exists a $*$-homomorphism $\psi_*:\mathcal{NT}_\mathbf{X}\rightarrow B$ such that $\psi_* \circ i_\mathbf{X}=\psi$. Since $i_\mathbf{X}$ generates $\mathcal{NT}_\mathbf{X}$, it follows that $\mathcal{NT}_\mathbf{X}={\overline{\mathrm{span}}}\left\{i_\mathbf{X}(x)i_\mathbf{X}(y)^*:x,y\in \mathbf{X}\right\}$. Proposition 4.7 of [@MR1907896] shows that there exists a coaction $\delta_\mathbf{X}:\mathcal{NT}_\mathbf{X}\rightarrow \mathcal{NT}_\mathbf{X}\otimes C^*(G)$ (we use an unadorned $\otimes$ to denote the minimal tensor product of $C^*$-algebras), which we call the canonical gauge coaction, such that $\delta_\mathbf{X}(i_{\mathbf{X}_p}(x))=i_{\mathbf{X}_p}(x)\otimes i_G(p)$ for each $p\in P$ and $x\in \mathbf{X}_p$. For those readers interested in learning more about coactions in general, we suggest [@MR2203930 Appendix A]. Lemma 1.3 of [@MR1375586] shows that there exists a conditional expectation $E_{\delta_\mathbf{X}}$ of $\mathcal{NT}_\mathbf{X}$ onto the generalised fixed-point algebra $\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}}:=\{b\in \mathcal{NT}_\mathbf{X}:\delta_\mathbf{X}(b)=b\otimes i_G(e)\}$ defined by $E_{\delta_\mathbf{X}}:=(\mathrm{id}_{ \mathcal{NT}_\mathbf{X}}\otimes \rho)\circ \delta_\mathbf{X}$, where $\rho:C^*(G)\rightarrow \C$ is the canonical trace. It can be shown that $\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}}={\overline{\mathrm{span}}}\{i_{\mathbf{X}_p}(\mathbf{X}_p)i_{\mathbf{X}_p}(\mathbf{X}_p)^*:p\in P\}$, and, for any $p,q\in P$, $x\in \mathbf{X}_p$, $y\in \mathbf{X}_q$, we have $E_{\delta_\mathbf{X}}\left(i_{\mathbf{X}_p}(x)i_{\mathbf{X}_q}(y)^*\right)=\delta_{p,q}i_{\mathbf{X}_p}(x)i_{\mathbf{X}_q}(y)^*$. We are particularly interested in the situation where the expectation $E_{\delta_\mathbf{X}}$ is faithful on positive elements, i.e. $E_{\delta_\mathbf{X}}(b^*b)=0 \Rightarrow b=0$ for any $b\in \mathcal{NT}_\mathbf{X}$. Inspired by [@MR1907896 Definition 7.1], we say that a compactly aligned product system $\mathbf{X}$ is amenable if $E_{\delta_\mathbf{X}}$ is faithful on positive elements. The argument of [@MR1402771 Lemma 6.5] shows that if $G$ is an amenable group, then $\mathbf{X}$ is an amenable product system. A uniqueness theorem for Nica–Toeplitz algebras {#proof of the uniqueness theorem} =============================================== Firstly, we fix some notation. \[basic projections\] Let $(G,P)$ be a quasi-lattice ordered group, $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$, and $\psi:\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H})$ a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. We define a collection $\{P_p^\psi:p\in P\}$ of projections in $\mathcal{B}(\mathcal{H})$ by $P_e^\psi:=\mathrm{id}_\mathcal{H}$ and $P_p^\psi:=\mathrm{proj}_{\overline{\psi_p(\mathbf{X}_p)\mathcal{H}}}$ for each $p\in P\setminus\{e\}$. We also set $P_\infty^\psi:=0$. The purpose of this article is to prove the following result: \[uniqueness theorem for NT algebras\] Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Suppose $\psi:\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H})$ is a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. 1. If the product system $\mathbf{X}$ is amenable, and, for any finite set $K\subseteq P\setminus \{e\}$, the representation $$\begin{aligned} A\ni a \mapsto \psi_e(a)\prod_{t\in K}\big(\mathrm{id}_\mathcal{H}-P_t^\psi\big)\in \mathcal{B}(\mathcal{H})\end{aligned}$$ is faithful, then the induced $*$-homomorphism $\psi_*:\mathcal{NT}_\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H})$ is faithful. 2. If $\psi_*$ is faithful and $\phi_p(A)\subseteq \mathcal{K}_A(\mathbf{X}_p)$ for each $p\in P$, then the representation $$\begin{aligned} A\ni a \mapsto \psi_e(a)\prod_{t\in K}\big(\mathrm{id}_\mathcal{H}-P_t^\psi\big)\in \mathcal{B}(\mathcal{H})\end{aligned}$$ is faithful for any finite set $K\subseteq P\setminus \{e\}$. The main step in the proof of the uniqueness theorem is to show that the expectation $E_{\delta_\mathbf{X}}$ is also implemented spatially, i.e. there is a compatible expectation $E_\psi$ of $\psi_*(\mathcal{NT}_\mathbf{X})$ onto $\psi_*(\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}})$. To get this compatible expectation we need to be able to calculate the norms of elements of $\psi_*(\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}})$. To do this we will make use of the following well-known fact about operators on Hilbert spaces: if $P_1,\ldots, P_n\in \mathcal{B}(\mathcal{H})$ are mutually orthogonal projections that commute with $T\in \mathcal{B}(\mathcal{H})$ and satisfy $\sum_{i=1}^n P_i=\mathrm{id}_\mathcal{H}$, then ${\left\|T\right\|}_{\mathcal{B}(\mathcal{H})}=\max_{1\leq i\leq n}{\left\|P_i T\right\|}_{\mathcal{B}(\mathcal{H})}$. We now work towards showing that there exists a collection of mutually orthogonal projections in $\mathcal{B}(\mathcal{H})$ that decompose the identity and commute with everything in $\psi_*(\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}})$. We begin by showing that the $*$-homomorphism $\psi^{(p)}:\mathcal{K}_A(\mathbf{X}_p)\rightarrow \mathcal{B}(\mathcal{H})$ has a canonical extension to all of $\mathcal{L}_A(\mathbf{X}_p)$ (for each $p\in P$), and establish some properties of this extension. \[existence of rho map\] Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. Then 1. For each $p\in P$, there exists a representation $\rho_p^\psi:\mathcal{L}_A\left(\mathbf{X}_p\right)\rightarrow \mathcal{B}(\mathcal{H})$ such that for each $S\in \mathcal{L}_A(\mathbf{X}_p)$, $$\rho_p^\psi(S)\left(\psi_p(x)h\right)=\psi_p(Sx)h \quad \text{for each $x\in \mathbf{X}_p$, $h\in \mathcal{H}$}$$ and $\rho_p^\psi(S)$ is zero on $\left(\psi_p\left(\mathbf{X}_p\right)\mathcal{H}\right)^\perp$. 2. $\rho_p^\psi|_{\mathcal{K}_A\left(\mathbf{X}_p\right)}=\psi^{(p)}$. 3. For any $q\in P$ and $a\in A\cong \mathcal{K}_A(\mathbf{X}_e)$, we have $\rho_{q}^\psi\left(\iota_e^{q}(a)\right)=\rho_{e}^\psi(a)P_{q}^\psi$. Furthermore, if $p\in P\setminus \{e\}$, then $\rho_{pq}^\psi\left(\iota_p^{pq}(S)\right)=\rho_{p}^\psi(S)P_{pq}^\psi$ for any and $S\in \mathcal{L}_A(\mathbf{X}_p)$. 4. If $\mathcal{K}\subseteq \mathcal{H}$ is a $\psi_e$-invariant subspace of $\mathcal{H}$, then the subspace $\mathcal{M}:=\overline{\psi_p\left(\mathbf{X}_p\right)\mathcal{K}}$ is $\rho_p^\psi$-invariant. Furthermore, if $\psi_e|_\mathcal{K}$ is faithful, then $\rho_p^\psi|_\mathcal{M}$ is also faithful. Observe that for any $p\in P$ and $x,y\in \mathbf{X}_p$ and $h,k\in \mathcal{H}$, we have $$\label{existence of U} \begin{aligned} \langle \psi_p(x)h,\psi_p(y)k\rangle_\C \hspace{-0.1em}=\hspace{-0.1em} \langle h, \psi_p(x)^*\psi_p(y)k\rangle_\C \hspace{-0.1em}=\hspace{-0.1em} \langle h, \psi_e\hspace{-0.1em}\left(\hspace{-0.1em}\langle x, y \rangle_A\hspace{-0.1em}\right)k\rangle_\C &\hspace{-0.1em}=\hspace{-0.1em} \langle x\hspace{-0.1em}\otimes_A \hspace{-0.1em}h, y\hspace{-0.1em}\otimes_A\hspace{-0.1em} k\rangle_\C. \end{aligned}$$ Thus, there exists a linear isometry $U:\mathbf{X}_p \otimes_A \mathcal{H}\rightarrow \mathcal{H}$ such that $U\left(x\otimes_A h\right)=\psi_p(x)h$ for each $x\in \mathbf{X}_p$ and $h\in \mathcal{H}$. Equation  shows that $U^*\left(\psi_p(x)h\right)=x\otimes_A h$ for each $x\in \mathbf{X}_p$ and $h\in \mathcal{H}$. We claim that $U^*|_{\left(\psi_p\left(\mathbf{X}_p\right)\mathcal{H}\right)^\perp}=0$. To see this, observe that for any $f\in \left(\psi_p\left(\mathbf{X}_p\right)\mathcal{H}\right)^\perp$ and $y\in \mathbf{X}_p$, $h\in \mathcal{H}$ we have $$\begin{aligned} \langle U(y\otimes_A k),f\rangle_\C=\langle \psi_p(y)k, f\rangle_\C=0,\end{aligned}$$ and hence $U^*(f)=0$. Since $ \mathbf{X}_p\otimes_A \mathcal{H}=\left(\mathbf{X}_p\cdot A\right)\otimes_A \mathcal{H}=\mathbf{X}_p\otimes_A \overline{\psi_e(A)\mathcal{H}}, $ we may assume that the representation $\psi_e$ is nondegenerate without loss of generality. With this in mind, define $\rho_p^\psi:\mathcal{L}_A(\mathbf{X}_p)\rightarrow \mathcal{B}(\mathcal{H})$ by $$\rho_p^\psi(S):=U\circ \left(\mathbf{X}_p\text{-}\mathrm{Ind}_A^{\mathcal{L}_A\left(\mathbf{X}_p\right)}\psi_e(S)\right)\circ U^*$$ for each $S\in \mathcal{L}_A\left(\mathbf{X}_p\right)$. Thus, for each $S\in \mathcal{L}_A\left(\mathbf{X}_p\right)$, the restriction $\rho_p^\psi(S)|_{\left(\psi_p\left(\mathbf{X}_p\right)\mathcal{H}\right)^\perp}$ is zero, whilst for any $x\in \mathbf{X}_p$ and $h\in \mathcal{H}$ we have $$\begin{aligned} \rho_p^\psi(S)(\psi_p(x)h) &=\left(U\circ \left(\mathbf{X}_p\text{-}\mathrm{Ind}_A^{\mathcal{L}_A\left(\mathbf{X}_p\right)}\psi_e(S)\right)\circ U^*\right)\left(\psi_p(x)h\right)\\ &=\left(U\circ\left(\mathbf{X}_p\text{-}\mathrm{Ind}_A^{\mathcal{L}_A\left(\mathbf{X}_p\right)}\psi_e(S)\right)\right)(x\otimes_A h) =U\left(Sx\otimes_A h\right) =\psi_p(Sx)h. \end{aligned}$$ This completes the proof of part (i). Since both $\psi^{(p)}$ and $\rho_p^\psi$ are $*$-homomorphisms, to prove (ii) it suffices to show that $\psi^{(p)}$ and $\rho_p^\psi$ agree on rank-one operators. Fix $x,y \in \mathbf{X}_p$. Firstly, we check that $\psi^{(p)}\left(\Theta_{x,y}\right)$ and $\rho_p^\psi\left(\Theta_{x,y}\right)$ agree on $\overline{\psi_p\left(\mathbf{X}_p\right)\mathcal{H}}$. For any $z\in \mathbf{X}_p$ and $h\in \mathcal{H}$, we have $$\begin{aligned} \rho_p^\psi\left(\Theta_{x,y}\right)\left(\psi_p(z)h\right) &=\psi_p\left(\Theta_{x,y}(z)\right)h =\psi_p\left(x\cdot \langle y,z \rangle_A\right)h\\ &=\psi_p(x)\psi_p(y)^*\psi_p(z)h =\psi^{(p)}\left(\Theta_{x,y}\right)\left(\psi_p(z)h\right).\end{aligned}$$ Since both $\psi^{(p)}\left(\Theta_{x,y}\right)$ and $\rho_p^\psi\left(\Theta_{x,y}\right)$ are linear and continuous, we conclude that they agree on $\overline{\psi_p\left(\mathbf{X}_p\right)\mathcal{H}}$. It remains to check that $\psi^{(p)}\left(\Theta_{x,y}\right)$ and $\rho_p^\psi\left(\Theta_{x,y}\right)$ agree on the orthogonal complement $\left(\psi_p\left(\mathbf{X}_p\right)\mathcal{H}\right)^\perp$. Making use of part (i), we see that the restriction $\rho_p^\psi\left(\Theta_{x,y}\right)|_{\left(\psi_p\left(\mathbf{X}_p\right)\mathcal{H}\right)^\perp}=0$. Since $$\big\langle \psi^{(p)}\left(\Theta_{x,y}\right)h,k\big\rangle_\C=\langle \psi_p(x)\psi_p(y)^*h,k\rangle_\C=\langle h, \psi_p(y)\psi_p(x)^*k\rangle_\C=0$$ for any $h\in \left(\psi_p\left(\mathbf{X}_p\right)\mathcal{H}\right)^\perp$ and $k\in \mathcal{H}$, we conclude that $\psi^{(p)}(\Theta_{x,y})|_{\left(\psi_p\left(\mathbf{X}_p\right)\mathcal{H}\right)^\perp}=0$ as well. This completes the proof of part (ii). We now prove part (iii). Let $q\in P$ and $a\in A$. If $q=e$, then $P_q^\psi=\mathrm{id}_\mathcal{H}$, and so $$\rho_e^\psi(a)P_q=\rho_e^\psi(a)=\rho_e^\psi\left(\iota_e^e(a)\right)=\rho_e^\psi\left(\iota_e^q(a)\right).$$ On the other hand, if $q\neq e$, then $P_q^\psi=\mathrm{proj}_{\overline{\psi_q\left(\mathbf{X}_q\right)\mathcal{H}}}$. Hence, both $\rho_e^\psi(a)P_q$ and $\rho_e^\psi(\iota_e^q(a))$ are zero on $\left(\psi_q\left(\mathbf{X}_q\right)\mathcal{H}\right)^\perp$. Since $\rho_q^\psi\left(\iota_e^q(a)\right)$ and $\rho_e^\psi(a)P_q^\psi$ are linear and continuous, whilst $$\begin{aligned} \rho_q^\psi\left(\iota_e^q(a)\right)\left(\psi_q(x)h\right) &=\psi_q\left(\iota_e^q(a)x\right)h =\psi_q\left(a\cdot x\right)h\\ &=\rho_e^\psi(a)\left(\psi_q(x)h\right) =\rho_e^\psi(a)P_q^\psi\left(\psi_q(x)h\right)\end{aligned}$$ for any $x\in \mathbf{X}_q$ and $h\in \mathcal{H}$, we see that $\rho_q^\psi\left(\iota_e^q(a)\right)$ and $\rho_e^\psi(a)P_q^\psi$ also agree on $ \overline{\psi_q\left(\mathbf{X}_q\right)\mathcal{H}}$. Thus, $\rho_q^\psi\left(\iota_e^q(a)\right)=\rho_e^\psi(a)P_q^\psi$. Now fix $p\in P\setminus \{e\}$ and $S\in \mathcal{L}_A\left(\mathbf{X}_p\right)$. Since $pq\neq e$, both $\rho_{pq}^\psi\left(\iota_p^{pq}(S)\right)$ and $\rho_{p}^\psi(S)P_{pq}^\psi$ are zero on the orthogonal complement $\left(\psi_{pq}\left(\mathbf{X}_{pq}\right)\mathcal{H}\right)^\perp$. Observe that for any $x\in \mathbf{X}_p$, $y\in \mathbf{X}_q$, and $h\in \mathcal{H}$, we have $$\begin{aligned} \rho_{pq}^\psi\left(\iota_p^{pq}(S)\right)\left(\psi_{pq}(xy)h\right) =\psi_{pq}\left(\iota_p^{pq}(S)(xy)\right)h =\psi_{pq}\left((Sx)y\right)h =\psi_p(Sx)\psi_q(y)h,\end{aligned}$$ which by part (i) is the same as $$\begin{aligned} \rho_p^\psi(S)\left(\psi_p(x)\psi_q(y)h\right) =\rho_p^\psi(S)\left(\psi_{pq}(xy)h\right) =\rho_p^\psi(S)P_{pq}^\psi\left(\psi_{pq}(xy)h\right).\end{aligned}$$ Since $\overline{\psi_{pq}\left(\mathbf{X}_{pq}\right)\mathcal{H}}=\overline{\psi_p\left(\mathbf{X}_p\right)\psi_q\left(\mathbf{X}_q\right)\mathcal{H}}$, whilst $\rho_{pq}^\psi\left(\iota_p^{pq}(S)\right)$ and $\rho_p^\psi(S)P_{pq}^\psi$ are linear and continuous, we conclude that $\rho_{pq}^\psi\left(\iota_p^{pq}(S)\right)$ and $\rho_p^\psi(S)P_{pq}^\psi$ are also equal on $\overline{\psi_{pq}\left(\mathbf{X}_{pq}\right)\mathcal{H}}$. Finally, we prove part (iv). Firstly, observe that the subspace $\mathcal{M}$ is $\rho_p^\psi$-invariant, since $\rho_p^\psi(S)\left(\psi_p(x)k\right)=\psi_p(Sx)k\in \mathcal{M}$ for any $S\in \mathcal{L}_A\left(\mathbf{X}_p\right)$, $x\in \mathbf{X}_p$, and $k\in \mathcal{K}$. Now suppose that $\psi_e|_\mathcal{K}$ is faithful. Since $\mathcal{L}_A\left(\mathbf{X}_p\right)$ acts faithfully on $\mathbf{X}_p$, the induced representation $\mathbf{X}_p\text{-}\mathrm{Ind}_A^{\mathcal{L}_A\left(\mathbf{X}_p\right)}\left(\psi_e|_\mathcal{K}\right):\mathcal{L}_A\left(\mathbf{X}_p\right)\rightarrow \mathcal{B}\left(\mathbf{X}_p\otimes_A \mathcal{K}\right)$ is faithful by [@MR1634408 Corollary 2.74]. Since $U$ implements a unitary equivalence between $\mathbf{X}_p\text{-}\mathrm{Ind}_A^{\mathcal{L}_A\left(\mathbf{X}_p\right)}\left(\psi_e|_\mathcal{K}\right)$ and $\rho_p^\psi|_\mathcal{M}$, and unitary equivalence preserves the faithfulness of representations, we conclude that $\rho_p^\psi|_\mathcal{M}$ is faithful. We now show what a product of projections from the collection $\{P_p^\psi:p\in P\}$ looks like. Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. Then for each $p,q\in P$, we have $$P_p^\psi P_q^\psi=P_{p\vee q}^\psi.$$ In particular, the projections $\{P_p^\psi:p\in P\}$ commute. Firstly, observe that part (i) of Proposition \[existence of rho map\] implies that $P_p^\psi =\rho_p^\psi\left(\mathrm{id}_{\mathbf{X}_p}\right)$ for any $p\in P\setminus \{e\}$. Next, we show that if $p\in P$ and $(e_i)_{i\in I}$ is the canonical approximate identity for the $C^*$-algebra $\mathcal{K}_A(\mathbf{X}_p)$, then 1. $\lim_{i\in I}(e_i x)=x$ for each $x\in \mathbf{X}_p$; 2. $\lim_{i\in I}\big(\rho_p^\psi(e_i)\big)=\rho_p^\psi\left(\mathrm{id}_{\mathbf{X}_p}\right)$ (converging in the strong operator topology). To see (i), fix $x\in \mathbf{X}_p$ and $\varepsilon>0$. Choose $x'\in \mathbf{X}_p$ so that $x=x'\cdot \langle x',x'\rangle_A$ by the Hewitt–Cohen–Blanchard factorisation theorem. Choose $i\in I$ such that for all $j\geq i$, $${\left\|e_j \Theta_{x',x'}-\Theta_{x',x'}\right\|}_{\mathcal{K}_A(\mathbf{X}_p)}<\frac{\varepsilon}{{\left\|x'\right\|}_{\mathbf{X}_p}+1}.$$ Thus, for all $j\geq i$, we have $$\begin{aligned} {\left\|e_j x-x\right\|}_{\mathbf{X}_p} &={\left\|e_j x'\cdot \langle x',x'\rangle_A -x'\cdot \langle x',x'\rangle_A\right\|}_{\mathbf{X}_p}\\ &={\left\|\left(e_j \Theta_{x',x'}-\Theta_{x',x'}\right)x'\right\|}_{\mathbf{X}_p} \leq {\left\|e_j \Theta_{x',x'}-\Theta_{x',x'}\right\|}_{\mathcal{K}_A(\mathbf{X}_p)}{\left\|x'\right\|}_{\mathbf{X}_p} <\varepsilon.\end{aligned}$$ Since $\varepsilon>0$ was arbitrary, we conclude that $\lim_{i\in I}(e_i x)=x$ for each $x\in \mathbf{X}_p$. We now move on to proving (ii). Fix $h\in \mathcal{H}$ and $\varepsilon>0$. If $h\in \left(\psi_p(\mathbf{X}_p)\mathcal{H}\right)^\perp$, then $$\rho_p^\psi(e_i)h=0=\rho_p^\psi\left(\mathrm{id}_{\mathbf{X}_p}\right)h$$ for each $i\in I$. Thus, $\lim_{i\in I}\big\|\rho_p^\psi(e_i)h-\rho_p^\psi\left(\mathrm{id}_{\mathbf{X}_p}\right)h\big\|_{\mathcal{H}}=0$. On the other hand, if $h\in \overline{\psi_p(\mathbf{X}_p)\mathcal{H}}$, then we can choose $x_1,\ldots, x_n\in \mathbf{X}_p$ and $h_1,\ldots, h_n\in \mathcal{H}$ such that $$\bigg\|h-\sum_{i=1}^n \psi_p(x_i)h_i\bigg\|_\mathcal{H}<\varepsilon/4.$$ Since ${\left\|e_i\right\|}_{\mathcal{L}_A(\mathbf{X}_p)}\leq 1$ for each $i\in I$ and $\rho_p^\psi$ is norm-decreasing, we see that $$\begin{aligned} \bigg\|\rho_p^\psi\left(e_i-\mathrm{id}_{\mathbf{X}_p}\right)\bigg(h-\sum_{i=1}^n \psi_p(x_i)h_i\bigg)\bigg\|_{\mathcal{H}} &\leq {\left\|\rho_p^\psi\left(e_i-\mathrm{id}_{\mathbf{X}_p}\right)\right\|}_{\mathcal{B}(\mathcal{H})}\bigg\|h-\sum_{i=1}^n \psi_p(x_i)h_i\bigg\|_{\mathcal{H}}\\ &\leq {\left\|e_i-\mathrm{id}_{\mathbf{X}_p}\right\|}_{\mathcal{L}_A(\mathbf{X}_p)}\bigg\|h-\sum_{i=1}^n \psi_p(x_i)h_i\bigg\|_{\mathcal{H}}\\ &\leq 2\bigg\|h-\sum_{i=1}^n \psi_p(x_i)h_i\bigg\|_{\mathcal{H}}\\ &<\varepsilon/2.\end{aligned}$$ By (i), for each $1\leq i \leq n$, we can choose $j_i\in I$ such that whenever $k\geq j_i$, $${\left\|e_k x_i-x_i\right\|}_{\mathbf{X}_p}<\frac{\varepsilon}{2n\left(\max_{1\leq i\leq n}{\left\|h_i\right\|}_\mathcal{H}+1\right)}.$$ As $I$ is directed, we can choose $m\in I$ such that $m\geq j_i$ for each $1\leq i \leq n$. Since $\psi_p$ is norm-decreasing, we see that for any $k\geq m$, $$\begin{aligned} \bigg\|\rho_p^\psi\left(e_k-\mathrm{id}_{\mathbf{X}_p}\right)\bigg(\sum_{i=1}^n \psi_p(x_i)h_i\bigg)\bigg\|_{\mathcal{H}} &=\bigg\|\sum_{i=1}^n \psi_p\left(\left(e_k-\mathrm{id}_{\mathbf{X}_p}\right)x\right)h_i\bigg\|_{\mathcal{H}}\\ &\leq \sum_{i=1}^n{\left\| \psi_p\left(\left(e_k-\mathrm{id}_{\mathbf{X}_p}\right)x\right)\right\|}_{\mathcal{B}(\mathcal{H})}{\left\|h_i\right\|}_{\mathcal{H}}\\ &\leq \sum_{i=1}^n{\left\|\left(e_k-\mathrm{id}_{\mathbf{X}_p}\right)x\right\|}_{\mathbf{X}_p}{\left\|h_i\right\|}_{\mathcal{H}}\\ &< \sum_{i=1}^n \frac{\varepsilon{\left\|h_i\right\|}_{\mathcal{H}}}{2n\left(\max_{1\leq i\leq n}{\left\|h_i\right\|}_\mathcal{H}+1\right)}\\ &\leq \varepsilon/2.\end{aligned}$$ Thus, for each $k\geq m$, $$\begin{aligned} \big\|\rho_p^\psi&(e_k)h-\rho_p^\psi\left(\mathrm{id}_{\mathbf{X}_p}\right)h\big\|_{\mathcal{H}}\\ &=\big\|\rho_p^\psi\left(e_k-\mathrm{id}_{\mathbf{X}_p}\right)h\big\|_{\mathcal{H}}\\ &\leq \bigg\|\rho_p^\psi\left(e_k-\mathrm{id}_{\mathbf{X}_p}\right)\bigg(h-\sum_{i=1}^n \psi_p(x_i)h_i\bigg)\bigg\|_{\mathcal{H}}+ \bigg\|\rho_p^\psi\left(e_k-\mathrm{id}_{\mathbf{X}_p}\right)\bigg(\sum_{i=1}^n \psi_p(x_i)h_i\bigg)\bigg\|_{\mathcal{H}}\\ &<\varepsilon. \end{aligned}$$ Since $\varepsilon>0$ was arbitrary, we conclude that $\lim_{i\in I}\big\|\rho_p^\psi(e_i)h-\rho_p^\psi\left(\mathrm{id}_{\mathbf{X}_p}\right)h\big\|_{\mathcal{H}}=0$ for each $h\in \mathcal{H}$. Thus, $\lim_{i\in I} \rho_p^\psi(e_i)=\rho_p^\psi\left(\mathrm{id}_{\mathbf{X}_p}\right)$ in the strong operator topology. Finally, we are ready to prove that $P_p^\psi P_q^\psi=P_{p\vee q}^\psi$ for every $p,q\in P$. Since $P_e^\psi=\mathrm{id}_\mathcal{H}$, the result is trivial when $p=e$ or $q=e$. Thus, we may as well suppose that $p,q\neq e$. Let $(e_i)_{i\in I}$ and $(f_j)_{j\in J}$ be the canonical approximate identities for $\mathcal{K}_A(\mathbf{X}_p)$ and $\mathcal{K}_A(\mathbf{X}_q)$ respectively. Then for any $i\in I$ and $j\in J$, Proposition \[existence of rho map\] and the Nica covariance of $\psi$ tell us that $$\begin{aligned} \rho_p^\psi(e_i)\rho_q^\psi(f_j) =\psi^{(p)}(e_i)\psi^{(q)}(f_j) &=\begin{cases} \psi^{(p\vee q)}\left(\iota_p^{p\vee q}(e_i)\iota_q^{p\vee q}(f_j)\right) & \text{if $p\vee q<\infty$}\\ 0 & \text{otherwise} \end{cases}\\ &=\begin{cases} \rho_{p\vee q}^\psi\left(\iota_p^{p\vee q}(e_i)\iota_q^{p\vee q}(f_j)\right) & \text{if $p\vee q<\infty$}\\ 0 & \text{otherwise} \end{cases}\\ &=\begin{cases} \rho_{p\vee q}^\psi\left(\iota_p^{p\vee q}(e_i)\right)\rho_{p\vee q}^\psi\left(\iota_q^{p\vee q}(f_j)\right) & \text{if $p\vee q<\infty$}\\ 0 & \text{otherwise} \end{cases}\\ &=\begin{cases} \rho_{p}^\psi\left(e_i\right)P_{p\vee q}^\psi\rho_{q}^\psi\left(f_j\right)P_{p\vee q}^\psi & \text{if $p\vee q<\infty$}\\ 0 & \text{otherwise.} \end{cases}\end{aligned}$$ Hence, by (ii), we have $$\begin{aligned} P_p^\psi P_q^\psi =\rho_p^\psi\left(\mathrm{id}_{\mathbf{X}_p}\right)\rho_q^\psi\left(\mathrm{id}_{\mathbf{X}_q}\right) &=\begin{cases} \rho_{p}^\psi\left(\mathrm{id}_{\mathbf{X}_p}\right)P_{p\vee q}^\psi\rho_{q}^\psi\left(\mathrm{id}_{\mathbf{X}_q}\right)P_{p\vee q}^\psi & \text{if $p\vee q<\infty$}\\ 0 & \text{otherwise} \end{cases}\\ &=\begin{cases} \rho_{p\vee q}^\psi\left(\iota_p^{p\vee q}\left(\mathrm{id}_{\mathbf{X}_p}\right)\right)\rho_{p\vee q}^\psi\left(\iota_q^{p\vee q}\left(\mathrm{id}_{\mathbf{X}_q}\right)\right) & \text{if $p\vee q<\infty$}\\ 0 & \text{otherwise} \end{cases}\\ &=\begin{cases} \rho_{p\vee q}^\psi\left(\iota_p^{p\vee q}\left(\mathrm{id}_{\mathbf{X}_p}\right)\iota_q^{p\vee q}\left(\mathrm{id}_{\mathbf{X}_q}\right)\right) & \text{if $p\vee q<\infty$}\\ 0 & \text{otherwise} \end{cases}\\ &=\begin{cases} \rho_{p\vee q}^\psi\left(\mathrm{id}_{\mathbf{X}_{p\vee q}}\right) & \text{if $p\vee q<\infty$}\\ 0 & \text{otherwise} \end{cases}\\ &=\begin{cases} P_{p\vee q}^\psi & \text{if $p\vee q<\infty$}\\ 0 & \text{otherwise.} \end{cases}\end{aligned}$$ Thus, $P_p^\psi P_q^\psi=P_{p\vee q}^\psi$ for each $p,q\in P$. Our aim is to use the projections defined in Definition \[basic projections\] to construct a collection of mutually orthogonal projections in $\mathcal{B}(\mathcal{H})$ that decompose the identity and commute with everything in $\psi_*(\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}})$. Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. Let $F$ be a finite subset of $P$. For each $C\subseteq F$, define $$Q_{C,F}^\psi:= P_{\bigvee C}^\psi\prod_{p\in F\setminus C}\left(\mathrm{id}_\mathcal{H}-P_p^\psi\right),$$ where, by convention, the product over the empty set is $\mathrm{id}_\mathcal{H}$. Whilst we have defined the projections $Q_{C,F}^\psi$, for every subset $C$ of $F$, we are particularly interested in the projections corresponding to so called initial segments of $F$. Let $(G,P)$ be a quasi-lattice ordered group. Let $F\subseteq P$ be a finite set. A subset $C\subseteq F$ is said to be an initial segment of $F$ if $\bigvee C<\infty$ and $C=\left\{t\in F:t\leq \bigvee C\right\}$. The next result shows how the projections $\{Q_{C,F}^\psi: \text{$C$ is an initial segment of $F$}\}$ and $\{P_p^\psi:p\in P\}$ interact with the operators $\{\psi_p(x):p\in P, \, x\in \mathbf{X}_p\}$. \[projections and psi\] Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. 1. Let $p,q\in P$ and $x\in \mathbf{X}_p$. Then $$P_q^\psi \psi_p(x)= \begin{cases} \psi_p(x)P_{p^{-1}(p\vee q)}^\psi & \text{if $p\vee q<\infty$}\\ 0 &\text{otherwise.} \end{cases}$$ 2. If $F\subseteq P$ is finite and $p\in F$, then $$Q_{C,F}^\psi \psi_p(x)= \begin{cases} Q_{C,F}^\psi \psi_p(x)P_{p^{-1}\left(\bigvee C\right)}^\psi & \text{if $p\leq\bigvee C$}\\ 0 &\text{otherwise} \end{cases}$$ for any initial segment $C$ of $F$. Fix $p,q\in P$ and $x\in \mathbf{X}_p$. If $p\vee q=\infty$, then $\psi_q(y)^*\psi_p(x)=0$ for any $y\in \mathbf{X}_q$. Hence, for any $h,g\in \mathcal{H}$, it follows that $$\begin{aligned} \langle \psi_q(y)h, \psi_p(x)g\rangle_\C =\langle h, \psi_q(y)^*\psi_p(x)g \rangle_\C=0.\end{aligned}$$ Thus, $\psi_p(\mathbf{X}_p)\mathcal{H}\subseteq \left(\psi_q(\mathbf{X}_q)\mathcal{H}\right)^\perp$, and so $P_q^\psi \psi_p(x)=0$. Now suppose that $p\vee q<\infty$. If $p=p\vee q$, then $p\geq q$, and so $$P_q^\psi\psi_p(x)=\psi_p(x)=\psi_p(x)P_e^\psi=\psi_p(x)P_{p^{-1}(p\vee q)}^\psi.$$ If $p\neq p\vee q$, then for any $y\in \mathbf{X}_{p^{-1}(p \vee q)}$ and $h\in \mathcal{H}$, we have $$\begin{aligned} \psi_p(x)P_{p^{-1}(p \vee q)}^\psi\psi_{p^{-1}(p\vee q)}(y)h &=\psi_p(x)\psi_{p^{-1}(p\vee q)}(y)h\\ &=\psi_{p\vee q}(xy)h =P_q^\psi\psi_{p\vee q}(xy)h =P_q^\psi\psi_p(x)\psi_{p^{-1}(p\vee q)}(y)h.\end{aligned}$$ Consequently, $\psi_p(x)P_{p^{-1}(p \vee q)}^\psi$ and $P_q^\psi\psi_p(x)$ agree on $\overline{\psi_{p^{-1}(p\vee q)}(\mathbf{X}_{p^{-1}(p\vee q)})\mathcal{H}}$. It remains to check that they agree on the orthogonal complement $\left(\psi_{p^{-1}(p\vee q)}(\mathbf{X}_{p^{-1}(p\vee q)})\mathcal{H}\right)^\perp$. Let $f\in \left(\psi_{p^{-1}(p\vee q)}(\mathbf{X}_{p^{-1}(p\vee q)})\mathcal{H}\right)^\perp$. We need to show that $P_q^\psi\psi_p(x)f=0$. It suffices to show that $\psi_p(x)f\in \left(\psi_q(\mathbf{X}_q)\mathcal{H}\right)^\perp$: for any $y\in \mathbf{X}_q$ and $h\in \mathcal{H}$, we have $$\begin{aligned} \langle \psi_p(x)f,\psi_q(y)h \rangle_\C &=\langle f, \psi_p(x)^*\psi_q(y)h \rangle_\C\\ &\in \left\langle f, {\overline{\mathrm{span}}}\{\psi_{p^{-1}(p\vee q)}(\mathbf{X}_{p^{-1}(p\vee q)})\psi_{q^{-1}(p\vee q)}(\mathbf{X}_{q^{-1}(p\vee q)})^*\}\mathcal{H}\right\rangle_\C\\ &\subseteq \left\langle f, \psi_{p^{-1}(p\vee q)}(\mathbf{X}_{p^{-1}(p\vee q)})\mathcal{H}\right\rangle_\C\\ &=\{0\}.\end{aligned}$$ This completes the proof of (i). We now prove part (ii). Fix a finite set $F\subseteq P$ with $p\in F$. Let $C$ be an initial segment of $F$. If $p\leq \bigvee C$, then $p \vee \left(\bigvee C\right)=\bigvee C<\infty$. By part (i) it follows that $$Q_{C,F}^\psi \psi_p(x)=Q_{C,F}^\psi P_{\bigvee C}^\psi \psi_p(x)=Q_{C,F}^\psi \psi_p(x)P_{p^{-1}\left(p\vee \left(\bigvee C\right)\right)}^\psi=Q_{C,F}^\psi \psi_p(x)P_{p^{-1} \left(\bigvee C\right)}^\psi.$$ On the other hand, suppose that $p\not \leq \bigvee C$. Since $p\in F$, this implies that $C\neq F$. Moreover, since $C$ is an initial segment of $F$, this forces $p\in F\setminus C$. Therefore, $$Q_{C,F}^\psi \psi_p(x)=Q_{C,F}^\psi \big(\mathrm{id}_\mathcal{H}-P_p^\psi\big) \psi_p(x)=0.$$ This completes the proof of part (ii). Using the previous result we can show that every element of $\psi_*(\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}})$ commutes with the projections $\{P_p^\psi:p\in P\}$ and $\{Q_{C,F}^\psi: C\subseteq F\}$. \[projections commute with core\] For any $q\in P$, the projection $P_q^\psi$ commutes with every element of $\psi_*(\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}})$. In particular, if $F\subseteq P$ is finite and $C\subseteq F$, then $Q_{C,F}^\psi$ commutes with every element of $\psi_*(\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}})$. Since $\psi_*(\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}})={\overline{\mathrm{span}}}\{\psi_p(\mathbf{X}_p)\psi_p(\mathbf{X}_p)^*\}$, it suffices to show that $$P_q^\psi\psi_p(x)\psi_p(y)^*=\psi_p(x)\psi_p(y)^*P_q^\psi$$ for each $x,y\in \mathbf{X}_p$. Via two applications of Lemma \[projections and psi\], we see that $$\begin{aligned} P_q^\psi\psi_p(x)\psi_p(y)^* &= \begin{cases} \psi_p(x)P_{p^{-1}(p\vee q)}^\psi\psi_p(y)^* & \text{if $p\vee q<\infty$}\\ 0 &\text{otherwise} \end{cases}\\ &= \begin{cases} \psi_p(x)\big(\psi_p(y)P_{p^{-1}(p\vee q)}^\psi\big)^* & \text{if $p\vee q<\infty$}\\ 0 &\text{otherwise} \end{cases}\\ &=\psi_p(x)\big(P_q^\psi\psi_p(y)\big)^*\\ &=\psi_p(x)\psi_p(y)^*P_q^\psi,\end{aligned}$$ as required. We now show that the projections $\{Q_{C,F}^\psi: \text{$C$ is an initial segment of $F$}\}$ are mutually orthogonal and decompose the identity operator on $\mathcal{H}$. Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$. Let $F\subseteq P$ be finite. Then 1. if $C\subseteq F$ is not an initial segment of $F$, then $Q_{C,F}^\psi=0$; 2. $\{Q_{C,F}^\psi:C\subseteq F \text{ is an initial segment of } F\}$ is a decomposition of the identity on $\mathcal{H}$ into mutually orthogonal projections. Suppose $C\subseteq F$ is not an initial segment of $F$. If $\bigvee C=\infty$, then $$Q_{C,F}^\psi=Q_{C,F}^\psi P_{\bigvee C}^\psi=Q_{C,F}^\psi P_\infty^\psi=0.$$ Alternatively, $\bigvee C<\infty$ and $C\neq \{t\in F:t\leq \bigvee C\}$. Thus, $C\neq F$. Choose $t\in F\setminus C$ with $t\leq \bigvee C$. Since $t \vee \left(\bigvee C\right)=\bigvee C$, we see that $$\begin{aligned} Q_{C,F}^\psi =Q_{C,F}^\psi P_{\bigvee C}^\psi(\mathrm{id}_\mathcal{H}-P_t^\psi) &=Q_{C,F}^\psi\big(P_{\bigvee C}^\psi- P_{\bigvee C}^\psi P_t^\psi\big)\\ &=Q_{C,F}^\psi\big(P_{\bigvee C}^\psi-P_{t\vee \left(\bigvee C\right)}^\psi\big) =Q_{C,F}^\psi\big(P_{\bigvee C}^\psi-P_{\bigvee C}^\psi\big) =0.\end{aligned}$$ Thus, $Q_{C,F}^\psi=0$, which proves part (i). We now prove part (ii). Since $Q_{C,F}^\psi=0$ whenever $C$ is not an initial segment of $F$, it suffices to show that $\{Q_{C,F}^\psi:C\subseteq F\}$ is a decomposition of the identity into mutually orthogonal projections. Firstly, we show orthogonality. Suppose $C,D\subseteq F$ are distinct. Without loss of generality, we may assume that $D\setminus C\neq \emptyset$. Thus, $C\neq F$ and we can choose $t\in D\setminus C$. Since $t \vee \left(\bigvee D\right)=\bigvee D$, we have $$\begin{aligned} Q_{C,F}^\psi Q_{D,F}^\psi =Q_{C,F}^\psi P_{\bigvee C}^\psi \big(\mathrm{id}_\mathcal{H} -P_t^\psi\big)P_{\bigvee D}^\psi Q_{D,F}^\psi &=Q_{C,F}^\psi \big(P_{\bigvee C}^\psi -P_{t\vee \left(\bigvee C\right)}^\psi\big)P_{\bigvee D}^\psi Q_{D,F}^\psi\\ &=Q_{C,F}^\psi \big(P_{\left(\bigvee C\right)\vee \left(\bigvee D\right)}^\psi -P_{t\vee \left(\bigvee C\right) \vee \left(\bigvee D\right)}^\psi\big) Q_{D,F}^\psi\\ &=Q_{C,F}^\psi \big(P_{\left(\bigvee C\right)\vee \left(\bigvee D\right)}^\psi -P_{\left(\bigvee C\right) \vee \left(\bigvee D\right)}^\psi\big) Q_{D,F}^\psi\\ &=0. \end{aligned}$$ It remains to check that $\sum_{C\subseteq F}Q_{C,F}^\psi=\mathrm{id}_\mathcal{H}$. To prove this, we will use induction on $|F|$. When $|F|=0$ we have $$\begin{aligned} \sum_{C\subseteq F}Q_{C,F}^\psi=Q_{\emptyset,\emptyset}^\psi=P_{\bigvee \emptyset}^\psi=P_e^\psi=\mathrm{id}_\mathcal{H}.\end{aligned}$$ Now let $n\geq 0$ and suppose that $\sum_{C\subseteq F}Q_{C,F}^\psi=\mathrm{id}_\mathcal{H}$ whenever $F\subseteq P$ and $|F|=n$. Fix $F'\subseteq P$ with $|F'|=n+1$. Then, for any $y\in F'$, we have $$\begin{aligned} \sum_{C\subseteq F'}Q_{C,F'}^\psi &=\sum_{C\subseteq F', \, y\in C}Q_{C,F'}^\psi+\sum_{C\subseteq F', \, y\not\in C}Q_{C,F'}^\psi\\ &=\sum_{C\subseteq F', \, y\in C}P_{\bigvee C}^\psi\prod_{p\in F'\setminus C}\big(\mathrm{id}_\mathcal{H}-P_p^\psi\big)+\sum_{C\subseteq F', \, y\not\in C}P_{\bigvee C}^\psi\prod_{p\in F'\setminus C}\big(\mathrm{id}_\mathcal{H}-P_p^\psi\big)\\ &=\sum_{C\subseteq F'\setminus \{y\}}P_{\bigvee (C\cup \{y\})}^\psi\prod_{p\in (F'\setminus \{y\})\setminus C}\big(\mathrm{id}_\mathcal{H}-P_p^\psi\big)\\ & \quad \quad \quad \quad \quad \quad +\sum_{C\subseteq F'\setminus \{y\}}P_{\bigvee C}^\psi\prod_{p\in (F'\setminus \{y\})\setminus C}\big(\mathrm{id}_\mathcal{H}-P_p^\psi\big)\big(\mathrm{id}_\mathcal{H}-P_y^\psi\big)\\ &=\sum_{C\subseteq F'\setminus \{y\}}P_{\bigvee C}^\psi P_y^\psi\prod_{p\in (F'\setminus \{y\})\setminus C}\big(\mathrm{id}_\mathcal{H}-P_p^\psi\big)\\ & \quad \quad \quad \quad \quad \quad +\sum_{C\subseteq F'\setminus \{y\}}P_{\bigvee C}^\psi\prod_{p\in (F'\setminus \{y\})\setminus C}\big(\mathrm{id}_\mathcal{H}-P_p^\psi\big)\big(\mathrm{id}_\mathcal{H}-P_y^\psi\big)\\ &=\big(P_y^\psi+\big(\mathrm{id}_\mathcal{H}-P_y^\psi\big)\big)\sum_{C\subseteq F'\setminus \{y\}}P_{\bigvee C}^\psi \prod_{p\in (F'\setminus \{y\})\setminus C}\big(\mathrm{id}_\mathcal{H}-P_p^\psi\big)\\ &=\sum_{C\subseteq F'\setminus \{y\}}P_{\bigvee C}^\psi \prod_{p\in (F'\setminus \{y\})\setminus C}\big(\mathrm{id}_\mathcal{H}-P_p^\psi\big)\\ &=\sum_{C\subseteq F'\setminus \{y\}}Q_{C,F'\setminus \{y\}}^\psi\\ &=\mathrm{id}_\mathcal{H},\end{aligned}$$ where the last equality follows from applying the inductive hypothesis to $F'\setminus \{y\}$. Putting these results together we get an expression for the norm of an element in $\psi_*(\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}})$ that does not depend on the representation $\psi$. \[norm of things in core\] Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. If $Z:=\sum_k \psi_{p_k}(x_k)\psi_{p_k}(y_k)^*\in \psi_*(\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}})$ is a finite sum, then $${\left\|Z\right\|}_{\mathcal{B}(\mathcal{H})}=\max\bigg\{\bigg\|Q_{C,F}^\psi \rho_{\bigvee C}^\psi\bigg(\sum_k \iota_{p_k}^{\bigvee C}\left(\Theta_{x_k,y_k}\right)\bigg)\bigg\|_{\mathcal{B}(\mathcal{H})}: \text{$C$ is an initial segment of $F$}\bigg\}$$ for any finite set $F\subseteq P$ containing each $p_k$. Furthermore, if the representation $$\begin{aligned} A\ni a \mapsto \psi_e(a)\prod_{t\in K}\big(\mathrm{id}_\mathcal{H}-P_t^\psi\big)\in \mathcal{B}(\mathcal{H})\end{aligned}$$ is faithful for any finite set $K\subseteq P\setminus \{e\}$, then $${\left\|Z\right\|}_{\mathcal{B}(\mathcal{H})}=\max\bigg\{\bigg\|\sum_k \iota_{p_k}^{\bigvee C}\left(\Theta_{x_k,y_k}\right)\bigg\|_{\mathcal{L}_A\left(\mathbf{X}_{\bigvee C}\right)}: \text{$C$ is an initial segment of $F$}\bigg\}.$$ Let $F$ be a finite subset of $P$ containing each $p_k$. Since $Q_{C,F}^\psi$ commutes with $Z$ for each $C\subseteq F$ (by Proposition \[projections commute with core\]) and $\{Q_{C,F}^\psi: \text{$C$ is an initial segment of $F$}\}$ is an orthogonal decomposition of the identity, we have that $${\left\|Z\right\|}_{\mathcal{B}(\mathcal{H})}=\max\left\{{\left\|Q_{C,F}^\psi Z\right\|}_{\mathcal{B}(\mathcal{H})}: \text{$C$ is an initial segment of $F$}\right\}.$$ However, for any initial segment $C$ of $F$, Lemma \[projections and psi\] shows that $$\begin{aligned} Q_{C,F}^\psi Z =Q_{C,F}^\psi \sum_k \psi_{p_k}(x_k)\psi_{p_k}(y_k)^* =Q_{C,F}^\psi \sum_{k:p_k\leq \bigvee C} \psi_{p_k}(x_k)\psi_{p_k}(y_k)^*P_{\bigvee C}^\psi\end{aligned}$$ Using parts (ii) and (iii) of Proposition \[existence of rho map\], this is equal to $$\begin{aligned} Q_{C,F}^\psi \sum_{k:p_k\leq \bigvee C} \rho_{p_k}^\psi\left(\Theta_{x_k,y_k}\right)P_{\bigvee C}^\psi &=Q_{C,F}^\psi \sum_{k:p_k\leq \bigvee C} \rho_{\bigvee C}^\psi\big(\iota_{p_k}^{\bigvee C}\left(\Theta_{x_k,y_k}\right)\big)\\ &=Q_{C,F}^\psi \rho_{\bigvee C}^\psi\bigg(\sum_k\iota_{p_k}^{\bigvee C}\left(\Theta_{x_k,y_k}\right)\bigg).\end{aligned}$$ Now suppose that for any finite set $K\subseteq P\setminus \{e\}$, the representation $$\begin{aligned} A\ni a \mapsto \psi_e(a)\prod_{t\in K}\big(\mathrm{id}_\mathcal{H}-P_t^\psi\big)\in \mathcal{B}(\mathcal{H})\end{aligned}$$ is faithful. To complete the proof we will show that the representation $$\mathcal{L}_A\left(\mathbf{X}_{\bigvee C}\right)\ni T \mapsto Q_{C,F}^\psi \rho_{\bigvee C}^\psi(T)\in \mathcal{B}(\mathcal{H})$$ is faithful, and hence $$\begin{aligned} {\left\|Z\right\|}_{\mathcal{B}(\mathcal{H})}&=\max\left\{{\left\|Q_{C,F}^\psi Z\right\|}_{\mathcal{B}(\mathcal{H})}: \text{$C$ is an initial segment of $F$}\right\}\\ &=\max\bigg\{\bigg\|Q_{C,F}^\psi \rho_{\bigvee C}^\psi\bigg(\sum_k\iota_{p_k}^{\bigvee C}\left(\Theta_{x_k,y_k}\right)\bigg)\bigg\|_{\mathcal{B}(\mathcal{H})}: \text{$C$ is an initial segment of $F$}\bigg\}\\ &=\max\bigg\{\bigg\|\sum_k\iota_{p_k}^{\bigvee C}\left(\Theta_{x_k,y_k}\right)\bigg\|_{\mathcal{L}_A\left(\mathbf{X}_{\bigvee C}\right)}: \text{$C$ is an initial segment of $F$}\bigg\}.\end{aligned}$$ Let $\mathcal{K}:=\prod_{\left\{t\in F\setminus C:t\vee \left(\bigvee C\right) <\infty\right\}}\left(\mathrm{id}_\mathcal{H}-P_{\left(\bigvee C\right)^{-1}\left(t\vee \left(\bigvee C\right)\right)}^\psi\right)\mathcal{H}$. Since $\psi_e(a)P_p^\psi=P_p^\psi \psi_e(a)$, for each $a\in A$ and $p\in P$, by Lemma \[projections and psi\], we see that $\mathcal{K}$ is a $\psi_e$-invariant subspace of $\mathcal{H}$. As $C$ is an initial segment of $F$, if $t\in F\setminus C$ with $t\vee \left(\bigvee C\right) <\infty$, then $t\not\leq \bigvee C$, and so $\left(\bigvee C\right)^{-1}\left(t\vee\left(\bigvee C\right)\right)\neq e$. Thus, $\psi_e |_\mathcal{K}$ is faithful. Therefore, by Proposition \[existence of rho map\] it follows that $\mathcal{M}:=\overline{\psi_{\bigvee C}(\mathbf{X}_{\bigvee C})\mathcal{K}}$ is a $\rho_{\bigvee C}^\psi$-invariant subspace and $\rho_{\bigvee C}^\psi|_\mathcal{M}$ is faithful. To show that the map $\mathcal{L}_A\left(\mathbf{X}_{\bigvee C}\right)\ni T \mapsto Q_{C,F}^\psi \rho_{\bigvee C}^\psi(T)\in \mathcal{B}(\mathcal{H})$ is faithful, it remains to check that $\mathcal{M}\subseteq Q_{C,F}^\psi\mathcal{H}$. Lemma \[projections and psi\] tells us that for any $x\in \mathbf{X}_{\bigvee C}$, we have $$\begin{aligned} Q_{C,F}^\psi\psi_{\bigvee C}(x) =P_{\bigvee C}^\psi\hspace{-0.2em}\prod_{t\in F\setminus C}\big(\mathrm{id}_\mathcal{H}-P_t^\psi\big)\psi_{\bigvee C}(x) =\psi_{\bigvee C}(x)\hspace{-0.7em}\prod_{\substack{t\in F\setminus C,\\ t\vee \hspace{-1em}\left(\bigvee C\right)<\infty}}\left(\mathrm{id}_\mathcal{H}-P_{\left(\bigvee C\right)^{-1}\left(t\vee \left(\bigvee C\right)\right)}^\psi\right).\end{aligned}$$ Therefore, $Q_{C,F}^\psi$ is the identity on $$\overline{\psi_{\bigvee C}(\mathbf{X}_{\bigvee C})\prod_{\substack{t\in F\setminus C,\\ t\vee \left(\bigvee C\right) <\infty}}\left(\mathrm{id}_\mathcal{H}-P_{\left(\bigvee C\right)^{-1}\left(t\vee \left(\bigvee C\right)\right)}^\psi\right)\mathcal{H}} =\overline{\psi_{\bigvee C}\left(\mathbf{X}_{\bigvee C}\right)\mathcal{K}} =\mathcal{M},$$ and so $\mathcal{M}\subseteq Q_{C,F}^\psi\mathcal{H}$. Now that we have an expression for the norm of elements in $\psi_*(\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}})$, we are ready to show that the expectation $E_{\delta_\mathbf{X}}$ can be implemented spatially. Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ a compactly aligned product system over $P$ with coefficient algebra $A$. Let $\psi:\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H})$ be a Nica covariant representation of $\mathbf{X}$ on a Hilbert space $\mathcal{H}$. Suppose that for any finite set $K\subseteq P\setminus \{e\}$, the representation $$\begin{aligned} A\ni a \mapsto \psi_e(a)\prod_{t\in K}\big(\mathrm{id}_\mathcal{H}-P_t^\psi\big)\in \mathcal{B}(\mathcal{H})\end{aligned}$$ is faithful. Then 1. $\psi_*|_{\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}}}$ is faithful; and 2. there exists a linear map $E_\psi:\psi_*\left(\mathcal{NT}_\mathbf{X}\right)\rightarrow \psi_*(\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}})$ such that $$E_\psi \circ \psi_*= \psi_* \circ E_{\delta_\mathbf{X}}.$$ Firstly, we prove that the restriction of $\psi_*$ to $\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}}$ is faithful. Fix a finite sum $Z:=\sum_k i_{\mathbf{X}_{p_k}}(x_k)i_{\mathbf{X}_{p_k}}(y_k)^*\in \mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}}$. Let $\sigma:\mathcal{NT}_\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H}')$ be a faithful representation. Thus, $\sigma\circ i_\mathbf{X}$ is a Nica covariant representation of $\mathbf{X}$ on $\mathcal{H}'$. For any finite set $F\subseteq P$ containing each $p_k$, two applications of Lemma \[norm of things in core\] show that $$\begin{aligned} \|&Z\|_{\mathcal{NT}_\mathbf{X}} = {\left\|\sigma(Z)\right\|}_{\mathcal{B}(\mathcal{H}')} = \bigg\|\sum_k (\sigma\circ i_\mathbf{X})_{p_k}(x_k)(\sigma\circ i_\mathbf{X})_{p_k}(y_k)^*\bigg\|_{\mathcal{B}(\mathcal{H}')}\\ &= \max\bigg\{\bigg\|Q_{C,F}^{\sigma \circ i_\mathbf{X}} \rho_{\bigvee C}^{\sigma \circ i_\mathbf{X}}\bigg(\sum_k \iota_{p_k}^{\bigvee C}\left(\Theta_{x_k,y_k}\right)\bigg)\bigg\|_{\mathcal{B}(\mathcal{H}')}:\text{$C$ is an initial segment of $F$}\bigg\}\\ &\leq \max\bigg\{\bigg\|\sum_k \iota_{p_k}^{\bigvee C}\left(\Theta_{x_k,y_k}\right)\bigg\|_{\mathcal{L}_A\left(\mathbf{X}_{\bigvee C}\right)}: \text{$C$ is an initial segment of $F$}\bigg\}\\ &=\bigg\|\sum_k \psi_{p_k}(x_k)\psi_{p_k}(y_k)^*\bigg\|_{\mathcal{B}(\mathcal{H})}\\ &={\left\|\psi_*(Z)\right\|}_{\mathcal{B}(\mathcal{H})},\end{aligned}$$ where we used the fact that each $Q_{C,F}^{\sigma \circ i_\mathbf{X}}$ is a projection and each $*$-homomorphism $\rho_{\bigvee C}^{\sigma \circ i_\mathbf{X}}$ is norm-decreasing. Thus, $\psi_*|_{\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}}}$ is faithful. Next we prove part (ii). We first show that for any finite sum $\sum_k \psi_{p_k}(x_k)\psi_{q_k}(y_k)^*$, we have $$\label{norm estimate onto core} \begin{aligned} \bigg\|\sum_{k:p_k=q_k}\psi_{p_k}(x_k)\psi_{p_k}(y_k)^*\bigg\|_{\mathcal{B}(\mathcal{H})}\leq \bigg\|\sum_k \psi_{p_k}(x_k)\psi_{q_k}(y_k)^*\bigg\|_{\mathcal{B}(\mathcal{H})}. \end{aligned}$$ Let $F\subseteq P$ be the finite set consisting of each $p_k$ and $q_k$. By Lemma \[norm of things in core\], there exists an initial segment $C$ of $F$ such that $$\bigg\|\sum_{k:p_k=q_k}\psi_{p_k}(x_k)\psi_{p_k}(y_k)^*\bigg\|_{\mathcal{B}(\mathcal{H})}=\bigg\|\sum_{k:p_k=q_k}\iota_{p_k}^{\bigvee C}(\Theta_{x_k,y_k})\bigg\|_{\mathcal{L}_A\left(\mathbf{X}_{\bigvee C}\right)}.$$ For each $s,t\in C$ with $s\neq t$ and $\left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)<\infty$ define $$\begin{aligned} \beta_{s,t} := \begin{cases} s\left(\left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)\right) &\text{if $s^{-1}\left(\bigvee C\right)<\left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)$}\\ t\left(\left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)\right) &\text{otherwise.} \end{cases}\end{aligned}$$ Observe that for any $s,t\in C$ with $s\neq t$ and $\left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)<\infty$ we have $$s\big(\big(s^{-1}\big(\bigvee C\big)\big)\vee \big(t^{-1}\big(\bigvee C\big)\big)\big)\geq s\big(s^{-1}\big(\bigvee C\big)\big)=\bigvee C$$ and $$t\big(\big((s^{-1}\big(\bigvee C\big)\big)\vee \big(t^{-1}\big(\bigvee C\big)\big)\big)\geq t\big(t^{-1}\big(\bigvee C\big)\big)=\bigvee C.$$ Thus, $\beta_{s,t}\geq \bigvee C$. Hence, $P_{\bigvee C}^\psi P^\psi_{\beta_{s,t}}=P^\psi_{\left(\bigvee C\right) \vee \beta_{s,t}}=P^\psi_{\beta_{s,t}}$, and we can define a projection $$R_{C,F}^\psi:=Q_{C,F}^\psi\hspace{-0.7em}\prod_{\substack{s,t\in C, \, s\neq t, \\ \left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)<\infty}} \hspace{-1em}\big(P^\psi_{\bigvee C}-P^\psi_{\beta_{s,t}}\big).$$ We claim that $$R_{C,F}^\psi\bigg(\sum_k \psi_{p_k}(x_k)\psi_{q_k}(y_k)^*\bigg)R_{C,F}^\psi=R_{C,F}^\psi\sum_{k:p_k=q_k}\psi_{p_k}(x_k)\psi_{p_k}(y_k)^*.$$ Since $R_{C,F}^\psi$ commutes with $\sum_{k:p_k=q_k}\psi_{p_k}(x_k)\psi_{p_k}(y_k)^*$ by Proposition \[projections commute with core\], it suffices to show that $R_{C,F}^\psi\psi_p(x)\psi_q(y)^*R_{C,F}^\psi$=0 whenever $x\in \mathbf{X}_p$, $y\in \mathbf{X}_q$, with $p,q\in F$ and $p\neq q$. Firstly, if $p\not \in C$ or $q\not \in C$, then $p\not \leq \bigvee C$ or $q\not \leq \bigvee C$ (since $C$ is an initial segment of $F$), and so by Lemma \[projections and psi\] we have $Q_{C,F}^\psi\psi_p(x)=0$ or $Q_{C,F}^\psi\psi_q(y)=0$. Consequently, $$\begin{aligned} R_{C,F}^\psi\psi_p(x)\psi_q(y)^*R_{C,F}^\psi &=R_{C,F}^\psi Q_{C,F}^\psi\psi_p(x)\psi_q(y)^*Q_{C,F}^\psi R_{C,F}^\psi\\ &=R_{C,F}^\psi Q_{C,F}^\psi\psi_p(x)(Q_{C,F}^\psi\psi_q(y))^*R_{C,F}^\psi\\ &=0.\end{aligned}$$ Alternatively, if $p,q\leq \bigvee C$ and $\left(p^{-1}\left(\bigvee C\right)\right)\vee \left(q^{-1}\left(\bigvee C\right)\right)=\infty$, then $$P^\psi_{p^{-1}\left(\bigvee C\right)}P^\psi_{q^{-1}\left(\bigvee C\right)} =0.$$ Hence, Lemma \[projections and psi\] tells us that $$\begin{aligned} R_{C,F}^\psi\psi_p(x)\psi_q(y)^*R_{C,F}^\psi &=R_{C,F}^\psi Q_{C,F}^\psi\psi_p(x)\psi_q(y)^*Q_{C,F}^\psi R_{C,F}^\psi\\ &=R_{C,F}^\psi Q_{C,F}^\psi\psi_p(x)P^\psi_{p^{-1}\left(\bigvee C\right)}P^\psi_{q^{-1}\left(\bigvee C\right)} \psi_q(y)^*Q_{C,F}^\psi R_{C,F}^\psi =0.\end{aligned}$$ With this in mind, suppose that $p,q\in C$ and $\left(p^{-1}\left(\bigvee C\right)\right)\vee \left(q^{-1}\left(\bigvee C\right)\right)<\infty$. Since $p$ and $q$ are distinct, it follows that either $p^{-1}\left(\bigvee C\right)<\left(p^{-1}\left(\bigvee C\right)\right)\vee \left(q^{-1}\left(\bigvee C\right)\right)$ or $q^{-1}\left(\bigvee C\right)<\left(p^{-1}\left(\bigvee C\right)\right)\vee \left(q^{-1}\left(\bigvee C\right)\right)$. By taking adjoints, we may assume, without loss of generality, that $p^{-1}<\left(p^{-1}\left(\bigvee C\right)\right)\vee \left(q^{-1}\left(\bigvee C\right)\right)$. Therefore, $$\beta_{p,q}=p\big(\big(p^{-1}\big(\bigvee C\big)\big)\vee \big(q^{-1}\big(\bigvee C\big)\big)\big).$$ Consequently, an application of Lemma \[projections and psi\] shows that $$\begin{aligned} R_{C,F}^\psi&\psi_p(x)\psi_q(y)^*R_{C,F}^\psi\\ &=R_{C,F}^\psi \big(P^\psi_{\bigvee C}-P^\psi_{\beta_{p,q}}\big)\psi_p(x)\psi_q(y)^* P^\psi_{\bigvee C}R_{C,F}^\psi\\ &=R_{C,F}^\psi\psi_p(x)\big(P^\psi_{p^{-1}\left(\bigvee C\right)}-P^\psi_{p^{-1}\beta_{p,q}}\big)P^\psi_{q^{-1}\left(\bigvee C\right)}\psi_q(y)^*R_{C,F}^\psi\\ &=R_{C,F}^\psi\psi_p(x)\big(P^\psi_{p^{-1}\left(\bigvee C\right)}-P^\psi_{\left(p^{-1}\left(\bigvee C\right)\right)\vee \left(q^{-1}\left(\bigvee C\right)\right)}\big)P^\psi_{q^{-1}\left(\bigvee C\right)}\psi_q(y)^*R_{C,F}^\psi\\ &=0. \end{aligned}$$ Additionally, we claim that the representation $$\mathcal{L}_A\left(\mathbf{X}_{\bigvee C}\right)\ni T \mapsto R_{C,F}^\psi\rho_{\bigvee C}^\psi(T)\in \mathcal{B}(\mathcal{H})$$ is faithful. Let $$\mathcal{K}:=\hspace{-1em}\prod_{\substack{p\in F\setminus C, \, s,t\in C,\\ p\vee \left(\bigvee C\right)<\infty, \, s\neq t,\\ \left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)<\infty}}\hspace{-1em}\left(\mathrm{id}_\mathcal{H}-P_{\left(\bigvee C\right)^{-1}\left(p\vee \left(\bigvee C\right)\right)}^\psi\right) \left(\mathrm{id}_\mathcal{H}-P^\psi_{\left(\bigvee C\right)^{-1}\beta_{s,t}}\right)\mathcal{H}.$$ As $\psi_e(a)P_q^\psi=P_q^\psi \psi_e(a)$, for each $a\in A$ and $q\in P$, by Lemma \[projections and psi\], we see that $\mathcal{K}$ is a $\psi_e$-invariant subspace of $\mathcal{H}$. Since $C$ is an initial segment of $F$, if $p\in F\setminus C$ with $p\vee\left(\bigvee C\right) <\infty$, then $p\not\leq \bigvee C$, and so $\left(\bigvee C\right)^{-1}\left(p\vee \left(\bigvee C\right)\right)\neq e$. We also claim that for any $s,t\in C$ with $s\neq t$ and $\left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)<\infty$, we have $\left(\bigvee C\right)^{-1}\beta_{s,t}\neq e$. Firstly, if $s^{-1}\left(\bigvee C\right)< \left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)$, then $$\begin{aligned} \big(\bigvee C\big)^{-1}\beta_{s,t} &=\big(\bigvee C\big)^{-1}s\big(\big(s^{-1}\big(\bigvee C\big)\big)\vee \big(t^{-1}\big(\bigvee C\big)\big)\big)\\ &=\big(s^{-1}\big(\bigvee C\big)\big)\big(\big(s^{-1}\big(\bigvee C\big)\big)\vee \big(t^{-1}\big(\bigvee C\big)\big)\big) \neq e.\end{aligned}$$ On the other hand, if $s^{-1}\left(\bigvee C\right)=\left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)$, then $$\big(\bigvee C\big)^{-1}\beta_{s,t}=\big(\bigvee C\big)^{-1}ts^{-1}\big(\bigvee C\big)\neq e$$ since $s\neq t$. Thus, $\psi_e |_\mathcal{K}$ is faithful. Hence, by Proposition \[existence of rho map\] it follows that the subspace $\mathcal{M}:=\overline{\psi_{\bigvee C}(\mathbf{X}_{\bigvee C})\mathcal{K}}$ is $\rho_{\bigvee C}^\psi$-invariant and $\rho_{\bigvee C}^\psi|_\mathcal{M}$ is faithful. To see that the map $\mathcal{L}_A\left(\mathbf{X}_{\bigvee C}\right)\ni T \mapsto R_{C,F}^\psi \rho_{\bigvee C}^\psi(T)\in \mathcal{B}(\mathcal{H})$ is faithful, it remains to show that $\mathcal{M}\subseteq R_{C,F}^\psi\mathcal{H}$. Suppose $s,t\in C$ with $s\neq t$ and $\left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)<\infty$. Since $\bigvee C\leq \beta_{s,t}$, Lemma \[projections and psi\] tells us that for any $x\in \mathbf{X}_{\bigvee C}$ we have $$P_{\beta_{s,t}}\psi_{\bigvee C}(x)=\psi_{\bigvee C}(x)P_{\left(\bigvee C\right)^{-1}\beta_{s,t}}.$$ Thus, $$\begin{aligned} R_{C,F}^\psi&\psi_{\bigvee C}(x)\\ &= Q_{C,F}^\psi\prod_{\substack{s,t\in C, \, s\neq t, \\ \left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)<\infty}} \left(P^\psi_{\bigvee C}-P^\psi_{\beta_{s,t}}\right)\psi_{\bigvee C}(x)\\ &= Q_{C,F}^\psi\psi_{\bigvee C}(x)\prod_{\substack{s,t\in C, \, s\neq t, \\ \left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)<\infty}} \left(\mathrm{id}_\mathcal{H}-P^\psi_{\left(\bigvee C\right)^{-1}\beta_{s,t}}\right)\\ &= \psi_{\bigvee C}(x)\prod_{\substack{p\in F\setminus C, \, s,t\in C,\\ p\vee \left(\bigvee C\right) <\infty, \, s\neq t,\\ \left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)<\infty}}\left(\mathrm{id}_\mathcal{H}-P_{\left(\bigvee C\right)^{-1}\left(p\vee \left(\bigvee C\right)\right)}^\psi\right) \left(\mathrm{id}_\mathcal{H}-P^\psi_{\left(\bigvee C\right)^{-1}\beta_{s,t}}\right).\end{aligned}$$ Hence, $R_{C,F}^\psi$ is the identity on $$\begin{aligned} \mathcal{M} &= \overline{\psi_{\bigvee C}\left(\mathbf{X}_{\bigvee C}\right)\mathcal{K}}\\ &= \overline{\psi_{\bigvee C}\left(\mathbf{X}_{\bigvee C}\right) \hspace{-0.9em} \prod_{\substack{p\in F\setminus C, \, s,t\in C,\\ p\vee \left(\bigvee C\right) <\infty, \, s\neq t,\\ \left(s^{-1}\left(\bigvee C\right)\right)\vee \left(t^{-1}\left(\bigvee C\right)\right)<\infty}} \hspace{-0.9em} \left(\mathrm{id}_\mathcal{H}-P_{\left(\bigvee C\right)^{-1}\left(p\vee \left(\bigvee C\right)\right)}^\psi\right) \left(\mathrm{id}_\mathcal{H}-P^\psi_{\left(\bigvee C\right)^{-1}\beta_{s,t}}\right)\mathcal{H}},\end{aligned}$$ and so $\mathcal{M}\subseteq R_{C,F}^\psi\mathcal{H}$. Putting all of this together, we see that $$\begin{aligned} \bigg\|\sum_{k:p_k=q_k}\psi_{p_k}(x_k)\psi_{p_k}(y_k)^*\bigg\|_{\mathcal{B}(\mathcal{H})} &=\bigg\|\sum_{k:p_k=q_k}\iota_{p_k}^{\bigvee C}(\Theta_{x_k,y_k})\bigg\|_{\mathcal{L}_A\left(\mathbf{X}_{\bigvee C}\right)}\\ &=\bigg\|R_{C,F}^\psi\rho_{\bigvee C}^\psi\bigg(\sum_{k:p_k=q_k}\iota_{p_k}^{\bigvee C}(\Theta_{x_k,y_k})\bigg)\bigg\|_{\mathcal{B}(\mathcal{H})}\\ &=\bigg\|R_{C,F}^\psi\sum_{k:p_k=q_k}\psi_{p_k}(x_k)\psi_{p_k}(y_k)^*\bigg\|_{\mathcal{B}(\mathcal{H})}\\ &=\bigg\|R_{C,F}^\psi\bigg(\sum_k \psi_{p_k}(x_k)\psi_{q_k}(y_k)^*\bigg)R_{C,F}^\psi\bigg\|_{\mathcal{B}(\mathcal{H})}\\ &\leq \bigg\|\sum_k \psi_{p_k}(x_k)\psi_{q_k}(y_k)^*\bigg\|_{\mathcal{B}(\mathcal{H})}.\end{aligned}$$ Since the norm estimate (\[norm estimate onto core\]) holds, the formula $\psi_p(x)\psi_q(y)^*\mapsto \delta_{p,q}\psi_p(x)\psi_q(y)^*$ extends to a map on $\psi_*(\mathcal{NT}_\mathbf{X})={\overline{\mathrm{span}}}\left\{\psi_p(x)\psi_q(y)^*:p,q\in P, \ x\in \mathbf{X}_p, \ y \in \mathbf{X}_q\right\}$ by linearity and continuity, which we denote by $E_\psi$. Furthermore, for any $p,q\in P$, $x\in \mathbf{X}_p$, $y \in \mathbf{X}_q$, we have $$\begin{aligned} \left(E_\psi \circ \psi_*\right)\left(i_{\mathbf{X}_p}(x)i_{\mathbf{X}_q}(y)^*\right) &=E_\psi\left(\psi_p(x)\psi_q(y)^*\right) =\delta_{p,q}\psi_p(x)\psi_q(y)^*\\ &=\psi_*\left(\delta_{p,q} i_{\mathbf{X}_p}(x)i_{\mathbf{X}_q}(y)^*\right) =\left(\psi_* \circ E_{\delta_\mathbf{X}}\right)\left(i_{\mathbf{X}_p}(x)i_{\mathbf{X}_q}(y)^*\right). \end{aligned}$$ Since $\mathcal{NT}_\mathbf{X}={\overline{\mathrm{span}}}\left\{i_{\mathbf{X}_p}(x)i_{\mathbf{X}_q}(y)^*:p,q\in P, x\in \mathbf{X}_p, y \in \mathbf{X}_q\right\}$, whilst the maps $E_\psi \circ \psi_*$ and $\psi_* \circ E_{\delta_\mathbf{X}}$ are linear and norm-decreasing, we conclude that $E_\psi \circ \psi_*=\psi_* \circ E_{\delta_\mathbf{X}}$. This completes the proof of part (ii). Finally, we prove the uniqueness theorem for Nica–Toeplitz algebras. We remind the reader that if $(G,P)$ is a quasi-lattice ordered group with $G$ amenable, then any compactly aligned product system $\mathbf{X}$ over $P$ is automatically amenable (in the sense that the expectation $E_{\delta_\mathbf{X}}$ is faithful on positive elements). Suppose that the representation $$\begin{aligned} A\ni a \mapsto \psi_e(a)\prod_{t\in K}\big(\mathrm{id}_\mathcal{H}-P_t^\psi\big)\in \mathcal{B}(\mathcal{H})\end{aligned}$$ is faithful for any finite set $K\subseteq P\setminus \{e\}$. Let $b\in \mathcal{NT}_\mathbf{X}$ be such that $\psi_*(b)=0$. Thus, $$\psi_*(E_{\delta_\mathbf{X}}(b^*b))=E_\psi(\psi_*(b^*b))=E_\psi(\psi_*(b)^*\psi_*(b))=0.$$ Since $\psi_*$ is faithful on $\mathcal{NT}_\mathbf{X}^{\delta_\mathbf{X}}=E_{\delta_\mathbf{X}}(\mathcal{NT}_\mathbf{X})$, we must have $E_{\delta_\mathbf{X}}(b^*b)=0$. As $E_{\delta_\mathbf{X}}$ is faithful on positive elements, we conclude that $b=0$. Hence, $\psi_*$ is faithful. We now prove (ii). Suppose that $\psi_*$ is faithful and $\phi_p(A)\subseteq \mathcal{K}_A(\mathbf{X}_p)$ for each $p\in P$. Fix a finite set $K\subseteq P\setminus \{e\}$. For each $a\in A$, define $$T_a:=\sum_{J\subseteq K, \, \bigvee J<\infty}(-1)^{|J|}i_\mathbf{X}^{(\bigvee J)}\left(\phi_{\bigvee J}(a)\right)\in \mathcal{NT}_\mathbf{X}.$$ We claim that for any Nica covariant representation $\varphi:\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H}')$ we have $$\label{something about representations} \begin{aligned} \varphi_*(T_a)=\varphi_e(a)\prod_{t\in K}\big(\mathrm{id}_{\mathcal{H}'}-P_t^\varphi\big). \end{aligned}$$ To see this, firstly observe that for any $t\in P$ and $a\in A$, we have $$\varphi_*\big(i_\mathbf{X}^{(t)}(\phi_t(a))\big)=\rho_t^\varphi(\phi_p(a))=\varphi_e(a)P_t^\varphi.$$ Therefore, $$\begin{aligned} \varphi_*(T_a) &=\sum_{J\subseteq K, \, \bigvee J<\infty}(-1)^{|J|}\varphi_*\big(i_\mathbf{X}^{(\bigvee J)}\left(\phi_{\bigvee J}(a)\right)\big)\\ &=\varphi_e(a)\sum_{J\subseteq K, \, \bigvee J<\infty}(-1)^{|J|}P_{\bigvee J}^\varphi\\ &=\varphi_e(a)\sum_{J\subseteq K}(-1)^{|J|}P_{\bigvee J}^\varphi.\end{aligned}$$ Hence, to prove that Equation (\[something about representations\]) holds, it suffices to show that $$\label{realising product as sum} \begin{aligned} \sum_{J\subseteq K}(-1)^{|J|}P_{\bigvee J}^\varphi=\prod_{t\in K}\big(\mathrm{id}_{\mathcal{H}'}-P_t^\varphi\big). \end{aligned}$$ To prove this we use induction on $|K|$. When $|K|=0$ we have $$\begin{aligned} \sum_{J\subseteq K}(-1)^{|J|}P_{\bigvee J}^\varphi =(-1)^{|\emptyset|}P_{\bigvee \emptyset}^\varphi =P_e^\varphi =\mathrm{id}_{\mathcal{H}'} =\prod_{t\in \emptyset}\left(\mathrm{id}_{\mathcal{H}'}-P_t^\varphi\right) =\prod_{t\in K}\left(\mathrm{id}_{\mathcal{H}'}-P_t^\varphi\right). \end{aligned}$$ Now let $n\in \N$ and suppose we have equality whenever $K\subseteq P$ and $|K|=n$. Fix $K'\subseteq P$ with $|K'|=n+1$. Let $s\in K'$. Then $$\begin{aligned} \sum_{J\subseteq K'}(-1)^{|J|}P_{\bigvee J}^\varphi &=\sum_{J\subseteq K'\setminus \{s\}}(-1)^{|J|}P_{\bigvee J}^\varphi+\sum_{\{J\subseteq K': s\in J\}}(-1)^{|J|}P_{\bigvee J}^\varphi\\ &=\sum_{J\subseteq K'\setminus \{s\}}(-1)^{|J|}P_{\bigvee J}^\varphi+\sum_{J\subseteq K'\setminus \{s\}}(-1)^{|J\cup \{s\}|}P_{\bigvee (J\cup \{s\})}^\varphi\\ &=\sum_{J\subseteq K'\setminus \{s\}}(-1)^{|J|}P_{\bigvee J}^\varphi-\sum_{J\subseteq K'\setminus \{s\}}(-1)^{|J|}P_{\bigvee J}^\varphi P_{s}^\varphi\\ &=\left(\mathrm{id}_{\mathcal{H}'}-P_{s}^\varphi\right)\sum_{J\subseteq K'\setminus \{s\}}(-1)^{|J|}P_{\bigvee J}^\varphi\\ &=\left(\mathrm{id}_{\mathcal{H}'}-P_{s}^\varphi\right)\prod_{t\in K'\setminus \{s\}}\left(\mathrm{id}_{\mathcal{H}'}-P_t^\varphi\right)\\ &=\prod_{t\in K'}\left(\mathrm{id}_{\mathcal{H}'}-P_t^\varphi\right).\end{aligned}$$ This proves that Equation (\[realising product as sum\]) holds, and so Equation (\[something about representations\]) follows. Now let $\pi:A\rightarrow \mathcal{B}(\mathcal{H}')$ be a faithful nondegenerate representation of $A$. Define $\Psi:\mathbf{X}\rightarrow \mathcal{B}(\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}')$ by $$\Psi:=\big(\mathcal{F}_\mathbf{X}\text{-}\mathrm{Ind}_A^{\mathcal{L}_A(\mathcal{F}_\mathbf{X})}\pi\big)\circ l$$ where $l:\mathbf{X}\rightarrow \mathcal{L}_A(\mathcal{F}_\mathbf{X})$ is the Fock representation of $\mathbf{X}$. Since $l$ is a Nica covariant representation of $\mathbf{X}$ and $\mathcal{F}_\mathbf{X}\text{-}\mathrm{Ind}_A^{\mathcal{L}_A(\mathcal{F}_\mathbf{X})}\pi$ is a $*$-homomorphism, $\Psi$ is a Nica covariant representation of $\mathbf{X}$. We claim that the representation $$A\ni a \mapsto \Psi_e(a)\prod_{t\in K}\big(\mathrm{id}_{\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}'}-P_t^\Psi\big)\in \mathcal{B}(\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}')$$ is faithful. To see this, suppose that $a\in A\setminus\{0\}$, so that $aa^*\neq 0$. As $\pi$ is faithful, we can find $h\in \mathcal{H}'$ such that $\pi(aa^*)h\neq 0$. For any $t\in P\setminus \{e\}$ we have $$\begin{aligned} \overline{\Psi_t(\mathbf{X}_t)\left(\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}'\right)} &=\overline{\big(\mathcal{F}_\mathbf{X}\text{-}\mathrm{Ind}_A^{\mathcal{L}_A(\mathcal{F}_\mathbf{X})}\pi\left(l_t(\mathbf{X}_t)\right)\big)\left(\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}'\right)}\\ &=\overline{l_t(\mathbf{X}_t)(\mathcal{F}_\mathbf{X})\otimes_A \mathcal{H}'}\\ &=\overline{\bigoplus_{s\geq t}\mathbf{X}_{s}\otimes_A \mathcal{H}'}.\end{aligned}$$ Hence, it follows that $P_t^\Psi=\mathrm{proj}_{\overline{\Psi_t(\mathbf{X}t)\left(\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}'\right)}}$ is zero on $A\otimes_A \mathcal{H}'\subseteq \mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}'$. Thus, as $e\not \in K$ we see that $$\begin{aligned} \Psi_e(a)\prod_{t\in K}\left(\mathrm{id}_{\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}'}-P_t^\Psi\right)\left(a^*\otimes_A h\right) =\Psi_e(a)(a^*\otimes_A h) =l_e(a)(a^*)\otimes_A h &=aa^*\otimes_A h,\end{aligned}$$ which is nonzero because $$\begin{aligned} {\left\|aa^*\otimes_A h\right\|}_{\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}'}^2 &=\langle aa^*\otimes_A h, aa^*\otimes_A h \rangle_\C =\langle h, \pi((aa^*)^*aa^*)h \rangle_\C\\ &=\langle \pi(aa^*)h, \pi(aa^*)h \rangle_\C ={\left\| \pi(aa^*)h\right\|}_{\mathcal{H}'}^2 \neq 0. \end{aligned}$$ Therefore, $A\ni a \mapsto \Psi_e(a)\prod_{t\in K}\left(\mathrm{id}_{\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}'}-P_t^\Psi\right)\in \mathcal{B}(\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}')$ is faithful. Putting all of this together, and using that $\psi_*$ is faithful at the penultimate equality, we see that for any $a\in A$, $$\begin{aligned} {\left\|a\right\|}_A =\bigg\|\Psi_e(a)\prod_{t\in K}\big(\mathrm{id}_{\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}'}-P_t^\Psi\big)\bigg\|_{\mathcal{B}(\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}')} &={\left\|\Psi_*(T_a)\right\|}_{\mathcal{B}(\mathcal{F}_\mathbf{X}\otimes_A \mathcal{H}')}\\ &\leq {\left\|T_a\right\|}_{\mathcal{NT}_\mathbf{X}}\\ &={\left\|\psi_*(T_a)\right\|}_{\mathcal{B}(\mathcal{H})}\\ &=\bigg\|\psi_e(a)\prod_{t\in K}\big(\mathrm{id}_\mathcal{H}-P_t^\psi\big)\bigg\|_{\mathcal{B}(\mathcal{H})}.\end{aligned}$$ Hence, $A\ni a \mapsto \psi_e(a)\prod_{t\in K}\big(\mathrm{id}_\mathcal{H}-P_t^\psi\big)\in \mathcal{B}(\mathcal{H})$ is faithful. In practice, we are often interested in representations of product systems in more general $C^*$-algebras, rather than on Hilbert spaces. The following corollary shows that provided the coefficient algebra acts compactly on each fibre of the product system, we can still characterise the faithfulness of the induced representation. \[representations in C\*-algebras\] Let $(G,P)$ be a quasi-lattice ordered group and $\mathbf{X}$ an amenable compactly aligned product system over $P$ with coefficient algebra $A$. Suppose that $A$ acts compactly on each $\mathbf{X}_p$. Let $\psi:\mathbf{X}\rightarrow B$ be a Nica covariant representation of $\mathbf{X}$ in a $C^*$-algebra $B$. Then the induced $*$-homomorphism $\psi_*:\mathcal{NT}_\mathbf{X}\rightarrow B$ is faithful if and only if for every $a\in A\setminus \{0\}$ and every finite set $K\subseteq P\setminus \{e\}$, we have $$\prod_{t\in K}\big(\psi_e-\psi^{(t)}\circ\phi_t\big)(a)\neq 0.$$ Fix a faithful representation $\pi:B\rightarrow \mathcal{B}(\mathcal{H})$. Then $\pi\circ \psi:\mathbf{X}\rightarrow \mathcal{B}(\mathcal{H})$ is a Nica covariant representation with induced representation $(\pi\circ \psi)_*=\pi\circ \psi_*$. If $a\in A$ and $t\in P\setminus\{e\}$, then Proposition \[existence of rho map\] implies that $$\pi(\psi_e(a))P_t^{\pi\circ \psi}=\rho_t^{\pi\circ \psi}(\phi_t(a))=(\pi\circ \psi)^{(t)}(\phi_t(a))=\pi\big(\psi^{(t)}(\phi_t(a))\big).$$ Hence, for any $a\in A$ and any finite set $K\subseteq P\setminus \{e\}$, we have that $$\begin{aligned} \pi\bigg(\prod_{t\in K}\big(\psi_e-\psi^{(t)}\circ\phi_t\big)(a)\bigg) &=\prod_{t\in K}\Big(\pi(\psi_e(a))-\pi\big(\psi^{(t)}(\phi_t(a))\big)\Big)\\ &=\prod_{t\in K}\Big(\pi(\psi_e(a))-\pi(\psi_e(a))P_t^{\pi\circ \psi}\Big)\\ &=(\pi\circ \psi)_e(a)\prod_{t\in K}\big(\mathrm{id}_\mathcal{H}-P_t^{\pi\circ \psi}\big),\end{aligned}$$ and so the result follows from Theorem \[uniqueness theorem for NT algebras\]. Acknowledgements ================ The results in this article are from my PhD thesis. Thank you to my supervisors Adam Rennie and Aidan Sims at the University of Wollongong for their advice and encouragement during my PhD and during the writing of this article. [^1]: This research was supported by an Australian Government Research Training Program (RTP) Scholarship and by the Marsden grant 15-UOO-071 from the Royal Society of New Zealand.

--- abstract: 'We show that if a polarised manifold admits an extremal metric then it is K-polystable relative to a maximal torus of automorphisms.' address: - 'Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK' - 'Columbia University, Department of Mathematics, 2990 Broadway New York, NY 10027, USA' author: - 'J. Stoppa and G. Székelyhidi^^' title: 'Relative K-stability of extremal metrics' --- Introduction ============ Calabi [@Cal82] introduced the notion of extremal metrics as candidates for canonical representatives of Kähler classes on compact Kähler manifolds. Unfortunately not all Kähler manifolds admit extremal metrics (eg. Levine [@Lev]) and even if they do, they may not admit them in all Kähler classes (see eg. Apostolov, Calderbank, Gauduchon, Tønnesen-Friedman [@ACGT3]). This makes the question of existence of extremal metrics quite delicate and there is now a vast literature on the topic. We refer to Phong-Sturm [@PS] for a recent survey and an extensive bibliography. By definition an extremal metric is a Kähler metric whose scalar curvature has holomorphic gradient vector field. Thus, special cases are constant scalar curvature Kähler (or cscK) metrics and Kähler-Einstein metrics. While one can study these metrics in arbitrary Kähler classes, perhaps the most interesting case is when the Kähler class is the first Chern class of an ample line bundle. Indeed, existence of a cscK metric on a manifold $M$ in the Kähler class $c_1(L)$ for an ample line bundle $L$, is expected to be closely related to algebro-geometric properties of the polarised manifold $(M,L)$. This is expressed by the following. \[conj:1\] The manifold $M$ admits a cscK metric in the class $c_1(L)$ if and only if the pair $(M,L)$ is K-polystable. The notion of K-polystability will be recalled below. Building on the K-semistability proved by Donaldson [@Don05] and on the work of Arezzo-Pacard [@AP06] on blowing up cscK metrics, the first named author completed the proof of one direction of this conjecture, under the assumption that the automorphism group of $(M,L)$ is discrete. If $M$ admits a cscK metric in $c_1(L)$ and $\mathrm{Aut}(M,L)$ is discrete, then $(M,L)$ is K-polystable. Using a different approach, this was recently extended to manifolds with not necessarily discrete automorphism groups by Mabuchi [@Mab1], [@Mab2]. The aim of the present paper is to generalise this theorem to the case of extremal metrics. In this case the conjecture analogous to Conjecture \[conj:1\] was formulated by the second named author in [@GSz04]. \[conj:2\] The manifold $M$ admits an extremal metric in the class $c_1(L)$ if and only if the pair $(M,L)$ is K-polystable relative to a maximal torus of automorphisms of $(M,L)$. By generalising the approach in [@Sto08] we obtain the following, which is the main result of this paper. \[thm:main\] If $M$ admits an extremal metric in $c_1(L)$ then $(M,L)$ is K-polystable relative to a maximal torus of automorphisms of $(M,L)$. In particular the theorem applies when $M$ admits a cscK metric and has continuous automorphisms, proving that $M$ is K-polystable with respect to all test- configurations that commute with a maximal torus of automorphisms, but note that this is a priori a weaker condition than K-polystability (see the next section for the detailed definitions). Note that by an example in [@ACGT3] relative K-polystability may not be sufficient to ensure the existence of an extremal metric, so it is likely that the conjectures \[conj:1\] and \[conj:2\] have to be refined. Acknowledgements {#acknowledgements .unnumbered} ---------------- We would like to thank Julius Ross and Richard Thomas for helpful discussions. The second named author would also like to thank D. H. Phong for his encouragement and support. Relative K-polystability ======================== In this section we recall the notion of relative K-polystability following [@GSz04]. This is a modification of the notion of K-polystability introduced by Donaldson [@Don02]. Suppose that $(V,L)$ is a polarised scheme of dimension $n$, with a $\mathbf{C}^*$ action $\alpha$. Let us write $A_k$ for the infinitesimal generator of the action of $\alpha$ on $H^0(V,L^k)$, and write $d_k$ for the dimension of $H^0(V,L^k)$. Then $d_k$ is a polynomial of degree $n$ and $\mathrm{Tr}(A_k)$ is a polynomial of degree $n+1$ for sufficiently large $k$, so we can write $$\begin{aligned} d_k &= c_0k^n + c_1k^{n-1} + O(k^{n-2}),\\ \mathrm{Tr}(A_k) &= a_0k^{n+1} + a_1k^n + O(k^{n-1}),\\ \end{aligned}$$ Donaldson’s Futaki invariant is defined to be $$F(\alpha) = \frac{c_1}{c_0}a_0 - a_1.$$ Sometimes we will write $F(V,L,\alpha)$ to emphasize the space that $\alpha$ is acting on. Suppose in addition that we have a $\mathbf{C}^*$-action $\beta$ acting on $(V,L)$ which commutes with $\alpha$, and write $B_k$ for the infinitesimal generator of the action on $H^0(V,L^k)$. Then $\mathrm{Tr}(A_kB_k)$ is a polynomial of degree $k+2$ for sufficiently large $k$, and we define the inner product $\langle\alpha,\beta\rangle$ to be the leading coefficient in the expansion $$\mathrm{Tr}(A_kB_k) - \frac{\mathrm{Tr}(A_k)\mathrm{Tr}(B_k)}{d_k} = \langle \alpha,\beta\rangle k^{n+2} + O(k^{n+1}).$$ When $V$ is a smooth manifold, then this inner product can also be computed differential geometrically. It was originally introduced in this form by Futaki-Mabuchi [@FM]. To define the relative Futaki invariant, suppose that we have a torus action $T$ on $(V,L)$ commuting with $\alpha$. Let us write $\overline{\alpha}$ for the projection of $\alpha$ orthogonal to $T$, with respect to the inner product we defined. Then we define the relative Futaki invariant $F_T(\alpha)$ by $$F_T(\alpha) = F(\overline{\alpha}).$$ Equivalently if $\beta_1,\ldots,\beta_d$ is a basis of $\mathbf{C}^*$-actions generating the torus $T$, then $$F_T(\alpha) = F(\alpha) - \sum_{i=1}^d \frac{\langle \alpha,\beta_i\rangle}{\langle \beta_i,\beta_i\rangle} F(\beta_i).$$ It will be convenient for us to extend these definitions to $\mathbf{Q}$-line bundles using the relation $$F(V,L^r,\alpha) = r^n F(V,L,\alpha),$$ which the reader can readily verify. It will also be useful to allow rational multiples of $\mathbf{C}^*$-actions. For this we use the relation $$F(V,L,r\alpha) = r F(V,L,\alpha).$$ We next recall the notion of a test-configuration from [@Don02] with the necessary modification for relative stability. A *test-configuration for $(X,L)$* consists of a $\mathbf{C}^*$-equivariant flat family of schemes $\pi:\mathcal{X}\to\mathbf{C}$ (where $\mathbf{C}^*$ acts on $\mathbf{C}$ by multiplication) and a $\mathbf{C}^*$-equivariant, relatively ample $\mathbf{Q}$-line bundle $\mathcal{L}$ over $\mathcal{X}$. We require that the fibres $(\mathcal{X}_t,\mathcal{L}|_{\mathcal{X}_t})$ are isomorphic to $(X,L)$ for $t\not=0$, where $\mathcal{X}_t=\pi^{-1}(t)$. The test-configuration is called a *product configuration* if $\mathcal{X} = X\times\mathbf{C}$. We say that the test-configuration is *compatible with a torus $T$ of automorphisms of $(X,L)$*, if there is a torus action on $(\mathcal{X},\mathcal{L})$ which preserves the fibres of $\pi:\mathcal{X}\to\mathbf{C}$, commutes with the $\mathbf{C}^*$-action, and restricts to $T$ on $(\mathcal{X}_t,\mathcal{L}|_{\mathcal{X}_t})$ for $t\not=0$. Note that given a test-configuration $(\mathcal{X},\mathcal{L})$, there is an induced $\mathbf{C}^*$-action $\alpha$ on the central fibre $(\mathcal{X}_0,\mathcal{L}|_{\mathcal{X}_0})$. We will write $F(\mathcal{X},\mathcal{L})$ for the Futaki invariant of this induced action $\alpha$. With these preliminaries we can state the main definition. \[def:kstable\] A polarised variety $(X,L)$ is *K-semistable relative to a torus $T$ of automorphisms* if $F_T(\mathcal{X},\mathcal{L})\geqslant 0$ for all test-configurations compatible with the torus. If in addition equality holds only for the product configuration, then $(X,L)$ is *K-polystable* relative to the torus $T$. If we have two tori $T'\subset T$ acting on $(X,L)$, then K-polystability relative to $T$ is a weaker condition than relative to $T'$, since there are fewer test-configurations compatible with a larger torus. Thus, the weakest notion is K-polystability relative to a maximal torus of automorphisms. The strongest notion is K-polystability relative to the extremal $\mathbf{C}^*$-action. This is a $\mathbf{C}^*$-action $\chi$ defined by Futaki-Mabuchi [@FM] as follows. Fix a maximal torus of automorphisms $T$, and write $\mathfrak{t}$ for its Lie algebra. The Futaki invariant gives a linear map $\mathfrak{t}\mapsto\mathbf{C}$, and $\chi$ is dual to this map under the inner product on $\mathfrak{t}$. This gives a $\mathbf{C}^*$-action on $(X,L)$, unique up to conjugation. In particular if the Futaki invariant of any $\mathbf{C}^*$-action on $(X,L)$ vanishes, then $\chi=0$, and K-polystability relative to $\chi$ is simply K-polystability. It would be interesting to strengthen the conclusion of Theorem \[thm:main\] to K-polystability relative to the extremal $\mathbf{C}^*$-action. Note that the analogous statement is true in finite dimensional geometric invariant theory, by Theorem 3.5 in [@GSz04] (the same proof works if we replace the maximal torus with any torus containing the extremal $\mathbf{C}^*$-action). We next recall the two theorems that we will use in the next section. \[thm:semistab\] If $M$ admits an extremal metric in $c_1(L)$ then $(M,L)$ is K-semistable relative to a maximal torus of automorphisms. This follows easily from Donaldson’s lower bound for the Calabi functional [@Don05]. For details see [@GSzThesis]. For the convenience of the reader we outline the argument here. Donaldson’s lower bound tells us that for any test-configuration, if $\alpha$ is the induced $\mathbf{C}^*$-action on the central fiber, then $$\label{eq:skd} \inf_{\omega\in c_1(L)} c_n\Vert S(\omega) - \hat{S}\Vert_{L^2} \geqslant \frac{-F(\alpha)}{\Vert\alpha\Vert},$$ where $c_n$ is a constant depending only on the dimension, $\Vert\alpha\Vert=\langle\alpha,\alpha\rangle^{1/2}$ using the inner product defined above, and $\hat{S}$ is the average of the scalar curvature $S(\omega)$. Moreover, if $\omega$ is an extremal metric, then $$\label{eq:ext} c_n\Vert S(\omega) - \hat{S}\Vert_{L^2} = \frac{ F(\chi) }{\Vert \chi\Vert} = \Vert\chi\Vert,$$ where $\chi$ is the extremal vector field on $(M,L)$. We are using here that $F(\chi)=\langle\chi,\chi\rangle$ by definition of the extremal vector field. It follows from (\[eq:skd\]) and (\[eq:ext\]) that if $M$ admits an extremal metric in $c_1(L)$ then $$\label{eq:lower} \frac{F(\alpha)}{\Vert \alpha\Vert} \geqslant -\Vert\chi\Vert$$ for all test-configurations. Suppose now that $M$ admits an extremal metric in $c_1(L)$, and we have a test-configuration for $(M,L)$ which is compatible with a maximal torus of automorphisms $T$. Write $\alpha$ for the induced $\mathbf{C}^*$-action on the central fiber. By twisting the $\mathbf{C}^*$-action on the total space by the projection of $\alpha$ onto $T$ if necessary, we can assume that $\alpha$ is orthogonal to $T$. We want to show that $F(\alpha)\geqslant 0$. Suppose on the contrary that $F(\alpha) < 0$, and let $\mu>0$ satisfy $F(\mu\alpha) = -\Vert\mu\alpha\Vert^2$. By pulling back the test-configuration under a base change $z\mapsto z^r$, and twisting the action on the total space by the inverse of $\chi$, we obtain a test-configuration for $(M,L)$ such that the action on the central fiber is $r(\mu\alpha-\chi)$, where $r$ is large enough to make this a genuine $\mathbf{C}^*$-action. From (\[eq:lower\]) we know that $$\frac{F(\mu\alpha - \chi)}{\Vert \mu\alpha-\chi\Vert} = \frac{F(r(\mu\alpha-\chi))}{\Vert r(\mu\alpha-\chi)\Vert} \geqslant -\Vert\chi\Vert.$$ But at the same time $$F(\mu\alpha - \chi) = -\Vert\mu\alpha\Vert^2 - \Vert\chi\Vert^2 = -\Vert \mu\alpha - \chi\Vert^2,$$ since $\alpha$ is orthogonal to $\chi$. So $$\frac{F(\mu\alpha - \chi)}{\Vert\mu\alpha - \chi\Vert} = -\Vert \mu\alpha - \chi\Vert < -\Vert\chi\Vert.$$ This contradiction shows that $(M,L)$ is K-polystable relative to $T$. The same argument also shows that $(M,L)$ is K-polystable relative to the extremal $\mathbf{C}^*$-action. \[thm:APS\] Suppose that $M$ admits an extremal metric in $c_1(L)$, and let $T$ be a maximal torus of automorphisms of $(M,L)$. If $p\in M$ is a fixed point of $T$, then the blowup $\mathrm{Bl}_p M$ of $M$ at $p$ admits an extremal metric in the class $c_1(\pi^*L - {\varepsilon}E)$ for sufficiently small ${\varepsilon}>0$. Here $\pi$ is the blowdown map, and $E$ is the exceptional divisor. This follows from [@APS] Theorem 2.1. Indeed we can choose an extremal metric $\omega$ on $M$ such that the isometry group of $\omega$ contains a compact maximal torus $T_\mathbf{R}$, which is contained in the complex torus $T$. In the notation of [@APS] we let $K=T_\mathbf{R}$, and let $\mathfrak{k}$ be its Lie algebra. Since $K$ is a maximal torus, any $K$-invariant holomorphic hamiltonian vector field lies in $\mathfrak{k}$. Moreover if we write $S(\omega)$ for the scalar curvature then by Calabi’s theorem [@Cal85] the vector field $J\nabla S(\omega)$ lies in the center of the Lie algebra of Killing fields, so it also lies in $\mathfrak{k}$. This allows us to apply [@APS] Theorem 2.1, and we get the stated result. Proof of Theorem \[thm:main\] ============================= Let us suppose that $M$ admits an extremal metric in $c_1(L)$ and choose a maximal torus $T\subset \mathrm{Aut}(M,L)$. From Theorem \[thm:semistab\] we know that if $(\mathcal{X},\mathcal{L})$ is a test-configuration for $(M,L)$ compatible with $T$, then the relative Futaki invariant satisfies $F_T(\mathcal{X})\geqslant0$. Suppose then that $F_T(\mathcal{X})=0$. We can assume that $M\subset\mathbf{P}(V)$, where $V=H^0(M,L)^*$. Moreover the torus $T$ acts on $\mathbf{P}(V)$, preserving $M$. In addition there is an extra $\mathbf{C}^*$-action $\alpha$ on $\mathbf{P}(V)$, commuting with the $T$-action and such that the flat closure of the family $t\mapsto \alpha(t)\cdot M$ across $t=0$ is the test-configuration $\mathcal{X}$. Let us write $(M_0,L_0)$ for the central fiber of the test-configuration. Then we have both $\alpha$ and the torus $T$ acting on $(M_0,L_0)$. By twisting the action on the total space by the orthogonal projection of $\alpha$ onto $T$ (which does not change the relative Futaki invariant), we can assume that $\langle\alpha, T\rangle=0$. In this case $$F_T(\mathcal{X},\mathcal{L}) = F(M_0,L_0,\alpha).$$ We now look at the weight decomposition under $\alpha$ given by $$V = \bigoplus_i V_{m_i},$$ where $m_0 < m_1 <\ldots < m_L$ for some $L>0$, and consider the least $l\geqslant 0$ such that $$\mathrm{red}(M_0)\subset \mathbf{P}\big(\bigoplus_{i\leqslant l} V_{m_i}\big).$$ It is proved in [@Sto08] section 3 that if $l=0$, so that $\alpha$ acts trivially on $\mathrm{red}(M_0)$, then either $\mathcal{X}$ is a product test-configuration, or $F(M_0,L_0,\alpha)>0$, which is a contradiction. On the other hand, if $l>0$, then consider the repulsive fixed point set $$M_0' = \mathrm{red}(M_0)\cap \mathbf{P}(V_{m_l}).$$ The set of points $p\in M$ for which the limit $$q = \lim_{t\to 0} \alpha(t)p$$ is in $M_0'$ is precisely $$M' = M \cap \mathbf{P}\left(\bigoplus_{i\geqslant l} V_{m_i}\right).$$ This is a closed $T$-invariant set, so it contains a point $p$ fixed by $T$. To see this, we can take a basis of $\mathbf{C}^*$-actions $\beta_i$ generating the torus $T$, and then given any point $p$ in $M'$ we can inductively move it to a fixed point of $\beta_i$ by taking the limit of $\beta_i(t)p$ as $t\to 0$. Doing this for each $i$, we end up with a fixed point of $T$. The corresponding limit $q$ will then be a $T$-invariant, repulsive fixed point of $\alpha$ in $\mathrm{red}(M_0)$. Letting $Z\subset \mathcal{X}$ be the closure of the orbit of $p$ under $\alpha$, we obtain a test-configuration $$(\widehat{\mathcal{X}}, \widehat{\mathcal{L}}) = (\mathrm{Bl}_Z\mathcal{X}, \phi^*\mathcal{L} - {\varepsilon}E)$$ for the polarised manifold $(\mathrm{Bl}_p M, \phi^*L - {\varepsilon}E)$, where $\phi\!: \widehat{\mathcal{X}}\to\mathcal{X}$ is the blowdown. The only nontrivial thing to check is flatness of the composition $\pi\circ \phi\!:\widehat{\mathcal{X}}\to \mathcal{X}\to \mathbf{C}$. This holds because blowing up $Z \subset \mathcal{X}$ does not introduce new associated points (i.e. embedded schemes) of $\mathcal{X}$, only the Cartier exceptional divisor $E$ (for details see the proof of Proposition 2.13 of [@Sto08]). For suitably small ${\varepsilon}> 0$ the test-configuration $(\widehat{\mathcal{X}}, \widehat{\mathcal{L}})$ will have negative Futaki invariant, and in fact it will even have negative Futaki invariant relative to $T$. This follows from the lemma below and its corollary. At the same time from Theorem \[thm:APS\] we know that $\mathrm{Bl}_pM$ admits an extremal metric in the class $c_1(\phi^*L - {\varepsilon}E)$ for suitably small ${\varepsilon}$ since $p$ is fixed by the torus $T$, which is a maximal torus of automorphisms of $M$. This contradicts Theorem \[thm:semistab\], and completes the proof of the main theorem. Let $(\mathcal{X},\mathcal{L})$ be a test-configuration for $(M,L)$ compatible with a torus $T$ of automorphisms, and suppose that the induced action $\alpha$ on the central fiber satisfies $\langle\alpha,T\rangle = 0$. Let $\widehat{\mathcal{X}}$ be given by the blowup of a $T$-invariant section as described above. Then $$F(\widehat{\mathcal{X}},\widehat{\mathcal{L}}) = F(\mathcal{X},\mathcal{L}) + \left(\lambda(q) - \frac{b_0}{a_0}\right) \frac{{\varepsilon}^{n-1}}{2(n-2)!} + O({\varepsilon}^n),$$ and $$\langle\hat{\alpha}, \hat{T}\rangle = O({\varepsilon}^n),$$ where we use the $\mathbf{Q}$-polarization $\widehat{\mathcal{L}}=\phi^*\mathcal{L} - {\varepsilon}E$ on $\widehat{\mathcal{X}}$ for some small rational ${\varepsilon}>0$, and $\hat{\alpha}, \hat{T}$ are the actions of $\alpha$ and $T$ lifted to the blowup. It follows that the relative Futaki invariants satisfy $$F_{T}(\widehat{\mathcal{X}},\widehat{\mathcal{L}}) = F_T(\mathcal{X},\mathcal{L}) + \left(\lambda(q) - \frac{b_0}{a_0}\right) \frac{{\varepsilon}^{n-1}}{2(n-2)!} + O({\varepsilon}^n).$$ Here $\lambda(q)$ is the weight of $\alpha$ on the fiber $L_{0,q}$, and $a_0, b_0$ are defined by the expansions of the dimension and weight on $H^0(M_0,L^k_0)$ calculated at the central fiber of $\mathcal{X}$ as usual: $$\begin{aligned} d_k &= a_0k^n + a_1k^{n-1} + \ldots, \\ w_k &= b_0k^{n+1} + b_1k^n + \ldots. \end{aligned}$$ The central fibre of $\widehat{\mathcal{X}}$ will not in general be isomorphic to $\widehat{M}_0 := \mathrm{Bl}_{q} M_0$. In fact it will contain another large component $P$ glued to $\widehat{M}_0$ along the exceptional divisor $E'$ for the morphism $\widehat{M}_0 \to M_0$, as we now explain. By [@hart], II Corollary 7.15, there is a closed immersion $\widehat{M}_0 \hookrightarrow \widehat{\mathcal{X}}_0$ induced by the closed immersion $M_0 \subset \mathcal{X}$ under blowing up $Z$. Let $\mathcal{I}_q \subset \mathcal{O}_{M_0}$ denote the ideal sheaf of $q \in M_0$. By the algebraic definition of blowing up we have $\widehat{M}_0 \cong \mathrm{Proj} \bigoplus_{k {\geqslant}0} \mathcal{I}^k_q$. On the other hand the generic fibre of $\widehat{\mathcal{X}}$ is $\mathrm{Proj} \bigoplus_{k {\geqslant}0} \mathcal{I}^k_p$, where $\mathcal{I}_p$ is the ideal of the smooth point $p \in M$. Thus by the numerical criterion for flatness when the Hilbert-Samuel polynomial for $p \in M$ is larger that that of $q \in M_0$ (i.e. when $q$ is singular enough as a point of $M_0$) there will be an additional component $P$ in the central fibre, given by the closure of $\widehat{\mathcal{X}}_0\setminus\widehat{M}_0$. A simple example has been suggested by S. Donaldson: when $q$ is an isolated threefold ordinary double point inside the central fibre one has $P \cong \mathbf{P}^3$ glued in along a smooth quadric. Note that this is different from the situation described in [@jag] section 2, where the central fibre of the original test configuration is smooth (isomorphic to $M$), but one blows up $0-$cycles instead of just a point. In any case the restriction $\widehat{\mathcal{L}}_{0|_{\widehat{M}_0}}$ is just $\phi^*L_0 - {\varepsilon}E'$. Taking this information into account we now compute the Donaldson-Futaki invariant for the action $\alpha$ on the central fiber $\widehat{\mathcal{X}}_0$. In the calculations that follow ${\varepsilon}$ is a fixed positive rational number, and we tacitly restrict to those $k \gg 1$ for which ${\varepsilon}k$ is an integer. We also suppress pullbacks like $\pi^*$ or $\phi^*$ when this causes no confusion. By flatness, using the Riemann-Roch theorem we have $$\label{eq:RR} \begin{aligned} h^0(\widehat{\mathcal{X}}_0, \widehat{\mathcal{L}}^k_0) &= h^0(\mathrm{Bl}_p M, L^k - k{\varepsilon}E)\\ &= h^0(M, L^k) - \frac{{\varepsilon}^n}{n!}k^n -\frac{{\varepsilon}^{n-1}}{2(n-2)!}k^{n-1} + \ldots. \end{aligned}$$ Using the restriction $\mathbf{C}^*$-equivariant exact sequence $$\label{eq:restrict} 0 \longrightarrow H^0_P(\mathcal{I}^{k{\varepsilon}}_{E'}\widehat{\mathcal{L}}^k_{0|_P})\longrightarrow H^0_{\widehat{\mathcal{X}}_0}(\widehat{\mathcal{L}}^k_0) \longrightarrow H^0_{\widehat{M}_0}(L^k_0 - k{\varepsilon}E')\longrightarrow 0$$ which holds for large $k \gg 1$, we find $$\mathrm{Tr}(H^0_{\widehat{\mathcal{X}}_0}(\widehat{\mathcal{L}}^k_0)) = \mathrm{Tr}(H^0_{\widehat{M}_0}(L^{k}_0 - k{\varepsilon}E')) + \mathrm{Tr}(H^0_P(\mathcal{I}^r_{E'}\widehat{\mathcal{L}}^k_{0|_P})).$$ Note that $H^0_{\widehat{M}_0}(L^{k}_0 - k{\varepsilon}E') \cong H^0_{M_0}(\mathcal{I}^{k {\varepsilon}}_{q} L^{k}_0)$ so the first term in the formula above equals $\mathrm{Tr}(H^0_{M_0}(L^k_0)) -\mathrm{Tr}(H^0(\mathcal{O}_{k {\varepsilon}q} \otimes L^{k}_0|_q))$. From the exact sequence $$0\longrightarrow \mathcal{I}_q^{k{\varepsilon}}L_0^k\longrightarrow L_0^k\longrightarrow \mathcal{O}_{k{\varepsilon}q}\otimes L_0^k|_q\longrightarrow 0,$$ together with (\[eq:RR\]) and (\[eq:restrict\]) we see that the length of the $\mathcal{O}_{M_0}-$module $\mathcal{O}_{k{\varepsilon}q}$ is given by $$h^0_{P}(\mathcal{I}^{k{\varepsilon}}_{E'}\widehat{\mathcal{L}}^k_{0|_P}) + \frac{{\varepsilon}^n}{n!}k^n + \frac{{\varepsilon}^{n-1}}{2(n-2)!}k^{n-1} + O(k^{n-2}).$$ It follows that the weight of the action on $\mathcal{O}_{k{\varepsilon}q}\otimes L^k|_q$ is given by $$\begin{aligned} w(\mathcal{O}_{k{\varepsilon}q}\otimes L^k|_p) &= w(\mathcal {O}_{k{\varepsilon}q}) + k\lambda(q)\mathrm{len}(\mathcal{O}_{k{\varepsilon}q}) \\ &= k\lambda(q)h^0_{P}(\mathcal{I}^{k{\varepsilon}}_{E'}\widehat{\mathcal{L}}^k_{0|_P}) \\&+ \left(c_0{\varepsilon}^{n+1} + \lambda(q) \frac{{\varepsilon}^n}{n!}\right)k^{n+1} + \left(c_1{\varepsilon}^n + \lambda(q)\frac{{\varepsilon}^{n-1}}{2(n-2)!}\right)k^n + \ldots, \end{aligned}$$ where $c_0, c_1$ are given by the expansion $$w(\mathcal{O}_{k{\varepsilon}q}) = c_0(k{\varepsilon})^{n+1} + c_1(k{\varepsilon})^n + \ldots.$$ Similarly $\mathcal{I}^{k{\varepsilon}}_{E'}\widehat{\mathcal{L}}^k_{0|_P} \cong \mathcal{L}_0^{k}|_q \otimes \mathcal{I}^{k{\varepsilon}}_{E'}\mathcal{O}(-k E)|_P$ and the action on the latter factor has vanishing weight, so one has $$\mathrm{Tr}(H^0_{P}(\mathcal{I}^{k{\varepsilon}}_{E'}\widehat{\mathcal{L}}^k_{0|_P})) = k\lambda(q)h^0_{P}(\mathcal{I}^{k{\varepsilon}}_{E'}\widehat{\mathcal{L}}^k_{0|_P}).$$ After a simple cancellation we find $$\begin{aligned} \hat{a}_0 &= a_0 + O({\varepsilon}^n) \\ \hat{a}_1 &= a_1 - \frac{{\varepsilon}^{n-1}}{2(n-2)!} \\ \hat{b}_0 &= b_0 + O({\varepsilon}^n) \\ \hat{b}_1 &= b_1 - \lambda(q)\frac{{\varepsilon}^{n-1}}{2(n-2)!}, \end{aligned}$$ where $\hat{a}_i, \hat{b}_i$ are computed on $\widehat{\mathcal{X}}$. Using the formula $F(\widehat{\mathcal{X}}) = \displaystyle{\frac{\hat{a}_1}{\hat{a}_0}}\hat{b}_0 - \hat{b}_1$ we get $$F(\widehat{\mathcal{X}}) = F(\mathcal{X}) + \left(\lambda(q) - \frac{b_0}{a_0}\right) \frac{{\varepsilon}^{n-1}}{2(n-2)!} + O({\varepsilon}^n).$$ Now let $\beta$ be any $\mathbf{C}^*$-action in the torus $T$. To compute the inner product $\langle \hat{\alpha}, \hat{\beta}\rangle$, let us write $A_k, B_k$ for the infinitesimal generators of the actions $\alpha,\beta$ on $H^0(M_0, L^k_0)$, and write $\hat{A}_k, \hat{B}_k$ for the infinitesimal actions of the corresponding actions on $H^0(\widehat{\mathcal{X}}_0, \widehat{\mathcal{L}}^k_0)$. The inner product $\langle\hat{\alpha},\hat{\beta}\rangle$ is the leading order term in $$\label{product} \mathrm{Tr}(\hat{A}_k\hat{B}_k) - \frac{\mathrm{Tr}(\hat{A}_k) \mathrm{Tr}(\hat{B}_k)}{\hat{d}_k}.$$ Since the actions $\alpha, \beta$ commute, we can use precisely the same exact sequences as before to compute $$\begin{aligned} \mathrm{Tr}(A_kB_k) - \mathrm{Tr}(\hat{A}_k\hat{B}_k) &= \lambda_{\alpha}(p)\lambda_{\beta}(p)(\mathrm{len}(\mathcal{O}_{ k{\varepsilon}p}) - h^0_{P}(\mathcal{I}^{k{\varepsilon}}_{E'}\widehat{\mathcal{L}}^k_{0|_P})) + d_0 k^{n+2}\\ &+ \mathrm{Tr}(A'_{k{\varepsilon}}B'_{k{\varepsilon}}),\end{aligned}$$ where $A'_{k{\varepsilon}}$ and $B'_{k{\varepsilon}}$ are the infinitesimal generators of the actions $\alpha,\beta$ on $\mathcal{O}_{k{\varepsilon}p}$. We have an expansion $$\mathrm{Tr}(A'_{k{\varepsilon}}B'_{k{\varepsilon}}) = c_0' ({\varepsilon}k)^{n+2} + O(k^{n+1}).$$ So up to terms of order ${\varepsilon}^n$, the leading order term in is the same as that in $$\mathrm{Tr}(A_kB_k) - \frac{\mathrm{Tr}(A_k)\mathrm{Tr}(B_k)}{ d_k},$$ which is just $\langle\alpha,\beta\rangle = 0$. This shows $\langle\hat{\alpha},\hat{\beta}\rangle = O({\varepsilon}^n)$. A similar computation of the inner product on the blowup is in [@DV]. The statement about the relative Futaki invariants now follows from the definition $$F_T(\widehat{\mathcal{X}},\hat{\alpha}) = F(\widehat{\mathcal{X}},\hat{\alpha}) - \sum_{i=1}^d\frac{\langle \hat{\alpha},\hat{\beta}_i\rangle}{\langle\hat{\beta}_i, \hat{\beta}_i\rangle} F(\widehat{\mathcal{X}},\hat{\beta}_i),$$ where the $\mathbf{C}^*$-actions $\beta_i$ generate the torus $T$. Following the notation above, if $q \in M_0$ is a repulsive fixed point for $\alpha$ then $F(\widehat{\mathcal{X}}) < F(\mathcal{X})$ for ${\varepsilon}$ small enough. It remains to prove that the highest order correction term $$\left(\lambda(q) - \frac{b_0}{a_0}\right) \frac{{\varepsilon}^{n-1}}{2(n-2)!}$$ is negative. It is proved is [@jag] section 4 that, possibly after a fixed basechange of the test-configuration, the coefficient $\lambda(q) - \frac{b_0}{a_0}$ is integral and equals minus the Hilbert-Mumford weight of $q$ under the induced action of $\alpha$ on $\mathbf{P}(V)$. The Hilbert-Mumford criterion combined with a local computation then shows that the weight of such a repulsive fixed point must be positive (for details see the proof of Theorem 1.2 in [@Sto08]). Alternatively we can give a self-contained proof as follows. Let $M_0$ be the central fiber of our test-configuration and suppose that $q$ is a repulsive fixed point with weight $m_l$ and also let $r$ be a point in $\mathrm{red}(M_0)\cap \mathbf{P}(V_{m_0})$, ie. a lowest weight invariant point. Then as in the Futaki invariant calculation we have the exact sequence $$0\longrightarrow \mathcal{I}^{k{\varepsilon}}_{r} L^k_0 \longrightarrow L^k_0 \longrightarrow \mathcal{O}_{{\varepsilon}k r}\otimes L^k_0|_r \longrightarrow 0.$$ Write $-\lambda$ for the weight $m_l$, so $\lambda(q)=\lambda$ and $m_0\leqslant-\lambda-1$. The weights on $L_0$ are the opposite by duality and they are all at least $\lambda$. Using the notation from the proof of Theorem \[thm:main\], from the exact sequence we have $$\label{eq:weight}\begin{aligned} w_k &= w(\mathcal{I}^{k{\varepsilon}}_{r}L^k) + w(\mathcal{O}_{{\varepsilon}k r}) -km_0\,\mathrm{len}(\mathcal{O}_{{\varepsilon}k r}) \\ &\geqslant k\lambda \big(d_k - \mathrm{len}(\mathcal{O}_{{\varepsilon}kr})\big) + w(\mathcal{O}_{{\varepsilon}k r}) + k(\lambda+1)\mathrm{len}(\mathcal{O}_{{\varepsilon}kr}) \\ &= k\lambda d_k + k\mathrm{len}(\mathcal{O}_{{\varepsilon}kr}) + w(\mathcal{O}_{{\varepsilon}kr}). \end{aligned}$$ Now we need the expansions $$\begin{gathered} \mathrm{len}(\mathcal{O}_{{\varepsilon}kr}) = c({\varepsilon}k)^n + O(k^{n-1}) \\ w(\mathcal{O}_{{\varepsilon}kr}) = c'({\varepsilon}k)^{n+1} + O(k^n). \end{gathered}$$ It is important here that $c > 0$. This follows from [@hart], III Corollary 9.6. 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--- abstract: 'We study the zero temperature ground state of a two-dimensional atomic Fermi gas with chemical potential and population imbalance in the mean-field approximation. All calculations are performed in terms of the two-body binding energy $\epsilon_B$, whose variation allows to investigate the evolution from the BEC to the BCS regimes. By means of analytical and exact expressions we show that, similarly to what is found in three dimensions, at fixed chemical potentials, BCS is the ground state until the critical imbalance $h_c$ after which there is a first-order phase transition to the normal state. We find that $h_c$, the Chandrasekhar-Clogston limit of superfluidity, has the same value as in three dimensional systems. We show that for a fixed ratio $\epsilon_B/\epsilon_F$, where $\epsilon_F$ is the two-dimensional Fermi energy, as the density imbalance $m$ is increased from zero, the ground state evolves from BCS to phase separation to the normal state. At the critical imbalance $m_c$ phase separation is not supported and the normal phase is energetically preferable. The BCS-BEC crossover is discussed in balanced and imbalanced configurations. Possible pictures of what may be found experimentally in these systems are also shown. We also investigate the necessary conditions for the existence of bound states in the balanced and imbalanced normal phase.' author: - Heron Caldas - 'A. L. Mota' - 'R. L. S. Farias' - 'L. A. Souza' title: 'Superfluidity in Two-Dimensional Imbalanced Fermi Gases' --- Introduction ============ In the last few years, great experimental advances have permitted the manipulation of trapped neutral ultracold two-spin-components atomic Fermi gases with tremendous accuracy. The essential technics under full domain are the cooling, trapping, the control of the number of atoms in the sample, and the tunning of the inter-atomic (s-wave) interactions via the application of an external uniform magnetic (Feshbach resonance) field [@Inouye]. This allowed the investigation of the crossover from the Bardeen-Cooper-Schrieffer (BCS) phase of weakly bound Cooper pairs to the Bose-Einstein condensate (BEC) phase of strongly bound diatomic molecules in three-dimensional (3D) trapped Fermi gases [@Exp1; @Exp2; @Exp3]. These many-body quantum gases are even more exciting when the two-components have mass, chemical potential or population imbalance, since this situation can be met in many areas of physics, from condensed matter to high-density quark matter. Many exotic phases have been proposed as the ground state of these imbalanced Fermi gases, such as the homogeneous Sarma [@BP1] or breached-pair state [@BP2], the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state with modulated order parameter [@FFLO], phase separation of superfluid and normal components in real space [@Caldas1; @Caldas2], deformed Fermi surfaces [@Armen], and magnetized superfluid in the BEC side of the resonance [@Sheehy]. Recent experiments in imbalanced atomic Fermi gases confined in 3D harmonic traps have observed that these systems phase separate into a unpolarized superfluid core surrounded by a normal outer region [@Exp4; @Exp5; @Exp6; @Exp66; @Exp7]. The observation of the FFLO state is an experimental challenge, since it is predicted to exist only in a narrow window of asymmetry between the particle’s chemical potentials. Indeed recent experiments of Shin et al. [@Exp7] have not observed any evidence of the exotic FFLO state in 3D. However, lower dimensions may favor FFLO due to the nesting of the Fermi surfaces, as in the recent experiments with two spin mixture of ultracold $^{6}Li$ atoms trapped in an array of one dimensional (1D) tubes, where FFLO-like correlations were found for a range of polarizations [@Hulet]. Thus, another very important parameter under control is the trap geometry, which permits the experimental investigation of fermionic atoms in lower dimensions. A two-dimensional (2D) trapped Fermi gas may be achieved in experiments by flattening magnetic or dipolar confinements [@Exp8], by trapping atoms in specially designed pancake potentials [@Exp9], by radio frequency (rf)-induced two-dimensional traps [@Exp10], by gravito-optical surface traps [@Exp11]. Due to these experimental advances, recent studies have reported various features of 2D balanced Fermi gases: a 2D Fermi gas of atoms has been prepared and directly observed [@prl_detect_2d_gas], the first realization of a strongly interacting 2D Fermi gas of atoms has been reported [@Michael1], the pairing pseudogap in a 2D Fermi gas in the strong coupling regime has been explored [@Michael2], the biding energy of fermion pairs was measured along the dimensional crossover from 3D to 2D [@prlzwierlein], and in Ref. [@baur] the contributions of the effective-range corrections to two-body bound-state energies were considered. Inspired by these experimental progresses, some recent papers have investigated two-component Fermi gases both in 2D [@Tempere; @Duan; @Zhuang; @Simons; @Silva; @Du; @Wan], as well as in one-dimensional (1D) Fermi systems as, for instance, in Refs. [@Orso; @Drumond1; @Drumond2; @Guan; @Paata]. Besides, spin imbalance has also been investigated in the context of metallic systems as, for example, electron-hole bilayer [@Pieri; @Kazuo] and multi-band [@Mucio] systems. In the imbalanced scenario, much effort has been dedicated to study the Fermi polaron problem in which a single spin-$\downarrow$ atom interacts strongly with a Fermi sea of spin-$\uparrow$ atoms [@zollner; @parish1; @klawnn; @schmidt]. Fermi polarons were observed in a tunable Fermi liquid of ultracold atoms [@Polaron], and experimental evidence for the polaron-molecule transition was recently found [@nature_khol]. In this paper we study the zero temperature ($T$) ground state of a 2D atomic Fermi gas with chemical potential and population imbalance in the mean-field approximation. In this analysis we do not consider the Berezinskii-Kosterlitz-Thouless (BKT) corrections [@BKT] to the mean-field results in 2D. BKT corrections to mean-field results are more important close (and below) a certain critical temperature $T_c$, while at $T = 0$, as considered in the present paper, all the vortices and antivortices are bound in molecules, so BKT corrections are not relevant. The (topological) BKT transition has been investigated in 2D balanced [@Sa; @Duan2], as well as imbalanced systems [@Devreese], by taking into account the phase fluctuation effect. We show that in the grand canonical ensemble the fundamental state is BCS up to a critical value of the chemical potentials asymmetry $h_c$, after which there is a quantum phase transition to the normal state. In the canonical ensemble we study in detail the issue of phase separation. We show that the normal phase is always unstable to phase separation when the density asymmetry $m$ is less than the critical imbalance $m_c$. All calculations are performed in terms of the two-body binding energy $\epsilon_B$, whose variation allows to investigate the evolution from the BEC to the BCS regimes. The possibility of the FFLO state is ignored in this work, and we only consider pairing between atoms with equal and opposite momentum. The main results of this paper may be summarized as: We show that in a zero $T$ imbalanced 2D gas of fermionic atoms, BCS is the ground state up to chemical potentials imbalance $h_c$ at which there is a first-order phase transition to the normal state. We find that $h_c$ is the same as in 3D, namely, the Chandrasekhar-Clogston limit of superfluidity. We demonstrate that, as found theoretically and experimentally in 3D, for a fixed two-body binding energy $\epsilon_B$, as the imbalance $m$ is increased from zero the stable states are BCS, phase separation and normal phase. For balanced systems, the BCS-BEC crossover is governed only by $\epsilon_B$. However, for imbalanced systems, the BCS-BEC crossover will depend on $\epsilon_B$ and the imbalance $m$. For a fix value of $\epsilon_B$, the crossover is possible only for asymmetries supported by phase separation. Above a critical value of the number densities imbalance $m_c$, phase separation is destroyed (in favor of the normal phase) as well as the path to BEC regime. We investigate the conditions for the existence of bound states in the balanced and imbalanced normal phase. We find that in the balanced normal phase at zero temperature, there will be bound states for any value of the two-body binding energy. For the imbalanced normal configuration, we find that stable (and real) bound states exist when $2\mu+|\epsilon_B|> 2h$, where $h$ is the chemical potential imbalance. As we point out in Section \[comp\], the results we obtain in the Canonical ensemble agree with previous works which employed the Grand Canonical ensemble, as it should be, since the phase diagram should be independent of this choice. The paper is organized as follows: In Sec. \[Int\] we present the model Hamiltonian and derive an analytical expression for the zero temperature thermodynamical potential in the mean-field approximation for the two-component gas of fermionic atoms in 2D. In Sec. \[P\] we employ this expression to investigate the stability of the possible phases of 2D imbalanced Fermi gases, both at fixed chemical potentials and densities. We also discuss the BCS-BEC crossover in balanced and imbalanced systems. In Sec. \[bs\] we study the necessary conditions for the formation of bound states in the normal phase. In Sec. \[comp\] we perform a comparison with solid zero temperature results obtained in the recent literature. Finally, Sec. \[conc\] is devoted to the conclusions. The Model Hamiltonian and the Mean-Field Theory {#Int} =============================================== We start by considering a 2D nonrelativistic dilute system of fermionic atoms of mass $M$, with two hyperfine states labeled as $\sigma = \uparrow, \downarrow$. This spin $\uparrow$ and $\downarrow$ mixture can be done with the two lowest hyperfine states of $^{6}{\rm Li}$ atoms, as in the 3D experiments [@Exp5; @Exp6; @Exp7]. This 2D system may be experimentally realized confining a 3D Fermi gas into a single layer (or a stack of layers) by a tight transverse harmonic oscillator potential $V(z)=\frac{m}{2} \omega_z^2z^2$, where $\omega_z$ is the trapping frequency along the confined direction. For very low temperatures and densities such that $k_B T, \epsilon_F<<\hbar \omega_z$, where $\epsilon_F$ is the Fermi energy, and weak longitudinal trapping, the collisions can be considered to be quasi-2D [@prl_detect_2d_gas]. The single-particle dispersion relations are given by $\xi_k=\frac{\hbar^2 k^2}{2 M}$. Throughout the paper we set $\hbar=1$. Since at low densities (appropriate to describe the ultracold trapped Fermi gases we are interested) the form of the potential is not probed, it can be considered that the atoms interact via a contact interaction which is modeled by the following pairing Hamiltonian: $$\label{H0} H=H_{0} + H_{\rm int},$$ where $$\label{H} H_{0}=\sum_{k,\sigma = \uparrow, \downarrow} \epsilon_k^{\sigma} \psi_{\sigma}^{\dagger}(k) \psi_{\sigma} (k),$$ is the kinetic (free) part of $H$ and $H_{\rm int}$ is given below. $\psi_{\sigma}^{\dagger}(k)$ and $\psi_{\sigma}(k)$ in Eq. (\[H\]) are the creation and annihilation operators, respectively, for the $\uparrow, \downarrow$ particles. To assure population imbalance in the system, we have introduced different chemical potential for the species $\sigma$ as $\mu_{\sigma}=\mu + \sigma h$, where $\sigma h \equiv \pm h$. Then, the chemical potential $\mu_{\sigma}$ fix the number densities $n_{\sigma}$ of the different fermions. The new dispersions for the free species $\sigma$, relative to their Fermi energies, are $\epsilon_k^{\sigma} \equiv \xi_k -\mu_{\sigma}$. The interaction Hamiltonian is given by $$H_{int}= g \sum_{k, k'} \psi_{\uparrow}^{\dagger}(k) \psi_{\downarrow}^{\dagger}(-k) \psi_{\downarrow} (-k') \psi_{\uparrow} (k'),$$ where the bare coupling constant $g$ is negative, to express the attractive (s-wave) interaction between the spin $\uparrow$ and $\downarrow$ fermionic atoms. After the mean field (MF) approximation and the subsequent diagonalization of the expression for $H_{MF}$, we arrive at the following expression for thermodynamic potential: $$\begin{aligned} \Omega &=&-\frac{\Delta^2}{g}+\int_{k<k_1,~k>k_2} \frac{d^2 k}{(2\pi)^2} (\epsilon_k^b - {\cal E} _k^{\beta}) \nonumber \\ &+& \int_{k_1}^{k_2} \frac{d^2 k}{(2\pi)^2} \epsilon_k^b \nonumber \\ &=&-\frac{\Delta^2}{g}+\int_{k<k_1,~k>k_2} \frac{d^2 k}{(2\pi)^2} (\epsilon_k^+ - E_k) \nonumber \\ &+& \int_{k_1}^{k_2} \frac{d^2 k}{(2\pi)^2} \epsilon_k^b, \label{Omega1}\end{aligned}$$ where, for simplicity of notation we have labeled $\downarrow=a$, $\uparrow=b$. Here we have defined ${\cal{E}}_k^{a,b}= E_k \pm \epsilon_k^{-}$ are the quasiparticle excitations, with $E_k=\sqrt{ {\epsilon_k^{+}}^2+\Delta^2 }$, $\epsilon_k^{\pm} = \frac {\epsilon_k^a \pm \epsilon_k^b}{2}$ and the constant pairing gap is given by $\Delta = - g \int \frac{d^2 k}{(2\pi)^2} \langle \psi_{\downarrow}^{\dagger}(-k) \psi_{\uparrow}^{\dagger}(k) \rangle = \Delta^*$. In Eq. (\[Omega1\]) $$\label{roots} k_{1,2}^2 = \frac{k_a^2+k_b^2}{2} \pm \frac{1}{2}\sqrt{(k_b^2-k_a^2)^2-16M^2 \Delta^2}$$ are the roots of ${\cal E} _k^{\beta}$, and $k_{\alpha}=\sqrt{2M \mu_{\alpha}}$ is the Fermi momentum of the species ${\alpha=a,b}$. From the equation above one sees that the condition for $\Delta$ such that $k_{1,2}$ is real is $\Delta \leq \frac{k_b^2-k_a^2}{4M}$. To regulate the ultraviolet divergence associated with the first integral in Eq. (\[Omega1\]) we introduce [@Randeria]: $$\label{reg} \frac{1}{U}= \int \frac{d^2 k}{(2 \pi)^2} \frac{1}{2\xi_k+|\epsilon_B|},$$ where $U \equiv - g > 0$, and $\epsilon_B$ is the 2D two-body binding energy. In order to make contact with current experiments, it is convenient to relate $\epsilon_B$ to the three dimensional scattering length $a_s$. In the scattering of atoms confined in the axial direction by a harmonic potential with characteristic frequency $\omega_L$ they are related by [@Petrov; @Tempere]: $$\label{EB} |\epsilon_B|=\frac{C \omega_L}{\pi} exp\left( \sqrt{2 \pi} \frac{l_L}{a_s} \right),$$ where $a_s$ is the 3D s-wave scattering length, $\omega_L=\sqrt{8 \pi^2 V_0/ (M \lambda^2)}$, $l_L=1/\sqrt{M\omega_L}$ is the axial ground state, and $C \approx 0.915$. $V_0$ is the amplitude of the periodic potential $V_0 \sin^2(2 \pi z/ \lambda)$ generated by two counter-propagating laser beans with length $\lambda$ parallel to the $z$-axis [@Tempere]. It is worth to mention that if $l_L>a_s$, the relative wave function of molecules in quasi-2D Fermi gases explores higher transverse harmonic oscillator modes [@baur]. In this case, i.e., in the limit of large binding energies $(|\epsilon_B|>\omega_L)$, effective corrections to the zero-range usual result which gives Eq. (\[EB\]) are required [@baur]. Thus we obtain an analytical expression for the grand canonical thermodynamic potential at $T=0$: $$\begin{aligned} \bar\Omega &=& \Omega(\Delta, \mu)+ \Theta_h \Omega(\Delta, h)\\ \nonumber &=& \Delta^2\left[ \ln \left(\frac{\sqrt{\mu^2+\Delta^2}-\mu}{|\epsilon_B|}\right) -\frac{1}{2} \right]-\mu\left(\sqrt{\mu^2+\Delta^2}+\mu \right) - \Theta_h \left[2h \sqrt{h^2-\Delta^2}- \Delta^2 \ln \left(\frac{h+\sqrt{h^2-\Delta^2}}{h-\sqrt{h^2-\Delta^2}} \right) \right], \label{LagTLM}\end{aligned}$$ where $\bar\Omega \equiv \frac{4 \pi}{M}\Omega$, $\Theta_h \equiv \Theta(h^2-\Delta^2)$, $\mu=\frac{\mu_a+\mu_b}{2}$, $h=\frac{\mu_b-\mu_a}{2}$, and $\Theta(x)$ is the Heaviside step function, defined as $1$ if $x\geq0$, and $0$ if $x<0$. With these definitions, Eq. (\[roots\]) may be written as $k_{1,2}^2 = 2M(\mu \pm \sqrt{h^2-\Delta^2})$ and the condition for real $k_{1,2}^2$ is now $\Delta \leq h$. Phases of 2D Imbalanced Fermi Systems and Their Stabilities {#P} =========================================================== Fixed Chemical Potentials ------------------------- In the grand canonical ensemble the chemical potentials of the two-components are held fixed. This happens when the system is connected to reservoirs of species $a$ and $b$ such that the particle densities are allowed to change. We analyze the ground state of this case for balanced and imbalanced configurations. ### Balanced Systems The gap and number equations are obtained by $\partial \Omega/ \partial \Delta =0$ and $n_{\alpha}=- \partial \Omega / \partial \mu_{\alpha}$, respectively. For the balanced system where $h=0$, which implies $\Theta_h=0$, we find $$\sqrt{\mu^2+\Delta^2}-\mu = |\epsilon_B|, \label{gap}$$ and $$\sqrt{\mu^2+\Delta^2}+ \mu = 2 \epsilon_F, \label{mu}$$ where the two dimensional Fermi energy is defined as $\epsilon_F = \frac{\pi n_T}{M}$, with $ n_T=n_a+n_b$, see Eq. (\[eq2\]). In the balanced configuration $n_{a}=n_{b} \equiv n = \frac{n_T}{2}$. Solving these two equations self-consistently we arrive at $$\Delta_0=\sqrt{2\epsilon_F | \epsilon_B |}, \label{gap0}$$ and $$\label{mu0} \mu_0=\epsilon_F - \frac{|\epsilon_B|}{2}.$$ The value of the free energy at the minimum is $$\Omega(h=0, \Delta=\Delta_0) \equiv \Omega_0 = - \kappa \left(\mu_0 + \frac{|\epsilon_B|}{2} \right)^2, \label{min1}$$ where $\kappa \equiv \frac{M}{2 \pi}$, whereas the energy of the balanced normal state is given by $$\Omega(h=\Delta=0) \equiv \Omega_b^N = - \kappa \mu_0^2. \label{min2}$$ A direct comparison between Eqs. (\[min1\]) and (\[min2\]) shows that the superfluid state is energetically preferable to the normal state for any $\epsilon_B \neq 0$. Since a two-body bound state exists even for an arbitrarily small attraction in 2D [@Randeria], the pairing instability will always happens in 2D balanced two-component Fermi systems at $T=0$. ### BCS-BEC Crossover in Balanced Systems {#CrossoverBalanced} The two well known [@Randeria; @Loktev] equations (\[gap0\]) and (\[mu0\]) reveal that the BCS-BEC crossover can be accessed by varying $\epsilon_B$ from weak to strong interaction regimes. The BCS state is reached in the weak attraction limit or high density ($\epsilon_F$), $|\epsilon_B| << \epsilon_F$, where $\mu_0 \cong \epsilon_F$ and the energy gap is found to be $$\label{evalgap} \Delta_0 = 2 \sqrt{\epsilon_F \rm{E}_{\Lambda}}~e^{-\frac{2 \pi}{M U}} << \epsilon_F,$$ where $\rm{E}_{\Lambda} \equiv \frac{\Lambda^2}{2M}$, and $\Lambda$ is a momentum cutoff. To obtain Eq. (\[evalgap\]) from Eq. (\[gap0\]) we have made use of Eq. (\[reg\]). The opposite limit, of very strong attraction or low density, $|\epsilon_B| >> \epsilon_F$, results in the formation of composite bosons with $\mu_0 \cong - |\epsilon_B|/2$. Since the BCS and BEC regions are characterized by positive and negative chemical potentials, respectively, a clear distinction between these two regimes is the value of the binding energy at which $\mu_0$ changes sign i.e., $|\epsilon_B|=2\epsilon_F$. ### Imbalanced Systems We now turn our attention to the cases where $h \neq 0$. The free energy of the imbalanced normal state, $\Omega(h,\Delta=0) \equiv \Omega_i^N$, is found to be $$\begin{aligned} \Omega_i^N&=&\int_{k>k_2, k<k_1} \frac{d^2 k}{(2\pi)^2} (\epsilon_k^+ - |\epsilon_k^+|)\\ \nonumber &+& \int_{k_1}^{k_2} \frac{d^2 k}{(2\pi)^2} \epsilon_k^b.\end{aligned}$$ In the limit $\Delta \to 0$ we find $k_1(\Delta=0)=\sqrt{2M(\mu-h)}=k_F^a < k_F$ and $k_2(\Delta=0)=\sqrt{2M(\mu+h)}=k_F^b > k_F$, where $k_F=\sqrt{2M \mu}$. We are considering $\mu_{\alpha}$ and $h$ positive. Thus the equation above is written as: $$\begin{aligned} \Omega_i^N= \int_{0}^{k_F^a} \frac{d^2 k}{(2\pi)^2} \epsilon_k^a + \int_{0}^{k_F^b} \frac{d^2 k}{(2\pi)^2} \epsilon_k^b,\end{aligned}$$ which gives, after the integration in $k$, the free energy of a (normal) two species gas of fermionic atoms in two dimensions $$\begin{aligned} \label{fn} \Omega_i^N (\mu_a, \mu_b) &=& -\frac{\kappa}{2} [(\mu-h)^2 + (\mu+h)^2] \nonumber \\ &=& -\frac{\kappa}{2}[\mu_a^2 + \mu_b^2].\end{aligned}$$ The gap and number equation now read $$\label{gap2} \ln \left (\frac{ \sqrt{\mu^2+\Delta^2}-\mu}{ |\epsilon_B|} \right) + \Theta_h \ln \left(\frac{h+\sqrt{h^2-\Delta^2}}{h-\sqrt{h^2-\Delta^2}} \right)=0,$$ $$\sqrt{\mu^2+\Delta^2}+ \mu = 2 \epsilon_F. \label{n2}$$ Note that for $h<\Delta$ Eq. (\[gap2\]) is reduced to Eq. (\[gap\]). Seeking now solutions where $h>\Delta$ we then have to solve $$\label{gap3} \frac{ \sqrt{\mu^2+\Delta^2}-\mu}{ |\epsilon_B|} = \frac{h-\sqrt{h^2-\Delta^2}}{h+\sqrt{h^2-\Delta^2}}.$$ Solving these equations self-consistently we find $$\label{gaph} \Delta_S(h)=\sqrt{\Delta_0(2h-\Delta_0)}.$$ From the equation above we see that the “Sarma state” (as explained below) gap and the chemical potential imbalance exist in the ranges $$\label{Deltah} 0 \leq \Delta_S(h) \leq \Delta_0,$$ $$\label{rangeh} \frac{\Delta_0}{2} \leq h \leq \Delta_0,$$ where the upper limit is imposed by the existence of real $k_{1,2}^2$. From the graphical analysis of $\Omega$ as a function of $\Delta$ for various asymmetries, shown in Fig. \[omega\], one sees that the first curve, from top to bottom, is the usual balanced system with its minimum at $\Delta_0$. Increasing the imbalance $h$ and keeping $\mu$ positive (to guarantee that we are in the BCS regime), the minimum is still located at $\Delta_0$ up to a maximum or critical imbalance $h_c$, after which there is a quantum phase transition to the normal state with $\Delta=0$. $h_c$ is find through the equality $\Omega_0=\Omega_i^N$, which yields: $$h_c^2= \mu_0 |\epsilon_B| + \left( \frac{\epsilon_B}{2} \right)^2,$$ from which one easily finds plugging in the equation above $\mu_0$ from Eq. (\[mu0\]) in the BCS limit ($|\epsilon_B| << \epsilon_F$): $$h_c=\frac{\Delta_0}{\sqrt{2}},$$ which is the same ${\rm 3D}$ result known as the Chandrasekhar-Clogston limit of superfluidity [@Chandrasekhar; @Clogston]. We have seen above that, as happens in 3D [@Caldas1; @Caldas2; @JSTAT1], the BCS phase turns from stable, while $h < h_c$, to metastable, when $h > h_c$, in which case the normal phase is stable. Besides these two phases, there is an unstable phase, known as Sarma state, corresponding to a local maximum of $\Omega$ versus $\Delta$, that is located between the BCS minimum at $\Delta_0$ and the normal phase with $\Delta=0$. Fixed Number Densities ---------------------- In the previous subsection we discussed the stability of several phases with fixed chemical potentials $\mu_a$ and $\mu_b$. We investigate now the situations where the number densities $n_a$ and $n_b$ are fixed, since it is appropriate to describe trapped gases in current experiments. It is convenient to introduce now the “magnetization” defined as $m=-\frac{d \Omega}{d h}=n_b-n_a$, which will be needed to describe the energy of the Sarma phase in terms of the number densities. The total derivative of $\Omega=\Omega(\mu_a,\mu_b,\Delta)$ is written as $$d \Omega = \frac{\partial \Omega}{\partial \mu_a} d \mu_a + \frac{\partial \Omega}{\partial \mu_b} d \mu_b + \frac{\partial \Omega}{\partial \Delta} d \Delta. \label{tot}$$ Since the densities have to evaluated at the minimum of $\Omega$, i.e., at $\Delta_0$, the last term vanishes. The particle densities are given by $n_{\alpha}=- \frac{\partial \Omega}{\partial \mu_{\alpha}}$. Then, using $\mu_a=\mu-h$ and $\mu_b=\mu+h$ in Eq. (\[tot\]) we find $$m=n_{b}-n_{a}=-\frac{d \Omega}{dh}, \label{eq1}$$ and $$n_T=n_{b}+n_{a}=-\frac{d \Omega}{d \mu}. \label{eq2}$$ Then the magnetization can be written as $$m=\frac{M}{\pi} \sqrt{h^2-\Delta^2}. \label{mag}$$ Solving equations (\[n2\]), (\[gap3\]) and (\[mag\]) self-consistently, we find the gap $\Delta_{S}(m)$ and the “average” chemical potential $\mu_S$ of the Sarma phase $$\Delta_{S}(m)=\sqrt{\Delta_0^2-\frac{2 \pi \Delta_0}{M}m}, \label{gapS}$$ where $\Delta_0$, the gap parameter of the balanced system, is given by Eq. (\[gap0\]), and $$\mu_S=\epsilon_F - \frac{{\Delta_S}^2}{4 \epsilon_F}. \label{mu_S}$$ Note that the equations above reduce to Eqs. (\[gap0\]) and (\[mu0\]), respectively, since in the limit $m \to 0$, $\Delta_S \to \Delta_0$, as it should be. From Eq. (\[gapS\]) it is easy to obtain the windows for the Sarma gap and $m$: $$0 \leq \Delta_{S}(m) \leq \Delta_0, \label{gapS2}$$ $$0 \leq m \leq m_{max}, \label{m_max}$$ where $m_{max}=\frac{M \Delta_0}{2 \pi}$ is the maximum value for the density imbalance in the Sarma phase, which is easily obtained from Eq. (\[gapS\]). The expression for the energy $E$ (not the free energy or thermodynamic potential) of the homogeneous Sarma phase is obtained as: $$E(n_T,m,\Delta)= \Omega(\mu,h,\Delta) + \mu_a n_a + \mu_b n_b. \label{eq3}$$ To write $E=E(n_T,m)$, we express $\mu_a n_a + \mu_b n_b=\mu n_T + h m$, with $\mu=\mu(n_T)$, $h=h(m)$ to find: $$\begin{aligned} \label{eq33} E(n_T,m,\Delta) &\equiv& E^S(n_a,n_b)= \Omega(n_T,m,\Delta) + \mu n_T + h m \nonumber \\ &=& \frac{M}{4 \pi} \left\{ \Delta_S^2 \left[ \ln \left( \frac{\Delta_S^2}{\Delta_0^2} \right) -\frac{1}{2} \right] - 2\epsilon_F \left( \epsilon_F - \frac{\Delta_S^2}{4\epsilon_F} \right) -2 \sqrt{\bar m^2 + \Delta_S^2} ~\bar m + \Delta_S^2 \ln \left( \frac{\sqrt{\bar m^2 + \Delta_S^2} + \bar m}{\sqrt{\bar m^2 + \Delta_S^2} - \bar m} \right) \right\} \nonumber \\ &+& \left( \epsilon_F - \frac{\Delta_S^2}{4\epsilon_F} \right) n_T + \sqrt{\bar m^2 + \Delta_S^2}~m ,\end{aligned}$$ where $\bar m \equiv \frac{\pi m}{M}$. To investigate which one is the ground state of an imbalanced system with different densities $n_a$ and $n_b$, we have to see which energy of the following states we are considering is smaller, the normal $E^N(n_a, n_b)$, see Eq. (\[EN\]) below, the homogeneous Sarma $E^S(n_a,n_b)$, or the phase separation (PS) $E^{PS}(n_a, n_b)$ state. The PS is an inhomogeneous phase where given $n_a$ and $n_b$ densities in a trap, a fraction $1-x$ of the 2D (real) space is in the BCS phase where both species have a common density $n_a^{BCS}= n_b^{BCS} = n$ and the rest of particles are in the normal phase occupying the fraction $x$ around [^1] the BCS core, with densities $\tilde n_a$ and $\tilde n_b$. The most favored composition minimizes: $$\label{PS} E^{PS}=Min_{x, \tilde n}[(1-x)E(n)+ xE^N(\tilde n_a, \tilde n_b)],$$ where the space fraction is obviously limited to the range $0 \leq x \leq 1$. The energy of the balanced superfluid phase is found as $$\begin{aligned} \label{BCS(n)} E(n) &=& \Omega(\Delta, h=0, \mu) + \mu n_T \nonumber \\ &=& \frac{M}{4 \pi} \left\{ \Delta^2 \left[ \ln \left( \frac{\Delta^2}{\Delta_0^2} \right) -1 \right] + 2 \epsilon_F^2 \right\} \nonumber \\ &=& \frac{M \Delta^2}{4 \pi} \left[ \ln \left( \frac{\Delta^2}{\Delta_0^2} \right) -1 \right] + \frac{\pi (2n)^2}{2M}.\end{aligned}$$ Since $E(n)$ which enters $E^{PS}$ has to be written at its minimum, the equation above turns out to be: $$\begin{aligned} \label{BCS(n)2} E(n) &=& \frac{\pi {n_T}^2}{2M} -\frac{M \Delta_0^2(n)}{4 \pi} \nonumber \\ &=& \frac{2 \pi}{M} n^2 - |\epsilon_B| n,\end{aligned}$$ where the gap parameter of the BCS phase is given by Eq. (\[gap0\]) and written as $\Delta_0(n)=2\sqrt{\frac{\pi}{M} | \epsilon_B | n}$. The energy of the normal state entering in Eq. (\[PS\]) is easy to find from Eq. (\[fn\]) as $$\begin{aligned} \label{EN} E^N(n_a, n_b)&=& \Omega^N(n_a, n_b) + \mu_a n_b + \mu_a n_b \nonumber \\ &=& \frac{1}{2 \kappa} (n_a^2 + n_b^2).\end{aligned}$$ Note that since $n_a= \frac{n_T - m}{2}$ and $n_b= \frac{n_T + m}{2}$, the equation above for $m=0$ gives $E^N=\frac{\pi n_T^2}{2 M}$ agreeing with the result of Eq. (\[BCS(n)\]) in the limit $\Delta \to 0$. As we mentioned before, the $n_a$ and $n_b$ particle densities are accommodated in the trap as $$\begin{aligned} \label{n} n_a &=& x \tilde n_a + (1-x) \tilde n, \nonumber \\ n_b &=& x \tilde n_b + (1-x) \tilde n.\end{aligned}$$ Then we rewrite Eq. (\[EN\]) as $$\begin{aligned} \label{EN2} && E^N(\tilde n_a, \tilde n_b) = \nonumber \\ && \frac{1}{2 \kappa} \left[ \left(\frac{n_a - (1-x) \tilde n}{x} \right)^2 + \left(\frac{n_b - (1-x) \tilde n}{x} \right)^2 \right],\end{aligned}$$ It is interesting to note that Eqs. (\[BCS(n)2\]) and (\[EN\]) are easily obtained from Eq. (\[eq33\]) in the limits $m \to 0$ and $m \to m_{max}$, respectively. In the minimization of $E^{PS}$ we shall find $x \approx 0$ and $\tilde n \approx n_a$ ($~\rm{or}$ $ n_b$) if $n_b \approx n_a$, which means that the entire system is in the superfluid phase. In the other limit, if $n_b >> n_a$, the system will be in the normal state, so we shall find in the minimization of $E^{PS}$ $x \approx 1$ and $\tilde n \approx 0$. In the intermediate cases the system will phase separate, i.e., if the imbalance is less than the critical value, as experiments have shown for 3D gases. ### Phase Competition There are some limiting cases where the comparison between the phase separation and Sarma states can be done analytically and exactly. The first one is obtained when $m=0$, giving $E^S(n)=E^{PS}(n)=E^{BCS}(n)$. The second case happens when $m=m_{max}$, which reduces the Sarma phase to the normal phase with $\Delta=0$. This is easily seen from Eqs. (\[m\_max\]) and (\[gapS2\]). Then we define $\Delta E = E^{PS}-E^N$, which is written as $$\begin{aligned} \label{DeltaE} \Delta E &=& \left[\frac{\tilde n^2}{\kappa}-E(\tilde n)\right](x-1) \nonumber \\ &+& \frac{1}{2 \kappa} \left[ (n_T -\tilde n)^2 -2 n_a n_b +\tilde n^2 \right]\left(\frac{1}{x} -1 \right).\end{aligned}$$ Minimization with respect to $x$ yields: $$\label{xmin1} x_{min}=\sqrt{\frac{(n_T-\tilde n)^2 -2n_an_b +\tilde n^2}{2(\tilde n^2 - \kappa E(\tilde n))}}.$$ An upper bound on $\Delta E$ can be obtained by setting the density of the BCS component of the PS state $\tilde n=n_a$ and writing $n_b= n_a + m$, where for low imbalance, $m <~n_a, n_b$, Eq. (\[xmin1\]) gives: $$\label{xmin} x_{min}= \frac{m}{ \sqrt{ \frac{M^2 \Delta_0^2(n_a)}{4 \pi^2} + \frac{3}{2} n_a^2 }},$$ and $$\begin{aligned} \label{deltaE} \Delta E(x_{min}) &=& - \left( \frac{M \Delta_0^2(n_a)}{4 \pi} + \frac{3 \pi}{2 M} n_a^2 \right) \nonumber \\ &\times& (1-x_{min})^2 < 0.\end{aligned}$$ Unlike the three-dimensional case where the result obtained is an approximation, valid only for small asymmetries $m << n_a, n_b$, where terms of order ${m}^3/{na}^{4/3}$ have been neglected [@Caldas1; @Caldas2], the result above is exact and valid for any $m < n_a, n_b$. These results agree with the numerical analysis made to construct the phase diagram depicted in Fig. (\[pd\]). Let us now show the results of the minimization of Eq (\[PS\]) with respect to $x$ and $n$ for a given $n_a$ and $n_b$. The results are shown in Fig. (\[pd\]). =5.5cm We show in Fig. (\[cartoon\]) an illustration of what may be found experimentally in an imbalanced $2D$ gas of femionic atoms. The first picture represents the balanced system with $m=0$. The following pictures represent a set of experiments with fixed $\epsilon_B/\epsilon_F=0.0318$ with increasing imbalance. From the second picture to the fifth the system phase separates and the sixth picture is for the critical imbalance $m_c$ representing the normal phase. ### BCS-BEC Crossover in Imbalanced Systems Now we investigate the BCS-BEC crossover in an imbalanced gas of fermionic atoms. As we have seen, there are three possible stable phases with fixed number densities. For zero imbalance between the densities, the only phase present is BCS (for any value of the interaction between the spin up and down fermionic atoms) and the crossover is that discussed in Subsection \[CrossoverBalanced\], which is governed only by $\epsilon_B$. For imbalances where phase separation is possible, the BCS-BEC crossover can be realized in the superfluid core of the separated phases, again varying $\epsilon_B$ in $\mu_0=\frac{2 \pi \tilde n}{M} - \frac{|\epsilon_B|}{2}$, where $\tilde n$ is the BCS density that minimizes the PS formation, as described above Eq. (\[PS\]). For large enough asymmetries, phase separation is not supported and the entire system is in the normal phase with no possibility of pairing formation and the crossover to the BEC regime. Then, when $m \neq 0$ the BCS-BEC crossover can not be accessed only by varying the biding energy since the imbalance plays an important role now. Bound States in the Normal Phase {#bs} ================================ In this section we investigate the necessary conditions for the formation of bound states in the normal phase. We expand the action up to second order in the order parameter $|\Delta_{\vec{q}}|$, and obtain $$S_{\text{eff}}=\sum_{\vec{q},\omega_0}\alpha({|\vec{q}|}, \omega_0) |\Delta_{\vec{q}}|^{2}+ \mathcal{O}\left(|\Delta|^4\right)\,,$$ where $\alpha(|{\vec{q}}|,\omega_0)=\frac{1}{g}-\chi({|\vec{q}|}, \omega_0)$, and $\chi(|{\vec{q}}|, \omega_0)$ is the generalized pair susceptibility: $$\begin{aligned} \label{chi1-1} \chi(|{\vec{q}}|, \omega_0)= \sum_{\vec{k}} \frac{1-n(\xi_{\vec{k}-\vec{q}/2,\uparrow})-n(\xi_{\vec{k}+\vec{q}/2,\downarrow})}{\xi_{\vec{k}-\vec{q}/2,\uparrow}+\xi_{\vec{k}+\vec{q}/2,\downarrow}-\omega_0}.\end{aligned}$$ Evaluation of the equation above is straightforward. At zero temperature we find that $\chi(|{\vec{q}}|, \omega_0)$ is given by: $$\begin{aligned} \label{chi1-2} \chi(|{\vec{q}}|, \omega_0) &=& N(0) \left[\ln\left(\frac{2 \omega_c}{h} \right) - \ln \left(1 - \frac{\omega_0}{2h}+ \sqrt{ \left(1 - \frac{\omega_0}{2h} \right)^2 -{\bar q}^2} \right) \right],\\ \nonumber &&{\rm for}~ \bar q \leq 1 - \frac{\omega_0}{2h},\end{aligned}$$ where $\omega_c$ is an energy cutoff, $N(0)=\frac{m}{2 \pi}$ is the density of states at the Fermi level, $\bar q \equiv \frac{v_F |q|}{2h}$, and $v_F$ is the Fermi velocity. The spectrum of bound states is given by $1/g-\chi(|{\vec{q}}|=0, \omega_0)=0$, which corresponds to the Thouless criterion $1-g \chi=0$, signalizing the divergence of the T-matrix [@Loktev]. Using Eq. (\[reg\]) to trade the cutoff in favor of the 2D two-body binding energy $\epsilon_B$, we obtain the following real (and unique) equation for the energies of these states in the normal phase at zero $T$: $$\label{bond} \omega_0=2h-(2\mu+|\epsilon_B|).$$ In the balanced normal phase with $h=0$ we obtain $\omega_0=-(2\mu+|\epsilon_B|)$ that, for $\mu>0$, corresponds to the energy of Cooper pairs, agreeing with previous work [@Loktev]. In the imbalanced case, bound states exist for $2\mu+|\epsilon_B|> 2h$. Comparison With Some Previous Zero Temperature Results {#comp} ====================================================== In this paper using the mean-field approximation we have studied the zero temperature ground state of a 2D imbalanced gas of fermonic atoms. We performed our calculations in the Grand Canonical and Canonical ensembles, considering chemical potential and population imbalance, respectively. It is very interesting to note that our results corroborate with solid results obtained in the recent literature. In particular, the result of our Fig. \[pd\] is equivalent to the diagram (bottom panel of Fig. 2) in the work of He and Zhuang [@Zhuang]. In this work we have shown that if you change the ensemble the results are the same, as expected, since it is known that the physics can not depend on the ensemble used, and we have verified this in detail. Similar studies and results can be found in other very interesting and recent papers [@Tempere; @Simons], and we have obtained the same results of these papers where the order of the phase transition from BCS to normal state at a critical chemical potential imbalanced is first order. Results II mentioned in our introduction also agree with the results found in [@Simons]. In Ref. [@Du] the mean-field zero $T$ phase diagram of an imbalanced Fermi gas was investigated when there is also mass imbalance. Although they did not consider phase separation, they studied the influence of population and mass imbalance on the different Fermi surfaces topologies, associated to stable and unstable phases. Spin-orbit coupling (SOC) $\lambda$ has been taken into account in Ref. [@Wan] in a imbalanced 2D Fermi gas, and it was found that for large values of the SOC there is a topological phase transition from phase separation to a nontrivial superfluid phase. Our phase diagram of Fig. 2 is in complete agreement with the ($\lambda=0$ i.e., the vertical axes of the) zero $T$ mean-field phase diagrams of Fig. 1 shown in Ref. [@Wan]. Conclusions {#conc} =========== We have investigated superfluidity in two-dimensional imbalanced Fermi gases. With exact expressions, we have constructed the zero temperature mean-field phase diagram of these systems. We show that in the grand canonical ensemble, BCS is the ground state up to a critical chemical potential imbalance $h_c$ at which there is a first-order phase transition to the normal state. In the canonical ensemble, relevant to current experiments, we also show that, as found theoretically and experimentally in 3D, for a fixed two-body binding energy $\epsilon_B$, as the imbalance $m$ is increased from zero the stable states are BCS, phase separation and normal phase. Regarding the BCS-BEC crossover, in balanced systems it is governed only by $\epsilon_B$. However, for imbalanced systems, the BCS-BEC crossover will depend on $\epsilon_B$ and the imbalance $m$. For a fix value of $\epsilon_B$, the crossover is possible only for asymmetries supported by phase separation. Above a critical value of the number densities imbalance $m_c$, phase separation is destroyed in favor of the normal phase, as well as the path to BEC regime. Thus, for a given $m<m_c$ such that the system phase separates into BCS and normal phases, the BEC regime can be accessed by varying $\epsilon_B$ from the weak to strong couplings. A smaller and smaller circular core shall be visibly apparent, as $\epsilon_B$ is increased, signalizing the condensation of the Cooper pairs. This is a direct prediction of our results that could be tested experimentally. We have investigated the conditions for the existence of bound states in the normal phase. As we have seen, since a two-body bound state exists even for an arbitrarily small attraction in 2D, the normal balanced state at $T=0$ corresponds to the superfluid case. In the imbalanced normal phase, bound states are likely to be formed provided $2\mu+|\epsilon_B|> 2h$. As we mentioned in the introduction Section, the Fermi polaron problem appears in a very large imbalance $m>m_c$ or, in the unities of Fig. \[pd\], $m/n_T \approx 1$. 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--- author: - | M. Krech\ Fachbereich Physik, BUGH Wuppertal, 42097 Wuppertal, Germany title: 'Short - time scaling behavior of growing interfaces' --- Introduction ============ Analytic Theory =============== Monte-Carlo Results =================== Acknowledgment ============== The author gratefully acknowledges partial financial support of this work through the Heisenberg program of the Deutsche Forschungsgemeinschaft.

--- author: - Xiaojia Li - 'and Da-Xin Zhang' title: 'Proton decay suppression in a supersymmetric SO(10) model' --- Introduction ============ Grand Unified Theory (GUT)[@gut1; @gut2] is one of the most attractive candidates for the physics beyond the Standard Model (SM). The Supersymmetric (SUSY) GUT models based on SO(10)[@clark1982; @aulakh1983] are especially interesting for several reasons. Firstly, each generation of fermion superfields are unified in a single **16**-plet spinor representation which contains the right-handed neutrino, so that sub-eV neutrino masses can be generated naturally by the seesaw mechanism[@seesawi1; @seesawi2; @seesawi3; @seesawi4; @seesawi5; @seesawii1; @seesawii2; @seesawii3; @seesawii4]. Secondly, in the renormalizable versions of SUSY SO(10) models[@rp1; @rp2; @rp3], R-parity is conserved automatically which eliminates the most dangerous dimension-4 operators of proton decay. The elegant running behaviors of the coupling constants in MSSM strongly suggest that the unification scale should be taken at $2\times 10^{16}$GeV [@Einhorn; @Marciano; @MSSM1; @MSSM2; @MSSM3; @MSSM4] which we call the GUT scale $M_G$. Not only the three coupling constants are unified at $M_G$, but also the masses of the gauge superfields G(SO(10))/G(SM) are taken at the same scale. It has been recognized recently[@dlz] that instead of an intermediate seesaw scale, in SUSY SO(10) models with several pairs of **126** $+\overline{\textbf{126}}$, only an intermediate vacuum expectation value (VEV) of the SM singlet in one $\overline{\textbf{126}}$ is needed which couples with the matter superfields. Consequently, the spectra of this kind of models do not contain particles at intermediate scale so that gauge coupling unification is maintained, meanwhile the seesaw mechanism still works. This mechanism is further incorporated in models aiming at sufficiently suppressing proton decay[@dlz2], where the seesaw VEV is related to the VEV of an SO(10) singlet which breaks an extra global U(1) symmetry. Proton decay amplitudes are found to be suppressed in [@dlz2] by a factor $\frac{M_I}{M_G}$, where $M_I\sim 10^{14}$GeV is the seesaw VEV which is much smaller than the GUT scale $M_G\sim 10^{16}$GeV. This suppression of proton decay is archived by the enhancements of the effective triplet masses through an inversely analogue to the mass texture in the seesaw mechanism, or a lever mechanism. In the present work we will extend the observation made by the previous study that the seesaw VEV might be related to the suppression of proton decay in other models. Instead of using **210** to break SO(10), we will use **45**+**54**. The global U(1) will be replaced by an anomalous U(1) whose breaking is generated by an SO(10) singlet through the Green-Shwarz mechanism[@green1; @green2]. The enhancements of the effective triplet masses responsible for proton decay are through an inversely analogue to the mass texture in the double-seesaw mechanism[@double; @double2]. We will not improve on either the running behavior of the SO(10) gauge coupling above the GUT scale or on the minimal fine-tuning for the weak doublets which is implicitly assumed. In the next section, we will give a simple overview on proton decay suppression. Then, we will propose in Section \[model\] a renormalizable model and show its consistency with high energy supersymmetry. Proton decay suppression mechanism in this model is shown in Section \[proton\]. The discussion on the weak doublets of the MSSM and the prediction of small $\textrm{tan}\beta$ are followed in Section \[DT\]. We will summarize in Section \[Sum\]. General consideration on proton decay suppression ================================================= Consider a simplified model with two pairs of color triplets-anti-triplets, with only one pair of them couple with fermions. The mass term for the triplets can be written as $(\varphi_{\overline{T}})_i(M)_{ij}(\varphi_T)_j$, where $i$, $j$ run from 1 to 2. We need to rotate to the mass eigenstates in order to calculate proton decay amplitudes. Two $2\times2$ unitary matrices $U$ and $V$ are introduced as $$M'_{ij}=U_{ik}M_{kl}{V}^{\dagger}_{lj},$$ where $M'=\textrm{diag}(M_1, M_2).$ The mass eigenstates $\varphi'_{\overline{T}}$s and $\varphi'_{T}$s are $$(\varphi'_{\overline{T}})_i=(\varphi_{\overline{T}})_j{U}^{\dagger}_{ji},\ \ (\varphi'_{T})_i={V}_{ij}(\varphi_{T})_j.$$ Then the dimension-5 operators mediated by the color triplet higgsinos are proportional to [@Nath] $$\label{m-1} \sum_i{V}^{\dagger}_{1i}\frac{1}{M'_{ii}}{U}_{i1}=({V}^{\dagger}\cdot M'^{-1}\cdot{U})_{11}=[({U}^{\dagger}\cdot M'\cdot{V})^{-1}]_{11}=(M^{-1})_{11}.$$ The inverse of $(M^{-1})_{11}$ is called the effective triplet mass which mimics the role of the color triplet higgsino in the simplest models with only one pair of color triplet-anti-triplet. Eq.(\[m-1\]) is easy to be generalized to models with more pairs of color triplets-anti-triplets. The proton decay amplitudes are proportional to sums of specific elements in the inverse of the triplet mass matrix. Algebraically, these matrix elements in the inverse mass matrix can be written as $$\label{m-1ij} (M^{-1})_{ij}=(-1)^{i+j}\frac{\textrm{Det}(M^*_{ji})}{\textrm{Det}(M)},$$ where $M^*_{ij}$ represents $M$ with the $i$th row and the $j$th column eliminated, whose determinant is called as the algebraic complement, and $i$ and $j$ are the labels of those color triplets-anti-triplets which can couple with the fermions. There are two possible ways to get small $(M^{-1})_{ij}$’s following (\[m-1ij\]). We can construct a mass matrix either with all small algebraic complements for the elements which couple with fermions, or with a large determinant of the entire mass matrix. In the previous work [@dlz2] the first approach is used where the color triplet mass matrix can be symbolically expressed as $$\label{m2p2} M_T=\left( \begin{array}{cc} 0 &M_{G} \\ M_{G} & M_I \end{array} \right).$$ Here $M_G$ stands for a GUT scale mass while $M_I$ is the intermediate seesaw VEV. Only the up-left block couples with matter fields, so it is clear that $M^*_{11}=M_I$ is smaller than $M_G$. In this work, we are trying to realize the second possibility. The mass matrix for the color triplets is written as $$\label{m3p3} M_T=\left( \begin{array}{ccc} 0 &M_{G} &0\\ M_{G} & 0 &M_{G} \\ 0 &M_{G} & M_{P} \end{array} \right).$$ Again, the matter fields couple with the up-left block only. Here $M_{P}$ represents a mass at a scale higher than the GUT scale, or at the Plank scale. Then $\textrm{Det}(M_T)\sim M_{P}M_{G}^2$ is enhanced to give large effective triplet masses. As the texture in (\[m2p2\]) is analogue to the neutrino mass matrix in the seesaw mechanism, the present texture in (\[m3p3\]) is analogue to the neutrino mass matrix in the double-seesaw mechanism. The mass texture either in (\[m2p2\]) or in (\[m3p3\]) is sufficient to suppress proton decay. The model and SUSY preserving {#model} ============================= The particle content of the present model is as follows. First, it contains three generations of fermion fields which are embedded into three **16**-plet ($\psi_{1,2,3}$) superfields as usual. Second, **45**+**54** ($A$, $E$) are introduced to break SO(10). Third, in order to give satisfied fermion masses and mixing [@Bajc], Higgs in **120** ($D$) is introduced, which is also needed to couple through **45**+**54** with those in **10** ($H$) and in **126**/$\overline{\textbf{126}}$ ($\Delta/\overline{\Delta}$). Forth, the **45** is further copied ($A'$) to generate a small VEV for the seesaw mechanism, and to generate the structure (\[m3p3\]) for suppression of proton decay. Three sets of Higgs are needed with the first two sets contain $H+ \Delta/\overline{\Delta} +D$ while the third set contains $\Delta/\overline{\Delta}$. An extra U(1) symmetry, whose breaking is realized by the SO(10) singlets $S_1$ and $S_2$, is introduced to distinguish these Higgs. All the fields and their U(1) charges are listed in Table \[Qnumbers\]. Note that the different U(1) charges of the first set of Higgs ($H_1...$) and the third set of Higgs ($\Delta_3...$) also require different fields ($A+E$ and $A'$) to couple the first two and the last two sets of Higgs. $\psi_{i}$ $H_1,D_1,\Delta_1/\overline{\Delta}_1$ $H_2,D_2,\Delta_2/\overline{\Delta}_2$ $\Delta_3/\overline{\Delta}_3$ $A,E $ $A'$ $S_1$ $S_2$ ------------- ---------------- ---------------------------------------- ---------------------------------------- -------------------------------- -------- --------------- ------- ---------------- U(1) charge $-\frac{1}{2}$ 1 -1 $\frac{1}{2}$ 0 $\frac{1}{2}$ -1 $-\frac{1}{2}$ : SO(10) multiplets and their U(1) charges.[]{data-label="Qnumbers"} Only $H_1,D_1,\overline{\Delta}_1$ couple with matter fields due to the U(1) charges. The Yukawa sector is given as $$\label{superp1} W_Y=Y_{10}^{ij}\psi_i\psi_j H_1+Y_{120}^{ij}\psi_i\psi_j D_1+Y_{126}^{ij}\psi_i\psi_j \overline{\Delta}_1,$$ which is general enough to fit all fermion masses and mixing [@rp2; @fits1; @fits2; @fits3; @fits4; @Bajc2004; @Bajc2006; @revise1; @revise2; @revise3]. The general renormalizable Higgs superpotential is given by $$\begin{aligned} \label{superp2} W&=&m_H H_1H_2+{m_{\Delta}}_{12}\overline{\Delta}_1\Delta_2 +{m_{\Delta}}_{21}\overline{\Delta}_2\Delta_1+m_D D_1D_2+\frac{1}{2}m_{A}A^2+\frac{1}{2}m_{E}E^2 \nonumber \\ &+&H_1H_2(\lambda_1A +\lambda_2E)-i A(\lambda_{3}\overline{\Delta}_1\Delta_2+\lambda_{4}\overline{\Delta}_2\Delta_1) + E(\lambda_{5}{\Delta}_1\Delta_2+\lambda_{6}\overline{\Delta}_1\overline{\Delta}_2)\nonumber \\ &+&D_1 A(\lambda_7H_2+\lambda_{8}\Delta_2+\lambda_{9}\overline{\Delta}_2) +D_2 A(\lambda_{10}H_1+\lambda_{11}\Delta_1+\lambda_{12}\overline{\Delta}_1)\nonumber \\ &+&D_1D_2(\lambda_{13}A +\lambda_{14}E)+\lambda_{15} E^3+\lambda_{16} AE^2 -i A'(\alpha_{1}\overline{\Delta}_2\Delta_3+\alpha_{2}\overline{\Delta}_3\Delta_2)\nonumber \\ &+&D_2 A'(\alpha_{3}\Delta_3+\alpha_{4}\overline{\Delta}_3)+ \frac{1}{2}S_1(2\beta_{1}\overline{\Delta}_3\Delta_3+\beta_2{A'}^2)+\beta_3 S_2 AA'.\end{aligned}$$ Labeled by the representations under the $SU(4)_C\times SU(2)_L\times SU(2)_R$ subgroup of SO(10), the following components get VEVs responsible for the SO(10) symmetry breaking $$\begin{aligned} \label{allvev} A_1^{(\prime)}&=&\langle A^{(\prime)}(1,1,3)\rangle, {}~A_2^{(\prime)}=\langle A^{(\prime)}(15,1,1)\rangle, {}~E=\langle E(1,1,1)\rangle; \nonumber \\ v_{(1,2,3)}&=&\langle \Delta_{(1,2,3)} (\overline{10},1,3)\rangle,{}~\overline{v}_{(1,2,3)} =\langle\overline{\Delta}_{(1,2,3)}(10,1,3)\rangle.\end{aligned}$$ Inserting these VEVs into (\[superp2\]), we get $$\begin{aligned} \label{superp2VEV} \langle W\rangle &=&{m_{\Delta}}_{12}\overline{v}_{1}v_2 +{m_{\Delta}}_{21}\overline{v}_{2}v_1 +\frac{1}{2}m_A(A_1^2+A_2^2) +\frac{1}{2}m_E E^2+A_0 (\lambda_{3}\overline{v}_1v_2+\lambda_{4}\overline{v}_2v_1)\nonumber\\ &+&\frac{\lambda_{15}}{2\sqrt{15}} E^3 +\lambda_{16}E(\frac{\sqrt{3}}{2\sqrt{5}}A_1^2-\frac{1}{\sqrt{15}}A_2^2)+A'_0 (\alpha_{1}\overline{v}_2v_3+\alpha_{2}\overline{v}_3v_2)\nonumber \\ &+&\frac{1}{2}S_1(2\beta_{1}\overline{v}_3v_3+\beta_2{A'_1}^2+\beta_2{A'_2}^2 )+S_2(\beta_3A_1A'_1+\beta_3A_2A'_2),\end{aligned}$$ where we have defined $$\begin{aligned} \label{A0} A_0 \equiv \left(-\frac{1}{5}A_1-\frac{3}{5\sqrt{6}}A_2\right), \quad A'_0 \equiv \left(-\frac{1}{5}A'_1-\frac{3}{5\sqrt{6}}A'_2\right),\end{aligned}$$ for later convenience. To preserve SUSY at high energy, the F- and D-flatness conditions are required. The D-flatness condition requires $$\label{dterm} |v_{1}|^2+|v_{2}|^2+|v_{3}|^2=|\overline{v}_{1}|^2+|\overline{v}_{2}|^2+|\overline{v}_{3}|^2,$$ which constrains only the sum of $|v|^2$s and $|\overline{v}|^2$s, so that an intermediate valued VEV of $\overline{v}_1$ can be generated without breaking gauge coupling unification, if both sides in (\[dterm\]) are of the order $M^2_{G}$. The F-flatness conditions $$\left\{ \frac{\partial }{\partial v_{1}}, \frac{\partial }{\partial v_{2}}, \frac{\partial }{\partial v_{3}}, \frac{\partial }{\partial \overline{v}_{1}}, \frac{\partial }{\partial \overline{v}_{2}}, \frac{\partial }{\partial \overline{v}_{3}}, \frac{\partial }{\partial A' _{1}}, \frac{\partial }{\partial A' _{2}}, \frac{\partial }{\partial S_{1}}, \frac{\partial }{\partial S_{2}}, \frac{\partial }{\partial A_{1}}, \frac{\partial }{\partial A_{2}}, \frac{\partial }{\partial E } \right\} \langle W\rangle =0,\nonumber \label{partiale}$$ are explicitly $$\begin{aligned} 0&=&M_{21}\overline{v}_{2}, \label{equv1}\\ 0&=&M_{12}\overline{v}_{1}+\alpha_2 A'_0\overline{v}_{3} , \label{equv2} \\ 0&=&\alpha_1 A'_0 \overline{v}_{2}+\beta_{1}S_1\overline{v}_{3}, \label{equv3} \\ 0&=&M_{12}{v}_{2}, \label{equv1bar}\\ 0&=&M_{21}{v}_{1}+\alpha_1 A'_0{v}_{3} , \label{equv2bar} \\ 0&=&\alpha_2 A'_0 {v}_{2}+\beta_{1}S_1{v}_{3}, \label{equv3bar} \\ 0&=&\beta_2S_1A'_1+\beta_3S_2A_1-\frac{1}{5}(\alpha_{1}\overline{v}_2v_3+\alpha_{2}\overline{v}_3v_2), \label{equAp1} \\ 0&=&\beta_2S_1A'_2+\beta_3S_2A_2-\frac{3}{5\sqrt6}(\alpha_{1}\overline{v}_2v_3+\alpha_{2}\overline{v}_3v_2), \label{equAp2} \\ 0&=&\beta_{1}\overline{v}_3v_3+\beta_2{A'_1}^2+\beta_2{A'_2}^2 \label{equS1} \\ 0&=&\beta_3A_1A'_1+\beta_3A_2A'_2 \label{equS2}\\ 0&=&m_{A}A_1-\frac{1}{5}(\lambda_{3}\overline{v}_1v_2+\lambda_{4}\overline{v}_2v_1)+\frac{\sqrt3}{\sqrt5}\lambda_{16}EA_1+\beta_3A'_1S_2, \label{equA1} \\ 0&=&m_{A}A_2-\frac{3}{5\sqrt6}(\lambda_{3}\overline{v}_1v_2+\lambda_{4}\overline{v}_2v_1)-\frac{2}{\sqrt{15}}\lambda_{16}EA_2+\beta_3A'_2S_2, \label{equA2} \\ 0&=&m_E E+\frac{3}{2\sqrt{15}}\lambda_{15}E^2+\lambda_{16}(\frac{\sqrt{3}}{2\sqrt{5}}A_1^2-\frac{1}{\sqrt{15}}A_2^2), \label{equE}\end{aligned}$$ where $$\begin{aligned} \label{M12M21} M_{12}\equiv {m_{\Delta}}_{12}+\lambda_3 A_0, \quad M_{21}\equiv {m_{\Delta}}_{21}+\lambda_4 A_0.\end{aligned}$$ From (\[dterm\]) to (\[equE\]), there are 13 variables and 14 equations in total, but only 12 of the equations are independent. One of the VEVs, $S_1$ for example, can be assigned to any scale. First, (\[equv1\])-(\[equv3\]) are linear equations about the $\overline{v}$s, which can be rewritten as $$\label{detvb} \left( \begin{array}{ccc} \overline{v}_1, &\overline{v}_2, &\overline{v}_3 \end{array} \right) \left( \begin{array}{ccc} 0 &M_{12} &0\\ M_{21} & 0 &\alpha_1 A'_0 \\ 0 & \alpha_2 A'_0 &\beta_{1}S_1 \end{array} \right)=0.$$ Similarly, equations (\[equv1bar\])-(\[equv3bar\]) can be rewritten as $$\label{detv} \left( \begin{array}{ccc} 0 &M_{12} &0\\ M_{21} & 0 &\alpha_1 A'_0 \\ 0 & \alpha_2 A'_0 &\beta_{1}S_1 \end{array} \right) \left( \begin{array}{c} {v}_1 \\ {v}_2 \\ {v}_3 \end{array} \right)=0.$$ Both (\[detvb\]) and (\[detv\]) require $$\label{det} \beta_{1}S_1 M_{12} M_{21}=0,$$ which corresponds to three different possibilities as following. If $S_1=0$, a lot of particles cannot get masses through couplings with $S_1$. Thus this case is simply excluded. If $M_{12}=0$ is taken, then from (\[detvb\]-\[detv\]) it follows that $\overline{v}_{1}=M_{G}$ and $\overline{v}_{2}=\overline{v}_{3}=0$ which gives too small neutrino masses and is thus excluded as well. We thus have the last possibility, $$\label{M21} M_{21}=0,$$ which gives $$\label{vbratio} v_1\sim M_G, ~v_2=v_3=0, ~~~\frac{\overline{v}_{1}}{\overline{v}_{3}}=-\frac{\alpha_2 A_0'}{M_{12}}, ~\frac{\overline{v}_{2}}{\overline{v}_{3}}=-\frac{\beta_1 S_1}{\alpha_1 A_0'},$$ following (\[detvb\]) and (\[detv\]). Then, $\overline{v}_{1,2,3}$ are expressed by $v_1$ through (\[dterm\]). Furthermore, substituting $v_2=v_3=0$ into (\[equAp1\])-(\[equAp2\]), $A'_1$ and $A'_2$ can be expressed by $A_1$ and $A_2$, respectively, and (\[equS2\]) is now identical to (\[equS1\]). Equation $M_{21}=0$ in (\[M21\]) itself gives the dependence of $A_2$ on $A_1$ through (\[M12M21\]). Taking $S_1$ as free, the remaining variables are now $A_1, E, S_2, v_1$ with equations (\[equS1\]), (\[equA1\]),(\[equA2\]) and (\[equE\]) left. Given the parameters in (\[superp2\]), all the VEVs are now determined. Numerically, $A_1$, $A_2$ and $E$ are taken as GUT scale VEVs in order to break SO(10) down to MSSM. According to the analysis in [@missingvev], the extra U(1) symmetry is naturally related to string theory, and it is appropriate to take the VEV of breaking this U(1) at $$S_1\sim 10^{17}\textrm{GeV}\sim 10M_G.$$ After inserting (\[equAp1\]-\[equAp2\]) into (\[equA1\]-\[equA2\]), the last terms will change into $$\beta_3A'_1S_2 \rightarrow -\frac{\beta_3^2 S_2^2}{\beta_2 S_1}A_1, \quad \beta_3A'_2S_2 \rightarrow -\frac{\beta_3^2 S_2^2}{\beta_2 S_1}A_2.$$ They are naturally at the same scale as other terms, i.e. $M_G^2$, which indicates that $$S_2\sim \frac{1}{\sqrt{10}}S_1 \sim \sqrt{10}M_G, \quad A'_1 \sim A'_2 \sim\frac{1}{\sqrt{10}} M_G.$$ Thus we get from (\[vbratio\]) $$\label{vvev} \left( \begin{array}{ccc} {v}_1, &{v}_2, &{v}_3 \end{array} \right)= \left( \begin{array}{ccc} O(M_G), &0, &0 \end{array} \right) \quad \left( \begin{array}{ccc} \overline{v}_1, &\overline{v}_2, &\overline{v}_3 \end{array} \right)= O\left( \begin{array}{ccc} 10^{-2}M_G, &M_G, &10^{-\frac{3}{2}}M_G \end{array} \right).$$ Now that all the constrains on SUSY preserving have been satisfied, all the VEVs can be determined and all their scales are known. The seesaw VEV $\overline{v}_1\sim M_I$ is naturally generated at $10^{-2}M_G$, which differs from [@dlz2] where it introduced a VEV of an SO(10) singlet which broke a global U(1) symmetry. Consequence of the large masses of the third set of Higgs and $A'$ given by the VEV $S_1\sim 10M_G$ is that these Higgs are to be integrated out above the GUT scale $M_G$, so that they do not affect the running behaviors of gauge couplings of MSSM. Proton decay suppression {#proton} ======================== To demonstrate the effectiveness of the present model on solving the proton decay problem, we need to write down the color triplet mass matrix. The color triplets are ordered as $$\label{phit} \varphi_{T}=(H_{1T},D_{1T},D'_{1T},\Delta_{1T}, \overline{\Delta}_{1T}, \overline{\Delta}'_{1T}; H_{2T},D_{2T},D'_{2T},\Delta_{2T}, \overline{\Delta}_{2T}, \overline{\Delta}'_{2T}; \Delta_{3T}, \overline{\Delta}_{3T}, \overline{\Delta}'_{3T}),$$ while the color anti-triplets are $$\label{antiphit} \varphi_{\overline{T}}=(H_{1\overline{T}},D_{1\overline{T}},D'_{1\overline{T}}, \overline{\Delta}_{1\overline{T}}, \Delta_{1\overline{T}}, \Delta'_{1\overline{T}}; H_{2\overline{T}},D_{2\overline{T}},D'_{2\overline{T}}, \overline{\Delta}_{2\overline{T}}, \Delta_{2\overline{T}}, \Delta'_{2\overline{T}}; \overline{\Delta}_{3\overline{T}}, \Delta_{3\overline{T}}, \Delta'_{3\overline{T}}).$$ The mass term of the Higgs color triplets is given by $(\varphi_{\overline{T}})_a(M_T)_{ab}(\varphi_T)_b$, with the $15\times 15$ matrix $M_T$ written as $$\label{triplet} M_T=\left( \begin{array}{ccc} 0_{(6\times 6)} &B_{12(6\times 6)} &0_{(6\times 3)}\\ B_{21(6\times 6)} & 0_{(6\times 6)} &B_{23(6\times 3)} \\ 0_{(3\times 6)} & B_{32(3\times 6)} &B_{33(3\times 3)} \\ \end{array} \right),$$ where $$\label{MT12} B_{12}=\left( \begin{array}{cccccc} B_{H12} &-\frac{i\lambda_{10}}{\sqrt3}A_1 & -\frac{i\sqrt2\lambda_{10}}{3}A_2 & 0 &0 &0 \\ \frac{i\lambda_{7}}{\sqrt3}A_1 & B_{D12} & 0 & -\frac{i\lambda_{8}}{2\sqrt5}A_1 & -\frac{i\lambda_{9}}{2\sqrt5}A_1 & -\frac{i\lambda_{9}}{\sqrt{15}}A_2\\ -\frac{i\sqrt2\lambda_{7}}{3}A_2 & 0 & B'_{D12} &\frac{i\lambda_{8}}{\sqrt{30}} A_2 &-\frac{i\lambda_{9}}{\sqrt{30}} A_2 &-\frac{i\lambda_{9}}{\sqrt{10}} A_1 \\ 0 & \frac{i\lambda_{12}}{2\sqrt5}A_1 & -\frac{i\lambda_{12}}{\sqrt{30}} A_2 & {m_{\Delta}}_{12}+\frac{\lambda_3}{5\sqrt6}A_2 & \frac{2\lambda_6}{\sqrt{15}}E & 0\\ 0 & \frac{i\lambda_{11}}{2\sqrt5}A_1 & \frac{i\lambda_{11}}{\sqrt{30}} A_2 & \frac{2\lambda_5}{\sqrt{15}}E & {m_{\Delta}}_{21}-\frac{\lambda_4}{5\sqrt6}A_2 & 0\\ 0 & \frac{i\lambda_{11}}{\sqrt{15}}A_2 & \frac{i\lambda_{11}}{\sqrt{10}} A_1 & 0 & 0 & {m_{\Delta}}_{21}-\frac{\lambda_4}{5\sqrt6}A_2 \end{array} \right),$$ $$\label{MT21} B_{21}=\left( \begin{array}{cccccc} B_{H21}&-\frac{i\lambda_{7}}{\sqrt3}A_1 & -\frac{i\sqrt2\lambda_{7}}{3}A_2 & 0 &0 &0 \\ \frac{i\lambda_{10}}{\sqrt3}A_1 & B_{D21} & 0 & -\frac{i\lambda_{11}}{2\sqrt5}A_1 & -\frac{i\lambda_{12}}{2\sqrt5}A_1 & -\frac{i\lambda_{12}}{\sqrt{15}}A_2\\ -\frac{i\sqrt2\lambda_{10}}{3}A_2 & 0 & B'_{D21} &\frac{i\lambda_{11}}{\sqrt{30}} A_2 &-\frac{i\lambda_{12}}{\sqrt{30}} A_2 &-\frac{i\lambda_{12}}{\sqrt{10}} A_1 \\ 0 & \frac{i\lambda_{9}}{2\sqrt5}A_1 & -\frac{i\lambda_{9}}{\sqrt{30}} A_2 & {m_{\Delta}}_{21}+\frac{\lambda_4}{5\sqrt6}A_2 & \frac{2\lambda_6}{\sqrt{15}}E & 0\\ 0 & \frac{i\lambda_{8}}{2\sqrt5}A_1 & \frac{i\lambda_{8}}{\sqrt{30}} A_2 & \frac{2\lambda_5}{\sqrt{15}}E & {m_{\Delta}}_{12}-\frac{\lambda_3}{5\sqrt6}A_2 & 0\\ 0 & \frac{i\lambda_{8}}{\sqrt{15}}A_2 & \frac{i\lambda_{8}}{\sqrt{10}} A_1 & 0 & 0 & {m_{\Delta}}_{12}-\frac{\lambda_3}{5\sqrt6}A_2 \end{array} \right),$$ $$\label{MT23} B_{23}=\left( \begin{array}{ccc} 0 &0 &0 \\ -\frac{i\alpha_{3}}{2\sqrt5}A'_1 & -\frac{i\alpha_{4}}{2\sqrt5}A'_1 & -\frac{i\alpha_{4}}{\sqrt{15}}A'_2\\ \frac{i\alpha_{3}}{\sqrt{30}} A'_2 &-\frac{i\alpha_{4}}{\sqrt{30}} A'_2 &-\frac{i\alpha_{4}}{\sqrt{10}} A'_1 \\ \frac{\alpha_1}{5\sqrt6}A'_2 & 0 & 0\\ 0 & -\frac{\alpha_2}{5\sqrt6}A'_2 & 0\\ 0 & 0 & -\frac{\alpha_2}{5\sqrt6}A'_2 \end{array} \right),$$ $$\label{MT32} B_{32}=\left( \begin{array}{cccccc} 0 & \frac{i\alpha_{4}}{2\sqrt5}A'_1 & -\frac{i\alpha_{4}}{\sqrt{30}} A'_2 & \frac{\alpha_2}{5\sqrt6}A'_2 & 0 & 0\\ 0 & \frac{i\alpha_{3}}{2\sqrt5}A'_1 & \frac{i\alpha_{3}}{\sqrt{30}} A'_2 & 0 & -\frac{\alpha_1}{5\sqrt6}A'_2 & 0\\ 0 & \frac{i\alpha_{3}}{\sqrt{15}}A'_2 & \frac{i\alpha_{3}}{\sqrt{10}} A'_1 & 0 & 0 & -\frac{\alpha_1}{5\sqrt6}A'_2 \end{array} \right),$$ and $$\label{MT33} B_{33}=\left( \begin{array}{ccc} \beta_1 S_1 &0 &0 \\ 0& \beta_1 S_1 &0 \\ 0& 0 & \beta_1 S_1 \end{array} \right).$$ Here $$\begin{aligned} B_{H12}&\equiv &m_H+\frac{i\lambda_1}{\sqrt{6}}A_2+\frac{2\lambda_2}{\sqrt{15}}E, \quad \ \ B_{H21}\equiv m_H-\frac{i\lambda_{1}}{\sqrt{6}}A_2+\frac{2\lambda_2}{\sqrt{15}}E, \nonumber\\ B_{D12}&\equiv &m_D+\frac{i\lambda_{13}}{3\sqrt{6}}A_2+\frac{4\lambda_{14}}{3\sqrt{15}}E, \quad B_{D21}\equiv m_D-\frac{i\lambda_{13}}{3\sqrt{6}}A_2+\frac{4\lambda_{14}}{3\sqrt{15}}E, \nonumber\\ B'_{D12}&\equiv &m_D+\frac{i\lambda_{13}}{3\sqrt{6}}A_2-\frac{2\lambda_{14}}{\sqrt{15}}E, \quad B'_{D21}\equiv m_D-\frac{i\lambda_{13}}{3\sqrt{6}}A_2-\frac{2\lambda_{14}}{\sqrt{15}}E.\nonumber\end{aligned}$$ The mass matrix can be also expressed symbolically as $$\label{trisym1} M_T=\left( \begin{array}{ccc} 0_{(6\times 6)} &M_{G(6\times 6)} &0_{(6\times 3)}\\ M_{G(6\times 6)} & 0_{(6\times 6)} &\frac{1}{\sqrt{10}}M_{G(6\times 3)} \\ 0_{(3\times 6)} & \frac{1}{\sqrt{10}}M_{G(3\times 6)} & 10M_{G(3\times 3)} \\ \end{array} \right).$$ Note that the texture in (\[trisym1\]), constrained by the F- and D-flatness conditions, differs slightly from that in (\[m3p3\]). However, as will be seen in the rest of this Section, the mechanism of suppressing proton decay following (\[m3p3\]) will not change. In SUSY GUTs, the dominant channels inducing proton decay are through the dimension-5 operators [@ETM1; @ETM2] $$\label{dimension5} -W_5=C_L^{ijkl}\frac{1}{2}q_iq_jq_kl_l+C_R^{ijkl}u_i^cd_j^cu_k^c e_l^c,$$ which are called the $LLLL$ and $RRRR$ operators, respectively, obtained by integrating out the color triplet and anti-triplet Higgs superfields in the interactions in (\[superp1\]). Both $C_L^{ijkl}$ and $C_R^{ijkl}$ are inversely proportional to the effective mass of the colored Higgsino. Since only $B_{11}$ part couples with fermions, we can get the effective mass by integrating out the uncoupled parts. From (\[trisym1\]), such a mass matrix is similar to the mass matrix in the double seesaw models for neutrino masses [@double; @double2] which is used to generate the small neutrino masses. In the present model, the effective masses are large instead of small because $B_{23}\sim B_{32}\ll B_{12}\sim B_{21}\ll B_{33}$. Similarly, this proton decay suppression mechanism requires two steps of integrations. Since $S_1$ is ten times of the GUT scale, the $B_{33}$ part can be integrate out first. Then the mass matrix becomes $$\label{trisym2} M_T=\left( \begin{array}{cc} 0_{(6\times 6)} &M_{G(6\times 6)}\\ M_{G(6\times 6)} & M_{I(6\times 6)} \end{array} \right),$$ where $$\label{seesawmass} M_{I(6\times 6)}=-B_{23}\cdot B_{33}^{-1} \cdot B_{32},$$ is a matrix with all elements of the order $10^{-2}M_G$. Then after the second step, $$\label{trisym3} M_{eff}=-B_{21}\cdot M_{I(6\times 6)}^{-1} \cdot B_{12}\sim\frac{M_{G}^2}{M_I}\sim 2\times 10^{18}\textrm{GeV},$$ which is of the order of the Plank scale, a hundred times heavier than those color triplet Higgs masses in SUSY GUT models. The proton decay rates will be suppressed by a factor of $10^{-4}$, which is small enough surviving all the current experimental limits. One may have found that $M_{I(6\times 6)}$ is a rank 3 matrix and is thus not reversible. This is because that we have not introduced the third **10**+**120**-plet $H_3+D_3$ for simplicity. But, as was discussed in [@dlz; @dlz2], each rank contributes one eigenvalue in the effective masses. The diagonal form of the effective mass matrix is $$\label{effmass} M_{eff}=\textrm{diag} \ O(\frac{M_G^2}{M_I}, \frac{M_G^2}{M_I}, \frac{M_G^2}{M_I}, \infty,\infty, \infty).$$ Note that it is the lightest eigenvalues that dominates the proton decay rates, while the three infinitely heavy masses do not contribute. The suppression can be better understood if we write down the dimension-5 operators explicitly. The coefficients $C_L$s at the GUT scale $M_G$ are [@fukuyamageneral] $$\label{cl} C_L^{ijkl}(M_G)=\left( \begin{array}{cccc} Y^{ij}_{10}, & Y^{ij}_{120}, & Y^{ij}_{120}, & Y^{ij}_{126} \end{array} \right) \left( \begin{array}{cccc} (M_T^{-1})_{11} &(M_T^{-1})_{12} & (M_T^{-1})_{13} & (M_T^{-1})_{14}\\ (M_T^{-1})_{21} & (M_T^{-1})_{22}& (M_T^{-1})_{23} & (M_T^{-1})_{24}\\ (M_T^{-1})_{31} & (M_T^{-1})_{32}& (M_T^{-1})_{33} & (M_T^{-1})_{34}\\ (M_T^{-1})_{51} & (M_T^{-1})_{52}& (M_T^{-1})_{53} & (M_T^{-1})_{54} \end{array} \right) \left( \begin{array}{c} Y^{kl}_{10}\\ Y^{kl}_{120} \\ Y^{kl}_{120} \\ Y^{kl}_{126} \end{array} \right).$$ Here the Yukawa couplings are strongly constrained by fitting the fermion masses and mixing [@fits1; @fits2; @fits3; @fits4]. The elements of $M_T^{-1}$ are of the order $\frac{1}{M_G}$ in usual SUSY GUT models, but in our model, $$\label{triinv} M_T^{-1}=\frac{1}{M_G}\left( \begin{array}{ccc} 10^{-2}_{(6\times 6)} &1_{(6\times 6)} &10^{-\frac{3}{2}}_{(6\times 3)}\\ 1_{(6\times 6)} & 0_{(6\times 6)} &0_{(6\times 3)} \\ 10^{-\frac{3}{2}}_{(3\times 6)} & 0_{(3\times 6)} & \frac{1}{10}_{(3\times 3)} \\ \end{array} \right).$$ We can see clearly that the elements contributing to dimension-5 operators, i.e. elements in the up-left most block, are of the order $\frac{10^{-2}}{M_G}$. This is the same conclusion drawn in (\[trisym3\]). This conclusion applies for both the LLLL and RRRR operators. The weak doublets {#DT} ================= Like in [@dlz2], the doublet-triplet splitting (DTS) problem requires a minimal fine-tuning, and similar results can be reached. The up-type doublets are ordered as $$\varphi_{u}=(H_{1u}, D_{1u}, D'_{1u}, \Delta_{1u}, \overline{\Delta}_{1u}; H_{2u}, D_{2u}, D'_{2u}, \Delta_{2u}, \overline{\Delta}_{2u}; \Delta_{3u}, \overline{\Delta}_{3u}),$$ while down-type doublets are $$\varphi_{d}=(H_{1d}, D_{1d}, D'_{1d}, \overline{\Delta}_{1d}, \Delta_{1d}; H_{2d}, D_{2d}, D'_{2d}, \overline{\Delta}_{2d}, \Delta_{2d}; \overline{\Delta}_{3d}, \Delta_{3d}).$$ The mass terms of the weak doublets are given by $(\varphi_{d})_a(M_D)_{ab}(\varphi_u)_b$, with the $12\times 12$ matrix $M_D$ written as $$M_D=\left( \begin{array}{ccc} 0_{(5\times 5)} &A_{12(5\times 5)} &0_{(5\times 2)}\\ A_{21(5\times 5)} & 0_{(5\times 5)} &A_{23(5\times 2)} \\ 0_{(2\times 5)} & A_{32(2\times 5)} &A_{33(2\times 2)} \end{array} \right)=\left( \begin{array}{ccc} 0_{(5\times 5)} &M_{G(5\times 5)} &0_{(5\times 2)}\\ M_{G(5\times 5)} & 0_{(5\times 5)} &\frac{1}{\sqrt{10}}M_{G(5\times 2)} \\ 0_{(2\times 5)} & \frac{1}{\sqrt{10}}M_{G(2\times 5)} & 10M_{G(2\times 2)} \end{array} \right).$$ For general parameters, there is no massless doublet. The DTS, which requires a zero determinant of $M_D$, can be obtained if $$\label{detmd} \textrm{Det}(M_D)=\textrm{Det}(A_{12})*\textrm{Det}(A_{21})*\textrm{Det}(A_{33})=0.$$ $\textrm{Det}(A_{33})$ is obviously nonzero, leaving us two choices. $\textrm{Det}(A_{12})=0$ is not acceptable because the large top quark mass would not be generated for perturbative Yukawa couplings. If we chose $\textrm{Det}(A_{21})=0$, we will further get the massless doublet can be expressed as $$\label{phiud} H_{u} =\sum_{i=1}^{13} \alpha_u^{i\ast} \varphi_{u}^i,\quad H_{d} =\sum_{i=1}^{13} \alpha_d^{i\ast} \varphi_{d}^i.$$ and the components are, up to a normalization factor, $$\begin{aligned} \alpha_u^\ast&=&O(\underbrace{1,...,1}_\textrm{five};\underbrace{0,...,0}_\textrm{seven}),\label{phiu}\nonumber\\ \alpha_d^\ast&=&O(\underbrace{10^{-2},...,10^{-2}}_\textrm{five}; \underbrace{1,...,1}_\textrm{five}; \underbrace{10^{-\frac{3}{2}},...,10^{-\frac{3}{2}}}_\textrm{two}).\end{aligned}$$ The large ratio of $\frac{\alpha_u^i}{\alpha_d^i}$ $(i\le 5)$ is consistent with the ratio of $\frac{m_t}{m_b}\sim 100$ at high energy [@fits1; @fits2; @fits3; @fits4]. It also gives the constrain on $\textrm{tan}\beta$ $$\label{mtmb} \textrm{tan}\beta =\frac{v_u}{v_d} \approx \frac{m_t}{m_b}10^{-2}\sim O(1).$$ Equation (\[mtmb\]) suggests that a small $\textrm{tan}\beta$ is favored in the present model, which is also the same conclusion drawn in [@dlz2]. Summery and conclusions {#Sum} ======================= As in [@dlz2], we do not perform explicitly the fine-tuning in the weak doublets which takes only one free parameter in the superpotential. As was pointed out in [@Bajc], threshold effects can be big in the minimal SUSY SU(5) theory, and can be even bigger due to the more super-heavy particles in SO(10) models. In this work, we have focused mainly on the new method of proton decay suppression and this method does not require explicit threshold effect calculations. The other reason is that there are more than enough free parameters in the superpotential that can be adjusted in calculating the threshold effects to fulfill the gauge coupling unification. In the present work we have presented a renormalizable SUSY SO(10) model with sufficient suppression of proton decay. Similar to [@dlz2], gauge coupling unification is maintained due to the absence of intermediate scales, and the seesaw VEV, proton decay and $\textrm{tan}\beta$ are found to be all related, Thus the main conclusions are quite general in a class of models which follow the mechanisms of suppressing proton decay through constructing seesaw-like textures in the color triplet mass matrices. Different from the previous study, we use **45**+**54** instead of **210** to break SO(10). Instead of a global U(1) used in [@dlz2], we use an anomalous U(1) to generate the seesaw VEV through Green-Schwarz mechanism. We have also included **120**-plet Higgs to couple with fermions so that the model is highly realistic. We have, however, two main problems untouched. The first is the DTS problem which we simply use an assumed fine-tuning in the weak doublets. The second is the perturbative difficulty for the gauge coupling above the GUT scale which is also common to all realistic SUSY GUT models. We can compare our work with [@Aulakh1; @Aulakh2] where Higgs in $\textbf{10}+\textbf{120}+\textbf{126}/\overline{\textbf{126}}$ are used to fit fermion masses and mixing while **210** is used to break SO(10). Proton decay suppression is carried out by raising the GUT scale up to the Planck scale or even higher so that the color-triplet Higgs masses are also enhanced accordingly, otherwise proton lifetime is around $10^{28}yr$ only. This picture conflicts with the most important results supporting SUSY GUT which suggest the GUT scale to be $M_G\sim 2\times 10^{16}$GeV [@MSSM1; @MSSM2; @MSSM3; @MSSM4]. 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-27pt 8.5in PURD-TH-94-11\ November 1994\ hep-ph/9411427 [The current of fermions scattered off a bubble wall ]{}\ S.Yu. Khlebnikov[^1]\ \ \ Proceeding from WKB quantization conditions, we derive a semiclassical expression for the current of fermions scattered off a propagating bubble wall in the presence of longitudinal gauge field. It agrees with the expression used by Nasser and Turok in semiclassical analysis of instability of electroweak bubble walls with respect to longitudinal $Z$ condensation. We discuss the resulting dispersion relation for longitudinal $Z$ field and show that light species are important for the analysis of stability, because of their large contribution to plasma frequency. Introduction ============ In a recent preprint [@NT], Nasser and Turok have found that interactions of bubble walls propagating during a first order electroweak phase transition with fermions can lead to a $CP$ violating instability on the walls, a condensate of longitudinal $Z$ bosons. This result, if confirmed by comprehensive study, is very important because it bears upon the main question in the theory of electroweak baryogenesis, namely, whether successful electroweak baryogenesis requires physics beyond the standard model or it does not. Because of deviations from equilibrium that occur during the wall propagation, the fermionic current, which enters the equation of motion for the $Z$ field and contributes to its effective mass, cannot be obtained from thermodynamic relations and should be computed by direct averaging of a microscopic expression. As a first approach to the problem, Nasser and Turok considered the WKB approximation for fermions and the thin-wall limit. (They have also checked results against a fully quantum-mechanical solution.) The expression for the current density they used in the WKB case would be obviously correct if the local value of canonical momentum of fermion remained unchanged during adiabatic switching on of the $Z$ field. For a non-uniform Higgs field (bubble wall), this is not so, hence, a justification for that expression is required. In this note we shall show how that expression can be derived from WKB quantization conditions for fermions via certain manipulations with partial derivatives, similar to those used in thermodynamics. We shall also discuss the resulting dispersion relation for longitudinal $Z$ field and show that, in realistic case, not only the top quark but also light species are important for the analysis of stability, because of their large contribution to plasma frequency. Derivation of the formula for the current ========================================= The microscopic expression for the current density of a given species in the thin-wall case is J() = \_n f\_n j\_n() , \[tot\] where $j_n$ is the current density for the $n$-th single-particle mode and $f_n$ is the corresponding filling factor. The use of single-particle modes is specific of the thin-wall approximation that assumes that particles do not interact with each other while scattering off the wall. Though for a planar wall the non-trivial coordinate dependence is only that for the direction orthogonal to the wall, we keep the full three-dimensional coordinate in (\[tot\]) because some of the expressions below are more general than that special situation. The partial current densities $j_n$ are found by averaging the current density operator over eigenfunctions of the corresponding Dirac equation. If the Dirac hamiltonian is $H$, the current operator is j()=- . \[ope\] where $Z$ is some component of the field (in our case, that orthogonal to the wall). The eigenfunctions $\psi_n$ satisfy H\_n = E\_n \_n . \[eig\] By a variant of the Feynman-Hellmann theorem (see for example ref.[@Griffiths]) j\_n()=\_n\^\* j() \_n = - . \[adi\] Hence, the partial currents are related to adiabatic, fixed $n$ variations of the energy eigenvalues with respect to the field. We can transform the fixed $n$ variations to fixed energy variations via =-( )\_Z ( )\_E , \[ene\] which is derived by using jacobians (cf. ref.[@LL]) =   . \[jac\] The resulting expression for the current is quite general (once the thin-wall limit is assumed) but it is especially useful in the WKB approximation, because in that case the quantum numbers have simple representation through WKB quantization conditions. Like ref.[@NT], we assume the potential barrier created for particles by the wall and the $Z$ field together to be monotonic. The WKB quantization conditions in the direction orthogonal to the wall (the only non-trivial ones) are $$\begin{aligned} \int_{-L_z/2}^{L_z/2} p_z(z) dz & = & 2\pi n_z \; , \label{qua1} \\ 2 \int_{-L_z/2}^{z^*(n)} p_z(z) dz & = & 2\pi n_z + C \label{qua2}\end{aligned}$$ for transmitted and reflected fermions, respectively. Here $p_z(z)$ is the longitudinal component of fermion momentum at a given spatial point, as determined from the classical dispersion law by conservation of single-particle energy and transverse momentum. $n_z$ is a positive or negative integer. The classical turning point $z^*$ of reflected particles depends on state $n=(n_x,n_y,n_z)$. A constant (up to exponential in $-L_z$ terms) phase shift $C$ depends on the boundary condition for reflected particles at $-L_z/2$. The dispersion laws are listed in ref.[@NT]. A single-particle mode can be specified by the value of momentum at one of the infinities (more precisely, at $\pm L_z/2$), $\mbox{\bf p}_{\infty}$. That incident momentum is related to the energy of the mode by the usual, field-independent expression. So, at given $p_{x,\infty}$, $p_{y,\infty}$, the variation at fixed energy is variation at fixed $p_{z,\infty}$. Next, we use $n_z$ instead of $n$ in (\[ene\]). Going over to one-dimensional variations we obtain =-( )\_Z ( )\_[p\_[z,]{}]{} =- ( )\_Z ( )\_[p\_[z,]{}]{} \[one\] where $\eta$ is 1 for transmitted states, 0 for reflected states with $z^*<z$, and 2 for reflected states with $z^*>z$. Note that the variational derivative with respect to $Z$ is converted into ordinary one. The derivatives are taken at fixed $p_{x,\infty}$, $p_{y,\infty}$, which is not indicated explicitly in (\[one\]). Using (\[one\]) and another transformation of derivatives ( )\_[p\_[z,]{}]{} =- ( )\_[Z]{} ( )\_[p\_z]{} , \[ano\] we transform the current density as follows $$\begin{aligned} J(z) & = & \int dn_xdn_ydn_z f_n j_n(z) \nonumber \\ & = & - \int \eta \frac{dp_{x,\infty}dp_{y,\infty}}{(2\pi)^3} dn_z f(\mbox{\bf p}_{\infty}) \left( {\frac{\partial E_n}{\partial n_z}} \right)_Z \left( {\frac{\partial p_z(z)}{\partial p_{z,\infty}}} \right)_{Z} \left( {\frac{\partial p_{z,\infty}}{\partial Z}} \right)_{p_z} \nonumber \\ & = & - \int \eta \frac{dp_{x,\infty}dp_{y,\infty}}{(2\pi)^3} dn_z f(\mbox{\bf p}_{\infty}) {\frac{\partial E}{\partial p_{z,\infty}}} \left( {\frac{\partial p_{z,\infty}}{\partial n_z}} \right)_Z \left( {\frac{\partial p_z(z)}{\partial p_{z,\infty}}} \right)_{Z} \left( {\frac{\partial p_{z,\infty}}{\partial Z}} \right)_{p_z} \nonumber \\ & = & - \int \eta \frac{d^3p_{\infty}}{(2\pi)^3} f(\mbox{\bf p}_{\infty}) \left( {\frac{\partial p_z(z)}{\partial p_{z,\infty}}} \right)_{Z} \left( {\frac{\partial E}{\partial Z}} \right)_{p_z} \; . \label{fin}\end{aligned}$$ The last line is the expression used in ref.[@NT]. We have thus shown how it can be derived from WKB quantization conditions. Dispersion relation and plasma frequency ======================================== The formula for the current allows us to compute the response of the non-equilibrium plasma to longitudinal $Z$ field. Because in the WKB calculation of the current both the wall profile and the gauge field were assumed slowly varying, we are actually considering the zero-momentum limit of the response. Accordingly, the relation (\[fin\]) between the current and the field is local in space. The linearized equation of motion — the dispersion relation for the $Z$ field is (\^2 - m\^2\_Z(z), \^2) - \_p\^2(z) = 0 . \[dis\] This is a matrix equation because of the mixing between $Z$ and photon field $A$ due to plasma effects; ${\hat \omega}_p^2(z)$ is the non-equilibrium plasma frequency matrix, \^2\_p(z) = -( [rr]{} J(z)/Z(z) & J(z)/A(z)\ J\_[em]{}(z)/Z(z) & J\_[em]{}(z)/A(z) )\_[Z=A=0]{} , \[ome\] where $J$ now denotes the total, summed over all species, current coupled to $Z$ and $J_{em}$ is the total electromagnetic current. Non-equilibrium effects in the plasma frequency are important only for the heaviest fermion, the top quark, which interacts effectively with the bubble wall. Evaluation of the integral (\[fin\]) shows [@NT] that for non-zero wall velocity, fermions produce a contribution to the current which has a square-root singularity in $m_{\infty}-m(z)$, where $m_{\infty}$ is the mass of a fermion in the broken phase at infinity and $m(z)$ is its mass locally (so the singularity is behind the wall). This singular term corresponds to a negative contribution to the upper-left entry of the plasma frequency matrix, \^2\_[p11]{}(z) \~- , \[sin\] where $\alpha_W=g^2/(4\pi)$ is the weak interaction constant, $u$ is the wall velocity. One can see that the typical value of $p_z(z)$ that give rise to the singular term (\[sin\]) is of order $[m_{\infty}-m(z)]^{1/2}$. For the WKB approximation to be applicable, this should be large compared to the inverse thickness of the wall given roughly by the Higgs mass $m_H$ (all masses are those at the phase transition temperature). So, within the WKB domain, the singularity provides an enhancement factor of order m\_/\[m\_\^2 - m\^2(z)\]\^[1/2]{} m\_/m\_H , \[enh\] which can be considerable for top quarks. However, to decide if non-equilibrium contribution of top quarks to the dispersion relation, whether obtained by WKB or fully quantum-mechanical means, leads to an instability of the $Z=0$ state, we should consider it together with contributions of other particles.[^2] In particular, numerous light species (those with masses much smaller than temperature) give an essentially equilibrium contribution to ${\hat \omega}_p^2$. The equilibrium contribution of fermions and [*transverse*]{} $W$ bosons to ${\hat \omega}_p^2$ can be obtained simply by orthogonal transformation of the diagonal matrix of plasma frequencies of the $SU(2)\times U(1)$ basis. With $N_g$ light fermionic generations, light transverse $W$, and small wall velocity, this contribution is (\_p\^2)\_[eq]{} ( [rr]{} N\_g ( 1-2s\^2+s\^4) + 2c\^4 & N\_g ( 1-s\^2)sc + 2sc\^3\ N\_g ( 1-s\^2)sc + 2sc\^3 & ( N\_g + 2 ) s\^2 c\^2 ) , \[equ\] where $c\equiv\cos\theta_W$, $s\equiv\sin\theta_W$. For our estimates, we took $N_g=3$ but subtracted the top-quark contribution from (\[equ\]), considered $W$ bosons as light, and neglected the role of longitudinal $W$ and Higgs bosons. In the WKB approximation, the quantity of interest is the magnitude of the non-equilibrium top-quark contribution to ${\hat \omega}_p^2$, the most significant part of which is (\[sin\]), corresponding to the onset of instability. It is determined approximately from ( [cc]{} \^2\_[p11]{}(z) + m\^2\_Z(z) + A &      B\ B &      C ) = 0 , \[det\] where $A$, $B$, $C$ are the entries of the equilibrium plasma frequency matrix of light species. (In a fully quantum-mechanical, non-WKB calculation, the relation between the current and the field becomes non-local [@NT] and so does the criterion for instability.) Estimating the plasma frequency of light species as described above, we find from (\[det\]) that the onset of instability is at - \^2\_[p11]{}(z) = m\^2\_Z(z) + 0.25 g\^2 T\^2 /\^2\_W . \[ons\] For realistic values of $\phi_{\infty}/T$, where $\phi_{\infty}$ is the expectation value of the Higgs field in the broken phase, the second term on the right-hand side is of the same order as $m_Z^2$ in the broken phase, for example, for $\phi_{\infty}=1.5 T$ used in ref.[@NT], about 45% of it. We conclude that light species are in general important for the analysis of stability, because of their large contribution to the dispersion relation. The author is grateful to N. Turok for a lot of useful correspondence and sending parts of the longer paper prior to publication, and to T. Clark, G. Giuliani, S. Love and M. Shaposhnikov for discussions. This work was supported in part by the Alfred P. Sloan Foundation and in part by the U.S. Department of Energy. [99]{} S. Nasser and N. Turok, Princeton preprint PUPT-94-1456 (1994), hep-ph/9406270. D.J. Griffiths, [*Introduction to Quantum Mechanics*]{}, Prentice Hall, 1994; Problem 6.27. L.D. Landau and E.M. Lifshitz, [*Statistical Physics*]{}, Part 1, Pergamon, 1980; Sect.16. [^1]: Alfred P. Sloan Foundation Fellow; DOE Outstanding Junior Investigator [^2]: The author is grateful to M. Shaposhnikov for discussion that led to these considerations.

--- abstract: 'We have analyzed in detail a set of Rossi X-ray Timing Explorer (RXTE) observations of the galactic microquasar GRS 1915+105 corresponding to times when quasi-periodic oscillations in the infrared have been reported. From time-resolved spectral analysis, we have estimated the mass accretion rate through the (variable) inner edge of the accretion disk. We compare this accretion rate to an estimate of the mass/energy outflow rate in the jet. We discuss the possible implications of these results in terms of disk-instability and jet ejection, and in particular note an apparent anti-correlation between the accretion and ejection rates, implying that the gas expelled in the jet must leave the accretion disk before reaching its innermost radius.' author: - 'Belloni, T., Migliari, S., Fender, R.P.' date: 'Received ; accepted 15 May 2000' title: Disk mass accretion rate and infrared flares in GRS 1915+105 --- =1.7 true cm Introduction ============ GRS 1915+105 is a transient X-ray source discovered in 1992 with WATCH (Castro-Tirado, Brandt & Lund 1992). Since then it has probably never switched off completely and it has remained as a highly variable bright X-ray source (see Sazonov et al. 1994; Paciesas et al. 1996; Bradt et al. 2000). It is the first Galactic object that was found to show superluminal expansion in the radio (Mirabel & Rodríguez 1994). The interpretation of this phenomenon in terms of relativistic jets (Rees 1966) implies bulk velocities of the ejecta of $\geq 0.9c$ at an angle of 60–70 degrees to the line of sight (Mirabel & Rodríguez 1994, Fender et al. 1999, Rodríguez & Mirabel 1999). Because of the high value of the extinction on the line of sight, no optical counterpart is available, but an infrared counterpart has been found (Mirabel et al. 1994). The source is suspected to host a black hole because of its high X-ray luminosity and its similarity with another Galactic superluminal source GRO J1655-40 (Zhang et al. 1994), for which a dynamical estimate of the mass is available (Orosz & Bailyn 1997). Four years of monitoring with the All-Sky Monitor (ASM) on board RXTE showed that the 2-10 keV flux of GRS 1915+105 is extremely variable, considerably more than any other known X-ray source (see Bradt et al. 2000). See Belloni et al. (2000) for a complete reference list of RXTE observations of the source. Belloni et al. (1997a,b), from the analysis of selected X-ray spectra, showed that the X-ray variability of the source can be interpreted as the repeated appearance/disappearance of the inner portion of the accretion disk, caused by a thermal-viscous instability. During the low-flux intervals, when the source spectrum hardens considerably, the inner disk up to a certain radius becomes unobservable and is slowly re-filled again. A more complete picture of these variations, where the observations were classified into twelve different classes and another type of (soft) low-flux intervals was presented, was shown by Belloni et al. (2000). Additional spectral analysis has been presented by Markwardt et al. (1999) and Muno et al. (1999), who analyzed in detail the connection between QPOs and energy spectra in GRS 1915+105. One of the problems caused by the exceptional variability of the source is that it is difficult to estimate the accretion rate through the disk or even to rate observations according to accretion rate. Quasi-periodic variability in the radio, infrared and millimetre bands has been discovered (Pooley 1995, Pooley & Fender 1997; Fender et al. 1997; Fender & Pooley 2000). Fender et al. (1997) suggested that these oscillations could correspond to small ejections of material from the system. Indeed, these oscillations have been found to correlate with the disk-instability as observed in the X-ray band (Pooley & Fender 1997; Eikenberry et al. 1998,2000; Mirabel et al. 1998). This suggests that (some of) the gas is ejected from the inner disk during each low-flux interval. On longer time scales an analogous pattern is observed in the form of major relativistic ejections occurring at the end of a 20-day X-ray dip or ‘plateau’ (Fender et al. 1999). In this Letter we present the results of detailed time-resolved spectral analysis of RXTE/PCA data of observations when (quasi-)simultaneous infrared data are available. We estimate the value of the accretion rate through the disk for each observation and show that it is anticorrelated with the estimated jet power. Data analysis ============= The published infrared observations of GRS 1915+105 for which there are simultaneous or quasi-simultaneous (ie. within 2 days) RXTE/PCA data are those from Mirabel et al. (1998), Eikenberry et al. (1998), Fender et al. (1998), Eikenberry et al. (2000), Fender & Pooley,(2000). All observations reveal very variable X-ray light curves (see Table 1), corresponding to classes $\beta$, $\nu$ and $\theta$ in the classification by Belloni et al. (2000). Date Obs\# Class T$_{\rm start}$(UT) $\Delta$ t (s) R$_{\rm max}$(km) $\dot{M}_{\rm disk}$ (M$_\odot$/yr) $\dot{M}_{\rm J} $ (M$_\odot$/yr) $P_{\rm J}$ (erg s$^{-1}$) --------- ---------------- ---------- --------------------- ---------------- ------------------- ------------------------------------- ----------------------------------------- ---------------------------- 14/8/97 20186-03-03-01 $\beta$ 4:02 530-690 170$\pm$ 14 1.3$\times 10^{-7}$ 6$\times 10^{-7}$(a) $9\times 10^{37}$ 09/9/97 20402-01-45-03 $\beta$ 6:00 500-720 128$\pm$ 13 7.1$\times 10^{-8}$ 3$\times 10^{-7}$(b) $1\times 10^{38}$ 15/9/97 20186-03-02-00 $\theta$ 12:31 600-1000 —$^c$ —$^c$ 5$\times 10^{-7}$(d\*) $9\times 10^{37}$ 10/7/98 30182-01-03-00 $\nu$ 5:05 2250-3500$^e$ 288$\pm$ 27 2.7$\times 10^{-7}$ $10^{-7}$(f) $4 \times 10^{37}$ 22/5/99 40702-01-02-00 $\nu$ 20:41 1100-1370 55$\pm$ 13 8.0$\times 10^{-9}$ 2$\times 10^{-6}$(g\*) $3 \times 10^{38}$ from Eikenberry et al. (1998); $^{\mathrm{b}}$ from Mirabel et al. (1998); $^{\mathrm{c}}$ not measurable; $^{\mathrm{d}}$ from Fender & Pooley (1998) determined from IR data; $^{\mathrm{f}}$ from Eikenberry et al. (2000); $^{\mathrm{g}}$ from Fender & Pooley (2000); $^{\mathrm{*}}$ quasi-simultaneous For each observation, we produce light curves at 1s time resolution (from [Standard1]{} data) and isolated the long hard low-flux intervals corresponding to state C (unobservable inner disk) of Belloni et al. (2000). For each interval, we measured its length from the light curve (see Table 1). Then we accumulated spectra on a time scale of 16 seconds from [Standard2]{} data, thus retaining the full energy resolution and coverage of the PCA. From each spectrum, we subtracted the background estimated with [pcabackest]{} vers. 2.1b. We did not correct for deadtime effects, but we do not expect this effect to be too important. For each observation in PCA epoch 3 we produced a detector response matrix using [pcarsp]{}, while for epoch 4 we used the response provided on line by K. Yahoda [^1]. We fitted each spectrum with the “standard” model used for black-hole candidates, consisting of the superposition of a multicolor disk-blackbody and a power law. By assuming a distance of 12.5 kpc and a disk inclination of 70$^{\circ}$ (Mirabel & Rodríguez 1994), we can derive from the fits the inner radius of the accretion disk. Correction for interstellar absorption (fixed to $6\times 10^{22}$cm$^{-2}$, see Belloni et al. 2000) and an additional emission line (fixed at 6.4 keV) were also included. A systematic error of 1% was added. The value of the reduced $\chi^2$ was usually around 1, although some fits were slightly worse. The resulting interesting parameters (inner disk radius and temperature, slope of the power law) as a function of time are shown in Fig. 1 for three of the five observations, for which this automated procedure gave good results. The remaining two observations had to be treated more carefully. The observation from 1997 Sep 15th, the only one from class $\theta$, resulted in an extremely strong power law component, with a photon index steeper than 3. The softness and intensity of this component made it impossible to obtain sensible values for the disk parameters, although there is evidence of its presence. This enhanced power law is probably the reason of the difference between this class and the others (see Belloni et al. 2000). The observation from 1998 July 10th did not include full state-C intervals: in this case, we measured the length of the intervals from the infrared (Eikenberry et al. 2000). Also, the inner disk radius resulted to be larger and therefore more difficult to measure as this component is softer. In order to estimate the disk parameters, we produced a 32s spectrum corresponding to the bottom of the dip only and obtained the best fit parameters, corresponding to the largest inner radius. This is the reason why there is only one point for this observation in Fig. 2. Results ======= In principle, from each spectrum the accretion rate through the measured inner radius of the disk could be measured from the values of kT$_{\rm in}$ and R$_{\rm in}$ (see Belloni et al. 1997a) by using the expression from a standard thin accretion disk. However, given the errors on these parameters, this measurement is too uncertain. In order to obtain an improved estimate of the disk accretion rate or, better, a ranking of the observations in terms of accretion rate (since the actual values of the inner disk radii obtained with the multicolor disk-blackbody model are probably underestimates, see Merloni, Fabian & Ross 1999), we plotted the values corresponding to the deepest parts of the X-ray light curves in a kT$_{\rm in}$ vs. R$_{\rm in}$ plane (see Fig. 2). If for each observation the disk accretion rate was constant, the points should lie on the diagonal lines corresponding to a slope $-$3/4 (as, for a given $\dot{M}$, $T \propto R^{-3/4}$ – Belloni et al. 1997a). Their actual distribution is flatter, showing that there is a deviation from the expected law, but it is interesting to note that the distributions lie on parallel curves in the log-log plane. This indicates different values of the disk accretion rate. Lines corresponding to the larger measured radius for each of the four observations are shown in Fig. 2 with their associated accretion rate value. Typical 1$\sigma$ errors are also shown. Although the actual values for the accretion rate are probably not accurate, on the basis of this plot we can rank the observations by accretion rate. It is important to note that the accretion rate measured this way correspond to matter passing [*through*]{} the observed inner radius of the disk only: if some matter leaves the disk before that radius, its presence cannot be detected with this procedure. This estimate of accretion rate can be double checked by considering the length of the state C intervals, which Belloni et al. (1997a,b) interpreted as the viscous time scale of the disk at the edge of the unobservable region which is refilled. The observation from 1999 May 22nd has a smaller inner disk radius (see Fig. 2) than the 1997 ones and a longer re-fill time (Tab. 1), indicating a lower value of the accretion rate. The 1998 July 10th observation has a much larger inner disk radius than the 1997 ones, by a factor of 1.7 and 2.3, which would correspond to a re-fill time longer by a factor 6.4 and 18 respectively, while it is much shorter, indicating a higher accretion rate. Discussion ========== The results of our analysis indicate that, at least for observations of class $\nu$ and $\beta$ (which have many similar traits), we have a way to estimate the disk accretion rate during an instability event, when the inner disk radius grows from its “minimum” value of $\sim$30 km and slowly moves back to it. Although we know that the measured value is only an underestimate, it is natural to associate this minimum value with the innermost stable orbit. It is interesting to compare these values, or at least their ranking, with the rate of ejection in the jets. As we mentioned above, the accretion rate measured through this procedure is associated to matter flowing [*through*]{} the observable inner edge of a geometrically thin accretion disk. Some of the accreting gas must leave the accretion disk to form the jet, unless it is entirely composed of pairs generated by photon-photon interactions. and how this happens is basically unknown. There are two extreme possibilities: either matter ejected in the jet leaves the accretion disk before entering the innermost regions, thus not contributing to our measured disk accretion rate (case 1), or it leaves it after passing through our measured inner disk radius, in which case it is a fraction of the accretion rate we measure (case 2). In case 1, if the fraction of matter in the jet is constant and the total external accretion rate (disk+jet) is variable, we expect a positive correlation between disk accretion rate (from X rays) and disk ejection rate (from the infrared). If the fraction is variable and the total is constant, these quantities should be anticorrelated. In case 2, if the fraction of matter in the jet is constant, we expect a positive correlation, while the constant total is in this case not possible as the total would be what we measure, which is not observed to be constant. If both fraction and total vary, the situation is complicated. Of course, there is a spectrum of intermediate possibilities, where the jet production is connected to the inner region of the disk in a way that would not allow to dissociate the two processes. With the paucity of existing data, we limit ourselves to the extreme cases. Notice that measuring an anti-correlation would be an indication against case 2. Table 1 also lists an estimate of the mass ejection rate $\dot{M}_{\rm J}$. This is based upon an equipartition calculation for one proton for each electron, negligible kinetic energy associated with the repeated ejection events, and an average over the repetition period of the oscillations. Note that there is a systematic uncertainty in these numbers due to lack of knowledge of the intrinsic electron spectrum which corresponds to the observed flat-spectrum radio–infrared emission. However, unless the spectral form of the distribution changes between observations then the effect is the same for all data sets and the ranking remains the same. Of course we may be observing synchrotron emission from a pair plasma with no baryonic content, in which case the amount of power being supplied to the jet, $P_{\rm J}$, makes more useful comparison with the accretion rate; this value is also listed in Table 1. For more details of how these quantities are calculated, see Fender & Pooley (2000). Either way, there appears to be an [*anticorrelation*]{} between accretion rate inferred from the X-ray spectral fits and the outflow rate of mass/energy in the jet. The low number of points in our sample prevents us from saying something more firm. Notice that an anticorrelation is also suggested by the strong flat-spectrum radio emission observed during long ‘plateau’ intervals; periods when Belloni et al. (2000) estimate that the accretion rate must be very low. We also note that the faint infrared flares reported by Eikenberry et al. (2000) do not appear to be different from the others in other respects, as the X-ray light curves are too undersampled to allow a detailed correlation. If future observations show that disk accretion rate and jet ejection rate are indeed anti-correlated, the following scenario could be speculated. A fraction of the accreting gas leaves the geometrically thin accretion disk before reaching the inner edge (from which it would fall into the black hole) and goes into a hot corona. The details are not known, but our results indicate that this does not happen after the inner edge. As the disk refills, the inner radius moves inwards, more soft photons from the disk reach the corona, which causes its Comptonization emission to soften gradually. At the end of the instability period, when the disk is refilled down to the innermost stable orbit, this “reservoir” of hot gas is expelled to produce the jet, resulting in the observed infrared / mm / radio emission, causing the power-law component to steepen dramatically and to cause the sudden change in the X-ray count rate and spectral parameters. Notice that, as we remarked earlier, the distributions of points in Fig. 2 are flatter than the expected curve for a constant disk accretion rate according to a standard thin disk: in other words, as the inner disk radius decreases, the disk accretion rate seems to decrease as well. This could mean that the process that re-routes some gas from the disk to the corona becomes more efficient closer to the central object, and therefore the fraction of matter going into the corona increases as the disk refills. We thank G. Ghisellini and M. Tagger for useful discussions. Belloni, T., Méndez, M., King, A.R., van der Klis, M, & van Paradijs, J., 1997, ApJ, 479, L145 Belloni, T., Méndez, M., King, A.R., van der Klis, M, & van Paradijs, J., 1997, ApJ, 488, L109 Belloni, T., Klein-Wolt, M., Méndez, M., van der Klis, M., van Paradijs, J., 2000, A&A, 355, 271 Bradt, H., Levine, A.M., Remillard, R.A., Smith, D.A., 2000, Mem SAIt, Vol. 71, in press. Castro-Tirado, A. 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--- abstract: 'For a nonnegative weakly irreducible tensor $\mathcal{A}$, we give some characterizations of the spectral radius of $\mathcal{A}$, by using the digraph of tensors. As applications, some bounds on the spectral radius of the adjacency tensor and the signless Laplacian tensor of the $k$-uniform hypergraphs are shown.' address: - 'School of Science, Harbin Institute of Technology, Harbin 150001, PR China' - 'School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, PR China' - 'College of Science, Harbin Engineering University, Harbin 150001, PR China' author: - Lizhu Sun - Baodong Zheng - Yimin Wei - Changjiang Bu title: Characterizations of the spectral radius of nonnegative weakly irreducible tensors via digraph --- Nonnegative tensor, Spectral radius, Digraph, $k$-uniform hypergraph\ *AMS classification:* 15A69, 15A18 Introduction {#Introduction } ============ An order $m$ dimension $n$ tensor $\mathcal{A}$ consist of $n^{m}$ complex entries, $$\mathcal{A}=\left( {a_{i_1i_2\cdots i_m } } \right),~{1 \leqslant i_j\leqslant n}~ \left( {j = 1,2,\ldots ,m} \right).$$ The tensor $A=(a_{i_1i_2\cdots i_m})$ is called symmetric if $a_{i_1i_2\cdots i_m}=a_{\sigma(i_1)\sigma(i_2)\cdots \sigma(i_m)}$, where $\sigma$ is any permutation of the indices. The tensor $\mathcal{A}$ is called nonnegative if all the entries $a_{i_1\ldots i_m } \geqslant 0$. Let $\mathbb{R}^{[m,n]}_+$ be the set of order $m$ dimension $n$ nonnegative tensors. And let $\mathbb{R}^{n}_{++}$ be the set of the dimension $n$ positive vectors (all the entries positive). In 2005, the *eigenvalue of tensors* is defined by Qi [@Qi2005] and Lim [@Lim; @2005], respectively. For a complex tensor $\mathcal{A}$ of order $m$ dimension $n$, a complex number $\lambda $ and a nonzero complex vector $x=(x_1,x_2,\ldots,x_n)^{\rm T}$ is called an eigenvalue and an eigenvector (corresponding to $\lambda$) of the tensor $\mathcal{A}$, respectively, if they satisfy $$\mathcal{A}x^{m - 1} = \lambda x^{\left[ {m - 1} \right]},$$ where $\mathcal{A}x^{m - 1}$ is a dimension $n$ vector with entry $$\left( {\mathcal{A}x^{m - 1} } \right)_i = \sum\limits_{i_2 ,...,i_m = 1}^n {a_{ii_2 ...i_m } x_{i_2 } ...x_{i_m } }, i=1,2,\ldots,n,$$ and $x^{\left[ {m - 1} \right]} = \left( {x_1^{m - 1} ,x_2^{m - 1},\ldots,x_n^{m - 1} } \right)^\mathrm{T}$ (see [@Qi2005]). Let $\rho(\mathcal{A})=\sup\{|\lambda|:\lambda \in {\rm spec}(\mathcal{A})\}$ be the spectral radius of tensor $\mathcal{A}$, where ${\rm spec}(\mathcal{A})$ is the set of all the eigenvalues of $\mathcal{A}$. Recently, the spectral theory of tensors has attracted much attention [@Bu2015; @Chang; @Chang; @2011; @Yang; @2010; @Yang; @2013]. In 2005, Lim [@Lim; @2005] proposes the definition of *irreducible tensors*. In 2008, Chang et al. [@Chang] give the Perron-Frobenius Theorem for nonnegative irreducible tensors. It is shown that $\rho(\mathcal{A})$ is an eigenvalue of nonnegative irreducible tensors, and $\rho(\mathcal{A})$ is the only eigenvalue with nonnegative eigenvectors [@Chang]. Similarly as the *Collatz-Wielandt Theorem* of matrices, the *Minimax Theorem* of the spectral radius of nonnegative irreducible tensors $\mathcal{A}$ is given as follows (see [@Chang]) $$\min \limits_{x \in \mathbb{R}_{++} ^n }\max \limits_{1 \leqslant i \leqslant n} \frac{{\left( {\mathcal{A}x^{m - 1} } \right)_i }} {{x_i^{m - 1} }} = \rho \left( \mathcal{A} \right) = \max \limits_{x \in \mathbb{R}_ {++} ^n} \min \limits_{1 \leqslant i \leqslant n} \frac{{\left( {\mathcal{A}x^{m - 1} } \right)_i }} {{x_i^{m - 1} }} .$$ Scholars pay much attention to find the largest eigenvalue of nonnegative irreducible tensors [@Chen2013; @Ng; @Ni2014; @Zhou; @2013]. The *weakly irreducible tensors* are defined by associated with tensors a digraph [@SF; @2013; @Pearson2014]. \[def1\] An order $m$ dimension $n$ tensor $\mathcal{A}$ over real field is called weakly irreducible if the digraph $G_{\mathcal{A}}$ is strongly connected. Digraph $G_{\mathcal{A}}$ and the strongly connectivity are introduced in Section 2. If tensor $\mathcal{A}$ is irreducible then it is weakly irreducible [@SF; @2013; @Pearson2014]. For the nonnegative weakly irreducible tensor $\mathcal{A}$, there exists a positive eigenvector corresponding to the eigenvalue $\rho(\mathcal{A})$ (see [@SF; @2013]). A hypergraph $\mathcal{H}=(V(\mathcal{H}), E(\mathcal{H}))$ is called *$k$-uniform* if each edge of $\mathcal{H}$ contains exactly $k$ distinct vertices [@Cooper2012]. The *adjacency tensor* of $\mathcal{H}$, denoted by $\mathcal{A}_{\mathcal{H}}=(a_{i_1i_2\cdots i_k})$, is an order $k$ dimension $|V(\mathcal{H})|$ tensor with entries $$a_{i_1i_2\cdots i_k}= \left\{ {\begin{array}{*{20}{c}} \frac{1}{(k-1)!}, ~ \{i_1i_2\cdots i_k\}\in E(\mathcal{H}), \\ 0 ,~~~~~~~\mbox{otherwises} .~~~~~ \\ \end{array}} \right.$$ The *degree tensor* of the $k$-uniform uniform hypergraph $\mathcal{H}$, denoted by $\mathcal{D}_{\mathcal{H}}$, is an order $k$ dimension $|V(\mathcal{H})|$ diagonal tensor whose $(i\cdots i)$-diagonal entry is the degree of vertex $i$, $i=1,2,\ldots,|V(\mathcal{H})|$. The tensor $\mathcal{Q}_{\mathcal{H}}=\mathcal{D}_{\mathcal{H}}+\mathcal{A}_{\mathcal{H}}$ is called the *signless Laplacian tensor* of the $\mathcal{H}$. Recently, the spectral theory of hypergraphs developed rapidly [@Cooper2012; @Hu2015; @Qi2014; @Shao2015; @Zhou2014]. It is well-known that the nonnegative irreducible matrices are closely related to the digraphs [@RA1991]. For a nonnegative irreducible matrices $M$ with all diagonal entries zero, Brualdi [@RA1982] gives the characterizations of the spectral radius of $M$ by using the associated digraph. In this paper, we use the digraph of tensors to characterize the spectral radius of the nonnegative weakly irreducible tensors, which generalize the results of matrices to tensors [@RA1982]. By applying the characterizations, some bounds on the spectral radius of the adjacency tensor and signless Laplacian tensor of a $k$-uniform hypergraph are shown. Preliminary =========== For an order $m$ dimension $n$ tensor $A=(a_{i_1i_2\cdots i_m})$, let $G_\mathcal{A} = \left({V(G_\mathcal{A} ),E( G_\mathcal{A} )} \right)$ be the digraph of the tensor $\mathcal{A }$ with vertex set $V (G_\mathcal{A} )= \left\{ {1,2,\ldots,n} \right\}$ and arc set $E( G_\mathcal{A} ) = \{ (i,j)| a_{ii_2 ...i_m } \neq 0,j \in \{ i_2 ,...,i_m \}\}$ (see [@SF; @2013; @Pearson2014]). If there exist directed paths from $i$ to $j$ and $j$ to $i$ for each $i, ~j\in V(G_\mathcal{A} )$ ($i\neq j$), then $ G_\mathcal{A}$ is called strongly connected. Denote the set of the circuits in $G_\mathcal{A} $ by $C( G_\mathcal{A} )$ (Loops in the circuits are allowed). Let $G_\mathcal{A} ^ + \left( v \right): = \left\{ {u\in V(G_\mathcal{A} ):\left( {v,u} \right) \in E( G_\mathcal{A} )} \right\}$. Define a map $f$ from the vertex set of $G_\mathcal{A} $ to the real field, $f:V(G_\mathcal{A} )\rightarrow \mathbb{R}$, $f$ is called a vertex labelling of $G_\mathcal{A} $. By the Lemma 2.6 of [@RA1982], we can get the following result. [[@RA1982]]{}\[lem1\] Let $G_\mathcal{A} $ be the digraph of $\mathcal{ A}$ with a vertex labelling $f$ on $V(G_\mathcal{A} )$. If $G_\mathcal{A} ^ + \left( v \right)$ is nonempty for each $v \in V(G_\mathcal{A} )$, then there exist circuits $\{v_{i_1 },v_{i_2 } ,...,v_{i_k } ,v_{i_{k + 1} } \\= v_{i_1 }\}$ and $\{v_{t_1 },v_{t_2 } ,...,v_{t_s } ,v_{t_{s + 1} } = v_{t_1} \}$ (Loops in the circuits are allowed) such that $f(v_{i_{j + 1} })=\max\{f(v):v\in G_\mathcal{A} ^ + ( v_{i_j} )\}$ and $f(v_{t_{l + 1} })=\min\{f(v):v\in G_\mathcal{A}^ + ( v_{t_l} )\}$ $j = 1,2,\ldots,k$, $ l = 1,2,\ldots,s$, respectively. [[@SF; @2013]]{}\[lem3\] Let $\mathcal{A}\in \mathbb{R}^{[m,n]}_+$ be a weakly irreducible tensor. Then $\rho(\mathcal{A})$ is an eigenvalue of $\mathcal{A}$, and there exists a unique positive eigenvector corresponding to $\rho(\mathcal{A})$ up to a multiplicative constant. [[@shao; @2013]]{}\[lem5\] For an order $m$ dimension $n$ tensor $\mathcal{A}={(a_{i_1i_2\ldots i_m})}$ and an invertible diagonal matrix $D={\rm diag}(d_{11}, d_{22},\ldots, d_{nn})$, $\mathcal{B}=D^{-(m-1)}\mathcal{A}D$ is an order $m$ dimension $n$ tensor with entries $$b_{i_1i_2\ldots i_m}= d_{i_1 i_1}^{-(m-1)}a_{i_1i_2\ldots i_m}d_{i_2i_2}\cdots d_{i_mi_m}.$$ In this case, $\mathcal{A}$ and $\mathcal{B}$ are called diagonal similar, and $\mathcal{A}$ and $\mathcal{B}$ have the same spectrum. \[lem6\][[@Pearson2014]]{} Let $\mathcal{H}$ be an $k$-uniform hypergraph. Then $\mathcal{A}_{\mathcal{H}}\in \mathbb{R}^{[m,n]}_+$ (and $\mathcal{Q}_{\mathcal{H}}\in \mathbb{R}^{[m,n]}_+$) is weakly irreducible if and only if $\mathcal{H}$ is connected. Main results ============ For a tensor $\mathcal{A}=\left( {a_{i_1 i_2 \cdots i_m } } \right)\in \mathbb{R}^{[m,n]}_+$, we denote the sum of $i$-th slice of $\mathcal{A}$ by $K_i= \sum\limits^n_{i_2,\ldots,i_m =1}{a_{ii_2 \cdots i_m } } $, $i=1,2,\ldots,n$. Let $|\gamma|$ be the length of the circuit $\gamma\in C (G_\mathcal{A} )$. We first give two results on the bounds of spectral radius for nonnegative weakly irreducible tensors, which extend Theorem 4.7 and Corollary 4.6, 4.8 of [@RA1982] to tensors. \[thm2\] Let $\mathcal{A} = \left( {a_{i_1 i_2 ...i_m } } \right) \in \mathbb{R}^{[m,n]}_+$ be a weakly irreducible tensor. Then $$\min_{\gamma \in C(G_\mathcal{ A})}\left(\prod\limits_{i\in \gamma } {K_i }\right)^{\frac{1}{|\gamma|}}\leqslant \rho (\mathcal{A}) \leqslant \max_{\gamma \in C(G_\mathcal{ A})} \left(\prod\limits_{i\in \gamma } {K_i }\right)^{\frac{1}{|\gamma|}}.$$ By Lemma \[lem3\], we suppose that $x = \left( {x_1 ,x_2, \ldots,x_n } \right)^{\rm T}$ is a positive eigenvector corresponding to the eigenvalue $\rho(\mathcal{A})$. Let $f(i)=x_i$ ($i \in V(G_\mathcal{A})$) be the vertex labelling of digraph $G_\mathcal{A}$. From Definition \[def1\], we have $G_\mathcal{A}^+(i)$ is nonempty for each $i\in V(G_\mathcal{A})$. Lemma \[lem1\] gives that there exists at least one circuit $\gamma_1 = \{i_1 ,i_2,\ldots,i_p ,i_{p + 1} = i_1\}$ such that $x_{i_{j + 1} } \geqslant x_k $, for each $ k \in G_\mathcal{A} ^ + \left( {i_j } \right)$, $j = 1,2,\ldots,p$. Hence, $$\begin{aligned} \rho(\mathcal{A}) x_{i_j }^{m - 1} & = \sum\limits_{ k_2 ,\cdots,k_m =1}^n a_{i_j k_2 ...k_m } x_{k_2} \cdots x_{k_m } \hfill \\ &\leqslant \left( {\sum\limits_{ k_2 ,\cdots,k_m =1}^n {a_{i_j k_2 ...k_m } }} \right)x_{i_{j + 1} }^{m - 1} \hfill \\ & = K_{i_j } x_{i_{j + 1} }^{m - 1} \hfill ,\end{aligned}$$ for $j = 1,2,\ldots,p$. Thus $$\left(\rho (\mathcal{A})\right)^p \prod\limits_{j = 1}^p x_{i_j }^{m - 1} \leqslant \prod\limits_{j = 1}^p {K_{i_j } x_{i_{j + 1} }^{m - 1} } .$$ Note that $x = \left( {x_1 ,x_1,\ldots,x_n } \right)^{\rm T}$ is positive, so we can get $$\rho (\mathcal{A}) \leqslant \left(\prod\limits_{j = 1}^p {K_{i_j } }\right)^{\frac{1}{p}} ,$$ that is $$\rho (\mathcal{A}) \leqslant \left(\prod\limits_{i\in \gamma_1 } {K_i }\right)^{\frac{1}{p}}.$$ Lemma \[lem1\] also shows that there exists at least one circuit $\gamma_2=\{v_{t_1 } ,v_{t_2 },\ldots,v_{t_s } ,v_{t_{s + 1} } = v_{t_1 }\}$ such that $x_{t_{l + 1} } \leqslant x_k $ for each $ k \in G_\mathcal{A} ^ + ( t_l )$, $l = 1,2,\ldots,s$. Similarly as the above proof, we can get $$\rho (\mathcal{A}) \geqslant \left(\prod\limits_{i\in \gamma_2 } {K_i }\right)^{\frac{1}{s}}.$$ Thus $$\min_{\gamma \in C(G_\mathcal{ A})}\left(\prod\limits_{i\in \gamma } {K_i }\right)^{\frac{1}{|\gamma|}} \leqslant \rho (\mathcal{A}) \leqslant \max_{\gamma \in C(G_\mathcal{ A})} \left(\prod\limits_{i\in \gamma } {K_i }\right)^{\frac{1}{|\gamma|}}.$$ **Remark.** It is easy to see that if $K_1=K_2=\cdots=K_n$, the equalities in the above theorem hold. Lemma \[lem6\] gives that the adjacency tensor (and signless Laplacian tensor) of a connected hypergraph $\mathcal{A}_{\mathcal{H}}$ (and $\mathcal{Q}_{\mathcal{H}}$) is nonnegative weakly irreducible, so $\rho(\mathcal{A}_{\mathcal{H}})$ (and $\rho(\mathcal{Q}_{\mathcal{H}})$) is an eigenvalue of $\mathcal{A}_{\mathcal{H}}$ (and $\mathcal{Q}_{\mathcal{H}}$). In [@Cooper2012], it is shown that the largest eigenvalue $\lambda_{\max}$ of $\mathcal{A}_{\mathcal{H}}$ is between the maximum degree $d_{\max}$ and the average degree $\overline{d}$ of $\mathcal{H}$, $\overline{d} \leqslant\lambda_{\max}\leqslant d_{\max} $. By using Theorem \[thm2\], we also show the bounds on the largest eigenvalue of adjacency tensor and signless Laplacian tensor in terms of the degrees. \[thm5\] Let $\mathcal{H}$ be a connected $k$-uniform hypergraph with $n$ vertices. Suppose that $d_i$ is the degree of vertex $i$, $i=1,2,\ldots,n$. Then $$\min_{\gamma \in C(G_{\mathcal{ A}_{\mathcal{H}}})}\left(\prod\limits_{i\in \gamma } {d_i }\right)^{\frac{1}{|\gamma|}}\leqslant \rho(\mathcal{A}_{\mathcal{H}}) \leqslant \max_{\gamma \in C(_{\mathcal{ A}_{\mathcal{H}}})} \left(\prod\limits_{i\in \gamma } {d_i }\right)^{\frac{1}{|\gamma|}},$$ and $$\min_{\gamma \in C(G_{\mathcal{ Q}_{\mathcal{H}}})}\left(\prod\limits_{i\in \gamma } {2d_i }\right)^{\frac{1}{|\gamma|}}\leqslant \rho(\mathcal{Q}_{\mathcal{H}}) \leqslant \max_{\gamma \in C(_{\mathcal{ Q}_{\mathcal{H}}})} \left(\prod\limits_{i\in \gamma } {2d_i }\right)^{\frac{1}{|\gamma|}}.$$ The shortest length of the circuits in $G_\mathcal{ A}$ is called the *girth* of $G_\mathcal{ A}$. If we bring an order to the slice sums $K_i$ $(i=1,2,\ldots,n)$ of tensor $\mathcal{A}$, the following result can be obtained. \[thm1\] Let $\mathcal{A} = \left( {a_{i_1 i_2 ...i_m } } \right)\in \mathbb{R}^{[m,n]}_+ $ be a weakly irreducible tensor. Suppose that $K_1 \leqslant K_2\leqslant \cdot\cdot\cdot \leqslant K_n$ and the girth of $G_\mathcal{ A}$ is $g$. Then $$\left ({K_1K_2\cdots K_g }\right)^{\frac{1}{g}}\leqslant \rho \left( \mathcal{A} \right) \leqslant \left({K_{n - g + 1} K_{n - g + 2}\cdots K_n }\right)^{\frac{1}{g}} .$$ By Theorem \[thm2\], we have there exist circuits $\gamma_1,\gamma_2\in C(G_\mathcal{ A})$ of length $|\gamma_1|\geqslant g,~|\gamma_2|\geqslant g$ such that $$\left({\prod\limits_{i \in \gamma_2 } {K_i } }\right)^{\frac{1}{|\gamma_2|}}\leqslant \rho \left( \mathcal{A} \right) \leqslant \left({\prod\limits_{i \in \gamma_1 } {K_i } }\right)^{\frac{1}{|\gamma_1|}} .$$ Since $$\left({\prod\limits_{i \in \gamma_1 } {K_i } }\right)^{\frac{1}{|\gamma_1|}} \leqslant \left({K_{n - |\gamma_1| + 1} K_{n - |\gamma_1| + 2}\cdots K_n }\right)^{\frac{1}{|\gamma_1|}} \leqslant \left({K_{n - g + 1} K_{n - g + 2}\cdots K_n }\right)^{\frac{1}{g}}$$ and $$\left({\prod\limits_{i \in \gamma_2 } {K_i } }\right)^{\frac{1}{|\gamma_2|}}\geqslant \left(K_1K_2\cdots K_{|\gamma_2|} \right)^{\frac{1}{|\gamma_2|}}\geqslant \left(K_1K_2\cdots K_g \right)^{\frac{1}{g}},$$ we obtain the theorem holds. For a connected $k$-uniform hypergraph $\mathcal{H}$, According to the definition of hypergraph, there are no loops in $G_{\mathcal{A}_{\mathcal{H}}}$. And since $\mathcal{A}_{\mathcal{H}}$ is symmetric tensor, we have the girth of $G_{\mathcal{A}_{\mathcal{H}}}$ is $2$. Clearly, the girth of $G_{\mathcal{Q}_{\mathcal{H}}}$ is $1$. Hence, it follows from Theorem \[thm5\] and \[thm1\] that the follow result can be obtained, which are shown in [@Cooper2012] and [@Hu2015+], respectively. For a connected $k$-uniform hypergraph $\mathcal{H}$, if $\mathcal{H}$ is regular with degree $d$, then $\rho(\mathcal{A}_{\mathcal{H}})=d$ and $\rho(\mathcal{Q}_{\mathcal{H}})=2d$. As we introduce in the first section, there are some results to find the spectral radius (largest eigenvalue) of nonnegative irreducible tensors. Here, we also give two theorems on the minimum and the maximum characterizations of the spectral radius of nonnegative weakly irreducible tensors. \[thm3\] Let $\mathcal{A} = \left( {a_{i_1 i_2 ...i_m } } \right) \in \mathbb{R}^{[m,n]}_+$ be a weakly irreducible tensor. Then $$\min_{x\in \mathbb{R}_{++}^n} \max_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{|\gamma|}}=\rho(\mathcal{A})=\max_{x\in \mathbb{R}_{++}^n} \min_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{|\gamma|}}.$$ Let $\mathcal{B}=X^{-(m-1)}\mathcal{A}X$, where the matrix $X={\rm diag}(x_1,x_2,\ldots,x_n)$, $x_i>0$, $i=1,2,\ldots,n$. It is easy to see $\mathcal{B}$ is nonnegative weakly irreducible, and $G_\mathcal{ A}$ and $G_\mathcal{ B}$ are the same digraph. So by Theorem \[thm2\], we have $$\min_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} K_i(\mathcal{B})\right)^{\frac{1}{|\gamma|}}\leqslant \rho(\mathcal{B})\leqslant\max_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} K_i(\mathcal{B})\right)^{\frac{1}{|\gamma|}}.$$ Calculation gives that $$K_i(\mathcal{B})= \frac{1}{{x_i^{m - 1} }} \sum\limits^n_{ i_2,\cdots,i_m =1 } { {a_{ii_2 ...i_m } } x_{i_2 } ...x_{i_m } }=\frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}},$$ and since Lemma \[lem5\] shows that $\rho(\mathcal{A})=\rho(\mathcal{B})$, we obtain $$\min_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{|\gamma|}}\leqslant \rho(\mathcal{A})\leqslant \max_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{|\gamma|}}.$$ When $(x_1,x_2,\ldots, x_n)^{\rm T}$ is a positive eigenvector of $\mathcal{A}$ corresponding to $\rho(\mathcal{A})$, the equalities in the above equation hold, so we can get $$\min_{x\in \mathbb{R}_{++}^n} \max_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{|\gamma|}}=\rho(\mathcal{A})=\max_{x\in \mathbb{R}_{++}^n} \min_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{|\gamma|}}.$$ By the proof of the above theorem, we get the following result. Let $\mathcal{A} = \left( {a_{i_1 i_2 ...i_m } } \right) \in \mathbb{R}^{[m,n]}_+$ be a weakly irreducible tensor. Then $$\min_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{|\gamma|}}\leqslant \rho(\mathcal{A})\leqslant \max_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{|\gamma|}},$$ where $x=(x_1,x_2,\ldots,x_n)$ is a positive vector. From Theorem \[thm1\], the following result can be obtained. \[thm4\] Let $\mathcal{A}\in \mathbb{R}^{[m,n]}_+$ be a weakly irreducible tensor. Let the girth of $G_\mathcal{ A}$ is $g$. Then $$\min_{X\in D^n} \left({\prod_{i=n-g+1}^n K_i(X^{-(m-1)}\mathcal{A}X)}\right)^{\frac{1}{g}}=\rho(\mathcal{A}) =\max_{X\in D^n} \left({\prod_{i=1}^gK_i(X^{-(m-1)}\mathcal{A}X)}\right)^{\frac{1}{g}},$$ where the slice sums of tensor $X^{-(m-1)}\mathcal{A}X$ are in the order $K_1 \leqslant K_2 \leqslant\cdot\cdot\cdot \leqslant K_n$ and $D^n$ is the set of all the $n\times n$ positive diagonal matrices. **Remark.** Brualdi gives the characterizations of the spectral radius of a nonnegative irreducible matrices with all diagonal entries zero ( Corollary 4.10 and 4.11 of [@RA1982]). Theorem \[thm3\] and \[thm4\] generalize them to general nonnegative weakly irreducible tensors without the condition that diagonal entries are zero. **References** [00]{} R.A. Brualdi, H.J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, Cambridge, 1991. R.A. Brualdi, Matrices, eigenvalues, and directed graphs, Linear and Multilinear Algebra 11(1982) 143-165. C. Bu, Y. Wei, L. Sun, J. 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--- abstract: | Cosmic acceleration is investigated through a kink-like expression for the deceleration parameter ($q$). The new parametrization depends on the initial ($q_i$) and final ($q_f$) values of $q$, on the redshift of the transition from deceleration to acceleration ($z_{t}$) and the width of such transition ($\tau $). We show that although supernovae (SN) observations (*Gold182* and SNLS data samples) indicate, at high confidence, that a transition occurred in the past ($z_{t}>0$) they do not, by themselves, impose strong constraints on the maximum value of $z_{t}$. However, when we combine SN with the measurements of the ratio between the comoving distance to the last scattering surface and the SDSS+2dfGRS BAO distance scale ($S_{k}/D_{v}$) we obtain, at $95.4\%$ confidence level, $z_{t}=0.84\pm _{0.17}^{0.13}$ and $\tau =0.51\pm _{0.17}^{0.23}$ for ($S_{k}/D_{v}$+*Gold182*), and $% z_{t}=0.88\pm _{0.10}^{0.12}$ and $\tau =0.35\pm _{0.10}^{0.12}$ for ($% S_{k}/D_{v}$ + SNLS), assuming $q_i=0.5$ and $q_f=-1$. We also analyze the general case, $q_f\in(-\infty,0)$ finding the constraints that the combined tests ($% S_{k}/D_{v}$ + SNLS) impose on the present value of the deceleration parameter ($q_0$). address: 'Universidade Federal do Rio de Janeiro, Instituto de Física, CEP 21941-972, Rio de Janeiro, RJ, Brazil' author: - 'Émille E. O. Ishida' - 'Ribamar R. R. Reis' - 'Alan V. Toribio' - Ioav Waga title: 'When did cosmic acceleration start? How fast was the transition?' --- , , , Introduction ============ Since the discovery of the accelerated expansion of the universe in 1998 [@riess98; @perlm99], considerable effort in cosmology has been devoted to determine the source of this acceleration. The two most common possibilities discussed in the literature are: the existence of an exotic component with sufficiently negative pressure (dark energy) and proper modifications of general relativity at cosmological scales (for recent reviews see [@review]). One way of making progress in determining the cosmic expansion history is through a model by model analysis. Another is to carry out a phenomenological analysis with the use of different parameterizations of the dark energy equation of state [@eosparam], the Hubble parameter [@hubble] or the dark energy density [@wang01]. This procedure may provide interesting pieces of information, but in general a parametrization assumes the existence of dark matter and dark energy as different substances (barring a few exceptions no interaction in the dark sector is considered) and general relativity is in most cases assumed. In this framework, an important question regards the number of parameters necessary to get reliable conclusions. If too many are used, the allowed region in the parameter space could be so large that it would not be possible to get firm conclusions [@linder05]. Otherwise, if not enough parameters are used, the obtained results may be strongly dependent on the particular parametrization choice and misleading conclusions could be reached [@bassett04]. The strategy we follow here is to use a large (four) number of parameters in order to be quite general, but, based on physical arguments, fix two of them from the start. We then relax the condition on one of the fixed parameters and obtain the confidence surface on the other three. In this work we are mainly interested in the following questions: what is the redshift of the transition from decelerated to accelerated expansion? How fast was it? We investigate these by introducing a new parametrization for the deceleration parameter ($q$) that depends on four parameters: the initial ($q_i$) and final ($q_f$) values of $q$, the redshift of the transition from deceleration to acceleration ($z_{t}$) and a quantity related to the width in redshift of such transition ($\tau $). With this formulation we aim to answer the above questions with the minimum amount of assumptions about the dark sector and the fundamental gravitation theory. This paper is organized as follows: in Section 2, we present the new $q$ parametrization and discuss some of its properties. In Section 3 the outcomes of the confrontation of this parametrization with two supernovae samples, the new *Gold182* SNe Ia from [@gold] and the first year data set of the Supernova Legacy Survey (SNLS) [@snls], are obtained (first assuming $q_f=-1$ and $q_i=1/2$). We show that current supernovae observations alone are not able to satisfactorily constrain the transition redshift. To break the SN Ia degeneracy, we combine this observable with the ratio of the comoving distance to the last scaterring surface ($S_{k}(z_{ls}=1098)$) to the baryon acoustic oscillations (BAO) distance scale ($D_{v}(z)$) at $ z_{BAO}=0.2$ and $z_{BAO}=0.35$ as estimated in [@percival07]. We show that the $\Lambda$CDM model is within the region allowed by SNLS+$S_k/D_v$ results but is excluded for *Gold182*+$S_k/D_v$ data, at $95\%$ confidence level. We then discuss the broader case with arbitrary $q_f$ exhibiting the $95\%$ confidence surface in the parameter space $(z_t,\tau,q_0)$ ($q_0$ is the present value of $q$), obtained using the SNLS+ $S_{k}/D_{v}$ data. Our conclusions are presented in Section 4. The Model ========= At large scales, it is a good approximation to consider a spatially homogeneous and isotropic universe. With this assumption we are lead to the Friedman-Robertson-Walker metric: $$ds^{2}=dt^{2}-a(t)^2\left[ \frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^2 \right] ,$$where $a(t)$ is the scale factor and $k=-1,0,+1$ characterizes the curvature of the spatial sections of space-time. From now on we will assume a flat universe ($k=0$), which is in agreement with CMB results [@spergel]. In terms of the Hubble parameter ($H\equiv \frac{\dot{a}}{a}$), the deceleration parameter can be written as: $$q=-\frac{\ddot{a}}{aH^{2}}=\frac{d}{dt}\left( \frac{1}{H}\right) -1.$$Therefore, $$H=H_{0}\exp {\left[ \int_{0}^{z}(q(\tilde{z})+1)d\ln {(1+\tilde{z})}\right] }. \label{H}$$ In this work, we propose the following phenomenological functional dependence with redshift for the deceleration parameter: $$q(z)\equiv q_{f}+\frac{(q_{i}-q_{f})}{1-\frac{q_{i}}{q_{f}}\left( \frac{% 1+z_{t}}{1+z}\right) ^{1/\tau }}, \label{novoq}$$where $q_{i}>0$ (deceleration) and $q_{f}<0$ (acceleration) are the initial $% (z\gg z_{t})$ and final $(z=-1)$ values of the deceleration parameter, respectively. The parameter $z_{t}$ denotes the redshift of the transition ($% q(z_{t})=0$) and $\tau >0$ is associated with the width of the transition. It is related to the derivative of $q$ with respect to the redshift at $z=z_{t}$. More precisely, $$\tau ^{-1}=\left(\frac{1}{q_{i}}-\frac{1}{q_{f}}\right)\left[\frac{dq(z)}{d\ln (1+z)}% \right]_{z=z_{t}}. \label{tau}$$The influence of parameters $z_t$ and $\tau$ are demonstrated in Fig. (\[fig3\]). Expression (\[novoq\]) is similar in spirit to the one suggested in [bassett]{} (see also [@linder05; @bassett04; @kink]), but here we parametrize $q(z)$ instead of $w(z)$. One of the advantages of using the above kink-like parametrization for the deceleration parameter is that $z_{t}$ has a very clear physical meaning. Different physical aspects and parameterizations of $q(z)$ were also investigated in [@qparam]. With the above definition, equation (\[H\]) is now integrated to give, $$\begin{aligned} \left( \frac{H(z)}{H_{0}}\right) ^{2}&=&\left( 1+z\right) ^{2(1+q_{i})} \nonumber \\ & & \times \left( \frac{q_{i}\left( \frac{1+z_{t}}{1+z}\right) ^{1/\tau }-q_{f}}{q_{i}\left( 1+z_{t}\right) ^{1/\tau }-q_{f}}\right) ^{2\tau (q_{i}-q_{f})}. \label{hq0}\end{aligned}$$ We now define an effective matter density parameter ($\Omega _{m\infty }$) as $$\Omega _{m\infty } \equiv \lim_{z\rightarrow \infty }\left( \frac{H(z)}{% H_{0}}\right) ^{2}\left( 1+z\right) ^{-2(1+q_i)},$$where the limit should be understood as $z>>z_t$. In most (and simplest) scenarios, in order to form large scale structures, the universe passes trough a kind of matter dominated phase such that, at early times (but after radiation domination), $H^{2}\propto (1+z)^{3}$, which implies $q=1/2$. In this work we fix $q_{i}=1/2$ reducing to three the number of free parameters. In principle, with this assumption we are losing generality but the question is: how much $q_{i}$ can deviate from $1/2$ during large scale structure formation? In the general relativity framework, in some models with a constant coupling ($\delta$) between dark matter and dark energy this condition ($q_{i}= 1/2$) is not satisfied. In this case we have $H^{2}\propto (1+z)^{(3+\delta)}$ during matter domination. But, what are the allowed values for $\delta$? In [@guo07] it has been shown that background cosmological tests, impose |$\delta| < 0.1$. Taking into account matter perturbations stronger constraints on the coupling can be obtained [@fabris]. Another possibility is to consider models in which matter has pressure, such that when it dominates $q_{i}\neq 1/2$. However, if matter perturbations are adiabatic, due to a finite speed of sound, the mass power spectrum will present instabilities ruling out these models unless $p=0$ ($q=1/2$) or very close to it. In principle, it is possible to circumvent this kind of problem by assuming entropy perturbations such that $\delta p=0$ [@reis]. However, in this case the models may have problems with lensing skewness as pointed out in [@reis2]. Although there are some indications that by fixing $q_{i}=1/2$ we are not losing much in our description of the majority of the viable models, relaxing this condition requires further investigation, and we leave it for future work. In the specific case $q_{i}=1/2$ we have , $$\Omega _{m\infty }=\left( 1-\frac{1}{2q_{f}}\left( 1+z_{t}\right) ^{1/\tau }\right) ^{-\tau (1-2q_{f})}. \label{omeff}$$With the above definition, we can eliminate $z_{t}$ from equation (\[hq0\]) and rewrite it as $$\begin{aligned} &&\left( \frac{H(z)}{H_{0}}\right) ^{2}=\left( 1+z\right) ^{3} \notag \\ \times &&\left( \Omega _{m\infty }^{\frac{1}{\tau (1-2q_{f})}}+(1-\Omega _{m\infty }^{\frac{1}{\tau (1-2q_{f})}})(1+z)^{-\frac{1}{\tau }}\right) ^{\tau (1-2q_{f})}. \label{hq1b}\end{aligned}$$The above expression for $H(z)$ in terms of $\Omega _{m\infty }$ is very useful to make connections with models already discussed in the literature. For instance, it is simple to verify from (\[hq1b\]) that the parametrization (\[novoq\]), in the special case $q_{i}=1/2$, is related to the Modified Polytropic Cardassian (MPC) model [@gondolo]. This model depends on three parameters: $m$ (denoted by $q$ in [@gondolo]), $n$ and $\Omega _{m0}$. If we identify, $% \Omega _{m0}=\Omega _{m\infty }$, $m=1/(\tau (1-2q_{f}))$ and $n=2/3(1+q_{f}) $, it follows that the two models have the same kinematics. Note that, since $% q_{f}<0$, the condition $n<2/3$ follows naturally. We remark that in the MPC model, $% \Omega _{m0}$ is the present value of the matter density parameter, while $\Omega _{m\infty }$ is defined at high redshift. These two quantities do not necessarily have the same value in the general case if, for instance, dark matter and dark energy are coupled [@coup]. As an example, consider models with a variable coupling between dark matter and dark energy (assumed to have constant equation of state $w_x$), and such that $\rho_X/\rho_m=\rho_{X0}/\rho_{m0} a^{\xi}$ [@dalal]. These models can be described by Eqn. (\[hq1b\]) if we identify $\Omega_{m0}=\Omega _{m\infty }^{1/\tau(1-2q_f)}$, $\xi=1/\tau$ and $w_X=-(1-2q_f)/3$. As remarked before, in our formulation it is not necessary to make strong assumptions about the dark sector or gravity theory. In MPC model the universe components are specified to be matter and radiation; there is no dark energy. The parametrization (\[novoq\]) includes the MPC model (and the coupling models above) as special cases. Neglecting baryons, the quartessence Chaplygin model $(p=-M^{4(\alpha +1)}/\rho ^{\alpha })$ [@quartessence], is obtained if we assume $q_{i}=1/2$, $q_{f}=-1$, identify $1/\tau =3(1+\alpha )$ and $\Omega _{m\infty }=(1-w_{0})^{1/(1+\alpha )}$, where $w_{0}=-M^{4}/\rho _{0}^{\alpha +1}$ is the present value of the equation of state parameter. The conventional dark energy model with constant equation of state ($w_{X}$) is obtained if we identify $\Omega _{m\infty }=\Omega _{m0}$ and impose the condition $-3w_{X}=1/\tau =(1-2q_{f})$ in Eqn. (\[hq1b\]). In particular, if $q_{f}=-1$ and $\tau =1/3$, $\Lambda$CDM is recovered. For this model the transition redshift is equal to $(2(1-\Omega _{m0})/$ $\Omega _{m0})^{1/3}-1$. Identifying $\Lambda$CDM in the parameter space is very convenient; it fits current data quite well and we should expect the true cosmology not to be far from this limit. We remark that, in the framework of general relativity with non-interacting dark matter and dark energy, if $\tau <1/3$ and $q_{f}=-1$, the dark energy component will present a transient phantom ($% w<-1$) behavior, that could either have started in the past or in the future ($z<0$). Models with $\tau >1/3$ are always non-phantom. It is curious that if we apply the definition of $\tau $, given by Eqn. ([tau]{}) (assuming $q_{i}=1/2$ and $q_{f}=-1$), to the flat DGP brane-world model [@dgp] we obtain $\tau =1/2$, independent of $\Omega _{m0}$. Therefore q-models with $z_{t}=(2(1-\Omega _{m0})^{2}/$ $\Omega _{m0})^{1/3}-1$ (the DGP redshift transition) and $\tau \approx 1/2$ are expected to be a good approximation for flat DGP models. Observational Constraints ========================= One of the main questions today in cosmology is to know if cosmic acceleration is generated by a cosmological constant or not. The data seem to indicate that models close to $\Lambda$CDM are favored. In our analysis we first consider the special case of models that have a final de Sitter phase ($q_{f}=-1$). In this case, the flat $\Lambda$CDM model is more easily identified in the parameter space allowing a simple test of the $\Lambda$CDM paradigm. The more general case, with arbitrary $q_{f}$, will also be briefly considered. Assuming $q_f=-1$ we now derive constraints on the parameters $\tau $ and $z_{t}$ by combining supernovae measurements with the ratio of the comoving distance to the last scaterring surface, $S_{k}(z_{ls}=1098)$, to the BAO distance scale, $D_{v}(z)$, at $z_{BAO}=0.2$ and $z_{BAO}=0.35$, as estimated in [@percival07]. In fact, the ratio $S_{k}/D_{v}$ times $z_{BAO}$ is equal to the ratio of the CMB shift parameter ($\mathcal{R}$) [@shift] at $z_{ls}$ to the BAO parameter $\mathcal{A}(z_{BAO})$ [@eisenstein]. This observable is appropriate for our purpose for two reasons. First, it does not explicitly depend on the exotic dark constituents of the universe and neither on the gravity theory. It is essentially controlled by the function $H(z)/H_{0}$. Second, complementarity with the SN constraints is generated because the $S_{k}/D_{v}$ ratio and SN are sensitive to distances to objects (events) in different redshift range; with supernovae we are measuring distances up to $z\sim 1-2$, while $S_{k}$ depends on the comoving distance to $z\sim 1100$. In the SN Ia analysis we considered both, the *Gold182* [@gold] and the SNLS [@snls] samples. To determine the likelihood of the parameters we follow the same procedure described in these two references. In our computations, when marginalizing over the Hubble parameter, we use a Gaussian prior such that $h=0.72\pm 0.08$ [@hst]. In Fig.\[fig1\] (left-panel), assuming $q_f=-1$, we display constant confidence contours (68% and 95%) in the ($\arctan z_{t},\tau)$ plan allowed by SN experiments. Notice that, for the two SN data sets $z_{t}<0$ is not allowed at a high confidence level, indicating that a transition occurred in the past. We remark that this is expected in $\Lambda$CDM models (or other models) that have a non-null transition time, but our results indicate that this is true even if the transition is instantaneous ($\tau =0$). This conclusion also applies if $q_f\neq-1$. Furthermore, it is also clear in Fig.\[fig1\] (left-panel) that current SN observations cannot impose strong constraints on the maximum value allowed for $z_{t}$. Since SN observations prove the universe only up to redshift $z\sim 1-2$, in a model in which the transition is slow ($\tau \gtrsim 1$), even if $z_{t}$ is high, the distance to an object, let say, at $z\lesssim 1$, can be similar to the distance to the same object in another model in which $z_{t}\lesssim 1$ with a faster transition (smaller $\tau $). This explains the shape of the SN contours. By comparing the confidence contours for the two data sets, we observe that those from *Gold182* are shifted to lower $z_t$ with respect to those from SNLS. We remark that even in the region of more interest ($z_{t}\lesssim 1$), the difference between the outcome of the two SN Ia data sets, although not so severe, exists and is important. Similar results were obtained in [@nesseris07], which can be related to possible inhomogeneities present in the *Gold182* sample and should be further investigated. To obtain the constraints on the parameters from the $S_{k}/D_{v}$ test, we use a $\chi ^{2}$ statistics taking into account the correlation matrix and the ratio $r_{s}/S_{k}$ given in [@percival07]. Since we are assuming flat space we have, $D_{v}(z_{BAO})=[z_{BAO}H^{-1}( \int_{0}^{z_{BAO}} d\tilde{z}{H)^{2}}] ^{1/3}$ and $ S_{k}=\int_{0}^{1098}d\tilde{z}H_{e}^{-1}$. The $H_e$ term in the definition of $S_k$ incorporates the necessity of taking into account the contribution of radiation at very early times, which is not included in our parametrization Eqn. (\[novoq\]). One may argue that by introducing radiation we are losing generality. However, any late time modification of the standard cosmological model should satisfy Big-Bang nucleosynthesis (BBN) constraints at very early time. We know that radiation exists and what should be the dependence of the Hubble parameter with redshift at very early times (when radiation dominates) in order not to spoil BBN’s success. During this phase $H^2\propto (1+z)^{4}$ and $q=1$. Therefore, to take radiation into account, we add the term $\Omega _{r0}(1+z)^{4}$ to the right hand side of (\[hq0\]) when applying it to calculate $S_k$. If we do not consider it, we would have a $\sim 18\%$ error in estimating ${S}_{k}$. We marginalize the likelihood over $h$ with the same Gaussian prior used in the supernovae analysis. In fact, $S_{k}$ is almost independent of $h$; the dependence entering only through the radiation term. In Fig.\[fig1\] (right-panel) we show constant confidence contours (68% and 95%) in the ($z_{t},\tau $) plan allowed by the $S_{k}/D_{v}$ test. It is worth to be mentioned that the shape of the $95\%$ contour is similar to contours of constant $\Omega _{m\infty }$ and we can think that this test essentially constrains this quantity. The same kind of behavior also appears in flat, constant $w$ models [@percival07]. Furthermore, it is clear from the figure the complementarity between the SN and this test. To get the combined (SN+$S_{k}/D_{v}$) results we multiply the marginalized likelihood functions. In Fig.\[fig2\] (left-panel) we show the results ($68\%$ and $95\%$ c.l.) of the $S_{k}/D_{v}$ test with the *Gold182* (green dashed contours) and with the SNLS (blue dot-dashed contours) data set. The red horizontal line in the figure represents the $\Lambda$CDM limit ($\tau =1/3$). It shows that this model is in good agreement with the $S_{k}/D_{v}$ + SNLS data. After marginalizing over the extra parameter [@betocs] we find for $S_{k}/D_{v}$+*Gold182* (at $95.4\%$ confidence level), $z_{t}=0.84\pm _{0.17}^{0.13}$ and $\tau =0.51\pm _{0.17}^{0.23}$, while for $S_{k}/D_{v}$+SNLS we have, $z_{t}=0.88\pm _{0.10}^{0.12}$ and $\tau =0.35\pm _{0.10}^{0.12}$. For a model with $q_f=-1$, $\tau = 0.35$ and $z_t=0.88$, we obtain from Eqn. (\[omeff\]) that $\Omega _{m\infty }=0.23$. It is also simple to show that the age of the universe in this particular model (assuming $h=0.72$) is $14.0$ $Gyr$ and that cosmic acceleration started $7.2$ $Gyr$ ago. Notice that the $\Lambda$CDM cosmology ($\tau =1/3$) is in good agreement with $S_{k}/D_{v}$ + SNLS, but excluded at $95.4\%$ confidence level by the $S_{k}/D_{v}$+*Gold182* data. This discrepancy reveals tension between the two SN data samples and reinforces the necessity of better SN data to clarify the issue. We also display in the same figure (solid contour) what should be expected from future surveys when combining SN Ia+$S_{k}/D_{v}$ measurements. In our Monte Carlo simulations we used as fiducial model a flat $\Lambda$CDM model with $\Omega _{m0}=0.23$ ($\tau =1/3,z_{t}\simeq 0.88$). For SN Ia we considered a SNAP-like survey assuming that the intercept is known. For the $S_{k}/D_{v}$ test we used a conservative (but somewhat arbitrary) assumption that the uncertainties will be reduced to 2/3 of their current values. We also assumed that the correlation coefficient would remain the same. In the figure we show the $95\%$ confidence contour. We also analyzed the broader case with arbitrary $q_f$. Our parametrization (\[novoq\]) allows us to determine the present value of $q$ ($q_0$) in terms of $z_t$, $\tau$ and $q_f$. Although the considered data sets do not impose a lower bound for $q_f$, they do constrain $q_0$ (we found $-1.4\lesssim q_0\lesssim -0.3$). In Fig.\[fig2\] (right-panel) we show, in the parameter space $(z_t,\tau,q_0)$, the $95\%$ confidence surface for the general case ($q_f \in (-\infty,0)$), obtained using SNLS+$S_{k}/D_{v}$ data. Conclusion ========== In this work, with a formulation that avoids strong assumptions about the dark sector and/or the metric theory of gravity, we showed that by using only SN data the transition redshift (from decelerated to accelerated expansion) could be very large ($z_t>10$). We demonstrated the importance of combining the SN test with the $S_k/D_v$ test to better constrain the parameters $z_t$ and $\tau$. We introduced the parameter $\tau$ and showed its relevance to characterize models like flat $\Lambda$CDM, DGP and others. We confirmed that there is a tension between *Gold182* and SNLS data sets with a quite general formulation. We also exhibited what should be expected from future $SN + S_{k}/D_{v}$ observations and, relaxing the condition $q_f = -1$, obtained current constraints on the parameters $z_t$, $\tau$ and $q_0$. A more detailed analysis of the consequences of our parametrization in the case $q_f \neq -1$ is still necessary. For instance, our parametrization is not able to describe, in all their redshift range, models that are now accelerating but that decelerates again in the future [@frieman]. The description of the expansion history of these models is more complicated since it requires a second transition. It would be interesting to investigate under what conditions it would be possible to describe, with our parametrization, the behavior of these kind of models for $z>0$. The case $q_i \neq 1/2$ should also be further investigated. 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--- address: | Department of Mathematics, The University of Texas at San Antonio\ San Antonio, TX 78249, USA author: - Gelu Popescu date: 'December 27, 2005' title: ' Free holomorphic functions on the unit ball of $B({{\mathcal H}})^n$' --- [^1] Contents {#contents .unnumbered} ======== ** Introduction 1. Free holomorphic functions and Hausdorff derivations 2. Cauchy, Liouville, and Schwartz type results for free holomorphic functions 3. Algebras of free holomorphic functions 4. Free analytic functional calculus and noncommutative Cauchy transforms 5. Weierstrass and Montel theorems for free holomorphic functions 6. Free pluriharmonic functions and noncommutative Poisson transforms 7. Hardy spaces of free holomorphic functions References Introduction {#introduction .unnumbered} ============ The Shilov-Arens-Calderon theorem ([@S], [@AC]) states that if $a_1,\ldots, a_n$ are elements of a commutative Banach algebra $A$ with the joint spectrum included in a domain $\Omega\subset {{\mathbb C}}^n$, then the algebra homomorphism $${{\mathbb C}}[z_1,\ldots,z_n]\ni p\mapsto p(a_1,\ldots, a_n)\in A$$ extends to a continuous homomorphism from the algebra $Hol(\Omega)$, of holomorphic functions on $\Omega$, to the algebra $A$. This result was greatly improved by the pioneering work of J.L. Taylor ([@T1], [@T2], [@T3]) who introduced a “better” notion of joint spectrum for $n$-tuples of commuting operators, which is now called Taylor spectrum, and developed an analytic functional calculus. Stated for the open unit ball of ${{\mathbb C}}^n$, $${{\mathbb B}}_n:=\{(\lambda_1,\ldots, \lambda_n)\in {{\mathbb C}}^n: \ |\lambda|^2+\cdots +|\lambda_n|^2<1\},$$ his result states that if $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is an $n$-tuple of commuting bounded linear operators on a Hilbert space ${{\mathcal H}}$ with Taylor spectrum $\sigma(T_1,\ldots, T_n)\subset {{\mathbb B}}_n$, then there is a unique continuous unital algebra homomorphism $$Hol({{\mathbb B}}_n)\ni f\mapsto f(T_1,\ldots, T_n)\in B({{\mathcal H}})$$ such that $z_i\mapsto T_i$,  $i=1,\ldots,n$. Due to a result of V. M" uller [@M], the condition that $\sigma(T_1,\ldots, T_n)\subset {{\mathbb B}}_n$ is equivalent to the fact that the joint spectral radius $$r(T_1,\ldots, T_n):=\lim_{k\to\infty}\left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2k}<1.$$ F.H. Vasilescu introduced and studied, in [@Va] and a joint paper with R.E. Curto [@CV], operator-valued Cauchy and Poisson transforms on the unit ball ${{\mathbb B}}_n$ associated with commuting operators with $r(T_1,\ldots, T_n)<1$, in connection with commutative multivariable dilation theory. In recent years, there has been exciting progress in noncommutative multivariable operator theory regarding noncommutative dilation theory ([@F], [@B], [@Po-models], [@Po-isometric], [@Po-charact], [@DKS], [@BB], [@BBD], [@Po-unitary], [@Po-varieties], etc.) and its applications concerning interpolation in several variables ([@Po-charact], [@Po-analytic], [@Po-interpo], [@ArPo2], [@DP], [@BTV], [@Po-entropy], etc.) and unitary invariants for $n$-tuples of operators ([@Po-charact], [@Arv], [@Arv2], [@Po-curvature], [@Kr], [@Po-similarity], [@BT], [@Po-entropy], [@Po-unitary], etc.). Our program to develop a [*free*]{} analogue of Sz.-Nagy–Foiaş theory [@SzF-book], for row contractions, fits perfectly the setting of the present paper, which includes that of free holomorphic functions on the open operatorial unit ball $$[B({{\mathcal H}})^n]_1:=\left\{ [X_1,\ldots, X_n]\in B({{\mathcal H}})^n: \ \|X_1X_n^*+\cdots + X_nX_n^*\|<1\right\}.$$ The present work is an attempt to develop a theory of holomorphic functions in several noncommuting (free) variables and thus provide a framework for the study of arbitrary $n$-tuples of operators, and to introduce and study a free analytic functional calculus in connection with Hausdorff derivations, noncommutative Cauchy and Poisson transforms, and von Neumann inequalities. In Section 1, we introduce a notion of radius of convergence for formal power series in $n$ noncommuting indeterminates $Z_1,\ldots, Z_n$ and prove noncommutative multivariable analogues of Abel theorem and Hadamard formula from complex analysis ([@Co], [@R]). This enables us to define, in particular, the algebra $Hol(B({{\mathcal X}})^n_1)$  of free holomorphic functions on the open operatorial unit $n$-ball, as the set of all power series $\sum_{\alpha\in {{\mathbb F}}_n^+}a_\alpha Z_\alpha$ with radius of convergence $\geq 1$. When $n=1$, $Hol(B({{\mathcal X}})^1_1)$ coincides with the algebra of all analytic functions on the open unit disc  ${{\mathbb D}}:=\{z\in {{\mathbb C}}:\ |z|<1\}$. The algebra of free holomorphic functions  $Hol(B({{\mathcal X}})^n_1)$  has the following universal property. [*Any strictly contractive representation $\pi: {{\mathbb C}}[Z_1,\ldots, Z_n]\to B({{\mathcal H}})$, i.e., $\|[\pi(Z_1),\ldots, \pi(Z_n)]\|<1$, extends uniquely to a representation of $Hol(B({{\mathcal X}})^n_1)$.*]{} A free holomorphic function on the open operatorial unit ball  $[B({{\mathcal H}})^n]_1$  is the representation of an element $F\in Hol(B({{\mathcal X}})^n_1)$ on the Hilbert space ${{\mathcal H}}$, that is, the mapping $$[B({{\mathcal H}})^n]_1\ni (X_1,\ldots, X_n)\mapsto F(X_1,\ldots, X_n)\in B({{\mathcal H}}).$$ As expected, we prove that any free holomorphic function is continuous on $[B({{\mathcal H}})^n]_1$ in the operator norm topology. In the last part of this section, we show that the Hausdorff derivations $\frac{\partial}{\partial Z_i}$, $i=1,\ldots, n$, on the algebra of noncommutative polynomials ${{\mathbb C}}[Z_1,\ldots, Z_n]$ ([@MKS], [@RSS]) can be extended to the algebra of free holomorphic functions. In Section 2, we obtain Cauchy type estimates for the coefficients of free holomorphic functions and a Liouville type theorem for free entire functions. Based on a noncommutative version of Gleason’s problem [@R2], which is obtained here, and the noncommutative von Neumann inequality [@Po-von], we provide a free analogue of Schwartz lemma from complex analysis ([@Co], [@R]). In particular, we prove that if $f$ is a free holomorphic function on $[B({{\mathcal H}})^n]_1$ such that $\|f\|_\infty\leq 1$ and $f(0)=0$, then $$\|f(X_1,\ldots, X_n)\|\leq \|[X_1,\ldots, X_n]\|,\qquad r(f(X_1,\ldots, X_n))\leq r(X_1,\ldots, X_n),$$ and  $\sum_{i=1}^n \left|\frac{\partial f}{\partial X_i}(0)\right|^2\leq 1$. In Section 3, following the classical case ([@H], [@RR]), we introduce two Banach algebras of free holomorphic functions, $H^\infty(B({{\mathcal X}})^n_1)$ and $A(B({{\mathcal X}})^n_1)$, and prove that, together with a natural operator space structure, they are completely isometrically isomorphic to the noncommutative analytic Toeplitz algebra $F_n^\infty$ and the noncommutative disc algebra ${{\mathcal A}}_n$, respectively, which were introduced in [@Po-von] in connection with a multivariable von Neumann inequality. We recall that the algebra $F_n^\infty$ (resp. ${{\mathcal A}}_n$) is the weakly (resp. norm) closed algebra generated by the left creation operators $S_1,\ldots, S_n$ on the full Fock space with $n$ generators, $F^2(H_n)$, and the identity. These algebras have been intensively studied in recent years by many authors ([@Po-charact], [@Po-multi], [@Po-von], [@Po-funct], [@Po-analytic], [@Po-disc], [@Po-poisson], [@Po-curvature], [@Po-similarity], [@ArPo], [@ArPo2], [@DP1], [@DP2], [@DKP], [@PPoS], [@Po-unitary]). The results of this section are used to obtain a maximum principle for free holomorphic functions. In Section 4, we provide a free analytic functional calculus for $n$-tuples $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. We show that there is a continuous unital algebra homomorphism $$\Phi_T:Hol(B({{\mathcal X}})^n_1)\to B({{\mathcal H}}), \quad \Phi_T(f)=f(T_1,\ldots, T_n),$$ which is uniquely determined by the mapping $z_i\mapsto T_i$,  $i=1,\ldots,n$. (The continuity and uniqueness of $\Phi_T$ are proved in Section 5.) We introduce a noncommutative Cauchy transform ${{\mathcal C}}_T:B(F^2(H_n))\to B({{\mathcal H}})$ associated with any $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. The definition is based on the [*reconstruction operator*]{} $$S_1\otimes T_1^*+\cdots + S_n\otimes T_n^*,$$ which has played an important role in noncommutative multivariable operator theory ([@Po-entropy], [@Po-unitary], [@Po-varieties]). We prove that $$f(T_1,\ldots, T_n)=C_T(f(S_1,\ldots, S_n)),\quad f\in H^\infty (B({{\mathcal X}})^n_1),$$ where $f(S_1,\ldots, S_n)$ is the boundary function of $f$. Hence, we deduce that $$\|f(T_1,\ldots, T_n)\|\leq M \|f\|_\infty$$ where $M:=\sum_{k=0}^\infty \left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2}$. Similar Cauchy representations are obtained for the $k$-order Hausdorff derivations of $f$. Finally, we show that the noncommutative Cauchy transform commutes with the action of the unitary group ${{\mathcal U}}({{\mathbb C}}^n)$. More precisely, we prove that $${{\mathcal C}}_T(\beta_U(f))={{\mathcal C}}_{\beta_U(T)}(f)\quad \text{ for any } \ U\in {{\mathcal U}}({{\mathbb C}}^n), \, f\in {{\mathcal A}}_n,$$ where $\beta_U$ denotes a natural isometric automorphism (generated by $U$) of the noncommutative disc algebra ${{\mathcal A}}_n$, or the open unit ball $[B({{\mathcal H}})^n]_1$. In Section 5, we obtain Weierstrass and Montel type theorems [@Co] for the algebra of free holomorphic functions on the open operatorial unit $n$-ball. This enables us to introduce a metric on $Hol(B({{\mathcal X}})^n_1)$ with respect to which it becomes a complete metric space, and the Hausdorff derivations $$\frac{\partial}{\partial Z_i}:Hol(B({{\mathcal X}})^n_1)\to Hol(B({{\mathcal X}})^n_1),\quad i=1,\ldots,n,$$ are continuous. In the end of this section, we prove the continuity and uniqueness of the free functional calculus for $n$-tuples of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. Connections with the $F_n^\infty$-functional calculus for row contractions [@Po-funct] and, in the commutative case, with Taylor’s functional calculus [@T2] are also discussed. Given an operator $A\in B(F^2(H_n))$, the noncommutative Poisson transform [@Po-poisson] generates a function $$P[A]: [B({{\mathcal H}})^n]_1\to B({{\mathcal H}}).$$ In Section 6, we provide classes of operators $A\in B(F^2(H_n))$ such that $P[A]$ is a free holomorphic (resp. pluriharmonic) function on $[B({{\mathcal H}})^n]_1$. In this case, the operator $A$ can be regarded as the boundary function of the Poisson extension $P[A]$. Using some results from [@Po-von], [@Po-funct], and [@Po-poisson], we characterize the free holomorphic functions $u$ on the open unit ball $[B({{\mathcal H}})^n]_1$ such that $u=P[f]$ for some boundary function $f$ in the noncommutative analytic Toeplitz algebra $F_n^\infty$, or the noncommutative disc algebra ${{\mathcal A}}_n$. For example, we prove that there exists $f\in F_n^\infty$ such that $u=P[f]$ if and only if $$\sup_{0\leq r<1}\|u(rS_1,\ldots, rS_n)\|<\infty.$$ We also obtain noncommutative multivariable versions of Herglotz theorem and Dirichlet extension problem ([@Co], [@H]) for free pluriharmonic functions. In Section 7, we define the radial maximal Hardy space $H^p(B({{\mathcal X}})^n_1)$, $p\geq 1$, as the set of all free holomorphic function $F$ such that $$\|F\|_p:=\left( \int_0^1\|F(rS_1,\ldots, rS_n)\|^p dr \right)^{1/p}<\infty,$$ and prove that it is a Banach space. Moreover, we show that $$\|f(T_1,\ldots, T_n)\|\leq \frac{1}{(1-\|[T_1,\ldots, T_n]\|)^{1/p}} \|f\|_p$$ for any $[T_1,\ldots, T_n]\in [B({{\mathcal H}})^n]_1$ and $f\in H^p(B({{\mathcal X}})^n_1)$. Finally, we introduce the symmetrized Hardy space $H^\infty_{\text{\rm sym}}({{\mathbb B}}_n)$ as the set of all holomorphic function on ${{\mathbb B}}_n$ such that $ \|f\|_{\text{\rm sym}}:= \|f_{\text{\rm sym}}\|_\infty<\infty, $ where $f_{\text{\rm sym}}\in Hol(B({{\mathcal X}})^n_1)$ is the symmetrized functional calculus of $f\in Hol({{\mathbb B}}_n)$. We prove that $H^\infty_{\text{\rm sym}}({{\mathbb B}}_n)$ is a Banach space and $$\|f(T_1,\ldots, T_n)\|\leq M \|f_{\text{\rm sym}}\|_\infty,$$ for any commuting $n$-tuple of operators with $r(T_1,\ldots, T_n)<1$. Several classical results from complex analysis are extended to our noncommutative multivariable setting. The present paper exhibits, in particular, a “very good” free analogue of the algebra of analytic functions on the open unit disc ${{\mathbb D}}$. This claim is also supported by the fact that numerous results in noncommutative multivariable operator theory ([@Po-von], [@Po-funct], [@Po-disc], [@Po-poisson], [@Po-unitary]) fit perfectly our setting and can be seen in a new light. We strongly believe that many other results in the theory of analytic functions have free analogues in our noncommutative multivariable setting. In a forthcoming paper [@Po-Bohr], we consider operator-valued Wiener and Bohr type inequalities for free holomorphic (resp. pluriharmonic) functions on the open operatorial unit $n$-ball. As consequences, we obtain operator-valued Bohr inequalities for the noncommutative Hardy algebra $H^\infty(B({{\mathcal X}})^n_1)$ and the symmetrized Hardy space $H^\infty_{\text{\rm sym}}({{\mathbb B}}_n)$. Free holomorphic functions {#free holomorphic} =========================== We introduce a notion of radius of convergence for formal power series in $n$ noncommuting indeterminates $Z_1,\ldots, Z_n$ and prove noncommutative multivariable analogues of Abel theorem and Hadamard formula. This enables us to define algebras of free holomorphic functions on open operatorial $n$-balls. We show that the Hausdorff derivations $\frac{\partial}{\partial Z_i}$, $i=1,\ldots, n$, on the algebra of noncommutative polynomials ${{\mathbb C}}[Z_1,\ldots, Z_n]$ (see [@MKS], [@RSS]) can be extended to algebras of free holomorphic functions. Let ${{\mathbb F}}_n^+$ be the unital free semigroup on $n$ generators $g_1,\ldots, g_n$ and the identity $g_0$. The length of $\alpha\in {{\mathbb F}}_n^+$ is defined by $|\alpha|=0$ if $\alpha=g_0$ and $|\alpha|:=k$ if $\alpha=g_{i_1}\cdots g_{i_k}$, where $i_1,\ldots, i_k\in \{1,\ldots, n\}$. We consider formal power series in $n$ noncommuting indeterminates $Z_1,\ldots, Z_n$ and coefficients in $B({{\mathcal K}})$, the algebra of all bounded linear operators on the Hilbert space ${{\mathcal K}}$, of the form $$\sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha,\quad A_{(\alpha)}\in B({{\mathcal K}}),$$ where $Z_\alpha:=Z_{i_1}\cdots Z_{i_k}$ if $\alpha=g_{i_1}\cdots g_{i_k}$ and $Z_{g_0}:=I$. If $F=\sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha $ and $G=\sum_{\alpha\in {{\mathbb F}}_n^+} B_{(\alpha)}\otimes Z_\alpha $ are such formal power series, we define their sum and product by setting $$F+G:=\sum_{\alpha\in {{\mathbb F}}_n^+} (A_{(\alpha)}+B_{(\alpha)})\otimes Z_\alpha \quad \text{ and }\quad FG:=\sum_{\alpha\in {{\mathbb F}}_n^+} C_{(\alpha)}\otimes Z_\alpha,$$ respectively, where $C_{(\alpha)}:=\sum\limits_{\sigma, \beta\in {{\mathbb F}}_n^+:\ \alpha=\sigma \beta} A_{(\sigma)} B_{(\beta)}$. By abuse of notation, throughout this paper, we will denote by $[T_1,\ldots,T_n]$ either the $n$-tuple of operators $(T_1,\ldots, T_n)\in B({{\mathcal H}})^n$ or the row operator matrix $[T_1\,\cdots \,T_n]\in B({{\mathcal H}}^{(n)}, {{\mathcal H}})$ acting as an operator from ${{\mathcal H}}^{(n)}$, the direct sum of $n$ copies of the Hilbert space ${{\mathcal H}}$, to ${{\mathcal H}}$. We also denote by $[T_\alpha:\ |\alpha|=k]$ the row operator matrix acting from ${{\mathcal H}}^{n^k}$ to ${{\mathcal H}}$, where the entries are arranged in the lexicographic order of the free semigroup ${{\mathbb F}}_n^+$. In what follows we show that given a sequence of operators $A_{(\alpha)}\in B({{\mathcal K}})$, $\alpha\in {{\mathbb F}}_n^+$, there is a unique $R\in [0,\infty]$ such that the series $$\sum_{k=0}^\infty\sum_{|\alpha|=k} A_{(\alpha)}\otimes X_\alpha$$ converges in the operator norm of $B({{\mathcal K}}\otimes {{\mathcal H}})$ (${{\mathcal K}}\otimes {{\mathcal H}}$ is the Hilbert tensor product) for any Hilbert space ${{\mathcal H}}$ and any $n$-tuple $[X_1,\ldots, X_n]\in B({{\mathcal H}})^n$ with $\|[X_1,\ldots, X_n]\|<R$, and it is divergent for some $n$-tuples $[Y_1,\ldots, Y_n]$ of operators with $\|[Y_1,\ldots, Y_n]\|>R$. The result can be regarded as a noncommutative multivariable analogue of Abel theorem and Hadamard’s formula from complex analysis. \[Abel\] Let ${{\mathcal H}}$, ${{\mathcal K}}$ be Hilbert spaces and let $A_{(\alpha)}\in B({{\mathcal K}})$, $\alpha\in {{\mathbb F}}_n^+$, be a sequence of operators. Define $R\in [0,\infty]$ by setting $$\frac {1} {R}:= \limsup_{k\to\infty} \left\|\sum_{|\alpha|=k} A^*_{(\alpha)} A_{(\alpha)}\right\|^{\frac{1} {2k}}.$$ Then the following properties hold: 1. For any $n$-tuple of operators $[X_1,\ldots, X_n]\in B({{\mathcal H}})^n$, the series $\sum\limits_{k=0}^\infty \left\| \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right\| $ converges if  $\|[X_1,\ldots,X_n]\|<R$. Moreover, if $0\leq \rho<R$, then the convergence is uniform for $[X_1,\ldots, X_n]$ with $\|[X_1,\ldots,X_n]\|\leq \rho$. 2. If $R<R'<\infty$ and ${{\mathcal H}}$ is infinite dimensional, then there is an $n$-tuple $[X_1,\ldots, X_n]\in B({{\mathcal H}})^n$ of operators with $$\|X_1X_1^*+\cdots +X_nX_n^*\|^{1/2}=R'$$ such that $\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right)$ is divergent in the operator norm of $B({{\mathcal K}}\otimes {{\mathcal H}})$. Moreover, the number $R$ satisfying properties (i) and (ii) is unique. Assume that $R>0$ and $[X_1,\ldots, X_n]$ is an $n$-tuple of operators on ${{\mathcal H}}$ such that $\|[X_1,\ldots,X_n]\|<R$. Let $\rho',\rho>0$ be such that $\|[X_1,\ldots,X_n]\|<\rho'<\rho<R$. Since $\frac{1}{\rho}> \frac{1}{R}$, we can find $m_0\in{{\mathbb N}}:=\{1,2,\ldots\}$ such that $$\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^*A_{(\alpha)}\right\|^{1/2k}< \frac{1}{\rho}\quad \text{ for any }\ k\geq m_0.$$ Hence, we deduce that $$\begin{split} \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha \right\|&= \left\|\left[ I\otimes X_\alpha:\ |\alpha|=k\right] \left[\begin{matrix} A_{(\alpha)}\otimes I\\ :\\|\alpha|=k \end{matrix}\right]\right\|\\ &=\left\|\sum\limits_{|\alpha|=k}X_{\alpha}X_\alpha^*\right\|^{1/2}\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^*A_{(\alpha)}\right\|^{1/2}\\ &\leq \left\|\sum_{i=1}^nX_iX_i^*\right\|^{k/2}\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^*A_{(\alpha)}\right\|^{1/2}\\ &\leq \left(\frac{\rho'}{\rho}\right)^k \end{split}$$ for any $k\geq m_0$. This proves the convergence of the series $\sum\limits_{k=0}^\infty \left\| \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right\|$. Assume now that $0\leq \rho<R$ and $\|[X_1,\ldots, X_n]\|\leq \rho$. Choose $\gamma$ such that $0\leq \rho< \gamma<R$ and notice that, due to similar calculations as above, there exists $n_0\in {{\mathbb N}}$ such that $$\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha \right\| \leq \left(\frac{\rho}{\gamma}\right)^k$$ for any $(X_1,\ldots, X_n)$ with $\|[X_1,\ldots, X_n]\|\leq \rho$, and $k\geq n_0$, which proves the uniform convergence of the above series. The case $R=\infty$, can be treated in a similar manner. To prove part (ii), assume that $R<\infty$ and ${{\mathcal H}}$ is infinite dimensional. Let $R', \rho>0$ be such that $R<\rho< R'$ and define the operators $X_i:= R' V_i$,  $i=1,\ldots, n$, where $V_1,\ldots, V_n$ are isometries with orthogonal ranges. Notice that $\|[X_1,\ldots, X_n]\|=R'$ and $$\begin{split} \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha \right\|&={R'}^k \left\|\left(\sum\limits_{|\alpha|=k}A_{(\alpha)}^*\otimes V_\alpha^*\right) \left(\sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes V_\alpha\right)\right\|^{1/2}\\ &= {R'}^k \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^*A_{(\alpha)} \otimes I\right\|^{1/2}\\ &= {R'}^k \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^* A_{(\alpha)} \right\|^{1/2}. \end{split}$$ On the other hand, since $\frac{1}{\rho}<\frac{1}{R}$, there are arbitrarily large $k\in {{\mathbb N}}$ such that $$\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^* A_{(\alpha)} \right\|^{1/2}>\left(\frac{1}{\rho}\right)^k.$$ Consequently, we deduce that $$\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha \right\|>\left(\frac{R'}{\rho}\right)^k,$$ which proves that the series $\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right)$ is divergent in the operator norm. The uniqueness of the number $R$ satisfying properties (i) and (ii) is now obvious. As expected, the number $R$ in the above theorem is called the radius of convergence of the power series $\sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha.$ Let us consider the full Fock space $$F^2(H_n)={{\mathbb C}}1\oplus\ \oplus_{m\ge1}H_n^{\otimes m}$$ where $H_n$ is an $n$-dimensional complex Hilbert space with orthonormal basis $\{e_1,\dots,e_n\}$. Setting $e_\alpha:=e_{i_1}\otimes\cdots e_{i_k}$ if $\alpha=g_{i_1}\cdots g_{i_k}$, and $e_{g_0}=1$, it is clear that $\{ e_\alpha:\ \alpha\in {{\mathbb F}}_n^+\}$ is an orthonormal basis of the full Fock space $F^2(H_n)$. For each $i=1,2,\dots$, we define the left creation operator $\ S_i\in B(F^2(H_n))$ by $$S_i\xi=e_i\otimes\xi,\qquad \xi\in F^2(H_n).$$ We can now obtain the following characterization of the radius of convergence, which will be useful later. \[Cs\] Let $\sum\limits_{\alpha\in {{\mathbb F}}_n^+}A_{(\alpha)}\otimes Z_\alpha$ be a formal power series with radius of convergence $R$. 1. If $R>0$ and $0<r<R$, then there exists $C>0$ such that $$\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^* A_{(\alpha)} \right\|^{1/2}\leq \frac {C}{r^k}\quad \text{for any } \ k=0,1,\ldots.$$ 2. The radius of convergence of the power series satisfies the relations $$R=\sup\left\{ r\geq 0:\ \text{ the sequence }\ \left\{r^k\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^* A_{(\alpha)} \right\|^{1/2} \right\}_{k=0}^\infty \text{ is bounded }\right\}$$ and $$R=\sup\left\{ r\geq 0: \ \sum_{k=0}^\infty \sum_{|\alpha|=k}r^{|\alpha|} A_{(\alpha)}\otimes S_\alpha\ \text{ is convergent in the operator norm }\right\}.$$ Setting $X_i:= rS_i$,  $i=1,\ldots, n$, where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space, we have $\|[X_1,\ldots, X_n]\|=r<R$. According to Theorem \[Abel\], the series $ \sum_{k=0}^\infty \left\|r^k\sum_{|\alpha|=k} A_{(\alpha)} \otimes S_\alpha\right\|$ is convergent. Since $S_1,\ldots, S_n$ are isometries with orthogonal ranges, the above series is equal to $\sum_{k=0}^\infty r^k \left\|\sum_{|\alpha|=k}A_{(\alpha)}^* A_\alpha \right\|^{1/2}. $ Consequently, there is a constant $C>0$ such that $$r^k \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^* A_\alpha \right\|^{1/2}\leq C\ \text{ for any } k=0,1,\ldots.$$ Now, the second part of this corollary follows easily from part (i) and Theorem \[Abel\]. This completes the proof. We establish terminology which will be used throughout the paper. Denote by $[B({{\mathcal H}})^n]_{\gamma}$ the open ball of $B({{\mathcal H}})^n$ of radius $\gamma> 0$, i.e., $$[B({{\mathcal H}})^n]_{\gamma}:=\{[X_1,\ldots, X_n]:\ \|X_1X_1^*+\cdots +X_nX_n^*\|^{1/2}<\gamma\}.$$ We also use the notation $[B({{\mathcal H}})^n]_1^-$ for the closed ball. A formal power series $F:=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha$ represents a free holomorphic function on the open operatorial $n$-ball of radius $\gamma$ with coefficients in $B({{\mathcal K}})$, if for any Hilbert space ${{\mathcal H}}$ and any representation $$\pi:{{\mathbb C}}[Z_1,\ldots, Z_n]\to B({{\mathcal H}})\quad \text{ such that } \quad [\pi(Z_1),\ldots, \pi(Z_n)]\in [B({{\mathcal H}})^n]_{\gamma}$$ the series $$F(\pi(Z_1),\ldots, \pi(Z_n)):=\sum\limits_{k=0}^\infty \sum\limits_{|\alpha|=k} A_{(\alpha)} \otimes \pi(Z_\alpha)$$ converges in the operator norm of $B({{\mathcal K}}\otimes {{\mathcal H}})$. Due to Theorem \[Abel\], we must have $\gamma\leq R$, where $R$ is the radius of convergence of $F$. The mapping $$[B({{\mathcal H}})^n]_{\gamma}\ni [X_1,\ldots, X_n]\mapsto F(X_1,\ldots X_n)\in B({{\mathcal K}}\otimes {{\mathcal H}}).$$ is called the representation of $F$ on the Hilbert space ${{\mathcal H}}$. Given a Hilbert space ${{\mathcal H}}$, we say that a function $G:[B({{\mathcal H}})^n]_{\gamma}\to B({{\mathcal K}}\otimes {{\mathcal H}})$ is a [*free holomorphic function*]{} on $[B({{\mathcal H}})^n]_{\gamma }$ with coefficients in $B({{\mathcal K}})$ if there exist operators $A_{(\alpha)}\in B({{\mathcal K}})$, $\alpha\in {{\mathbb F}}_n^+$, such that the power series $\sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha$ has radius of convergence $\geq \gamma$ and $$G(X_1,\ldots, X_n)=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right),$$ where the series converges in the operator norm for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{\gamma}$. We remark that the coefficients of a free holomorphic function are uniquely determined by its representation on an infinite dimensional Hilbert space. Indeed, let $0<r<\gamma$ and assume $F(rS_1,\ldots, rS_n)=0$, where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space $F^2(H_n)$. Taking into account that $S_i^* S_j=\delta_{ij} I$, we have $$\left< F(rS_1,\ldots, rS_n)(x\otimes 1), (I_{{\mathcal K}}\otimes S_\alpha)(y\otimes 1)\right>=\left<A_{(\alpha)}x,y\right>=0$$ for any $x,y\in {{\mathcal K}}$ and $\alpha\in {{\mathbb F}}_n^+$. Therefore $A_{(\alpha)}=0$ for any $\alpha\in {{\mathbb F}}_n^+$. We establish now the continuity of free holomorphic functions on the open ball $[B({{\mathcal H}})^n]_{\gamma}$. \[continuous\] Let $ f(X_1,\ldots, X_n)=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right) $ be a free holomorphic function on $[B({{\mathcal H}})^n]_{\gamma}$ with coefficients in $B({{\mathcal K}})$. If $X:=[X_1,\ldots, X_n]$, $Y:=[Y_1,\ldots, Y_n]$ are in the closed ball $[B({{\mathcal H}})^n]_r^-$, $0<r<\gamma$, then $$\|f(X)-f(Y)\|\leq \|X-Y\|\sum _{k=1}^\infty kr^{k-1}\left\|\sum_{|\alpha|=k} A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2}.$$ In particular, $f$ is continuous on $[B({{\mathcal H}})^n]_{\gamma}$ and uniformly continuous on $[B({{\mathcal H}})^n]_r^-$ in the operator norm topology. Let $X^{[k]}:=[X_\alpha:\ \alpha\in {{\mathbb F}}_n^+, \ |\alpha|=k]$,  $k=1,2,\ldots$, be the row operator matrix with entries arranged in the lexicographic order of the free semigroup ${{\mathbb F}}_n^+$. First, we prove that if $\|X\|\neq \|Y\|$, then $$\label{[k]} \frac{\|X^{[k]}-Y^{[k]}\|}{\|X-Y\|}\leq \frac{\|X\|^k-\|Y\|^k}{\|X\|-\|Y\|}.$$ Notice that $$\begin{split} X^{[k]}-Y^{[k]}&= \left[(X_1-Y_1)X^{[k-1]},\ldots, (X_n-Y_n) X^{[k-1]}\right]\\ &\qquad + \left[ Y_1(X^{[k-1]}-Y^{[k-1]}),\ldots, Y_n(X^{[k-1]}-Y^{[k-1]})\right]\\ &= (X-Y)\text{\rm diag}_n(X^{[k-1]})+Y\text{\rm diag}_n(X^{[k-1]}-Y^{[k-1]}), \end{split}$$ where $\text{\rm diag}_n(A)$ is the $n\times n$ block diagonal operator matrix with $A$ on the diagonal and $0$ otherwise. Hence, we deduce that $$\|X^{[k]}-Y^{[k]}\|\leq \|X-Y\|\|X^{[k-1]}\|+\|Y\| \|X^{[k-1]}-Y^{[k-1]}\|$$ for any $k\geq 2$. Iterating this relation and taking into account that $\|X^{[k]}\|\leq \|X\|^k$ for $k=1,2,\ldots$, we obtain $$\begin{split} \|X^{[k]}-Y^{[k]}\|&\leq \|X-Y\|\left(\|X^{[k-1]}\| +\|\|X^{[k-2]}\|\|Y^{[1]}\|+\cdots + \|Y^{[k-1]}\|\right)\\ &\leq \|X-Y\|\left(\|X\|^{k-1}+\|X\|^{k-2}\|Y\|+\cdots+ \|Y\|^{k-1}\right), \end{split}$$ which proves inequality . Assuming that $\|X\|\leq r$ and $\|Y\|\leq r$, we deduce that $$\|X^{[k]}-Y^{[k]}\|\leq kr^{k-1}\|X-Y\|,\quad k=1,2,\ldots.$$ Hence, we obtain $$\begin{split} \|f(X)-f(Y)\|&\leq \sum_{k=1}^\infty\left\|\sum_{|\alpha|=k} A_{(\alpha)}\otimes (X_\alpha-Y_\alpha)\right\|\\ &\leq \sum_{k=1}^\infty\left\|\sum_{|\alpha|=k} A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2} \|X^{[k]}-Y^{[k]}\|\\ &\leq \|X-Y\| \sum_{k=1}^\infty kr^{k-1} \left\|\sum_{|\alpha|=k} A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2}. \end{split}$$ Let $\rho$ be a constant such that $r<\rho<\gamma$. Since $\gamma\leq R$ ($R$ is the radius of convergence of $f$) and $\frac{1}{\rho}>\frac{1}{\gamma}\geq \frac{1}{R}$, we can find $m_0\in {{\mathbb N}}$, such that $$\left\|\sum_{|\alpha|=k} A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2k}<\frac{1}{\rho}\quad \text{ for any } \ k\geq m_0.$$ Combining this with the above inequality, we deduce that $$\|f(X)-f(Y)\|\leq \|X-Y\|\left(\sum_{k=1}^{m_0-1} kr^{k-1} \left\|\sum_{|\alpha|=k} A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2}+ \sum_{k=m_0}^\infty \frac{k}{r} \left(\frac{r}{\rho}\right)^k\right).$$ Since $r<\rho$, the above series is convergent. Consequently, there exists a constant $M>0$ such that $$\|f(X)-f(Y)\|\leq M \|X-Y\|\qquad \text{ for any }\ X,Y\in [B({{\mathcal H}})^n]_r^-.$$ This implies the uniform continuity of $f$ on any closed ball $[B({{\mathcal H}})^n]_r^-$, $0<r<\gamma$, in the norm topology and, consequently, the continuity of $f$ on $[B({{\mathcal H}})^n]_{\gamma}$. \[operations\] Let $F$ and $G$ be formal power series such that $$\begin{split} F(X_1,\ldots, X_n)&=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right)\text{ and }\\ G(X_1,\ldots, X_n)&=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}B_{(\alpha)}\otimes X_\alpha\right) \end{split}$$ are free holomorphic functions on $[B({{\mathcal H}})^n]_{\gamma}$, and let $a, b\in {{\mathbb C}}$. Then the power series $aF+bG$, and $FG$ generate free holomorphic functions on $[B({{\mathcal H}})^n]_{\gamma}$. Moreover, $$\begin{split} aF(X_1,\ldots, X_n)+bG(X_1,\ldots, X_n)&=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}(aA_{(\alpha)}+bB_{(\alpha)})\otimes X_\alpha\right) \text{ and }\\ F(X_1,\ldots, X_n)G(X_1,\ldots, X_n)&=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k} C_{(\alpha)}\otimes X_\alpha \right) \end{split}$$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_\gamma$, where $C_{(\alpha)}:= \sum\limits_{\alpha=\sigma\beta}A_{(\sigma)} B_{(\beta)}$,  $\alpha\in {{\mathbb F}}_n^+$. According to the hypotheses, both power series $F$ and $G$ have radius of convergence $\geq \gamma$. Due to Theorem \[Abel\], we deduce that, given any $\epsilon>0$, there exists $k_0\in {{\mathbb N}}$ such that $$\left\|\sum_{|\alpha|=k} A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2k}\leq \frac{1}{\gamma} +\epsilon \ \text{ and }\ \left\|\sum_{|\alpha|=k} B_{(\alpha)}^* B_{(\alpha)}\right\|^{1/2k}\leq \frac{1}{\gamma} +\epsilon$$ for any $k\geq k_0$. Assume that $|a|+|b|\neq 0$. Since the left creation operators $S_1,\ldots, S_n$ are isometries with orthogonal ranges, we have $$\begin{split} \Biggl\|\sum_{|\alpha|=k} (aA_{(\alpha)}+bB_{(\alpha)})^* &(aA_{(\alpha)}+ bB_{(\alpha)})\Biggr\|^{1/2}\\ &= \left\|\sum_{|\alpha|=k} (aA_{(\alpha)} +bB_{(\alpha)})\otimes S_\alpha \right\|\\ &\leq \left\|\sum_{|\alpha|=k} aA_{(\alpha)} \otimes S_\alpha \right\|+\left\|\sum_{|\alpha|=k} bB_{(\alpha)}\otimes S_\alpha\right\|\\ &=\left\|\sum_{|\alpha|=k} |a|^2A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2}+\left\|\sum_{|\alpha|=k} |b|^2B_{(\alpha)}^* B_{(\alpha)}\right\|^{1/2}\\ &=(|a|+|b|)\left( \frac{1}{\gamma}+\epsilon\right)^k \end{split}$$ for any $k\geq k_0$. Hence, we deduce that $$\limsup_{k\to\infty} \left\|\sum_{|\alpha|=k} (aA_{(\alpha)}+bB_{(\alpha)})^* (aA_{(\alpha)}+ bB_{(\alpha)})\right\|^{1/2k}\leq \frac{1}{\gamma}+\epsilon$$ for any $\epsilon>0$. Taking $\epsilon\to 0$, we deduce that the power series $aF+bG$ has the radius of convergence $\geq \gamma$. Now, we prove that the power series $FG$ has radius of convergence $\geq \gamma$. If $0<r<\gamma$, then, due to Corollary \[Cs\], there is a constant $M>0$ such that $$\begin{split} \left\|\sum_{|\sigma|=k} C_{(\sigma)}^* C_{(\sigma)}\right\|^{1/2}&= \left\| \sum_{|\sigma|=k}C_{(\sigma)}\otimes S_\sigma\right\|\\ &= \left\| \sum_{p+q=k} \left( \sum_{|\alpha|=p}A_{(\alpha)}\otimes S_\alpha\right) \left( \sum_{|\beta|=q}B_{(\beta)}\otimes S_\beta\right)\right\|\\ &\leq \sum_{p+q=k}\left\| \sum_{|\alpha|=p}A_{(\alpha)}^* A_{(\alpha)} \right\|^{1/2} \left\|\sum_{|\beta|=q}B_{(\beta)}^*B_{(\beta)} \right\|^{1/2}\\ &\leq \sum_{p+q=k} \frac{M}{r^p}\cdot\frac{M}{r^q}\\ &= (k+1) \frac{M^2}{r^k} \end{split}$$ for any $k=0,1,\ldots$. Hence, we obtain $$\limsup_{k\to\infty} \left\|\sum_{|\sigma|=k} C_{(\sigma)}^* C_{(\sigma)}\right\|^{1/2k}\leq \frac {1}{r}$$ for any $r$ such that $0<r<\gamma$. Consequently, the radius of convergence of the power series $FG$ is $\geq \gamma$. The last part of the theorem follows easily using Theorem \[Abel\]. We are in position to give a characterization as well as models for free holomorphic functions on the open operatorial $n$-ball of radius $\gamma$. \[caract-shifts\] A power series $F:=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha$ represents a free holomorphic function on the open operatorial $n$-ball of radius $\gamma$ with coefficients in $B({{\mathcal K}})$ if and only if the series $$\sum\limits_{k=0}^\infty \sum\limits_{|\alpha|=k} r^{|\alpha|} A_{(\alpha)}\otimes S_\alpha$$ is convergent for any $r\in [0,\gamma)$, where $S_1,\ldots, S_n$ are the left creation operators on the Fock space $F^2(H_n)$. Moreover, in this case, the series $$\label{cre-seri} \sum\limits_{k=0}^\infty\left\| \sum\limits_{|\alpha|=k} r^{|\alpha|} A_{(\alpha)}\otimes S_\alpha \right\|=\sum_{k=0}^\infty r^k\left\| \sum_{|\alpha|=k} A_{(\alpha)}^*A_{(\alpha)}\right\|^{1/2}$$ are convergent for any $r\in [0,\gamma)$. Assume that $F$ represents a free holomorphic function on the open operatorial $n$-ball of radius $\gamma$. According to Theorem \[Abel\], $\gamma\leq R$, where $R$ is the radius of convergence of $F$, and $\sum\limits_{k=0}^\infty \left\|\sum\limits_{|\alpha|=k} A_{(\alpha)}\otimes X_\alpha\right\|$ converges for any $n$-tuple $[X_1,\ldots, X_n]$ with $\|[X_1,\ldots, X_n]\|=r<\gamma$. Since $\|[rS_1,\ldots, rS_n]\|=r<\gamma$, we deduce that the series is convergent for any $r\in [0,\gamma)$. Now, assume that the series is convergent for any $r\in [0,\gamma)$. According to the noncommutative von Neumann inequality [@Po-von], we have $$\sum\limits_{k=0}^\infty\left\| \sum\limits_{|\alpha|=k} r^{|\alpha|} A_{(\alpha)}\otimes T_\alpha \right\|\leq \sum\limits_{k=0}^\infty\left\| \sum\limits_{|\alpha|=k} r^{|\alpha|} A_{(\alpha)}\otimes S_\alpha \right\|$$ for any $n$-tuple $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ with $T_1T_1^*+\cdots T_nT_n^*\leq I$ and any $r\in [0,\gamma)$. Hence, we deduce that the series $$\sum_{k=0}^\infty \left\|\sum_{|\alpha|=k} A_{(\alpha)}\otimes X_\alpha\right\|$$ converges for any $n$-tuple of operators $[X_1,\ldots, X_n]$ with $\|[X_1,\ldots, X_n]\|<\gamma$. Due to Theorem \[Abel\], the power series $F=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha$ represents a free holomorphic function on the open operatorial $n$-ball of radius $\gamma$. This completes the proof. Let $\{a_k\}_{k=0}^\infty$ be a sequence of complex numbers. Then the following statements are equivalent: 1. $f(z):=\sum_{k=0}^\infty a_k z^k$ is an analytic function on the open unit disc ${{\mathbb D}}:=\{z\in {{\mathbb C}}:\ |z|<1\}$. 2. $f_r(S):=\sum_{k=0}^\infty r^ka_k S^k$ is convergent in the operator norm for each $r\in [0,1)$, where $S$ is the unilateral shift on the Hardy space $H^2$. 3. $f(Z):=\sum_{k=0}^\infty a_k Z^k$ is a free holomorphic function on the open operatorial unit $1$-ball. If $f(z)=\sum\limits_{k=0}^\infty a_k z^k$ is an analytic function on the open unit disc, then Hadamard’s theorem implies $\limsup\limits_{k\to\infty} |a_k|^{1/k}\leq 1$. Hence $\sum\limits_{k=0}^\infty r^k|a_k|<\infty$ for any $r\in [0,1)$ and, consequently, the series $\sum\limits_{k=0}^\infty r^k a_k S^k$ is convergent in the operator norm. Conversely, if the latter series is norm convergent, then, due to von Neumann inequality [@vN], the series $\sum\limits_{k=0}^\infty r^k a_k z$ converges for any $r\in[0,1)$ and $z\in {{\mathbb D}}$. Hence, we deduce (i). The equivalence (ii)$\Longleftrightarrow$ (iii) is a particular case of Theorem \[caract-shifts\]. If $\lambda:=(\lambda_1,\ldots, \lambda_n)\in{{\mathbb C}}^n$ and $\alpha=g_{i_1}\cdots g_{i_k}\in {{\mathbb F}}_n^+$, then we set $\lambda_\alpha:=\lambda_{i_1}\cdots \lambda_{i_k}$ and $\lambda_0=1$. \[part-case\] If $f =\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha Z_\alpha$, $a_\alpha\in {{\mathbb C}}$, is a free holomorphic function on the open operatorial unit $n$-ball, then its representation on ${{\mathbb C}}$, $$f(\lambda_1,\ldots, \lambda_n)=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha \lambda_\alpha,$$ is a holomorphic function on ${{\mathbb B}}_n$, the open unit ball of ${{\mathbb C}}^n$. Due to Theorem \[caract-shifts\], we have $$\begin{split} \sum_{k=0}^\infty\sum_{|\alpha|=k}|a_\alpha||\lambda_\alpha|&\leq \sum_{k=0}^\infty\left(\sum_{|\alpha|=k}|a_\alpha|^2\right)^{1/2}\left(\sum_{|\alpha|=k}|\lambda_\alpha|^2\right)^{1/2}\\ &\leq \sum_{k=0}^\infty\left(\sum_{|\alpha|=k}|a_\alpha|^2\right)^{1/2}\left( \sum_{i=1}^n |\lambda_i|^2\right)^{k/2}<\infty \end{split}$$ for any $(\lambda_1,\ldots, \lambda_n)\in {{\mathbb B}}_n$. Hence, the result follows. In the last part of this section, we show that the Hausdorff derivations on the algebra of noncommutative polynomials ${{\mathbb C}}[Z_1,\ldots, Z_n]$ (see [@MKS], [@RSS]) can be extended to the algebra of free holomorphic functions. For each $i=1,\ldots, n$, we define the free partial derivation $\frac{\partial } {\partial Z_i}$ on ${{\mathbb C}}[Z_1,\ldots, Z_n]$ as the unique linear operator on this algebra, satisfying the conditions $$\frac{\partial I} {\partial Z_i}=0, \quad \frac{\partial Z_i} {\partial Z_i}=I, \quad \frac{\partial Z_j} {\partial Z_i}=0\ \text{ if } \ i\neq j,$$ and $$\frac{\partial (fg)} {\partial Z_i}=\frac{\partial f} {\partial Z_i} g +f\frac{\partial g} {\partial Z_i}$$ for any $f,g\in {{\mathbb C}}[Z_1,\ldots, Z_n]$ and $i,j=1,\ldots n$. The same definition extends to formal power series in the noncommuting indeterminates $Z_1,\ldots, Z_n$. Notice that if $\alpha=g_{i_1}\cdots g_{i_p}$, $|\alpha|=p$, and $q$ of the $g_{i_1},\ldots, g_{i_p}$ are equal to $g_j$, then $\frac{\partial Z_\alpha} {\partial Z_j}$ is the sum of the $q$ words obtained by deleting each occurence of $Z_j$ in $Z_\alpha:=Z_{i_1}\cdots Z_{i_p}$. For example, $$\frac{\partial (Z_1 Z_2 Z_1^2)} {\partial Z_1}= Z_2 Z_1^2+ Z_1Z_2Z_1+ Z_1Z_2Z_1.$$ One can easily show that $\frac{\partial } {\partial Z_i}$ coincides with the Hausdorff derivative. If $\beta:=g_{i_1}\cdots g_{i_k}\in {{\mathbb F}}_n^+$, $i_1,\ldots, i_k\in \{1,2,\ldots, n\}$, we denote $Z_\beta:=Z_{i_1}\cdots Z_{i_k}$ and define the $k$-order free partial derivative of $G\in {{\mathbb C}}[Z_1,\ldots, Z_n]$ with respect to $Z_{i_1},\ldots, Z_{i_k}$ by $$\frac {\partial^k G}{\partial Z_{i_1}\cdots \partial Z_{i_k}}:= \frac{\partial} {\partial Z_{i_1}}\left(\frac{\partial} {\partial Z_{i_2}}\cdots \left( \frac{\partial G} {\partial Z_{i_k}}\right)\cdots \right).$$ These definitions can easily be extended to formal power series. If $F:=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)} \otimes Z_\alpha$ is a power series with operator-valued coefficients, then we define the $k$-order free partial derivative of $F$ with respect to $Z_{i_1}, \ldots, Z_{i_k}$ to be the power series $$\frac {\partial^k F}{\partial Z_{i_1}\cdots \partial Z_{i_k}} := \sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)} \otimes \frac {\partial^k Z_\alpha}{\partial Z_{i_1}\cdots \partial Z_{i_k}}.$$ \[deriv-comu\] If $i,j\in \{1,\ldots, n\}$, then $$\frac{\partial^2 F}{\partial Z_i \partial Z_j}= \frac{\partial^2 F}{\partial Z_j \partial Z_i}$$ for any formal power series $F$. Due to linearity, it is enough to prove the result for monomials. Let $\alpha:=g_{i_1}\cdots g_{i_k}$ be a word in $ {{\mathbb F}}_n^+$ and $Z_\alpha:=Z_{i_1}\cdots Z_{i_k}$. Let $i,j\in \{1,\ldots, n\}$ be such that $i\neq j$. Assume that $Z_i$ occurs $q$ times in $Z_\alpha$, and $Z_j$ occurs $p$ times in $Z_\alpha$. Then $\frac{\partial Z_\alpha}{\partial Z_i}$ is the sum of the $q$ words obtained by deleting each occurence of $Z_i$ in $Z_\alpha$. Notice that $Z_j$ occurs $p$ times in each of these $q$ words. Therefore, $\frac{\partial ^2 Z_\alpha}{\partial Z_j \partial Z_i}$ is the sum of the $qp$ words obtained by deleting each occurence of $Z_i$ in $Z_\alpha$ and then deleting each occurence of $Z_j$ in the resulting words. Similarly, $\frac{\partial ^2 Z_\alpha}{\partial Z_i \partial Z_j}$ is the sum of the $qp$ words obtained by deleting each occurence of $Z_j$ in $Z_\alpha$ and then deleting each occurence of $Z_i$ in the resulting words. Hence, it is clear that $$\frac{\partial^2 Z_\alpha}{\partial Z_i \partial Z_j}= \frac{\partial^2 Z_\alpha}{\partial Z_j \partial Z_i}.$$ This completes the proof. \[derivation\] Let $F=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)} \otimes Z_\alpha$ be a power series with radius of convergence $R$ and let $R'$ be the radius of convergence of the power series $ \frac{\partial^k F} {\partial Z_{j_1}\cdots \partial Z_{j_k}}$, where $j_1,\ldots, j_k\in \{1,\ldots, n\}$. Then $R'\geq R$ and, in general, the inequality is strict. It is enough to prove the result for first order free partial derivatives. For any word $\omega:=g_{i_1}\cdots g_{i_k}$, $|\omega|=k\geq 1$, and $0\leq m\leq k$, we define the insertion mapping of $g_j$, $j=1,\ldots, n$, on the $m$ position of $\omega$ by setting $$\chi(g_j, m,\omega):= \begin{cases} g_j\omega & \text{ if } m=0,\\ g_{i_1}\cdots g_{i_m} g_j g_{i_{m+1}}\cdots g_{i_k} & \text{ if } 1\leq m\leq k-1,\\ \omega g_j & \text{ if } m=|\omega|=k, \end{cases}$$ and $\chi(g_j, 0, g_0):=g_j$. Let $$\frac{\partial F} {\partial Z_j}= \sum_{\beta\in {{\mathbb F}}_n^+} B_{(\beta)} \otimes Z_\beta.$$ Using the definition of the Hausdorff derivation and the insertion mapping, we deduce that $$B_{(\beta)}=\sum_{m=0}^k A_{(\chi(g_j,m,\beta))}$$ for any $\beta\in {{\mathbb F}}_n^+$ with $|\beta|=k$. This is the case, since the monomial $Z_\beta$ comes from free differentiation with respect to $Z_j$ of the monomials $Z_{\chi(g_j,m,\beta)}$, $m=0,1,\ldots, |\beta|$. Therefore, we have $$\begin{split} \sum_{|\beta|=k} B_{(\beta)}^* B_{(\beta)} &= \sum_{|\beta|=k} \left( \sum_{m=0}^k A_{(\chi(g_j,m,\beta))}^*\right)\left( \sum_{m=0}^k A_{(\chi(g_j,m,\beta))}\right)\\ &\leq (k+1)\sum_{|\beta|=k} \sum_{m=0}^k A_{(\chi(g_j,m,\beta))}^*A_{(\chi(g_j,m,\beta))} \\ &\leq (k+1)^2 \sum_{|\alpha|=k+1} A_{(\alpha)}^* A_{(\alpha)}. \end{split}$$ The last inequality holds since, for each $j=1,\ldots,n$, each $\alpha\in {{\mathbb F}}_n^+$ with $|\alpha|=k+1$, and each $\beta\in {{\mathbb F}}_n^+$ with $|\beta|=k$, the cardinal of the set $$\{(g_j,m,\beta):\ \chi(g_j,m,\beta)=\alpha, \text{ where } m=0,1,\ldots, k\}$$ is $\leq k+1$. Hence, we deduce that $$\left(\sum_{|\beta|=k} B_{(\beta)}^* B_{(\beta)} \right)^{1/2k}\leq (k+1)^{1/k} \left( \sum_{|\alpha|=k+1} A_{(\alpha)}^* A_{(\alpha)}\right)^{1/2k}.$$ Consequently, due to Theorem \[Abel\], we have $\frac {1}{R'}\leq \frac {1}{R}$. Therefore, $R'\geq R$. To prove the last part of the theorem, let $R_1, R_2>0$ be such that $R_1<R_2$. Let us consider two power series $$F=\sum_{k=0}^\infty a_k Z_1^k \ \text { and } \ G=\sum_{k=0}^\infty b_k Z_2^k$$ with radius of convergence $R_1$ and $R_2$, respectively. We shall show that the power series $$F+G=\sum_{k=0}^\infty (a_kZ_1^k+b_k Z_2^k)$$ has the radius of convergence equal to $R_1$. First, since $$\sup_k\left( |a_k|^2+|b_k|^2\right)^{1/2k}\geq \sup|a_k|^{1/k}=\frac{1}{R_1},$$ we deduce that the radius of convergence of $F+G$ is $\leq R_1$. On the other hand, if $r<R_1$, Corollary \[Cs\] shows that both sequences $\{r^k |a_k|\}_{k=0}^\infty$ and $\{r^k |b_k|\}_{k=0}^\infty$ are bounded. This implies that the sequence $\{r^k \left(|a_k|^2+|b_k|^2\right)^{1/2}\}_{k=0}^\infty$ is bounded. Applying again Corollary \[Cs\], we can conclude that $F+G$ has radius of convergence $R_1$. Since $$\frac{\partial (F+G)}{\partial Z_2}=\sum_{k=1}^\infty k b_kZ_2^{k-1},$$ the power series $ \frac{\partial (F+G)}{\partial Z_2}$ has radius of convergence $R_2$, which is strictly larger than the radius of convergence of $F+G$. This completes the proof. Cauchy, Liouville, and Schwartz type results for free holomorphic functions {#Liouville} =========================================================================== In this section , we obtain Cauchy type estimates for the coefficients of free holomorphic functions and a Liouville type theorem for free entire functions. Based on a noncommutative version of Gleason’s problem [@R2] and the noncommutative von Neumann inequality [@Po-von], we provide a free analogue of Schwartz lemma. First, we obtain Cauchy type estimates for the coefficients of free holomorphic functions on the open ball $[B({{\mathcal H}})^n]_{\gamma}$ with coefficients in $B({{\mathcal K}})$. \[Cauchy-est\] Let $F:[B({{\mathcal H}})^n]_{\gamma}\to B({{\mathcal K}})\bar\otimes B({{\mathcal H}})$ be a free holomorphic function on $[B({{\mathcal H}})^n]_{\gamma}$ with the representation $$F(X_1,\ldots, X_n)=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k} A_{(\alpha)}\otimes X_\alpha\right),$$ and define $$M(\rho):= \|F(\rho S_1,\ldots, \rho S_n)\|\quad \text{for any } \ \rho\in (0,\gamma),$$ where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space. Then, for each $k=0,1,\ldots,$ $$\left\|\sum_{|\alpha|=k} A_\alpha^* A_\alpha\right\|^{1/2}\leq \frac{1} {\rho^k} M(\rho).$$ Let $\{Y_{(\alpha)}\}_{|\alpha|=k}$ be an arbitrary sequence of operators in $B({{\mathcal K}})$. Using Theorem \[caract-shifts\], we have $$\begin{split} \left|\left<\left(\sum_{|\alpha|=k} Y_{(\alpha)}^*\otimes S_\alpha^*\right)F(\rho S_1,\ldots, \rho S_n) h\otimes 1, h\otimes 1\right>\right| &\leq \left\|\sum_{|\alpha|=k} Y_{(\alpha)}^*\otimes S_\alpha^*\right\| M(\rho) \|h\|^2\\ &=\left\|\sum_{|\alpha|=k} Y_{(\alpha)}^* Y_{(\alpha)}\right\|^{1/2} M(\rho) \|h\|^2 \end{split}$$ for any $h\in {{\mathcal K}}$. On the other hand, since $S_1,\ldots, S_n$ are isometries with orthogonal ranges, we have $$\begin{split} \Bigl|\Bigl<\Bigl(\sum_{|\alpha|=k} Y_{(\alpha)}^*\otimes S_\alpha^*\Bigr)F(\rho S_1,\ldots, &\rho S_n) h\otimes 1, h\otimes 1\Bigr>\Bigr|\\ &= \rho^k\left|\left<\left(\sum_{|\alpha|=k} Y_{(\alpha)}^* A_{(\alpha)}\otimes I\right) h\otimes 1, h\otimes 1\right>\right|\\ &= \rho^k\left|\left<[Y_{(\alpha)}^*: |\alpha|=k] \left[\begin{matrix}A_{(\alpha)}\\:\\|\alpha|=k\end{matrix} \right]h,h\right>\right|. \end{split}$$ Combining these relations and taking $Y_{(\alpha)}:=A_{(\alpha)}$, $|\alpha|=k$, we deduce that $$\rho^k\left\|\left[\begin{matrix}A_{(\alpha)}\\:\\|\alpha|=k\end{matrix} \right]h\right\|^2\leq \left\|\left[\begin{matrix}A_{(\alpha)}\\:\\|\alpha|=k\end{matrix} \right]\right\| M(\rho) \|h\|^2$$ for any $h\in {{\mathcal K}}$. Therefore, $$\left\|\sum_{|\alpha|=k} A_\alpha^* A_\alpha\right\|^{1/2}=\|[A_{(\alpha)}^*:\ |\alpha|=k]\|\leq \frac{1} {\rho^k} M(\rho),$$ which completes the proof. A free holomorphic function with radius of convergence $R=\infty$ is called free entire function. We can prove now the following noncommutative multivariable generalization of Liouville’s theorem. \[Liou\] Let $F$ be an entire function and let $$F(X_1,\ldots, X_n)=\sum_{k=0}^\infty\sum_{|\alpha|=k} A_{(\alpha)}\otimes X_\alpha$$ be its representation on an infinite dimensional Hilbert space ${{\mathcal H}}$. Then $F$ is a polynomial of degree $\leq m$, $m=0, 1, \ldots$, if and only if there are constants $M>0$ and $C>1$ such that $$\|F(X_1,\ldots, X_n)\|\leq M\|[X_1,\ldots, X_n]\|^m$$ for any $[X_1,\ldots, X_n]\in B({{\mathcal H}})^n$ such that $\|[X_1,\ldots, X_n]\|\geq C$. If $F=\sum_{|\alpha|\leq m} A_{(\alpha)}\otimes X_\alpha$ is a polynomial, then $$\begin{split} \|F\|&\leq \sum_{k=0}^m \left\|\sum_{|\alpha|=m} A_{(\alpha)}\otimes X_\alpha \right\|\\ &\leq \sum_{k=0}^m \left\| \sum_{|\alpha|=k} A_{(\alpha)}^* A_{\alpha)} \right\|^{1/2}\|[X_1,\ldots, X_n]\|^k \end{split}$$ if $\|[X_1,\ldots, X_n]\|\geq 1$. Therefore, there exists $M>0$ and $R>1$ such that $$\label{f-norm} \|F(X_1,\ldots, X_n)\|\leq M \|[X_1,\ldots, X_n]\|^k$$ for any $n$-tuple of operators $[X_1,\ldots, X_n]$ with $\|[X_1,\ldots, X_n]\|\geq R$. Conversely, if the inequality holds, then $$\|F(\rho S_1,\ldots, \rho S_n)\|\leq M \rho^m,\quad \text{ as }\ \rho\to\infty.$$ According to Theorem \[Cauchy-est\], we have $$\left\|\sum_{|\alpha|=k} A_\alpha^* A_\alpha\right\|^{1/2}\leq \frac{1} {\rho^k} M(\rho),$$ where $M(\rho):=\|F(\rho S_1,\ldots, \rho S_n)\|$. Combining these inequalities, we deduce that $$\left\|\sum_{|\alpha|=k} A_\alpha^* A_\alpha\right\|^{1/2}\leq M\frac{1} {\rho^{k-m}}.$$ Consequently, if $k>m$ and $\rho\to\infty$, we obtain $\sum_{|\alpha|=k} A_\alpha^* A_\alpha=0$. This shows that $A_{(\alpha)}=0$ for any $\alpha\in {{\mathbb F}}_n^+$ with $|\alpha|>m$. We say that a free holomorphic function $F$ on the open operatorial $n$-ball of radius $\gamma$ is bounded if $$\|F\|_\infty:=\sup \|F(X_1,\ldots, X_n)\|<\infty,$$ where the supremum is taken over all $n$-tuples of operators $[X_1,\ldots, X_n]\in (B({{\mathcal H}})^n)_\gamma$ and any Hilbert space ${{\mathcal H}}$. In the particular case when $m=0$, Theorem \[Liou\] implies the following free analogue of Liouville’s theorem from complex analysis (see [@R], [@Co]). If $F$ is a bounded free entire function, then it is constant. We recall that the joint spectral radius of an $n$-tuple of operators $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$, $$r(T_1,\ldots,T_n):=\lim_{k\to\infty} \left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{\frac{1} {2k}},$$ is also equal to the spectral radius of the reconstruction operator $S_1\otimes T_1^*+\cdots + S_n\otimes T_n^*$ (see [@Po-unitary]). Consequently, $r(T_1,\ldots,T_n)<1$ if and only if $$\sigma(S_1\otimes T_1^*+\cdots + S_n\otimes T_n^*)\subset {{\mathbb D}}.$$ Moreover, the joint right spectrum  $\sigma_r(T_1,\ldots, T_n)$  is included in the closed ball of ${{\mathbb C}}^n$ of radius equal to $r(T_1,\ldots,T_n)$. We recall that $\sigma_r(T_1,\ldots, T_n)$ is the set of all $n$-tuples $(\lambda_1,\ldots, \lambda_n)\in {{\mathbb C}}^n$ such that the right ideal of $B({{\mathcal H}})$ generated by $\lambda_1 I-T_1,\ldots, \lambda_n I-T_n$ does not contain the identity. Now, we prove an analogue of Schwartz lemma, in our multivariable operatorial setting. \[Schwartz\] Let $F(X_1,\ldots, X_n)=\sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes X_\alpha$,  $A_{(\alpha)}\in B({{\mathcal K}})$, be a free holomorphic function on $[B({{\mathcal H}})^n]_1$ with the properties: 1. $\|F\|_\infty\leq 1$ and 2. $A_{(\beta)}=0$ for any $\beta\in {{\mathbb F}}_n^+$ with $|\beta|\leq m-1$, where $m=1,2,\ldots$. Then $$\|F(X_1,\ldots, X_n)\| \leq \|[X_1,\ldots, X_n]\|^m \quad \text{ and }\quad r(F(X_1,\ldots, X_n))\leq r(X_1,\ldots, X_n)^m$$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Moreover, $$\left\|\sum_{|\alpha|=k} A_\alpha A_\alpha^*\right\|^{1/2} \leq 1\quad \text{ for any } \ k\geq m.$$ For each $\beta\in {{\mathbb F}}_n^+$ with $|\beta|\leq m$, define the formal power series $$\Phi_{(\beta)}(Z_1,\ldots, Z_n):=\sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\beta\alpha)}\otimes Z_\alpha.$$ Since $$\left\|\sum_{|\alpha|=k} A_{(\beta \alpha)}^* A_{(\beta\alpha)}^*\right\|\leq \left\| \sum_{|\gamma|=m+k} A_{(\gamma)}^* A_{(\gamma)} \right\|,$$ we deduce that $$\limsup_{k\to\infty}\left\|\sum_{|\alpha|=k} A_{(\beta \alpha)}^* A_{(\beta\alpha)}^*\right\|^{1/2k} \leq \limsup_{k\to\infty} \left\| \sum_{|\gamma|=m+k} A_{(\gamma)}^* A_{(\gamma)} \right\|^{\frac{1}{2(m+k)}}.$$ Consequently, due to Theorem \[Abel\], the radius of convergence of $\Phi_{(\beta)}$ is greater than the radius of convergence of $F$. Therefore, $\Phi_{(\beta)}$ represents a free holomorphic function on the open operatorial unit $n$-ball. Since $A_{(\beta)}=0$ for any $\beta\in {{\mathbb F}}_n^+$ with $|\beta|\leq m-1$, and due to Theorem \[operations\], we have the following Gleason type decomposition $$F(Z_1,\ldots, Z_n)=\sum_{|\beta|=m}\left[(I_{{\mathcal K}}\otimes Z_\beta)\sum_{\alpha\in {{\mathbb F}}_n^+} A_{\beta \alpha)} \otimes Z_\alpha\right]= \sum_{|\beta|=m}(I_{{\mathcal K}}\otimes Z_\beta)\Phi_{(\beta)}(Z_1,\ldots, Z_n).$$ Therefore, $$\label{F-Phi} F(rS_1,\ldots, rS_n) =\sum_{|\beta|=m} (I_{{\mathcal K}}\otimes r^{|\beta|} S_\beta )\Phi_{(\beta)}(rS_1,\ldots, rS_n)$$ for any $r\in [0,1)$. Since $S_1,\ldots, S_n$ are isometries with orthogonal ranges, $S_\beta$, $|\beta|=m$, are also isometries with orthogonal ranges and we have $$F(rS_1,\ldots, rS_n)^*F(rS_1,\ldots, rS_n) =r^{2m}\sum_{|\beta|=m} \Phi_{(\beta)}(rS_1,\ldots, rS_n)^*\Phi_{(\beta)}(rS_1,\ldots, rS_n).$$ Now, due to the noncommutative von Neumann inequality [@Po-von] and Theorem \[caract-shifts\], we deduce that $$\label{M-P} \left\|\left[\begin{matrix} \Phi_{(\beta)}(rX_1,\ldots, rX_n)\\ :\\ |\beta|=m \end{matrix} \right]\right\|\leq \left\|\left[\begin{matrix} \Phi_{(\beta)}(rS_1,\ldots, rS_n)\\ :\\ |\beta|=m \end{matrix} \right]\right\|$$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Consequently, using relations and , we obtain $$\begin{split} \|F(rX_1,\ldots, rX_n)\| &= \left\|\sum_{|\beta|=m} (I_{{\mathcal K}}\otimes r^{|\beta|} X_\beta )\Phi_{(\beta)}(rX_1,\ldots, rX_n)\right\| \\ &\leq \left\|[r^m X_\beta:\ |\beta|=m]\right\| \left\|\left[\begin{matrix} \Phi_{(\beta)}(rX_1,\ldots, rX_n)\\ :\\ |\beta|=m \end{matrix} \right]\right\| \\ &\leq r^m\left\|\sum_{|\beta|=m} X_\beta X_\beta^*\right\|^{1/2}\left\|\left[\begin{matrix} \Phi_{(\beta)}(rS_1,\ldots, rS_n)\\ :\\ |\beta|=m \end{matrix} \right]\right\| \\ &= r^m\left\|\sum_{|\beta|=m} X_\beta X_\beta^*\right\|^{1/2} \left\| \sum_{|\beta|=m} \Phi_{(\beta)}(rS_1,\ldots, rS_n)^*\Phi_{(\beta)}(rS_1,\ldots, rS_n) \right\|^{1/2}\\ &= r^m\left\|\sum_{|\beta|=m} X_\beta X_\beta^*\right\|^{1/2} \left\|F(rS_1,\ldots, rS_n)^*F(rS_1,\ldots, rS_n)\right\|\\ &\leq r^m\left\|\sum_{|\beta|=m} X_\beta X_\beta^*\right\|^{1/2} \|F\|_\infty\\ &\leq r^m\left\|\sum_{|\beta|=m} X_\beta X_\beta^*\right\|^{1/2} \leq r^m\|[X_1,\ldots, X_n]\|^m. \end{split}$$ Taking $r\to 1$ and using the continuity of the free holomorphic function $F$ on $[B({{\mathcal H}})^n]_1$ (see Theorem \[continuous\]), we infer that $$\|F(X_1,\ldots, X_n)\|\leq \left\|\sum_{|\beta|=m} X_\beta X_\beta^*\right\|^{1/2} \leq \|[X_1,\ldots, X_n]\|^m$$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Due to Theorem \[operations\], the power series $F^k =\sum_{\alpha\in {{\mathbb F}}_n^+} B_{(\alpha)}\otimes Z_\alpha$ represents a free holomorphic function on the open operatorial unit $n$-ball, with $B_{(\alpha)}=0$ for any $\alpha\in {{\mathbb F}}_n^+$ with $|\alpha|\leq mk$. Applying the above inequality to $F^k$, we obtain $$\|F(X_1,\ldots, X_n)^k\|\leq \left\|\sum_{|\beta|=mk} X_\beta X_\beta^*\right\|^{1/2}\leq \left\|\sum_{|\beta|=k} X_\beta X_\beta^*\right\|^{m/2}.$$ Hence, and using the definition of the joint spectral radius, we deduce that $r(F(X_1,\ldots, X_n))\leq r(X_1,\ldots, X_n)^m. $ To prove the last part of the theorem, notice that, according to Theorem \[Cauchy-est\], we have $$\left\|\sum_{|\alpha|=k} A_{(\alpha)} A_{(\alpha)}^*\right\|^{1/2}\leq \frac{1}{\rho^k} M(\rho)$$ for any $\rho\in (0,1)$, where $M(\rho)=\|F(\rho S_1,\ldots, \rho S_n)\|$. Since $M(\rho)\leq \|F\|_\infty\leq 1$, we take $\rho\to 1$ and deduce that $\left\|\sum_{|\alpha|=k} A_{(\alpha)} A_{(\alpha)}^*\right\|^{1/2}\leq 1 $ for any $k\geq m$. The proof is complete. In the scalar case we get a little bit more. Let $f(X_1,\ldots, X_n)=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha X_\alpha$, $a_\alpha\in {{\mathbb C}}$, be a free holomorphic function on $[B({{\mathcal H}})^n]_1$ with scalar coefficients and the properties: 1. $\|f\|_\infty\leq 1$ and 2. $f(0)=0$. Then 1. $\|f(X_1,\ldots, X_n)\|\leq \left\|[X_1,\ldots, X_n]\right\| $ and  $r(f(X_1,\ldots, X_n))\leq r(X_1,\ldots, X_n) $ for any $n$-tuple $ \ [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$; 2. $ \sum\limits_{i=1}^n \left|\frac{\partial f}{\partial X_i}(0)\right|^2\leq 1. $ Moreover, if $ \sum\limits_{i=1}^n \left|\frac{\partial f}{\partial X_i}(0)\right|^2= 1, $ then $\|f\|_\infty=1$. The first part of this corollary is a particular case of Theorem \[Schwartz\], when $m=1$ and ${{\mathcal K}}={{\mathbb C}}$. To prove the second part, assume that $ \sum\limits_{i=1}^n \left|\frac{\partial f}{\partial X_i}(0)\right|^2= 1 $ Consequently, we have $\sum_{i=1}^n |a_i|^2=1$. Hence, and due to Theorem \[Cauchy-est\], we have $$1\leq \sum_{i=1}^n |a_i|^2\leq \frac{1}{\rho} \|f\|_\infty$$ for any $0<\rho<1$. Therefore, $\|f\|_\infty=1$. This completes the proof. Algebras of free holomorphic functions {#algebras} ======================================== In this section, we introduce two Banach algebras of free holomorphic functions, $H^\infty(B({{\mathcal X}})^n_1)$ and $A(B({{\mathcal X}})^n_1)$, and prove that they are isometrically isomorphic to the the noncommutative analytic Toeplitz algebra $F_n^\infty$ and the noncommutative disc algebra ${{\mathcal A}}_n$, respectively. The results of this section are used to obtain a maximum principle for free holomorphic functions. We denote by $Hol(B({{\mathcal X}})^n_\gamma)$ the set of all free holomorphic functions with scalar coefficients on the open operatorial $n$-ball of radius $\gamma$. Due to Theorem \[operations\] and Theorem \[Abel\], $Hol(B({{\mathcal X}})^n_\gamma)$ is an algebra and an element $F=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} a_\alpha Z_\alpha$ is in $Hol(B({{\mathcal X}})^n_\gamma)$ if and only if $$\limsup_{k\to\infty}\left(\sum_{|\alpha|=k} |a_\alpha|^2\right)^{1/2k}\leq 1.$$ Let $H^\infty(B({{\mathcal X}})^n_1)$ denote the set of all elements $F$ in $Hol(B({{\mathcal X}})^n_1)$ such that $$\|F\|_\infty:= \sup \|F(X_1,\ldots, X_n)\|<\infty,$$ where the supremum is taken over all $n$-tuples $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$ and any Hilbert space ${{\mathcal H}}$. We denote by $A(B({{\mathcal X}})^n_1)$ be the set of all elements $F$ in $Hol(B({{\mathcal X}})^n_1)$ such that, for any Hilbert space ${{\mathcal H}}$, the mapping $$[B({{\mathcal H}})^n]_1\ni (X_1,\ldots, X_n)\mapsto F(X_1,\ldots, X_n)\in B({{\mathcal H}})$$ has a continuous extension to the closed unit ball $[B({{\mathcal H}})^n]^-_1$. In this section, we will show that $H^\infty(B({{\mathcal X}})^n_1)$ and $A(B({{\mathcal X}})^n_1)$ are Banach algebras under pointwise multiplication and the norm $\|\cdot \|_\infty$, which can be identified with the noncommutative analytic Toeplitz algebra $F_n^\infty$ and the noncommutative disc algebra ${{\mathcal A}}_n$, respectively. Let us recall (see [@Po-von], [@Po-funct], [@Po-disc], [@Po-poisson]) a few facts about the Banach algebras ${{\mathcal A}}_n$ and $F_n^\infty$. Any element $~f$ in the full Fock space $ F^2(H_n)$ has the form $$f=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha,\quad\text{\ with\ }a_\alpha\in{{\mathbb C}}, \quad\text {\ such that\ }\ \|f\|_2:=\left(\sum\limits_{\alpha\in{{\mathbb F}}_n^+}|a_\alpha|^2\right)^{1/2}<\infty.$$ If $g=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} b_\alpha e_\alpha\in F^2(H_n)$, we define the product $f\otimes g$ to be the formal power series $$f\otimes g:=\sum_{\gamma\in {{\mathbb F}}_n^+} c_\gamma e_\gamma,\quad \text{ where }\quad c_\gamma :=\sum_{\stackrel{\alpha,\beta\in {{\mathbb F}}_n^+}{ \alpha\beta=\gamma}} a_\alpha b_\beta, \quad \gamma\in {{\mathbb F}}_n^+.$$ We also make the natural identification of $e_\alpha\otimes 1$ and $1\otimes e_\alpha$ with $e_\alpha$. Let ${{\mathcal P}}$ denote the set of all polynomials $p\in F^2(H_n)$, i.e., elements of the form $p=\sum_{|\alpha|\leq m} a_\alpha e_\alpha$, where $m=0,1,\ldots$. In [@Po-von], we introduced the noncommutative Hardy algebra $F_n^\infty$ as the set of all $f\in F^2(H_n)$ such that $$\label{norm} \|f\|_\infty:=\sup\{\|f\otimes p\|_2:p\in{{\mathcal P}}, \ \|p\|_2\le 1\}<\infty.$$ If $f\in F^2(H_n)$, then $f\in F_n^\infty$ if and only if $f\otimes g\in F^2(H_n)$ for any $g\in F^2(H_n)$. Moreover, if $f\in F_n^\infty$, then the left multiplication mapping $L_f:F^2(H_n)\to F^2(H_n)$ defined by $$L_fg:=f\otimes g, \quad g\in F^2(H_n),$$ is a bounded linear operator with $\|L_f\|=\|f\|_\infty$. The noncommutative Hardy algebra $F_n^\infty$ is isometrically isomorphic to the left multiplier algebra of the full Fock space $F^2(H_n)$, which is also called the noncommutative Toeplitz algebra. Under this identification, $F_n^\infty$ is the weakly closed algebra generated by the left creation operators $S_1,\ldots, S_n$ and the identity. The noncommutative disc algebra ${{\mathcal A}}_n$ was introduced in [@Po-von] as is the norm closed algebra generated by the left creation operators and the identity. Let $ f=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$ be an element in $F^2(H_n)$ and define $$f_r:=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} r^{|\alpha|} a_\alpha e_\alpha \quad \text{ for } \ ~0<r<1.$$ In [@Po-funct], [@Po-poisson], we proved that if $~f\in F_n^\infty~$ then $\|f_r\|_\infty\leq \|f\|_\infty$ for $ 0\leq r<1$, and $$\label{So} L_f=\text{\rm{SOT-}}\lim\limits_{r\to1}f_r(S_1,\dots,S_n),$$ where $f_r(S_1,\ldots, S_n):=\sum_{k=0}^\infty\sum_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha$. Moreover, if $f\in {{\mathcal A}}_n$ then the above limit exists in the operator norm topology. We identify $M_m(B({{\mathcal H}}))$, the set of $m\times m$ matrices with entries from $B({{\mathcal H}})$, with $B( {{\mathcal H}}^{(m)})$, where ${{\mathcal H}}^{(m)}$ is the direct sum of $m$ copies of ${{\mathcal H}}$. Thus we have a natural $C^*$-norm on $M_m(B({{\mathcal H}}))$. If $X$ is an operator space, i.e., a closed subspace of $B({{\mathcal H}})$, we consider $M_m(X)$ as a subspace of $M_m(B({{\mathcal H}}))$ with the induced norm. Let $X, Y$ be operator spaces and $u:X\to Y$ be a linear map. Define the map $u_m:M_m(X)\to M_m(Y)$ by $$u_m ([x_{ij}]):=[u(x_{ij})].$$ We say that $u$ is completely bounded ($cb$ in short) if $$\|u\|_{cb}:=\sup_{m\ge1}\|u_m\|<\infty.$$ If $\|u\|_{cb}\leq1$ (resp. $u_m$ is an isometry for any $m\geq1$) then $u$ is completely contractive (resp. isometric), and if $u_m$ is positive for all $m$, then $u$ is called completely positive. For each $m=1,2,\ldots$, we define the norms $\|\cdot \|_m:M_m\left(H^\infty(B({{\mathcal X}})^n_1)\right)\to [0,\infty)$ by setting $$\|[F_{ij}]_m\|_m:= \sup \|[F_{ij}(X_1,\ldots, X_n)]_m\|,$$ where the supremum is taken over all $n$-tuples $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$ and any Hilbert space ${{\mathcal H}}$. It is easy to see that the norms $\|\cdot\|_m$, $m=1,2,\ldots$, determine an operator space structure on $H^\infty(B({{\mathcal X}})^n_1)$, in the sense of Ruan (see [@ER]). \[f-infty\] Let $F:=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha Z_\alpha$ be a free holomorphic function on the open operatorial unit $n$-ball. Then the following statements are equivalent: 1. $F$ is in $H^\infty(B({{\mathcal X}})^n_1)$; 2. $f:=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$ is in $F_n^\infty$; 3. $\sup\limits_{0\leq r<1}\|F(rS_1,\ldots, rS_n)\|<\infty$; 4. The map $\varphi:[0,1)\to B(F^2(H_n))$ defined by $$\varphi(r):=F(rS_1,\ldots, rS_n) \quad \text{ for any } \ r\in [0,1)$$ has a continuous extension to $[0,1]$ with respect to the strong operator topology of $B(F^2(H_n))$. In this case, we have $$\label{many eq} \|L_f\|=\|f\|_\infty=\sup_{0\leq r<1}\|F(rS_1,\ldots, rS_n)\|= \lim_{r\to 1}\|F(rS_1,\ldots, rS_n)\|=\|F\|_\infty.$$ Moreover, the map $$\Phi:H^\infty((B({{\mathcal X}})^n_1)\to F_n^\infty\quad \text{ defined by } \quad \Phi(F):=f$$ is a completely isometric isomorphism of operator algebras. Assume (ii) holds. Since $f\in F_n^\infty$, we have $$\label{f-inf} \|F(rS_1,\ldots, rS_n)\|=\|f(rS_1,\ldots, rS_n)\|=\|L_{f_r}\|=\|f_r\|\leq \|f\|_\infty$$ for any $r\in [0,1)$. Therefore, (ii)$\implies$(iii). To prove that (iii)$\implies$(ii), assume that (iii) holds. Consequently, we have $$\begin{split} \sum_{\alpha\in {{\mathbb F}}_n^+} r^{2|\alpha|} |a_\alpha|^2&= \left\|\sum_{\alpha\in {{\mathbb F}}_n^+} r^{|\alpha|} a_\alpha S_\alpha(1)\right\|\\ &\leq \sup\limits_{0\leq r<1}\|F(rS_1,\ldots, rS_n)\|<\infty \end{split}$$ for any $0\leq r<1$. Hence, $\sum_{\alpha\in {{\mathbb F}}_n^+} |a_\alpha|^2<\infty$, which shows that $f:=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$ is in $F^2(H_n)$. Now assume that $f\notin F_n^\infty$. Due to the definition of $F_n^\infty$, given an arbitrary positive number $M$, there exists a polynomial $q\in {{\mathcal P}}$ with $\|q\|_2=1$ such that $$\|f\otimes q\|_2>M.$$ Since $\|f_r-f\|_2\to 0$ as $r\to 1$, we have $$\|f\otimes q-f_r\otimes q\|_2=\|(f-f_r)\otimes q\|_2\to 0, \quad \text{ as }\ r\to 1.$$ Therefore, there is $r_0\in (0,1)$ such that $ \|f_{r_0}\otimes q\|_2> M. $ Hence, $$\|f_{r_0}(S_1,\ldots, S_n)\|=\|L_{f_{r_0}}\|=\|f_{r_0}\|_\infty>M.$$ Since $M>0$ is arbitrary, we deduce that $$\sup_{0\leq r<1}\|f(rS_1,\ldots, rS_n)\|=\infty,$$ which is a contradiction. Consequently, (ii)$\Longleftrightarrow$(iii). Now, let us prove that (ii)$\implies$(iv). Assume (ii) and define the map $\tilde\varphi:[0,1]\to B(F^2(H_n)$ by setting $$\tilde\varphi(r):= \begin{cases} F(rS_1,\ldots, rS_n) &\quad \text{if } 0\leq r<1\\ L_f &\quad \text{if } r=1. \end{cases}$$ Since $f(rS_1,\ldots, rS_n)=F(rS_1,\ldots, rS_n)$,  $0\leq r<1$, the SOT-continuity of $\tilde\varphi$ at $r=1$ is due to relation , while the continuity of $\tilde\varphi$ on $[0,1)$ is a consequence of Theorem \[continuous\]. Therefore, the item (iv) holds. Assume now that (iv) holds. For each $x\in F^2(H_n)$, the map $[0,1)\ni \mapsto \|\varphi(r)x\|\in {{\mathbb R}}^+$ is bounded, i.e., $\sup\limits_{0\leq r<1}\|\varphi(r)x\|<\infty$. Due to the principle of uniform boundedness, we deduce condition (iii). The implication (i)$\implies$(iii) is obvious, and the implication (iii)$\implies$(i) is due to Theorem \[Abel\] and the noncommutative von Neumann inequality. Indeed, if $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$, ${{\mathcal H}}$ is an arbitrary Hilbert space, and $\|[X_1,\ldots, X_n]\|=r<1$, then $$\left\|\sum_{k=0}^m \sum_{|\alpha|=k} a_\alpha X_\alpha \right\|\leq \left\|\sum_{k=0}^m \sum_{|\alpha|=k} r^{|\alpha|}a_\alpha S_\alpha \right\|, \quad m=1,2,\ldots.$$ Hence, and taking into account Theorem \[Abel\], we deduce that $$\|F(X_1,\ldots, X_n)\|\leq \|F(rS_1, \ldots, rS_n)\|,\quad \text{ for any } \ r\in [0,1).$$ Consequently, $$\label{supsup} \sup_{[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1} \|F(X_1,\ldots, X_n)\|\leq \sup_{0\leq r<1} \|F(rS_1, \ldots, rS_n)\|<\infty,$$ whence (i) holds. We prove now the last part of the theorem. If $f\in F_n^\infty$ and $\epsilon>0$, then there exists a polynomial $q\in {{\mathcal P}}$ with $\|q\|_2=1$ such that $$\|f\otimes q\|_2>\|f\|_\infty-\epsilon.$$ Due to relation , there exists $r_0\in (0,1)$ such that $\|f_{r_0}(S_1,\ldots, S_n)q\|>\|f\|_\infty-\epsilon$. Using now relation , we deduce that $$\sup_{0\leq r<1}\|f(rS_1,\ldots, rS_n)\|=\|f\|_\infty.$$ Now, let $r_1,r_2\in [0,1)$ with $r_1<r_2$ and let $f:=\sum_{\alpha\in {{\mathbb F}}_n^+}a_\alpha e_\alpha $. Since $g:=\sum_{\alpha\in {{\mathbb F}}_n^+} r_2^{|\alpha|}a_\alpha e_\alpha $ is in the noncommutative disc algebra ${{\mathcal A}}_n$, we have $\|g_r\|_\infty\leq \|g\|_\infty$ for any $0\leq r<1$. In particular, when $r:=\frac {r_1}{r_2}$, we deduce that $$\|f_{r_1}(S_1,\ldots, S_n)\|\leq \|f_{r_2}(S_1,\ldots, S_n)\|.$$ Consequently, the function $[0,1]\ni r\to \|f(rS_1,\ldots, rS_n)\|\in {{\mathbb R}}^+$ is increasing. Hence, and using relation , we deduce . Using the same techniques, one can prove a matrix form of relation . In particular, we have $\|[F_{ij}]_m\|_m=\|[L_{f_{ij}}]_m\|$ for any $[F_{ij}]_m\in M_m\left(H^\infty(B({{\mathcal X}})^n_1)\right)$ and $m=1,2,\ldots$. Hence, we deduce that $\Phi$ is a complete isometry of $ H^\infty(B({{\mathcal X}})^n_1)$ onto $F_n^\infty$. The proof is complete. \[A-infty\] Let $F:=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha Z_\alpha$ be a free holomorphic function on the open operatorial unit $n$-ball. Then the following statements are equivalent: 1. $F$ is in $A(B({{\mathcal X}})^n_1)$; 2. $f:=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$ is in ${{\mathcal A}}_n$; 3. The map $\varphi:[0,1)\to B(F^2(H_n))$ defined by $$\varphi(r):=F(rS_1,\ldots, rS_n)$$ has a continuous extension to $[0,1]$, with respect to the operator norm topology of $B(F^2(H_n))$. Moreover, the map $$\Psi:A((B({{\mathcal X}})^n_1)\to {{\mathcal A}}_n\quad \text{ defined by } \quad \Psi(F):=f$$ is a completely isometric isomorphism of operator algebras. The implication (i)$\implies$(iii) is due to the definition of $A(B({{\mathcal X}})^n_1)$. Assume that item (ii) holds, i.e., $f\in {{\mathcal A}}_n$. The norm continuity of $\varphi$ on \[0,1) is due to Theorem \[continuous\], while the continuity of $\varphi$ at $r=1$ is due to the fact that $\lim_{r\to 1} f_r(S_1,\dots, S_n)=L_f$ in the operator norm for any $f\in {{\mathcal A}}_n$ (see the remarks preceeding this theorem). Therefore, the implication (ii)$\implies$(iii) is true. Conversely, assume item (iii) holds. Then $\lim_{r\to \infty} F(rS_1,\ldots, rS_n) $ exists in the operator norm. Since $F(rS_1,\ldots, rS_n)\in {{\mathcal A}}_n$ and ${{\mathcal A}}_n$ is a Banach algebra, there exists $g\in {{\mathcal A}}_n$ such that $L_g=\lim_{r\to \infty} F(rS_1,\ldots, rS_n)$ in the operator norm. On the other hand, due to Theorem \[f-infty\], we deduce that $f:=\sum_{\alpha\in {{\mathbb F}}_n} a_\alpha e_\alpha\in F_n^\infty$. Since $f(rS_1,\ldots, rS_n)=F(rS_1,\ldots, rS_n)$,  $0\leq r<1$, and $L_f=\text{\rm SOT}-\lim_{r\to \infty} f(rS_1,\ldots, rS_n)$, we conclude that $L_f=L_g$, i.e., $f=g$. Therefore, condition (ii) holds. It remains to prove that (ii)$\implies$(i). According to [@Po-funct] (see also [@Po-poisson]), if $f\in{{\mathcal A}}_n$ then, for any $n$-tuple $[Y_1,\ldots, Y_n]\in [B({{\mathcal H}})^n]_1^-$, $$\tilde F(Y_1,\ldots, Y_n):= \lim_{r\to 1} f(rY_1,\ldots, rY_n),$$ exists in the operator norm, and $$\|\tilde F(Y_1,\ldots, Y_n)\|\leq \|f\|_\infty\quad \text{ for any } \ [Y_1,\ldots, Y_n]\in [B({{\mathcal H}})^n]_1^-.$$ Notice also that $\tilde F$ is an extension of the free holomorphic function $F$ on $[B({{\mathcal H}})^n]_1$. Indeed, if $ [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$, then $$\begin{split} \tilde F(X_1,\ldots, X_n)&=\lim_{r\to 1}f(rX_1,\ldots, rX_n)\\ &=\lim_{r\to 1}F(rX_1,\ldots, rX_n)=F(X_1,\ldots, X_n). \end{split}$$ The last equality is due to Theorem \[continuous\]. Let us prove that $\tilde F:[B({{\mathcal H}})^n]_1^-\to B({{\mathcal H}})$ is continuous. Since $f\in {{\mathcal A}}_n$, for any $\epsilon>0$ there exists $r_0\in [0,1)$ such that $\|L_f-f(r_0S_1,\ldots, r_0 S_n)\|<\epsilon$. Applying the above mentioned result from [@Po-poisson] to $ f-\ f_{r_0}\in {{\mathcal A}}_n$, we deduce that $$\label{tild-f} \|\tilde F(T_1,\ldots, T_n)-f_{r_0}(T_1,\ldots, T_n)\|\leq \|L_f-L_{f_{r_0}} \|< \frac{\epsilon}{3}$$ for any $[T_1,\ldots, T_n]\in [B({{\mathcal H}})^n]_1^-$. Due to Theorem \[continuous\], $F$ is a continuous function on $[B({{\mathcal H}})^n]_1$. Therefore, there exists $\delta>0$ such that $$\|F_{r_0}(T_1,\ldots, T_n)-F_{r_0}(Y_1,\ldots, Y_n)\|<\frac{\epsilon}{3}$$ for any $n$-tuples $[T_1,\ldots, T_n]$ and $[Y_1,\ldots, Y_n]$ in $[B({{\mathcal H}})^n]_1^-$ such that $\|[T_1-Y_1,\ldots, T_n-Y_n]\|<\delta$. Hence, and using , we have $$\begin{split} \|\tilde F(T_1,\ldots, T_n)-\tilde F(Y_1,\ldots, Y_n)\| &\leq \|\tilde F(T_1,\ldots, T_n)-f_{r_0}(T_1,\ldots, T_n)\|\\ &\qquad + \| f_{r_0}(T_1,\ldots, T_n)- f_{r_0}(Y_1,\ldots, Y_n)\|\\ &\qquad + \|f_{r_0}(Y_1,\ldots, Y_n)-\tilde F(Y_1,\ldots, Y_n)\| <\epsilon, \end{split}$$ whenever $\|[T_1-Y_1,\ldots, T_n-Y_n]\|<\delta$. This proves the continuity of $\tilde F$ on $[B({{\mathcal H}})^n]_1^-$. Therefore, $F\in A(B({{\mathcal X}})^n_1)$. To prove the last part of the theorem, notice that if $f\in{{\mathcal A}}_n\subset F_n^\infty$, then by Theorem \[f-infty\] (see relation and its matrix form), we have $\|[L_{f_{ij}}]_m\|=\|[F_{ij}]_m\|_m$. Since ${{\mathcal A}}_n\subset B(F^2(H_n))$ is an operator algebra, we deduce that $\Psi$ is a completely isometric isomorphism of operator algebras. This completes the proof. Here is our version of the maximum principle for free holomorphic functions. \[max-mod1\] Let ${{\mathcal H}}$ be an infinite dimensional Hilbert space. Assume that $f:[B({{\mathcal H}})^n]_1^-\to B({{\mathcal H}})$ is a continuous function in the operator norm, and it is free holomorphic on $[B({{\mathcal H}})^n]_1$. Then $$\begin{split} \max\{\|f(X_1,\ldots, X_n)\|&:\ \|[X_1,\ldots, X_n]\|\leq 1\}\\ &= \max\{\|f(X_1,\ldots, X_n)\|:\ \|[X_1,\ldots, X_n]\|= 1\}. \end{split}$$ Due to the continuity of $f$, for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1^-$, $$\|f(X_1,\ldots, X_n)\|=\lim_{r\to 1} \|f(rX_1,\ldots, rX_n)\|.$$ On the other hand, the noncommutative von Neumann inequality implies $$\|f(rX_1,\ldots, rX_n)\|\leq \|f(rS_1,\ldots, rS_n)\| \quad \text{ for } \ 0\leq r<1.$$ By Theorem \[A-infty\], $f\in {{\mathcal A}}_n$ and, consequently, $$\lim_{r\to 1} \|f(rS_1,\ldots, rS_n)\|=\|L_f\|=\|f\|_\infty.$$ Combining these relations, we deduce that $$\label{ff} \|f(X_1,\ldots, X_n)\|\leq \|f\|_\infty\quad \text{ for any } \ [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1.$$ Since ${{\mathcal H}}$ is infinite dimensional, there exists a subspace ${{\mathcal K}}\subset {{\mathcal H}}$ and a unitary operator $U:F^2(H_n)\to {{\mathcal K}}$. Define the operators $$V_i:=\left(\begin{matrix} US_iU^*&0\\ 0&0 \end{matrix}\right), \quad i=1,\ldots,n,$$ with respect to the orthogonal decomposition ${{\mathcal H}}={{\mathcal K}}\oplus {{\mathcal K}}^{\perp}$, where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space $F^2(H_n)$. Notice that $\|[V_1,\ldots, V_n]\|=1$ and $$f(V_1,\ldots, V_n)=\lim_{r\to 1} \left(\begin{matrix} Uf_r(S_1,\ldots, S_n)U^*&0\\ 0&0 \end{matrix}\right)$$ in the operator norm. Consequently, $$\|f(V_1,\ldots, V_n)\|=\lim_{r\to 1}\|f_r(S_1,\ldots, S_n)\|=\|f\|_\infty.$$ Hence, and using inequality , we deduce that $$\begin{split} \max\{\|f(X_1,\ldots, X_n)\|&:\ \|[X_1,\ldots, X_n]\|\leq 1\}\\ &= \max\{\|f(X_1,\ldots, X_n)\|:\ \|[X_1,\ldots, X_n]\|= 1\}\\ &=\|f\|_\infty. \end{split}$$ This completes the proof. \[max-mod2\] Let $f$ be a free holomorphic function on $[B({{\mathcal H}})^n]_1$, where ${{\mathcal H}}$ is an infinite dimensional Hilbert space, and let $r\in [0,1)$. Then $$\begin{split} \max\{\|f(X_1,\ldots, X_n)\|&:\ \|[X_1,\ldots, X_n]\|\leq r\}\\ &= \max\{\|f(X_1,\ldots, X_n)\|:\ \|[X_1,\ldots, X_n]\|= r\}\\ &=\|f(rS_1,\ldots, rS_n)\|. \end{split}$$ In a forthcoming paper [@Po-Bohr], we obtain operator-valued multivariable Bohr type inequalities for free holomorphic functions on the open operatorial unit $n$-ball. As consequences, we obtain operator-valued Bohr inequalities for the noncommutative disc algebra ${{\mathcal A}}_n$ and the noncommutative analytic Toeplitz algebra $F_n^\infty$. Free analytic functional calculus and noncommutative Cauchy transforms {#free analytic} ======================================================================= In this section, we introduce a free analytic functional calculus for $n$-tuples $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. We introduce a noncommutative Cauchy transform ${{\mathcal C}}_T:B(F^2(H_n))\to B({{\mathcal H}})$ associated with any such $n$-tuple of operators and prove that $$f(T_1,\ldots, T_n)=C_T(f(S_1,\ldots, S_n)),\quad f\in H^\infty (B({{\mathcal X}})^n_1),$$ where $f(S_1,\ldots, S_n)$ is the boundary function of $f$. Similar Cauchy representations are obtained for the $k$-order Hausdorff derivations of $f$. Finally, we show that the noncommutative Cauchy transform commutes with the action of the unitary group ${{\mathcal U}}({{\mathbb C}}^n)$. \[abel\] Let $F:=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha$ be a free holomorphic function on the open operatorial $n$-ball of radius $\gamma$. Then, for any Hilbert space ${{\mathcal H}}$ and any $n$-tuple of operators $[X_1,\ldots, X_n]\in B({{\mathcal H}})^n$ with $r(X_1,\ldots,X_n)<\gamma$, the series $$F(X_1,\ldots, X_n)=\sum\limits_{k=0}^\infty \sum\limits_{|\alpha|=k} A_{(\alpha)}\otimes X_\alpha$$ is convergent in the operator norm of $B({{\mathcal K}}\otimes {{\mathcal H}})$. Moreover, if   $0<r<1$, then $$\label{lim-Fr} \lim_{r\to 1}F_r(X_1,\ldots, X_n)=F(X_1,\ldots, X_n)$$ and $$\label{lim-PFr} \lim_{r\to 1}\left(\frac{\partial^k F_r}{\partial Z_{i_1}\cdots Z_{i_k}}\right)(X_1,\ldots, X_n)=\left(\frac{\partial^k F}{\partial Z_{i_1}\cdots \partial Z_{i_k}}\right)(X_1,\ldots, X_n)$$ for $i_1,\ldots, i_k\in \{1,\ldots, n\}$, where the limits are in the operator norm. . Assume that $[X_1,\ldots, X_n]$ is an $n$-tuple of operators on ${{\mathcal H}}$ such that $r(X_1,\ldots,X_n)<R$, where $R$ is the radius of convergence of $F$. Let $\rho',\rho>0$ be such that $r(X_1,\ldots,X_n)<\rho'<\rho<R$. Due to the definition of $r(X_1,\ldots,X_n)$, there exists $k_0\in {{\mathbb N}}$ such that $$\label{ro'} \left\|\sum\limits_{|\alpha|=k}X_{\alpha}X_\alpha^*\right\|^{1/2k}< \rho'\quad \text{ for any }\ k\geq k_0.$$ Since $\frac{1}{\rho}> \frac{1}{R}$, we can find $m_0$ such that $$\label{ro} \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}A_{(\alpha)}^*\right\|^{1/2k}< \frac{1}{\rho}\quad \text{ for any }\ k\geq m_0.$$ If $k\geq \max\{k_0,m_0\}$, then relations and imply $$\begin{split} \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha \right\|&= \left\|\left[ I\otimes X_\alpha:\ |\alpha|=k\right] \left[\begin{matrix} A_{(\alpha)}\otimes I\\ :\\|\alpha|=k \end{matrix}\right]\right\|\\ &=\left\|\sum\limits_{|\alpha|=k}X_{\alpha}X_\alpha^*\right\|^{1/2}\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}A_{(\alpha)}^*\right\|^{1/2}\\ &\leq \left(\frac{\rho'}{\rho}\right)^k. \end{split}$$ This proves the convergence of the series $\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right)$ in the operator norm. Now, using the above inequalities, we obtain $$\begin{split} \left\|\sum_{k=0}^\infty \sum_{|\alpha|=k} (r^{|\alpha|}-1)A_\alpha\otimes X_\alpha\right\|&\leq \sum_{k=1}^\infty (r^k-1)\left\|\sum_{|\alpha |=k}A_\alpha \otimes X_\alpha\right\|\\ &\leq \sum_{k=1}^\infty (r^k-1) \left(\frac{\rho'}{\rho}\right)^k\\ &\leq (r-1)\sum_{k=1}^\infty k \left(\frac{\rho'}{\rho}\right)^k. \end{split}$$ Since $\rho'<\rho$, the latter series is convergent and therefore relation holds. Due to Theorem \[derivation\], $\frac{\partial F}{\partial Z_i}$ is a free holomorphic function on the open operatorial $n$-ball of radius $\gamma$, and $$\frac{\partial^k F_r}{\partial Z_{i_1}\cdots \partial Z_{i_k}}(X_1,\ldots, X_n)=r^k \frac{\partial^k F}{\partial Z_{i_1}\cdots \partial Z_{i_k}}(rX_1,\ldots, rX_n),\quad 0<r<1.$$ Applying relation to $\frac{\partial^k F}{\partial Z_{i_1}\cdots \partial Z_{i_k}}$, we deduce . The proof is complete. Let $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. We introduce the [*Cauchy kernel*]{} associated with $T$ to be the operator $C_T(S_1,\ldots, S_n)\in B(F^2(H_n)\otimes {{\mathcal H}})$ defined by $$\label{Cauc} C_T(S_1,\ldots, S_n):=\sum_{k=0}^\infty\sum_{|\alpha|=k} S_\alpha\otimes T_{\tilde\alpha}^*,$$ where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space $F^2(H_n)$, and $\tilde\alpha$ is the reverse of $\alpha$, i.e., $\tilde \alpha= g_{i_k}\cdots g_{i_k}$ if $\alpha=g_{i_1}\cdots g_{i_k}$. Applying Theorem \[Abel\], when $A_{(\alpha)}:=T_\alpha^*$, $\alpha\in {{\mathbb F}}_n^+$ and $X_i:=S_i$, $i=1,\ldots, n$, we deduce that $$\frac{1}{R}=\lim_{k\to\infty} \left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2k}=r(T_1,\ldots, T_n)<1$$ and $\|[S_1,\ldots, S_n]\|=1<R$. Consequently, the series in is convergent in the operator norm and $C_T(S_1,\ldots, S_n)\in {{\mathcal A}}_n\bar\otimes B({{\mathcal H}})\subset B(F^2(H_n)\otimes {{\mathcal H}})$. Now, one can easily see that $$\label{Cauc-inv} C_T(S_1,\ldots, S_n)=\left( I-S_1\otimes T_1^*-\cdots -S_n\otimes T_n^*\right)^{-1}.$$ We call the operator $$S_1\otimes T_1^*+\cdots +S_n\otimes T_n^*$$ the [*reconstruction operator*]{} associated with the $n$-tuple $[T_1,\ldots, T_n]$. We should mention that this operator plays an important role in noncommutative multivariable operator theory (see [@Po-varieties], [@Po-unitary]). We remark that if $1$ is not in the spectrum of the reconstruction operator, then the Cauchy kernel defined by makes sense. In this case, $C_T(S_1,\ldots, S_n)$ is in $F_n^\infty\bar\otimes B({{\mathcal H}})$, the $WOT$-closed operator algebra generated by the spatial tensor product, and not necessarily in ${{\mathcal A}}_n\bar\otimes B({{\mathcal H}})$. Morever, we can think of the series $\sum_{k=0}^\infty\sum_{|\alpha|=k} S_\alpha\otimes T_{\tilde \alpha}^*$ as the Fourier representation of the Cauchy kernel. In what follows we also use the notation $C_T:=C_T(S_1,\ldots, S_n)$. \[Prop-Cauc\] Let $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. Then: 1. $\|C_T\|\leq \sum\limits_{k=0}^\infty\left\|\sum\limits_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2}$. In particular, if $T:=[T_1,\ldots, T_n]\in[B({{\mathcal H}})^n]_1$, then $\|C_T\|\leq \frac{1}{1-\|T\|}$. 2. $C_T-C_X=C_T\left[\sum\limits_{i=1}^n S_i\otimes (T_i^*-X_i^*)\right] C_X$ and $$\|C_T-C_X\|\leq \|C_T\| \|C_X\|\|[T_1-X_1,\ldots, T_n-X_n]\|$$ for any $n$-tuple $X:=[X_1,\ldots, X_n]\in B({{\mathcal H}})^n$ with joint spectral radius $r(X_1,\ldots, X_n)<1$. Since $S_1,\ldots, S_n$ are isometries with orthogonal ranges, we have $$\|C_T\|\leq \sum_{k=0}^\infty\left\| \sum_{|\alpha|=k} S_\alpha\otimes T_{\tilde \alpha}^*\right\| = \sum_{k=0}^\infty\left\| \sum_{|\alpha|=k} T_\alpha T_{\alpha}^*\right\|^{1/2}.$$ If $\|[T_1,\ldots, T_n]\|<1$, then $$\sum_{k=0}^\infty\left\| \sum_{|\alpha|=k} T_\alpha T_{\alpha}^*\right\|^{1/2} \leq \sum_{k=0}^\infty\left\| \sum_{i=1}^n T_iT_i^*\right\|^{k/2} =\frac{1}{1-\|T\|}.$$ To prove (ii), notice that $$\begin{split} C_T-C_X&= \left(I-\sum_{i=1}^n S_i\otimes T_i^*\right)^{-1} \left[ I-\sum_{i=1}^n S_i\otimes X_i^*-\left(I-\sum_{i=1}^n S_i\otimes T_i^*\right)\right]\left(I-\sum_{i=1}^n S_i\otimes X_i^*\right)^{-1}\\ &=C_T\left[\sum\limits_{i=1}^n S_i\otimes (T_i^*-X_i^*)\right] C_X, \end{split}$$ and $$\begin{split} \|C_T-C_X\|&\leq \|C_T\| \|C_X\|\left\| \sum_{i=1}^n S_i\otimes (T_i^*-X_i^*)\right\|\\ &=\|C_T\| \|C_X\|\left\|\sum_{i=1}^n (T_i-X_i)(T_i-X_i)^*\right\|^{1/2}, \end{split}$$ which completes the proof. The [*Cauchy transform*]{} at $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is the mapping $${{\mathcal C}}_T:B(F^2(H_n))\to B({{\mathcal H}})$$ defined by $$\left< {{\mathcal C}}_T(A)x,y\right>:= \left<(A\otimes I_{{\mathcal H}})(1\otimes x), C_T(R_1,\ldots, R_n)(1\otimes y)\right>$$ for any $x,y\in {{\mathcal H}}$, where $R_1,\ldots, R_n$ are the right creation operators on the full Fock space $F^2(H_n)$. The operator ${{\mathcal C}}_T(A)$ is called the Cauchy transform of $A$ at $T$. Given $A\in B(F^2(H_n))$, the Cauchy transform generates a function (the Cauchy transform of $A$) $${{\mathcal C}}[A]:[B({{\mathcal H}})^n]_1\to B({{\mathcal H}})$$ by setting $${{\mathcal C}}[A](X_1,\ldots, X_n):={{\mathcal C}}_X(A) \quad \text{ for any } \ X:=[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1.$$ Indeed, it is enough to see that $r(X_1,\ldots, X_n)\leq \|[X_1,\ldots, X_n]\|<1$, and therefore ${{\mathcal C}}_X(A)$ is well-defined. This gives rise to an important question: when is ${{\mathcal C}}[A]$ a free holomorphic function on $[B({{\mathcal H}})^n]_1$. Due to Theorem \[abel\], if $f=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha Z_\alpha$ is a free holomorphic function on the open operatorial unit $n$-ball and $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is any $n$-tuple of operators with $r(T_1,\ldots, T_n)<1$ then, we can define a bounded linear operator $$f(T_1,\ldots, T_n):=\sum_{k=0}^\infty\sum_{|\alpha|=k} a_\alpha T_\alpha,$$ where the series converges in norm. This provides the [*free analytic functional calculus*]{}. If $F=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha Z_\alpha $ is in the Hardy algebra $H^\infty(B({{\mathcal X}})^n_1)$, we denote by $F(S_1,\ldots, S_n)$ the boundary function of $F$, i.e., $F(S_1,\ldots, S_n):=L_f\in B(F^2(H_n))$, where $f:= \sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$. \[an=cauch\] Let $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. Then, for any $f\in H^\infty(B({{\mathcal X}})^n_1)$, $$f(T_1,\ldots, T_n)={{\mathcal C}}_T(f(S_1,\ldots, S_n)),$$ where $f(T_1,\ldots, T_n)$ is defined by the free analytic functional calculus, and $f(S_1,\ldots, S_n)$ is the boundary function of $f$. Moreover, $$\|f(T_1,\ldots, T_n)\|\leq \left(\sum\limits_{k=0}^\infty\left\|\sum\limits_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2}\right) \|f\|_\infty.$$ First, we prove the above equality for monomials. Notice that $$\begin{split} \left<{{\mathcal C}}_T(S_\alpha)x,y\right>&= \left< (S_\alpha\otimes I_{{\mathcal H}})(1\otimes x), C_T(R_1,\ldots, R_n)(1\otimes y)\right>\\ &=\left< e_\alpha\otimes x, \left(\sum_{\beta\in {{\mathbb F}}_n^+} R_\beta\otimes T_{\tilde \beta}^*\right)(1\otimes y)\right>\\ &= \left< e_\alpha\otimes x, \sum_{\beta\in {{\mathbb F}}_n^+} e_{\tilde \beta}\otimes T_{\tilde \beta}^*y\right>\\ &=\left<T_\alpha x,y\right> \end{split}$$ for any $ x,y\in {{\mathcal H}}$. Now, assume that $f:=\sum_{k=0}^\infty\sum_{|\alpha|=k} a_\alpha Z_\alpha$ is in $H^\infty(B({{\mathcal X}})^n_1)$ and $0<r<1$. Then, due to Theorem \[abel\], we have $$\lim_{m\to\infty}\sum_{k=0}^m r^k \sum_{|\alpha|=k} a_\alpha S_\alpha=f_r(S_1,\ldots, S_n)\in {{\mathcal A}}_n$$ in the operator norm of $B(F^2(H_n))$, and $$\lim_{m\to\infty}\sum_{k=0}^m r^k \sum_{|\alpha|=k} a_\alpha T_\alpha=f_r(T_1,\ldots, T_n)$$ in the operator norm of $B({{\mathcal H}})$. Now, due to the continuity of the noncommutative Cauchy transform in the operator norm, we deduce that $$\label{f_r-C} f_r(T_1,\ldots, T_n)={{\mathcal C}}_T(f_r(S_1,\ldots, S_n)).$$ Since $f(S_1,\ldots, S_n)\in F_n^\infty$, we know that $\lim\limits_{r\to 1} f_r(S_1,\ldots, S_n)=f(S_1,\ldots, S_n)$ in the strong operator topology. Since $\|f_r(S_1,\ldots, S_n)\|\leq \|f\|_\infty$, we deduce that $$\text{\rm SOT}-\lim\limits_{r\to 1}f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}}=f(S_1,\ldots, S_n)\otimes I_{{\mathcal H}}.$$ On the other hand, by Theorem \[abel\], $\lim\limits_{r\to 1} f_r(T_1,\ldots, T_n)=f(T_1,\ldots, T_n)$ in the operator norm. Passing to the limit, as $r\to 1$, in the equality $$\left<f_r(T_1,\ldots, T_n)x,y\right>=\left<(f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x), C_T(R_1,\ldots, R_n)(1\otimes y)\right>, \quad x, y\in {{\mathcal H}},$$ we obtain $f(T_1,\ldots, T_n)={{\mathcal C}}_T(f(S_1,\ldots, S_n))$, which proves the first part of the theorem. Now, we can deduce the second part of the theorem using Proposition \[Prop-Cauc\]. This completes the proof. Using the Cauchy representation provided by Theorem \[an=cauch\], one can deduce the following result. \[conv-u-w\*\] Let $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with $r(T_1,\ldots, T_n)<1$. 1. If $\{f_k\}_{k=1}^\infty$ and $f$ are free holomorphic functions in $Hol(B({{\mathcal X}})^n_1)$ such that $\|f_k-f\|_\infty\to 0$, as $k\to \infty$, then $f_k(T_1,\ldots, T_n)\to f(T_1,\ldots, T_n)$ in the operator norm of $B({{\mathcal H}})$. 2. If $\{f_k\}_{k=1}^\infty$ and $f$ are in the algebra $H^\infty (B({{\mathcal X}})^n_1)$ such that $f_k(S_1,\ldots, S_n)\to f(S_1,\ldots, S_n)$ in the $w^*$-topology (or strong operator topology) and $\|f_k\|_\infty\leq M$ for any $k=1,2,\ldots$, then $f_k(T_1,\ldots, T_n)\to f(T_1,\ldots, T_n)$ in the weak operator topology. We can extend Theorem \[an=cauch\] and obtain Cauchy representations for the $k$-order Hausdorff derivations of bounded free holomorphic functions. \[cauc-dif\] Let $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with the joint spectral radius $r(T_1,\ldots, T_n)<1$ and let $f \in H^\infty(B({{\mathcal X}})^n_1)$. Then $$\label{deriv-Cau} \begin{split} \Bigl<\left(\frac{\partial^k f}{\partial Z_{i_1}\cdots \partial Z_{i_k}}\right)&(T_1,\ldots, T_n) x,y\Bigr>\\ &= \left<\left[\frac{\partial^k \left(C_T(R_1,\ldots, R_n)^*\right)}{\partial T_{i_1}\cdots \partial T_{i_k}}\right](f(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x), 1\otimes y\right> \end{split}$$ for any  $i_1,\ldots, i_k\in \{ 1,\ldots, n\}$ and $x,y\in {{\mathcal H}}$, where $f(S_1,\ldots, S_n)$ is the boundary function of $f$. Moreover, $$\label{deriv-est} \left\| \left(\frac{\partial f}{\partial Z_i}\right)(T_1,\ldots, T_n)\right\|\leq \|f\|_\infty \sum_{k=1}^\infty k^{3/2}\left\|\sum_{|\beta|=k-1}T_\beta T_\beta\right\|^{1/2}, \quad i=1,\ldots,n.$$ First, notice that $$C_X(R_1,\ldots, R_n)^*=\sum_{k=0}^\infty \sum_{|\alpha|=k} R_{\tilde \alpha}^*\otimes X_\alpha,$$ where the series is convergent in norm for each $n$-tuple $[X_1,\ldots, X_n] $ with $r(X_1,\ldots, X_n)<1$. Therefore, $$G:=\sum_{k=0}^\infty \sum_{|\alpha|=k} R_{\tilde \alpha}^*\otimes Z_\alpha$$ is a free holomorphic function on the open operatorial unit $n$-ball. Due to Theorem \[derivation\], $\frac{\partial^k G}{\partial Z_{i_1}\cdots \partial Z_{i_k}}$ is also a free holomorphic function. By Theorem \[abel\], $\frac{\partial^k G}{\partial Z_{i_1}\cdots \partial Z_{i_k}}(X_1,\ldots, X_n)$ is a bounded operator for any $n$-tuple $[X_1,\ldots, X_n]$ with spectral radius $r(X_1,\ldots, X_n)<1$. Now, notice that, for each $\alpha\in {{\mathbb F}}_n^+$, $i=1,\ldots, n$, and $ x,y\in {{\mathcal H}}$, we have $$\begin{split} \Bigl<\Bigl[\frac{\partial \left(C_T(R_1,\ldots, R_n)^*\right)}{\partial T_{i}}\Bigr]&(S_\alpha\otimes I_{{\mathcal H}}) (1\otimes x), 1\otimes y\Bigr>\\ &= \left< \left(\sum_{k=0}^\infty \sum_{|\beta|=k}R_\beta^*\otimes \frac{\partial T_{\tilde \beta}}{\partial T_i}\right)(S_\alpha\otimes I_{{\mathcal H}}) (1\otimes x), 1\otimes y\right>\\ &= \left<e_\alpha\otimes x, \sum_{k=0}^\infty \sum_{|\beta|=k} e_{\tilde \beta}\otimes \left(\frac{\partial T_{\tilde \beta}}{\partial T_i}\right)^*y\right>\\ &= \left< \frac{\partial T_\alpha}{\partial T_i}x,y\right> =\left<\left(\frac{\partial Z_\alpha}{\partial Z_i}\right)(T_1,\ldots, T_n) x,y\right>. \end{split}$$ Hence, we deduce relation for polynomials. Let $f=\sum_{k=1}^\infty \sum_{|\alpha|=k} a_\alpha Z_\alpha$ be in $H^\infty(B({{\mathcal X}})^n_1)$. Due to Theorem \[abel\], we have $$\left(\frac{\partial f_r}{\partial Z_i}\right)(T_1,\ldots, T_n) =\lim_{m\to\infty}\sum_{k=0}^m \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha \left(\frac{\partial Z_\alpha}{\partial Z_i}\right)(T_1,\ldots, T_n),$$ where the convergence is in the operator norm of $B({{\mathcal H}})$, and $$f_r(S_1,\ldots, S_n)=\lim_{m\to\infty}\sum_{k=0}^m \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha\in {{\mathcal A}}_n$$ where the convergence is in the operator norm of $B(F^2(H_n))$. Since holds for polynomials, the last two relations imply $$\begin{split} \Bigl<\left(\frac{\partial f_r}{\partial Z_{i}}\right)&(T_1,\ldots, T_n) x,y\Bigr>\\ &= \left<\left[\frac{\partial \left(C_T(R_1,\ldots, R_n)^*\right)}{\partial T_{i} }\right](f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x), 1\otimes y\right> \end{split}$$ for any $x,y\in {{\mathcal H}}$ and $0<r<1$. Using again Theorem \[abel\], we have $$\lim_{r\to 1} \left(\frac{\partial f_r}{\partial Z_{i}}\right)(T_1,\ldots, T_n)= \left(\frac{\partial f}{\partial Z_{i}}\right)(T_1,\ldots, T_n)$$ in the operator norm. Since $f(S_1,\ldots, S_n)\in F_n^\infty$ (see Theorem \[f-infty\]), as in the proof of Theorem \[an=cauch\], we deduce that $$\text{\rm SOT}-\lim_{r\to\infty} f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}}=f(S_1,\ldots, S_n)\otimes I_{{\mathcal H}}.$$ Passing to the limit, as $r\to\infty$, in the above equality, we deduce relation in the particular case when $k=1$. Repeating this argument, one can prove the general case when $\frac{\partial}{\partial T_i}$ is replaced by $\frac{\partial^k}{\partial T_{i_1}\cdots \partial T_{i_k}}$. Now, we prove the second part of the theorem. Notice that $$\begin{split} \left\|\frac{\partial G}{\partial Z_i}(X_1,\ldots, X_n)\right\| &\leq \sum_{k=0}^\infty\left\|\sum_{|\alpha|=k}R_{\tilde \alpha}\otimes \left(\frac{\partial X_\alpha}{\partial X_i}\right)^*\right\|\\ &\leq \sum_{k=0}^\infty\left\|\sum_{|\alpha|=k} \left(\frac{\partial X_\alpha}{\partial X_i}\right)\left(\frac{\partial X_\alpha}{\partial X_i}\right)^*\right\|^{1/2}. \end{split}$$ For each $\alpha\in {{\mathbb F}}_n^+$, $|\alpha|=k$, we can prove that $$\label{X_ga} \left(\frac{\partial X_\alpha}{\partial X_i}\right)\left(\frac{\partial X_\alpha}{\partial X_i}\right)^*\leq k^2\sideset{}{^\alpha_d}\sum_\gamma X_\gamma X_\gamma^*,$$ where the sum is taken over all distinct words $\gamma$ obtained by deleting each occurence of $g_i$ in $\alpha$. Indeed, notice first that $\frac{\partial X_\alpha}{\partial X_i}=\sideset{}{^\alpha}\sum\limits_\beta X_\beta$, where the sum is taken over all words $\beta$ obtained by deleting each occurence of $g_i$ in $\alpha$. Since the above some contains at most $k$ terms, one can show that $$\left(\frac{\partial X_\alpha}{\partial X_i}\right)\left(\frac{\partial X_\alpha}{\partial X_i}\right)^*\leq k\sideset{}{^\alpha}\sum_\beta X_\beta X_\beta^*.$$ Indeed, it enough to use the following result which is an easy consequence of the classical Cauchy inequality: if $A_1,\ldots, A_k\in B({{\mathcal H}})$, then $$\left(\sum_{i=1}^k A_i\right)\left(\sum_{i=1}^k A_i^*\right) \leq k\sum_{i=1}^k A_iA_i^*.$$ Now, the $X_\beta$’s in the above sum are not necessarily distinct but each of them can occur at most $k$ times. Consequently, $$\sideset{}{^\alpha}\sum\limits_\beta X_\beta X_\beta^*\leq k \sideset{}{^\alpha_d}\sum_\gamma X_\gamma X_\gamma^*.$$ Combining these inequalities, we deduce . (We remark that the inequality is sharp and the equality occurs, for example, when $\alpha=g_i^k$.) Therefore, we have $$\begin{split} \sum_{k=0}^\infty\left\|\sum_{|\alpha|=k}\left(\frac{\partial X_\alpha}{\partial X_i}\right)\left(\frac{\partial X_\alpha}{\partial X_i}\right)^*\right\|^{1/2} &\leq \sum_{k=0}^\infty\left\|\sum_{|\alpha|=k} k^2\sideset{}{^\alpha_d}\sum_\gamma X_\gamma X_\gamma^*\right\|^{1/2}. \end{split}$$ We remark that if $\beta\in {{\mathbb F}}_n^+$, $|\beta|=k-1$, then $X_\beta$ can come from free differentiation with respect to $X_i$ of the monomials $X_{\chi(g_i,m,\beta)}$, $m=0,1,\ldots, k-1$, where $\chi(g_i,m,\beta)$ is the insertion mapping of $g_i$ on the $m$ position of $\beta$ (see the proof of Theorem \[derivation\]). Consequently, we have $$\sum_{|\alpha|=k} \sideset{}{^\alpha_d}\sum_\gamma X_\gamma X_\gamma^*\leq k \sum_{|\beta|=k-1} X_\beta X_\beta^*.$$ Using the above inequalities, we obtain $$\left\|\frac{\partial G}{\partial Z_i}(X_1,\ldots, X_n)\right\|\leq \sum_{k=1}^\infty k^{3/2}\left\|\sum_{|\beta|=k-1}X_\beta X_\beta\right\|^{1/2}.$$ Hence, and due to relation , we deduce inequality . The proof is complete. We remark that inequalities of type can be obtained for $k$-order Housdorff derivations. On the other hand, a similar result to Corollary \[conv-u-w\*\] can be obtain for $k$-order Housdorff derivations, if one uses Theorem \[cauc-dif\]. In the last part of this section, we show that the noncommutative Cauchy transform commutes with certain classes of automorphisms. Let ${{\mathcal U}}(H_n)$ be the group of all unitaries on $H_n$ and let $U\in {{\mathcal U}}(H_n)$. If $U:=\left[\lambda_{ij}\right]_{i,j=1}^n$ and $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$, we define $$\beta_U(T_j):=\sum_{i=1}^n \lambda_{ij} T_i,\quad j=1,\ldots, n,$$ and the map $\beta_U: B({{\mathcal H}})^n\to B({{\mathcal H}})^n$ by setting $\beta_U(T):=[\beta_U(T_1),\ldots, \beta_U(T_n)]. $ \[auto\] If $U\in {{\mathcal U}}(H_n)$,  $U:=\left[\lambda_{ij}\right]_{i,j=1}^n$, then the map $\beta_U $ is an isometric automorphism of the open unit ball $[B({{\mathcal H}})^n]_1$ and also of the ball $$\{[T_1,\ldots, T_n]\in B({{\mathcal H}})^n:\ r(T_1,\ldots, T_n)<1\}.$$ Moreover, there is a unique completely isometric automorphism of the noncommutative disc algebra ${{\mathcal A}}_n$, denoted also by $\beta_U$, such that $$\beta_U(S_j):= \sum_{i=1}^n \lambda_{ij} S_i,\quad j=1,\ldots, n,$$ where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space. For each $j=1,\ldots, n$, we define the operators $${\bf U}_j:=\left[\begin{matrix} \lambda_{1j}I_{{\mathcal H}}\\ \vdots\\ \lambda_{nj}I_{{\mathcal H}}\end{matrix}\right]:{{\mathcal H}}\to {{\mathcal H}}^{(n)},$$ where ${{\mathcal H}}^{(n)}$ is the direct sum of $n$ copies of ${{\mathcal H}}$. Notice that $$\label{Cuntz} {\bf U}_i^*{\bf U}_j=\delta_{ij} I_{{{\mathcal H}}^{(n)}}\quad \text{ and } \quad \sum_{i=1}^n{\bf U}_i{\bf U}_i^*=I_{{{\mathcal H}}^{(n)}}.$$ We have $\beta_U(T)=[B_1,\ldots,B_n]$, where $B_i:=T{\bf U}_i$,  $i=1,\ldots, n$ and $T:=[T_1,\ldots, T_n]$. Now, it is clear that $\sum_{i=1}^n B_iB_i^*=\sum_{i=1}^n T_iT_i$. If $A\in B({{\mathcal H}})$ then $${\bf U}_iA=\text{diag}_n (A) {\bf U}_i,\qquad i=1,\ldots, n,$$ where $\text{diag}_n (A)$ is the $n\times n$ block diagonal operator matrix having $A$ on the diagonal and $0$ otherwise. Using this relation and , we deduce that $$\begin{split} \sum_{|\alpha|=2} B_\alpha B_\alpha^*&=\sum_{i=1}^n B_i\left(\sum_{|\alpha|=1} B_\alpha B_\alpha^*\right) B_i\\ &= T\left[ \sum_{i=1}^n {\bf U}_i(TT^*){\bf U}_i^*\right] T^*\\ &= T\text{diag}_n (TT^*) \left( \sum_{i=1}^n{\bf U}_i{\bf U}_i^* \right) T^*\\ &=T\text{diag}_n (TT^*)T=\sum_{|\alpha|=2} T_\alpha T_\alpha^*. \end{split}$$ By induction over $k$, one can similarly prove that $$\label{B-T} \sum_{|\alpha|=k} B_\alpha B_\alpha^*=\sum_{|\alpha|=k} T_\alpha T_\alpha^*, \quad k=1,2,\ldots.$$ Consequently, we have $$\|\beta_U(T)\|=\|T\|\quad \text{ and }\quad r(\beta_U(T))=r(T).$$ Hence, and since $\beta_U(T)=T{\bf U}$, where ${\bf U}:=[{\bf U}_1,\ldots, {\bf U}_n]$ is a unitary operator, we deduce that the map $\beta_U:[B({{\mathcal H}})^n]_1\to [B({{\mathcal H}})^n]_1$ is an isometric authomorphism of the open unit ball of $B({{\mathcal H}})^n$ and $$\beta_U^{-1}(Y)=Y{\bf U}^*,\quad Y\in [B({{\mathcal H}})^n]_1.$$ Moreover, $\beta_U$ is an isometric automorphism of the operatorial ball $$\{[T_1,\ldots, T_n]\in B({{\mathcal H}})^n:\ r(T_1,\ldots, T_n)<1\}.$$ Now, let us prove the second part of the theorem. Using the same notation for the unitary operator ${\bf U}$, when ${{\mathcal H}}:=F^2(H_n)$, we deduce that $[\beta_U(S_1),\ldots, \beta_U(S_n)]=S{\bf U}$, where $S:=[S_1,\ldots, S_n]$. Setting $V_i:=\beta_U(S_i)$, $i=1,\ldots, n$, one can easily see that $V_1,\ldots, V_n$ are isometries with orthogonal ranges. For any polynomial $p(S_1,\ldots, S_n)$ in the noncommutative disc algebra ${{\mathcal A}}_n$, we have $\beta_U(p(S_1,\ldots, S_n))=p(V_1,\ldots, V_n)$. According to [@Po-disc], we have $$\|[p_{ij}(S_1,\ldots, S_n)]_m\|=\|[p_{ij}(V_1,\ldots, V_n)]_m\|.$$ Since ${{\mathcal A}}_n$ is the norm closure of all polynomials in $S_1,\ldots, S_n$ and the identity, $\beta_U$ can be uniquely extended to a completely isometric homomorphism from ${{\mathcal A}}_n$ to ${{\mathcal A}}_n$. Define the $n$-tuple $[X_1,\ldots, X_n]:=[S_1,\ldots,S_n]{\bf U}^*$ and notice that each entry $X_i$ is a homogenous polynomial of degree one in $S_1,\ldots, S_n$. Since $$[\beta_U(X_1),\ldots, \beta_U(X_n)]= [X_1,\ldots, X_n] {\bf U}= [S_1,\ldots,S_n],$$ we deduce that $\beta_U(X_i)=S_i$,  $i=1,\ldots, n$, and consequently, $\beta_U(X_\alpha)=S_\alpha$,  $\alpha\in {{\mathbb F}}_n^+$. Hence, the range of $\beta_U:{{\mathcal A}}_n\to {{\mathcal A}}_n$ contains all polynomials in ${{\mathcal A}}_n$. Using again the norm density of polynomials in ${{\mathcal A}}_n$, we conclude that $\beta_U$ is a completely isometric automorphism of ${{\mathcal A}}_n$. In what follows we show that the noncommutative Cauchy transform commutes with the action of the unitary group ${{\mathcal U}}(H_n)$. \[Cau-inv\] Let $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$ and $U\in {{\mathcal U}}(H_n)$. Then $${{\mathcal C}}_T(\beta_U(f))={{\mathcal C}}_{\beta_U(T)}(f), \quad f\in {{\mathcal A}}_n,$$ where $\beta_U$ is the canonical automorphism generated by $U$. Remember that ${{\mathcal A}}_n$ is the norm closure of the polynomials in $S_1,\ldots, S_n$ and the identity. Due to the continuity of the noncommutative Cauchy transform in the operator norm, it is enough to prove the above relation for $f:=S_\alpha$, $\alpha\in {{\mathbb F}}_n^+$. By Theorem \[an=cauch\], we have $$\begin{split} \left<{{\mathcal C}}_T(\beta_U(S_\alpha))x,y\right>&=\left<C_T(R_1,\ldots, R_n)^*(\beta_U(S_\alpha)\otimes I_{{\mathcal H}})(1\otimes x),1\otimes y)\right>\\ &= \left<B_\alpha x,y\right> \end{split}$$ for any $x,y\in {{\mathcal H}}$, where $[B_1,\ldots, B_n]:=\beta_U(T)$. On the other hand, due to Theorem \[auto\], we have $r(\beta_U(T))<1$. Applying again Theorem \[an=cauch\], we obtain $$\begin{split} \left<{{\mathcal C}}_{\beta_U(T)}(S_\alpha) x,y\right>&=\left<C_{\beta_U(T)}(R_1,\ldots, R_n)^*(S_\alpha\otimes I_{{\mathcal H}})(1\otimes x),1\otimes y)\right>\\ &= \left<B_\alpha x,y\right>. \end{split}$$ Hence, ${{\mathcal C}}_T(\beta_U(S_\alpha))={{\mathcal C}}_{\beta_U(T)}(S_\alpha)$, and the result follows. The continuity and the uniqueness of the free analytic functional calculus for $n$-tuples of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$ will be proved in the next section. Weierstrass and Montel theorems for free holomorphic functions {#Weierstrass and Montel} ============================================================== In this section, we obtain Weierstrass and Montel type theorems for the algebra of free holomorphic functions with scalar coefficients on the open operatorial unit $n$-ball. This enables us to introduce a metric on $Hol(B({{\mathcal X}})^n_1)$ with respect to which it becomes a complete metric space, and the Hausdorff derivations are continuous. In the end of this section, we prove the continuity and uniqueness of the free functional calculus. Connections with the $F_n^\infty$-functional calculus for row contractions [@Po-funct] and, in the commutative case, with Taylor’s functional calculus [@T2] are also discussed. We say that a sequence $\{F_m\}_{m=1}^\infty\subset Hol(B({{\mathcal X}})^n_1)$ of free holomorphic functions converges uniformly on the closed operatorial $n$-ball of radius $r\in [0,1)$ if it converges uniformly on the closed ball $$[B({{\mathcal H}})^n]_{ r}^{-}:=\{ [X_1,\ldots, X_n]\in B({{\mathcal H}})^n:\ \|X_1X_1^*+\cdots+X_nX_n^*\|\leq r\},$$ where ${{\mathcal H}}$ is an infinite dimensional Hilbert space. According to the maximum principle of Corollary \[max-mod2\], this is equivalent to the fact that the sequence $\{F_m(rS_1,\ldots, rS_n)\}_{m=1}^\infty$ is convergent in the operator norm topology of $B(F^2(H_n))$. The first result of this section is a multivariable operatorial version of Weierstrass theorem ([@Co]). \[Weierstrass\] Let $\{F_m\}_{m=1}^\infty\subset Hol(B({{\mathcal X}})^n_1)$ be a sequence of free holomorphic functions which is uniformly convergent on any closed operatorial $n$-ball of radius $r\in [0,1)$. Then there is a free holomorphic function $F\in Hol(B({{\mathcal X}})^n_1)$ such that $F_m$ converges to $F$ on any closed operatorial $n$-ball of radius $r\in [0,1)$. Moreover, given $i_1,\ldots, i_k\in \{1,\ldots, n\}$, the sequence $\left\{\frac{\partial^k F_m} {\partial Z_{i_1}\cdots \partial Z_{i_k}}\right\}_{m=1}^\infty $ is uniformly convergent to $\frac{\partial^k F} {\partial Z_{i_1}\cdots \partial Z_{i_k}}$ on any closed operatorial $n$-ball of radius $r\in [0,1)$, where $\frac{\partial^k} {\partial Z_{i_1}\cdots \partial Z_{i_k}}$ is the $k$-order Hausdorff derivation. Let $F_m:=\sum\limits_{k=0}^\infty \sum\limits_{|\alpha|=k} a_{\alpha}^{(m)} Z_\alpha$ and fix $r\in (0,1)$. Then, due to Theorem \[caract-shifts\], $$F_m(rS_1,\ldots, rS_n)=\sum\limits_{k=0}^\infty\sum\limits_{|\alpha|=k} r^{|\alpha|} a_{\alpha}^{(m)} S_\alpha$$ is in the noncommutative disc algebra ${{\mathcal A}}_n$. Since $\{F_m\}_{m=1}^\infty$ is uniformly convergent on the closed operatorial $n$-ball of radius $r$, the sequence $\{F_m(rS_1,\ldots, rS_n)\}_{m=1}^\infty$ is convergent in the operator norm of $B(F^2(H_n))$. On the other hand, since the noncommutative disc algebra ${{\mathcal A}}_n$ is closed in the operator norm, there exists $g\in {{\mathcal A}}_n$ such that $$\label{Fm-to} F_m(rS_1,\ldots, rS_n)\to L_g, \quad \text{ as } \ m\to\infty.$$ Assume $g=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} b_\alpha(r) e_\alpha$, and notice also that $$b_\alpha(r)=\left< S_\alpha^* L_g(1),1\right>, \quad \alpha\in {{\mathbb F}}_n^+.$$ If $\lambda_{(\beta)}\in {{\mathbb C}}$ for $\beta\in {{\mathbb F}}_n^+$ with $|\beta|=k$, we have $$\left|\left<\sum_{|\beta|=k} \lambda_{(\beta)} S_\beta^*(F_m(rS_1,\ldots, rS_n)-L_g)1,1\right>\right| \leq \|F_m(rS_1,\ldots, rS_n)-L_g\| \left\|\sum_{|\beta|=k} \lambda_{(\beta)} S_\beta^*\right\|.$$ Since $S_1,\ldots, S_n$ are isometries with orthogonal ranges, we deduce that $$\left|\sum_{|\beta|=k}(r^ka_\beta^{(m)}-b_\beta(r))\lambda_{(\beta)}\right|\leq \|F_m(rS_1,\ldots, rS_n)-L_g\| \left(\sum_{|\beta|=k} |\lambda_{(\beta)}|^2\right)^{1/2}.$$ for any $\lambda_{(\beta)}\in {{\mathbb C}}$ with $|\beta|=k$. Consequently, we have $$\left(\sum_{|\beta|=k}|r^ka_\beta^{(m)}-b_\beta(r)|^2\right)^{1/2}\leq \|F_m(rS_1,\ldots, rS_n)-L_g\|$$ for any $k=0,1,\ldots$. Since $\|F_m(rS_1,\ldots, rS_n)-L_g\| \to 0$, as $m\to\infty$, we deduce that $r^ka_\beta^{(m)}\to b_\beta(r)$, as $m\to\infty$, for any $|\beta|=k$ and $k=0,1,\ldots$. Hence, $a_\beta:=\lim\limits_{m\to\infty}a_\beta^{(m)}$ exists and $b_\beta(r)=r^k a_\beta$ for any $\beta\in {{\mathbb F}}_n^+$ with $|\beta|=k$ and $k=0,1,\ldots$. Consider the formal power series $F:= \sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha Z_\alpha$. We show now that $F$ is a free holomorphic function on the open operatorial unit $n$-ball. Due to the above calculations, we have $$r^k \left|\left( \sum_{|\beta|=k}|a_\beta^{(m)}|^2\right)^{1/2}- \left(\sum_{|\beta|=k}|a_\beta|^2\right)^{1/2}\right| \leq \|F_m(rS_1,\ldots, rS_n)-L_g\|.$$ Therefore, $$\label{conv-coef} \sum_{|\beta|=k}|a_\beta^{(m)}|^2 \to \sum_{|\beta|=k}|a_\beta|^2,\quad \text{ as } \ m\to\infty,$$ uniformly with respect to $k=0,1,\ldots$. Let us show that the radius of convergence of $F$ is $\geq 1$. To this end, assume that $\gamma>1$ and $$\limsup_{k\to\infty} \left(\sum_{|\beta|=k}|a_\beta|^2\right)^{1/2k}>\gamma.$$ Then there is $k\in {{\mathbb N}}$ as large as we want such that $$\label{sup-ga} \left(\sum_{|\beta|=k}|a_\beta|^2\right)^{1/2}>\gamma^k.$$ Choose $\lambda$ such that $1<\lambda< \gamma$ and let $\epsilon>0$ be such that $\epsilon<\gamma-\lambda$. Notice that $\epsilon<\gamma^k-\lambda^k$ for any $k=1,2,\ldots$. Now, due to relation , there exists $N_\epsilon\in {{\mathbb N}}$ such that $$\left|\left(\sum_{|\beta|=k}|a_\beta^{(m)}|^2\right)^{1/2} - \left(\sum_{|\beta|=k}|a_\beta|^2\right)^{1/2}\right|< \epsilon$$ for any $m>N_\epsilon$ and any $k=0,1,\ldots$. Hence, and using inequality , we deduce that $$\left(\sum_{|\beta|=k}|a_\beta^{(m)}|^2\right)^{1/2}\geq \gamma^k-\epsilon>\lambda^k$$ for any $m>N_\epsilon$ and some $k$ as large as we want. Consequently, we have $$\limsup_{k\to \infty} \left(\sum_{|\beta|=k}|a_\beta^{(m)}|^2\right)^{1/2k}\geq \lambda>1$$ for $m\geq N_\epsilon$. Due to Theorem \[Abel\], this shows that the radius of convergence of $F_m$ is $<1$, which contradicts the fact that $F_m$ is a free holomorphic function with radius of convergence $\geq 1$. Therefore, $$\limsup_{k\to\infty}\left(\sum_{|\beta|=k}|a_\beta|^2\right)^{1/2k}\leq 1$$ and, consequently, Theorem \[Abel\] shows that $F$ is a free holomorphic function on the open operatorial unit ball. The same theorem implies that $F(rS_1,\ldots, rS_n)=\sum\limits_{k=0}^\infty \sum\limits_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha$ is convergent in norm. Since $L_g$ and $F(rS_1,\ldots, rS_n)$ have the same Fourier coefficients, we must have $L_g=F(rS_1,\ldots, rS_n)$. Due to relation , we have $$\|F_m(rS_1,\ldots, rS_n)-F(rS_1,\ldots, rS_n)\|\to 0, \quad \text{ as }\ m\to \infty.$$ If $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{1}$ and $\|[X_1,\ldots, X_n]\|=r<1$, the noncommutative von Neumann inequality implies $$\|F_m(X_1,\ldots, X_n)-F(X_1,\ldots, X_n)\|\leq \|F_m(rS_1,\ldots, rS_n)-F(rS_1,\ldots, rS_n)\|.$$ Taking $m\to\infty$, we deduce that $F_m$ converges to $F$ on any closed operatorial $n$-ball of radius $r\in [0,1)$. Now, we show that for each $\gamma\in (0,1)$ $$\label{co-de} \left(\frac{\partial F_m}{\partial Z_i}\right)(\gamma S_1,\ldots, \gamma S_n)\to \left(\frac{\partial F}{\partial Z_i}\right)(\gamma S_1,\ldots, \gamma S_n)$$ in the operator norm, as $m\to \infty$. Let $r,r'\in (0,1)$ such that $\gamma=r r'$. Since $(F_m)_r$ and $ F_r\in {{\mathcal A}}_n$ are in the noncommutative disc algebra ${{\mathcal A}}_n$, we can apply Theorem \[cauc-dif\] (see inequality ) and obtain $$\left\|\left(\frac{\partial ((F_m)_r-F_r)}{\partial Z_i}\right)(r' S_1,\ldots, r' S_n)\right\| \leq M\|(F_m)_r-F_r\|_\infty,$$ where $M$ is an appropriate constant which does not depend on $m$. Since $\|(F_m)_r-F_r\|_\infty\to 0$ as $m\to\infty$ and $$\left(\frac{\partial ((F_m)_r-F_r)}{\partial Z_i}\right)(r' S_1,\ldots, r' S_n)=r\left(\frac{\partial (F_m-F)}{\partial Z_i}\right)(\gamma S_1,\ldots, \gamma S_n),$$ we deduce relation . Using the result for $\frac{\partial}{\partial Z_i}$, one can obtain the general case for $k$-order Housdorff partial derivations. The proof is complete. We say that a set ${{\mathcal F}}\subset Hol(B({{\mathcal X}})^n_1)$ is normal if each sequence in ${{\mathcal F}}$ has a subsequence which converges to a function in $Hol(B({{\mathcal X}})^n_1)$ uniformly on any closed operatorial ball of radius $r\in [0,1)$. The set ${{\mathcal F}}$ is called locally bounded if, for any $r\in[0,1)$, there exists $M>0$ such that $\|f(X_1,\ldots, X_n)\|\leq M$ for any $f\in {{\mathcal F}}$ and $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_r$, where ${{\mathcal H}}$ is an infinite dimensional Hilbert space. We can prove now the following noncommutative version of Montel theorem (see [@Co]). \[Montel\] Let ${{\mathcal F}}\subset Hol(B({{\mathcal X}})^n_1)$ be a family of free holomorphic functions. Then the following statements are equivalent: 1. $ \sup_{f\in {{\mathcal F}}}\|f(rS_1,\ldots, rS_n)\|<\infty $ for each $r\in [0,1)$. 2. ${{\mathcal F}}$ is a normal set. 3. ${{\mathcal F}}$ is locally bounded. Assume that condition (i) holds. For each $f\in {{\mathcal F}}$, let $\{a_\alpha(f)\}_{\alpha\in {{\mathbb F}}_n^+}$ be the sequence of coefficients. Due to (i), for each $r\in [0,1)$, there exists $M_r>0$ such that $$\label{FrMr} \|f(rS_1,\ldots, rS_n)\|\leq M_r \quad \text{ for any } \ f\in {{\mathcal F}}. $$ By the Cauchy type estimate of Theorem \[Cauchy-est\], if $r\in(0,1)$, then $$\label{Cau-est} \left( \sum_{|\alpha|=k} |a_\alpha(f)|^2\right)^{1/2}\leq \frac{1}{r^k} M_r\quad \text{ for any } \ f\in {{\mathcal F}}, k=0,1,\ldots.$$ Let $\{F_m\}_{m=1}^\infty$ be a sequence of elements in ${{\mathcal F}}$. Then, relation implies $$|a_0(F_m)|\leq M_0\quad \text{ for any } \ m=1,2,\ldots.$$ Due to the classical Bolzano-Weierstrass theorem for bounded sequences of complex numbers, there is a subsequence $\{F_{m_k^{(0)}}\}_{k=1}^\infty$ of $ \{F_m\}_{m=1}^\infty$ such that the scalar sequence $\{a_0(F_{m_k^{(0)}})\}_{k=1}^\infty$ is convergent in ${{\mathbb C}}$, as $k\to\infty$. Inductively, using relation , we find, for each $\alpha\in {{\mathbb F}}_n^+$, $|\alpha|\geq 1$, a subsequence $\{F_{m_k^{(\alpha)}}\}_{k=1}^\infty$ of $\{F_{m_k^{(\beta)}}\}_{k=1}^\infty$, where $\alpha$ is the succesor of $\beta$ in the lexicographic order of ${{\mathbb F}}_n^+$, such that the sequence $\{a_\alpha(F_{m_k^{(\alpha)}})\}_{k=1}^\infty$ is convergent in ${{\mathbb C}}$, as $k\to\infty$. Using the diagonal process, we find a subsequence $\{F_{p_k}\}_{k=1}^\infty$ of $\{F_m\}_{m=1}^\infty$ such that $\{a_\alpha(F_{p_k})\}_{k=1}^\infty$ converges in ${{\mathbb C}}$ as $k\to\infty$, for any $\alpha\in {{\mathbb F}}_n^+$. Now let us prove that, if $\gamma>1$, then $\{F_{p_k}(\frac{r}{\gamma}S_1,\ldots, \frac{r}{\gamma}S_n)\}_{k=1}^\infty $ converges in the norm topology of $B(F^2(H_n))$. Indeed, if $N\in {{\mathbb N}}$, then relation implies $$\begin{split} &\Bigl\|F_{p_k}\Bigl(\frac{r}{\gamma}S_1,\ldots, \frac{r}{\gamma}S_n\Bigr)-F_{p_s}\Bigl(\frac{r}{\gamma}S_1,\ldots, \frac{r}{\gamma}S_n\Bigr)\Bigr\|\\ &\leq \sum_{j=1}^N \frac{r^j}{\gamma^j}\left(\sum_{|\alpha|=j} |a_\alpha(F_{p_k})-a_\alpha(F_{p_s})|^2\right)^{1/2} + \sum_{j=N+1} \frac{r^j}{\gamma^j}\left(\sum_{|\alpha|=j} |a_\alpha(F_{p_k})-a_\alpha(F_{p_s})|^2\right)^{1/2}\\ &\leq \sum_{j=1}^N \frac{r^j}{\gamma^j}\left(\sum_{|\alpha|=j} |a_\alpha(F_{p_k})-a_\alpha(F_{p_s})|^2\right)^{1/2} +\sum_{j=N+1}^\infty \frac{r^j}{\gamma^j} \frac{2M_r}{r^j}\\ &\leq \sum_{j=1}^N \frac{r^j}{\gamma^j}\left(\sum_{|\alpha|=j} |a_\alpha(F_{p_k})-a_\alpha(F_{p_s})|^2\right)^{1/2} +\frac{2M_r}{\gamma^N(\gamma-1)}. \end{split}$$ Given $\epsilon>0$, we choose $N\in {{\mathbb N}}$ such that $\frac{2M_r}{\gamma^N}<\frac{\epsilon}{2}$. On the other hand, since $\{a_\alpha(F_{p_k})\}_{k=1}^\infty$ is a Cauchy sequence in ${{\mathbb C}}$, there is $k_0\in {{\mathbb N}}$ such that $$\sum_{j=1}^N \frac{r^j}{\gamma^j}\left(\sum_{|\alpha|=j} |a_\alpha(F_{p_k})-a_\alpha(F_{p_s})|^2\right)^{1/2}<\frac{\epsilon}{2}\quad \text{ for any } \ k,s\geq k_0.$$ Summing up the above results, we deduce that $$\Bigl\|F_{p_k}\Bigl(\frac{r}{\gamma}S_1,\ldots, \frac{r}{\gamma}S_n\Bigr)-F_{p_s}\Bigl(\frac{r}{\gamma}S_1,\ldots, \frac{r}{\gamma}S_n\Bigr)\Bigr\| <\epsilon \quad \text{ for any } \ k,s\geq k_0.$$ This proves that the the sequence $\{F_{p_k}(\frac{r}{\gamma}S_1,\ldots, \frac{r}{\gamma}S_n)\}_{k=1}^\infty $ converges in the norm topology of $B(F^2(H_n))$, for any $r\in[0,1)$ and $\gamma>1$. Since the set $A:=\{\frac{r}{\gamma}:\ 0\leq r<1, \gamma>1\}$ is equal to $[0,1)$, one can choose an increasing sequence $\{t_q\}_{q=1}^\infty$ such that $t_q\in A$ and $t_q\to 1$ as $q\to\infty$. Now, if $\{F_m\}_{m=1}^\infty\subset {{\mathcal F}}$, then, using the above result, there is a subsequence $\{F_{n_k^{(1)}}\}_{k=1}^\infty$ of $\{F_m\}_{m=1}^\infty$ such that $\{ F_{n_k^{(1)}}(t_1S_1,\ldots, t_1S_n)\}$ is convergent in the norm topology of $B(F^2(H_n))$, as $k\to\infty$. Inductively, for each $q=2,3,\ldots$, we find a subsequence $\{F_{n_k^{(q)}}\}_{k=1}^\infty$ of $\{F_{n_k^{(q-1)}}\}_{k=1}^\infty$ such that $\{ F_{n_k^{(q)}}(t_qS_1,\ldots, t_qS_n)\}$ is convergent in the norm topology of $B(F^2(H_n))$, as $k\to\infty$. Using again the diagonal process, we find a subsequence $\{F_{m_k}\}_{k=1}^\infty$ of $\{F_m\}_{m=1}^\infty$ such that, for each $r\in [0,1)$, the subsequence $\{F_{m_k}(rS_1,\ldots, rS_n)\}$ is convergent in the norm topology of $B(F^2(H_n))$, as $k\to\infty$. Applying Theorem \[Weierstrass\], we deduce that ${{\mathcal F}}$ is a normal set. Therefore, the implication $(i)\implies (ii)$ is true. To prove the converse, assume that there is $r_0\in (0,1)$ such that $$\sup_{f\in {{\mathcal F}}}\|f(r_0S_1,\ldots, r_0S_n)\|=\infty.$$ Let $\{f_m\}_{m=1}^\infty\subset {{\mathcal F}}$ be such that $$\label{r_0} \|f_m(r_0S_1,\ldots, r_0S_n)\|\to\infty \quad \text{ as } \ m\to \infty.$$ Since (ii) holds, there exists a subsequence $\{f_{m_k}\}_{k=1}^\infty$ such that $\{f_{m_k}(rS_1,\ldots, r S_n)\}_{k=1}^\infty$ is convergent for any $r\in [0,1)$. This contradicts relation . The equivalence (i)$\Longleftrightarrow$(ii) follows from Corollary \[max-mod2\]. The proof is complete. Now, we can obtain the following Vitali type result in our setting. \[Vitali\] Let $\{F_m\}_{m=1}^\infty$ be a sequence of free holomorphic functions on $[B({{\mathcal H}})^n]_{1}$ with scalar coefficients such that, for each $r\in [0,1)$, $$\sup_{m} \|F_m(rS_1,\ldots, rS_n)\|<\infty.$$ If there exists $0<\gamma<1$ such that $F_m(\gamma S_1,\ldots, \gamma S_n)$ converges in norm as $m\to \infty$, then $F_m$ converges uniformly on $[B({{\mathcal H}})^n]_{ r}^{-}$ for any $r\in [0, 1)$. Suppose that $\{F_m\}_{m=1}^\infty$ does not converge uniformly on $[B({{\mathcal H}})^n]_{r_0}^-$ for some $r_0\in (0,1)$. Then there exist $\delta>0$, subsequences $\{F_{m_k}\}_{k=1}^\infty$ and $\{F_{n_k}\}_{k=1}^\infty$ of $\{F_{m}\}_{m=1}^\infty$, and $n$-tuples of operators $[X_1^{(k)},\ldots, X_n^{(k)}]\in [B({{\mathcal H}})^n]_{r_0}^-$ such that $$\label{Fnkmk} \|F_{n_k}(X_1^{(k)},\ldots, X_n^{(k)})- F_{m_k}(X_1^{(k)},\ldots, X_n^{(k)}\|\geq \delta$$ for any $k=1,2,\ldots$. By Theorem \[Montel\], we find a subsequence $\{k_p\}_{p=1}^\infty$ of $\{k\}_{k=1}^\infty$ such that $\{F_{m_{k_p}}\}_{k=1}^\infty$ and $\{F_{n_{k_p}}\}_{k=1}^\infty$ are uniformly convergent to $f$ and $g$, respectively, on any closed operatorial $n$-ball of radius $r\in [0,1)$. Using Theorem \[Weierstrass\], we deduce that $f,g$ are free holomorphic functions on $[B({{\mathcal H}})^n]_1$ Now, the inequality and the noncommutative von Neumann inequality imply $$\|F_{n_{k_p}}(r_0 S_1,\ldots, r_0S_n)-F_{m_{k_p}}(r_0 S_1,\ldots, r_0S_n)\|\geq \delta>0$$ for any $k=1,2,\ldots$. Consequently, we have $$\label{fr0gr0} \|f(r_0 S_1,\ldots, r_0S_n)-g(r_0 S_1,\ldots, r_0S_n)\|\geq \delta>0.$$ On the other hand, since $\{F_m(\gamma S_1,\ldots, \gamma S_n)\}_{m=1}^\infty$ converges in norm as $m\to\infty$, we must have $$f(\gamma S_1,\ldots, \gamma S_n)=g(\gamma S_1,\ldots, \gamma S_n).$$ Since $0<\gamma<1$ and $f,g$ are free holomorphic functions on $[B({{\mathcal H}})^n]_1$, we deduce that $f=g$, which contradicts inequality . The proof is complete. Let ${{\mathcal H}}$ be a Hilbert space and let $C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$ be the vector space of all continuous functions from the open operatorial unit ball $[B({{\mathcal H}})^n]_1$ to $B({{\mathcal H}})$. If $f,g\in C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$ and $0<r<1$, we define $$\rho_r(f,g):=\sup_{[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_r^-} \|f(X_1,\ldots, X_n)-g(X_1,\ldots, X_n)\|.$$ Let $0<r_m<1$ be such that $\{r_m\}_{m=1}^\infty$ is an increasing sequence convergent to $1$. For any $f,g\in C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$, we define $$\rho (f,g):=\sum_{m=1}^\infty \left(\frac{1}{2}\right)^m \frac{\rho_{r_m}(f,g)}{1+\rho_{r_m}(f,g)}.$$ Based on standard arguments, one can prove that $\rho$ is a metric on $C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$. Following the corresponding result (see [@Co]) for the set of all continuous functions from a set $G\subset {{\mathbb C}}$ to a metric space $\Omega$, one can easily obtain the following operator version. We leave the proof to the reader. \[Conway\] If $\epsilon>0$, then there exists $\delta>0$ and $m\in {{\mathbb N}}$ such that for any $f,g\in C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$ $$\sup_{[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{r_m}^-} \|f(X_1,\ldots, X_n)-g(X_1,\ldots, X_n)\|<\delta\implies \rho(f,g)<\epsilon.$$ Conversely, if $\delta>0$ and $m\in {{\mathbb N}}$ are fixed, then there is $\epsilon>0$ such that for any $f,g\in C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$ $$\rho(f,g)<\epsilon \implies \sup_{[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{r_m}^-} \|f(X_1,\ldots, X_n)-g(X_1,\ldots, X_n)\|<\delta.$$ An immediate consequence of Lemma \[Conway\] is the following: if $\{f_m\}_{k=1}^\infty$ and $f$ are in $C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$, then $f_k$ is convergent to $f$ in the metric $\rho$ if and only if $f_m\to f$ uniformly on any closed ball $[B({{\mathcal H}})^n]_{r_m}^-$,  $m=1,2,\ldots$. This result is needed to prove the following. \[cont-comp\] $\left( C(B({{\mathcal H}})^n_1, B({{\mathcal H}})), \rho\right)$ is a complete metric space. Suppose that $\{f_k\}_{k=1}^\infty$ is a Cauchy sequence in $\left(C(B({{\mathcal H}})^n_1, B({{\mathcal H}})),\rho\right)$. Due to Lemma \[Conway\], the sequence $\left\{f_k|_{[B({{\mathcal H}})^n]_{r}^-}\right\}_{k=1}^\infty$ is Cauchy in $C([B({{\mathcal H}})^n]_r^-, B({{\mathcal H}}))$. Consequently, for any $\epsilon>0$, there exists $N\in {{\mathbb N}}$, such that $$\label{unif} \sup_{[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{r}^-} \|f_m(X_1,\ldots, X_n)-f_k(X_1,\ldots, X_n)\|<\epsilon\quad \text{ for any } k,m\geq N.$$ In particular, $\{f_k(X_1,\ldots, X_n)\}_{k=1}^\infty$ is a Cauchy sequence in the operator norm of $B({{\mathcal H}})$. Therefore, there is an operator $f(X_1,\ldots, X_n)\in B({{\mathcal H}})$ such that $$\label{li} f(X_1,\ldots, X_n)=\lim_{k\to\infty} f_k(X_1,\ldots, X_n)$$ in the operator norm. This gives rise to a function $f:[B({{\mathcal H}})^n]_1\to B({{\mathcal H}})$. We need to show that $\rho(f_k,f)\to 0$, as $k\to\infty$, and that $f$ is continuous. If $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{r}^-$, then, due to relations and , there exists $m\geq N$ such that $$\|f(X_1,\ldots, X_n)-f_m(X_1,\ldots, X_n)\|<\epsilon \quad \text{ and } \quad \|f(X_1,\ldots, X_n)-f_k(X_1,\ldots, X_n)\|<\epsilon$$ for any $k\geq N$. Since $N$ does not depend on $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{r}^-$, we deduce that $\{f_k\}_{k=1}^\infty$ converges to $f$ uniformly on any closed ball $[B({{\mathcal H}})^n]_{r}^-$. Due to Lemma \[Conway\], this shows that $\rho(f_k,f)\to 0$, as $k\to\infty$. The continuity of $f$ can be proved using standard arguments in the theory of metric spaces. We leave it to the reader. Let ${{\mathcal H}}$ be an infinite dimensional Hilbert space and denote by $Hol(B({{\mathcal H}})^n_1)$ the algebra of free holomorphic functions on $[B({{\mathcal H}})^n]_1$. \[complete-metric\] $\left(Hol(B({{\mathcal H}})^n_1), \rho\right)$ is a complete metric space and the Hausdorff derivations $$\frac {\partial}{\partial Z_i}: \left(Hol(B({{\mathcal H}})^n_1), \rho\right) \to \left(Hol(B({{\mathcal H}})^n_1), \rho\right),\quad i=1,\ldots, n,$$ are continuous. First, note that Theorem \[continuous\] implies that $Hol(B({{\mathcal H}})^n_1)\subset C(B({{\mathcal H}})^n_1, B({{\mathcal H}})$. Due to Theorem \[cont-comp\], it is enough to show that $\left({{\mathcal H}}ol(B({{\mathcal H}})^n_1), \rho\right)$ is closed in $\left( C(B({{\mathcal H}})^n_1, B({{\mathcal H}})), \rho\right)$. Let $\{f_m\}_{m=1}^\infty\subset Hol(B({{\mathcal H}})^n_1)$ and $f\in C(B({{\mathcal H}})^n_1, B({{\mathcal H}})$ be such that $\rho(f_m,f)\to 0$, as $m\to\infty$. Due to Lemma \[Conway\], $f_m\to f$ uniformly on any closed ball $[B({{\mathcal H}})^n]_{r_m}^-$,  $m=1,2,\ldots$. Applying now Theorem \[Weierstrass\], we deduce that $f\in Hol(B({{\mathcal H}})^n_1)$ and that $$\frac{\partial f_m}{\partial Z_i}\to \frac{\partial f}{\partial Z_i}$$ uniformly on any closed ball $[B({{\mathcal H}})^n]_{r_m}^-$ and, therefore, in the metric $\rho$. This completes the proof of the theorem. Now, Theorem \[Montel\] implies the following compactness criterion for subsets of $Hol(B({{\mathcal H}})^n_1)$. A subset ${{\mathcal F}}$ of $(Hol(B({{\mathcal H}})^n_1), \rho)$ is compact if and only if it is closed and locally bounded. We return now to the setting of Section \[free analytic\], where we showed that if $f=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha Z_\alpha$ is a free holomorphic function on the open operatorial unit $n$-ball and $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is any $n$-tuple of operators with $r(T_1,\ldots, T_n)<1$, then we can define the bounded linear operator $$f(T_1,\ldots, T_n):=\sum_{k=0}^\infty\sum_{|\alpha|=k} a_\alpha T_\alpha,$$ where the series converges in norm. This provides a [*free analytic functional calculus*]{}, which now turns out to be continuous and unique. If $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is any $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$ then the mapping $\Phi_T: Hol(B({{\mathcal X}})^n_1) \to B({{\mathcal H}})$ defined by $$\Phi_T(f):=f(T_1,\ldots, T_n)$$ is a continuous unital algebra homomorphism. Moreover, the free analytic functional calculus is uniquely determined by the mapping $$Z_i\mapsto T_i,\qquad i=1,\ldots,n.$$ Due to Theorem \[abel\] and Theorem \[operations\], we deduce that $\Phi_T$ is a well-defined unital algebra homomorphism. To prove the continuity of $\Phi_T$, let $f_m$ and $f$ be in $Hol(B({{\mathcal X}})^n_1)$ such that $f_m\to f$ in the metric $\rho$ of $Hol(B({{\mathcal X}})^n_1)$, as $m\to\infty$. Due to Lemma \[Conway\] and Corollary \[max-mod2\], this is equivalent to the fact that, for each $r\in [0,1)$, $$\label{conv-S} f_m(rS_1,\ldots, rS_n)\to f(rS_1,\ldots, rS_n),\quad \text{ as }\ m\to\infty,$$ where the convergence is in the operator norm of $B(F^2(H_n))$. We shall prove that $$\label{conv-f_m} \|f_m(T_1,\ldots, T_n)-f(T_1,\ldots, T_n)\|\to 0, \quad \text{ as }\ m\to\infty.$$ Let $f:=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha Z_\alpha$ and $f_m:=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha^{(m)} Z_\alpha$. Due to Theorem \[abel\], the series defining $f_m(T_1,\ldots, T_n)$ and $f(T_1,\ldots, T_n)$ are norm convergent. Notice that $$\begin{split} \|f_m(T_1,\ldots, T_n)-f(T_1,\ldots, T_n)\|&= \left\|\sum_{k=0}^\infty \sum_{|\alpha|=k}(a_\alpha^{(m)}-a_\alpha)T_\alpha\right\| \\ &\leq \sum_{k=0}^\infty\left\|\sum_{|\alpha|=k}(a_\alpha^{(m)}-a_\alpha)T_\alpha\right\| \\ &\leq \sum_{k=0}^\infty \left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2}\left(\sum_{|\alpha|=k}|a_\alpha^{m)}-a_\alpha|^2\right)^{1/2}. \end{split}$$ If $r(T_1,\ldots, T_n)<\rho<r<1$, then there exists $k_0\in {{\mathbb N}}$ such that $$\left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2}\leq \rho^k \quad \text{ for any }\ k\geq k_0.$$ According to Theorem \[Cauchy-est\], we have $$\left(\sum_{|\alpha|=k}|a_\alpha^{m)}-a_\alpha|^2\right)^{1/2}\leq \frac{1}{r^k}\|f_m(rS_1,\ldots, rS_n)-f(rS_1,\ldots, rS_n)\|.$$ Combining this with the above inequalities, we obtain $$\begin{split} \|f_m(T_1,\ldots, T_n)-f(T_1,\ldots, T_n)\|&\leq M(T,\rho,r) \|f_m(rS_1,\ldots, rS_n)-f(rS_1,\ldots, rS_n)\|, \end{split}$$ where $$M(T,\rho,r):=\sum_{k=0}^{k_0}\left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2}\frac{1}{r^k}+\sum_{k=k_0+1}^\infty\left(\frac{\rho}{r}\right)^{k}.$$ Now, using relation , we deduce , which proves the continuity of $\Phi_T$. To prove the uniqueness of the free analytic functional calculus, let $\Phi:Hol(B({{\mathcal X}})_1^n)\to B({{\mathcal H}})$ be a continuous unital algebra homomorphism such that $\Phi(Z_i)=T_i$,  $i=1,\ldots, n$. Hence, we deduce that $$\label{pol2} \Phi_T(p(Z_1,\ldots, Z_n))=\Phi(p(Z_1,\ldots, Z_n))$$ for any polynomial $p(Z_1,\ldots, Z_n)$ in $Hol(B({{\mathcal X}})_1^n)$. Let $f=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha Z_\alpha$ be an element in $Hol(B({{\mathcal X}})_1^n)$ and let $p_m:=\sum_{k=0}^m\sum_{|\alpha|=k} a_\alpha Z_\alpha$,   $m=1,2,\ldots$. Since $$f(rS_1,\ldots, rS_n)=\sum_{k=0}^\infty \sum_{|\alpha|=k} r^k a_\alpha S_\alpha$$ and the series $\sum_{k=0}^\infty r^k\left\|\sum_{|\alpha|=k} a_\alpha S_\alpha\right\|$ converges due to Theorem \[caract-shifts\], we deduce that $$p_m(rS_1,\ldots, rS_n)\to f(rS_1,\ldots, rS_n)$$ in the operator norm, as $m\to\infty$. Therefore, $p_m\to f$ in the metric $\rho$ of $Hol(B({{\mathcal X}})^n_1)$. Hence, using and the continuity of $\Phi$ and $\Phi_T$, we deduce that $\Phi=\Phi_T$. This completes the proof. Using Theorem \[f-infty\], Theorem \[abel\], and the results from [@Po-funct] concerning the $F_n^\infty$ functional calculus for row contractions, one can make the following observation. For strict row contractions, i.e. $\|[T_1,\ldots, T_n]\|<1$, and $F\in H^\infty(B({{\mathcal X}})^n_1)$, the free analytic functional calculus $F(T_1,\ldots, T_n)$ coincides with the $F_n^\infty$-functional calculus for row contractions. Let $\{F_m\}_{m=1}^\infty$ and $F$ be in $Hol(B({{\mathcal X}})^n_1)$ and let $\{f_m\}_{m=1}^\infty$ and $f$ be the corresponding representations on ${{\mathbb C}}$, respectively (see Corollary \[part-case\]). Due to the noncommuting von Neumann inequality, we have $$\sup_{|\lambda_1|^2+\cdots +|\lambda_n|^2\leq r^2} |f_m(\lambda_1,\ldots, \lambda_n)-f(\lambda_1,\ldots, \lambda_n)|\leq \|F_m(rS_1,\ldots, rS_n)-F(rS_1,\ldots, rS_n)\|$$ for any $r\in [0,1)$. Hence, we deduce that if $F_m\to F$ in the metric $\rho$ of $Hol(B({{\mathcal X}})^n_1)$, then $f_m\to f$ uniformly on compact subsets of ${{\mathbb B}}_n$. Since there is a sequence of polynomials $\{p_m\}_{m=1}^\infty$ such that $p_m\to F$ in the metric $\rho$, one can use the continuity of Taylor’s functional calculus and the continuity of the free analytic functional calculus as well as the fact that they coincide on polynomials, to deduce the following result. If $f$ is the representation of a free holomorphic function $F\in Hol(B({{\mathcal X}})^n_1)$ on ${{\mathbb C}}$ and $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is an $n$-tuple of commuting operators with Taylor spectrum $\sigma(T_1,\ldots, T_n)\subset {{\mathbb B}}_n$, then the free analytic calculus $F(T_1,\ldots, T_n)$ coincides with Taylor’s functional calculus $f(T_1,\ldots, T_n)$. Free pluriharmonic functions and noncommutative Poisson transforms {#free harmonic} ================================================================== Given an operator $A\in B(F^2(H_n))$, the noncommutative Poisson transform [@Po-poisson] generates a function $$P[A]: [B({{\mathcal H}})^n]_1\to B({{\mathcal H}}).$$ In this section, we provide classes of operators $A\in B(F^2(H_n))$ such that $P[A]$ is a free holomorphic (resp. pluriharmonic) function on $[B({{\mathcal H}})^n]_1$. We characterize the free holomorphic functions $u$ on $[B({{\mathcal H}})^n]_1$ such that $u=P[f]$ for some boundary function $f$ in the noncommutative analytic Toeplitz algebra $F_n^\infty$, or the noncommutative disc algebra ${{\mathcal A}}_n$. We also obtain noncommutative multivariable versions of Herglotz theorem and Dirichlet extension problem (see [@Co], [@H]), for free pluriharmonic functions. We define the operator $K_T(S_1,\ldots, S_n)\in B(F^2(H_n)\otimes {{\mathcal H}})$ associated with a row contraction $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ by setting $$K_T(S_1,\ldots, S_n):=\sum_{k=0}^\infty\sum_{|\alpha|=k} S_\alpha \otimes \Delta_T T_\alpha^*,$$ where $\Delta_T:=(I_{{\mathcal H}}-\sum_{i=1}^n T_iT_i^*)^{1/2}$. Due to Theorem \[Abel\], when $A_{(\alpha)}:=\Delta_T T_\alpha^*$ and $X_i:=S_i$, $i=1,\ldots, n$, the above series is convergent in the operator norm if $$\label{cond-conv} \limsup_{k\to\infty} \left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*-\sum_{|\alpha|=k+1} T_\alpha T_\alpha^* \right\|^{1/2k}<1.$$ In particular, if $\|[T_1,\ldots, T_n]\|<1$, then relation holds and the operator $K_T(S_1,\ldots, S_n)$ is in ${{\mathcal A}}_n\bar \otimes B({{\mathcal H}})$. Notice also that $$(S_\alpha^*\otimes I_{{\mathcal H}})K_T(S_1,\ldots, S_n)=K_T(S_1,\ldots, S_n) (I_{F^2(H_n)} \otimes T_\alpha^*),\qquad \alpha\in {{\mathbb F}}_n^+.$$ Introduced in [@Po-poisson], the noncommutative Poisson transform at $T:=[T_1,\ldots, T_n]$ is the map $P_T:B(F^2(H_n))\to B({{\mathcal H}})$ defined by $$\begin{split} \left<P_T(A)x,y\right>&:= \left<K_T(S_1,\ldots, S_n)^* ( A\otimes I_{{\mathcal H}}) K_T(S_1,\ldots, S_n) (1\otimes x),1\otimes y\right>\\ &:=\left< K_T^*( A\otimes I_{{\mathcal H}}) K_Tx,y\right> \end{split}$$ for any $x,y\in B({{\mathcal H}})$, where $K_T:=K_T(S_1,\ldots, S_n)|_{1\otimes {{\mathcal H}}}:{{\mathcal H}}\to F^2(H_n)\otimes {{\mathcal H}}$. We recall that the Poisson kernel $K_T$ is an isometry if $\|T\|<1$, and $$\label{pol} p(T_1,\ldots, T_n)=K_T^*(p(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})K_T$$ for any polynomial $p$. We refer to [@Po-poisson], [@Po-curvature], [@Po-similarity], and [@Po-unitary] for more on noncommutative Poisson transforms on $C^*$-algebras generated by isometries. Given an operator $A\in B(F^2(H_n))$, the noncommutative Poisson transform generates a function $$P[A]:[B({{\mathcal H}})^n]_1\to B({{\mathcal H}})$$ by setting $$P[A](X_1,\ldots, X_n):=P_X(A)\quad \text{ for }\ X:=[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1.$$ In what follows, we provide classes of operators $A\in B(F^2(H_n))$ such that the mapping $P[A]$ is a free holomorphic function on $[B({{\mathcal H}})^n]_1$. In this case, the operator $A$ can be seen as the boundary function of the Poisson transform $P[A]$. As in the previous sections, we identify $f\in F_n^\infty$ with the multiplication operator $L_f\in B(F^2(H_n))$. \[behave\] Let ${{\mathcal H}}$ be a Hilbert space and $u$ be a free holomorphic function on $[B({{\mathcal H}})^n]_1$. 1. There exists $f\in F_n^\infty$ with $u=P[f]$ if and only if $\sup\limits_{0\leq r<1}\|u(rS_1,\ldots, rS_n)\|<\infty$. In this case, $u(rS_1,\ldots, rS_n)\to f$, as $ r\to 1$, in the $w^*$-topology (or strong operator topology). 2. There exists $f\in {{\mathcal A}}_n$ with $u=P[f]$ if and only if $\{u(rS_1,\ldots, rS_n)\}_{0\leq r<1}$ is convergent in norm as, $r\to 1$. In this case, $u(rS_1,\ldots, rS_n)\to f$ in the operator norm, as $r\to 1$. To prove (i), assume that $f\in F_n^\infty$ and $u=P[f]$, where $f$ is identified with the multiplication operator $L_f\in B(F^2(H_n)$. Then $$u(X_1,\ldots, X_n)=K_X^*(L_f\otimes I_{{\mathcal H}})K_X,\quad [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$$ and $\|u(X_1,\ldots, X_n)\|\leq \|L_f\|=\|f\|_\infty$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. In particular, $$\label{sup-u}\sup\limits_{0\leq r<1}\|u(rS_1,\ldots, rS_n)\|\leq \|f\|_\infty<\infty.$$ Conversely, assume that $u(X_1,\ldots, X_n):=\sum_{k=0}\sum_{|\alpha|=k} a_\alpha X_\alpha$ is a free holomorphic function on $[B({{\mathcal H}})^n]_1$ such that holds. By Theorem \[f-infty\], $f:=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$ is in $F_n^\infty$. Due to Theorem \[Abel\], we have that $u_r(X_1,\ldots, X_n):=\sum_{k=0}^\infty \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha X_\alpha$ is convergent in norm for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$ and $r\in [0,1]$. Similarly, we have that $f_r(S_1,\ldots, S_n):=\sum_{k=0}^\infty \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha$ is convergent in norm for any $r\in [0,1)$. Using relation , we deduce that $$\sum_{k=0}^m \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha X_\alpha =K_X^*\left(\sum_{k=0}^m \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha\otimes I_{{\mathcal H}}\right)K_X.$$ Taking $m\to \infty$ and using the above convergences, we get $$\label{u_r-f_r} u_r(X_1,\ldots, X_n)=K_X^*(f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}}) K_X,\qquad r\in [0,1).$$ By Theorem \[continuous\], we have $$\lim_{r\to 1} u_r(X_1,\ldots, X_n)=u(X_1,\ldots, X_n)$$ in the operator norm. On the other hand, due to relation , we have $$\label{So2} \text{\rm SOT-}\lim_{r\to 1} f_r(S_1,\ldots, S_n)=L_f.$$ Since $\|f_r(S_1,\ldots, S_n)\|\leq \|f\|_\infty$ and the map $A\mapsto A\otimes I_{{\mathcal H}}$ is SOT-continuous on bounded subsets of $B(F^2(H_n))$, we take $r\to 1$ in relation and deduce that $u(X_1,\ldots, X_n)=P_X(f)$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Since $u_r(S_1,\ldots, S_n)=f(rS_1,\ldots, rS_n)$ and the strong operator topology coincides with the $w^*$-topology on $F_n^\infty$ (see [@DP1]), one can use to complete the proof of part (i). To prove (ii), assume that $f=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$ is in ${{\mathcal A}}_n$ and $u=P[f]$, i.e., $$u(X_1,\ldots, X_n)= K_X^* (L_f\otimes I_{{\mathcal H}}) K_X$$ for any $X=[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Due to Theorem \[A-infty\], we have $\lim\limits_{r\to 1} f_r(S_1,\ldots, S_n)=L_f$ in the operator norm. Hence, using relation and Theorem \[continuous\], we deduce that $$\begin{split} K_X^* (L_f\otimes I_{{\mathcal H}}) K_X&=\lim_{r\to 1}K_X (f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}}) K_X\\ &=\lim_{r\to 1} f(rX_1,\ldots, rX_n)=f(X_1,\ldots, X_n). \end{split}$$ This proves that $u(X_1,\ldots, X_n)=f(X_1,\ldots, X_n)$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. In particular, we deduce that $$u(rS_1,\ldots, rS_n)=f_r(S_1,\ldots, S_n)\to L_f, \quad \text{ as } \ r\to1,$$ in the operator norm. Conversely, assume that $u:=\sum_{k=0}\sum_{|\alpha|=k} a_\alpha Z_\alpha$ is a free holomorphic function on the open operatorial unit $n$-ball, such that $\{u(rS_1,\ldots, rS_n)\}_{0\leq r<1}$ is convergent in norm, as $r\to 1$. By Theorem \[caract-shifts\], we have that $u(rS_1,\ldots, rS_n)\in {{\mathcal A}}_n$. Since ${{\mathcal A}}_n$ is a Banach algebra, there exists $f\in {{\mathcal A}}_n$ such that $ u(rS_1,\ldots, rS_n)\to f$ in norm, as $r\to 1$. Due to Theorem \[A-infty\], we must have $f=\sum_{k=0}^\infty\sum_{|\alpha|=k} a_\alpha e_\alpha$. As in the proof of part (i), we have $$u(X_1,\ldots, X_n)=\lim_{r\to 1} f_r(X_1,\ldots, X_n)= \lim_{r\to 1} K_X^*(f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})K_X$$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Now, since $\lim\limits_{r\to 1} f_r(S_1,\ldots, S_n)=L_f$ in norm, we deduce that $u=P[f]$. This completes the proof. We now turn our attention to a noncommutative generalization of the harmonic functions on the open unit disc ${{\mathbb D}}$. We say that $G$ is a self-adjoint free pluriharmonic function on $[B({{\mathcal H}})^n]_1$ if there exists a free holomorphic function $F$ on $[B({{\mathcal H}})^n]_1$ such that $$G(X_1,\ldots, X_n)=\text{\rm Re}\,F(X_1,\ldots, X_n):=\frac{1}{2}\left(F(X_1,\ldots, X_n)+ F(X_1,\ldots, X_n)^*\right)$$ We remark that if ${{\mathcal H}}$ be an infinite dimensional Hilbert space, then $G$ determines $F$ up to an imaginary complex number. Indeed, if we assume that $\text{\rm Re}\, F=0$ and take the representation on the full Fock space $F^2(H_n)$, we obtain $F(rS_1,\ldots, rS_n)=-F(rS_1,\ldots, rS_n)^*$, $0<r<1$. If $F(rS_1,\ldots, rS_n)$ has the representation $\sum_{k=0}^\infty\sum_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha$, $a_\alpha\in {{\mathbb C}}$, the above relation implies $$\sum_{k=0}^\infty\sum_{|\alpha|=k} r^{|\alpha|} a_\alpha e_\alpha=F(rS_1,\ldots, rS_n)1=-F(rS_1,\ldots, rS_n)^*1=-\overline{a}_0.$$ Hence, $a_\alpha=0$ if $|\alpha|\geq 1$ and $a_0+\overline{a}_0=0$. Therefore, $F=a_0$, where $a_0$ is an imaginary complex number. This proves our assertion. Due to Theorem \[Abel\], $$G(X_1,\ldots, X_n):=\sum_{k=1}^\infty \sum_{|\alpha|=k} \overline{a}_\alpha X_\alpha^* +a_0 I+ \sum_{k=1}^\infty \sum_{|\alpha|=k} {a_\alpha} X_\alpha$$ represents a self-adjoint free pluriharmonic function on $[B({{\mathcal H}})^n]_1$ if and only if $$\limsup_{k\to\infty}\left(\sum_{|\alpha|=k}|a_\alpha|^2\right)^{1/2k}\leq 1.$$ If $H_1$ and $H_2$ are self-adjoint free pluriharmonic functions on $[B({{\mathcal H}})^n]_1$, we say that $H:=H_1+iH_2$ is a free pluriharmonic function on $[B({{\mathcal H}})^n]_1$. Notice that any free holomorphic function on $[B({{\mathcal H}})^n]_1$ is a free pluriharmonic function. This is due to the fact that $f=\frac{f+f^*}{2}+i\frac{f-f^*}{2i}$. Let $g$ be a free pluriharmonic function on the open operatorial $n$-ball of radius $1+\epsilon$, $\epsilon>0$. Then $$g(X_1,\ldots, X_n)=P_X(g(S_1,\ldots, S_n)),\quad X:=[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1,$$ where $P_X$ is the noncommutative Poisson transform at $X$. Moreover, if ${{\mathcal H}}$ is an infinite dimensional Hilbert space, then $g(S_1,\ldots, S_n)\geq 0$ if and only if $g(X_1,\ldots, X_n)\geq 0$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Without loss of generality, we can assume that $g$ is a self-adjoint free pluriharmonic function and $g(X_1,\ldots, X_n)=f(X_1,\ldots, X_n)+f(X_1,\ldots, X_n)^*$ for any $[X_1,\ldots, X_n)]\in [B({{\mathcal H}})^n]_{1+\epsilon}$, where the function $ f(X_1,\ldots, X_n)=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha X_\alpha$ is free holomorphic on $[B({{\mathcal H}})^n]_{1+\epsilon}$. According to Theorem \[caract-shifts\], the series $\sum_{k=0}^\infty \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha$ converges in the operator norm for any $r\in [0,1+\epsilon )$. Due to relation \[pol\] and taking limits in the operator norm, we have $$\begin{split} f(X_1,\ldots, X_n)&=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha X_\alpha= P_X[f(S_1,\ldots, S_n)] \ \text{ and}\\ f(X_1,\ldots, X_n)^*&=\sum_{k=0}^\infty \sum_{|\alpha|=k} \overline{a}_\alpha X_\alpha^*= P_X[f(S_1,\ldots, S_n)^*]. \end{split}$$ Consequently, $$g(X_1,\ldots, X_n)=P_X[g(S_1,\ldots,S_n)], \quad [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1.$$ We prove now the last part of the proposition. One implication is obvious due to the above relation. Conversely, assume that $g(X_1,\ldots, X_n)\geq 0$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Then, since ${{\mathcal H}}$ is infinite dimensional, we deduce that $g(rS_1,\ldots, rS_n)\geq 0$ for any $r\in [0,1)$. On the other hand, due to Theorem \[continuous\], $\lim\limits_{r\to 1} g(rS_1,\ldots, rS_n)=g(S_1,\ldots, S_n)$ in the operator norm. Hence, $g(S_1,\ldots, S_n)\geq 0$, and the proof is complete. Now, we obtain a noncommutative multivariable version of Herglotz theorem (see [@H]). \[Herglotz\] Let $f\in (F_n^\infty)^*+ F_n^\infty$ and let $u=P[f]$ be its noncommutative Poisson transform. Then $u$ is a free pluriharmonic function on $[B({{\mathcal H}})^n]_1$, where ${{\mathcal H}}$ is a Hilbert space. Moreover, $u\geq 0$ on $[B({{\mathcal H}})^n]_1$, where ${{\mathcal H}}$ is an infinite dimensional Hilbert space, if and only if $f\geq 0$. First, notice that, without loss of generality, we can assume that $f=f^*$. Then, one can prove that $f=g^*+g$ for some $g\in F_n^\infty$. Indeed, if $f=h^*+g$ for some $h,g\in F_n^\infty$, the we must have $(g-h)^*=g-h$. Hence, $(g-h)^*1=(g-h)1$ and one can easily deduce that $g-h$ is a constant, which proves our assertion. According to Theorem \[behave\], $P[g]$ is a free holomorphic function on the open operatorial unit $n$-ball. On the other hand, due to [@Po-varieties], we have $$\text{\rm SOT}-\lim_{r\to 1} g_r(S_1,\ldots, S_n)^*=L_g^*.$$ Hence, using the properties of the Poisson tranform and Theorem \[continuous\], we deduce that $$\begin{split} \left< P[g^*] x,y\right>&= \lim_{r\to 1}\left< K_X(g_r(S_1,\ldots, S_n)^*\otimes I_{{\mathcal H}})K_X x,y\right>\\ &= \lim_{r\to 1}\left< g_r(X_1,\ldots, X_n)^*x,y\right>\\ &= \left< g(X_1,\ldots, X_n)^* x,y\right>\\ &=\left< P[g]^*x,y\right>. \end{split}$$ Hence, we have $P[g]^*=P[g^*]$. Consequently, $$u=P[f]=P[g^*]+P[g]=P[g]^*+P[g],$$ which proves that $u$ is a self-adjoint free pluriharmonic function on $[B({{\mathcal H}})^n]_1$. Now, it is clear that if $f\geq 0$ then $u=P[f]\geq 0$. Conversely, assume that $u(X_1,\ldots, X_n)\geq 0$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Since ${{\mathcal H}}$ is an infinite dimensional Hilbert space ${{\mathcal H}}$, we deduce that $$u(rS_1,\ldots, rS_n)=g(rS_1,\ldots, rS_n)^*+ g(rS_1,\ldots, rS_n)\geq 0,\qquad r\in [0,1).$$ Due to Theorem \[behave\], we have $$\text{\rm WOT}-\lim_{r\to 1} [g(rS_1,\ldots, rS_n)^*+ g(rS_1,\ldots, rS_n)]=L_g^*+L_g\geq 0.$$ Under the identification of $g$ with $L_g$, we deduce $f=g^*+g\geq 0$, and complete the proof. Here again, we remark that $ f$ plays the role of the boundary function from the classical complex analysis. Our version of the classical Dirichlet extension problem for the unit disc (see [@Co], [@H]) is the following extension of Theorem \[A-infty\]. \[Dirichlet\] If $f\in {{\mathcal A}}_n^*+{{\mathcal A}}_n$, then $u:=P[f]$ is a free pluriharmonic function on the open operatorial unit $n$-ball such that 1. $u$ has a continuous extension $\tilde u$ to $[B({{\mathcal H}})^n]_1^-$ for any Hilbert space ${{\mathcal H}}$, in the operator norm; 2. $\tilde u(S_1,\ldots, S_n)=f$. Without loss of generality, we can assume that $f$ is self-adjoint. As in the proof of Theorem \[Herglotz\], one can prove that $f=g^*+g$ for some $g\in {{\mathcal A}}_n$ and $u:=P[f]=P[g]^*+P[g]$ is a self-adjoint pluriharmonic function on the open operatorial unit $n$-ball. Since $g\in {{\mathcal A}}_n$, we know that $g_r(S_1,\ldots, S_n)\to L_g$ in norm, as $r\to 1$. Consequently, $$f_r(S_1,\ldots, S_n):=g_r(S_1,\ldots, S_n)^*+g_r(S_1,\ldots, S_n)\to L_f^*+ L_f,\quad \text{ as } \ r\to1,$$ in norm. As in the proof of Theorem \[Herglotz\], we have $$u(X_1,\ldots, X_n)=f(X_1,\ldots, X_n),\quad \text{ for } \ [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1.$$ Moreover, $v:=P[g]$ is a free holomorphic function such that $v(X_1,\ldots, X_n)=g(X_1,\ldots, X_n)$,  for any $ [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. For each $n$-tuple $[Y_1,\ldots, Y_n]\in [B({{\mathcal H}})^n]_1^-$, we define $$\tilde v(Y_1,\ldots, Y_n):=\lim_{r\to 1} P_{rY}[g],$$ where $rY:=[rY_1,\ldots, rY_n]$. Hence, we have $\tilde v(Y_1,\ldots, Y_n)=\lim_{r\to 1} g(rY_1,\ldots, rY_n)$. Now, as in the proof of Theorem \[A-infty\], we deduce that the map $\tilde v:[B({{\mathcal H}})^n]_1^-\to B({{\mathcal H}})$ is a continuous extension of $v$. Therefore, the map $\tilde u:={\tilde v}^*+\tilde v$ is a continuous extension of $u$ to $[B({{\mathcal H}})^n]_1^-$. To prove (ii), apply part (i) when ${{\mathcal H}}=F^2(H_n)$ and take into account Theorem \[A-infty\]. We obtain $$\tilde v(S_1,\ldots, S_n)=\lim_{r\to 1} g(rS_1,\ldots, rS_n)=g,$$ where we used the identification of $g$ with $L_g$, and the limit is in the operator norm. Therefore, $$\tilde u(S_1,\ldots, S_n)=\tilde v(S_1,\ldots, S_n)^*+\tilde v(S_1,\ldots, S_n)=g^*+g=f.$$ This completes the proof. Let $u$ and $v$ be two self-adjoint free pluriharmonic functions on $[B({{\mathcal H}})^n]_1$. We say that $v$ is the pluriharmonic conjugate of $u$ if $u+iv$ is a free holomorphic function on $[B({{\mathcal H}})^n]_1$. The pluriharmonic conjugate of a self-adjoint free pluriharmonic function on $[B({{\mathcal H}})^n]_1$ is unique up to an additive real constant. Let $f$ be a free holomorphic function on $[B({{\mathcal H}})^n]_1$ and $u=\text{\rm Re}\, f$. Assume that $v$ is a selfadjoint free pluriharmonic function such that $u+iv=g$ is a free holomorphic function on $[B({{\mathcal H}})^n]_1$. Hence, we have $$\label{v} v=\frac{2g-f-f^*}{2i}.$$ Since $v=v^*$, we must have $(g-f=(g-f)^*$, i.e., $\text{\rm Re}\, (g-f)=0$. Based on the remarks following Theorem \[behave\], we have $g-f=w$, where $w$ is an imaginary complex number. Consequently, relation , implies $v=\frac{f-f^*}{2i}-iw$. This proves the assertion. We remark that if $u=\text{\rm Re}\, f$ and $f(0)$ is real then $v=\frac{f-f^*}{2i}$ is the unique pluriharmonic conjugate of $u$ such that $v(0)=0$. \[cauch-conj\] Let $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. If $f\in H^\infty (B({{\mathcal X}})^n_1) $, $u=\text{\rm Re}\,f$, and $f(0)$ is real, then $$\left<f(T_1,\ldots, T_n)x,y\right>=\left<(u(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x), [2C_T(R_1,\ldots, R_n)-I](1\otimes y)\right>$$ for any $x,y\in {{\mathcal H}}$, where $u(S_1,\ldots, S_n)$ is the boundary function of $u$. Due to Theorem \[an=cauch\], we have $$\begin{split} \left<(f(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x)\right. &, \left. [2C_T(R_1,\ldots, R_n)-I](1\otimes y)\right>\\ &= 2\left<(f(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x), [C_T(R_1,\ldots, R_n)](1\otimes y)\right>\\ &\qquad-\left<(f(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x), 1\otimes y\right>\\ &= 2\left< f(T_1,\ldots, T_n)x,y\right>-f(0)\left< x,y\right>. \end{split}$$ On the other hand, it is easy to see that $$\begin{split} \left<(f(S_1,\ldots, S_n)^*\otimes I_{{\mathcal H}})(1\otimes x)\right.&,\left. [2C_T(R_1,\ldots, R_n)-I](1\otimes y)\right>\\ &= \left<(\overline{(f(0)}\otimes I_{{\mathcal H}})(1\otimes x), [2C_T(R_1,\ldots, R_n)-I](1\otimes y)\right>\\ &=\overline{(f(0)}\left<x,y\right>. \end{split}$$ If $f(0)\in {{\mathbb R}}$, then adding up the above relations, we complete the proof. We remark that under the conditions of Theorem \[cauch-conj\] and using the noncommutative Cauchy transform, one can express the pluriharmonic conjugate of $u$ in terms of $u$. In a forthcoming paper [@Po-Bohr], we will consider operator-valued Bohr type inequalities for classes of free pluriharmonic functions on the open operatorial unit $n$-ball with operator-valued coefficients. Hardy spaces of free holomorphic functions {#Banach} ============================================ In this section, we define the radial maximal Hardy space $H^p(B({{\mathcal X}})^n_1)$, $p\geq 1$, and the symmetrized Hardy space $H^\infty_{\text{\rm sym}}({{\mathbb B}}_n)$, and prove that they are Banach spaces with respect to some appropriate norms. In this setting, we obtain von Neumann type inequalities for $n$-tuples of operators. Let $F$ be a free holomorphic function on the open operatorial unit $n$-ball. The map $\varphi:[0,1)\to B(F^2(H_n))$ defined by $\varphi(r):= F(rS_1,\ldots, rS_n)$ is called the [*radial boundary function*]{} associated with $F$. Due to Theorem \[continuous\], $\varphi$ is continuous with respect to the operator norm topology of $B(F^2(H_n))$. When $\lim\limits_{r\to 1} \varphi(r)$ exists, in one of the classical topologies of $B(F^2(H_n))$, we call it the [*boundary function*]{} of $F$. Due to the maximum principle for free holomorphic functions (see Theorem \[max-mod1\]), we have $$\|\varphi(r)\|=\sup \|F(X_1,\ldots, X_n)\|,\quad 0\leq r<1,$$ where the supremum is taken over all $n$ tuples of operators $[X_1,\ldots, X_n]$ in either one of the following sets $[B({{\mathcal H}})^n]_r,\ [B({{\mathcal H}})^n]_r^-$, or $$\{[X_1,\ldots, X_n]\in B({{\mathcal H}})^n: \ \|[X_1,\ldots, X_n]\|=r\},$$ where ${{\mathcal H}}$ is an arbitrary infinite dimensional Hilbert space. The [*radial maximal function*]{} $M_F:[0,1)\to [0,\infty)$ associated with a free holomorphic function $F\in Hol(B({{\mathcal X}})^n_1)$ is defined by $$M_F(r):=\|\varphi(r)\|=\|F(rS_1,\ldots, rS_n)\|.$$ $M_F$ is an increasing continuous function (see the proof of Theorem \[f-infty\]). We define the [*radial maximal Hardy space*]{} $H^p(B({{\mathcal X}})^n_1)$,  $p\geq 1$, as the set of all free holomorphic functions $F\in Hol(B({{\mathcal X}})^n_1)$ such that $M_F$ is in the Lebesque space $ L^p[0,1]$. Setting $$\|F\|_p:=\|M_F\|_p:=\left(\int_0^1\|F(rS_1,\ldots, rS_n)\|^p dr \right)^{1/p},$$ it is easy to see that $\|\cdot \|_p$ is a norm on the linear space $H^p(B({{\mathcal X}})^n_1)$. \[radial-Banach\] If $p\geq 1$, then the radial maximal Hardy space $H^p(B({{\mathcal X}})^n_1)$ is a Banach space. First we prove the result for $p=1$. Let $\{F_k\}_{k=1}^\infty \subset H^1(B({{\mathcal X}})^n_1)$ be a sequence such that $$\label{ser-conv-1} \sum_{k=1}^\infty \|F_k\|_1\leq M<\infty.$$ We need to prove that $\sum_{k=1}^\infty F_k$ converges in $\|\cdot\|_1$. By , we have $$\sum_{k=1}^m \int_0^1 \|F_k(rS_1,\ldots, rS_n)\| dr\leq M,\quad \text{ for any } \ m\in {{\mathbb N}}.$$ Using Fatou’s lemma, we deduce that the function $\psi(r):=\sum_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\|$ is integrable on $[0,1]$. Notice that the series $\sum_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\|<\infty$ for any $r\in [0,1)$. Indeed, assume that there exists $r_0\in [0,1)$ such that $\sum_{k=1}^\infty \|F_k(r_0S_1,\ldots, r_0S_n)\|=\infty$. Since the radial maximal function is increasing, we have $$\sum_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\|\geq \sum_{k=1}^\infty \|F_k(r_0S_1,\ldots, r_0S_n)\|=\infty$$ for any $r\in [r_0,1)$. Hence, we deduce that $$\int_0^1\sum_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\| dr\geq (1-r_0) \sum_{k=1}^\infty \|F_k(r_0S_1,\ldots, r_0S_n)\|=\infty,$$ which contradicts the fact that $\psi$ is integrable on $[0,1]$. Therefore, we deduce that $\sum\limits_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\|$ is convergent for any $r\in [0,1)$. Hence, the series $\sum\limits_{k=1}^\infty F_k(rS_1,\ldots, rS_n)$ is convergent in the operator norm of $B(F^2(H_n))$ for each $r\in [0,1)$. For each $m\geq 1$, define $g_m:=\sum_{k=1}^m F_k$. Since $\{g_m\}_{m=1}^\infty$ is a sequence of free holomorphic functions such that $\{g_m(rS_1,\ldots, rS_n)\}_{m=1}^\infty$ is convergent in norm for each $r\in [0,1)$, we deduce that $\{g_m\}_{m=1}^\infty$ is uniformly convergent on any closed operatorial ball $[B({{\mathcal X}})^n]_r^-$, $r\in [0,1)$. According to our noncommutative Weierstrass type result, Theorem \[Weierstrass\], there is a free holomorphic function $g$ on the open operatorial unit $n$-ball such that $\|g_m(rS_1,\ldots, rS_n)-g(rS_1,\ldots, rS_n)\|\to 0$, as $m\to\infty,$ and therefore $$g(rS_1,\ldots, rS_n)=\sum_{k=1}^\infty F_k(rS_1,\ldots, rS_n)\quad \text{ for any } \ r\in [0,1).$$ Moreover, due to the fact that $\psi$ is integrable, we have $$\int_0^1 \|g(rS_1,\ldots, rS_n)\| dr \leq \int_0^1 \sum_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\| dr <\infty,$$ which shows that $g\in H^1(B({{\mathcal X}})^n_1)$. Now, notice that $$\begin{split} \|g-g_m\|_1&=\int_0^1 \|g(rS_1,\ldots, rS_n)-g_m(rS_1,\ldots, rS_n)\| dr\\ &=\int_0^1\left\|\sum_{k=m+1}^\infty F_k(rS_1,\ldots, rS_n)\right\| dr\\ &\leq \int_0^1 \sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\| dr \end{split}$$ Since $\sum_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\|<\infty$, we have $$\lim_{m\to\infty} \sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\|=0 \quad \text{ for any} \ r\in [0,1).$$ On the other hand, $\sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\|\leq \psi(r)$ for any $m\in {{\mathbb N}}$. Since $\psi$ is integrable on $[0,1]$, we can apply Lebesgue’s dominated convergence theorem and deduce that $$\lim_{m\to\infty} \int_0^1\sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\| dr=0.$$ Now, we deduce that $\|g-g_m\|_1\to 0$, as $m\to\infty$, which shows that the series $\sum_{k=1}^\infty F_k$ is convergent in $\|\cdot\|_1$. This completes the proof when $p=1$. Assume now that $p>1$ and let $\{F_k\}_{k=1}^\infty\subset H^p((B({{\mathcal X}})^n_1)$ be a sequence such that $\sum_{k=1}^\infty \|F\|_p\leq M<\infty$. Since $\|F_k\|_1\leq \|F_k\|_p$, we have $\sum_{k=1}^\infty \|F\|_1\leq M$. Applying the first part of the proof, we find $g\in H^1(B({{\mathcal X}})^n_1)$ such that, for each $r\in [0,1)$, $$g(rS_1,\ldots, rS_n)=\sum_{k=1}^\infty F_k(rS_1,\ldots, rS_n),$$ where the convergence is in the operator norm of $B(F^2(H_n))$. Moreover, we have $$\begin{split} \int_0^1\left\|\sum_{k=1}^m F_k(rS_1,\ldots, rS_n)\right\|^p dr &\leq \int_0^1\left( \sum_{k=1}^m \|F_k(rS_1,\ldots, rS_n)\|\right)^{p} dr\\ &\leq \left[\sum_{k=1}^m\left(\int_0^1 \|F_k(rS_1,\ldots, rS_n)\|^p\right)^{1/p}\right]^p\\ &= \left(\sum_{k=1}^m \|F_k\|_p\right)^p\leq M^p. \end{split}$$ Using Fatou’s lemma, we deduce that the function $r\mapsto \left\| \sum\limits_{k=1}^\infty F_k(rS_1,\ldots, rS_n)\right\|^p$ is integrable on $[0,1]$ and therefore $g\in H^p((B({{\mathcal X}})^n_1)$. Notice also that $$\label{norm-int} \|g-g_m\|_p\leq \left[\int_0^1 \left( \sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\|\right)^p\right]^{1/p}.$$ Since $ \sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\|\leq \psi$ for any $m\in {{\mathbb N}}$, and $$\lim\limits_{m\to\infty} \sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\|=0\quad \text{ for any }\ r\in [0,1),$$ we can apply again Lebesgue’s dominated convergence theorem and deduce that $$\lim_{m\to\infty} \left[\int_0^1 \left( \sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\|\right)^p\right]^{1/p}=0.$$ Hence and using inequality , we deduce that $\|g-g_m\|_p\to0$ as $m\to\infty$. Consequently, the series $\sum_{k=1}^\infty F_k$ converges in the norm $\|\cdot\|_p$. This completes the proof. \[prop-Hp\] Let $p\geq 1$. 1. If $f\in H^\infty(B({{\mathcal X}})^n_1)$, then $\|f\|_1\leq \|f\|_p\leq \|f\|_\infty$. Moreover, $$H^\infty(B({{\mathcal X}})^n_1)\subset H^p(B({{\mathcal X}})^n_1)\subset H^1(B({{\mathcal X}})^n_1)\subset Hol(B({{\mathcal X}})^n_1).$$ 2. If $f\in H^\infty(B({{\mathcal X}})^n_1)$, then $$\|f\|_\infty=\lim_{p\to\infty}\left(\int_0^1\|f(rS_1,\ldots, rS_n)\|^p dr\right)^{1/p}.$$ 3. If $f=\sum\limits_{k=0}^\infty \sum\limits_{|\alpha|=k} a_\alpha Z_\alpha$ is in $H^p(B({{\mathcal X}})^n_1)$, then $$\left(\sum_{|\alpha|=k}|a_\alpha|^2\right)^{1/2}\leq (pk+1)^{1/p} \|f\|_p.$$ Part (i) follows as in the classical theory of $L^p$ spaces. To prove (ii), define the function $G:[0,1]\to [0,\infty)$ by setting $G(r):=\|f(rS_1,\ldots, rS_n)\|$ if $r\in [0,1)$ and $G(1):=\lim\limits_{r\to 1}\|f(rS_1,\ldots, rS_n)\|$. Due to Theorem \[f-infty\], $G$ is an increasing continuous function and $G(1)=\|f\|_\infty$. Therefore, $$\begin{split} \lim_{p\to\infty}\left(\int_0^1\|f(rS_1,\ldots, rS_n)\|^p dr\right)^{1/p}&= \lim_{p\to\infty} \left( \int_0^1 G(r)^p\right)^{1/p}\\ &=\max_{r\in [0,1]} G(r)=G(1)=\|f\|_\infty. \end{split}$$ To prove (iii), notice that Theorem \[Cauchy-est\] implies $$r^k \left(\sum_{|\alpha|=k}|a_\alpha|^2\right)^{1/2}\leq \|f(rS_1,\ldots, rS_n)\|,\quad r\in [0,1).$$ Integrating over $[0,1]$, we complete the proof of (iii). The next result extends the noncommutative von Neumann inequality from $H^\infty(B({{\mathcal X}})^n_1)$ to the radial maximal Hardy space $H^p(B({{\mathcal X}})^n_1)$, $p\geq 1$. \[vN-Hp\] If $T:=[T_1,\ldots, T_n]\in [B({{\mathcal H}})^n]_1$ and $p\geq 1$, then the mapping $$\Psi_T:H^p(B({{\mathcal X}})^n_1)\to B({{\mathcal H}})\quad \text{ defined by } \ \Psi_T(f):=f(T_1,\ldots, T_n)$$ is continuous, where $f(T_1,\ldots, T_n)$ is defined by the free analytic functional calculus and $B({{\mathcal H}})$ is considered with the operator norm topology. Moreover, $$\|f(T_1,\ldots, T_n)\|\leq\frac{1}{(1-\|[T_1,\ldots, T_n]\|)^{1/p}} \|f\|_p$$ for any $f\in H^p(B({{\mathcal X}})^n_1)$. Assume that $\|[T_1,\ldots, T_n]\|=r_0<1$ and let $f\in H^p(B({{\mathcal X}})^n_1)$. Since the radial maximal function is increasing and and due to Corollary \[max-mod2\], we have $$\begin{split} \|f\|_p&\geq \left(\int_{r_0}^1 \|f(rS_1,\ldots, rS_n)\|^p dr\right)^{1/p}\\ &\geq (1-r_0)^{1/p} \|f(r_0S_1,\ldots, r_0S_n)\|\\ &\geq (1-r_0)^{1/p}\|f(T_1,\ldots, T_n)\|. \end{split}$$ Hence, we deduce the above von Neumann type inequality, which can be used to prove the continuity of $\Psi_T$. We remark that if $f\in H^\infty(B({{\mathcal X}})^n_1)$, then one can recover the noncommutative von Neumann inequality [@Po-von] for strict row contractions, i.e., $\|f(T_1,\ldots, T_n)\|\leq \|f\|_\infty$. Indeed, take $p\to\infty$ in the above inequality and use part (ii) of Proposition \[prop-Hp\]. In the last part of this paper, we introduce a Banach space of analytic functions on the open unit ball of ${{\mathbb C}}^n$ and obtain a von Neumann type inequality in this setting. We use the standard multi-index notation. Let ${\bf p}:=(p_1,\ldots, p_n)$ be a multi-index in ${{\mathbb Z}}_+^n$. We denote $|{\bf p}|:=p_1+\cdots + p_n$ and ${\bf p} !:={ p}_1 !\cdots { p}_n !$. If $\lambda:=(\lambda_1,\ldots,\lambda_n)$, then we set $\lambda^{\bf p}:=\lambda_1^{p_1}\cdots \lambda_n^{p_n}$ and define the symmetrized functional calculus $$(\lambda^{\bf p})_{\text{\rm sym}} (S_1,\ldots, S_n):=\frac {{\bf p}!} {|{\bf p}|! }\sum_{\alpha\in \Lambda_{\bf p}} S_\alpha,$$ where $$\Lambda_{\bf p}:=\{\alpha\in {{\mathbb F}}_n^+: \lambda_\alpha= \lambda^{\bf p} \text{ for any } \lambda\in {{\mathbb B}}_n\}$$ and $S_1,\ldots, S_n$ are the left creation operators on the Fock space $F^2(H_n)$. Notice that card$\Lambda_{\bf p}=\frac {|{\bf p}|!} {{\bf p}!}$. Denote by $H_{\text{\rm sym}}({{\mathbb B}}_n)$ the set of all analytic functions on ${{\mathbb B}}_n$ with scalar coefficients $$f(\lambda_1,\ldots,\lambda_n):=\sum\limits_{\bf p\in {{\mathbb Z}}_+^n} \lambda^{\bf p} a_{\bf p}, \quad a_{\bf p}\in {{\mathbb C}},$$ such that $$\label{sup-AA} \limsup_{k\to \infty}\left( \sum\limits_{{\bf p}\in {{\mathbb Z}}_+^n,|{\bf p}|=k} \frac {|{\bf p}|!}{{\bf p}!} |a_{\bf p}|^2\right)^{1/2k}\leq 1.$$ Then $$\begin{split} f_{\text{\rm sym}}(rS_1,\ldots, rS_n) &:=\sum_{k=0}^\infty \sum\limits_{{\bf p}\in {{\mathbb Z}}_+^n,|{\bf p}|=k} r^k a_{{\bf p}}[(\lambda^{\bf p})_{\text{\rm sym}} (S_1,\ldots, S_n)]\\ &=\sum_{k=0}^\infty \sum_{|\alpha|=k} r^{|\alpha|} c_{\alpha}S_\alpha, \end{split}$$ where $c_{0}:=a_{0}$ and $c_{\alpha}:= \frac {{\bf p}!}{|{\bf p}|!}a_{{\bf p}}$ for ${\bf p}\in {{\mathbb Z}}_+^n$, ${\bf p}\neq (0,\ldots, 0)$, and $\alpha\in \Lambda_{\bf p}$. It is clear that, for each $k=1,2,\ldots, $ we have $$\begin{split} \sum_{|\alpha|=k}|c_{\alpha}|^2&= \sum\limits_{{\bf p}\in {{\mathbb Z}}_+^n,|{\bf p}|=k}\left(\sum_{\alpha\in \Lambda_{\bf p}} |c_{\alpha}|^2\right)\\ &=\sum\limits_{{\bf p}\in {{\mathbb Z}}_+^n,|{\bf p}|=k} \frac {{\bf p}!}{|{\bf p}|!} |a_{\alpha}|^2. \end{split}$$ Due to Theorem \[Abel\], condition implies that $f_{\text{\rm sym}}(rS_1,\ldots, rS_n)$ is norm convergent for each $r\in [0,1)$, and $f_{\text{\rm sym}}(Z_1,\ldots, Z_n)$ is a free holomorphic function on the open operatorial unit $n$-ball. We define $H_{\text{\rm sym}}^\infty({{\mathbb B}}_n) $ as the set of all functions $f\in H_{\text{\rm sym}}({{\mathbb B}}_n)$ such that $$\|f\|_{\text{\rm sym}}:=\sup_{0\leq r<1}\left\| f_{\text{\rm sym}}(rS_1,\ldots, rS_n)\right\|<\infty.$$ \[sym\] $\left(H_{\text{\rm sym}}^\infty({{\mathbb B}}_n), \|\cdot\|_{\text{\rm sym}}\right)$ is a Banach space. First notice that if $f\in H_{\text{\rm sym}}^\infty({{\mathbb B}}_n)$ then $f_{\text{\rm sym}}(rS_1,\ldots, rS_n)$ is norm convergent and $f_{\text{\rm sym}}(Z_1,\ldots, Z_n)$ is a free holomorphic function on the open operatorial unit $n$-ball. Using Theorem \[operations\], it is easy to see that $H_{\text{\rm sym}}^\infty({{\mathbb B}}_n)$ is a vector space and $\|\cdot \|_{\text{\rm sym}}$ is a norm. Let $\{f_m\}_{m=1}^\infty$ be a Cauchy sequence of functions in $H_{\text{\rm sym}}^\infty({{\mathbb B}}_n)$. According to Theorem \[f-infty\], $(f_m)_{\text{\rm sym}}\in F_n^\infty$ and $\{(f_m)_{\text{\rm sym}}\}_{m=1}^\infty$ is a Cauchy sequence in $\|\cdot\|_\infty$, the norm of the Banach algebra $F_n^\infty$. Therefore, there exists $g\in F_n^\infty$ such that $\|(f_m)_{\text{\rm sym}}-L_g\|_\infty\to 0$, as $m\to\infty$. If $ f(\lambda_1,\ldots,\lambda_n)=\sum\limits_{\bf p\in {{\mathbb Z}}_+^n} a_{\bf p}^{(m)}\lambda^{\bf p}, \quad a_{\bf p}\in {{\mathbb C}}, $ then $(f_m)_{\text{\rm sym}}(S_1,\ldots, S_n)=\sum_{k=0}^\infty\sum_{|\alpha|=k} c_\alpha^{(m)} S_\alpha$, where $c_\alpha^{(m)}:=\frac {|{\bf p}|!}{{\bf p}!} a_{\bf p}^{(m)}$ for ${\bf p}\in {{\mathbb Z}}_+^n$, ${\bf p}\neq (0,\ldots,0)$ and $\alpha\in \Lambda_{\bf p}$. If $g=\sum_{\alpha\in {{\mathbb F}}_n^+} b_\alpha e_\alpha$ is the Fourier representation of $g$ as an element of $F^2(H_n)$, then we have $$\begin{split} |c_\alpha^{(m)}-b_\alpha|&= \left|\left<[(f_m)_{\text{\rm sym}}(S_1,\ldots, S_n)-L_g]1,1\right>\right|\\ &\leq \|(f_m)_{\text{\rm sym}}-L_g\|_\infty. \end{split}$$ Taking $m\to \infty$, we deduce that $c_\alpha^{(m)}\to b_\alpha$ for each $\alpha\in {{\mathbb F}}_n^+$. Since $c_\alpha^{(m)}=c_\beta^{(m)}$ for any $\alpha,\beta \in \Lambda_{\bf p}$, we get $b_\alpha=b_\beta$. Setting $h(\lambda_1,\ldots, \lambda_n):= \sum_{k=0}^\infty \sum_{|\alpha|=k} b_\alpha \lambda_\alpha$, one can see that $h$ is holomorphic in ${{\mathbb B}}_n$ and $h_{\text{\rm sym}}=L_g$. Moreover, $\|h\|_{\text{\rm sym}}=\|g\|_\infty<\infty$. This shows that $H_{\text{\rm sym}}^\infty({{\mathbb B}}_n)$ is a Banach space. Now, using Theorem \[abel\] in the scalar case, we can deduce the following. If $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is a commuting $n$-tuple of operators with the joint spectral radius $r(T_1,\ldots, T_n)<1$ and $f(\lambda_1,\ldots,\lambda_n):=\sum\limits_{\bf p\in {{\mathbb Z}}_+^n} a_{\bf p} \lambda^{\bf p} $ is in $ H_{\text{\rm sym}}({{\mathbb B}}_n)$, then $$f(T_1,\ldots, T_n):= \sum_{k=0}^\infty \sum\limits_{{\bf p}\in {{\mathbb Z}}_+^n,|{\bf p}|=k}a_{\bf p} T^{\bf p}$$ is a well-defined operator in $B({{\mathcal H}})$, where the series is convergent in the operator norm topology. 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--- abstract: 'This paper is concerned with the modeling errors appeared in the numerical methods of inverse medium scattering problems (IMSP). Optimization based iterative methods are wildly employed to solve IMSP, which are computationally intensive due to a series of Helmholtz equations need to be solved numerically. Hence, rough approximations of Helmholtz equations can significantly speed up the iterative procedure. However, rough approximations will lead to instability and inaccurate estimations. Using the Bayesian inverse methods, we incorporate the modelling errors brought by the rough approximations. Modelling errors are assumed to be some complex Gaussian mixture (CGM) random variables, and in addition, well-posedness of IMSP in the statistical sense has been established by extending the general theory to involve CGM noise. Then, we generalize the real valued expectation-maximization (EM) algorithm used in the machine learning community to our complex valued case to learn parameters in the CGM distribution. Based on these preparations, we generalize the recursive linearization method (RLM) to a new iterative method named as Gaussian mixture recursive linearization method (GMRLM) which takes modelling errors into account. Finally, we provide two numerical examples to illustrate the effectiveness of the proposed method.' address: - 'School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China' - 'School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China' - 'School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China' - 'School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China' author: - Junxiong Jia - Bangyu Wu - Jigen Peng - Jinghuai Gao bibliography: - 'references.bib' title: Recursive linearization method for inverse medium scattering problems with complex mixture Gaussian error learning --- [^1] Introduction ============ Scattering theory has played a central role in the field of mathematical physics, which is concerned with the effect that an inhomogeneous medium has on an incident particle or wave [@ColtonThirdBook]. Usually, the total field is viewed as the sum of an incident field and a scattered field. Then, the inverse scattering problems focus on determining the nature of the inhomogeneity from a knowledge of the scattered field [@Bleistein2001Book; @ColtonSIAMReview2000], which have played important roles in diverse scientific areas such as radar and sonar, geophysical exploration, medical imaging and nano-optics. Deterministic computational methods for inverse scattering problems can be classified into two categories: nonlinear optimization based iterative methods [@Bao2015TopicReview; @Metivier2016IP; @Natterer1995IP] and imaging based direct methods [@Cakoni2006Book; @Cheney2001IP]. Direct methods are called qualitative methods which need no direct solvers and visualize the scatterer by highlighting its boundary with designed imaging functions. Iterative methods are usually called quantitative methods, which aim at providing some functions to represent the scatterer. Because a sequence of direct and adjoint scattering problems need to be solved, the quantitative methods are computationally intensive. This paper is concerned with the nonlinear optimization based iterative methods, especially focus on the recursive linearization method (RLM) for inverse medium scattering problems [@Bao2015TopicReview]. Although the computational obstacle can be handled in some circumstances, the accuracy of the forward solver is still a critical topic, particularly for applications in seismic exploration [@Fichtner2011Book] and medical imaging [@Koponen2014IEEE]. A lot of efficient forward solvers based on finite difference method, finite element methods and spectral methods have been proposed [@Teresa2006IP; @Wang1997JASA]. Here, we will not propose a new forward solver to reduce the computational load, but attempt to reformulate the nonlinear optimization model based on Bayesian inverse framework which can incorporate statistical properties of the model errors induced by rough forward solvers. By using the statistical properties, we aim to reduce the computational load for the inverse procedure. In order to give a clear sketch of our idea, let us provide a concise review of the Bayesian inverse methods according to our purpose. Denote $X$ to be some separable Banach space, then the forward problem usually modeled as follows $$\begin{aligned} \label{forwardForm} d = \mathcal{F}(m) + \epsilon,\end{aligned}$$ where $d \in \mathbb{C}^{N_{d}}$ ($N_{d} \in \mathbb{N}^{+}$) stands for the measured data, $m \in X$ represents the interested parameter and $\epsilon$ denotes noise. For inverse scattering problems, $m$ is just the scatterer, $\mathcal{F}$ represents a Helmholtz equation combined with some measurement operator. The nonlinear optimization based iterative methods just formulate inverse problem as follows $$\begin{aligned} \label{optimiFormu} \min_{m \in X} \Bigg\{ \frac{1}{2}\big\|d - \mathcal{F}(m)\big\|_{2}^{2} + \mathcal{R}(m) \Bigg\},\end{aligned}$$ where $\mathcal{R}(\cdot)$ stands for some regularization operator and $\|\cdot\|_{2}$ represents the $\ell^{2}$-norm. Different to the minimization problem (\[optimiFormu\]), Bayesian inverse methods reformulate the inverse problem as a stochastic inference problem, which has the ability to give uncertainty quantifications [@inverse_fluid_equation; @Besov_prior; @Junxiong2016IP; @book_comp_bayeisn; @acta_numerica]. Bayesian inverse methods aim to provide a complete posterior information, however, it can also offer a point estimate. Up to now, there are usually two frequently used point estimators: maximum a posteriori (MAP) estimate and conditional mean (CM) estimate [@Tenorio2006Book]. For problems defined on finite dimensional space, MAP estimate is obviously just the solution of the minimization problem (\[optimiFormu\]), which is illustrated rigorously in [@book_comp_bayeisn]. Different to the finite dimensional case, only recently, serious results for relationships between MAP estimates and minimization problem (\[optimiFormu\]) are obtained in [@Burger2014IP; @MAPSmall2013; @Dunlop2016IP] when $X$ is an infinite dimensional space. Simply speaking, if minimization problem (\[optimiFormu\]) has been used to solve our inverse problem, then an assumption has been made that is the noise $\epsilon$ is sampled from some Gaussian distribution $\mathcal{N}(\bar{\epsilon},\Sigma_{\epsilon})$ with mean $\bar{\epsilon}$ and covariance operator $\Sigma_{\epsilon}$. In real world applications, we would like to use a fast forward solver (limited accuracy) to obtain an estimation as accurately as possible. Hence, the noise usually not only brought by inaccurate measurements but also induced by a rough forward solver and inaccurate physical assumptions [@Calvetti2017]. Let us denote $\mathcal{F}_{a}(\cdot)$ to be the forward operator related to some rough forward solver, then (\[forwardForm\]) can be rewrite as follows by following the methods used in [@Koponen2014IEEE] $$\begin{aligned} \label{forwardForm2} d = \mathcal{F}_{a}(m) + (\mathcal{F}(m) - \mathcal{F}_{a}(m)) + \epsilon.\end{aligned}$$ By denoting $\xi := (\mathcal{F}(m) - \mathcal{F}_{a}(m))$, we obtain that $$\begin{aligned} \label{forwardForm3} d = \mathcal{F}_{a}(m) + \xi + \epsilon.\end{aligned}$$ From the perspective of Bayesian methods, we can model $\xi$ as a random variable which obviously has the following two important features 1. $\xi$ depend on the unknown function $m$; 2. $\xi$ may distributed according to a complicated probability measure. For feature (1), we can relax this tough problem to assume that $\xi$ is independent of $m$ but the probability distribution of $\xi$ and the prior probability measure of $m$ are related with each other [@Lasanen2012IPI]. For feature (2), to the best of our knowledge, the existing literatures only provide a compromised methods that is assume $\xi$ sampled from some Gaussian probability distributions [@Junxiong2016; @Koponen2014IEEE]. Here, we attempt to provide a more realistic assumptions for the probability measures of the random variable $\xi$. Noticing that Bayes’ formula is also one of fundamental tools for investigations about statistical machine learning [@PR2006Book] which is a field attracts numerous researchers from various fields, e.g., computer science, statistics and mathematics. Notice that for problems such as background subtraction [@Yong2017IEEE], low-rank matrix factorization [@Zhao2015IEEE] and principle component analysis [@MENG2012487; @Zhao2014ICML], learning algorithms deduced by Bayes’ formula are useful and the errors brought by inaccurate forward modeling also appears. For modeling errors appeared in machine learning tasks, Gaussian mixture model is widely used since it can approximate any probability measure in some sense [@PR2006Book]. Gaussian mixture distributions usually have the following form of density function $$\begin{aligned} \sum_{k = 1}^{K}\pi_{k} \mathcal{N}(\cdot \,| \,\zeta_{k},\Sigma_{k}),\end{aligned}$$ where $\mathcal{N}(\cdot \,| \,\zeta_{k},\Sigma_{k})$ stands for a Gaussian probability density function with mean value $\zeta_{k}$ and covariance matrix $\Sigma_{k}$ and for every $k$, $\pi_{k}\in (0,1)$ satisfy $\sum_{k=1}^{K}\pi_{k} = 1$. In the following, we always assume that the measurement noise $\epsilon$ is a Gaussian random variable with mean $0$ and covariance matrix $\nu I$ ($\nu \in \mathbb{R}^{+}$ and $I$ is an identity matrix). For our problem, we can intuitively provide the following optimization problem if we assume $\xi$ sampled from some Gaussian mixture probability distributions $$\begin{aligned} \label{modelQ1} \min_{m\in X}\Bigg\{-\ln\Big( \sum_{k = 1}^{K}\pi_{k} \mathcal{N}(d-\mathcal{F}_{a}(m) \,| \, \zeta_{k},\Sigma_{k} + \nu I) \Big) + \mathcal{R}(m) \Bigg\}.\end{aligned}$$ In the machine learning field, there usually have a lot of sampling data and the forward problems are not computationally intensive compared with the inverse medium scattering problem. Hence, they use alternative iterative methods to find the optimal solution and estimate the modeling error simultaneously [@Zhao2015IEEE]. However, considering the lack of learning data and the high computational load of our forward problems, we can not trivially generalize their alternative iterative methods to our case. In order to employ Gaussian mixture distribution, we will meet the following three problems 1. Under which conditions, Bay’s formula and MAP estimate with Gaussian mixture distribution hold in infinite-dimensional space; 2. How to construct learning examples and how to learn the parameters in Gaussian mixture distributions. Firstly, since we can hardly have so many learning examples as for the usual machine learning problem, we will meet a situation that is the number of learning examples are smaller than the number of discrete points which is also an ill-posed problem. Secondly, the solution of Helmholtz equation is a complex valued function. Because of that, we should develop learning algorithms for complex valued variables which is different to the classical cases for machine learning tasks [@PR2006Book; @Yong2017IEEE]. 3. For the complicated minimization problem (\[modelQ1\]), how to construct a suitable iterative type method, i.e., some modified RLM. In this paper, we provide a primitive study about these three problems. Theoretical foundations for using Gaussian mixture distributions in infinite-dimensional space problems have been established. Learning algorithm has been designed based on the relationships between real Gaussian distribution and complex Gaussian distribution. By carefully calculations, a modified RLM name as Gaussian mixture recursive linearization method (GMRLM) has been proposed to efficiently solve the inverse medium problem with multi-frequencies data. Numerical examples are finally reported to illustrate the effectiveness of the proposed method. The outline of this paper is as follows. In Section 2, general Bayesian inverse method with Gaussian mixture noise model is established and the relationship between MAP estimators with classical regularization methods is also discussed. In Section 3, well-posedness of inverse medium scattering problem in the Bayesian sense is proved. Then, we propose the learning algorithm for Gaussian mixture distribution by generalizing the real valued expectation-maximization (EM) algorithm to complex valued EM algorithm. At last, we deduce the Gaussian mixture recursive linearization method. In Section 4, two typical numerical examples are given, which illustrate the effectiveness of the proposed methods. Bayesian inverse theory with Gaussian mixture distribution {#BayeTheoSection} ========================================================== In this section, we prove the well-posedness and illustrate the validity of MAP estimate of inverse problems under the Bayesian inverse framework when the noise is assumed to be a random variable sampled from a complex valued Gaussian mixture distribution. Before diving into the main contents, let us provide a brief notation list which will be used in all of the following parts of this paper. **Notations:** - For an integer $N$, denote $\mathbb{C}^{N}$ as $N$-dimensional complex vector space; $\mathbb{R}^{+}$ and $\mathbb{N}^{+}$ represent positive real numbers and positive integers respectively; - For a Banach space $X$, $\|\cdot\|_{X}$ stands for the norm defined on $X$ and, particularly, $\|\cdot\|_{2}$ represents the $\ell^{2}$-norm of $\ell^{2}$ space. - For a matrix $\Sigma$, denote its determinant as $\det(\Sigma)$; - Denote $B(m,R)$ as a ball with center $m$ and radius $R$. Particularly, denote $B_{R} := B(0,R)$ when the ball is centered at origin; - Denote $X$ and $Y$ to be some Banach space; For an operator $F :\, X \rightarrow Y$, denote $F'(x_{0})$ as the Fréchet derivative of $F$ at $x_{0} \in X$. - Denote $\text{Re}(\xi)$, $\text{Imag}(\xi)$, $\xi^{T}$, $\xi^{H}$ and $\bar{\xi}$ as the real part, imaginary part, transpose, conjugate transpose and complex conjugate of $\xi \in \mathbb{C}^{N}$ respectively; - The notation $\eta \sim p(\eta)$ stands for a random variable $\eta$ obeys the probability distribution with density function $p(\cdot)$. Let $\mathcal{N}_{c}(\eta \,|\, \zeta,\Sigma)$ represents the density function of $N_{d}$-dimensional complex valued Gaussian probability distribution [@Goodman1963Annals] defined as follows $$\begin{aligned} \mathcal{N}_{c}(\eta \, | \, \zeta,\Sigma) := \frac{1}{(\pi)^{N_{d}}\det(\Sigma)} \exp\left( -\Big\|\eta-\zeta\Big\|_{\Sigma}^{2} \right),\end{aligned}$$ where $\zeta$ is a $N_{d}$-dimensional complex valued vector, $\Sigma$ is a positive definite Hermitian matrix and $\|\cdot\|_{\Sigma}^{2}$ is defined as follow $$\begin{aligned} \big\|\eta-\zeta\big\|_{\Sigma}^{2} := \big(\eta-\zeta\big)^{H} \, \Sigma^{-1} \, \big(\eta-\zeta\big),\end{aligned}$$ with the superscript $H$ stands for conjugate transpose. Denote $\eta := \xi + \epsilon$, then formula (\[forwardForm3\]) can be written as follows $$\begin{aligned} d = \mathcal{F}_{a}(m) + \eta,\end{aligned}$$ where $$\begin{aligned} d \in \mathbb{C}^{N_{d}}, \quad \eta \sim \sum_{k = 1}^{K}\pi_{k}\mathcal{N}_{c}(\eta \,|\, \zeta_{k},\Sigma_{k} + \nu I),\end{aligned}$$ with $N_{d}$, $K$ denote some positive integers and $\nu \in \mathbb{R}^{+}$. Before going further, let us provide the following basic assumptions about the approximate forward operator $\mathcal{F}_{a}$. **Assumption 1.** 1. for every $\epsilon > 0$ there is $M = M(\epsilon) \in \mathbb{R}$, $C\in\mathbb{R}$ such that, for all $m \in X$, $$\begin{aligned} \|\mathcal{F}_{a}(m)\|_{2} \leq C \exp(\epsilon\|m\|_{X}^{2} + M). \end{aligned}$$ 2. for every $r > 0$ there is $K = K(r) > 0$ such that, for all $m \in X$ with $\|m\|_{X} < r$, we have $$\begin{aligned} \|\mathcal{F}_{a}'(m)\|_{op} \leq K, \end{aligned}$$ where $\|\cdot\|_{op}$ denotes the operator norm. At this stage, we need to provide some basic notations of the Bayesian inverse method when $m$ in some infinite-dimensional space. Following the work [@inverse_fluid_equation; @acta_numerica], let $\mu_{0}$ stands for the prior probability measure defined on a separable Banach space $X$ and denote $\mu^{d}$ to be the posterior probability measure. Then the Bayes’ formula may be written as follows $$\begin{aligned} \frac{d\mu^{d}}{d\mu_{0}}(m) & = \frac{1}{Z(d)} \exp\Big( \Phi(m;d) \Big), \label{DefineMuY} \\ Z(d) & = \int_{X} \exp\Big( \Phi(m;d) \Big)\mu_{0}(dm), \label{DefineOfZd}\end{aligned}$$ where $\frac{d\mu^{d}}{d\mu_{0}}(\cdot)$ represents the Radon-Nikodym derivative and $$\begin{aligned} \Phi(m;d) := \ln\Bigg\{\sum_{k = 1}^{K} \pi_{k} \frac{1}{\pi^{N_{d}}\det(\Sigma_{k} + \nu I)} \exp\left( -\Big\|d-\mathcal{F}_{a}(m)-\zeta_{k}\Big\|_{\Sigma_{k} + \nu I}^{2} \right) \Bigg\}.\end{aligned}$$ Well-posedness -------------- In this subsection, we prove the following results which demonstrate formula (\[DefineMuY\]) and (\[DefineOfZd\]) under some general conditions. \[wellPosedBaye\] Let Assumption 1 holds for some $\epsilon$, $r$, $K$ and $M$. Assume that $X$ is some separable Banach space, $\mu_{0}(X) = 1$ and that $\mu_{0}(X\cap B) > 0$ for some bounded set $B$ in $X$. In addition, we assume $\int_{X} \exp(2\epsilon \|m\|_{X}^{2}) \mu_{0}(dm) < \infty$. Then, for every $d \in \mathbb{C}^{N_{d}}$, $Z(d)$ given by (\[DefineOfZd\]) is positive and the probability measure $\mu^{d}$ given by (\[DefineMuY\]) is well-defined. In addition, there is $C = C(r) > 0$ such that, for all $d_{1}, d_{2} \in B(0,r)$ $$\begin{aligned} d_{\text{Hell}}(\mu^{d_{1}}, \mu^{d_{2}}) \leq C \|d_{1} - d_{2}\|_{2},\end{aligned}$$ where $d_{\text{Hell}}(\cdot,\cdot)$ denotes the Hellinger distance defined for two probability measures. In order to prove this theorem, we need to verify three conditions stated in Assumption 4.2 and Theorem 4.4 in [@Dashti2014]. Since $$\begin{aligned} \sum_{k = 1}^{K} \pi_{k} \frac{1}{\pi^{N_{d}}\det(\Sigma_{k}+\nu I)} \exp\left( -\Big\|d-\mathcal{F}_{a}(m)-\zeta_{k}\Big\|_{\Sigma_{k}+\nu I}^{2} \right) \leq 1,\end{aligned}$$ we know that $$\begin{aligned} \label{Cond1} \Phi(m;d) \leq 0.\end{aligned}$$ In the following, we denote $$\begin{aligned} f_{k}(d,m):= \big(d-\mathcal{F}_{a}(m)-\zeta_{k}\big)^{H} \, \big(\Sigma_{k}+\nu I\big)^{-1} \, \big(d-\mathcal{F}_{a}(m)-\zeta_{k}\big).\end{aligned}$$ Then, we have $$\begin{aligned} \nabla_{d}f_{k}(d,m) & = (d - \mathcal{F}_{a}(m) - \zeta_{k})^{H}(\Sigma_{k}+\nu I)^{-1} + \overline{(d-\mathcal{F}_{a}(m)-\zeta_{k})^{H}(\Sigma_{k}+\nu I)^{-1}} \\ & = 2\text{Re}\Big( (d - \mathcal{F}_{a}(m) - \zeta_{k})^{H}(\Sigma_{k}+\nu I)^{-1} \Big).\end{aligned}$$ Through some simple calculations, we find that $$\begin{aligned} \label{DdPhi1} \nabla_{d}\Phi(m;d) = - \sum_{k=1}^{K} 2 g_{k} \text{Re}\Big( (d - \mathcal{F}_{a}(m) - \zeta_{k})^{H}(\Sigma_{k}+\nu I)^{-1} \Big),\end{aligned}$$ where $$\begin{aligned} \label{gkDef} g_{k} := \frac{\pi_{k}\mathcal{N}_{c}(d-\mathcal{F}_{a}(m) \,|\, \zeta_{k},\Sigma_{k}+\nu I)} {\sum_{j=1}^{K}\pi_{j}\mathcal{N}_{c}(d-\mathcal{F}_{a}(m) \,|\, \zeta_{j},\Sigma_{j}+\nu I)}.\end{aligned}$$ From the expression (\[DdPhi1\]) and (i) of Assumption 1, we can deduce that $$\begin{aligned} \label{BoundDd} \|\nabla_{d}\Phi(m;d)\|_{2} \leq C\big( 1 + \|d\|_{2} + \exp(\epsilon\|m\|_{X}^{2}) \big).\end{aligned}$$ where the constant $C$ depends on $K$, $\{\Sigma_{k}\}_{k=1}^{K}$ and $\{\zeta_{k}\}_{k = 1}^{K}$. Considering (\[BoundDd\]), we obtain $$\begin{aligned} \label{Cond2} |\Phi(m;d_{1}) - \Phi(m;d_{2})| \leq C\big( 1 + r + \exp(\epsilon\|m\|_{X}^{2}) \big) \|d_{1} - d_{2}\|_{2}.\end{aligned}$$ By our assumptions, the following relation obviously hold $$\begin{aligned} \label{Cond3} C^{2}\big( 1 + r + \exp(\epsilon\|m\|_{X}^{2}) \big)^{2} \in L_{\mu_{0}}^{1}(X;\mathbb{R}).\end{aligned}$$ At this stage, estimates (\[Cond1\]), (\[Cond2\]) and (\[Cond3\]) verify Assumption 4.2 and conditions of Theorem 4.4 in [@Dashti2014]. Employing theories constructed in [@Dashti2014], we complete the proof. The assumptions of the prior probability measure are rather general, which include Gaussian probability measure and TV-Gaussian probability measure [@TGPrior2016] for certain space $X$. MAP estimate ------------ Through MAP estimate, Bayesian inverse method and classical regularization method are in accordance with each other. Because our aim is to develop an efficient optimization method, we need to demonstrate the validity of MAP estimate which provide theoretical foundations for our method. Firstly, let us assume that the prior probability measure $\mu_{0}$ is a Gaussian probability measure and define the following functional $$\begin{aligned} \label{MiniProForm} J(m) = \left \{\begin{aligned} & -\Phi(m;d) + \frac{1}{2} \|m\|_{E}^{2} \quad \text{if }m\in E, \text{ and} \\ & + \infty, \quad\quad\quad\quad\quad\quad\quad\,\,\, \text{else.} \end{aligned}\right.\end{aligned}$$ Here $(E,\|\cdot\|_{E})$ denotes the Cameron-Martin space associated to $\mu_{0}$. In infinite dimensions, we adopt small ball approach constructed in [@MAPSmall2013]. For $m \in E$, let $B(m,\delta) \in X$ be the open ball centred at $m \in X$ with radius $\delta$ in $X$. Then, we can prove the following theorem which encapsulates the idea that probability is maximized where $J(\cdot)$ is minimized. \[SmallBall\] Let Assumption 1 holds and assume that $\mu_{0}(X) = 1$. Then the function $J(\cdot)$ defined by (\[MiniProForm\]) satisfies, for any $m_{1}, m_{2} \in E$, $$\begin{aligned} \lim_{\delta\rightarrow 0}\frac{\mu(B(m_{1},\delta))}{\mu(B(m_{2},\delta))} = \exp\Big( J(m_{2}) - J(m_{1}) \Big).\end{aligned}$$ In order to prove this theorem, let us verify the following two conditions concerned with $\Phi(m;d)$, 1. for every $r > 0$ there exists $K = K(r) > 0$ such that, for all $m \in X$ with $\|m\|_{X} \leq r$ we have $\Phi(m;d) \geq K$. 2. for every $r > 0$ there exists $L = L(r) > 0$ such that, for all $m_{1}, m_{2} \in X$ with $\|m_{1}\|_{X}, \|m_{2}\|_{X} < r$ we have $|\Phi(m_{1};d) - \Phi(m_{2};d)| \leq L \|m_{1} - m_{2}\|_{X}$. For the first condition, by employing Jensen’s inequality, we have $$\begin{aligned} \Phi(m;d) & = \ln\Big( \sum_{k = 1}^{K}\pi_{k} \mathcal{N}_{c}\big(d - \mathcal{F}_{a}(m) \, | \, \zeta_{k}, \Sigma_{k}+\nu I\big) \Big) \\ & \geq \sum_{k = 1}^{K} \pi_{k} \ln\Bigg( \frac{1}{\pi^{N_{d}}|\Sigma_{k} + \nu I|} \exp\bigg( -\big\| d-\mathcal{F}_{a}(m) - \zeta_{k} \big\|_{\Sigma_{k}+\nu I}^{2} \bigg) \Bigg) \\ & \geq \sum_{k = 1}^{K} \pi_{k} \Big( -\big\| d - \mathcal{F}_{a}(m) - \zeta_{k} \big\|_{\Sigma_{k}+\nu I}^{2} - N_{d}\ln(\pi) - \ln(|\Sigma_{k}+\nu I|) \Big) \\ & \geq - C \big( 1 + \|d\|_{2}^{2} + \exp(\epsilon r^{2}) \big),\end{aligned}$$ where $C$ is a positive constant depends on $K$, $\{\pi_{k}\}_{k=1}^{K}$, $\{\Sigma_{k}\}_{k = 1}^{K}$, $\{\zeta_{k}\}_{k=1}^{K}$ and $N_{d}$. Now, the first condition holds true by choosing $K = - C \big( 1 + \|d\|_{2}^{2} + \exp(\epsilon r^{2}) \big)$. In order to verify the second condition, we denote $$\begin{aligned} f_{k}(d,m):= \big(d-\mathcal{F}_{a}(m)-\zeta_{k}\big)^{H} \, \big(\Sigma_{k}+\nu I\big)^{-1} \, \big(d-\mathcal{F}_{a}(m)-\zeta_{k}\big),\end{aligned}$$ then focus on the derivative of $f_{k}$ with respect to $m$. Through some calculations, we find that $$\begin{aligned} \nabla_{m}f_{k}(d,m) = - 2 \text{Re}\Big( (d-\mathcal{F}_{a}(m)-\zeta_{k})^{H}(\Sigma_{k}+\nu I)^{-1}\mathcal{F}_{a}'(m) \Big).\end{aligned}$$ Hence, we have $$\begin{aligned} \label{DePhi1} \nabla_{m}\Phi(m;d) = - \sum_{k=1}^{K}2 g_{k} \text{Re}\Big( (d-\mathcal{F}_{a}(m)-\zeta_{k})^{H}(\Sigma_{k}+\nu I)^{-1}\mathcal{F}_{a}'(m) \Big),\end{aligned}$$ where $g_{k}$ defined as in (\[gkDef\]). Using Assumption 1 and formula (\[DePhi1\]), we find that $$\begin{aligned} |\Phi(m_{1};d) - \Phi(m_{2};d)| \leq C K (1+\|d\|_{2} + \exp(\epsilon \, r^{2})) \|m_{1} - m_{2}\|_{X}.\end{aligned}$$ Let $L = C K (1+\|d\|_{2} + \exp(\epsilon \, r^{2}))$, obviously the second condition holds true. Combining these two conditions with (\[Cond1\]), we can complete the proof by using Theorem 4.11 in [@Dashti2014]. Now, if we assume $\mu_{0}$ is a TV-Gaussian probability measure, then we can define the following functional $$\begin{aligned} \label{MiniProFormTG} J(m) = \left \{\begin{aligned} & -\Phi(m;d) + \lambda \|m\|_{\text{TV}} + \frac{1}{2} \|m\|_{E}^{2} \quad \text{if }m\in E, \text{ and} \\ & + \infty, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\, \text{else.} \end{aligned}\right.\end{aligned}$$ Using similar methods as for the Gaussian case and the above functional (\[MiniProFormTG\]), we can prove a similar theorem to illustrate that the MAP estimate is also the minimal solution of $\min_{m\in X}J(m)$. Inverse medium scattering problem {#SecInverMedi} ================================= In this section, we will apply the general theory developed in Section \[BayeTheoSection\] to a specific inverse medium scattering problem. Then we construct algorithms to learn the parameters appeared in the complex Gaussian mixture distribution. At last, we give the model error compensation based recursive linearization method. Since the model errors are estimated by some Gaussian mixture distributions, the proposed iterative method is named as Gaussian mixture recursive linearization method (GMRLM). Now, let us provide some basic settings of the inverse scattering problem considered in this paper. In the following, we usually assume that the total field $u$ satisfies $$\begin{aligned} \label{zongEq} \Delta u + \kappa^{2}(1+q)u = 0 \quad \text{in }\mathbb{R}^{2},\end{aligned}$$ where $\kappa > 0$ is the wavenumber, and $q(\cdot)$ is a real function known as the scatterer representing the inhomogeneous medium. We assume that the scatterer has a compact support contained in the ball $B_{R} = \{ \mathbf{r}\in\mathbb{R}^{2} : \, |\mathbf{r}| < R \}$ with boundary $\partial B_{R} = \{ \mathbf{r}\in\mathbb{R}^{2} : \, |\mathbf{r}| = R \}$, and satisfies $-1 < q_{\text{min}} \leq q \leq q_{\text{max}} < \infty$, where $q_{\text{min}}$ and $q_{\text{max}}$ are two constants. The scatterer is illuminated by a plane incident field $$\begin{aligned} u^{\text{inc}}(\mathbf{r}) = e^{i\kappa\mathbf{r}\cdot\mathbf{d}},\end{aligned}$$ where $\mathbf{d} = (\cos\theta, \sin\theta) \in \{ \mathbf{r}\in\mathbb{R}^{2} \, : \, |\mathbf{r}| = 1 \}$ is the incident direction and $\theta \in (0,2\pi)$ is the incident angle. Obviously, the incident field satisfies $$\begin{aligned} \label{inEq} \Delta u^{\text{inc}} + \kappa^{2} u^{\text{inc}} = 0 \quad \text{in }\mathbb{R}^{2}.\end{aligned}$$ The total field $u$ consists of the incident field $u^{\text{inc}}$ and the scattered field $u^{s}$ $$\begin{aligned} \label{fenjieEq} u = u^{\text{inc}} + u^{s}.\end{aligned}$$ It follows from (\[zongEq\]), (\[inEq\]) and (\[fenjieEq\]) that the scattered field satisfies $$\begin{aligned} \label{scatterEq} \Delta u^{s} + \kappa^{2}(1+q)u^{s} = -\kappa^{2}qu^{\text{inc}} \quad \text{in }\mathbb{R}^{2},\end{aligned}$$ accompanied with the following Sommerfeld radiation condition $$\begin{aligned} \label{radiatEq} \lim_{|\mathbf{r}| \rightarrow \infty}r^{1/2}\big(\partial_{r}u^{s} - i\kappa u^{s}\big) = 0,\end{aligned}$$ where $r = |\mathbf{r}|$. Well-posedness in the sense of Bayesian formulation {#WellSubsec} --------------------------------------------------- In this subsection, we suppose that the scatterer $q(\cdot)$ appeared in (\[zongEq\]) has compact support and $\text{supp}(q) \subset \Omega \subset B_{R}$ where $\Omega$ is a square region. For the reader’s convenience, we provide an illustration of this relation in Figure \[illuFig\]. \ Because the scatterer $q(\cdot)$ is assumed to have compact support, the problem (\[scatterEq\]) and (\[radiatEq\]) defined on $\mathbb{R}^{2}$ can be reformulated to the following problem defined on bounded domain [@Bao2015TopicReview] $$\begin{aligned} \label{BoundedHelEq} \left \{\begin{aligned} & \Delta u^{s} + \kappa^{2}(1+q)u^{s} = -\kappa^{2}qu^{\text{inc}} \quad \text{in }B_{R}, \\ & \partial_{\mathbf{n}}u^{s} = \mathcal{T}u^{s} \quad \text{on }\partial B_{R}, \end{aligned}\right.\end{aligned}$$ where $\mathcal{T}$ is the Dirichlet-to-Neumann (DtN) operator defined as follows: for any $\varphi \in H^{1/2}(\partial B_{R})$, $$\begin{aligned} (\mathcal{T}\varphi)(R,\theta) = \kappa\sum_{n\in\mathbb{Z}}\frac{H^{(1)'}_{n}(\kappa R)}{H^{(1)}_{n}(\kappa R)}\hat{\varphi}_{n}e^{in\theta}\end{aligned}$$ with $H^{(1)}_{n}$ is the Hankel function of the first kind with order $n$ and $$\begin{aligned} \hat{\varphi}_{n} = (2\pi)^{-1}\int_{0}^{2\pi} \varphi(R,\theta)e^{-in\theta}d\theta.\end{aligned}$$ For problem (\[BoundedHelEq\]), we define the map $\mathcal{S}(q,\kappa)u^{\text{inc}}$ by $u^{s} = \mathcal{S}(q,\kappa)u^{\text{inc}}$ as in [@Bao2015TopicReview]. From [@Bao2010StochasticSource; @ColtonThirdBook], we easily know that the following estimate holds for equations (\[BoundedHelEq\]) $$\begin{aligned} \label{EstimateH} \|u^{s}\|_{H^{2}(\Omega)} \leq C \|q\|_{L^{\infty}(\Omega)}\|u^{\text{inc}}\|_{L^{2}(B(0,R))}.\end{aligned}$$ Considering Sobolev embedding theorem, we can define the following measurement operator $$\begin{aligned} \label{MeaOp1} \mathcal{M}(\mathcal{S}(q,\kappa)u^{\text{inc}})(x) = \big( u^{s}(x_{1}), \ldots, u^{s}(x_{N_{d}}) \big)^{T},\end{aligned}$$ where $x_{i} \in \partial\Omega$, $i = 1,2,\ldots,N_{d}$, are the points where the wave field $u^{s}$ is measured. In practice, we employ a uniaxial PML technique to transform the problem defined on the whole domain to a problem defined on a bounded rectangular domain, as seen in Figure \[illuFig2\]. \ Let $D$ be the rectangle which contain $\Omega = [x_{1},x_{2}] \times [y_{1},y_{2}]$ with $\text{supp} (q) \subset \Omega$ and let $d_{1}$ and $d_{2}$ be the thickness of the PML layers along $x$ and $y$, respectively. Let $s_{1}(x) = 1+i\sigma_{1}(x)$ and $s_{2}(y) = 1+i\sigma_{2}(y)$ be the model medium property and usually we can simply take $$\begin{aligned} \sigma_{1}(x) = \left\{\begin{aligned} & \sigma_{0}\left( \frac{x-x_{2}}{d_{1}} \right)^{p} \quad \text{for }x_{2} < x < x_{2} + d_{1} \\ & 0 \quad\quad\quad\quad\quad\quad\,\,\,\,\, \text{for }x_{1} \leq x \leq x_{2} \\ & \sigma_{0}\left( \frac{x_{1} - x}{d_{1}} \right)^{p} \quad \text{for }x_{1} - d_{1} < x < x_{1}, \end{aligned}\right.\end{aligned}$$ and $$\begin{aligned} \sigma_{2}(y) = \left\{\begin{aligned} & \sigma_{0}\left( \frac{y-y_{2}}{d_{2}} \right)^{p} \quad \text{for }y_{2} < y < y_{2} + d_{2} \\ & 0 \quad\quad\quad\quad\quad\quad\,\,\,\, \text{for }y_{1} \leq y \leq y_{2} \\ & \sigma_{0}\left( \frac{y_{1} - y}{d_{2}} \right)^{p} \quad \text{for }y_{1} - d_{2} < y < y_{1}, \end{aligned}\right.\end{aligned}$$ where the constant $\sigma_{0} > 1$ and the integer $p \geq 2$. Denote $$s = \text{diag}(s_{2}(x)/s_{1}(x), s_{1}(x)/s_{2}(y)),$$ then the truncated PML problem can be defined as follow $$\begin{aligned} \label{PMLBoundedHelEq} \left \{\begin{aligned} & \nabla\cdot(s \nabla u^{s}) + s_{1}s_{2}\kappa^{2}(1+q)u^{s} = -\kappa^{2}qu^{\text{inc}} \quad \text{in }D, \\ & u^{s} = 0 \quad \text{on }\partial D. \end{aligned}\right.\end{aligned}$$ Similar to the physical problem (\[BoundedHelEq\]), we introduce the map $\mathcal{S}_{a}(q,\kappa)$ defined by $u^{s}_{a} = \mathcal{S}_{a}(q,\kappa)u^{\text{inc}}$ where $u^{s}_{a}$ stands for the solution of the truncated PML problem (\[PMLBoundedHelEq\]). Through similar methods for equations (\[BoundedHelEq\]), we can prove that $u^{s}_{a}$ is a continuous function and satisfies $$\begin{aligned} \label{EstimateHDis} \|u^{s}_{a}\|_{L^{\infty}(D)} \leq C \|q\|_{L^{\infty}(D)} \|u^{\text{inc}}\|_{L^{2}(D)}.\end{aligned}$$ Now, we can define the measurement operator similar to (\[MeaOp1\]) as follow $$\begin{aligned} \label{MeaOp2} \mathcal{M}(\mathcal{S}_{a}(q,\kappa)u^{\text{inc}})(x) = \big( u^{s}_{a}(x_{1}), \ldots, u^{s}_{a}(x_{N_{d}}) \big)^{T},\end{aligned}$$ where $x_{i} \in \partial D$, $i = 1,2,\ldots,N_{d}$. In order to introduce appropriate Gaussian probability measures, we give the following assumptions related to the covariance operator. **Assumption 2.** Denote $A$ to be an operator, densely defined on the Hilbert space $\mathcal{H} = L^{2}(D;\mathbb{R}^{d})$, satisfies the following properties: 1. $A$ is positive-definite, self-adjoint and invertible; 2. the eigenfunctions $\{\varphi_{j}\}_{j\in\mathbb{N}}$ of $A$, form an orthonormal basis for $\mathcal{H}$; 3. the eigenvalues satisfy $\alpha_{j} \asymp j^{2/d}$, for all $j\in\mathbb{N}$; 4. there is $C > 0$ such that $$\begin{aligned} \sup_{j\in\mathbb{N}}\left( \|\varphi_{j}\|_{L^{\infty}} + \frac{1}{j^{1/d}}\text{Lip}(\varphi_{j}) \right) \leq C. \end{aligned}$$ At this moment, we can show well-posedness for inverse medium scattering problem with some Gaussian prior probability measures. For a constant $s > 1$, we consider the prior probability measure to be a Gaussian measure $\mu_{0} := \mathcal{N}(\bar{q},A^{-s})$ where $\bar{q}$ is the mean value and the operator $A$ satisfies Assumption 2. In addition, we take $X = C^{t}$ with $t < s - 1$. Then we know that $\mu_{0}(X) = 1$ by Example 2.19 shown in [@Dashti2014]. For the scattering problem, we can take $\mathcal{F}_{a}(q) = \mathcal{M}(\mathcal{S}_{a}(q,\kappa)u^{\text{inc}})$ and let the noise $\eta$ obeys a Gaussian mixture distribution with density function $$\sum_{k = 1}^{K}\pi_{k}\mathcal{N}_{c}(\eta \,|\, \zeta_{k},\Sigma_{k}+\nu I).$$ Then, the measured data $d \in \mathbb{C}^{N_{d}}$ are $$\begin{aligned} \label{DataSpecPro} d = \mathcal{F}_{a}(q) + \eta.\end{aligned}$$ \[BayeTheoScatter\] For the two dimensional problem (\[PMLBoundedHelEq\])(problem (\[BoundedHelEq\])), if we assume space $X$, $q \sim \mu_{0}$ and $\eta$ are specified as previous two paragraphes in this subsection. Then, the Bayesian inverse problems of recovering input $q \in X$ of problem (\[PMLBoundedHelEq\])(problem (\[BoundedHelEq\])) from data $d$ given as in (\[DataSpecPro\]) is well formulated: the posterior $\mu^{d}$ is well defined in $X$ and it is absolutely continuous with respect to $\mu_{0}$, the Radon-Nikodym derivative is given by (\[DefineMuY\]) and (\[DefineOfZd\]). Moreover, there is $C = C(r)$ such that, for all $d_{1},d_{2} \in \mathbb{C}^{N_{d}}$ with $|d_{1}|,|d_{2}| \leq r$, $$\begin{aligned} d_{Hell}(\mu^{d_{1}},\mu^{d_{2}}) \leq C \|d_{1} - d_{2}\|_{2}.\end{aligned}$$ From Section \[BayeTheoSection\], we easily know that Theorem \[BayeTheoScatter\] holds when Assumption 1 is satisfied. According to the estimates (\[EstimateHDis\]) and (\[MeaOp2\]), we find that $$\begin{aligned} \|\mathcal{F}_{a}(q)\|_{2} \leq C \|q\|_{L^{\infty}(D)},\end{aligned}$$ which indicates that statement (1) of Assumption 1 holds. In order to verify statement (2) of Assumption 1, we denote $u_{a}^{s} + \delta u = \mathcal{F}_{a}(q+\delta q)$. By simple calculations, we deduce that $\delta u$ satisfies $$\begin{aligned} \label{deltaUEq} \left \{\begin{aligned} & \nabla\cdot(s \nabla \delta u) + s_{1}s_{2}\kappa^{2}(1+q)\delta u = -\kappa^{2}\delta q(u^{\text{inc}} + s_{1}s_{2} u_{a}^{s}) \quad \text{in }D, \\ & \delta u = 0 \quad \text{on }\partial D. \end{aligned}\right.\end{aligned}$$ Now, denote $\mathcal{F}_{a}'(q)$ to be the Fréchet derivative of $\mathcal{F}_{a}(q)$, we find that $$\begin{aligned} \label{FDerQ} \mathcal{F}_{a}'(q)\delta q = \mathcal{M}(\delta u),\end{aligned}$$ where $\delta u$ is the solution of equations (\[deltaUEq\]). By using some basic estimates for equations (\[PMLBoundedHelEq\]), we obtain $$\begin{aligned} \label{FDerEst} \|\mathcal{F}_{a}'(q)\delta q\|_{2} \leq \|\delta u\|_{L^{\infty(D)}} \leq C(1+\|q\|_{L^{\infty}(\Omega)})\|\delta q\|_{L^{\infty}(D)},\end{aligned}$$ where $C$ depends on $\kappa$, $D$, $s_{1}$ and $s_{2}$. Estimate (\[FDerEst\]) ensures that statement (2) of Assumption 1 holds, and the proof is completed by employing Theorem \[wellPosedBaye\]. From the proof of Theorem \[BayeTheoScatter\], we can see that Theorem \[SmallBall\] holds true for inverse medium scattering problem considered in this subsection. Hence, we can compute the MAP estimate by minimizing functional defined in (\[MiniProForm\]) with the forward operator defined in (\[DataSpecPro\]). If we assume $\mu_{0}$ is a TV-Gaussian probability measure, similar results can be obtained. The posterior probability measure is well-defined and the MAP estimate can be obtained by solving $\min_{q\in X}J(q)$ with $J$ defined in (\[MiniProFormTG\]). Since there are no new ingredients, we omit the details. Learn parameters of complex Gaussian mixture distribution {#LearnSection} --------------------------------------------------------- How to estimate the parameters is one of the key steps for modeling noises by some complex Gaussian mixture distributions. This key step consists two fundamental elements: learning examples and learning algorithms. For the learning examples, they are the approximate errors $e := \mathcal{F}(q) - \mathcal{F}_{a}(q)$ that is the difference of measured values for slow explicit forward solver and fast approximate forward solver. In order to obtain this error, we need to know the unknown function $q$ which is impossible. However, in practical problems, we usually know some prior knowledge of the unknown function $q$. Relying on the prior knowledge, we can construct some probability measures to generate functions which we believe to maintain similar statistical properties as the real unknown function $q$. For this, we refer to a recent paper [@Iglesias2014IP]. Since this procedure depends on specific application fields, we only provide details in Section \[SecNumer\] for concrete numerical examples. For the learning algorithms, expectation-maximization (EM) algorithm is often employed in the machine learning community [@PR2006Book]. Here, we need to notice that the variables are complex valued and the complex Gaussian distribution are used in our case. This leads some differences to the classical real variable situation. In order to provide a clear explanation, let us recall some basic relationships between complex Gaussian distributions and real Gaussian distributions which are proved in [@Goodman1963Annals]. Denote $e = (e_{1}, \ldots, e_{N_{d}})^{T}$ is a $N_{d}$-tuple of complex Gaussian random variables. Let $\tau_{k} := \text{Re}(e_{k})$ and $\varsigma_{k} := \text{Imag}(e_{k})$ as the real and imaginary parts of $e_{k}$ with $k = 1,\ldots,N_{d}$, then define $$\begin{aligned} \label{defTau} \tau = (\tau_{1},\varsigma_{1},\ldots,\tau_{N_{d}},\varsigma_{N_{d}})\end{aligned}$$ is $2N_{d}$-tuple of random variables. From the basic theories of complex Gaussian distributions, we know that $\tau$ is $2N_{d}$-variate Gaussian distributed. Denote the covariance matrix of $e$ by $\Sigma$ and the covariance matrix of $\tau$ by $\tilde{\Sigma}$. As usual, we assume $\Sigma$ is a positive definite Hermitian matrix, then $\tilde{\Sigma}$ is a positive definite symmetric matrix by Theorem 2.2 and Theorem 2.3 in [@Goodman1963Annals]. In addition, we have the following lemma which is proved in [@Goodman1963Annals]. \[complexGauPro\] For complex Gaussian distributions, we have that the matrix $\Sigma$ is isomorphic to the matrix $2\tilde{\Sigma}$, $e^{H}\Sigma e = \tau^{T}\tilde{\Sigma}\tau$ and $\text{det}(\Sigma)^{2} = \text{det}(\tilde{\Sigma})$. Let $N_{s} \in \mathbb{N}^{+}$ stands for the number of learning examples. Let $e_{n} = (e_{1}^{n}, \ldots, e_{N_{d}}^{n})^{T}$ with $n = 1,\ldots,N_{s}$ represent $N_{s}$ learning examples. Then, for some fixed $K \in \mathbb{N}^{+}$, we need to solve the following optimization problem to obtain estimations of parameters $$\begin{aligned} \min_{\{\pi_{k}, \zeta_{k}, \Sigma_{k}\}_{k=1}^{K}} J_{G}(\{\pi_{k}, \zeta_{k}, \Sigma_{k}\}_{k =1}^{K}),\end{aligned}$$ where $$\begin{aligned} J_{G}(\{\pi_{k}, \zeta_{k}, \Sigma_{k}\}_{k =1}^{K}) := \sum_{n = 1}^{N_{s}} \ln \Bigg\{ \sum_{k = 1}^{K} \pi_{k} \mathcal{N}_{c}(e_{n} \, | \, \zeta_{k},\Sigma_{k}) \Bigg\}.\end{aligned}$$ In the following, we only show two different parts compared with the real variable Gaussian case. **Estimation of means**: Setting the derivatives of $J_{G}(\{\pi_{k}, \zeta_{k}, \Sigma_{k}\}_{k =1}^{K})$ with respect to $\zeta_{k}$ of the complex Gaussian components to zero and using Lemma \[complexGauPro\], we obtain $$\begin{aligned} 0 = -\sum_{n=1}^{N_{s}}\frac{\pi_{k}\mathcal{N}_{c}(e_{n}\,|\,\zeta_{k},\Sigma_{k})} {\sum_{j = 1}^{K}\pi_{j}\mathcal{N}_{c}(e_{j}\,|\,\zeta_{j},\Sigma_{j})}\tilde{\Sigma}_{k}^{-1}(\tau_{n} - \tilde{\zeta}_{k}),\end{aligned}$$ where $\tau_{n}$ defined as in (\[defTau\]) with $e$ replaced by $e_{n}$, $\tilde{\zeta}_{k}$ also defined as in (\[defTau\]) with $e$ replaced by $\zeta_{k}$ and $\tilde{\Sigma}_{k}$ is the covariance matrix corresponding to $\Sigma_{k}$. Hence, by some simple simplification, we find that $$\begin{aligned} \zeta_{k} = \frac{1}{\tilde{N}_{k}}\sum_{n=1}^{N_{s}}\gamma_{nk}e_{n},\end{aligned}$$ where $$\begin{aligned} \label{defineTilN} \tilde{N}_{k} = \sum_{n=1}^{N_{s}}\gamma_{nk}, \quad \gamma_{nk} = \frac{\pi_{k}\mathcal{N}_{c}(e_{n}\,|\,\zeta_{k},\Sigma_{k})} {\sum_{j = 1}^{K}\pi_{j}\mathcal{N}_{c}(e_{j}\,|\,\zeta_{j},\Sigma_{j})}.\end{aligned}$$ In the above formula, $\tilde{N}_{k}$ usually interpret as the effective number of points assigned to cluster $k$ and $\gamma_{nk}$ usually is a variable depend on latent variables [@PR2006Book]. **Estimation of covariances**: For the covariances, we need to use latent variables to provide the following complete-data log likelihood function as formula (9.40) shown in [@PR2006Book] $$\begin{aligned} \sum_{n = 1}^{N_{s}}\sum_{k = 1}^{K}\gamma_{nk}\Big\{ \ln\pi_{k} + \ln\mathcal{N}_{c}(e_{n} \, | \, \zeta_{k}, \Sigma_{k}) \Big\}.\end{aligned}$$ Now, for $k = 1,\dots,K$, we prove that $$\begin{aligned} \Sigma_{k} := \frac{1}{\tilde{N}_{k}}\sum_{n = 1}^{N_{s}}\gamma_{nk}(e_{n} - \zeta_{k})(e_{n} - \zeta_{k})^{H}\end{aligned}$$ solves the following maximization problem $$\begin{aligned} \label{ZuidaWen} \max_{\{\Sigma_{k}\}_{k=1}^{K}}\Bigg\{\sum_{n = 1}^{N_{s}}\sum_{k = 1}^{K}\gamma_{nk} \Big( \ln\pi_{k} + \ln\mathcal{N}_{c}(e_{n} \, | \, \zeta_{k}, \Sigma_{k}) \Big)\Bigg\}.\end{aligned}$$ Denote $$\begin{aligned} L = \sum_{n = 1}^{N_{s}}\sum_{k = 1}^{K}\gamma_{nk} \Big( \ln\pi_{k} + \ln\mathcal{N}_{c}(e_{n} \, | \, \zeta_{k}, \Sigma_{k}) \Big).\end{aligned}$$ Let $$\begin{aligned} B_{k} := \frac{1}{\tilde{N}_{k}}\sum_{n = 1}^{N_{s}}\gamma_{nk}(e_{n} - \zeta_{k})(e_{n} - \zeta_{k})^{H}\end{aligned}$$ and notice that $$\begin{aligned} \sum_{n=1}^{N_{s}}\sum_{k=1}^{K}\gamma_{nk}(e_{n}-\zeta_{k})^{H}\Sigma_{k}^{-1}(e_{n}-\zeta_{k}) & = \sum_{n=1}^{N_{s}}\sum_{k=1}^{K}\gamma_{nk}\text{tr}\Big( \Sigma_{k}^{-1}(e_{n}-\zeta_{k})(e_{n}-\zeta_{k})^{H} \Big) \\ & = \sum_{k=1}^{K}\text{tr}\Big( \Sigma_{k}^{-1}\sum_{n=1}^{N_{s}}\gamma_{nk}(e_{n}-\zeta_{k})(e_{n}-\zeta_{k})^{H} \Big) \\ & = \sum_{k=1}^{K}\tilde{N}_{k}\text{tr}\Big( \Sigma_{k}^{-1}B_{k} \Big),\end{aligned}$$ where $\tilde{N}_{k}$ defined as in (\[defineTilN\]). Then, using the explicit form of density function, we obtain $$\begin{aligned} \label{LDEF} L = - \sum_{k = 1}^{K}\tilde{N}_{k}\ln\text{det}(\Sigma_{k}) - \sum_{k=1}^{K}\tilde{N}_{k}\text{tr}(\Sigma_{k}^{-1}B_{k}) - N_{d}\ln\pi + \sum_{k=1}^{K}\tilde{N}_{k}\ln\pi_{k}.\end{aligned}$$ Define $p(\xi,\Sigma):= \frac{1}{\pi^{N_{d}}\text{det}(\Sigma)}\exp\left( -\xi^{H}\Sigma^{-1}\xi \right)$, then we have $$\begin{aligned} \label{JDEF} \begin{split} J & = \sum_{k=1}^{K} \tilde{N_{k}} \int_{\xi}p(\xi,\Sigma_{k}^{-1})\ln\Big( p(\xi,B_{k}^{-1})/p(\xi,\Sigma_{k}^{-1}) \Big) d\xi \\ & = \int_{\xi} \Bigg\{ \left( \ln\text{det}(B_{k}) - \xi^{H}B_{k}\xi \right)p(\xi,\Sigma_{k}^{-1}) \\ & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad - \left( \ln\text{det}(\Sigma_{k}) - \xi^{H}\Sigma_{k}\xi \right)p(\xi,\Sigma_{k}^{-1}) \Bigg\}d\xi \\ & = \sum_{k=1}^{K}\tilde{N}_{k}\ln\text{det}(B_{k}) + \sum_{k=1}^{K}\tilde{N}_{k}\text{tr}(I) \\ & \quad\quad\quad\quad\quad\quad\quad\quad\quad - \sum_{k=1}^{K}\tilde{N}_{k}\text{tr}(\Sigma_{k}^{-1}B_{k}) - \sum_{k=1}^{K}\tilde{N}_{k}\ln\text{det}(\Sigma_{k}), \end{split}\end{aligned}$$ where Corollary 4.1 in [@Goodman1963Annals] has been used for the last equality. On comparing the final result of (\[LDEF\]) with (\[JDEF\]) one observes that any series Hermitian positive definite matrixes $\{\Sigma_{k}\}_{k=1}^{K}$ that maximize $L$ maximize $J$ and conversely. Now, $\ln u \leq u-1$ with equality holding if and only if $u = 1$. Thus $$\begin{aligned} \label{Jleq1} \begin{split} J & = \sum_{k=1}^{K} \tilde{N_{k}} \int_{\xi}p(\xi,\Sigma_{k}^{-1})\ln\Big( p(\xi,B_{k}^{-1})/p(\xi,\Sigma_{k}^{-1}) \Big) d\xi \\ & \leq \sum_{k=1}^{K} \tilde{N_{k}} \int_{\xi} p(\xi,\Sigma_{k}^{-1})\Big( p(\xi,B_{k}^{-1})/p(\xi,\Sigma_{k}^{-1}) - 1 \Big) d\xi = 0. \end{split}\end{aligned}$$ If and only if $p(\xi,\Sigma_{k}) = p(\xi,B_{k})$ with $k=1,\ldots,K$, equality in (\[Jleq1\]) holds true. Hence, $\Sigma_{k} = B_{k}\, (k=1,\ldots,K)$ solves problem (\[ZuidaWen\]). With these preparations, we can easily construct EM algorithm following the line of reasoning shown in Chapter 9 of [@PR2006Book]. For concisely, the details are omitted and we provide the EM algorithm in Algorithm \[algComplexEM\]. In Algorithm \[algComplexEM\], if the parameters satisfy $N_{d} < N_{s}$, we can usually obtain nonsingular matrixes $\{\Sigma_{k}\}_{k=1}^{K}$. However, in our case, we can not generate so many learning examples $N_{s}$ and the number of measuring points $N_{d}$ is usually very large for real world applications. Hence, we will meet the situation $N_{d} > N_{s}$ which makes $\{\Sigma_{k}\}_{k=1}^{K}$ to be a series of singular matrixes. In order to solve this problem, we adopt a simple strategy that is replace the estimation of $\Sigma_{k}$ in Step 3 by the following formula $$\begin{aligned} \label{MStepReg} \Sigma_{k}^{\text{new}} = \frac{1}{\tilde{N}_{k}} \sum_{n=1}^{N_{s}}\gamma_{nk}(e_{n}-\zeta_{k}^{\text{new}}) (e_{n}-\zeta_{k}^{\text{new}})^{H} + \delta I,\end{aligned}$$ where $\delta$ is a small positive number named as the regularization parameter. Adjoint state approach with model error compensation ---------------------------------------------------- By Algorithm \[algComplexEM\], we obtain the estimated mixing coefficients, mean values and covariance matrixes. From the statements shown in Section \[BayeTheoSection\] and Subsection \[WellSubsec\], it is obviously that we need to solve optimization problems as follows $$\begin{aligned} \min_{q\in L^{\infty}(\Omega)} \Big\{ - \Phi(q;d) + \mathcal{R}(q) \Big\},\end{aligned}$$ where $$\begin{aligned} - \Phi(q;d)\! = \! - \ln\Bigg\{ \sum_{k=1}^{K}\pi_{k}\frac{1}{\pi^{N_{d}}\det(\Sigma_{k}+\nu I)} \exp\Big( -\frac{1}{2}\Big\| d-\mathcal{F}_{a}(q)-\zeta_{k} \Big\|_{\Sigma_{k}+\nu I}^{2} \Big) \Bigg\}, \\ \mathcal{R}(q) = \frac{1}{2}\|A^{s/2}q\|_{L^{2}(\Omega)}^{2}\, \quad \text{or} \quad \mathcal{R}(q) = \lambda \|q\|_{\text{TV}} + \frac{1}{2}\|A^{s/2}q\|_{L^{2}(\Omega)}^{2}. \label{DefFunR}\end{aligned}$$ Different form of functional $\mathcal{R}$ comes from different assumptions of the prior probability measures: Gaussian probability measure or TV-Gaussian probability measure. For the multi-frequency approach of inverse medium scattering problem, the forward operator in each optimization problem is related to $\kappa$. So we rewrite $\mathcal{F}_{a}(q)$ and $\Phi(q;d)$ as $\mathcal{F}_{a}(q,\kappa)$ and $\Phi(q,\kappa;d)$, which emphasize the dependence of $\kappa$. We have a series of wavenumbers $0 < \kappa_{1} < \kappa_{2} < \cdots \kappa_{N_{w}} < \infty$, and we actually need to solve a series optimization problems $$\begin{aligned} \label{opt1} \min_{q\in L^{\infty}(\Omega)} \Big\{ - \Phi(q,\kappa_{i};d) + \mathcal{R}(q) \Big\}\end{aligned}$$ with $i$ from $1$ to $N_{w}$ and the solution of the previous optimization problem is the initial data for the later optimization problem. Denote $F(q) = - \Phi(q,\kappa_{i};d)$. To minimize the cost functional by a gradient method, it is required to compute Fréchet derivative of functionals $F$ and $\mathcal{R}$. For functional $\mathcal{R}$ with form shown in (\[DefFunR\]), we can obtain the Fréchet derivatives as follows $$\begin{aligned} \label{DerR1} \mathcal{R}'(q) = A^{s}q, \quad \text{or} \quad \mathcal{R}'(q) = A^{s}q + 2\lambda \nabla\cdot\left( \frac{\nabla q}{\sqrt{|\nabla q|^{2} + \delta}} \right),\end{aligned}$$ where we used the following modified version of $\mathcal{R}$ $$\begin{aligned} \mathcal{R}(q) = \lambda\int_{\Omega}\sqrt{|\nabla q|^{2}+\delta} + \frac{1}{2}\|A^{s/2}q\|_{L^{2}(\Omega)}^{2}\end{aligned}$$ for the TV-Gaussian prior case and $\delta$ is a small smoothing parameter avoiding zero denominator in (\[DerR1\]). Next, we consider the functional $F$ with $\mathcal{F}_{a}$ is the forward operator related to problem (\[PMLBoundedHelEq\]). A simple calculation yields the derivative of $F$ at $q$; $$\begin{aligned} \label{zuihou0} F'(q)\delta q = \text{Re} \Big( \mathcal{M}(\delta u), \sum_{k = 1}^{K} \gamma_{k} (\Sigma_{k}+\nu I)^{-1}(d-\mathcal{F}_{a}(q,\kappa_{i})-\zeta_{k}) \Big),\end{aligned}$$ where $\delta u$ satisfy $$\begin{aligned} \label{deltaUEq2} \left \{\begin{aligned} & \nabla\cdot(s \nabla \delta u) + s_{1}s_{2}\kappa_{i}^{2}(1+q)\delta u = -\kappa^{2}\delta q(u^{\text{inc}} + s_{1}s_{2}u_{a}^{s}) \quad \text{in }D, \\ & \delta u = 0 \quad \text{on }\partial D, \end{aligned}\right.\end{aligned}$$ and $$\begin{aligned} \gamma_{k} = \frac{\pi_{k}\mathcal{N}_{c}(d-\mathcal{F}_{a}(q) \, | \, \zeta_{k},\Sigma_{k}+\nu I) }{\sum_{j = 1}^{K} \pi_{j}\mathcal{N}_{c}(d-\mathcal{F}_{a}(q) \, | \, \zeta_{j},\Sigma_{j}+\nu I). }\end{aligned}$$ To compute the Fréchet derivative, we introduce the adjoint system: $$\begin{aligned} \label{AdjSystem} \left \{\begin{aligned} & \nabla\cdot(\bar{s}\nabla v) + \bar{s}_{1}\bar{s}_{2}\kappa_{i}^{2}(1+q) v = - \kappa_{i}^{2} \sum_{j = 1}^{N_{d}}\delta(x-x_{j})\rho_{j} \quad \text{in }D, \\ & v = 0 \quad \text{on }D, \end{aligned}\right.\end{aligned}$$ where $\rho_{j} \, (j=1,\ldots,N_{d})$ denote the $j$th component of $\sum_{k = 1}^{K} \gamma_{k} (\Sigma_{k}+\nu I)^{-1}(d-\mathcal{F}_{a}(q,\kappa_{i})-\zeta_{k}) \in \mathbb{C}^{N_{d}}$. Multiplying equation (\[deltaUEq2\]) with the complex conjugate of $v$ on both sides and integrating over $D$ yields $$\begin{aligned} \int_{D}\nabla\cdot(s \nabla \delta u)\bar{v} + s_{1}s_{2}\kappa_{i}^{2}(1+q)\delta u \bar{v} = - \int_{D}\kappa^{2}\delta q(u^{\text{inc}} + s_{1}s_{2}u_{a}^{s})\bar{v}.\end{aligned}$$ By integration by parts formula, we obtain $$\begin{aligned} \int_{D}\delta u \Big( \nabla\cdot(s \nabla \bar{v}) + s_{1}s_{2}\kappa_{i}^{2}(1+q)\bar{v} \Big) = - \kappa_{i}^{2}\int_{D}\delta q (u^{\text{inc}} + s_{1}s_{2}u_{a}^{s})\bar{v}.\end{aligned}$$ Taking complex conjugate of equation (\[AdjSystem\]) and plugging into the above equation yields $$\begin{aligned} -\kappa_{i}^{2}\int_{D}\delta u \sum_{j = 1}^{N_{d}}\delta(x-x_{j})\bar{\rho}_{j} = - \kappa_{i}^{2}\int_{D}\delta q (u^{\text{inc}} + s_{1}s_{2}u_{a}^{s})\bar{v},\end{aligned}$$ which implies $$\begin{aligned} \label{zuihou1} \Big( \mathcal{M}(\delta u), \sum_{k = 1}^{K} \gamma_{k} \Sigma_{k}^{-1}(d-\mathcal{F}_{a}(q,\kappa_{i})-\zeta_{k}) \Big) = \int_{D}\delta q (u^{\text{inc}} + s_{1}s_{2}u_{a}^{s})\bar{v}.\end{aligned}$$ Considering both (\[zuihou0\]) and (\[zuihou1\]), we find that $$\begin{aligned} F'(q)\delta q = \text{Re}\int_{D}\delta q (u^{\text{inc}} + s_{1}s_{2}u_{a}^{s})\bar{v},\end{aligned}$$ which gives the Fréchet derivative as follow $$\begin{aligned} \label{Fdd1} F'(q) = \text{Re}\big( (\bar{u}^{\text{inc}} + \bar{s}_{1}\bar{s}_{2}\bar{u}_{a}^{s})v \big).\end{aligned}$$ With these preparations, it is enough to construct Gaussian mixture recursive linearization method (GMRLM) which is shown in Algorithm \[alg23\]. Notice that for the recursive linearization method (RLM) shown in [@Bao2015TopicReview], only one iteration of the gradient descent method for each fixed wavenumber can provide an acceptable recovery function. So we only iterative once for each fixed wavenumber. Numerical examples {#SecNumer} ================== In this section, we provide two numerical examples in two dimensions to illustrate the effectiveness of the proposed method. In the following, we assume that $\Omega = \{x\in\mathbb{R}^{2} \, : \, \|x\|_{2} \leq 1\}$ with $\Omega \subset D$ where $D$ is the PML domain with $d_{1} = d_{2} = 0.15$, $p = 2.5$ and $\sigma_{0} = 1.5$. For the forward solver, finite element method (FEM) has been employed and the scattering data are obtained by numerical solution of the forward scattering problem with adaptive mesh technique. For the following two examples, we choose $N_{w} = 20$ and $\textbf{d}_{j} \, (j = 1,\ldots,N_{w})$ are equally distributed around $\partial D$. Equally spaced wavenumbers are used, starting from the lowest wavenumber $\kappa_{\text{min}} = \pi$ and ending at the highest wavenumber $\kappa_{\text{max}} = 10\pi$. Denote by $\Delta\kappa = (\kappa_{\text{max}} - \kappa_{\text{min}})/9 = \pi$ the step size of the wavenumber; then the ten equally spaced wavenumbers are $\kappa_{j} = j\Delta\kappa$, $j = 1,\ldots,10$. We set $400$ receivers that equally spaced along the boundary of $\Omega$ as shown in Figure \[illuFig2\]. For the initial guess of the unknown function $q$, there are numerous strategies, i.e., methods based on Born approximation [@Bao2015TopicReview; @Bleistein2001Book]. Since the main point here is not on the initial gauss, we just set the initial $q$ to be a function always equal to zero for simplicity. In order to show the stability of the proposed method, some relative random noise is added to the data, i.e., $$\begin{aligned} u^{s}|_{\partial \Omega} := (1+\sigma \text{rand})u^{s}|_{\partial_{D}}.\end{aligned}$$ Here, rand gives uniformly distributed random numbers in $[-1,1]$ and $\sigma$ is a noise level parameter taken to be $0.02$ in our numerical experiments. Define the relative error by $$\begin{aligned} \text{Relative Error} = \frac{\|q-\tilde{q}\|_{L^{2}(\Omega)}}{\|q\|_{L^{2}(\Omega)}},\end{aligned}$$ where $\tilde{q}$ is the reconstructed scatterer and $q$ is the true scatterer. **Example 1**: For the first example, let $$\begin{aligned} \tilde{q}(x,y) = 0.3 (1-x)^{2}e^{-x^{2}-(y+1)^{2}} - (0.2x - x^{3} - y^{5})e^{-x^{2}-y^{2}} - 0.03 e^{-(x+1)^{2} - y^{2}}\end{aligned}$$ and reconstruct a scatterer defined by $$\begin{aligned} q(x,y) = \tilde{q}(3x,3y)\end{aligned}$$ inside the unit square $\{x\in\mathbb{R}^{2} \, : \, \|x\|_{2} < 1\}$. ![True scatterer and five typical learning examples[]{data-label="TrueLearnFig1"}](TrueLearnFig1.jpg "fig:"){width="100.00000%"}\ Denote $U[b_{1},b_{2}]$ to be a uniform distribution with minimum value $b_{1}$ and maximum value $b_{2}$. Now, we assume that some prior knowledge of this function $q$ have been known. According to the prior knowledge, we generate $200$ learning examples according to the following function $$\begin{aligned} q_{e}(x,y) := \sum_{k = 1}^{3}(1-x^{2})^{a_{k}^{1}}(1-y^{2})^{a_{k}^{2}} a_{k}^{3} \exp\bigg(-a_{k}^{4}(x - a_{k}^{5})^{2} - a_{k}^{6}(y - a_{k}^{7})^{2}\bigg),\end{aligned}$$ where $$\begin{aligned} & a_{k}^{1}, a_{k}^{2} \sim U[1,3], \quad a_{k}^{3} \sim U[-1,1], \\ & a_{k}^{4}, a_{k}^{6} \sim U[8,10], \quad a_{k}^{5}, a_{k}^{7} \sim U[-0.8,0.8].\end{aligned}$$ In order to provide an intuitional sense, we show the true scatterer and several learning examples in Figure \[TrueLearnFig1\]. We use $409780$ elements to obtain accurate solutions which we recognized as $\mathcal{S}(q)u^{\text{inc}}$. To test our approach, $16204$ elements will be used to obtain $\mathcal{S}_{a}(q)u^{\text{inc}}$. Learning algorithm with $K = 4$ proposed in Subsection \[LearnSection\] has been used to learn the statistical properties of differences $e_{n}^{i} := \mathcal{F}(q_{n},\kappa_{i}) - \mathcal{F}_{a}(q_{n},\kappa_{i})$ with $\kappa_{i} = i\cdot \pi \, (i = 1,\ldots 10)$ and $q_{n} \, (n = 1,\ldots 200)$ stands for the learning examples. Concerning the regularizing term, we take $A = 0.01 \Delta$, $s = 1.5$ and $\lambda = 0$, which can be computed by Fourier transform. Since regularization is not the main point of our paper, we will not discuss the strategies of choosing $A$ in details. ![Relative errors with different parameters: green dotted line are relative errors obtained by using the RLM with 16204 elements; cyan dotted line with circles are relative errors obtained by using the RLM with 183198 elements; blue solid line are relative errors obtained by using the GMRLM with 16204 elements.[]{data-label="RelaErrEx1"}](RelaErrEx1.jpg "fig:"){width="50.00000%"}\ \[table1\] [c|c|c|c]{} Algorithm & Element Number & Wavenumber & Relative Error\ RLM & 16204 & $7\pi$ & 7.10%\ RLM & 183198 & $10\pi$ & 0.26%\ GMRLM & 16204 & $4\pi$ & 0.49%\ RLM & 16204 & $4\pi$ & 30.58%\ RLM & 183198 & $4\pi$ & 20.82%\ RLM & 183198 & $9\pi$ & 0.42%\ Relative errors of RLM with small element number, RLM with large element number and GMRLM with small element number have been shown in Figure \[RelaErrEx1\], which illustrate the effectiveness of the proposed method. For the case of small element number, the RLM diverges when $\kappa\approx 7\pi$. The reason is that large number of elements are needed to ensure the convergence of finite element methods for Helmholtz equations with high wavenumber. Our error compensation method can not eliminate such errors, so it is also diverges when $\kappa\approx 7\pi$. However, when $\kappa \approx 4\pi$, our method provides a recovered function with relative error comparable to the result obtained by the RLM with more than eleven times of elements and $\kappa \approx 9\pi$. Hence, by learning process, the GMRLM can give an acceptable recovered function much faster than the traditional RLM. ![Recovered functions with different parameters. (a): true function; (b): minimum relative error estimate for the RLM with 16204 elements and the wavenumber computed to $7\pi$; (c): minimum relative error estimate for the RLM with 183198 elements and the wavenumber computed to $10\pi$; (d): minimum relative error estimate for the GMRLM with 16204 elements and the wavenumber computed to $4\pi$; (e): recovered function for the RLM with 16204 elements and the wavenumber computed to $4\pi$; (f): recovered function for the RLM with 183198 elements and the wavenumber computed to $4\pi$.[]{data-label="PcolorComEx1"}](PcolorComEx1.jpg "fig:"){width="100.00000%"}\ In addition, we show the accurate values of relative errors and element numbers in Table \[table1\]. In Figure \[PcolorComEx1\], the true scatterer function has been shown on the top left and five results obtained by RLM and GMRLM with different parameters have been given. From these, we can visually see the effectiveness of the proposed method. **Example 2**: For the second example, let $$\begin{aligned} q(x,y) := \left \{\begin{aligned} & 0.7 \qquad \text{for } -0.3 \leq x \leq 0.3 \text{ and }-0.3 \leq y \leq 0.3 \\ & -0.1 \quad\! \text{for } -0.1 < x < 0.1 \text{ and }-0.1 < y < 0.1 \\ & 0 \qquad\,\,\,\,\, \text{other areas in square } -1\leq x\leq 1 \text{ and } -1\leq y\leq 1. \end{aligned}\right.\end{aligned}$$ As in Example 1, we need to develop some learning examples. Here, we assume that there is a square in $[-1,1]^{2}$, but we did not know the position, size and height of the square. We assume that the position, size and height are all uniform random variables with height between $[-1,1]$ and the square supported in $[-1,1]^{2}$. As in Example 1, we generate 200 learning examples. To give the reader an intuitive idea, we show the true scatterer and five typical learning examples in Figure \[TrueLearnFig2\]. ![True scatterer and five typical learning examples[]{data-label="TrueLearnFig2"}](TrueLearnFig2.jpg "fig:"){width="100.00000%"}\ ![Relative errors with different parameters: green dotted line are relative errors obtained by using the RLM with 16204 elements; cyan dotted line with circles are relative errors obtained by using the RLM with 183198 elements; blue solid line are relative errors obtained by using the GMRLM with 16204 elements.[]{data-label="RelaErrEx2"}](RelaErrEx2.jpg "fig:"){width="50.00000%"}\ For this discontinuous scatterer, we take same values of parameters as in Example 1. Beyond our expectation, the proposed algorithm obviously converges even faster than the RLM with more than eleven times of elements, which is shown in Figure \[RelaErrEx2\]. By our understanding, the reason for such fast convergence is that the means and covariances learned by complex EM algorithm not only compensate numerical errors but also encode some prior information of the true scatterer by learning examples. Until the wavenumber is $9\pi \approx 28.26$, the RLM with 183198 elements provide a recovered function which has similar relative error as the recovered function obtained by the GMRLM. The RLM with only 16204 elements diverges as in Example 1 when wavenumber is too large, and the proposed algorithm still can not compensate the loss of physics as shown in Figure \[RelaErrEx2\]. For accurate value of relative errors and elements, we show them in Table \[table2\]. \[table2\] [c|c|c|c]{} Algorithm & Element Number & Wavenumber & Relative Error\ RLM & 16204 & $8\pi$ & 36.49%\ RLM & 183198 & $10\pi$ & 9.71%\ GMRLM & 16204 & $5\pi$ & 13.92%\ GMRLM & 16204 & $7\pi$ & 12.09%\ RLM & 16204 & $7\pi$ & 39.28%\ RLM & 183198 & $7\pi$ & 22.03%\ RLM & 183198 & $9\pi$ & 12.76%\ ![Recovered functions with different parameters. (a): true function; (b): minimum relative error estimate for the RLM with 16204 elements and the wavenumber computed to $8\pi$); (c): minimum relative error estimate for the RLM with 183198 elements and the wavenumber computed to $10\pi$; (d): minimum relative error estimate for the GMRLM with 16204 elements and the wavenumber computed to $8\pi$; (e): recovered function for the RLM with 16204 elements and the wavenumber computed to $8\pi$; (f): recovered function for the RLM with 183198 elements and the wavenumber computed to $8\pi$.[]{data-label="PcolorComEx2"}](PcolorComEx2.jpg "fig:"){width="100.00000%"}\ Finally, we provide the image of true scatterer on the top left in Figure \[PcolorComEx2\]. On the top middle, the best result obtained by the RLM with 16204 elements is given. From this image, we can see that it is failed to recover the small square embedded in the large square. The best result obtained by the RLM with 183198 elements is shown on the top right. It is much much better than the function obtained by algorithm with 16204 elements. At the bottom of Figure \[PcolorComEx2\], we show the best result obtained by the GMRLM with 16204 elements on the left and show the results obtained by the RLM (compute to the same wavenumber as the GMRLM) with 16204 elements and 183198 elements in the middle and on the righthand side respectively. The recovered function by the GMRLM is not as well as the recovered function obtained by the RLM with more than eleven times of elements and higher wavenumber. However, beyond our expectation, it is already capture the small square embedded in the large square, which is not incorporated in our 200 learning examples. In summary, the proposed GMRLM converges much faster than the classical RLM and it can provide a much better result at the same discrete level compared with the RLM. Conclusions =========== In this paper, we assume the modeling errors brought by rough discretization to be Gaussian mixture random variables. Based on this assumption, we construct the general Bayesian inverse framework and prove the relations between MAP estimates and regularization methods. Then, the general theory has been applied to a specific inverse medium scattering problem. Well-posedness in the statistical sense has been proved and the related optimization problem has been obtained. In order to acquire estimates of parameters in the Gaussian mixture distribution, we generalize the EM algorithm with real variables to the complex variables case rigorously, which incorporate the machine learning process into the classical inverse medium problem. Finally, the adjoint problem has been deduced and the RLM has been generalized to GMRLM based on the previous illustrations. Two numerical examples are given, which demonstrate the effectiveness of the proposed methods. This work is just a beginning, and there are a lot of problems need to be solved. For example, we did not give a principle of choosing parameter $K$ appeared in the Gaussian mixture distribution. In addition, in order to learn the model errors more accurately, we can attempt to design new algorithms to adjust the parameters in the Gaussian mixture distribution efficiently in the inverse iterative procedure. Acknowledgments {#acknowledgments .unnumbered} =============== This work was partially supported by the NSFC under grant Nos. 11501439, 11771347, 91730306, 41390454 and partially supported by the Major projects of the NSFC under grant Nos. 41390450 and 41390454, and partially supported by the postdoctoral science foundation project of China under grant no. 2017T100733. [^1]:

--- abstract: | Abstract ======== The studies of the electroweak symmetry breaking sector (EWSBS) at $\gamma\gamma$ colliders were considered previously in the loop processes of $\gamma\gamma \to w_Lw_L,\,z_Lz_L$, but they are suffered from the huge $W_T W_T$ and $Z_TZ_T$ backgrounds. Here we present another possible process that involves spectator $W$’s and $W_L$’s, the latter of which are scattered strongly by the interactions of the EWSBS. We also show that this process should be safe from the transverse backgrounds and it can probe the structure of the EWSBS. author: - Kingman Cheung --- [**Possibility of Studying Electroweak Symmetry Breaking at [$\gamma \gamma$]{} Colliders**]{} Dept. of Physics & Astronomy, Northwestern University, Evanston, Illinois 60208, USA\ Introduction {#intro} ============ So far very little is known about the electroweak symmetry-breaking-sector (EWSBS), except it gives masses to the vector bosons via the spontaneous symmetry breaking, and masses to fermions via the Yukawa couplings. In the minimal standard model (SM) one scalar Higgs boson is responsible for the electroweak symmetry breaking but its mass is not determined by the model. If in the future no Higgs boson is found below 800 GeV, the heavy Higgs scenario ($\approx 1$ TeV) will imply a strongly interacting Higgs sector because the Higgs self-coupling $\lambda\sim m_H^2$ becomes strong [@quigg]. However, there is no evidence to favor the model with a scalar Higgs, and so any models that can break the electroweak symmetry the same way as the single Higgs does can be a candidate for the EWSBS. One of the best ways to uncover the underlying dynamics of the EWSBS is to study the longitudinal vector boson scattering [@quigg; @chano]. The Equivalence Theorem (ET) recalls, at high energy, the equivalence between the longitudinal part $(W_L)$ of the vector bosons to the corresponding Goldstone bosons ($w_L$) that were “eaten" in the Higgs mechanism. These Goldstone bosons originate from the EWSBS so that their scattering must be via the interactions of the EWSBS, and therefore the $W_L W_L$ scattering can reveal the dynamics of the EWSBS. The strong $W_L W_L$ scattering have been studied quite seriously at the hadronic supercolliders [@bagger], but less at the $e^+e^-$ colliders, and very little at the $\gamma\gamma$ colliders. In hadronic colliders, only the “gold-plated" modes, the leptonic decays of the $W$ and $Z$ bosons, have been considered due to the messy hadronic backgrounds; whereas in $e^+e^-$ and $\gamma\gamma$ colliders one can make use of the hadronic decay mode or mixed decay mode of the final state $W$’s or $Z$’s. With the advance in the photon collider designs it is possible to construct an almost monochromatic $\gamma\gamma$ collider based on the next generation linear $e^+e^-$ colliders using the laser backscattering method [@teln]. The monochromaticity of the photon beams depends on the polarizations of the initial electron and the laser photon. The polarizations of the initial electron and the laser photon can be adjusted to maximize the monochromaticity of the photon beam [@teln] with a center-of-mass energy about 0.8 of the parent $e^+e^-$ collider. Hence, a 2 TeV $e^+e^-$ collider will give a 1.6 TeV $\gamma\gamma$ collider by the laser backscattering method. For the following we will assume a monochromatic $\gamma\gamma$ collider of energy 1.5 TeV with an integrated luminosity of 100 fb$^{-1}$. Studies of the strongly interacting EWSBS in $\gamma\gamma$ collision have been considered previously in Refs. [@previous]. They all concentrate on $\gamma\gamma \rightarrow W_L W_L$ or $Z_L Z_L$. Unfortunately, the $\gamma\gamma\to W_T W_T$ is almost three orders of magnitude larger than the $W_LW_L$ signal. Although we can improve the signal-to-background ratio by requiring the final state $W$’s away from the beam, it hardly reduces the $W_TW_T$ background to the level of the $W_LW_L$ signal. On the other hand, both the $\gamma\gamma\to Z Z$ signal and background are absent on tree level. But the box diagram contribution to $Z_TZ_T$ has been shown to be very large at high $m(ZZ)$ region, and so the $Z_T Z_T$ background is dominant over the $Z_L Z_L$ signal in the search of the SM Higgs with $m_H\agt 300$ GeV and in probing the other strong EWSB signals [@ZTZT]. As illustrated in Refs. [@previous], the central part of interest is the $w_L w_L \to w_L w_L$ or $z_Lz_L$, but the effects of the strong EWSBS only come in on loop level in these processes so that the effects might not be so significant. In the following we present a new type of processes involving $W_L W_L\to W_L W_L,\,Z_LZ_L$ at $\gamma\gamma$ colliders, schematically shown in Fig. \[one\] [@brodsky]. These $W_LW_L$ scattering processes will be in analogy to the $W_L W_L$ scattering considered at the hadronic supercolliders and $e^+e^-$ colliders. The advantages of the processes in Fig. \[one\] are that the $W_LW_L$ scattering is no longer on loop level, and additional vector bosons in the final state can be tagged on to eliminate the large $W_T W_T$ and $Z_T Z_T$ backgrounds. In addition, both the $W_L^+ W_L^-$ and $W_L^\pm W_L^\pm$ scattering can be studied in $\gamma\gamma$ collision but only one of them can be studied in the $e^+e^-$ or $e^-e^-$ collisions. Also any $Z_LZ_L$ pair in the final state must come from the $W_LW_L$ fusion because photon will not couple to $Z$ on tree level. Totally, we can study four scattering processes, $W_L^\pm W_L^\pm \to W^\pm_L W^\pm_L$, $W^+_L W^-_L \to W^+_L W^-_L,\, Z_L Z_L$. For simplification we will use the effective $W_L$ luminosity inside a photon in analogy to the effective $W$ approximation. This approximation will suffice for the purpose here for we will consider the kinematic region where the EWSBS will interact strongly, or in another words, in the large invariant mass region of the vector boson pair. The luminosity function of a $W_L$ inside a photon in the asymtotic energy limit is given by [@zerwas] $$\label{lum} f_{W_L/\gamma}(x) = \frac{\alpha}{\pi} \left [ \frac{1-x}{x} + \frac{x(1-x)}{2}\; \left ( \log \frac{s(1-x)^2}{m_W^2} - 2 \right ) \right ]\,,$$ which is in analogy to the luminosity function $f_{W_L/e}(x) = \frac{\alpha}{4\pi x_{\rm w}} \frac{1-x}{x}$ of $W_L$ inside an electron. The first term in Eq. (\[lum\]) is approximately equal to the luminosity of $W_L$ inside an electron, and the logarithm factor will enhance the luminosity at high energy. This is the reason why the signal rates can be achieved higher than those in the $e^+e^-$ colliders at the same energy. Models & Predictions ==================== In this section, we will calculate the number of signal events predicted by some of the models that have been proposed for the EWSBS. In $\gamma\gamma$ collision we can study the following subprocesses $$\begin{aligned} W_L^+ W_L^- &\rightarrow & W_L^+ W_L^- \,, Z_L Z_L \;, \\ W_L^\pm W_L^\pm & \rightarrow & W_L^\pm W_L^\pm \,.\end{aligned}$$ In analogy to the pion scattering in QCD, the scattering amplitudes of these processes can be expressed in terms of an amplitude function $A(s,t,u)$. Their scattering amplitudes are then expressed as $$\begin{aligned} % {\cal M}(W_L^\pm W_L^\pm \rightarrow W_L^\pm W_L^\pm)&=& A(t,s,u)+A(u,t,s) \,,\\ % {\cal M}( W_L^+ W_L^- \rightarrow W_L^+ W_L^- ) & = & A(s,t,u) + A(t,s,u) \,, \\ % {\cal M}( W_L^+ W_L^- \rightarrow Z_L Z_L ) & = & A(s,t,u) \,, %\end{aligned}$$ up to the symmetry factor of identical particles in the final state. The details of each model and the invariant amplitudes predicted by each model are summarized in Ref. [@bagger]. Here we only give a brief account of these models. The models can be classified according to the spin and isospin of the resonance fields, and there are scalar-like, vector-like, and nonresonant models. For scalar-like models we will employ the standard model with a 1 TeV Higgs, $O(2N)$ model with the cutoff $\Lambda=2$ TeV, and the model with a chirally-coupled scalar of mass $m_S=1$ TeV and width $\Gamma_S=350$ GeV. For the vector-like models we choose the chirally-coupled vector field (technirho) of masses $m_\rho=1$, 1.2, and 1.5 TeV, and $\Gamma_\rho=0.4$, 0.5, and 0.6 TeV respectively. In the case of no light resonances we use the amplitudes predicted by the Low Energy Theorem (LET) and extrapolate them to high energies. Each of the $W_L W_L$ scattering amplitudes grows with energy until reaching the resonances, [*e.g.*]{} SM Higgs boson of the minimal SM. The presence of the resonances (scalar or vector) is the natural unitarization to the scattering amplitudes, except there might be slight violation of unitarity around the resonance peak. After the resonance, the scattering amplitudes will stay below the unitarity limit. But for the nonresonant models the unitarity is likely to be saturated before reaching the lightest resonance. Here we employ the LET amplitude function, $A(s,t,u)=s/v^2$, for the nonresonant models. From the partial wave analysis, the only nonzero partial wave coefficients $a^I_J$ are $a^0_0$, $a^1_1$, and $a^2_0$. Among the nonzero $a^I_J$’s, $a^0_0$ saturates the unitarity ($|a^I_J|<1$) at the lowest energy $4\sqrt{\pi}v\approx 1.7$ TeV. So for the $\gamma\gamma$ colliders of 1.5 TeV, unitarity violation should not be a problem, therefore, we simply extrapolate the LET amplitudes without any unitarization. We show the number of signal events predicted by these models for each scattering channel in Table \[table1\], with $\sqrt{s_{\gamma\gamma}}=1.5$ TeV and integrated luminosity of 100 fb$^{-1}$, and under the acceptance cuts of $$M_{WW}\;{\rm or}\; M_{ZZ} > 500\;{\rm GeV}\quad {\rm and}\quad |y(W,Z)|<1.5\,.$$ One interesting thing to note here is that different channel is sensitive to different new physics. If the underlying dynamics of the EWSBS is scalar-like the signal is more likely to be found in the $W_L^+ W_L^-$ channel, and next at the $Z_LZ_L$ channel, due to the presence of $I=0,J=0$ scalars. But if the underlying dynamics is vector-like the signal in the $W_L^+W_L^-$ channel will be far more important that the $Z_L Z_L$ channel. On the other hand, if no light resonances are within reach the $Z_LZ_L$ channel has the largest signal rate, and next is the $W^\pm_L W^\pm_L$ channels. So by counting the number of $W^\pm_L W^\pm_L$, $W^+_L W^-_L$, and $Z_L Z_L$ pairs in the final state one can tell the different structure of the EWSBS [@han]. But to distinguish a $W$ from a $Z$ by the dijet mass measurement is not a trivial issue, though we can use the $B$-tagging to distinguish a $W$ from a $Z$ somehow. For a discussion on this subject please see, [*e.g.*]{}, Ref. [@han]. The number of signal events in Table \[table1\] does not include any detection efficiencies of the $W_L$’s coming out from the strong scattering region, nor the tagging efficiencies for the spectator $W$’s. The tagging efficiencies for the spectator $W$’s will be dealt with in the next section. The detection efficiencies of the $W_L$’s consist of the branching ratios of the $W_L$ into jets or leptons, and the tagging efficiencies of these decay products. The branching ratio BR($W\to jj)\approx$BR$(Z\to jj) \approx 0.7$. Assuming a 30% (reasonable to pessimistic) tagging efficiencies for the decay products, we have about 15% overall detection efficiencies for the $W_LW_L$ coming out from the strong scattering region. Tagging the Spectator $W$’s =========================== So far we have not considered any backgrounds nor background suppression techniques. In our calculation, we use the effective $W_L$ luminosity which does not predict the correct kinematics for the spectator $W$’s, and therefore any acceptance cuts on the spectator $W$’s will be unrealistic. However, we need to tag at least one or both of these spectator $W$’s in order to eliminate the enormous $\gamma\gamma\to W_T W_T,\,Z_T Z_T$ backgrounds. One way to remedy is to carry out an exact SM calculation of $\gamma\gamma\to WWWW$ or $WWZZ$ with a heavy Higgs boson, and estimate the acceptance efficiencies on tagging the spectator $W$’s, and then apply these efficiencies to the other models which can only be calculated using the effective $W_L$ luminosity. However, the calculations of the processes $\gamma\gamma\to WWWW$ or $WWZZ$ are non-trivial. Instead, we can obtain the tagging efficiencies by calculating a simpler process $\gamma\gamma \to WWH$ for $m_H\approx $1 TeV, with and without imposing acceptance cuts on the final state $W$’s. We will calculate the total cross section for $\gamma\gamma\to WWH$ without any cuts, and also the cross section with the acceptance cuts $$p_T(W) > 25\;{\rm GeV},\qquad |y(W)| < 1.5\;{\rm or}\;2$$ on either one or both of the $W$’s. The cross sections are presented in Table \[table2\] for $m_H=1$ TeV. There are two tagging efficiencies corresponding to tagging at least one or both of the spectator $W$’s. From Table \[table2\], if we require the spectator $W$’s within a rapidity of $|y(W)|<1.5$ the tagging efficiencies are 91% and 42% for tagging at least one or both the $W$’s respectively. To eliminate the $W_T W_T$ or $Z_T Z_T$ backgrounds we need only tag one of the spectator $W$’s plus the $W_L$’s from the strong scattering. A further confirmation by tagging two spectator $W$’s will result in an efficiency of only 42%. But if we tag both spectator $W$’s within the rapidity $|y(W)|<2$ the double-tag efficiency increases to 82%. This drastic difference of the double-tag efficiencies between rapidity cut of 1.5 and 2 demonstrates that it is likely (40% chance) to have at least one spectator $W$ in the forward rapidity region $1.5<|y(W)|<2$. Next we can multiply these efficiencies to the numbers in Table \[table1\] to get a more reliable number of signal events when the spectator $W$’s are tagged. Taking into account of the 15% (from the last section) detection efficiency for the $W_L W_L$ plus the tagging efficiency of at least one or both of the spectator $W$’s, we still have at least 10% overall efficiency. With 10% efficiency we still have a sizeable number of signal events. Scalar-type models will be shown up in the $W^+W^-\to W^+W^-$ channel with at least 47 events. The vector-like models will also be shown up in the $W^+W^-\to W^+W^-$ channel if the vector resonance is within reach of the energy of the $\gamma\gamma$ collider. For nonresonant models we have about 15 events for the $W^\pm W^\pm \to W^\pm W^\pm$ channels and 17 events for $W^+W^-\to ZZ$ channel. Background Discussions ====================== The continuum productions of $\gamma\gamma\to WWWW$ and $WWZZ$, together with the heavy quark production of $\gamma\gamma\to t\bar t t\bar t$ followed by the top decays into $W$’s, form the irreducible set of backgrounds. They are the SM predictions that any significant excess of $W_L W_L$ or $Z_L Z_L$ events will indicate some kinds of new physics for the EWSBS. The other reducible backgrounds include the productions of $W$’s with jets, $Z$’s with jets, and multi-jet. The $WWWW$ and $WWZZ$ productions are of order $\alpha_w^4$, and so should be at most the same level as our strong $W_L W_L$ signal. Although the $t\bar t t\bar t$ background is ${\cal O}(\alpha_s^2/\alpha_w^2)$ larger than the $WWWW$ background, we can to certain extent reduce it by reconstructing the top and by imposing the top-mass constraints. The other QCD backgrounds of $W$’s or $Z$’s with jets are reducible by the $W$ or $Z$ mass constraints. In addition, we can make use of the kinematics of the spectator $W$’s and the strongly scattered $W_L$’s [@kingman]. The $p_T$ of the spectator $W$’s should be of order $m_W/2$ after the photon emits an almost on-shell $W_L$, which then participates in the strong scattering. Also, as mentioned in the last section, at least one of the spectator $W$’s tend to go forward in the rapidity region $|y(W)|>1.5$. On the other hand, the $W_L W_L$ after the strong scattering come out in the central rapidity region with large $p_T$ and large invariant mass, and back-to-back in the transverse plane, which are all due to the strong interaction of the EWSBS. But it is hardly true for the backgrounds. Acceptance cuts can be formulated based on the above arguments to substantially reduce the backgrounds [@future]. In conclusions, we have demonstrated another type of processes in $\gamma\gamma$ collision that can probe the strongly interacting EWSBS scenario. The processes do not involve the indirect loop effects, and also are safe from the huge $W_T W_T$ or $Z_T Z_T$ backgrounds due to the presence of the spectator $W$’s. Even with only 10% overall efficiency we still have enough signal events with 100 fb$^{-1}$ luminosities. Irreducible backgrounds from $WWWW$, $WWZZ$, and $t\bar t t\bar t$ can be reduced by considering the special kinematics of the strongly scattered $W_L$’s and the spectator $W$’s. Other reducible backgrounds are reduced by the mass constraints. We are grateful to V. Barger, D. Bowser-Chao, and T. Han for useful discussions. This work was supported by the U. S. Department of Energy, Division of High Energy Physics, under Grant DE-FG02-91-ER40684. B. W. Lee, C. Quigg and H. Thacker, Phys. Rev. [**D16**]{}, 1519 (1977). M. S. Chanowitz and M. K. Gaillard, Nucl. Phys. [**B261**]{}, 379 (1985). J. Bagger, [*et al.*]{}, Fermilab Report number FERMILAB-PUB-93-040-T (to appear in Phys. Rev. D), and reference therein. V. Telnov, Nucl. Instr. & Methods [**A294**]{}, 72 (1990); I. Ginzburg, G. Kotkin, V. Serbo and V. Telnov, Nucl. Instr. & Methods [**205**]{}, 47 (1983); [*idem*]{} [**219**]{}, 5 (1984). E. Boos and G. Jikia, Phys. Lett. [**B275**]{}, 164 (1992); A. Abbasabadi, D. Bowser-Chao, D. Dicus, and W. Repko, Michigan State Univ. preprint MSUTH-92-03 (1992); R. Rosenfeld, Northeastern Univ. preprint NUB-3074/93-Th (1993). G. Jikia, Phys. Lett. [**B298**]{}, 224 (1993); M. S. Berger, UW-Madison preprint MAD/PH/771; C. Kao and D. Dicus, UT-Austin preprint DOE-ER-40757-024. This type of processes was pointed out briefly by S. Brodsky in his talk at the “2nd International Workshop on Physics and Experiments at $e^+e^-$ colliders", Waikoloa, Hawaii (April 1993), see also SLAC-PUB-6314. K. Hagiwara, I. Watanabe, and P. Zerwas, Phys. Lett. [**B278**]{}, 187 (1992). Plenary Talk by T. Han at the same workshop in Ref. [@brodsky], see also Fermilab-Conf-93/217-T. Talk by K. Cheung at the same workshop in Ref. [@brodsky] and at the “Workshop on Physics at Current Accelerator and the Supercolliders", Argonne, Illinois (June 1993), see also NUHEP-TH-93-16. K.Cheung, in preparation. $W^+_L W^+_L \to W^+_L W^+_L$ $W^+_L W^-_L \to W^+_L W^-_L$ $W^+_L W^-_L \to Z_L Z_L$ --------------------------------------- ------------------------------- ------------------------------- --------------------------- \(1) SM Higgs $m_H=1$ TeV 88 1600 760 \(2) chirally-coupled scalar $m_S=1$ TeV, $\Gamma_S=350$ GeV 100 570 430 \(3) O(2N) 90 470 350 \(4) chirally-coupled vector a\. $m_V=1$ TeV, $\Gamma_V=0.4$ TeV 180 2400 280 b\. $m_V=1.2$ TeV, $\Gamma_V=0.5$ TeV 52 590 29 c\. $m_V=1.5$ TeV, $\Gamma_V=0.6$ TeV 88 120 40 LET 150 110 170 : \[table1\] The number of the signal events for the strong $W_L W_L$ scattering predicted by various models at $\gamma\gamma$ collider of $\sqrt{s}=1.5$ TeV. The acceptance cuts on the final $W_LW_L$ or $Z_LZ_L$ are: $m(WW,ZZ)>500$ GeV and $|y(W,Z)|<1.5$. The luminosity is assumed 100 fb$^{-1}$. No efficiencies are included here. $|y(W)|<$ No cuts Tagging at least one $W$ Tagging both $W$’s ----------- --------- -------------------------- -------------------- - 14.7 - - 1.5 - 13.4 (91%) 6.16 (42%) 2.0 - 14.5 (98.5%) 12.1 (82%) : \[table2\] Table showing the cross sections (fb) for the process $\gamma\gamma\to WWH$ with a SM Higgs boson of mass $m_H=1$ TeV at $\sqrt{s_{\gamma\gamma}}=1.5$ TeV, with and without imposing acceptance cuts on the final state $W$’s. The acceptance cuts are $p_T(W)>25$ GeV and $|y(W)|<1.5$ or 2. The second column shows the total cross section without cuts. The third column corresponds to tagging at least one of the $W$’s, and the last column corresponds to tagging both. The percentages in the parentheses are the efficiencies.

--- abstract: 'We investigate two approaches to derive the proper Floquet-based quantum-classical Liouville equation (F-QCLE) for laser-driven electron-nuclear dynamics. The first approach projects the operator form of the standard QCLE onto the diabatic Floquet basis, then transforms to the adiabatic representation. The second approach directly projects the QCLE onto the Floquet adiabatic basis. Both approaches yield a form which is similar to the usual QCLE with two modifications: 1. The electronic degrees of freedom are expanded to infinite dimension. 2. The nuclear motion follows Floquet quasi-energy surfaces. However, the second approach includes an additional cross derivative force due to the dual dependence on time and nuclear motion of the Floquet adiabatic states. Our analysis and numerical tests indicate that this cross derivative force is a fictitious artifact, suggesting that one cannot safely exchange the order of Floquet state projection with adiabatic transformation. Our results are in accord with similar findings by Izmaylov *et al*., who found that transforming to the adiabatic representation must always be the last operation applied, though now we have extended this result to a time-dependent Hamiltonian. This paper and the proper derivation of the F-QCLE should lay the basis for further improvements of Floquet surface hopping.' author: - 'Hsing-Ta Chen' - Zeyu Zhou - 'Joseph E. Subotnik' bibliography: - 'ZoteroLibrary.bib' title: 'On the Proper Derivation of the Floquet-based Quantum Classical Liouville Equation Describing a Molecule or Material Subject to an External Field' --- Introduction\[sec:Introduction\] ================================ A computational understanding light-matter interactions for a molecular system in a laser field is useful key for interpreting spectroscopy and photochemistry, where the dynamical interplay between electronic non-adiabatic transitions and photon excitation plays an important role for many exciting phenomena, such as molecular photodissociation[@regan_ultraviolet_1999; @franks_orientation_1999; @hilsabeck_photon_2019; @hilsabeck_photon_2019-1] and coherent X-ray diffraction[@glownia_self-referenced_2016; @glownia_glownia_2017; @lemke_coherent_2017; @fuller_drop--demand_2017]. These phenomena usually involve dynamical processes in which electrons in a molecular system can make a transition through either (a) non-adiabatic coupling associated with the reorganization of nuclear configurations or (b) radiative coupling in conjunction with absorption or emission of photons. Thus, simulating these processes concurrently requires accurate theoretical treatments of both non-adiabatic molecular dynamics and light-matter interactions[@bajo_interplay_2014; @hoffmann_light-matter_2018; @abedi_exact_2010]. Over the past decades, many successful simulation schemes have been developed based on mixed quantumclassical frameworks in which the electronic wavefunction evolves according to quantum mechanics while the nuclear degrees of freedom and the laser excitation are treated as classical parameters in a time-dependent electronic Hamiltonian[@zhou_nonadiabatic_2020-1; @richter_sharc:_2011; @mitric_laser-field-induced_2009; @mai_nonadiabatic_2018; @thachuk_semiclassical_1996; @bajo_mixed_2012; @lisinetskaya_simulation_2011]. Among the myraid of semiclassical dynamics, Floquet-based fewest switch surface hopping (F-FSSH) has emerged as one of the most powerful methods especially for simulating photodissociation and ionization in a monochromatic laser field[@fiedlschuster_floquet_2016; @fiedlschuster_surface_2017; @horenko_theoretical_2001]. In a nutshell, F-FSSH integrates Floquet theory with Tully’s FSSH algorithm[@tully_perspective:_2012]. The general idea is to expand the electronic wavefunction in a Floquet state basis (with the electronic states dressed by $e^{im\omega t}$ for an integer $m$ and the laser frequency $\omega$), so that one can recast an explicitly time-dependent Hamiltonian into a time-independent Floquet Hamiltonian, albeit of infinite dimension. With the Floquet Hamiltonian, one can simply employ Tully’s FSSH method in the Floquet state representation with a minimal modification[@fiedlschuster_floquet_2016; @fiedlschuster_surface_2017]. In addition to the standard advantages of the usual surface hopping algorithmstability, efficiency, and ease of incorporation with electronic structure calculationsF-FSSH also yield a better estimate for both electronic and nuclear observables than other FSSH-based methods relying on instantaneous adiabatic surfaces[@zhou_nonadiabatic_2020-1]. Furthermore, given the time-independent nature of the Floquet Hamiltonian, many techniques designed to improve standard FSSH method, such as velocity reversal and decoherence[@bittner_quantum_1995; @jasper_electronic_2005; @jasper_improved_2003; @nelson_nonadiabatic_2013; @subotnik_understanding_2016; @jain_efficient_2016; @fang_improvement_1999; @bedard-hearn_mean-field_2005], should be applicable within the Floquet formalism. That being said, due to the fact that one cannot directly derive Tully’s FSSHbut rather only indirectly connect the equations of motion for FSSH dynamics with the quantum-classical Liouville equation (QCLE)[@subotnik_can_2013; @kapral_surface_2016]a proper understanding of the correct F-QCLE is crucial if we aim to make future progress in semiclassical modeling of light-matter interactions. Unfortunately, even without a light field, the proper derivation of the correct QCLE is trickyone can find two different versions of QCLE if one invokes slightly different formal derivations. Following Kapral’s approach[@kapral_progress_2006; @kapral_mixed_1999], the proper derivation of the standard QCLE includes two operations: (i) Wigner transformation and (ii) projection onto the adiabatic electronic state basis. First, one performs a partial Wigner transformation with respect to the nuclear degrees of freedom to obtain the operator form of the QCLE. Wigner transformation provides an exact framework to interpret the full quantum density matrix in terms of the joint electronic-nuclear probability density in the phase space of the nuclear configuration while retaining the quantum operator character of the electronic subsystem. Second, one projects the operator form of the QCLE onto the adiabatic electronic states basis obtained by diagonalizing the electronic Hamiltonian. This adiabatic representation allows the connection to electronic structure calculations in a mixed quantum-classical sense[@wong_solvent-induced_2002; @wong_dissipative_2002; @webster_nonadiabatic_1991; @kelly_efficient_2013; @kim_all-atom_2012]. This approach is called the Wigner-then-Adiabatic (WA) approach. As Izmaylov and co-workers have shown[@ryabinkin_analysis_2014], however, exchanging these two operations (the Adiabatic-then-Wigner (AW) approach) leads to a different QCLE that cannot capture geometric phase effects arising from a conical intersection. With this background in mind, the proper derivation of the F-QCLE is now even more challenging. In addition to the two operations above, there is a third step: one needs to dress the electronic states and expand the density matrix in the Floquet state basis. In the literature to date, the F-QCLE has been derived via the AW approach (projecting in the Floquet adiabatic representation, and then performing partial Wigner transformation)[@horenko_theoretical_2001]. Nevertheless, as shown by Izmaylov and co-workers[@ryabinkin_analysis_2014; @kapral_mixed_1999], even the limit of a time-independent Hamiltonian, such an (incorrect) AW approach will lead to a QCLE that neglects geometric phase related features in the nuclear dynamics or introduces artificial nuclear effects. Despite recent progress, the proper derivation of the F-QCLE is still an open question. In this paper, our goal is to explore different approaches to derive the F-QCLE as we shuffle the three key operations and quantify their differences in the context of driven electron-nuclear dynamics. By isolating the correct F-QCLE, our work will not only validate F-FSSH methods, it should also provide means to improve surface hopping methods. This paper is organized as follows. In Sec. \[sec:Three-operations\], we formulate the three operations that are required to derive the F-QCLE. In Sec. \[sec:Floquet-QCLE\], we derive F-QCLEs via approaches with different ordering of operations. In Sec. \[sec:Results:\], we implement F-FSSH calculations corresponding to these F-QCLEs and analyze their results for a modified avoided crossing model. We conclude with an outlook for the future in Sec. \[sec:Conclusion\]. For notation, we denote a quantum operator by $\hat{H}$ and use bold font for matrix $\mathbf{H}$. We use $\widetilde{\boldsymbol{H}}$ to denote the corresponding matrix in expanded Floquet basis (infinite dimensional). The nuclear position and momentum are $\vec{R}=\{R^{\alpha}\}$, $\vec{P}=\{P^{\alpha}\}$ where $\alpha$ is the nuclear coordinate index. We use a shorthand notation for dot product: $X^{\alpha}\cdot Y^{\alpha}=\sum_{\alpha}X^{\alpha}Y^{\alpha}$ . Three Operations\[sec:Three-operations\] ======================================== In the context of driven electron-nuclear dynamics, let us formulate the three necessary operations for deriving F-QCLE. Consider a coupled electron-nuclei system driven by an external field of frequency $\omega$, the total Hamiltonian takes the form of $\hat{H}=\hat{T}(\hat{P}^{\alpha})+\hat{V}(\hat{R}^{\alpha},t)$ where $\hat{T}(\hat{P}^{\alpha})=\sum_{\alpha}\frac{(\hat{P}^{\alpha})^{2}}{2M^{\alpha}}$ is the nuclear kinetic energy and $\hat{V}(\hat{R}^{\alpha},t)$ is the electronic Hamiltonian with explicit time periodicity $\hat{V}(t)=\hat{V}(t+\tau)$ with $\tau=2\pi/\omega$. Formally, the dynamics of the total system can be described by the time-dependent Schrödinger equation (TDSE) $i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat{H}|\Psi\rangle$ of the total wavefunction $|\Psi\rangle$ or the quantum Liouville equation (QLE) $\frac{\partial}{\partial t}\hat{\rho}=-\frac{i}{\hbar}[\hat{H},\hat{\rho}]$ of the total density matrix $\hat{\rho}=|\Psi\rangle\langle\Psi|$. To derive F-QCLE, we need to apply the following three operations to the QLE. Partial Wigner transformation of the nuclear degrees of freedom --------------------------------------------------------------- To describe the dynamics in a mixed quantum-classical sense, we will follow Kaprals approach and perform a partial Wigner transformation with respect to the nuclear degrees of freedom $$\hat{\rho}_{W}\left(\vec{R},\vec{P},t\right)=\frac{1}{(2\pi\hbar)^{N}}\int d\vec{S}\left\langle \vec{R}+\frac{\vec{S}}{2}\right|\hat{\rho}(t)\left|\vec{R}-\frac{\vec{S}}{2}\right\rangle e^{-i\vec{P}\cdot\vec{S}/\hbar}\label{eq:partial_Wigner_distribution}$$ where $N$ is the dimension of the nuclear coordinate. A nuclear position eigenstate is defined as $\hat{R}^{\alpha}|R^{\alpha}\rangle=R^{\alpha}|R^{\alpha}\rangle$. In Eq. , the density matrix operator has been transformed into a Wigner wavepacket in phase space with coordinates $(\vec{R},\vec{P})$. In what follows, we will denote the partial Wigner transformed operator by the subscript $W$ [\[]{}for example $\hat{V}(\hat{R}^{\alpha},t)\rightarrow\hat{V}_{W}(R^{\alpha},t)$[\]]{}. Note that, after the partial Wigner transformation, $\hat{\rho}_{W}$ and $\hat{V}_{W}$ remain electronic operators while $R^{\alpha}$ and $P^{\alpha}$ are parameters. The equation of motion of the Wigner wavepacket can be obtained by transforming the QLE by $\frac{\partial}{\partial t}\hat{\rho}_{W}=-\frac{i}{\hbar}[(\hat{H}\hat{\rho})_{W}-(\hat{\rho}\hat{H})_{W}]$. The Wigner transform of operator products can be expanded further by the WignerMoyal operator $(\hat{H}\hat{\rho})_{W}=\hat{H}_{W}e^{-i\hbar\overleftrightarrow{\Lambda}/2}\hat{\rho}_{W}$ with $\overleftrightarrow{\Lambda}=\sum_{\alpha}\overleftarrow{\frac{\partial}{\partial P^{\alpha}}}\overrightarrow{\frac{\partial}{\partial R^{\alpha}}}-\overleftarrow{\frac{\partial}{\partial R^{\alpha}}}\overrightarrow{\frac{\partial}{\partial P^{\alpha}}}$[@kapral_mixed_1999]. Then, if we expand the the WignerMoyal operator in Taylor series and truncate to the first order of $\hbar$, we obtain the operator form of the QCLE $$\frac{\partial}{\partial t}\hat{\rho}_{W}=-\frac{i}{\hbar}\left[\hat{V}_{W},\hat{\rho}_{W}\right]-\frac{P^{\alpha}}{M^{\alpha}}\frac{\partial\hat{\rho}_{W}}{\partial R^{\alpha}}+\frac{1}{2}\left\{ \frac{\partial\hat{V}_{W}}{\partial R^{\alpha}},\frac{\partial\hat{\rho}_{W}}{\partial P^{\alpha}}\right\} .\label{eq:QCLE-operator}$$ Here, the commutator is $\left[\hat{A},\hat{B}\right]=\hat{A}\hat{B}-\hat{B}\hat{A}$ and the anti-commutator is $\left\{ \hat{A},\hat{B}\right\} =\hat{A}\hat{B}+\hat{B}\hat{A}$. Note that Eq.  is exact if the partial Wigner transformed Hamiltonian is quadratic in $R^{\alpha}$, for example harmonic oscillators. To propagate the Wigner wavepacket in Eq. , one must project the operator form of the QCLE in an electronic basis. One straightforward choice is to use a complete set of diabatic states for the electronic subsystem $\{|\mu\rangle\}$; such a set does not depend on any nuclear configuration. With this electronic diabatic basis, one can derive equations of motion for the density matrix ($A_{\nu m}^{\text{dia}}=\langle\nu|\hat{\rho}_{W}|\mu\rangle$) using matrix elements of the electronic Hamiltonian, $V_{\nu\mu}(\vec{R},t)=\langle\mu|\hat{V}_{W}(\vec{R},t)|\mu\rangle$. However, for many realistic electron-nuclei systems (and certainly any ab initio calculations), this diabatic QCLE cannot be solved since finding a complete set of exactly diabatic electronic states over a large set of nuclear geometries is rigorously impossible and quite demanding in practice even for approximate diabats. Dress the electronic basis in the Floquet formalism --------------------------------------------------- Let us now focus on the Floquet formalism, according to which one solves the TDSE by transforming the time-dependent Hamiltonian into a time-independent Floquet Hamiltonian in an extended Hilbert space of infinite dimension. For the moment let us ignore all nuclear motion and focus on the electronic exclusively. According to Floquet theory, we utilize the time periodicity of the electronic Hamiltonian and dress the electronic diabatic states $\{|\mu\rangle\}$ by a time-periodic function $e^{im\omega t}$ where $m$ is an integer formally from $-\infty$ to $\infty$. We denote the dressed state as the the *Floquet diabatic state* $|\mu m\rangle\equiv e^{im\omega t}|\mu\rangle$. In terms of the Floquet diabatic basis, a time periodic electronic wavefunction can be expressed as $|\Psi\rangle=\sum_{\mu m}\tilde{c}_{\mu m}|\mu m\rangle$ where $\tilde{c}_{\mu m}$ is an infinite dimensional state vector. The electronic wavefunction coefficient must satisfy the electronic TDSE $$i\hbar\sum_{\mu m}\frac{\partial\tilde{c}_{\mu m}}{\partial t}|\mu m\rangle=\sum_{\mu m}\hat{V}^{\text{F}}(t)|\mu m\rangle\tilde{c}_{\mu m}\label{eq:TDSE-Floquet-operator}$$ where the Floquet Hamiltonian operator is defined as $$\hat{V}^{\text{F}}(t)\equiv\hat{V}(t)-i\hbar\frac{\partial}{\partial t}.\label{eq:define-Floquet_Hamiltonian}$$ Next, we close Eq. \[eq:TDSE-Floquet-operator\] by multiplying both sides by$\langle\nu|$ and write $\langle\nu|\hat{V}^{\text{F}}(t)|\mu m\rangle=\sum_{n}\widetilde{V}_{\nu n,\mu m}^{\text{F}}e^{in\omega t}$ as a Fourier series: $$i\hbar\sum_{m}\frac{\partial\tilde{c}_{\nu m}}{\partial t}e^{im\omega t}=\sum_{\mu m}\sum_{n}\widetilde{V}_{\nu n,\mu m}^{\text{F}}e^{in\omega t}\tilde{c}_{\mu m}.$$ Thus, the TDSE in Eq.  can be solved by grouping together all terms with the same time dependence, leading to the equation of motion for $\tilde{c}$ $$i\hbar\frac{\partial}{\partial t}\tilde{c}_{\nu n}=\sum_{\mu m}\widetilde{V}_{\nu n,\mu m}^{\text{F}}\tilde{c}_{\mu m}.\label{eq:TDSE-Floquet-matrix}$$ The matrix elements of the Floquet Hamiltonian can be obtained by performing a Fourier transformation on the matrix elements $$\widetilde{V}_{\nu n,\mu m}^{\text{F}}=\langle\langle\nu n|\hat{V}^{\text{F}}|\mu m\rangle\rangle=\frac{1}{\tau}\int_{0}^{\tau}dt\left\langle \nu\right|\hat{V}^{\text{F}}(t)\left|\mu\right\rangle e^{i(m-n)\omega t}.\label{eq:double-bracket}$$ Here, we define the double bracket projection by $\langle\langle\nu n|\cdots|\mu m\rangle\rangle=\frac{1}{\tau}\int_{0}^{\tau}dt\left\langle \nu\right|\cdots\left|\mu\right\rangle e^{i(m-n)\omega t}$. *Given that the electronic Hamiltonian operator $\hat{V}(t)$ is periodic in time, the double-bracket projection eliminates all time dependence* and the Floquet Hamiltonian matrix reads $$\widetilde{V}_{\nu n,\mu m}^{\text{F}}=\langle\langle\nu n|\hat{V}|\mu m\rangle\rangle+\delta_{\mu\nu}\delta_{mn}m\hbar\omega.\label{eq:Floquet_Hamiltonian_projection_notation}$$ In the end, with this time-independent Hamiltonian, Eq.  can be formally solved by the exponential operator $\exp(-i\widetilde{\mathbf{V}}^{\text{F}}t/\hbar)$ with an arbitrary initial state. At this point, we will allow nuclei to move and turn out attention to the equation of motion for the density matrix $\widehat{\rho}_{W}(\vec{R},\vec{P},t)$ within the Floquet diabatic basis. The Wigner-transformed density matrix in the Floquet diabatic representation is $$\widetilde{A}_{\nu n,\mu m}^{\text{dia}}(\vec{R},\vec{P},t)=\langle\nu n|\widehat{\rho}_{W}(\vec{R},\vec{P},t)|\mu m\rangle.$$ For a proper F-QCLE, we will need to calculate the time derivative of $\widetilde{\mathbf{A}}^{\text{dia}}$ $$\frac{\partial}{\partial t}\widetilde{A}_{\nu n,\mu m}^{\text{dia}}=\left\langle \nu n\right|\frac{\partial\widehat{\rho}_{W}}{\partial t}\left|\mu m\right\rangle -i\left(n-m\right)\hbar\omega\widetilde{A}_{\nu n,\mu m}^{\text{dia}}\label{eq:dAdt-diabatic}$$ where the Floquet diabatic states depends on time explicitly. We begin by using Eq.  to project $\frac{\partial}{\partial t}\hat{\rho}_{W}$ into a Floquet diabatic basis. For the commutator term in Eq. , we can divide the operator product into matrix multiplication: $$\begin{aligned} \langle\nu n|\left[\hat{V}_{W},\hat{\rho}_{W}\right]|\mu m\rangle & = & \sum_{\lambda l}\langle\langle\nu n|\hat{V}_{W}|\lambda l\rangle\rangle\widetilde{A}_{\lambda l,\mu m}^{\text{dia}}\nonumber \\ & & -\widetilde{A}_{\nu n,\lambda l}^{\text{dia}}\langle\langle\lambda l|\hat{V}_{W}|\mu m\rangle\rangle\label{eq:commutator-diabatic}\\ & = & [\widetilde{\mathbf{V}}_{W},\widetilde{\mathbf{A}}^{\text{dia}}]_{\nu n,\mu m}\nonumber \end{aligned}$$ by inserting the identity of the diabatic electronic basis: $\hat{1}=\sum_{\lambda}|\lambda\rangle\langle\lambda|$ and expanding the time-dependent coefficients in terms of a Fourier series; see Appendix \[sec:The-trick\] for more details. Furthermore, if we combine Eq.  with the second term on the RHS of Eq. , we can write the sum of both terms as $[\widetilde{\mathbf{V}}^{\text{F}},\widetilde{\mathbf{A}}^{\text{dia}}]$, i.e. we can replace $\widetilde{\mathbf{V}}_{W}$ with $\widetilde{\mathbf{V}}^{\text{F}}$. For the anti-commutator term, we can use the same procedure to divide the operator product $$\begin{aligned} \langle\nu n|\left\{ \frac{\partial\hat{V}_{W}}{\partial R^{\alpha}},\frac{\partial\hat{\rho}_{W}}{\partial P^{\alpha}}\right\} |\mu m\rangle & = & \sum_{\lambda l}\langle\langle\nu n|\frac{\partial\hat{V}_{W}}{\partial R^{\alpha}}|\lambda l\rangle\rangle\frac{\partial\widetilde{A}_{\lambda l,\mu m}^{\text{dia}}}{\partial P^{\alpha}}\nonumber \\ & & +\frac{\partial\widetilde{A}_{\nu n,\lambda l}^{\text{dia}}}{\partial P^{\alpha}}\langle\langle\lambda l|\frac{\partial\hat{V}_{W}}{\partial R^{\alpha}}|\mu m\rangle\rangle\end{aligned}$$ where $\langle\nu n|\frac{\partial\widehat{\rho}_{W}}{\partial P^{\alpha}}|\mu m\rangle=\frac{\partial}{\partial P^{\alpha}}\widetilde{A}_{\nu n,\mu m}^{\text{dia}}$. Note that, since the Floquet diabatic states do not depend on the nuclear coordinate, we can rewrite the derivative of the electronic Hamiltonian in terms of the Floquet Hamiltonian $$\langle\langle\nu n|\frac{\partial\hat{V}_{W}}{\partial R^{\alpha}}|\mu m\rangle=\frac{\partial}{\partial R^{\alpha}}\widetilde{V}_{\mu m,\nu n}^{\text{F}}.\label{eq:dVdR-diabatic}$$ In the end, we may combine the above expressions to write down a complete *diabatic* F-QCLE $$\begin{aligned} \frac{\partial}{\partial t}\widetilde{\mathbf{A}}^{\text{dia}} & = & -\frac{i}{\hbar}\left[\widetilde{\mathbf{V}}^{\text{F}},\widetilde{\mathbf{A}}^{\text{dia}}\right]-\frac{P^{\alpha}}{M}\frac{\partial\widetilde{\mathbf{A}}^{\text{dia}}}{\partial R^{\alpha}}\nonumber \\ & & +\frac{1}{2}\left\{ \frac{\partial\widetilde{V}^{\text{F}}}{\partial R^{\alpha}},\frac{\partial\widetilde{\mathbf{A}}^{\text{dia}}}{\partial P^{\alpha}}\right\} \label{eq:F-QCLE-diabatic}\end{aligned}$$ As a final remark, we emphasize that the Floquet Hamiltonian $\widetilde{\mathbf{V}}^{\text{F}}=\widetilde{\mathbf{V}}^{\text{F}}(\vec{R})$ is a time-independent matrix, so Eq.  is simply the diabatic QCLE corresponding to an infinitely large electronic Hamiltonian $\widetilde{\mathbf{V}}^{\text{F}}$. Transformation to the adiabatic representation ---------------------------------------------- To recast the diabatic F-QCLE in an adiabatic representation, we diagonalize the Floquet Hamiltonian matrix by solving the eigenvalue problem: $$\sum_{\nu n}\widetilde{V}_{\mu m,\nu n}^{\text{F}}(\vec{R})G_{\nu n}^{J}(\vec{R})={\cal E}_{J}^{\text{F}}(\vec{R})G_{\mu m}^{J}(\vec{R}).\label{eq:Floquet_adiabatic_eigenvalue}$$ The eigenvalues ${\cal E}_{J}^{\text{F}}={\cal E}_{J}^{\text{F}}(\vec{R})$ are the so-called Floquet quasi-energies with corresponding eigenvectors $G_{\mu m}^{J}(\vec{R}).$ Since $\widetilde{\mathbf{V}}^{\text{F}}$ is Hermitian, we can choose the eigenvectors $G_{\mu m}^{J}$ to be othornormal so that we have the identities $\mathbf{G}^{\dagger}\mathbf{G}=\mathbf{G}\mathbf{G}^{\dagger}=\mathbf{I}$, i.e. $\sum_{\lambda\ell}G_{\lambda\ell}^{J*}G_{\lambda\ell}^{K}=\delta_{JK}$ and $\sum_{L}G_{\mu m}^{L}G_{\nu n}^{L*}=\delta_{\mu m,\nu n}$. The Floquet adiabatic state corresponding to the quasi-energy ${\cal E}_{J}^{\text{F}}$ are $$|\phi^{J}(\vec{R},t)\rangle=\sum_{\mu m}G_{\mu m}^{J}(\vec{R})\left|\mu m\right\rangle .\label{eq:Floquet_adiabatic_eigenstate}$$ As a practical matter, although $\widetilde{\mathbf{V}}^{\text{F}}$ is infinite dimensional, we can truncate highly oscillating Floquet states by replacing $\sum_{m=-\infty}^{\infty}$ with $\sum_{m=-M}^{M}$. With this Floquet adiabatic state basis, the probability density can be obtained by a diabatic-to-adiabatic transformation $$\widetilde{A}_{JK}^{\text{adi}}(\vec{R},\vec{P},t)=\langle\Phi^{J}|\widehat{\rho}_{W}\left(\vec{R},\vec{P},t\right)|\Phi^{K}\rangle=\left(\mathbf{G}^{\dagger}\widetilde{\mathbf{A}}^{\text{dia}}\mathbf{G}\right)_{JK}$$ in the Floquet adiabatic representation. Note that, since the eigenvectors $G_{\mu m}^{J}(\vec{R})$ do not depend on time explicitly, the time-derivative of the adiabatic probability density can be calculated simply to be: $$\frac{\partial\widetilde{\mathbf{A}}^{\text{adi}}}{\partial t}=\mathbf{G}^{\dagger}\frac{\partial\widetilde{\mathbf{A}}^{\text{dia}}}{\partial t}\mathbf{G}.\label{eq:dAdt-adiabatic}$$ We are now ready to derive the adiabatic F-QCLE in the following section. Different Approaches to derive F-QCLE\[sec:Floquet-QCLE\] ========================================================= In this section, we present two approaches with different orders for the three operations above; as will be shown, different orders will result in different adiabatic F-QCLEs. We summarize these approaches and the corresponding F-QCLEs in Table \[tab:F-QCLEs-via-different\]. Note that, in practice, even more approaches are possible, but we will ignore all AW approaches given that Izmaylov and co-workers have shown that such an ordering is inappropriate[@ryabinkin_analysis_2014]. Wigner-Floquet-Adiabatic (WFA) approach --------------------------------------- Our first approach follows the order presented above: we first perform partial Wigner transformation, we second project to a Floquet diabatic basis, we third transform to an adiabatic representation. For the last step, following Eq. , we transform the diabatic F-QCLE by sandwiching the diabatic F-QCLE (Eq. ) with $\mathbf{G}^{\dagger}$ and $\mathbf{G}$. The first term of Eq.  (the commutator term) becomes $$\begin{aligned} \sum_{\nu n}\sum_{\mu m}G_{\nu n}^{J*}\left[\widetilde{V}^{\text{F}},\widetilde{A}^{\text{dia}}\right]_{\nu n,\mu m}G_{\mu m}^{K} & = & \left({\cal E}_{J}^{\text{F}}-{\cal E}_{K}^{\text{F}}\right)\widetilde{A}_{JK}^{\text{adi}}\end{aligned}$$ For the second term, since the Floquet adiabatic states depend on the nuclear coordinate $\vec{R}$, the $R^{\alpha}$ derivative of the density must yield $$\sum_{\nu n}\sum_{\mu m}G_{\nu n}^{J*}\frac{\partial\widetilde{A}_{\nu n,\mu m}^{\text{dia}}}{\partial R^{\alpha}}G_{\mu m}^{K}=\frac{\partial\widetilde{A}_{JK}^{\text{adi}}}{\partial R^{\alpha}}+\sum_{L}\left(D_{JL}^{\alpha}\widetilde{A}_{LK}^{\text{adi}}-\widetilde{A}_{JL}^{\text{adi}}D_{LK}^{\alpha}\right)$$ where the derivative coupling is $D_{JK}^{\alpha}=\langle\langle\Phi^{J}|\frac{\partial}{\partial R^{\alpha}}|\Phi^{K}\rangle\rangle=\sum_{\mu m}G_{\mu m}^{J*}\frac{\partial G_{\mu m}^{K}}{\partial R^{\alpha}}$ corresponding to the change of the Floquet adiabatic states with respect to the nuclear coordinate $R^{\alpha}$. Note that, if the Floquet Hamiltonian is real, the diagonal element of the derivative coupling is zero ($D_{JJ}^{\alpha}=0$). For the third term (the anti-commutator term), the $R^{\alpha}$ derivative of the Floquet Hamiltonian can be written in terms of the force matrix $$\sum_{\nu n}\sum_{\mu m}G_{\nu n}^{J*}\frac{\partial\widetilde{V}_{\nu n,\mu m}^{\text{F}}}{\partial R^{\alpha}}G_{\mu m}^{K}=-F_{JK}^{\alpha}\label{eq:Force-matrix}$$ explicitly, $$F_{JK}^{\alpha}=-\frac{\partial{\cal E}_{J}^{\text{F}}}{\partial R^{\alpha}}\delta_{JK}+({\cal E}_{J}^{\text{F}}-{\cal E}_{K}^{\text{F}})D_{JK}^{\alpha}.$$ The force matrix accounts for direct changes in the nuclear momentum associated with the electronic coupling. One can understand the diagonal element $F_{JJ}^{\alpha}=-\frac{\partial{\cal E}_{J}^{\text{F}}}{\partial R^{\alpha}}$ as the classical force for nuclear dynamics moving along the $J$-th Floquet quasi-energy surface in the phase space. Finally, the F-QCLE via the WFA approach reads $$\begin{aligned} \frac{\partial}{\partial t}\widetilde{A}_{JK}^{\text{adi}} & = & -\frac{i}{\hbar}\left({\cal E}_{J}^{\text{F}}-{\cal E}_{K}^{\text{F}}\right)\widetilde{A}_{JK}^{\text{adi}}\nonumber \\ & & -\frac{P^{\alpha}}{M^{\alpha}}\frac{\partial\widetilde{A}_{JK}^{\text{adi}}}{\partial R^{\alpha}}-\frac{P^{\alpha}}{M^{\alpha}}\sum_{L}\left(D_{JL}^{\alpha}\widetilde{A}_{LK}^{\text{adi}}-\widetilde{A}_{JL}^{\text{adi}}D_{LK}^{\alpha}\right)\nonumber \\ & & -\frac{1}{2}\sum_{L}\left(F_{JL}^{\alpha}\frac{\partial\widetilde{A}_{LK}^{\text{adi}}}{\partial P^{\alpha}}+\frac{\partial\widetilde{A}_{JL}^{\text{adi}}}{\partial P^{\alpha}}F_{LK}^{\alpha}\right)\label{eq:F-QCLE-WDA}\end{aligned}$$ We find that Eq. \[eq:F-QCLE-WDA\] takes exactly the same form as the standard QCLE in the adiabatic representation (for electron-nuclear dynamics without a driving laser). Wigner-Adiabatic-Floquet (WAF) approach --------------------------------------- For the second approach, we exchange the “adiabatic” and “Floquet” operations after the partial Wigner transformation. In this case, we directly project Eq.  onto the Floquet adiabatic basis $|\phi^{J}(\vec{R},t)\rangle$. Namely, we make the diabatic-to-adiabatic transform of the Floquet electronic basis prior to the projection onto the dressed electronic states. Thus, we consider this path as the *Wigner-Adiabatic-Floquet* (WAF) approach. Overall, we apply a procedure similar to what was used in Eq. \[eq:F-QCLE-WDA\]. For the commutator term, we divide operator products using the same technique as in Appendix \[sec:The-trick\]. The $R^{\alpha}$ derivative term yields a derivative coupling term as the Floquet adiabatic basis depends on $R^{\alpha}$ explicitly. In the end, the WAF approach includes the first three terms exactly as Eq. . However, from the anti-commutator term of Eq. , the WAF approach leads to an additional cross derivative force. To see this, we focus on the derivative of the electronic Hamiltonian in the adiabatic representation $$\langle\langle\Phi^{J}|\frac{\partial\hat{V}}{\partial R^{\alpha}}|\Phi^{K}\rangle\rangle=\langle\langle\Phi^{J}|\frac{\partial\hat{V}^{\text{F}}}{\partial R^{\alpha}}|\Phi^{K}\rangle\rangle+i\hbar\langle\langle\Phi^{J}|\frac{\partial}{\partial R^{\alpha}}\frac{\partial}{\partial t}|\Phi^{K}\rangle\rangle$$ where we have used the definition of the Floquet Hamiltonian $\frac{\partial\hat{V}^{\text{F}}}{\partial R^{\alpha}}=\frac{\partial\hat{V}}{\partial R^{\alpha}}-i\hbar\frac{\partial}{\partial R^{\alpha}}\frac{\partial}{\partial t}$. The derivative of the electronic Hamiltonian yields two terms: first, the same force matrix we obtained in Eq. : $$\langle\langle\Phi^{J}|\frac{\partial\hat{V}^{\text{F}}}{\partial R^{\alpha}}|\Phi^{K}\rangle\rangle=-F_{JK}^{\alpha}$$ second, a cross derivative force comes from the explicit dependence of the Floquet adiabatic states on both the nuclear coordinate and time $$i\hbar\langle\langle\Phi^{J}|\frac{\partial}{\partial R^{\alpha}}\frac{\partial}{\partial t}|\Phi^{K}\rangle\rangle=-\chi_{JK}^{\alpha},$$ and explicitly, $$\chi_{JK}^{\alpha}=\sum_{\mu m}m\hbar\omega G_{\mu m}^{J*}\frac{\partial G_{\mu m}^{K}}{\partial R^{\alpha}}.\label{eq:cross-derivative-force}$$ Note that, unlike the derivative coupling $D_{JK}^{\alpha}$, the diagonal element of $\chi_{JK}^{\alpha}$ is non-zero and real. Finally, the F-QCLE via the WAF approach reads: $$\begin{aligned} \frac{\partial}{\partial t}\widetilde{\mathbf{A}}^{\text{adi}} & = & -\frac{i}{\hbar}\left({\cal E}_{J}^{\text{F}}-{\cal E}_{K}^{\text{F}}\right)\widetilde{A}_{JK}^{\text{adi}}\nonumber \\ & & -\frac{P^{\alpha}}{M^{\alpha}}\frac{\partial\widetilde{A}_{JK}^{\text{adi}}}{\partial R^{\alpha}}-\frac{P^{\alpha}}{M^{\alpha}}\sum_{L}\left(D_{JL}^{\alpha}\widetilde{A}_{LK}^{\text{adi}}-\widetilde{A}_{JL}^{\text{adi}}D_{LK}^{\alpha}\right)\nonumber \\ & & -\frac{1}{2}\sum_{L}\left(F_{JL}^{\alpha}+\chi_{LJ}^{\alpha*}\right)\frac{\partial\widetilde{A}_{LK}^{\text{adi}}}{\partial P^{\alpha}}+\frac{\partial\widetilde{A}_{JL}^{\text{adi}}}{\partial P^{\alpha}}\left(F_{LK}^{\alpha}+\chi_{LK}^{\alpha}\right)\label{eq:F-QCLE-WAD}\end{aligned}$$ We observe that, while Eq.  and Eq. \[eq:F-QCLE-WAD\] take the same form, the “effective” force matrix (defined as the coefficients of $\frac{\partial\widetilde{\mathbf{A}}^{\text{adi}}}{\partial P^{\alpha}}$) includes an additional cross derivative force that indicates the difference between these two equations. The similarities suggest that the time evolution of the electronic degrees of freedom ($\frac{\partial\widetilde{\mathbf{A}}^{\text{adi}}}{\partial R^{\alpha}}=\frac{\partial\widetilde{\mathbf{A}}^{\text{adi}}}{\partial P^{\alpha}}=0$) should follow the same equation for both the WFA and WAF. However, the difference in the “effective” force matrix will affect the nuclear dynamics in phase space. Specifically, from Eq. \[eq:F-QCLE-WAD\], the classical force on the $J$-th quasi-energy surface is given by $F_{JJ}^{\alpha}+\chi_{JJ}^{\alpha}$, implying that the nuclear dynamics should experience the additional cross derivative force on top of the Floquet quasi-energy surface. F-QCLE effective force matrix ----- -------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- WFA Eq.  $-\frac{\partial{\cal E}_{J}^{\text{F}}}{\partial R^{\alpha}}\delta_{JK}+({\cal E}_{J}^{\text{F}}-{\cal E}_{K}^{\text{F}})D_{JK}^{\alpha}$ WAF Eq.  $-\frac{\partial{\cal E}_{J}^{\text{F}}}{\partial R^{\alpha}}\delta_{JK}+({\cal E}_{J}^{\text{F}}-{\cal E}_{K}^{\text{F}})D_{JK}^{\alpha}-\sum_{\mu m}m\hbar\omega G_{\mu m}^{J*}\frac{\partial G_{\mu m}^{K}}{\partial R^{\alpha}}$ : The F-QCLEs obtained via different approaches differ in the effective force matrices for nuclear motions.\[tab:F-QCLEs-via-different\] Results \[sec:Results:\] ======================== Shifted avoided crossing model ------------------------------ To analyze the difference between these approaches, we consider a shifted avoided crossing model composed of a two-level electronic system coupled to a 1D nuclear motion. The diabatic electronic states are denoted as $|g\rangle$ and $|e\rangle$ and the electronic Hamiltonian is given by $$\hat{V}(R,t)=\left(\begin{array}{cc} V_{gg}(R) & V_{ge}(R,t)\\ V_{eg}(R,t) & V_{ee}(R) \end{array}\right)$$ where the diabatic energy is given by $$V_{gg}(R)=\begin{cases} A(1-e^{-BR}) & R>0\\ -A(1-e^{BR}) & R<0 \end{cases}\label{eq:Vgg}$$ $$V_{ee}(R)=-V_{gg}(R)+\hbar\omega$$ and the diabatic coupling is periodic in time $$V_{ge}(R,t)=V_{eg}(R,t)=Ce^{-DR^{2}}\cos\omega t.$$ The parameters are $A=0.01$, **$B=1.6$**, $C=0.01$, $D=1.0$, and the nuclear mass is $M=2000$. Note that this model is Tullys simple avoided crossing model with two modifications: the diabatic coupling becomes time-periodic and the excited potential energy surface is shifted by $\hbar\omega$. For simplicity, we choose the laser frequency $\hbar\omega=0.024$ large enough ($\hbar\omega>2A$) so that the diabatic Floquet states $|gm\rangle$ have an avoided crossing only with $|e(m-1)\rangle$ (at $R=0$) and do not have any trivial crossings. We assume the initial wavepacket is a Gaussian centered at the initial position $R_{0}$ and momentum $P_{0}$ on diabat $|g\rangle$: $$|\Psi_{0}\rangle=\frac{1}{{\cal N}}\exp\left(-\frac{(R-R_{0})^{2}}{2\sigma^{2}}+\frac{i}{\hbar}P_{0}(R-R_{0})\right)|g\rangle$$ where the normalization factor is ${\cal N}^{4}=\pi\sigma^{2}$. The width of the Gaussian is chosen to be $\sigma=20\hbar/P_{0}$. The wavepacket can be propagated exactly in the diabatic representation. F-FSSH based on WFA and WAF --------------------------- To show the difference between the dynamics as obtained by the different F-QCLEs, we will simulate F-FSSH results for both the WFA and WFA approaches. Within F-FSSH, we describe the propagation of the Floquet wavepacket by a swarm of trajectories, each with its own electronic amplitudes $\tilde{c}_{J}$ satisfying $$\frac{\partial\tilde{c}_{J}}{\partial t}=-\frac{i}{\hbar}{\cal E}_{J}^{\text{F}}\tilde{c}_{J}-\frac{P}{M}\sum_{L}D_{JL}\tilde{c}_{L}$$ where ${\cal E}_{J}^{\text{F}}={\cal E}_{J}^{\text{F}}(R)$, $D_{JL}=D_{JL}(R)$ and $R=R(t)$, $P=P(t)$ representing nuclear trajectory. All nuclear trajectories move classically along an active Floquet state ($J$) obeying $$\frac{dR}{dt}=\frac{P}{M}$$ $$\frac{dP}{dt}=\begin{cases} -\frac{\partial{\cal E}_{J}^{\text{F}}}{\partial R} & \text{for WFA}\\ -\frac{\partial{\cal E}_{J}^{\text{F}}}{\partial R}-\chi_{JJ} & \text{for WAF} \end{cases}\label{eq:dPdt}$$ Here, based on the connection between the QCLE and FSSH, the nuclear force in Eq. \[eq:dPdt\] is determined according to the diagonal element of the effective force matrix in the F-QCLEs. Consistent with the standard FSSH technique, the hopping probability from active Floquet state $J$ to state $K$ is given by $$\text{Prob}(J\rightarrow K)=-2\text{Re}\left(\frac{P^{\alpha}}{M^{\alpha}}\cdot D_{KJ}^{\alpha}\frac{\tilde{c}_{J}\tilde{c}_{K}^{*}}{|\tilde{c}_{J}|^{2}}\right)dt$$ where $dt$ is the classical time step. After each successful hop, the velocity is adjusted to conserve the total Floquet quasi-energy. If a frustrated hop occurs, we implement velocity reversal[@jasper_improved_2003]. Note that we neglect the decoherence correction since the over-coherence problem should not be severe for a simple avoided crossing model[@landry_how_2012]. In the end, the probability to measure diabatic state $\mu$ can be evaluated by the density matrix interpretation[@landry_communication:_2013] $$P_{\mu}=\sum_{m}\frac{N(\mu m)}{N_{\text{traj}}}+\sum_{n\neq m}\frac{\sum_{l}^{N(\mu m)}\sum_{k}^{N(\mu n)}\tilde{c}_{\mu m}^{(l)}\tilde{c}_{\mu n}^{(k)*}e^{i(m-n)\omega t}}{N(\mu m)N(\mu n)}$$ where $N(\mu m)=\sum_{l}^{N_{\text{traj}}}\delta_{J^{(l)}\mu m}$ is the number of the trajectories that have the active surface $J^{(l)}$ end up on the Floquet state $|\mu m\rangle$. Here $l$ and $k$ are the trajectory indices. We propagate $N_{\text{traj}}$ trajectories for an amount of time long enough for each trajectory to pass through the coupling region ($|R|<3$ for this parameter set). Effective Floquet quasi-energy surfaces for nuclear dynamics ------------------------------------------------------------ First, we analyze the effective potential energy surfaces for nuclear dynamics by integrating Eq. \[eq:dPdt\] over $R$ for the WFA and WAF approaches respectively. For the WFA approach, the effective PES simply recovers the Floquet quasi-energy surfaces ${\cal E}_{J}^{\text{F}}$. For the WAF approach, the effective PES is $V_{eff}(R)={\cal E}_{J}^{\text{F}}(R)+\int_{-\infty}^{R}\chi_{JJ}(R')dR'$ where the quasi-energy surface is modified by the integration of the cross derivative force. We find that including the cross derivative force in the WAF approach increases the crossing barrier for the nuclear dynamics on the lower adiabat (see Fig. \[fig:PES\]). Note that, in terms of an F-FSSH calculation, these changes will have a direct effect on the nuclear dynamics, but not the electronic amplitudes. ![The effective potential energy surfaces for the WFA approach (solid lines) and the WAF approach (dash lines). The lower (upper) quasi-energy surface is in red (blue). The diabatic Floquet state energies $\widetilde{V}_{\mu m,\mu m}^{\text{F}}$ are plotted for $|g1\rangle$ (green) and $|e0\rangle$ (orange) in dotted lines. Note that the barrier height and the equilibrium energy ratio of the WAF surface is significantly modified (relative to the WFA approach) by the presence of the additional cross derivative force.\[fig:PES\]](fig_Floquet_eigen) Transmission and reflection --------------------------- Next, we turn our attention to the transmission and reflection probabilities produced by the F-FSSH calculations. Overall, the WFA results are more accurate than the WAF results. In Fig. \[fig:Transmission\], we find that, according to the WFA approach, there should be a rise in transmission on the lower adiabat around $P_{0}\approx5.3$, which is the momentum for which transmission should be allowed classically; see the barrier height ($\approx0.007$) in Fig. \[fig:PES\]. Indeed, such a threshold at $P_{0}\approx5.3$ is found according to exact wavefunction simulation as well. However, for the WAF approach with the cross derivative force, one find a higher crossing barrier energy ($\approx0.015$), and the transmission on the lower adiabat occurs (incorrectly) at $P_{0}\approx7.8$. This result suggests that the cross derivative force is a fictitious term: the WAF semiclassical derivation is spurious. Let us now focus on the WFA results in more detail. Several points are worth mentioning. First, the transmission to the upper adiabat occurs after $P_{0}\approx8.9$, which agrees with the classical energy difference $2A=0.02$ (see Eq.  for the definition of $A$). Second, for high initial momentum ($P_{0}>8.0$), the F-FSSH-WFA can almost recover the correct nuclear dynamics. Third, in the intermediate momentum region $P_{0}\in(6,8)$, unfortunately, the F-FSSH wavepacket exhibits less transmission than the exact calculation. This discrepancy may be attributed to FSSH’s incapability to capture nuclear tunneling effects. ![The probability of transmission (right) and reflection (left) on the upper and lower adiabats as a function of the initial momentum. The F-FSSH dynamics is implemented using the effective nuclear forces of the WFA (red) and WAF (blue). Overall, the WFA result is more accurate than the WAF result. The WFA result almost recover the correct nuclear dynamics. Due to the cross derivative force, the nuclear trajectory of the F-FSSH(WAF) experiences a much higher crossing barrier, requiring larger $P_{0}$ for transmission. \[fig:Transmission\]](fqcle_WDAvsWAD) Conclusion and Outlook\[sec:Conclusion\] ======================================== We have analyzed two approaches for deriving the QCLE within a Floquet formalism, and found two different F-QCLEs. While these F-QCLEs take similar forms, the difference in the effective force matrix can lead to large discrepancies in nuclear dynamics. As such, in the context of driven electron-nuclear dynamics, our results reiterate the fact that one cannot change the order of the operations in the derivation of the correct QCLE. Specifically, as opposed to the WFA approach, the WAF approach [\[]{}exchanging Floquet electronic basis dressing (F) and adiabatic transformation (A)[\]]{} is spurious. Overall, our results are very consistent with the results of Izmaylov and Kapral[@ryabinkin_analysis_2014] who find that one must be careful when deriving the QCLE even without time dependence; in the end, with or without a time-dependent Hamiltonian, it appears that one will always derive the correct semiclassical equation of motion provided that one moves to the adiabatic representation *as the very last step*. Looking forward, the derivation of the F-QCLE presented here validates the F-FSSH method and paves the way to further improvements in the future. With regard to coherence and decoherence, given the time-independent nature of the Floquet Hamiltonian, we can immediately apply many decoherence schemes, including augmented moment decoherence[@jain_efficient_2016; @landry_how_2012; @petit_how_2014; @schwartz_quantum_1996; @prezhdo_relationship_1998], to the F-FSSH algorithm. As far as geometric phase effect is concerned, it is known that Berry phases are already included within the QCLE[@subotnik_demonstration_2019] for a time-independent electronic Hamiltonian, and so we would expect that similar effects should already be included within this proper F-QCLE for periodic (time-dependent) electronic Hamiltonians. Nevertheless, however, there is one nuance which we have conveniently neglected in the present paper. Note that, according to Eq. , we have every reason to believe that the F-QCLE formalism (especially for a non-monocrhomatic driving field with more than one Fourier mode in the time-dependent Hamiltonian) will necessarily introduce a complex (i.e. not real) Floquet Hamiltonian. In such a case, we should find not just Berry phases, but also Berry force[@miao_extension_2019]. Future research into the nature of this intrinsic magnetic Berry forcehow or if it appears in the context of F-QCLE and surface hopping dynamicsis currently underway and represents an exciting new direction for non-adiabatic theory. Lastly, it has been recently reported that novel control schemes, such as Floquet engineering[@thanh_phuc_control_2018; @schwennicke_optical_2020], can enhance the excitation energy transfer rate even in the presence of strong fluctuations and dissipation. Given so many potential applications for F-QCLE simulations, we believe the present manuscript should find immediate use in the physical chemistry and chemical physics community. Acknowledgment {#acknowledgment .unnumbered} ============== This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0019397. This research also used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231. We thank Abraham Nitzan for very helpful discussions. Data Availablity {#data-availablity .unnumbered} ================ The data that support the findings of this study are available from the corresponding author upon reasonable request. Rewriting the Product of Wignerized Operators As Matrix Multiplication in the Floquet Representation \[sec:The-trick\] ====================================================================================================================== Throughout this paper, we have constantly used one trick. Namely, we have consistently rewritten the product of two Wigner transformed operators (one of which must be time-periodic) into a non-standard matrix product in the Floquet representation. To see how this trick works in practice, we consider (for example) the operator product $\hat{\rho}_{W}\hat{H}_{W}$. We insert the electronic identity operator $\hat{1}=\sum_{\lambda}|\lambda\rangle\langle\lambda|$ in between $\hat{\rho}_{W}$ and $\hat{H}_{W}$: $$\langle\nu n|\hat{\rho}_{W}\hat{H}_{W}|\mu m\rangle=\sum_{\lambda}\langle\nu n|\hat{\rho}_{W}|\lambda\rangle\langle\lambda|\hat{H}_{W}|\mu m\rangle$$ Next, as in Sec. \[sec:Three-operations\], we express $\langle\lambda|\hat{H}_{W}|\mu m\rangle$ in the form of Fourier series: $$\langle\lambda|\hat{H}_{W}|\mu m\rangle=\sum_{l}\langle\langle\lambda l|\hat{H}_{W}|\mu m\rangle\rangle e^{il\omega t}$$ where the double bracket projection is defined by Fourier transform. With the Fourier series, we write $|\lambda\rangle e^{il\omega t}=|\lambda l\rangle$ and obtain $$\langle\nu n|\hat{\rho}_{W}\hat{H}_{W}|\mu m\rangle=\sum_{\lambda l}\langle\nu n|\hat{\rho}_{W}|\lambda l\rangle\langle\langle\lambda l|\hat{H}_{W}|\mu m\rangle\rangle$$

--- abstract: | **Purpose:** : To develop a Breast Imaging Reporting and Data System (BI-RADS^^) breast density DL model in a multi-site setting for synthetic 2D mammography (SM) images derived from 3D DBT exams using FFDM images and limited SM data.\ **Materials and Methods:** A DL model was trained to predict BI-RADS breast density using FFDM images acquired from 2008 to 2017 (Site 1: 57492 patients, 187627 exams, 750752 images) for this retrospective study. The FFDM model was evaluated using SM datasets from two institutions (Site 1: 3842 patients, 3866 exams, 14472 images, acquired from 2016 to 2017; Site 2: 7557 patients, 16283 exams, 63973 images, 2015 to 2019). Adaptation methods were investigated to improve performance on the SM datasets and the effect of dataset size on each adaptation method is considered. Statistical significance was assessed using confidence intervals (CI), estimated by bootstrapping.\ **Results:** Without adaptation, the model demonstrated close agreement with the original reporting radiologists for all three datasets (Site 1 FFDM: linearly-weighted $\kappa_w$ = 0.75, 95% CI: \[0.74, 0.76\]; Site 1 SM: $\kappa_w$ = 0.71, CI: \[0.64, 0.78\]; Site 2 SM: $\kappa_w$ = 0.72, CI: \[0.70, 0.75\]). With adaptation, performance improved for Site 2 (Site 1: $\kappa_w$ = 0.72, CI: \[0.66, 0.79\], Site 2: $\kappa_w$ = 0.79, CI: \[0.76, 0.81\]) using only 500 SM images from each site.\ **Conclusion:** A BI-RADS breast density DL model demonstrated strong performance on FFDM and SM images from two institutions without training on SM images and improved using few SM images. author: - 'Thomas P. Matthews' - Sadanand Singh - Brent Mombourquette - Jason Su - 'Meet P. Shah' - Stefano Pedemonte - Aaron Long - David Maffitt - Jenny Gurney - Rodrigo Morales Hoil - Nikita Ghare - Douglas Smith - 'Stephen M. Moore' - 'Susan C. Marks' - 'Richard L. Wahl' title: 'A multi-site study of a breast density deep learning model for full-field digital mammography and digital breast tomosynthesis exams' --- Introduction {#sec:introduction} ============ Breast density is an important risk factor for breast cancer [@kerlikowske_breast_2010; @mccormack_breast_2006; @boyd_mammographic_2007] and areas of higher density can mask findings within mammograms leading to lower sensitivity [@mandelson_breast_2000]. Many states have passed breast density notification laws requiring clinics to inform women of their density [@kressin_content_2016]. Radiologists typically assess breast density using the Breast Imaging Reporting and Data System (BI-RADS^^) lexicon, which divides breast density into four categories: A, almost entirely fatty; B, scattered areas of fibroglandular density; C, heterogeneously dense; and D, extremely dense (examples are presented in Figure \[fig:example\_density\_cview\]) [@sickles_acr_2013]. Unfortunately, radiologists exhibit intra- and inter-reader variability in the assessment of BI-RADS breast density, which can result in differences in clinical care and estimated risk [@sprague_variation_2016; @spayne_reproducibility_2012; @berg_breast_2000]. [0.25]{} ![Example synthetic 2D mammography (SM) images derived from digital breast tomosynthesis (DBT) exams for each of the four Breast Imaging Reporting and Data System (BI-RADS) breast density categories: (a) A, almost entirely fatty, (b) B, scattered areas of fibroglandular density, (c) C, heterogeneously dense, and (d) D, extremely dense. Images are normalized so that the grayscale intensity windows found in their Digital Imaging and Communications in Medicine (DICOM) headers range from 0.0 to 1.0.[]{data-label="fig:example_density_cview"}](example_birads_a "fig:"){width="\textwidth"} [0.25]{} ![Example synthetic 2D mammography (SM) images derived from digital breast tomosynthesis (DBT) exams for each of the four Breast Imaging Reporting and Data System (BI-RADS) breast density categories: (a) A, almost entirely fatty, (b) B, scattered areas of fibroglandular density, (c) C, heterogeneously dense, and (d) D, extremely dense. Images are normalized so that the grayscale intensity windows found in their Digital Imaging and Communications in Medicine (DICOM) headers range from 0.0 to 1.0.[]{data-label="fig:example_density_cview"}](example_birads_b "fig:"){width="\textwidth"} [0.25]{} ![Example synthetic 2D mammography (SM) images derived from digital breast tomosynthesis (DBT) exams for each of the four Breast Imaging Reporting and Data System (BI-RADS) breast density categories: (a) A, almost entirely fatty, (b) B, scattered areas of fibroglandular density, (c) C, heterogeneously dense, and (d) D, extremely dense. Images are normalized so that the grayscale intensity windows found in their Digital Imaging and Communications in Medicine (DICOM) headers range from 0.0 to 1.0.[]{data-label="fig:example_density_cview"}](example_birads_c "fig:"){width="\textwidth"} [0.25]{} ![Example synthetic 2D mammography (SM) images derived from digital breast tomosynthesis (DBT) exams for each of the four Breast Imaging Reporting and Data System (BI-RADS) breast density categories: (a) A, almost entirely fatty, (b) B, scattered areas of fibroglandular density, (c) C, heterogeneously dense, and (d) D, extremely dense. Images are normalized so that the grayscale intensity windows found in their Digital Imaging and Communications in Medicine (DICOM) headers range from 0.0 to 1.0.[]{data-label="fig:example_density_cview"}](example_birads_d_6 "fig:"){width="\textwidth"} Deep learning (DL) has previously been employed to assess BI-RADS breast density for film [@yi_deep-learning_2018] and full-field digital mammography (FFDM) images [@gandomkar_bi-rads_2019; @mohamed_deep_2018; @ma_multi-path_2019; @lehman_mammographic_2019; @wu_breast_2017] with some models demonstrating closer agreement with consensus estimates than individual radiologists [@lehman_mammographic_2019]. To realize the promise of using these DL models in clinical practice, two key challenges must be met. First, as breast cancer screening is increasingly moving to digital breast tomosynthesis (DBT) [@richman_adoption_2019] due to improved reader performance [@friedewald_breast_2014; @skaane_comparison_2013; @rafferty_diagnostic_2014], DL models should be compatible with DBT exams. Figure \[fig:cview\_ffdm\_comparison\] shows the differences in image characteristics between 2D images for FFDM and DBT exams. However, the relatively recent adoption of DBT at many institutions means that the datasets available for training DL models are often fairly limited for DBT exams compared with FFDM exams. Second, DL models must offer consistent performance across sites, where differences in imaging technology, patient demographics, or assessment practices could impact model performance. To be practical, this should be achieved while requiring little additional data from each site. [0.25]{} ![Comparison between (a) a full-field digital mammography (FFDM) image and (b) a synthetic 2D mammography (SM) image of the same breast under the same compression. A zoomed-in region, whose original location is denoted by the white box, is shown for both (c) the FFDM image and (d) the SM image to highlight the differences in texture and contrast that can occur between the two image types. Images are normalized so that the grayscale intensity windows found in their Digital Imaging and Communications in Medicine (DICOM) headers range from 0.0 to 1.0.[]{data-label="fig:cview_ffdm_comparison"}](example_ffdm_image "fig:"){width="\textwidth"} [0.25]{} ![Comparison between (a) a full-field digital mammography (FFDM) image and (b) a synthetic 2D mammography (SM) image of the same breast under the same compression. A zoomed-in region, whose original location is denoted by the white box, is shown for both (c) the FFDM image and (d) the SM image to highlight the differences in texture and contrast that can occur between the two image types. Images are normalized so that the grayscale intensity windows found in their Digital Imaging and Communications in Medicine (DICOM) headers range from 0.0 to 1.0.[]{data-label="fig:cview_ffdm_comparison"}](example_cview_image "fig:"){width="\textwidth"} [0.25]{} ![Comparison between (a) a full-field digital mammography (FFDM) image and (b) a synthetic 2D mammography (SM) image of the same breast under the same compression. A zoomed-in region, whose original location is denoted by the white box, is shown for both (c) the FFDM image and (d) the SM image to highlight the differences in texture and contrast that can occur between the two image types. Images are normalized so that the grayscale intensity windows found in their Digital Imaging and Communications in Medicine (DICOM) headers range from 0.0 to 1.0.[]{data-label="fig:cview_ffdm_comparison"}](example_ffdm_image_inset "fig:"){width="\textwidth"} [0.25]{} ![Comparison between (a) a full-field digital mammography (FFDM) image and (b) a synthetic 2D mammography (SM) image of the same breast under the same compression. A zoomed-in region, whose original location is denoted by the white box, is shown for both (c) the FFDM image and (d) the SM image to highlight the differences in texture and contrast that can occur between the two image types. Images are normalized so that the grayscale intensity windows found in their Digital Imaging and Communications in Medicine (DICOM) headers range from 0.0 to 1.0.[]{data-label="fig:cview_ffdm_comparison"}](example_cview_image_inset "fig:"){width="\textwidth"} In this work, we present a BI-RADS breast density DL model that offers close agreement with the original reporting radiologists for both FFDM and DBT exams at two institutions. A DL model is first trained to predict BI-RADS breast density using a large-scale FFDM dataset from one institution. Then, the model is evaluated on a test set of FFDM exams as well as synthetic 2D mammography (SM) images generated as part of DBT exams (C-View, Hologic, Inc., Marlborough, MA), acquired from the same institution and from a separate institution. Adaptation techniques, requiring few SM images, are explored to improve performance on the two SM datasets. Materials and Methods {#sec:materials_and_methods} ===================== This retrospective study was approved by an institutional review board for each of the two sites where data were collected (Site 1: internal institutional review board, Site 2: Western Institutional Review Board, Puyallup, WA). Informed consent was waived and all data were handled according to the Health Insurance Portability and Accountability Act. Datasets {#sec:datasets} -------- Mammography exams were collected from two sites: Site 1, an academic medical center located in the mid-western region of the United States, and Site 2, an out-patient radiology clinic located in northern California (Site 1 FFDM: 187627, acquired from 2008 to 2017; Site 1 SM: 3866, 2016 to 2017; Site 2 SM: 16283, 2015 to 2019). The exams were interpreted by one of 11 radiologists with breast imaging experience ranging from 2 to 30 years for Site 1 and by one of 9 radiologists with experience ranging from 10 to 41 years for Site 2. The BI-RADS breast density assessments of the radiologists were obtained from each site’s mammography reporting software (Site 1: Magview 7.1, Magview, Burtonsville, Maryland; Site 2: MRS 7.2.0; MRS Systems Inc. Seattle, Washington). Patients were randomly selected for training (FFDM: 50700, 88%; Site 1 SM: 3169, 82%; Site 2 SM: 6056, 80%), validation (FFDM: 1832, 3%; Site 1 SM: 403, 10%; Site 2 SM: 757, 10%), or testing (FFDM: 4960, 9%; Site 1 SM: 270, 7%; Site 2 SM: 744, 10%). Since the split was performed at the patient-level, the images for a given patient appear in only one of these sets. All exams with a BI-RADS breast density assessment were included. For the test sets, exams were required to have all four standard screening mammography images (the mediolateral oblique and craniocaudal views of both breasts). The distribution of the BI-RADS breast density assessments for each set are presented in Table \[tab:wustl\_datasets\] (Site 1) and Table \[tab:pdi\_datasets\] (Site 2). The two sites serve different patient populations. The patient cohort from Site 1 is 59% Caucasian (34192/58397), 23% African American (13201/58397), 3% Asian (1630/58397), and 1% Hispanic (757/58397) while Site 2 is 58% Caucasian (4350/7557), 1% African American (110/7557), 21% Asian (1594/7557), and 7% Hispanic (522/7557). The distribution of ages is similar for the two sites (Site 1: 54.8 $\pm$ 15.7 yr, Site 2: 55.7 $\pm$ 11.2 yr). [lllllll]{}\   & FFDM Train & FFDM Val & FFDM Test & SM Train & SM Val & SM Test\ Patients & 50700 & 1832 & 4960 & 3169 & 403 & 270\ Exams & 168208 & 6157 & 13262 & 3189 & 407 & 270\ Images & 672704 & 25000 & 53048 & 11873 & 1519 & 1080\ BI-RADS A & 80459  (12.0%) & 3465 (13.9%) & 4948 (9.3%) & 1160 (9.8%) & 154 (10.1%) & 96 (8.9%)\ BI-RADS B & 348878 (51.9%) & 12925 (51.7%) & 27608 (52.0%) & 6121 (51.6%) & 771 (50.8%) & 536 (49.6%)\ BI-RADS C & 214465 (31.9%) & 7587 (30.3%) & 18360 (34.6%) & 3901 (32.9%) & 510 (33.6%) & 388 (35.9%)\ BI-RADS D & 28902  (4.3%) & 1023 (4.1%) & 2132 (4.0%) & 691 (5.8%) & 84 (5.5%) & 60 (5.6%)\ \[tab:wustl\_datasets\] [llll]{}\   & Train & Val & Test\ Patients & 6056 & 757 & 744\ Exams & 13061 & 1674 & 1548\ Images & 51241 & 6540 & 6192\ BI-RADS A & 7866 (15.4%) & 865 (13.2%) & 948 (15.3%)\ BI-RADS B & 20731 (40.5%) & 2719 (41.6%) & 2612 (42.2%)\ BI-RADS C & 15706 (30.7%) & 2139 (32.7%) & 1868 (30.2%)\ BI-RADS D & 6938 (13.5%) & 817 (12.5%)& 764 (12.3%)\ \[tab:pdi\_datasets\] Deep Learning Model {#sec:deep_learning_model} ------------------- The DL model and training procedure were implemented using the `pytorch` DL framework (pytorch.org, version 1.0). The base model architecture is a pre-activation Resnet-34 [@he_identity_2016; @he_deep_2016; @wu_group_2018], which accepts as input a single image corresponding to one of the views from a mammography exam, and produces estimated probabilities that the image belongs to each of the BI-RADS breast density categories. The model was trained using the FFDM dataset following the procedure described in Appendix \[sec:training\_procedure\]. Domain Adaptation Methods {#sec:domain_adaptation} ------------------------- The goal of domain adaptation is to take a model trained on a dataset from one domain (source domain) and transfer its knowledge to a dataset in another domain (target domain), which is typically much smaller in size. Features learned by DL models in the early layers can be *general*, i.e. domain and task agnostic [@yosinski_how_2014]. Depending on the similarity of domains and tasks, even deeper features learned from one domain can be reused for another domain or task. In this work, we explore approaches for adapting the DL model trained on FFDM images (source domain) to SM images (target domain) that reuse all the features learned from the FFDM domain. First, inspired by the work of Guo et al. [@guo_calibration_2017], we consider the addition of a small linear layer following the final fully-connected layer where either the 44 matrix is diagonal (vector calibration) or the 44 matrix is allowed to freely vary (matrix calibration). Second, we retrain the final fully-connected layer of the Resnet-34 model on samples from the target domain (fine-tuning). To investigate the impact of the target domain dataset size, the adaptation techniques were repeated for different SM training sets across a range of sizes. The adaptation process was repeated 10 times for each dataset size with different training data in order to investigate the uncertainty arising from the selection of the training data. For each realization, the training images were randomly selected, without replacement, from the full training set. As a reference, a Resnet-34 model was trained from scratch, i.e. random initialization, for the largest number of training samples for each SM dataset. Further details on these methods are provided in Appendix \[sec:training\_procedure\]. Statistical Analysis {#sec:evaluation_methods} -------------------- To obtain an exam-level assessment, each image within an exam was processed by the DL model and the resulting probabilities were averaged. Several metrics were computed from these average probabilities for the 4-class BI-RADS breast density task and the binary dense (BI-RADS C+D) vs. non-dense (BI-RADS A+B) task: (1) accuracy, estimated based on concordance with the original reporting radiologists, (2) the area under the receiver operating characteristic curve (AUC), and (3) Cohen’s kappa (https://scikit-learn.org, version 0.20.0). Confidence intervals (CI) were computed by use of non-Studentized pivotal bootstrapping of the test sets for 8000 random samples [@carpenter_bootstrap_2000]. For the 4-class problem, macroAUC (the average of the four AUC values from the one vs. others tasks) and Cohen’s kappa with linear weighting are reported. For the binary density tasks, the predicted dense and non-dense probabilities were computed by summing the probabilities for the corresponding BI-RADS density categories. Results ======= Performance on FFDM Exams {#sec:performance_source_domain} ------------------------- The trained model was first evaluated on a large held-out test set of FFDM exams from Site 1 (4960 patients, 13262 exams, 53048 images, mean age: 56.9, age range: 23-97). In this case, the images were from the same institution and of the same image type (FFDM) as employed to train the model. The BI-RADS breast density distribution predicted by the DL model (A: 8.5%, B: 52.2%, C: 36.1%, D: 3.2%) was similar to that of the original reporting radiologists (A: 9.3%, B: 52.0%, C: 34.6%, D: 4.0%). The DL model exhibited close agreement with the radiologists for the BI-RADS breast density task across a variety of performance measures (see Table \[tab:wustl\_ffdm\_performance\]), including accuracy (82.2%, 95% CI: \[81.6%, 82.9%\]) and linearly-weighted Cohen’s kappa ($\kappa_w$ = 0.75, CI: \[0.74, 0.76\]). A high-level of agreement was also observed for the binary breast density task (accuracy = 91.1%, CI: \[90.6%, 91.6%\], AUC = 0.971, CI: \[0.968, 0.973\], $\kappa$ = 0.81, CI: \[0.80, 0.82\]). As demonstrated by the confusion matrices shown in Figure \[fig:confusion\_matrics\_wustl\_ffdm\], the DL model is rarely off by more than one breast density category (e.g. calls an extremely dense breast scattered; 0.03%, 4/13262). To place the results in the context of prior work, the performance on the FFDM test set is compared with results evaluated on other large FFDM datasets acquired from academic centers [@lehman_mammographic_2019; @wu_breast_2017] and with commercial breast density software [@brandt_comparison_2016] (see Table \[tab:wustl\_ffdm\_performance\]). While there are limitations to comparing results evaluated on different test sets (see Section \[sec:discussion\]), our FFDM DL model appears to offer competitive performance. [lcccccc]{}\   & & & & & &\ Ours & & & & & &\ Lehman et al. [@lehman_mammographic_2019] & & & &\ Wu et al. [@wu_breast_2017] & 76.7 & 0.916 & & 86.5 & & 0.65\ Volpara v1.5.0 [@brandt_comparison_2016] & 57 & & & 78 & &\ Quantra v2.0 [@brandt_comparison_2016] & 56 & & & 83 & &\ \[tab:wustl\_ffdm\_performance\] [0.5]{} ![Confusion matrices for the (a) Breast Imaging Reporting and Data System (BI-RADS) breast density task and the (b) binary density task (dense, BI-RADS C+D vs. non-dense, BI-RADS A+B) evaluated on the full-field digital mammography (FFDM) test set. The number of test samples (exams) within each bin is shown in parentheses.[]{data-label="fig:confusion_matrics_wustl_ffdm"}](confusion_matrix_4class_wustl_ffdm_bw "fig:"){width="\textwidth"} [0.5]{} ![Confusion matrices for the (a) Breast Imaging Reporting and Data System (BI-RADS) breast density task and the (b) binary density task (dense, BI-RADS C+D vs. non-dense, BI-RADS A+B) evaluated on the full-field digital mammography (FFDM) test set. The number of test samples (exams) within each bin is shown in parentheses.[]{data-label="fig:confusion_matrics_wustl_ffdm"}](confusion_matrix_2class_wustl_ffdm_bw "fig:"){width="\textwidth"} Performance on DBT Exams ------------------------ ### Site 1 Results Results are first reported for the Site 1 SM test set (270 patients, 270 exams, 1080 images, mean age: 54.6, age range: 28-72) as this avoids any differences that may occur between the two sites. Without adaptation, the model still demonstrates close agreement with the original reporting radiologists for the BI-RADS breast density task (accuracy = 79%, CI: \[74%, 84%\]; $\kappa_w$ = 0.71, CI: \[0.64, 0.78\]; see Table \[tab:adaptation\_performance\]). The DL model slightly underestimates breast density for SM images (see Figure \[fig:confusion\_matrices\_wustl\_cview\]), producing a BI-RADS breast density distribution (A: 10.4%, B: 57.8%, C: 28.9%, D: 3.0%) with more non-dense cases and fewer dense cases relative to the radiologists (A: 8.9%, B: 49.6%, C: 35.9%, D: 5.6%). Agreement for the binary density task is also quite high without adaptation (accuracy = 88%, CI: \[84%, 92%\]; $\kappa$ = 0.75, CI: \[0.67, 0.83\]; AUC = 0.97, CI: \[0.96, 0.99\]). [llcccccc]{}\ Datasets & Methods & & & & & &\ MM & & 82.2 & 0.952 & 0.75 & 91.1 & 0.971 & 0.81\ MM $\rightarrow$ S1 & None & & & & & &\ & Vector & & & & & &\ & Matrix & & & & & &\ & Fine-tune & & & & & &\ MM $\rightarrow$ S2 & None & & & & & &\ & Vector & & & & & &\ & Matrix & & & & & &\ & Fine-tune & & & & & &\ \[tab:adaptation\_performance\] After adaptation by matrix calibration with 500 Site 1 SM images, the density distribution is more similar to that of the radiologists (A: 5.9%, B: 53.7%, C: 35.9%, D: 4.4%), while overall agreement is about the same (accuracy = 80%, CI: \[76%, 85%\]; $\kappa_w$ = 0.72, CI: \[0.66, 0.79\]). Accuracy for the two dense classes is improved at the expense of the two non-dense classes (see Figure \[fig:confusion\_matrices\_wustl\_cview\]). A larger improvement is seen for the binary density task, where Cohen’s kappa rose from 0.75 \[0.67, 0.83\] to 0.82 \[0.76, 0.90\] (accuracy = 91%, CI: \[88%, 95%\]; AUC = 0.97, CI: \[0.96, 0.99\]). [0.5]{} ![Confusion matrices, evaluated on the Site 1 SM test set, for the (a) Breast Imaging Reporting and Data System (BI-RADS) breast density task and (b) the binary density task (dense, BI-RADS C+D vs. non-dense, BI-RADS A+B) without adaptation and for the (c) BI-RADS breast density task and (d) the binary density task (dense vs. non-dense) with adaptation by matrix calibration for 500 training samples. The number of test samples (exams) within each bin is shown in parentheses.[]{data-label="fig:confusion_matrices_wustl_cview"}](confusion_matrix_4class_wustl_cview_noadapt_bw "fig:"){width="\textwidth"} [0.5]{} ![Confusion matrices, evaluated on the Site 1 SM test set, for the (a) Breast Imaging Reporting and Data System (BI-RADS) breast density task and (b) the binary density task (dense, BI-RADS C+D vs. non-dense, BI-RADS A+B) without adaptation and for the (c) BI-RADS breast density task and (d) the binary density task (dense vs. non-dense) with adaptation by matrix calibration for 500 training samples. The number of test samples (exams) within each bin is shown in parentheses.[]{data-label="fig:confusion_matrices_wustl_cview"}](confusion_matrix_2class_wustl_cview_noadapt_bw "fig:"){width="\textwidth"} \ [0.5]{} ![Confusion matrices, evaluated on the Site 1 SM test set, for the (a) Breast Imaging Reporting and Data System (BI-RADS) breast density task and (b) the binary density task (dense, BI-RADS C+D vs. non-dense, BI-RADS A+B) without adaptation and for the (c) BI-RADS breast density task and (d) the binary density task (dense vs. non-dense) with adaptation by matrix calibration for 500 training samples. The number of test samples (exams) within each bin is shown in parentheses.[]{data-label="fig:confusion_matrices_wustl_cview"}](confusion_matrix_4class_wustl_cview_matcal500_bw "fig:"){width="\textwidth"} [0.5]{} ![Confusion matrices, evaluated on the Site 1 SM test set, for the (a) Breast Imaging Reporting and Data System (BI-RADS) breast density task and (b) the binary density task (dense, BI-RADS C+D vs. non-dense, BI-RADS A+B) without adaptation and for the (c) BI-RADS breast density task and (d) the binary density task (dense vs. non-dense) with adaptation by matrix calibration for 500 training samples. The number of test samples (exams) within each bin is shown in parentheses.[]{data-label="fig:confusion_matrices_wustl_cview"}](confusion_matrix_2class_wustl_cview_matcal500_bw "fig:"){width="\textwidth"} ### Site 2 Results Close agreement between the DL model and the original reporting radiologists was also observed for the Site 2 SM test set (744 patients, 1548 exams, 6192 images, mean age: 55.2, age range: 30-92) without adaptation (accuracy = 76%, CI: \[74%, 78%\]; $\kappa_w$ = 0.72 CI: \[0.70, 0.75\]; see Table \[tab:adaptation\_performance\]). The BI-RADS breast density distribution predicted by the DL model (A: 5.7%, B: 48.8%, C: 36.4%, D: 9.1%) was similar to the distribution found in the Site 1 datasets. The model could have learned a prior density distribution from the Site 1 FFDM dataset that may not be optimal for Site 2 where patient demographics are different. The predicted density distribution does not appear to be skewed towards low density estimates as seen for Site 1 (see Figure \[fig:confusion\_matrices\_pdi\_cview\]). Agreement for the binary density task was especially strong (accuracy = 92%, CI: \[91%, 93%\]; $\kappa$ = 0.84, CI: \[0.81, 0.87\]; AUC = 0.980, CI: \[0.976, 0.986\]). [0.5]{} ![Confusion matrices, evaluated on the Site 2 SM test set, for the (a) Breast Imaging Reporting and Data System (BI-RADS) breast density task and (b) the binary density task (dense, BI-RADS C+D vs. non-dense, BI-RADS A+B) without adaptation and for the (c) BI-RADS breast density task and (d) the binary density task (dense vs. non-dense) with adaptation by matrix calibration for 500 training samples. Number of test samples (exams) within each bin are shown in parentheses.[]{data-label="fig:confusion_matrices_pdi_cview"}](confusion_matrix_4class_pdi_cview_noadapt_bw "fig:"){width="\textwidth"} [0.5]{} ![Confusion matrices, evaluated on the Site 2 SM test set, for the (a) Breast Imaging Reporting and Data System (BI-RADS) breast density task and (b) the binary density task (dense, BI-RADS C+D vs. non-dense, BI-RADS A+B) without adaptation and for the (c) BI-RADS breast density task and (d) the binary density task (dense vs. non-dense) with adaptation by matrix calibration for 500 training samples. Number of test samples (exams) within each bin are shown in parentheses.[]{data-label="fig:confusion_matrices_pdi_cview"}](confusion_matrix_2class_pdi_cview_noadapt_bw "fig:"){width="\textwidth"} \ [0.5]{} ![Confusion matrices, evaluated on the Site 2 SM test set, for the (a) Breast Imaging Reporting and Data System (BI-RADS) breast density task and (b) the binary density task (dense, BI-RADS C+D vs. non-dense, BI-RADS A+B) without adaptation and for the (c) BI-RADS breast density task and (d) the binary density task (dense vs. non-dense) with adaptation by matrix calibration for 500 training samples. Number of test samples (exams) within each bin are shown in parentheses.[]{data-label="fig:confusion_matrices_pdi_cview"}](confusion_matrix_4class_pdi_cview_matcal500_bw "fig:"){width="\textwidth"} [0.5]{} ![Confusion matrices, evaluated on the Site 2 SM test set, for the (a) Breast Imaging Reporting and Data System (BI-RADS) breast density task and (b) the binary density task (dense, BI-RADS C+D vs. non-dense, BI-RADS A+B) without adaptation and for the (c) BI-RADS breast density task and (d) the binary density task (dense vs. non-dense) with adaptation by matrix calibration for 500 training samples. Number of test samples (exams) within each bin are shown in parentheses.[]{data-label="fig:confusion_matrices_pdi_cview"}](confusion_matrix_2class_pdi_cview_matcal500_bw "fig:"){width="\textwidth"} With adaptation by matrix calibration with 500 Site 2 training samples, performance for the BI-RADS breast density task on the Site 2 SM dataset substantially improved (accuracy = 80%, CI: \[78%, 82%\]; $\kappa_w$ = 0.79, CI: \[0.76, 0.81\]). After adaptation, the predicted BI-RADS breast density distribution (A: 16.9%, B: 43.3%, C: 29.4%, D: 10.4%) was more similar to that of the radiologists (A: 15.3%, B: 42.2%, C: 30.2%, D: 12.3%). Less improvement was seen for the binary breast density task (accuracy = 92%, CI: \[91%, 94%\]; $\kappa$ = 0.84, CI: \[0.82, 0.87\]; AUC = 0.983, CI: \[0.978, 0.988\]). ### Impact of Dataset Size on Adaptation The preferred adaptation method will depend on the number of training samples available for the adaptation, with more training samples benefiting methods with more parameters. Figure \[fig:adaptation\_data\_size\] shows the impact of the amount of training data on the performance of the adaptation methods, as measured by linearly weighted Cohen’s kappa and macroAUC, for both the Site 1 and Site 2 SM datasets. Each adaptation method has a range of number of samples where it offers the best performance, with the regions ordered by the corresponding number of parameters for the adaptation methods (vector calibration: 8 parameters, matrix calibration: 20, fine-tuning: 2052). This demonstrates the trade-off between the performance of the adaptation method and the amount of new training data that must be acquired. When the number of training samples is very small (e.g. $<$ 100 images), some adaptation methods negatively impact performance. Even at the largest dataset sizes, the amount of training data was too limited for the Resnet-34 model trained from scratch on SM images to exceed the performance of the models adapted from FFDM. [0.5]{} ![Impact of the number of training samples in the target domain on the performance of the adapted model for the Site 1 synthetic 2D mammography (SM) test set, as measured by (a) macroAUC and (b) linearly weighted Cohen’s kappa, and for the Site 2 SM test set, as measured by (c) macroAUC and (d) linearly weighted Cohen’s kappa. Results are shown for vector and matrix calibration, and retraining the last fully-connected layer (fine-tuning). Error bars indicate the standard error of the mean computed over 10 random realizations of the training data. Performance prior to adaptation (none) and training from scratch are shown as references. For the Site 1 SM studies, the full-field digital mammography (FFDM) performance serves as an additional reference. Note that each graph is shown with its own full dynamic range in order to facilitate comparison of the different adaptation methods for a given metric and dataset.[]{data-label="fig:adaptation_data_size"}](wustl_cview_adaptation_num_samples_macroAUC "fig:"){width="\textwidth"} [0.5]{} ![Impact of the number of training samples in the target domain on the performance of the adapted model for the Site 1 synthetic 2D mammography (SM) test set, as measured by (a) macroAUC and (b) linearly weighted Cohen’s kappa, and for the Site 2 SM test set, as measured by (c) macroAUC and (d) linearly weighted Cohen’s kappa. Results are shown for vector and matrix calibration, and retraining the last fully-connected layer (fine-tuning). Error bars indicate the standard error of the mean computed over 10 random realizations of the training data. Performance prior to adaptation (none) and training from scratch are shown as references. For the Site 1 SM studies, the full-field digital mammography (FFDM) performance serves as an additional reference. Note that each graph is shown with its own full dynamic range in order to facilitate comparison of the different adaptation methods for a given metric and dataset.[]{data-label="fig:adaptation_data_size"}](wustl_cview_adaptation_num_samples_linear_kappa "fig:"){width="\textwidth"} \ [0.5]{} ![Impact of the number of training samples in the target domain on the performance of the adapted model for the Site 1 synthetic 2D mammography (SM) test set, as measured by (a) macroAUC and (b) linearly weighted Cohen’s kappa, and for the Site 2 SM test set, as measured by (c) macroAUC and (d) linearly weighted Cohen’s kappa. Results are shown for vector and matrix calibration, and retraining the last fully-connected layer (fine-tuning). Error bars indicate the standard error of the mean computed over 10 random realizations of the training data. Performance prior to adaptation (none) and training from scratch are shown as references. For the Site 1 SM studies, the full-field digital mammography (FFDM) performance serves as an additional reference. Note that each graph is shown with its own full dynamic range in order to facilitate comparison of the different adaptation methods for a given metric and dataset.[]{data-label="fig:adaptation_data_size"}](pdi_cview_adaptation_num_samples_macroAUC "fig:"){width="\textwidth"} [0.5]{} ![Impact of the number of training samples in the target domain on the performance of the adapted model for the Site 1 synthetic 2D mammography (SM) test set, as measured by (a) macroAUC and (b) linearly weighted Cohen’s kappa, and for the Site 2 SM test set, as measured by (c) macroAUC and (d) linearly weighted Cohen’s kappa. Results are shown for vector and matrix calibration, and retraining the last fully-connected layer (fine-tuning). Error bars indicate the standard error of the mean computed over 10 random realizations of the training data. Performance prior to adaptation (none) and training from scratch are shown as references. For the Site 1 SM studies, the full-field digital mammography (FFDM) performance serves as an additional reference. Note that each graph is shown with its own full dynamic range in order to facilitate comparison of the different adaptation methods for a given metric and dataset.[]{data-label="fig:adaptation_data_size"}](pdi_cview_adaptation_num_samples_linear_kappa "fig:"){width="\textwidth"} Discussion {#sec:discussion} ========== Breast Imaging Reporting and Data System (BI-RADS) breast density can be an important indicator of breast cancer risk and radiologist sensitivity, but intra- and inter-reader variability may limit the effectiveness of this measure. Deep learning (DL) models for estimating breast density can reduce this variability while still providing accurate assessments. However, in order to serve as a useful clinical tool, DL models need to demonstrate that they can be applied to digital breast tomosynthesis (DBT) exams and generalize across institutions. To overcome the limited training data for DBT exams, a DL model was trained on a large set of full-field digital mammography (FFDM) images. The model showed close agreement with the radiologists reported BI-RADS breast density for a test set of FFDM images (Site 1: $\kappa_w$ = 0.75, 95% confidence interval (CI): \[0.74, 0.76\]) and for two datasets of synthetic 2D mammography (SM) images, which are generated as part of DBT exams (Site 1:  = 0.71, CI: \[0.64, 0.78\]; Site 2: = 0.72, CI: \[0.70, 0.75\]). The strong performance on the SM datasets from different institutions suggests that the DL model may generalize to DBT exams and multiple sites. Further adaptation of the model for the SM datasets led to a limited improvement for Site 1 ($\kappa_w$ = 0.72, CI: \[0.66, 0.79\]) and a more substantial improvement for Site 2 ($\kappa_w$ = 0.79, CI: \[0.76, 0.81\]). The investigation of the impact of dataset size suggests that these adaptation methods could serve as practical approaches for adapting deep learning models if a model must be updated to account for site-specific differences. When radiologists’ assessments are accepted as the ground truth, inter-reader variability may limit the performance that can be achieved for a given dataset. For example, the performance obtained on the Site 2 SM dataset following adaptation was higher than that obtained on the FFDM dataset used to train the model. This is likely a result of limited inter-reader variability for the Site 2 SM dataset due to over 80% of the exams having been read by only two readers. Unlike previous works, our BI-RADS breast density DL model was evaluated on SM images from DBT exams and on data from multiple institutions. Further, as shown in Section \[sec:performance\_source\_domain\], when evaluated on the FFDM images, the model appeared to offer competitive performance to previous DL models and commercial breast density software ($\kappa_w$ = 0.75, CI: \[0.74, 0.76\] vs. Lehman et al. 0.67, CI: \[0.66, 0.68\]; Volpara 0.57, CI: \[0.55, 0.59\], Quantra 0.46, CI: \[0.44, 0.47\]) [@lehman_mammographic_2019; @brandt_comparison_2016]. For each work, results are reported on their respective test sets, which may be more or less challenging due to varying levels of inter-reader variability or other factors. Other measures of breast density, such as volumetric breast density, have been previously estimated by automated software for DBT exams . Thresholds can be chosen to translate these measures to BI-RADS breast density, but this may result in lower levels of agreement than direct estimation of BI-RADS breast density (e.g. $\kappa_w~=~0.47$ ). Here, BI-RADS breast density is estimated from 2D SM images instead of the 3D tomosynthesis volumes as this simplifies transfer learning from the FFDM images and mirrors the manner in which breast radiologists assess density. This study has several limitations. First, the proposed domain adaptation approaches may be less effective when the differences between domains are larger. In this work, adaptation is from two types of mammography images produced by the same manufacturer. Second, the data from Site 1 was collected over a time period covering the transition from BI-RADS version 4 to BI-RADS version 5, during which the criteria for assessing BI-RADS breast density changed. Third, when a DL model is adapted to a new institution, adjustments may be made for differences in image content, patient demographics, or the interpreting radiologists. This last adjustment may result in a degree of inter-reader variability between the original and adapted DL models, though likely lower than the individual inter-reader variability if the model learns the consensus of each group of radiologists. As a result, the improved performance following adaptation for the Site 2 SM dataset could be due to differences in patient demographics or radiologist assessment practices compared with the FFDM dataset. The weaker improvement for the Site 1 SM dataset could be due to similarities in these same factors. Still, the broad use of Breast Imaging Reporting and Data System (BI-RADS) breast density deep learning (DL) models holds great promise for improving clinical care. The success of the DL model without adaptation suggests that the features learned by the model are largely applicable to both full-field digital mammography (FFDM) images and synthetic 2D mammography (SM) images from digital breast tomosynthesis (DBT) exams as well as to different readers and institutions. A BI-RADS breast density DL model that can generalize across sites and image types could lead to fast, low-cost, and more consistent estimates of breast density for women. Training Procedure {#sec:training_procedure} ================== The deep learning (DL) model, described in Section \[sec:deep\_learning\_model\], was a pre-activation Resnet-34 network, where the batch normalization layers were replaced with group normalization layers . It was trained using the full-field digital mammography (FFDM) dataset (see Table \[tab:wustl\_datasets\]) by use of the Adam optimizer [@kingma_adam:_2015] with a learning rate of $10^{-4}$ and a weight decay of $10^{-3}$. Weight decay not was applied to the parameters belonging to the normalization layers. The input was resized to 416$\times$320 pixels and the pixel intensity values were normalized so that the grayscale window denoted in the Digital Imaging and Communications in Medicine (DICOM) header ranged from 0.0 to 1.0. Training was performed using mixed precision [@micikevicius_mixed_2017] and gradient checkpointing [@chen_training_2016] with batch sizes of 256 distributed across two NVIDIA GTX 1080 Ti graphics processing units (Santa Clara, CA). Each batch was sampled such that the probability of selecting a BI-RADS B or BI-RADS C sample was four times that of selecting a BI-RADS A or BI-RADS D sample, which roughly corresponds to the distribution of densities observed nationally in the United States [@lehman_national_2017]. Horizontal and vertical flipping were employed for data augmentation. In order to obtain more frequent information on the training progress, epochs were capped at 100k samples compared with a total training set size of over 672k samples. The model was trained for 100 such epochs. Results are reported for the epoch that had the lowest cross entropy loss on the validation set, which occurred after 93 epochs. The parameters for the vector and matrix calibration methods were chosen by minimizing a cross-entropy loss function by use of the BFGS optimization method (https://scipy.org, version 1.1.0). The parameters were initialized such that the linear layer corresponded to the identity transformation. Training was stopped when the $\ell_2$ norm of the gradient was less than $10^{-6}$ or when the number of iterations exceeded 500. Retraining the last fully-connected layer for the fine-tuning method was performed by use of the Adam optimizer with a learning rate of $10^{-4}$ and weight decay of $10^{-5}$. The batch size was set to 64. The fully-connected layer was trained from random initialization for 100 epochs and results were reported for the epoch with the lowest validation cross entropy loss. Training from scratch on the synthetic 2D mammography (SM) datasets was performed following the same procedure as for the base model. For fine-tuning and training from scratch, the size of an epoch was set to the number of training samples. Acknowledgements {#acknowledgements .unnumbered} ================ This work was supported in part by funding from Whiterabbit AI, Inc. WU has equity interests in Whiterabbit AI, Inc. and may receive royalty income and milestone payments from a “Collaboration and License Agreement” with Whiterabbit AI, Inc. to develop a technology evaluated in this research. The following authors are employed by and/or have equity interests in Whiterabbit AI, Inc.: T.P.M., S.S., B.M., J.S., M.P.S., S.P., A.L., R.M.H., N.G., D.S., and S.C.M. The authors would like to thank Drs. Mark A. 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--- author: - 'A. Cappi' - 'F. Marulli' - 'J. Bel' - 'O. Cucciati' - 'E. Branchini' - 'S. de la Torre' - 'L. Moscardini' - 'M. Bolzonella' - 'L. Guzzo' - 'U. Abbas' - 'C. Adami' - 'S. Arnouts' - 'D. Bottini' - 'J. Coupon' - 'I. Davidzon' - 'G. De Lucia' - 'A. Fritz' - 'P. Franzetti' - 'M. Fumana' - 'B. Garilli' - 'B. R. Granett' - 'O. Ilbert' - 'A. Iovino' - 'J. Krywult' - 'V. Le Brun' - 'O. Le Fèvre' - 'D. Maccagni' - 'K. Ma[ł]{}ek' - 'H. J. McCracken' - 'L. Paioro' - 'M. Polletta' - 'A. Pollo' - 'M. Scodeggio' - 'L. A. .M. Tasca' - 'R. Tojeiro' - 'D. Vergani' - 'A. Zanichelli' - 'A. Burden' - 'C. Di Porto' - 'A. Marchetti' - 'C. Marinoni' - 'Y. Mellier' - 'R. C. Nichol' - 'J. A. Peacock' - 'W. J. Percival' - 'S. Phleps' - 'C. Schimd' - 'H. Schlagenhaufer' - 'M. Wolk' - 'G. Zamorani' bibliography: - 'References.bib' date: 'Received ..., 2015; accepted ..., 2015' subtitle: 'Hierarchical scaling and biasing [^1] ' title: 'The VIMOS Public Extragalactic Redshift Survey (VIPERS)' --- [Building on the two-point correlation function analyses of the VIMOS Public Extragalactic Redshift Survey (VIPERS), we investigate the higher-order correlation properties of the same galaxy samples to test the hierarchical scaling hypothesis at $z \sim 1$ and the dependence on galaxy luminosity, stellar mass, and redshift. With this work we also aim to assess possible deviations from the linearity of galaxy bias independently from a previously performed analysis of our survey. ]{} [We have measured the count probability distribution function in spherical cells of varying radii ($3 \le R \le 10 h^{-1}$ Mpc), deriving $\sigma_{8g}$ (the galaxy rms at 8 $h^{-1}$ Mpc), the volume–averaged two-, three-, and four–point correlation functions and the normalized skewness $S_{3g}$ and kurtosis $S_{4g}$ for different volume–limited subsamples, covering the following ranges: $-19.5 \le M_B(z=1.1) - 5 \log(h) \le -21.0$ in absolute magnitude, $9.0 \le \log(M_*/M_\odot h^{-2}) \le 11.0$ in stellar mass, and $0.5 \le z < 1.1$ in redshift.]{} [We have performed the first measurement of high–order correlation functions at $z \sim 1$ in a spectroscopic redshift survey. Our main results are the following. 1) The hierarchical scaling between the volume–averaged two- and three-point and two- and four–point correlation functions holds throughout the whole range of scale and redshift we could test. 2) We do not find a significant dependence of $S_{3g}$ on luminosity (below $z=0.9$ the value of $S_{3g}$ decreases with luminosity, but only at $1 \sigma$–level). 3) We do not detect a significant dependence of $S_{3g}$ and $S_{4g}$ on scale, except beyond $z \sim 0.9$, where $S_{3g}$ and $S_{4g}$ have higher values on large scales ($R \ge 10 h^{-1}$ Mpc): this increase is mainly due to one of the two CFHTLS Wide Fields observed by VIPERS and can be explained as a consequence of sample variance, consistently with our analysis of mock catalogs. 4) We do not detect a significant evolution of $S_{3g}$ and $S_{4g}$ with redshift (apart from the increase of their values with scale in the last redshift bin). 5) $\sigma_{8g}$ increases with luminosity, but does not show significant evolution with redshift. As a consequence, the linear bias factor $b=\sigma_{8g}/\sigma_{8m}$, where $\sigma_{8m}$ is the rms of matter at a scale of 8 $h^{-1}$ Mpc, increases with redshift, in agreement with the independent analysis of VIPERS and of other surveys such as the VIMOS–VLT Deep Survey (VVDS). We measure the lowest bias $b=1.47 \pm 0.18$ for galaxies with $M_B(z=1.1)-5\log(h) \le -19.5$ in the first redshift bin ($0.5 \le z < 0.7$) and the highest bias $b=2.12 \pm 0.28$ for galaxies with $M_B(z=1.1)-5\log(h) \le -21.0$ in the last redshift bin ($0.9 \le z < 1.1$). 6) We quantify deviations from the linear bias by means of the Taylor expansion parameter $b_2$. We obtain $b_2 = -0.20 \pm 0.49$ for $0.5 \le z < 0.7$ and $b_2 = -0.24 \pm 0.35$ for $0.7 \le z < 0.9$, while for the redshift range $0.9 \le z < 1.1$ we find $b_2 = +0.78 \pm 0.82$. These results are compatible with a null non-linear bias term, but taking into account another analysis for VIPERS and the analysis of other surveys, we argue that there is evidence for a small but non-zero non-linear bias term. ]{} Introduction ============ In the standard model of structure formation, the growth of density fluctuations from a primordial Gaussian density field is driven by gravity; it is possible to follow the evolution of these fluctuations through analytical and numerical approaches and predict the statistical properties for the dark matter field and dark matter haloes. Galaxies form in a complex process following the baryonic infall into dark matter halos: this means that the comparison between theory and observations is not straightforward, but it also implies that the spatial distribution of galaxies contains a wealth of information relevant for both cosmology and the physics of galaxy formation. Extracting and exploiting this information from the data requires a number of different and complementary statistical approaches. For example, while the two–point correlation function $\xi_2({\bf r})$ is the simplest and most widely used statistical indicator of galaxy clustering, a complete description of a distribution is only given by the full $J$–point correlation functions $\xi_J$, or equivalently, by the volume–averaged correlation functions $\overline{\xi}_J$, which are related to the $J$–order moments of the count probability distribution function (PDF)[^2]. The count PDF gives the probability of counting $N$ objects as a function of volume $V$. High–order correlations are particularly interesting because perturbation theory and numerical simulations can describe their behaviour for the gravitational evolution of matter density fluctuations. The first estimates of the two– and three–point galaxy correlations functions on angular catalogues of galaxies were made by [@1977ApJ...217..385G], who found that these estimates were well described by the hierarchical relation $\xi_3(r_{12},r_{13},r_{23}) = Q [\xi_2(r_{12}) \xi_2(r_{13}) + \xi_2(r_{13}) \xi_2(r_{23}) + \xi_2(r_{12}) \xi_2(r_{23})]$. The three–point correlation function has subsequently become a standard statistical tool for the analysis of clustering and has been applied to simulations and recent surveys of galaxies (see e.g. [@2008ApJ...672..849M], [@2014MNRAS.443.2874M]), while its Fourier transform, the bispectrum, has also been applied to the analysis of the Ly$_\alpha$ forest ([@2003MNRAS.344..776M], [@2004MNRAS.347L..26V]) and of the cosmic microwave background (CMB) ([@2013arXiv1303.5084P]). The scaling relation between the two– and three–point correlation functions was soon generalized to higher orders ([@1978ApJ...221...19F] up to $J=4$, up to $J=5$) and was mathematically described by the so–called hierarchical models, where the $J$–point correlation functions are expressed as a function of products of the two–point correlation function. Different versions of these models were suggested, but showed that all of them belong to the general class of scale–invariant models, which are defined by the scaling property: $$\xi_J(\lambda r_1, ..., \lambda r_J) = \lambda ^{-(J-1)\gamma} \xi_J(r_1, ..., r_J).$$ From a physical point of view, the hierarchical scaling of the correlation functions is expected in the highly non-linear regime (the BBGKY hierarchy, see [@1977ApJS...34..425D], [@1984ApJ...277L...5F], [@1988ApJ...332...67H]) and in the quasi–linear regime (from perturbation theory, see [@1980lssu.book.....P], [@1984ApJ...279..499F], [@1992ApJ...392....1B], [@2002PhR...367....1B] and references therein). Another prediction of the hierarchical models is that the normalized high–order reduced moments $S_J \equiv \overline{\xi}_J / \overline{\xi}_2 ^{J-1}$ should be constant. In the present paper we focus on the normalized skewness $S_3$ and kurtosis $S_4$. [@1980lssu.book.....P] showed that in second–order perturbation theory, assuming Gaussian primordial density fluctuations and an Einstein-de Sitter model, $S_{3m}$, the normalized skewness of matter fluctuations assumes the value $34/7$. Subsequent works have shown that the smoothed $S_{3m}$ depends on the slope of the power spectrum and has a very weak dependence on the cosmological model (see [@2002PhR...367....1B]). While in standard models with Gaussian primordial fluctuations the skewness and higher–order moments assume non–zero values as a consequence of gravitational clustering, scenarios with non–Gaussian primordial perturbations also predict a primordial non–zero skewness, particularly at large scales ($\ge 10 h^{-1}$ Mpc) ([@1993ApJ...408...33L], [@1994ApJ...429...36F], [@1996ApJ...462L...1G], [@1998MNRAS.301..524G], [@2000PhRvD..62b1301D]); therefore these scenarios can in principle be constrained by measuring the high–order moments ([@2014MNRAS.443.1402M]). Moreover, it has been shown that the hierarchy of the $J$–point functions and the measurement of $S_3$ and $S_4$ can be used as a cosmological test to distinguish between the standard $\Lambda CDM$ and models including long-range scalar interaction between dark matter particles (“fifth force” DM models), as shown by [@2010PhRvD..82j3536H], who found the largest deviations in the redshift range $0.5 < z < 2$. However, the comparison between the theoretical predictions for the matter distribution and the observed galaxy distribution is not trivial, as a consequence of bias. One of the first results derived from the analysis of the first redshift surveys was that the amplitude of the two–point correlation function depends on galaxy luminosity and galaxy colour (see and references therein); therefore, the galaxy distribution must generally differ from the underlying matter distribution. A common assumption is that the galaxy and matter density fields are related by a linear relation, $\delta_g = b \delta_m$, where $\delta_g \equiv \Delta\rho_g/\rho_g$ and $\delta_m \equiv \Delta\rho_m/\rho_m$ are the galaxy and matter density contrast, respectively. This relation is a consequence of the scenario of biased galaxy formation, where galaxies form above a given threshold of the linear density field, in the limit of high threshold and low variance. Of course, this relation cannot have general validity: when $b>1$ and $\delta_m < 0$, the linear relation gives an unphysical value $\delta_g < -1$. A simple prediction of linear biasing is that the two–point correlation function is amplified by a factor $b^2$, while $S_3$ is inversely proportional to $b$. The analysis of the first redshift surveys revealed instead that different classes of galaxies selected in the optical and infrared bands, while differing in the amplitude of the two–point correlation function, have similar values of $S_3$ ([@1992ApJ...398L..17G], [@1993ApJ...417...36B], [@1999ApJ...514..563B]); the same also holds for galaxy clusters ([@1995ApJ...438..507C]). In particular, [@1999ApJ...514..563B] analysed volume–limited samples of the Southern Sky Redshift Survey 2 (SSRS2, [@1994ApJ...424L...1D]) and found that, while the two–point correlation amplitude increases significantly with galaxy luminosity when $L>L_*$ ([@1996ApJ...472..452B]), the value of $S_3$ does not scale with the inverse of the bias parameter $b$ and is independent of luminosity and scale within the errors: this implies that the bias is non-linear. Similar results were obtained in the Durham/UKST and Stromlo-APM redshift surveys ([@2000MNRAS.317L..51H]) and in the larger and deeper 2dF Galaxy Redshift Survey (2dFGRS, [@2004MNRAS.351L..44B], [@2004MNRAS.352..828C]), which enabled a more detailed analysis: for example, [@2004MNRAS.352.1232C] found evidence for a weak dependence of $S_3$ on luminosity, while according to [@2007MNRAS.379.1562C] the $S_J$ of red galaxies depends on luminosity, while blue galaxies do not show any dependence. In an analysis of the Sloan Digital Sky Survey (SDSS) [@2006ApJ...649...48R] found that the values of $S_J$ are lower for late–type than for early–types galaxies. In more recent years, deeper surveys enabled exploring the effects of the evolution of gravitational clustering and bias, thus placing stronger constraints on models of galaxy formation and evolution. [@2013MNRAS.tmp.2056W] measured the hierarchical clustering of the CFHTLS–Wide from photometric redshifts. They found an indication that at small scales the hierarchical moments increase with redshift, while at large scales their results are still consistent with perturbation theory for $\Lambda CDM$ cosmology with a linear bias, but suggest the presence of a small non-linear term. &gt;From the analysis of the VIMOS–VLT Deep Survey, based on spectroscopic redshifts, (see also ) found that the value of $S_3$ for luminous ($M_B < -21$) galaxies is consistent with the local value at $z < 1$ while decreasing beyond $z\sim 1$, and that the bias is non-linear. In this paper we analyse the high–order correlations and moments of the first release of the VIMOS Public Extragalactic Redshift Survey (VIPERS[^3]) in the redshift range $0.5 < z \le 1.1$ as a function of luminosity and stellar mass. We also derive an estimate of the non-linear bias. Our analysis extends those presented in a number of recent works that have investigated various aspects of galaxy clustering in the VIPERS sample. Some works have focused on two–point statistics, like the standard galaxy-galaxy two–point correlation function to estimate redshift space distortions () and its evolution and dependence on galaxy properties (). A different type of two–point statistics, the clustering ratio, has been introduced by and applied to VIPERS galaxies () to estimate the mass density parameter $\Omega_M$. have searched the VIPERS survey for galaxy voids and characterized their properties by means of the galaxy-void cross-correlation. Bel et al. (2015, in preparation) have proposed a method to infer the one–point galaxy probability function from counts in cells that [@2014arXiv1406.6692D] have exploited to search for and detect deviations from linear bias; a result that we directly compare our results with. Finally, studied different methods for accounting for gaps in the VIPERS survey and assessing their impact on galaxy counts. As cosmological parameters we have adopted $H_0 = 70$ km/s/Mpc, $\Omega_M = 0.25$, $\Omega_\Lambda = 0.75$, but all cosmology–dependent quantities are given in $H_0=100$ km/s/Mpc units associated with the corresponding power of $h = H_0/100$. High–order statistics ===================== In this section we resume the formalism and define the statistical quantities measured in our work. The volume–averaged $J$–point correlation functions are given by $$\overline{\xi}_J(V) = \frac{1}{V^J} \int _V \xi_J dV_1 ... dV_J ,$$ where for spherical cells (used in this work) $\overline{\xi}_J$ is a function of the cell radius $R$ and $V = 4 \pi R^3 /3$. The volume–averaged two–point correlation function gives the variance of the density contrast: $$\sigma^2 (R) = \overline{\xi}_2(R). \label{eq:variance}$$ The volume–averaged $J$–point correlation functions can be easily derived from the moments of the count PDF $P(N,R)$, that is, the probability of counting $N$ objects in a randomly chosen spherical volume of radius $R$ (see [@1980lssu.book.....P]). For simplicity, in the following we omit the dependence on $R$. At a fixed scale $R$, the centred moments of order $J$ are $$\mu_J = \sum _N P(N) \left( \frac{N - \overline{N}}{\overline{N}} \right)^J ,$$ where $\overline{N} = nV = \sum N P(N)$ is the mean number of objects in a cell of radius $R$. The volume–averaged correlation functions correspond to the reduced moments and up to the fourth order are given by the following relations: $$\begin{aligned} \overline{\xi}_2 & = \mu _2 - \frac{1}{\overline{N}} \nonumber \\ \overline{\xi}_3 & = \mu _3 - 3 \frac{\mu _2}{\overline{N}} + \frac{2}{\overline{N}^2} \nonumber \\ \overline{\xi}_4 & = \mu _4 - 6 \frac{\mu _3}{\overline{N}} + 11 \frac{\mu_2}{\overline{N}^2} - 3 \mu_2 ^2 - \frac{6}{\overline{N}^3}.\end{aligned}$$ An alternative way to estimate the high–order correlations is through the factorial moments $m_k$: $$m_k = \sum _N P(N) N^{\underline{k}} ,$$ where $$N^{\underline{k}} \equiv N(N-1)...(N-k+1)$$ is the falling factorial power of order $k$ (see e.g. [@1994cm.book.....G]). In fact, for a local Poisson process the moments about the origin of a stochastic field are given by the factorial moments of $N$; as our variable is the number density contrast $(N-\overline{N})/\overline{N}$, we have to convert the factorial moments $m_k$ into the moments about the mean (central moments) $\mu _k ^\prime$ through the standard relations $$\begin{aligned} \mu _2 ^\prime & = m_2 - m_1^2 \nonumber \\ \mu _3 ^\prime & = m_3 - 3 m_1 m_2 + 2 m_1^3 \nonumber \\ \mu _4 ^\prime & = m_4 - 4 m_1 m_3 + 6 m_1^2 m_2 - 3 m_1^4. \end{aligned}$$ We can finally derive the volume–averaged $J$–point correlation functions $$\overline{\xi_J} = \frac{\mu _J ^\prime}{\overline{N}^J}$$ and the normalized moments $S_J$ $$S_J = \frac{\overline{\xi}_J}{\overline{\xi}_2 ^{J-1}}. \label{eq:sjclassic}$$ The normalized moments can also be obtained through a recursive formula ([@1993ApJ...408...43S], [@2000MNRAS.313..711C]): $$S_J = \frac{\overline{\xi}_2 N^{\underline{J}}}{N^J_c} - \frac{1}{J} \sum _{k=1} ^{J-1} \frac{J!}{(J-k)! k!} \frac{(J-k) S_{J-k} m_k}{N ^k _c}, \label{eq:sjrecursive}$$ where $$N_c \equiv \overline{N} \overline{\xi}_2.$$ The values given in this paper were calculated using factorial moments. At a fixed scale $R$, the deterministic bias parameter $b$ can be directly measured through the square root of the ratio of the galaxy variance $\sigma_g ^2$ to the matter variance $\sigma_m ^2$: $$b(z) = \frac{\sigma _g (z)}{\sigma _m (z)}. \label{eq:bias}$$ In the case of linear biasing, the galaxy density contrast $\delta_g$ is proportional to the matter density contrast $\delta_m$ by a constant factor $b$, $\delta_g = b \delta_m$: there is no dependence on scale, and $b$ is the only parameter that completely defines the relation between the galaxy and matter distribution. As we have noted in the introduction, the linear biasing cannot have a general validity. It is more general and realistic to assume a local, deterministic non–linear bias $b(z,\delta_m,R)$, which can be written as a Taylor expansion ([@1993ApJ...413..447F]): $$\delta_g = \sum _{k=0} \frac{b_k}{k!} \delta ^k ,$$ where $b_1 \equiv b$. [@1993ApJ...413..447F] have shown that such a local bias transformation preserves the hierarchical properties of the underlying matter distribution in the limit of small fluctuations (large scales). In the case of linear bias, $b_k = 0$ for $k>1$, and the galaxy and matter normalized moments are then related by the following equation: $$S_{Jg} = \frac{S_{Jm}}{b^{J-1}}. \label{eq:slinear}$$ In general, the deviation from linear biasing is measured by taking the second order of the expansion. In this case, the galaxy normalized skewness is given by the following relation: $$S_{3g} = \frac{1}{b} \left (S_{3m} + 3 \frac{b_2}{b} \right). \label{eq:secondorderbias}$$ VIPERS survey {#data-section} ============= ------------------- ---------------------------- ------------- ------------------ ------------------ ------------------- Redshift range Limiting magnitude $N_g$ $\sigma_{8g}$ $S_{3g}$ $S_{4g}$ $M_B (z=1.1)- 5 \log(h)$ W1 + W4 $R=8 h^{-1}$ Mpc $R=8 h^{-1}$ Mpc $0.5 \le z < 0.7$ &lt;-19.5 8670 + 6863 $0.95 \pm 0.06 $ $1.81 \pm 0.20$ $8.13 \pm 2.03 $ $0.5 \le z < 0.7$ &lt;-20.0 6101 + 4963 $1.00 \pm 0.06 $ $1.82 \pm 0.22$ $8.12 \pm 2.05 $ $0.5 \le z < 0.7$ &lt;-20.5 3671 + 3025 $1.07 \pm 0.07 $ $1.83 \pm 0.27$ $7.93 \pm 2.54 $ $0.5 \le z < 0.7$ &lt;-21.0 1787 + 1478 $1.16 \pm 0.15 $ $1.78 \pm 0.30$ $6.29 \pm 2.96 $ $0.7 \le z < 0.9$ &lt;-20.0 7455 + 5384 $1.01 \pm 0.05 $ $1.74 \pm 0.14$ $ 7.19 \pm 1.34 $ $0.7 \le z < 0.9$ &lt;-20.5 4979 + 3475 $1.05 \pm 0.04 $ $1.66 \pm 0.16$ $ 6.02 \pm 1.38 $ $0.7 \le z < 0.9$ &lt;-21.0 2457 + 1664 $1.10 \pm 0.06 $ $1.59 \pm 0.22$ $ 5.50 \pm 1.76 $ $0.9 \le z < 1.1$ &lt;-20.5 2751 + 1805 $1.12 \pm 0.07 $ $2.50 \pm 0.28$ $14.11 \pm 3.10 $ $0.9 \le z < 1.1$ &lt;-21.0 1752 + 1067 $1.16 \pm 0.08 $ $2.54 \pm 0.38$ $12.70 \pm 3.81 $ Redshift range Limiting stellar mass $N_g$ $\sigma_{8g}$ $S_{3g}$ $S_{4g}$ $\log(M/M_{\odot} h^{-2})$ W1 + W4 $R=8 h^{-1}$ Mpc $R=8 h^{-1}$ Mpc $0.5 \le z < 0.7$ &gt; 9.0 8745 + 6544 $0.97 \pm 0.10 $ $1.88 \pm 0.15$ $ 8.51 \pm 1.45$ $0.5 \le z < 0.7$ &gt; 9.5 6091 + 4318 $1.03 \pm 0.10 $ $1.94 \pm 0.15$ $ 8.66 \pm 1.43$ $0.5 \le z < 0.7$ &gt; 10.0 3654 + 2581 $1.16 \pm 0.11 $ $2.02 \pm 0.16$ $ 8.61 \pm 1.45$ $0.5 \le z < 0.7$ &gt; 10.5 1292 + 713 $1.34 \pm 0.11 $ $1.90 \pm 0.18$ $ 6.62 \pm 1.39$ $0.7 \le z < 0.9$ &gt; 9.5 6159 + 4009 $1.09 \pm 0.08 $ $1.88 \pm 0.14$ $ 7.59 \pm 1.37$ $0.7 \le z < 0.9$ &gt; 10.0 3746 + 2428 $1.18 \pm 0.08 $ $1.87 \pm 0.14$ $ 7.29 \pm 1.36$ $0.7 \le z < 0.9$ &gt; 10.5 1467 + 819 $1.41 \pm 0.09 $ $2.04 \pm 0.20$ $ 7.81 \pm 1.80$ $0.9 \le z < 1.1$ &gt; 10.0 1644 + 964 $1.23 \pm 0.08 $ $2.70 \pm 0.21$ $ 13.28 \pm 2.43$ $0.9 \le z < 1.1$ &gt; 10.5 738 + 456 $1.43 \pm 0.09 $ $3.19 \pm 0.29$ $ 16.18 \pm 3.88$ ------------------- ---------------------------- ------------- ------------------ ------------------ ------------------- The VIMOS Public Extragalactic Redshift Survey (VIPERS) is an ongoing ESO Large Programme aimed at determining redshifts for $\sim 10^5$ galaxies in the redshift range $0.5 < z < 1.2$, to accurately and robustly measure clustering, the growth of structure (through redshift-space distortions) and galaxy properties at an epoch when the Universe was about half its current age ([@2013Msngr.151...41G]; ). The survey is divided into two separate areas and will cover $\sim 24$ deg$^2$ when completed. The two areas are the so–called $W1$ and $W4$ fields of the Canada-France-Hawaii Telescope Legacy Survey Wide (CFHTLS-Wide); the CFHTLS optical photometric catalogues[^4] constitute the parent catalogue from which VIPERS spectroscopic targets were selected. The VIPERS survey strategy is optimized to achieve a good completeness in the largest possible area ([@2009Msngr.135...13S]). Galaxies are selected to a limit of $i_{AB}<22.5$, further applying a simple and robust $gri$ colour pre-selection to effectively remove galaxies at $z<0.5$. In this way, only one pass per field is required, allowing us to double the galaxy sampling rate in the redshift range of interest with respect to a pure magnitude-limited sample ($\sim 40\%$). The final volume of the survey will be $5 \times 10^{7} h^{-3}$ Mpc$^{3}$, comparable to that of the 2dFGRS at $z\sim0.1$. VIPERS spectra are obtained using the VLT Visible Multi–Object Spectrograph (VIMOS, [@2002Msngr.109...21L], [@2003SPIE.4841.1670L]) at moderate resolution ($R=210$), with the LR Red grism at $R=210$ and a wavelength coverage of 5500-9500$\rm{\AA}$. The typical radial velocity error is $140 (1+z)$ km sec$^{-1}$. A discussion of the survey data reduction and the first management infrastructure were presented in [@2012PASP..124.1232G] and the detailed description of the survey was given by . The data set used in this and the other published papers is the VIPERS Public Data Release 1 (PDR-1) catalogue, made available to the public in 2013 (). It includes about $47,000$ reliable spectroscopic redshifts of galaxies and active galactic nuclei (AGNs). We here only selected galaxies with reliable redshift, that is, with spectroscopic quality flags $2, 3, 4$, or $9$ (see for the definition). To avoid regions dominated by large gaps, we here selected a subset of the total area covered by VIPERS: our limits are $02^h 01^m 00^s \le RA \le 02^h 34^m 50^s$, $-5.08^o \le DEC \le -4.17^o$ (7.67 square degrees) in $W1$ and $22^h 01^m 12^s \le RA \le 22^h 18^m 00^s$, $0.865^o \le DEC \le 2.20^o$ (5.60 square degrees) in $W4$. We defined volume–limited subsamples with different absolute magnitude and stellar mass limits, following the same criteria as in . The choice of these particular samples is discussed in detail in that paper; here we recall their main properties. The rest–frame B–band absolute magnitude and the stellar mass were estimated through the HYPERZMASS program (, ), which applies a spectral energy distribution (SED) fitting technique. To take into account luminosity evolution, we fixed as a reference limit the luminosity at our maximum redshift ($z = 1.1$) and assumed an evolution $M(z) = M(0) - z$ (see and also , ). We did not correct the mass limit of the stellar-mass-limited subsamples; this limit was therefore kept fixed within each redshift bin because the evolution of $M_*$ is negligible in our redshift range (, , ) {width="17cm"} The respective numbers of galaxies for the different subsamples are given in Table \[table:1\]. We note that these numbers are slightly different from those in because we applied more stringent angular limits to avoid regions nearby prominent gaps that might affect the counts in spherical cells (while the direct estimate of the two–point correlation function through counts of galaxy pairs can be easily corrected for by using a random catalogue with the same survey geometry). Analysis of mock catalogues =========================== We used mock catalogues derived from cosmological simulations to estimate not only the statistical errors and the uncertainty related to cosmic variance, but also the systematic errors that are due to the inhomogeneous spectroscopic completeness and the specific geometry of the two fields. A detailed description of the way these mocks were built was given by . We analysed a set of 26 independent mock catalogues based on the dark matter halo catalogue of the MultiDark simulation ([@2012MNRAS.423.3018P]), which assumes a flat $\Lambda CDM$ cosmology with ($\Omega_M$, $\Omega_\Lambda$, $\Omega_b$, $h$, $n$, $\sigma_{8m}$) = (0.27, 0.73, 0.0469, 0.7, 0.95, 0.82). This catalogue was populated with galaxies using halo occupation distribution prescriptions, as described in . In particular, the original halo catalogue was repopulated with halos below the resolution limit with the new technique of [@2013MNRAS.435..743D], which enables reproducing the range in stellar mass and luminosity probed by VIPERS data. For luminosity–limited subsamples, galaxy luminosities were calibrated using VIPERS data, while for stellar mass–limited subsamples masses were assigned to galaxies using the stellar-to-halo mass relation (SHMR) of [@2013MNRAS.428.3121M]. From the parent mock catalogues, a set of spectroscopic catalogues was derived by applying the same angular, photometric, and spectroscopic selection functions as were applied to the real data. For a more detailed and complete description of the mock catalogues see . ![Fractional difference of the average $\overline{\xi_2}$, $\overline{\xi_3}$, $\overline{\xi_3}$, $S_3$ and $S_4$ (from top to bottom) for the same set of mock catalogues as defined in Fig. 1, i.e. with 100% sampling rate and without gaps, and with sampling rate and gaps as in VIPERS. The subsamples are limited at $M_B(z=1.1)-5\log(h) \le -20.5$. Red triangles: $0.5 \le z < 0.7$; blue squares: $0.7 \le z < 0.9$; green hexagons: $0.9 \le z < 1.1$.[]{data-label="fig:diffmocks"}](fig02.eps){width="9.5cm"} &gt;From the mock spectroscopic catalogues we derived volume–limited subsamples with cuts in blue absolute magnitude and stellar mass corresponding to the observed ones. First of all, these mocks were used to test the effect of the gaps in the survey. As VIMOS is made of four quadrants $7\times8$ separated by 2 arcmin, characteristic cross–shaped gaps are left in the survey; a further gap is present between the rows of pointings at different declination; finally, there are a few missing quadrants due to failed pointings. Cells whose projection on the sky includes a gap can potentially miss some galaxies, which affects final counts. These gaps might be avoided by conservatively only counting galaxies in the cells that are completely included in one single quadrant, but in this way, only small scales would be sampled (the exact value obviously depends on the cell distance but it is generally lower than $R \sim 5 h^{-1}$ Mpc). Alternatively, the counts in each cell might be associated with the effective volume of the cell, subtracting the volume falling into the gaps; but this less drastic choice, which would slightly alter the shape of the cells, would still limit the range of the sampled scales. Another option would be filling the gaps. applied two algorithms that use the photometric redshift information and assign redshifts to galaxies based upon the spectroscopic redshifts of the nearest neighbours. In this way, it is also possible to take into account the varying completeness from field to field. Tests on mocks have shown that these algorithms are successful in reconstructing the lowest and highest density environments at a scale of 5 $h^{-1}$ Mpc, but not in recovering the count PDF and its moments due to systematic biases. We therefore here adopted another solution. The tests on mocks have shown that when cells are not allowed to cross the gaps by more than 40% of their volume, the non–observed regions and the varying sampling rate can be approximated by a random Poisson sampling, and the original count PDF can be recovered with good precision (Bel et al. 2015, in preparation). This means that to obtain good estimates of the quantities we discuss here ($J$–point correlations and normalized moments), which depend on the density contrast $\Delta \rho / \rho$, it is sufficient to implement the restriction on the volume of the cells falling into the gaps. In our analysis, we conservatively only considered spherical cells for which no more than 30% of the volume falls in a gap. Moreover, to improve the statistics, we combined the counts of the $W1$ and $W4$ fields. In Fig. \[fig:mocks\] we show the results obtained from the analysis of mock subsamples limited at $M_B(z=1.1)-5\log(h) \le -20.5$ in the three redshift bins \[0.5,0.7\], \[0.7,0.9\], \[0.9,1.1\]. We compare the ideal case with 100% completeness and no gaps to the more realistic case with gaps and the same spectroscopic incompleteness as in our observed catalogue, that is, including the effects of the target sampling rate, $TSR(Q)$, and the spectroscopic sampling rate, $SSR(Q)$, where $Q$ indicates the quadrant dependence. Two other selection effects were not taken into account: the colour sampling rate, $CSR(z)$, and the small-scale bias due to the constraints in the spectroscopic target selection (slits cannot overlap). The first effect depends on redshift but it is weak in our redshift range (see Fig. 5 of ), while the second effect is negligible because the angular radii of our cells are generally larger than the size of one quadrant. We note that other sources of systematic errors, as discussed by [@1999ApJ...519..622H], are the integral constraint bias, affecting the $J$–point correlation functions, and the ratio bias, affecting the estimate of $S_J$. Given the large size of our volumes, such systematic effects are weaker than the other errors, however, and can be neglected. Figure \[fig:mocks\] shows that the original values are recovered with good precision (within 1$\sigma$ error), particularly in the scale range between $4$ and 10 $h^{-1}$ Mpc. A more detailed analysis of the differences is possible with Fig. \[fig:diffmocks\], which gives the fractional difference for $\overline{\xi_2}$, $\overline{\xi_3}$, $\overline{\xi_4}$, $S_3$ , and $S_4$ as a function of scale for the same mock subsamples as in Fig. \[fig:mocks\]: it shows that in most cases we can retrieve the $J$–point correlation functions and $S_J$ with only a small systematic difference. In the first redshift bin ($0.5 \le z < 0.7$) at a radius $R = 8 h^{-1}$ Mpc, $\overline{\xi_2}$ is overestimated by  8%, while $\overline{\xi_3}$ is underestimated by  3% and $\overline{\xi_4}$ by  6%: this translates into an underestimate of $S_3$ by  16% and of $S_4$ by  26%. We have similar values in the second redshift bin ($0.7 \le z < 0.9$). In the last redshift bin ($0.9 \le z < 1.1$) the $J$–point correlation functions show the largest difference, increasing with order $J$: but these deviations at different orders are correlated, so that finally the values of $S_3$ at 8$h^{-1}$ Mpc is underestimated by only 10% and of $S_4$ by 20%, which is comparable to what is found for the other two redshift bins. The cause of the larger deviations in the last redshift bin is the lower density of the subsample; we take these systematics into account in the discussion of our results. It is interesting to point out that we find values between 1.8 and 2.1 for $S_3$ and between 8 and 10 for $S_4$ for mocks; as an example, the analysis of the mock subsamples limited at $M_B(z=1.1)-5\log(h) \le -20.5$ in the redshift bin \[0.7,0.9\] gives $S_3 \sim 2.13 \pm 0.16 $ and $S_4 \sim 9.8 \pm 1.6$ at $R = 8 h^{-1}$ Mpc. $S_3$ and $S_4$ show no significant redshift evolution, and their values are also comparable within the errors to the value measured in local redshift surveys for galaxies in a similar luminosity range. Because we know both the cosmological and the “observed” redshift for galaxies in the mock samples, including the peculiar velocity and measurement error, we can estimate the conversion factor from redshift to real space from the mock samples. We need this factor to compare our results with second–order perturbation theory predictions. Figure \[fig:realspace\] shows the difference between the estimates in real and redshift space for the subsamples limited at $M_B(z=1.1)-5\log(h) \le -20.5$ in the three redshift ranges. The redshift space correlation functions show the expected loss of power at small scales and the reverse trend at large scales. The estimate of the volume–averaged two–point correlation function in redshift space is flatter than the corresponding estimate in real space; the difference becomes significant on scales smaller than $\sim 4 h^{-1}$ Mpc. While the real space values of $S_3$ and $S_4$ increase at smaller scales, the increase is suppressed in redshift space; the difference becomes small beyond $\sim 4 h^{-1}$ Mpc. However, at small scales we have large errors due to the small number of objects in the cells. For these reasons we focus our analysis on the 4–10 $h^{-1}$ Mpc range, and particularly at 8 $h^{-1}$ Mpc, where we expect to be in the quasi–linear regime and predictions of second–order perturbation theory should hold. We recall here another bias affecting mass–selected galaxy samples, which has been discussed and tested with mock catalogues by . The lowest stellar mass subsamples suffer from incompleteness because VIPERS is magnitude limited ($i_{AB} < 22.5$); as a consequence, we can miss high mass–to–light ratio galaxies. &gt;From the analysis of mocks, found that these galaxies are faint and red and that the clustering amplitude can be suppressed up to 50% on scales below 1 $h^{-1}$ Mpc. However, as discussed by , the abundance of red and faint galaxies is overpredicted by the semi–analytic model used for the tests, and the clustering of red galaxies appears to be overestimated with respect to real data (, ), so that the amplitude of the effect might be overestimated. As we have previously noted, we did not analyse small scales and did not correct for stellar mass incompleteness. Results ======= Volume–averaged correlation functions ------------------------------------- In this section we present the results of our statistical analysis on the combined $W1$ and $W4$ samples. {width="17cm"} Figure \[fig:xi2\] shows the volume–averaged two–point correlation function obtained from counts in cells for luminosity- and stellar mass–limited subsamples in the three different redshift bins. In the same figure, as a reference for comparing the results in the different redshift bins, we plot the expected [**]{} power–law $\overline{\xi_2}$ in the redshift bin \[0.5,0.7\] for the $M_B(z=1.1)-5\log(h) \le -20.5$ subsamples (top panels) and $M_* \ge 10.0 M_\odot$ (bottom panels), derived from the $\xi_2$ estimate of ; we converted their two–point correlation function to the volume–averaged correlation function through the formula (): $$\overline{\xi}_2 = \frac{72}{2^\gamma (3-\gamma)(4-\gamma)(6-\gamma)} \xi_2 .$$ The line shows the effects of redshift space distortions, which lower the value of $\overline{\xi}_2$ on small scales and increase it on large scales. It is clear that the amplitude of $\overline{\xi}_2$ increases with both luminosity and stellar mass at all redshifts. $\overline{\xi}_2$ appears to have a stronger dependence on stellar mass than on luminosity, in agreement with the results of : see their Fig.3 for the redshift space two–point correlation functions. There are some fluctuations: for example, the dependence on luminosity appears to be sligthly weaker in the intermediate and distant redshift bins. However, these variations are consistent when taking into account statistical errors and sample variance, which are included in error bars. We conclude that the dependence of the two–point correlation function on luminosity and stellar mass does not evolve significantly up to $z \sim 1$. {width="17cm"} {width="17cm"} In Figs. \[fig:xi3\] and \[fig:xi4\] we show the volume–averaged three- and four–point correlation functions. Their behaviour reflects the two–point correlation functions, showing a stronger dependence of the correlation amplitude on stellar mass than on luminosity. {width="17cm"} {width="17cm"} The specific signature of the hierarchical scaling is the power–law relation between high–order correlation functions (Eq. \[eq:sjclassic\]). In Figs. \[fig:x2x3\] and \[fig:x2x4\] we show the three- and four–point volume–averaged correlation functions as a function of the two–point volume–averaged correlation functions. The data clearly follow the hierarchical scaling relations $\overline{\xi}_3 \propto \overline{\xi}_2 ^2$ and $\overline{\xi}_4 \propto \overline{\xi}_2 ^3$. These relations appear to hold at all luminosities and masses in the the first two redshift bins, but some systematic differences appear in the last redshift bin, particularly for the stellar–mass limited subsamples, where points are systematically higher than the reference scaling law, but in this case the values are also consistent with the same scaling relation observed at lower redshifts. As we have previously discussed, the existence of these scaling relations has been verified in the local Universe: they are expected for the matter distribution in the quasi–linear regime, as a consequence of gravitational clustering. In this case, it is natural that they do not evolve with redshift: however, it is not an obvious result to observe the same hierarchical behaviour for the galaxy distribution at all redshifts, given the evolution of bias. Skewness and kurtosis --------------------- &gt;From the counts in cells we derived the rms $\sigma$ (Eq. \[eq:variance\]), the normalized skewness $S_3$ and kurtosis $S_4$ (Eq. \[eq:sjrecursive\]) for the different VIPERS subsamples. Their values at $R = 8 h^{-1}$ Mpc are given in Cols. (4), (5), and (6) of Table \[table:1\]. The $R=8 h^{-1}$ Mpc reference radius is nearly optimal because it is large enough to enter into the quasi–linear regime, and at the same time it is in the scale range for which we have a good sampling. In Figs. \[fig:s3\] and \[fig:s4\] we show $S_3$ and $S_4$ as a function of luminosity and stellar mass in the three redshift bins. We also show the predictions of second–order perturbation theory in real space for the matter distribution and the corresponding predictions for galaxies, derived from the matter value assuming the linear bias estimated from $\overline{\xi_2}$, and corrected for redshift space distortion using the factors obtained from mocks. This derivation is described in the next subsection. The theoretical curves for $S_3$ and $S_4$ are shown for radii larger than $\sim 6 h^{-1} $ Mpc, as they are calculated in the quasi–linear regime. {width="17cm"} {width="17cm"} In the first redshift bin, both for luminosity and stellar mass limited samples, the value of $S_3$ is constant and around $2$ at small and intermediate scales, but it starts decreasing beyond $R \sim 8 h^{-1}$ Mpc. In principle, variations of $S_3$ with scale can be due to changes in the slope of the power spectrum or to a scale–dependent bias. However, such a systematic effect can be ascribed to the small number of independent cells at large scales, as shown by mocks and reflected in the large error bars. In the same redshift bin, $S_4$ shows a small decrease at large scales and is consistent with a constant value of $\sim 7.3$ between 4 and 10 $h^{-1}$ Mpc. In the range $6-8 h^{-1}$ Mpc, the best scales to compare with perturbation theory (on larger scales the errors increase significantly), the theoretical predictions for $S_3$ and $S_4$ are slightly higher than the observed values corresponding to the $M_B(z=1.1)-5\log(h) \le -20.5$ subsample, but only at $1 \sigma$ level. In the second redshift bin the value of $S_3$ for luminosity–limited subsamples is around $1.8$, sligthly lower than in the first bin, but still consistent within the errors; moreover, it is consistent with a constant value in the whole range of scales. The value of $S_3$ for mass–limited subsamples is also constant in the whole range of scales and is consistent with the value in the first redshift bin. $S_4$ has an analogous behaviour: while showing a systematic decrease, particularly in luminosity–limited subsamples, it is still consistent with a constant value in the range $4-16 h^{-1}$ Mpc. As in the case of the first redshift bin, in the range $6-8 h^{-1}$ Mpc the theoretical predictions for $S_3$ and $S_4$ are slightly higher than the corresponding observed values. In the third redshift bin the values of $S_3$ and $S_4$ for luminosity- and stellar-mass-limited subsamples increase systematically with scale. Moreover, in contrast with the two previous redshift bins, in the range $6-8 h^{-1}$ Mpc, the theoretical predictions for $S_3$ and $S_4$ are [**]{} than the observed values. To better appreciate the significance of these deviations, we note that of 26 mocks, 3 show an increase of the values of $S_3$ and $S_4$ similar to what we find in the last redshift bin. In fact, higher–order statistics are very sensitive to large–scale structure, and the correlated variations in the measured values of $S_3$ and $S_4$ probably indicate genuine fluctuations in the galaxy distribution (see e.g. the discussion in [@2004MNRAS.352.1232C]). In our case, this interpretation is suggested by checking the $W1$ and $W4$ fields separately: we find that in the outermost redshift shell, both $S_3$ and $S_4$ are larger in $W1$ than in $W4$. For example, for the $M_B(z=1.1)-5\log(h) \le -21.0$ subsample, at R=8 $h^{-1}$ Mpc, we find $S_3 = 2.7 \pm 0.5$ in $W1$ and $S_3=1.6 \pm 0.3 $ in $W4$. Analogously, for the $\log(M/M_\odot h^{-2}) \ge 10.5$ subsample at R=8 $h^{-1}$ Mpc, we find $S_3 = 3.4 \pm 0.5$ in the $W1$ field and $S_3=2.0 \pm 0.3$ in the $W4$ field. This difference might be regarded as the imprint of spatially coherent structures more prominent in $W1$. In conclusion, the values of $S_3$ and $S_4$ do not show any significant dependence on luminosity or on stellar mass: the points corresponding to different subsamples are consistent within the error bars (we discuss a possible weak dependence on luminosity in the next subsection). There is no evidence of evolution in redshift either, apart from the systematic increase of $S_3$ and $S_4$ with scale in the last redshift bin. Taking into account the behaviour of mocks, the observed systematic variations in the values of high–order moments are consistent with the fluctuations expected for comparable volumes randomly extracted from a $\Lambda$CDM universe. It is possible to compare our results on $S_3$ and $S_4$ with those obtained by [@2013MNRAS.tmp.2056W] for the four CFHTLS-Wide fields. They have divided the galaxies in the photometric catalogue into four redshift bins through the estimated photometric redshifts; for galaxies with $M_g < -20.7$, they have estimated $S_J$ as a function of angular scale and the corresponding $3D$ values through deprojection, which, as they discussed, rely on some approximations. Their work is therefore complementary to ours: they have a larger area and number of objects, but we can directly estimate the 3D (redshift space) $S_J$; they can sample smaller, highly non–linear scales where we do not have enough statistics, but we can better sample the quasi–linear scales; finally, we can also test the dependence of $S_3$ on luminosity and stellar mass. A comparison with their Fig. 12 shows that, as expected (see our Fig.\[fig:realspace\]), their deprojected values for $S_3$ and $S_4$ on small scales ($R < 5$ $h^{-1}$ Mpc) are higher than our redshift space values. On larger scales, the redshift space effect on $S_3$ and $S_4$ becomes negligible, and their estimate is consistent with ours. We note that [@2013MNRAS.tmp.2056W] found significant deviations in the results for the $W3$ field, while we have found differences between $W1$ and $W4$ in our last redshift bin: this shows that sample variance is still significant for high–order statistics on the scale of CFHTLS Wide Fields. Implications for biasing ------------------------ We now discuss the implications of our analysis for biasing. We concentrate on the reference scale $R=8 h^{-1}$ Mpc, where second–order perturbation theory predictions can be applied and results are still reliable (errors and systematic deviations increase on larger scales). Because we aim to compare our results with the matter density field, statistical quantities referring to galaxies are indicated with a subscript $g$ and those relative to matter with a subscript $m$. Figure \[fig:s83z\] shows the values of $\sigma_{8g}$ (top panel) and $S_{3g}$ (bottom panel) at $R=8 h^{-1}$ Mpc for the VIPERS volume–limited subsamples with different limiting absolute magnitudes and in the different redshift bins. In the same figure we also show the corresponding VVDS estimates () and the 2dFGRS estimates for the local Universe ([@2004MNRAS.352.1232C]) for galaxies with a similar luminosity as ours. At a given redshift, VIPERS subsamples with a brighter absolute magnitude limit have higher values of $\sigma_{8g}$, but there is no significant evolution of $\sigma_{8g}$ with redshift. The same holds when combining our results with those of the 2dFGRS in the local Universe and those of the VVDS at higher redshift: $\sigma_{8g}$ shows no significant evolution from $z=0$ to $z=1.4$ (VVDS points are systematically lower but at the $1\sigma$ level). This implies (see e.g. the discussion in ) a strong evolution of the linear bias $b$ with redshift because $\sigma_{8m}$ increases with time (see Eq. \[eq:bias\]). There are various models that describe the evolution of $b(z)$ and explain its decrease with time (see e.g.[@2000ApJ...531....1B]); from an empirical point of view, we note that the available data can be fitted by the simple relation $b(z) \propto 1 / \sigma_{8m}$. The skewness $S_{3g}$ of the VIPERS subsamples measured at 8 $h^{-1}$ Mpc and plotted as a function of redshift has more fluctuations than $\sigma_{8g}$, with a minimum value in the redshift bin \[0.7,0.9\], but it does not show a significant dependence on luminosity and is still consistent with a constant value independent of redshift. The values of $S_{3g}$ in the VVDS below $z=1.2$ are lower than VIPERS values, but are consistent within the errors, while they start to decrease beyond $z \sim 1.1$. The absence of a significant evolution of $S_{3g}$ with redshift is not limited to our redshift range: the values of $S_{3g}$ measured in VIPERS are similar to those measured in the 2dFGRS, that is, $S_3 \sim 2.0 \pm 0.2$, where depending on the subsample $S_{3g}$ varies from 1.95 to 2.58 (while not shown in the figure, the values of $S_4$ are also consistent with the 2dFGRS ones). Therefore, taking into account all data points, starting from the local value for the 2dFGRS up to $z=1.1$ (VIPERS and VVDS data), $S_{3g}$ is consistent with a constant value $\sim 2$: in VIPERS the strongest but marginal deviations of the $S_{3g}$ value are for $M_B(z=1.1)-5\log(h) \le -20.0$ galaxies in the nearest redshift range \[0.5,0.7\] and for $M_B(z=1.1)-5\log(h) \le -21.0$ galaxies in the most distant redshift interval \[0.9,1.1\], both giving a value of $S_{3g}$ that is 15% higher. Figure \[fig:sigma8\] shows $\sigma_{8g}$ (top panel) and $S_{3g}$ at $8 h^{-1}$Mpc (bottom panel) as a function of absolute magnitude for the three redshift bins. $\sigma_{8g}$ shows a systematic increase with luminosity (reflecting the dependence of the correlation amplitude on luminosity), but at a given absolute luminosity its value is similar in the three redshift bins. $S_{3g}$ appears to be independent of absolute magnitude, with fluctuations from sample to sample. However, if we exclude the points relative to the last redshift bin, where $S_{3g}$ has a higher value, the data might suggest a small decrease of $S_{3g}$ with increasing luminosity, reminiscent of the results of [@2004MNRAS.352.1232C] for the 2dFGRS. A trend of $S_{3g}$ with luminosity is interesting because in the hypothesis of linear biasing, $S_{3g}$ is inversely proportional to the bias factor $b$: knowing from the two–point correlation function of our samples that $b$ increases with luminosity, we expect a corresponding decrease of $S_{3g}$. To test whether our results are consistent with the linearity of bias, we therefore estimated the bias of galaxies with respect to the underlying matter density field at $R=8 h^{-1}$ Mpc, using the observed $\sigma_{8g}$ and $S_{3g}$ of the galaxy distribution and estimating $\sigma _m$ and $S_{3m}$ of the matter distribution through perturbation theory. [@1993ApJ...412L...9J] and (see also [@2002PhR...367....1B] and references therein) have shown that for a smoothed density field with primordial Gaussian fluctuations, Peebles’ unsmoothed value of $S_{3m} = 34/7$ ([@1980lssu.book.....P]) has to be corrected according to the expression $$S_{3m} = 34/7 + d\ln \sigma_m ^2 / d \ln R , \label{eq:s3ps}$$ where $d\ln \sigma_m ^2 / d \ln R$ is the logarithmic slope of the linear variance of the matter density field smoothed with a spherical top–hat function of radius $R$, $$\sigma_m ^2(R) = \frac {1}{2 \pi^2} \int _0 ^{\infty} dk k^2 P(k) W^2(kR) . \label{eq:sigma2}$$ For a power–law spectrum $P(k) \propto k^n$, Eq. \[eq:s3ps\] becomes $S_{3m} = 34/7 - (n+3)$. Similar relations hold for higher orders, involving higher–order derivatives. The values obtained from perturbation theory have been tested with numerical simulations, and it has been shown that in the range we are studying, that is, at $R=8 h^{-1}$ Mpc and for $\sigma_{8m} \sim 1$, they are very accurate: for example, the difference in the $S_3$ values is smaller than a few percent ([@1995MNRAS.274.1049B], [@1998MNRAS.301..503F], [@2002PhR...367....1B]). Applying Eqs. (\[eq:s3ps\]) and (\[eq:sigma2\]) and using the software [CAMB]{} ([@2002PhRvD..66j3511L]), we have computed the values of $\sigma_{8m}$ and $S_{3m}$ for a power spectrum with the new cosmological parameters derived from the Planck mission ([@2013arXiv1303.5076P]) and with the old Millennium parameters (first year WMAP data and 2dFGRS, with $\Omega_M=0,25$, $\Omega_\Lambda=075$, $n=1$ and $\sigma_{8m}=0.9$). We here assumed that the standard $\Lambda CDM$ model is correct. With other assumptions, such as a dark energy component with an evolving equation of state or modified gravity, the clustering and bias evolution would be affected (see e.g. [@2004MNRAS.349..281M]), as would the redshift distortions ([@2013MNRAS.435.2806H]). This dependence on cosmology will be studied in a future work. We also converted the observed $\sigma_{8g}$ and $S_{3g}$ to real space values by applying correction factors directly derived from the mocks. For the subsample limited at $M_B (z=1.1) - 5 \log(h) \le -20.50$, we give in Table 2 the redshift range (Col. 1), the values of $\sigma_{8g}$ (Col. 2), $\sigma_{8m}$ (Col. 3), $b=\sigma_{8g}/\sigma_{8M}$ (Col. 4), $S_{3g}$ (Col. 5), $S_{3m}$ (Col. 6), all measured at a scale of $R=8 h^{-1}$ Mpc. ------------------- ----------------- --------------- ----------------- ------------------------------- ------------------------------- Redshift range $\sigma_{8g}$ $\sigma_{8m}$ $b$ $S_{3g}$ ($R = 8 h^{-1}$ Mpc) $S_{3m}$ ($R = 8 h^{-1}$ Mpc) (real space) (WMAP/Planck) (linear bias) (real space) (WMAP/Planck) $0.5 \le z < 0.7$ $1.00 \pm 0.12$ 0.61 / 0.63 $1.65 \pm 0.20$ $1.91 \pm 0.29$ 3.52 / 3.48 $0.7 \le z < 0.9$ $0.98 \pm 0.08$ 0.55 / 0.57 $1.77 \pm 0.14$ $1.76 \pm 0.18$ 3.52 / 3.48 $0.9 \le z < 1.1$ $1.04 \pm 0.13$ 0.51 / 0.52 $2.04 \pm 0.27$ $2.28 \pm 0.25$ 3.52 / 3.48 ------------------- ----------------- --------------- ----------------- ------------------------------- ------------------------------- In Fig. \[fig:bz\] we plot our estimates for the linear bias term $b$ as a function of redshift, with the correponding estimates for VIPERS of and [@2014arXiv1406.6692D]. As expected, the estimates are fully consistent, with $b$ increasing with luminosity and redshift. As discussed by [@2014arXiv1406.6692D], there is only a difference in the last redshift bin where the estimate of Marulli et al. is lower than that of [@2014arXiv1406.6692D]. The difference is probably due to the way $b$ is estimated (counts in cells in our case and in [@2014arXiv1406.6692D], pair counts in ). Our estimate is consistent with both the other two estimates at the $1 \sigma$ level, however. In Fig. \[fig:linearbias\] we compare the linear bias directly measured from the ratio of the galaxy and matter rms, $b = \sigma_{8g} / \sigma_{8m}$, with the ratio of the galaxy and matter skewness, $S_{3m} / S_{3g}$. Under the hypothesis of linear biasing, the two ratios should have the same value. For the first two redshift bins we find slightly different values: the skewness ratio is systematically higher than the bias directly computed from the variance. The third redshift bin shows the largest discrepancy, but with the opposite behaviour, that is, the skewness ratio is lower than the bias directly computed from the variance. This different behaviour is a consequence of the fact that the value of $S_{3g}$ in the last redshift bin increases with scale and becomes higher than at lower redshifts. We can quantify the degree of non–linearity by directly estimating the second–order term $b_2$ from Eq. \[eq:secondorderbias\]: $$b_2 = b (b S_{3g} - S_{3m})/3,$$ where we used the real space values $S_{3g}$ and $b$ obtained from the redshift space values by using the conversion factor calculated from the mocks. We note that this correction is small (a few percent) at our scale of $R=8 h^{-1}$ Mpc, because this scale is at the transition from the regime of small-scale velocity dispersion (where redshift space correlation functions are lower than real space ones) to the regime of infall where redshift space correlations are higher than real space ones (see Fig. \[fig:realspace\]). In this formalism, if $b > 0$, $b_2$ is negative when $\sigma_{8g}/\sigma_{8m} < S_{3m}/S_{3g}$. This is what happens in the first two redshift bins, where at nearly all magnitudes $b_2$ is negative: for example, for the subsample limited at $M \le -20.5$(z=1.1) - 5 $\log$(h), we find $b_2 = -0.20 \pm 0.49$ in the first redshift bin and $b_2 = -0.24 \pm 0.35$, in the second redshift bin. In contrast, we find a positive $b_2$ in the third bin, with $b_2 = +0.78 \pm 0.82$. As we have noted above when discussing the results of our tests on mocks, the assumption that masked regions and inhomogeneities can be described as a Poissonian random sampling gives a small bias with an overestimate of $b$ of a few percent and an underestimate of $S_3$ around 10-15%. Using the correction factors derived from the average of the mocks, we find for the subsample limited at $M \le -20.5$(z=1.1) - 5 $\log$(h) $b_2 = -0.03 \pm 0.49$ in the first redshift, $b_2 = -0.25 \pm 0.35$ in the second redshift bin, and $b_2 = +0.72 \pm 0.82$ in the third bin. The differences are well within $1 \sigma$ error. It would be tempting to interpret these results as suggesting a possible evolution of the non–linear bias $b_2$ with redshift, with a similar trend, for example, as for the model of [@2007PhRvD..76h3004S]. Unfortunately, the problem is the extreme sensitivity of $b_2$ to the errors on $b$ and $S_{3g}$, amplified by a factor $b^2$, and we have seen that subsamples in the last redshift bin are affected by larger errors and systematic trends. With these caveats, we can check the consistency of our results with other works in the same redshift and luminosity ranges. In this comparison, one has to take into account the sensitivity of $b_2$ to the different methods and, as pointed out by [@2011ApJ...731..102K], to sample variance. In fact, even local measurements of the non–linear term have given different values (see e.g. [@2002MNRAS.335..432V] and [@2005MNRAS.362.1363P], and the discussion in [@2005MNRAS.364..620G] and ). First of all, [@2014arXiv1406.6692D] have analysed the VIPERS data reconstructing the bias relation from the estimate of the probability distribution function: they found a small (&lt; 3%) but significant deviation from linear bias. In their analysis of the four CFHTLS Wide fields, [@2013MNRAS.tmp.2056W] have found that perturbation theory predictions agree well with their measurements when taking into account the linear bias, but note that there is still a small discrepancy that can be explained by the presence of a non–linear bias term. This is also consistent with what we found. have analysed VVDS volume–limited samples limited at $M_B < -20 + 5 \log(h)$ (this limit was fixed and did not take into account luminosity evolution) in the redshift bins $0.7 < z < 0.9$ and $0.9 < z < 1.1$, finding $b_2 = -0.20 \pm 0.08$ and $b_2 = -0.12 \pm 0.08$ (here the errors do not include sample variance): these values are consistent with ours below $z=1$. [@2011ApJ...731..102K] analysed the zCOSMOS galaxy overdensity field and estimated the mean biasing function between the galaxy and matter density fields and its second moment, finding a small non–linearity, with the nonlinearity parameter $\tilde{b}/\hat{b}$ (defined in the formalism of [@1999ApJ...520...24D]) at most 2% with an uncertainty of the same order. [@2005MNRAS.364..620G] have found $b= 0.93 ^{+0.10} _{-0.08}$ and $c_2=b_2/b=-0.34 ^{+0.11} _{-0.08}$from the measurement of the $Q_3$ parameter in the three–point correlation function of the 2dFGRS for the local Universe. The non–linear term we have measured in the redshift interval between $z=0.5$ and $z=0.9$ is therefore similar to what has been measured in the above surveys. We conclude that there is general evidence for a small but non–zero non–linear $b_2$ term. It is also clear that no evolution of $b_2$ with redshift can be detected in the available data, in contrast to the linear bias term. Conclusions =========== We have analysed the high–order clustering of galaxies in the first release of VIPERS, using counts in cells to derive the volume–averaged correlation functions and normalized skewness $S_{3g}$ and kurtosis $S_{4g}$. We have analysed volume–limited subsamples with different cuts in absolute magnitude and stellar mass in three redshift bins; these subsamples are the same as in . Errors were estimated through a set of mock catalogues, derived from dark matter halo catalogues repopulated with the method of [@2013MNRAS.435..743D]. The mocks were built to reproduce the properties of VIPERS, including masks and selection effects. Our analysis has shown that the high–order statistical properties of these mocks are consistent with observations. We also studied the dependence of the second– and third–order statistics of galaxy counts on the bias, deriving the linear bias term $b$ and the first non–linear term $b_2$, and comparing our results with predictions from perturbation theory and with other works in the literature. Here are our main conclusions. - We showed that the hierarchical scaling relations $\overline{\xi}_3 \propto \overline{\xi}_2 ^2$ and $\overline{\xi}_4 \propto \overline{\xi}_2 ^3$ hold in the range of scales and redshifts we could sample, that is, $3 \le R \le 10 h^{-1}$ Mpc and $0.5 \le z < 1.1$. These relations are consistent with predictions from gravitational clustering and with the scaling observed in local surveys. - $S_{3g}$ and $S_{4g}$ appear to be independent of luminosity; however, if we do not not take the last redshift bin into account, there is a slight decrease of $S_{3g}$ with increasing luminosity, an effect previously detected locally in the 2dFGRS by [@2004MNRAS.352.1232C]. - The values of $S_{3g}$ and $S_{4g}$ are scale–independent within the errors and do not evolve significantly at least up to $z=0.9$. We detected a systematic increase with scale in the last redshift bin (beyond $\sim 10 h^{-1}$ Mpc), mainly due to one of the two CFHTLS fields ($W1$); this deviation is consistent with what can be expected from the sample variance shown by mock catalogues. - The observed values of $S_{3g} \sim 2 \pm 0.2$ and $S_{4g} \sim 8 \pm 0.4$ are similar to those measured in local surveys for galaxies in the same luminosity range. This confirms the substantial absence of evolution of $S_{3g}$ in the redshift range $0 < z < 1$ at the level of $\sim 10$%. This result is expected for $S_{3m}$ , but is not trivial for $S_{3g}$, given the evolution of bias. - At second order, galaxies with higher luminosity or stellar mass have a larger amplitude (greater linear bias parameter) of the volume–averaged two–point correlation function, consistently with the direct analysis of the two–point correlation function by . We showed that our estimate of the linear bias parameter $b=\sigma_{8g}/\sigma_{8m}$ is consistent within $1 \sigma$ with those of and [@2014arXiv1406.6692D]. The linear bias increases both with luminosity and with redshift: in our redshift range, we measured the lowest bias $b=1.47 \pm 0.18$ for galaxies with $M_B(z=1.1)-5\log(h) \le -19.5$ in the redshift bin $0.5 \le z < 0.7$ and the largest bias $b=2.12 \pm 0.28$ for galaxies with $M_B(z=1.1)-5\log(h) \le -21.0$ in the redshift bin $0.9 \le z < 1.1$. - For a given luminosity class, $\sigma_{8g}$ does not evolve with redshift. For example, comparing our values for $M_B(z=1.1)-5\log(h) \le -20.5$ to the corresponding value measured in the 2dFGRS, we found that $\sigma_{8g}$ is consistent with a constant value  1.0 (our $1 \sigma$ error is 10%), from $z=0$ to $z \sim 1$. Given that $\sigma_{8m}$ increases with time, we have the empirical relation $b(z) \propto 1/ \sigma_{8m}(z)$. - The value of the non–linear bias parameter $b_2$ measured below $z \sim 1$ at the scale $R = 8 h^{-1}$ Mpc, that is, in the quasi–linear regime, is negative but not statistically different from zero when taking into account the error; however, taking into account the ensemble of results coming from this and other surveys in the redshift range $0.5 \le z < 1$ (, [@2011ApJ...731..102K], [@2013MNRAS.tmp.2056W], [@2014arXiv1406.6692D]), there is evidence for a small but non–zero non–linear term. Including the results from local surveys as well, no evolution of $b_2$ with redshift can be detected in the available data. - The comparison with the properties of mocks and with the predictions of perturbation theory shows that our results are consistent with the general scenario of biased galaxy formation and gravitational clustering evolution in a standard $\Lambda CDM$ cosmology. In conclusion, we have provided an independent check on the second–order statistical studies of the galaxy distribution through our analysis; we explored the galaxy bias with an independent technique; finally, we determined the higher–order statistical properties of the galaxy distribution in the redshift range between 0.5 and 1.1, thanks to the combination of volume and density of galaxies in the VIPERS survey. When VIPERS is complete, it will be possible to perform a more general analysis, which will allow us not only to decrease error bars, but also to include the dependence of high–order statistics on galaxy colour, to apply other high–order statistical tools such as the void probability function, and to give better constraints on the non–linear bias. This work is based on observations collected at the European Southern Observatory, Cerro Paranal, Chile, using the Very Large Telescope under programs 182.A-0886 and partly 070.A-9007. Also based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. The VIPERS web site is http://www.vipers.inaf.it/. We acknowledge the crucial contribution of the ESO staff for the management of service observations. In particular, we are deeply grateful to M. Hilker for his constant help and support of this program. Italian participation to VIPERS has been funded by INAF through PRIN 2008 and 2010 programs. DM gratefully acknowledges financial support of INAF-OABrera. LG, AJH, and BRG acknowledge support of the European Research Council through the Darklight ERC Advanced Research Grant (\#291521). AP, KM, and JK have been supported by the National Science Centre (grants UMO-2012/07/B/ST9/04425 and UMO-2013/09/D/ST9/04030), the Polish-Swiss Astro Project (co-financed by a grant from Switzerland, through the Swiss Contribution to the enlarged European Union), and the European Associated Laboratory Astrophysics Poland-France HECOLS. KM was supported by the Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation (\#R2405). OLF acknowledges support of the European Research Council through the EARLY ERC Advanced Research Grant (\#268107). GDL acknowledges financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement \# 202781. WJP and RT acknowledge financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007- 2013)/ERC grant agreement \#202686. WJP is also grateful for support from the UK Science and Technology Facilities Council through the grant ST/I001204/1. EB, FM and LM acknowledge the support from grants ASI-INAF I/023/12/0 and PRIN MIUR 2010-2011. LM also acknowledges financial support from PRIN INAF 2012. YM acknowledges support from CNRS/INSU (Institut National des Sciences de l’Univers) and the Programme National Galaxies et Cosmologie (PNCG). CM is grateful for support from specific project funding of the Institut Universitaire de France and the LABEX OCEVU. SdlT acknowledges the support of the OCEVU Labex (ANR-11-LABX-0060) and the A\*MIDEX project (ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir” French government program managed by the ANR. [^1]: based on observations collected at the European Southern Observatory, Cerro Paranal, Chile, using the Very Large Telescope under programs 182.A-0886 and partly 070.A-9007. Also based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. The VIPERS web site is http://www.vipers.inaf.it. [^2]: However, there is the important exception of the lognormal distribution, see [@1991MNRAS.248....1C] and [@2011ApJ...738...86C]. [^3]: http://vipers.inaf.it [^4]: Mellier, Y., Bertin, E., Hudelot, P., et al. 2008, http://terapix.iap.fr/cplt/oldSite/Descart/CFHTLS-T0005-Release.pdf.

--- abstract: 'Measurements have been made of the probability distribution of total transmission of microwave radiation in waveguides filled with randomly positioned scatterers which would have values of the dimensionless conductance $g$ near unity. The distributions are markedly non-Gaussian and have exponential tails. The measured distributions are accurately described by diagrammatic and random matrix calculations carried out for nonabsorbing samples in the limit $g \gg 1$ when $g$ is expressed in terms of the variance of the distribution, which equals the degree of long-range intensity correlation across the output face of the sample.' address: 'New York State Center for Advanced Technology for Ultrafast Photonic Materials and Applications, Dept. of Physics, Queens College of CUNY, Flushing, NY 11367' author: - 'M. Stoytchev and A. Z. Genack' title: Measurement of the Probability Distribution of Total Transmission in Random Waveguides --- Nonlocal correlation in the flux transmitted through mesoscopic samples leads to enhanced fluctuations of local and spatially averaged transmission for both classical and quantum waves.[@Sheng90; @Alt91] Such fluctuations increase dramatically as the ensemble average of the dimensionless conductance, $g$, approaches unity. Low values of $g$ can be achieved in quasi-one-dimensional samples such as conducting wires or multimode waveguides with lengths much greater than the transverse dimensions. In this Letter, we report measurements of the probability distribution of total transmission of microwave radiation in long waveguides filled with randomly positioned scatterers which in the absence of absorption would have values of $g$ near unity. The distributions observed are markedly non-Gaussian. They are compared to recent diagrammatic and random-matrix calculations for nonabsorbing samples in the limit $g \gg 1$ [@vanRos95; @Kog95]. This is done by reexpressing the distribution, which is a function of the single parameter g, as a function of the variance of the normalized transmission using the relation between these parameters. This result is in excellent agreement with the measured transmission distributions and indicates that the variance of the normalized transmission, which equals the degree of long-range intensity correlation across the output face of the sample, is the essential parameter describing fluctuations in random media. Key transmission quantities in order of increasing spatial averaging are the intensity, $T_{ab}$, which is the transmission coefficient for incoming mode $a$ into mode $b$, the total transmission for incoming mode $a$, $T_a = \sum_{b}T_{ab} \sim \ell/L$, and the total transmittance $T = \sum_{ab}T_{ab} \sim N\ell/L$, where $\ell$ is the transport mean free path, $L$ is the sample length, and $N$ is the number of modes. The total transmittance is equivalent to the dimensionless conductance in electronic systems, $T = G/(e^2\!/h)$, where $G$ is the conductance [@Land70] and $g\, =\, <T>\, =\, N\ell /L$. Though the variances of the transmission quantities normalized to their ensemble average values, $s_{ab} = T_{ab}/\!\!\!<\!\!T_{ab}\!\!>$, $s_a = T_a/\!\!\!<\!\!T_a\!\!>$ and $s = T/\!\!\!<\!\!T\!\!>$, are reduced as the extent of spatial averaging increases, fluctuations in these quantities do not self average, as they would if spatial correlation were absent. To leading order in $1/g$, the enhancement of the variances of $s_{ab}$, $s_a$, and $s$ arising from nonlocal correlation is 1, $L/\ell$ [@Steph87], and $(L/\ell)^2$ [@Lee85; @Alt85], respectively, which results in values of the variances of 1, $1/g$, and $(1/g)^2$. [@Feng88; @Mello88] To examine the scaling and the universality of transport, it is important to measure the full distribution of key transmission quantities as the sample size, and hence $g$, changes. In previous work, nonlocal correlation has been shown to lead to higher probabilities at large values of the intensity, leading to a deviation from negative exponential statistics for polarized microwave radiation when $g \sim 10$ [@Gar89; @Gen93], as well as to discernable deviations from a Gaussian distribution and enhanced variance for the total optical transmission when $g > 10^3$ [@deBoer94]. Recently, an expression for $P(s_a)$ in terms of $g$ for nonabsorbing samples was obtained by Nieuwenhuizen and van Rossum using diagrammatic techniques combined with random matrix theory [@vanRos95] and subsequently by Kogan and Kaveh within the framework of random matrix theory [@Kog95]. The diagrammatic calculations neglect some terms of order higher than $1/g$, whereas computations based on random matrix theory neglect sample-to-sample fluctuations in the probability distributions of eigenvalues of the transmission matrix and are expected to be accurate only to order $1/g$. More recently, van Langen, Brouwer and Beenakker carried out a nonperturbative calculation of the total transmission distribution in the absence of absorption [@vanLang96]. An analytic solution is obtained for the case in which time reversal invariance is broken $(\beta = 2)$ but not for the case of time reversal symmetry $(\beta = 1)$ considered here. However, good agreement is found between the $\beta$-independent result for $P(s_a)$ obtained in Refs.  [@vanRos95; @Kog95] and the result for $\beta = 2$ in Ref. [@vanLang96] for $g \geq 10$. The distribution of total transmission has been measured previously by de Boer et al. in optical measurements in slabs of titania particles [@deBoer94]. Samples with $g > 10^3$ were studied and the distribution was found to be Gaussian to within $1 \%$. A measure of the deviations of the distribution from a Gaussian is the value of the third cumulant $<\!\!s_a^3\!\!>_c$ which gives the skewness of the distribution and vanishes for a Gaussian distribution. For the samples studied, $<\!\!s_a^3\!\!>_c$ was of order of $10^{-6}$. It was found that $<\!\!s_a^3\!\!>_c = \gamma _g <\!\!s_a^2\!\!>_c^2$ with $\gamma _g = 2.9 \pm 0.4$ which is consistent with the value calculated for Gaussian beam excitation of 3.20 [@vanRos95]. In the present work, low values of $g$ are achieved by placing the sample in a cylindrical copper tube in order to restrict transverse diffusion and thus the number of modes $N$. The samples consist of randomly positioned polystyrene spheres with diameters of 1.27 cm at a volume filling fraction $f = 0.55$. Transmission spectra were taken at tube diameters of 7.5 and 5.0 cm and various sample lengths in the frequency range 16.8 - 17.8 GHz. The microwave radiation is coupled to the sample by a 0.4 cm wire antenna placed 0.5 cm from the front surface of the sample. The frequency is incremented in 4 MHz steps. The sample tube is rotated between successive measurements to produce new scatterer configurations. The total transmission is measured by use of a single Schottky diode detector positioned inside an integrating sphere which rotates at 2 Hz around the sample axis. The integrating sphere has a diameter of 40 cm and is comprised of two concentric plastic spherical shells separated by a distance of 2 cm. The outer shell is covered with aluminum foilt to form an irregular reflecting surface. The region between the shells is filled with thin-walled aluminum cylinders with diameters of 0.75 cm and typical lengths of 1 cm. The cylinders tumble as the integrating sphere rotates, resulting in fluctuations of the intensity at the detector with a correlation time of $\sim$ 2 ms for the sample with a length of 100 cm. The signal is averaged for 1 s at each frequency, giving an uncertainty of $2.5\ \%$ in the measurement of transmission. The signal is normalized by its ensemble average to give $s_a$. The transmission distributions $P(s_a)$ are obtained by using the data from at least 1000 sample configurations. Distributions obtained using different intervals of the frequency range coincide within experimental error. In the frequency range of the measurements, $\ell \approx 5$ cm and $N = k^2d^2/8 \approx 200$ and 90 for samples in tubes with diameters $d\, =\, 7.5$ and 5.0 cm, respectively. The wave number $k = 2\pi/\lambda = 2\pi\nu n/c$ is calculated using an effective medium index of refraction $n \approx 1.4$. The transmission distributions for three samples with dimensions (a) $d = 7.5$ cm, $L = 66.7$ cm, (b) $d = 5.0$ cm, $L = 50.0$ cm, and (c) $d = 5.0$ cm, $L = 200$ cm are shown in Fig. 1. In the absence of absorption, the dimensionless conductance for these samples without localization corrections, $g = N\ell /L$, would be approximately 15.0, 9.0, and 2.25 for samples $a$, $b$, and $c$, respectively. The distribution broadens and the deviation from a Gaussian becomes more pronounced as either the sample length increases or the cross-sectional area decreases. A value of $<\!\!s_a^3\!\!>_c$ as large as $0.112\, \pm\, 0.003$ is observed for sample c. Deviations from a Gaussian distribution in the tail of the distribution for this sample can be seen in the semilog plot of $P(s_a)$ in Fig. 2. For values of $s_a\, \geq\, 2$ the distribution is nearly exponential. In Fig. 3, we present a plot of $<\!\!s_a^3\!\!>_c$ versus $<\!\!s_a^2\!\!>_c^2$. The solid line is a least square linear fit to the data which gives $\gamma = 2.38\, \pm\, 0.05$. Within experimental error this equals the value $\gamma = 2.40$ calculated for an incident plane wave in the lowest order of a diagrammatic perturbation expansion in the small parameter $1/g$ [@vanRos95]. The results are compared to calculations for a plane wave since $d\, \ll\, L$ and there is a nearly complete mixing of modes in the sample, giving a uniform average intensity along a cross section of the sample. The agreement between theory and experiment is surprising, however, since $1/g \gtrsim 0.1$ for all samples, reaching a value of approximately 0.3 for sample c, and is by no means small. Furthermore, the influence of absorption was not included in the calculations, whereas the samples used in the experiment are strongly absorbing with $L > L_a \approx 40$ cm, where $L_a$ is the exponential absorption length [@Gen93a]. We now consider the full transmission distribution. The theoretical expressions for the full distribution function in Refs.  [@vanRos95; @Kog95; @vanLang96] are given as functions of $g$ for nonabsorbing samples. In the present case of strong absorption, the photon number is not conserved, and $g$ cannot be defined in terms of the steady state transmission, while serving as a useful measure of the proximity to the localization transition. This can be seen by noting that the reduction of the average transmission due to absorption would lead to a reduced value of $g$ even though the presence of absorption tends to lessen the degree of correlation in the sample and to push the system farther from the localization threshold. On the other hand, a parameter which characterizes the transmission distribution as well as the closeness to the localization threshold, even in the presence of absorption, is the degree of correlation of intensity in different coherence areas of the transmitted speckle pattern, $<\!\!\delta s_{ab}\delta s_{ab'}\!\!>$. Were this correlation to vanish, fluctuations in different coherence areas would be independent and the transmission distribution would be Gaussian by the central limit theorem with $var(s_a)\, \equiv\, <\!\!s_a^2\!\!>_c = 1/N$. As a result of nonlocal correlation, however, the variance of the transmission is enhanced. It is given by $<\!\!s_a^2\!\!>_c\, =\, (<\!\!s_{ab}^2\!\!>_c - 1)/2\, =\, <\!\!\delta s_{ab}\delta s_{ab'}\!\!>$ [@Kog95; @Feng88; @Mello88; @Gar93]. The last equality is consistent with the results of Ref. [@Gar93] when the cumulant intensity correlation function is properly normalized to the renormalized average transmission [@Shap97]. In that case, the crossing parameter found by Shnerb and Kaveh [@Shn91] which determines the intensity distribution is found experimentally to be equal to $<\!\!\delta s_{ab}\delta s_{ab'}\!\!>$ [@Gen93; @Gar93]. The connection of $<\!\!\delta s_{ab}\delta s_{ab'}\!\!>$ to the full transmission distribution can be seen by considering the expression of Refs. [@vanRos95; @Kog95] for $P(s_a)$ in the absence of absorption in the limit $g\ \gg\ 1$, $$\label{psa} P(s_a) = \int_{-i\infty}^{+i\infty}\frac{dx}{2\pi i}exp[xs_a - \Phi(x)],$$ where $$\Phi(x) = g ln^2(\sqrt{1 + x/g} + \sqrt{x/g})$$ is the generating function. From Eq. (\[psa\]) one obtains the expression for $<\!\!s_a^2\!\!>_c$ in terms of $g$, $$\label{secondc} <\!\!s_a^2\!\!>_c = \frac{2}{3g}.$$ From these expressions, a general relation for $P(s_a)$ in terms of $<\!\!s_a^2\!\!>_c$, or equivalently $<\!\!\delta s_{ab}\delta s_{ab'}\!\!>$, can be found by using Eq. (\[secondc\]) to define a new parameter $g' = 2/3\!\!<\!\!s_a^2\!\!>_c$ which is substituted for $g$ into Eq. (\[psa\]). Plots of $P(s_a)$ obtained by following this procedure with $g'$ determined from the measured values of $<\!\!s_a^2\!\!>_c$ are shown as the solid lines in Figs. 1 and 2. We find that $P(s_a)$ is accurately given even for the lowest value of $g'$ of 3.06 (sample c). The distribution of Eq. (\[psa\]) with $g'$ substituted for $g$ gives the exponential tail, $P(s_a) \sim exp(-g's_a)$ in the limit $s_a \gg 1$. For $s_a\, \geq\, 2.0$, the linear fit to the logarithm of the measured transmission distribution for sample c gives a slope of $2.71\, \pm\, 0.06$ in accord with the exponential fit of the theoretical curve of 2.70 in this range and is quite close to its predicted asymptotic value of 3.06 for $s_a \gg 1$. The extent of the agreement of Eq. (\[psa\]) when $g'$ is substituted for $g$ can also be gauged from the comparison between the calculated (circles) and the measured (squares) moments of the transmission distribution shown in Fig. 4 for samples with $g' = 10.2\, \pm\, 0.1$ and $g' = 3.06\, \pm\, 0.04$. The moments calculated from the theory are close to those obtained from the measured distributions. At $n\, =\, 10$, these defer by approximately $10\, \%$ which is within the experimental error. Thus it appears that $P(s_a)$ can be expressed as a function of the parameter $<\!\!s_a^2\!\!>_c$. The agreement between theory and experiment indicates that the ratio of moments is accurately reflected in Eq. (\[psa\]). The dependence of the second cumulant itself upon sample dimensions is shown in Fig. 5. In the limit $g \gg 1$, in the absence of absorption, . The straight line in the figure is drawn through the first data point and represents $<\!\!s_a^2\!\!>_c\, \sim\, L/N$. As $g \rightarrow 1$, and the localization threshold is approached, the scaling theory of localization [@gang4] suggests that $g$ falls more rapidly and hence $<\!\!s_a^2\!\!>_c$ should increase superlinearly with sample length. Instead, we find that $<\!\!s_a^2\!\!>_c$ depends sublinearly upon $L$. This is presumably due to the presence of absorption which diminishes the degree of nonlocal correlation. This raises the question of whether the transmission distribution continues to broaden as $L$ increases or, instead, it reaches a limiting distribution for particular sample parameters. In conclusion, we have measured the total transmission distribution of microwave radiation in quasi-one-dimensional absorbing samples with small values of $g$. We find that the distribution can be described using an expression originally derived for nonabsorbing samples in the limit $g \gg 1$ when this expression is reformulated as a function of the single parameter $g'\, =\, 2/3<\!\!s_a^2\!\!>_c$ determined from the measurements. The validity of the expression for values of $g'$ as small as 3, well beyond the limits assumed in the calculations, may well be associated with the identification of $<\!\!s_a^2\!\!>_c$ with $<\!\!\delta s_{ab}\delta s_{ab'}\!\!>$, the degree of spatial correlation in the sample, which is the key microscopic parameter in mesoscopic physics. We are pleased to acknowledge stimulating discussions with E. Kogan, M. C. W. van Rossum and B. Shapiro. We thank E. Kuhner, Z. Ozimkowski and D. Genack for constructing and testing the integrating sphere as well as W. Polkosnik for his help in automating the experiment. We are grateful to N. Garcia for his encouragement and fruitful discussions. This work was supported by a National Science Foundation Grant No. 9632789 and by a PSC- CUNY award. [99]{} , edited by P. Sheng (World Scientific, Singapore, 1990). , edited by B. L. Altshuler, P. A. Lee, and R. A. Webb (North-Holland, Amsterdam, 1991). Th. M. Nieuwenhuizen and M. C. W. van Rossum, Phys. Rev. Lett. [**74**]{}, 2674 (1995). E. Kogan and M. Kaveh, Phys. Rev. B [**52**]{}, R3813 (1995). R. Landauer, Phil. Mag. [**21**]{}, 863 (1970). M. J. Stephen and G. Cwilich, Phys. Rev. Lett. [**59**]{}, 285 (1987). P. A. Lee and A. D. Stone, Phys. Rev. Lett. [**55**]{}, 1622 (1985). B. L. Altshuler and D. E. Khmelnitskii, Sov. Phys. JETP Lett. [**61**]{}, 359 (1985). S. Feng, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. [**61**]{}, 834 (1988). P. A. Mello, E. Akkermans, and B. Shapiro, Phys. Rev. Lett. [**61**]{}, 459 (1988). N. Garcia and A. Z. Genack, Phys. Rev. Lett. [**63**]{}, 1678 (1989). A. Z. Genack and N. Garcia, Europhys. Lett. [**21**]{}, 753 (1993). J. F. de Boer, M. C. W. van Rossum, M. P. van Albada, Th. M. Nieuwenhuizen, and A. Lagendijk, Phys. Rev. Lett. [**73**]{}, 2567 (1994). S. A. van Langen, P. W. Brouwer, and C. W. J. Beenakker, Phys. Rev. E [**53**]{}, 1344 (1996). A. Z. Genack, N. Garcia, and A. A. Lisyansky, in [ *Photonic Band Gaps and Localization*]{}, edited by C. M. Soukoulis (Plenum Press, New York, 1993). N. Garcia, A. Z. Genack, R. Pnini, and B. Shapiro, Phys. Lett. A [**176**]{}, 458 (1993). B. Shapiro, privite communication. N. Shnerb and M. Kaveh, Phys. Rev. B [**43**]{}, 1279 (1991). E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. [**42**]{}, 673 (1979). [**FIGURES**]{}:\ Fig. 1. Distribution function of the normalized transmission $P(s_a)$ for three samples with dimensions (a) $d = 7.5$ cm, $L = 66.7$ cm, (b) $d = 5.0$ cm, $L = 50.0$ cm, and (c) $d = 5.0$ cm, $L = 200$ cm.\ Fig. 2. Semilogarithmic plot of the transmission distributions for the same samples as in .\ Fig. 3. Plot of $<\!\!s_a^3\!\!>_c$ versus $<\!\!s_a^2\!\!>_c^2$. The solid line represents a least square linear fit to the data.\ Fig. 4. Comparison of the calculated (circles) and measured (squares) moments of the transmission distribution for samples with (a) $g' = 10.2$ and (b) $g' = 3.06$.\ Fig. 5. Dependence of the second cumulant $<\!\!s_a^2\!\!>_c$ upon sample dimensions.\

--- abstract: 'Analysis of the saturation of the Kelvin-Helmholtz (KH) instability is undertaken to determine the extent to which the conjugate linearly stable mode plays a role. For a piecewise-continuous mean flow profile with constant shear in a fixed layer, it is shown that the stable mode is nonlinearly excited, providing an injection-scale sink of the fluctuation energy similar to what has been found for gyroradius-scale drift-wave turbulence. Quantitative evaluation of the contribution of the stable mode to the energy balance at the onset of saturation shows that nonlinear energy transfer to the stable mode is as significant as energy transfer to small scales in balancing energy injected into the spectrum by the instability. The effect of the stable mode on momentum transport is quantified by expressing the Reynolds stress in terms of stable and unstable mode amplitudes at saturation, from which it is found that the stable mode can produce a sizable reduction in the momentum flux.' author: - 'A.E. Fraser' - 'P.W. Terry' - 'E.G. Zweibel' - 'M.J. Pueschel' title: Coupling of Damped and Growing Modes in Unstable Shear Flow --- Introduction ============ Shear flows are encountered in a variety of different systems. In the atmosphere, shear-flow instabilities are observed in cloud patterns[@Browning]. In fusion devices, turbulence generates shearing zonal flows whose potential for instability can significantly impact confinement[@Dimits; @Rogers]. Shear-flow instabilities are especially important in astrophysics. There, differential velocities are produced by a host of processes in a variety of settings, including jets driven by accretion of mass onto compact objects such as protostars or supermassive black holes, intergalactic clouds falling into a galaxy, and galaxies plowing through the intracluster medium. In astrophysical systems, it is thought that shear-flow instabilities induce formation of a turbulent shear layer, resulting in entrainment of material through turbulent momentum transport[@Churchwell], thermal and chemical mixing[@Kwak], and the possibility of acceleration of particles to high energy[@Rieger]. Shear-flow instability in a plasma with a uniform magnetic field perpendicular to both the flow and shear directions has the same growth rate as hydrodynamic shear flow with the same profile, illustrating that strong connections exist between hydrodynamic and plasma shear-flow instabilities. The number of potential applications in both systems makes quantitative models of turbulence driven by sheared flows highly desirable. Analytical models that describe spectral properties are important because both the separation between scales and Reynolds numbers found in astrophysical systems are much larger than what can typically be obtained in converged hydrodynamic and magnetohydrodynamic (MHD) simulations[@Palotti; @Lecoanet]. Efforts to characterize the nonlinear state of turbulent systems like those mentioned above commonly employ the growth rate and mode structure of the dominant linearly unstable eigenmode, which, after all, drives the turbulence. Examples are mixing-length estimates of transport, which for unstable systems are built on the linear growth rate and an unstable wavenumber, and the quasilinear transport approximation, which uses the cross phase of the unstable eigenmode to approximate the fluctuating correlation responsible for transport. Such approximations are straightforward to construct because they rely on well-understood linear properties of instabilities. However, as unstable systems move into the turbulent regime, there can be no saturation if fluctuations and transport are not modified from the linear state in some essential fashion. The precise nature of such modifications is not well understood. The standard assumption is that the modifications can be treated as a cascade to smaller scales that eventually become damped, in analogy to externally forced Navier-Stokes turbulence. This type of approach overlooks stable eigenmodes at the same scales as the instability, which invariably exist as other roots of the instability dispersion relation, and may modify the dynamics at the largest scales. In gyroradius-scale instability-driven turbulence relevant to fusion devices, it has been recognized for more than a decade that stable modes are important in turbulence and should not be neglected[@Baver; @Hatch2012]. Such modes can be represented as eigenmodes of the linearized system, and occur at the same length scale as the driving instability. Both stable modes, which have a negative linear growth rate,[@Terry2006] and subdominant modes, which can have a growth rate that is positive but smaller than that of the most unstable mode[@MJ], are difficult to detect in initial value simulations. When perturbations are small and only the linear dynamics are considered, these modes are negligible compared to the most unstable mode. However, as the most unstable mode grows in amplitude, nonlinear three-wave interactions between it and the stable modes can pump energy into the latter, causing them to grow and have a significant impact on the turbulence. In collisionless trapped electron mode turbulence, for example, stable modes radically change the dynamics of the system, including changing the direction of particle flux[@Terry2006; @TerryGatto]. In recent studies of plasma microturbulence in stellerators, quasilinear calculations of energy transport cannot reproduce the results of nonlinear simulations without including every subdominant unstable mode[@MJ]. While it has been demonstrated that stable modes are universally excited and can have significant impacts on turbulence in the context of gyroradius-scale instabilities in fusion plasmas, their effects have not been studied in global-scale hydrodynamic or MHD instabilities. This paper presents an analysis of a hydrodynamic system with global-scale eigenmodes, demonstrating the nonlinear excitation of stable modes and quantifying their impact on the turbulence using techniques that were successful in plasma microturbulence. An important aspect of this paper is that tools developed in previous analytic calculations for homogeneous systems are extended for analysis of nonlinear excitation in the inhomogeneous environment of unstable shear flows. In previous calculations, the PDEs that describe relevant dynamical quantities were Fourier-transformed to obtain a system of ODEs describing the time-dependence of the Fourier amplitudes. The ODEs were then linearized about an unstable equilibrium to obtain a system of equations of the form $\dot{\mathbf{f}}=\mathcal{D}\mathbf{f}$, where $\mathbf{f}(\mathbf{k},t)$ is a vector describing the state of the system at wavenumber $\mathbf{k}$, and $\mathcal{D}$ is the matrix of linear coupling coefficients. The eigenvectors of $\mathcal{D}$ are the eigenmodes of the system, and their eigenvalues are the frequencies and growth rates. The nonlinear excitation of linearly stable modes was then demonstrated by expanding the nonlinearities of the ODEs in the basis of the linear eigenmodes. With inhomogeneous systems, eigenmodes are not obtained by Fourier-transforming the PDEs and diagonalizing a matrix. Consequently, constructing an invertible transformation between dynamical quantities and linear eigenmodes, and expanding nonlinearities in terms of the eigenmodes, requires appropriate conditioning of the problem. The paper is organized as follows. In Sec. II we consider an unstable shear flow and discuss its unstable and stable eigenmodes. In Sec. III we develop a mapping of the fluctuating flow onto the linear eigenmodes that allows a quantitative description of the energy transfer between the unstable and stable modes. In Sec. IV we use the tools of previous calculations to assess the level to which stable modes are excited relative to unstable modes in saturation. In Sec. V we consider the impact of stable modes on turbulent momentum transport. Conclusions are presented in Sec. VI. Though we start from equations that describe a two-dimensional, unmagnetized shear flow, this system is identical to a magnetized shear flow where the equilibrium magnetic field is uniform and in the direction perpendicular to both the flow and the gradient of the flow[@Chandra]. Future work will consider the case of a magnetic field in the direction of flow. Linear Modes ============ We investigate a piecewise linear equilibrium flow in the $x$ direction with variation in the $z$ direction within a finite region of width $2d$, referred to as the shear layer. The equilibrium flow is $\mathbf{v}_0 = (U(z),0,0)$, where $$U(z) = \begin{cases} 1 & z\geq 1 \\ z & -1\leq z\leq 1 \\ -1 & z\leq -1. \end{cases}$$ Here, $U = U^*/U_0$ is the flow normalized to the flow speed $U_0$ outside the layer, $(x,z) = (x^*/d, z^*/d)$ are coordinates normalized to the layer half-width $d$, and time will be normalized by $t = t^*U_0/d$. Constant shear in a shear layer provides the simplest shear-flow instability for which the nonlinear driving of stable modes can be described analytically. The vortex sheet[@Chandra] is an even simpler manifestation of shear-flow instability, but the discontinuous equilibrium flow leads to a discontinuous eigenmode structure. Consequently, the eigenmode projection of the nonlinearity, which is calculated in the following section and involves a product of derivatives of the eigenmodes \[see Eq. \], is not well-defined. Here, flow is assumed to be 2D ($\partial/\partial y \equiv 0$), inviscid, and incompressible. It has been shown that for unmagnetized shear flows, 2D perturbations are the most unstable[@Drazin], so it suffices to restrict this analysis to the 2D system. The inviscid assumption simplifies the calculation, although in physical systems at scales much smaller than those considered here, viscosity acts to remove energy from perturbations. The assumption of incompressibility is convenient because of the stabilizing effect of compressibility on shear flow instabilities[@Gerwin]. These assumptions allow the perturbed velocity to be written in terms of a stream function $\mathbf{v}_1 = \hat{y}\times\nabla \Phi(x,z) = (\partial \Phi / \partial z, 0, -\partial \Phi / \partial x)$. The perturbed vorticity is then entirely in the $-\hat{y}$ direction and is governed by the equation[@Drazin], $$\label{NLvorticity} \frac{\partial}{\partial t}\nabla^2\Phi + U\frac{\partial}{\partial x}\nabla^2\Phi - \frac{\partial\Phi}{\partial x}\frac{d^2 U}{dz^2} + \frac{\partial\Phi}{\partial z}\frac{\partial}{\partial x}\nabla^2\Phi - \frac{\partial\Phi}{\partial x}\frac{\partial}{\partial z}\nabla^2\Phi = 0.$$ This equation follows either from vorticity evolution in hydrodynamics or in MHD when the mean field is perpendicular to the flow and the fluctuations are electrostatic. We drop terms nonlinear in $\Phi$ and investigate normal modes of the form $\Phi(x,z,t) = \phi(k,z)\exp[ikx + i\omega(k)t]$, where $k = k^*d$ and $\omega = \omega^* d/U_0$. If we find that $\mathrm{Im}(\omega(k)) < 0$ for some mode at wavenumber $k$, then the mode is unstable and grows exponentially in time. If $\mathrm{Im}(\omega(k)) > 0$, the mode is stable and decays exponentially. If $\mathrm{Im}(\omega(k)) = 0$, the mode is marginally stable. We take Fourier modes in $x$ because Eq.  is homogeneous in $x$, but the dependence of $U$ on $z$ implies that Fourier modes in $z$ are not solutions to the linear equation. This significantly complicates the analysis of stable mode interactions, as discussed in the following section. The linearized equation for the normal modes is[@DrazinHoward] $$\label{Lvorticity} (\omega+kU)\left(\frac{d^2}{dz^2} - k^2\right)\phi - k\phi \frac{d^2U}{dz^2} = 0.$$ Solutions of this system are well known[@Chandra], but usually only the growth rate of the unstable mode and its eigenfunction are considered. We reexamine the problem to keep track of both the unstable and stable modes, in order to investigate their interaction through the nonlinearities in Eq. . Note that for the shear layer, $d^2U/dz^2$ is singular at $z = \pm 1$. For $|z| \neq 1$ however, $d^2U/dz^2 = 0$, so $$(\omega+kU)\left(\frac{d^2}{dz^2} - k^2\right)\phi = 0$$ (for $|z| \neq 1$). Solutions are given by either $\omega+kU = 0$ or $(d^2/dz^2 - k^2)\phi = 0$. While modes that satisfy the former are solutions of the system, we are interested in stable and unstable modes, which require $\mathrm{Im}(\omega) \neq 0$. Therefore we construct eigenmodes from $(d^2/dz^2 - k^2)\phi = 0$. It has been shown that for shear flow instabilities, the initial value calculation admits additional modes that decay algebraically[@Case]. While these modes potentially play a role in saturation of the instability and should be considered eventually, it makes sense to focus first on the interaction between the exponentially growing and decaying modes. Both the exponentially and algebraically decaying modes are ignored in quasilinear models of turbulence, so to show that these models overlook important, driving-scale features of the system it suffices to demonstrate the importance of stable modes. Focusing on solutions of $(d^2/dz^2 - k^2)\phi = 0$, modes are given by $$\label{unsolved phi} \phi(z) = \begin{cases} ae^{-|k|z} & z > 1 \\ e^{|k|z}+be^{-|k|z} & -1 < z < 1 \\ ce^{|k|z} & z < -1, \end{cases}$$ with the constants $a, b,$ and $c$ to be determined. The flow profile $U(z)$ is continuous at the boundaries of the shear layer, which we assume to be fixed at $z = \pm 1$. Therefore $\phi$ must be continuous[@Chandra], so $a$ and $c$ can each be written in terms of $b$. Although $U(z)$ and $\phi$ are continuous at $z = \pm 1$, the discontinuities in $dU/dz$ lead to discontinuities in $d\phi/dz$. The jump conditions that determine these discontinuities are obtained by integrating Eq.  from $-1-\epsilon$ to $-1+\epsilon$ and from $1-\epsilon$ to $1+\epsilon$, then taking $\epsilon \to 0$: $$\label{linearjump} \lim_{\epsilon \to 0} (\omega \pm k)\frac{d\phi}{dz}\Big|_{\pm 1-\epsilon}^{\pm 1+\epsilon} \pm k\phi(\pm 1) = 0.$$ After inserting Eq. , these form two constraints on $b$ in terms of $\omega(k)$, which can be solved to obtain the dispersion relation, $$\label{dispersion} \omega = \pm \frac{e^{-2|k|}}{2}\sqrt{e^{4|k|}(1-2|k|)^2-1}.$$ ![Growth rate $\mathrm{Im}(\omega)$ and frequency $\mathrm{Re}(\omega)$ of the two modes for the inviscid shear layer. For $|k| \lesssim 0.64$ one mode is unstable and the other is stable, while for $|k| \gtrsim 0.64$ both modes are marginally stable.[]{data-label="dispersion_plot"}](Fig1.eps){width="16cm"} Figure \[dispersion\_plot\] shows how the growth rates and frequencies depend on wavenumber. Note that $\omega^2 < 0$ for $0 < |k| < k_c$, where $k_c \approx 0.64$. For $k > k_c$, we shall refer to the negative and positive branches of $\omega$ as $\omega_1$ and $\omega_2$ respectively, noting that the reality condition requires $\omega_j(-k) = \omega_j^*(k)$. For $|k| < k_c$, we choose $\omega_1$ to be the unstable root and $\omega_2$ the stable one. Because $b$ depends on $\omega$ through Eq.  and the eigenmode structure $\phi(z)$ depends on $b$ through Eq. , the two solutions $\omega_j$ correspond to two different eigenmodes $\phi_j(z)$. We identify $b_j$ and $\phi_j$ as the $b$ and $\phi$ corresponding to $\omega_j$. The eigenmodes are then $$\label{solved phi} \phi_j(k,z) = \begin{cases} \left(e^{2|k|}+b_j\right)e^{-|k|z} & z > 1 \\ e^{|k|z}+b_je^{-|k|z} & -1 < z < 1 \\ \left(1+b_je^{2|k|}\right)e^{|k|z} & z < -1, \end{cases}$$ where $$\label{bj} b_j = e^{2|k|}\frac{2|k|(\omega_j+k)-k}{k}$$ satisfies $b_1(k) = b_2(-k) = b_2^*(k)$ for $|k|<k_c$, and $b_j(k) = b_j(-k) = b_j^*(k)$ for $|k|>k_c$. ![Equilibrium (left column) compared with velocity profiles of the unstable $\phi_1$ (middle column) and the stable $\phi_2$ (right column) at wavenumbers $k = 0.4$ (top row) and $k=1$ (bottom row) plotted over one wavelength in $x$ and from $z=-2$ to $z=2$. Streamlines are plotted with color indicating flow speed. The first row is in the unstable range, where $\phi_1$ grows exponentially while $\phi_2$ decays exponentially. The second row is a marginally stable wavenumber, where both $\phi_1$ and $\phi_2$ oscillate without any growth \[see Fig. \[dispersion\_plot\]\].[]{data-label="eigenmode_plot"}](Fig2.eps){width="16cm"} For $\omega^2<0$, the eigenmodes are nearly identical but satisfy $\phi_1(k,z) = \phi^*_2(k,z)$. Figure \[eigenmode\_plot\] shows the flows corresponding to these eigenmodes for four wavenumbers sampling the unstable and stable ranges. Previous work has shown that the physical mechanisms for instability of $\phi_1$ and stability of $\phi_2$ can be understood in terms of resonant vorticity waves in both the hydrodynamic[@Baines] and MHD[@Heifetz] systems. In standard descriptions of turbulence and quasilinear transport calculations, it is common practice to neglect stable modes given their exponential decay from a small initial value. In this paper we account for the nonlinear drive of the stable mode by the unstable mode and investigate its impact on the evolution of the system. Eigenmode Projection ==================== In previous calculations of stable mode excitation[@Terry2006; @Makwana], fluctuations from equilibrium were represented by a vector $\mathbf{f}(\mathbf{k},t)$ whose components were the Fourier-transformed dynamical quantities. Because the systems were homogeneous, the linearized, Fourier-transformed PDEs became ODEs of the form $\dot{\mathbf{f}} = \mathcal{D}\mathbf{f}$ with the dynamics at each wavenumber $\mathbf{k}$ linearly decoupled. Thus, the eigenmodes of the system were the eigenvectors $\mathbf{f}_j$ of the $N\times N$ matrix $\mathcal{D}$, and arbitrary states could be expanded as linear combinations of the eigenvectors: $$\label{2006 expansion} \mathbf{f}(\mathbf{k},t) = \sum_{j=1}^{N}\beta_j(\mathbf{k},t)\mathbf{f}_j(\mathbf{k},t),$$ where $\beta_j(\mathbf{k},t)$ is the component of $\mathbf{f}$ in the eigenmode basis. Also called eigenmode amplitudes, the functions $\beta_j$ are not specified by the solutions of the linearized equations except through an initial condition. Under linear evolution the stable modes subsequently decrease to insignificance. However, the full nonlinear ODEs can readily be written in terms of the eigenmodes by substituting the eigenmodes for the dynamical quantities using Eq. . From there, separate equations for each $\dot{\beta}_j$ can be derived. These equations for $\dot{\beta}_j$ are equivalent to the original PDEs, but they describe the nonlinear evolution of the system in terms of the eigenmode amplitudes. We refer to this process, both the expansion of the perturbations and the manipulation of their governing equations, as an eigenmode decomposition. The equations for $\dot{\beta}_j$ provide powerful insight into the system. The nonlinearities that couple the dynamical fields at different scales become nonlinearities that couple eigenmodes at different wavelengths. Thus, it was shown (and borne out by many simulations[@Terry2006]) that despite decaying in the linear regime, the stable modes are nonlinearly driven by the unstable modes. In these previous calculations, the homogeneous nature of the system made the set of linear eigenmodes a complete basis: at every time $t$ and wavevector $\mathbf{k}$, the state vector $\mathbf{f}$ could be expanded in a basis of the eigenmodes \[i.e. Eq. \]. Due to the inhomogeneity of the present system, the linear solutions are not simply Fourier modes in $z$, so this system does not readily lend itself to the vector representation of Ref. [@Terry2006]. Moreover, Eq.  admits only two eigenmodes which are expected not to span arbitrary perturbations that satisfy the boundary conditions[@Case]. So the true state of the system cannot be written exactly in the form of Eq.  with $N = 2$. In order to properly describe the evolution of the system given an arbitrary initial condition, the system could be expanded in appropriate orthogonal polynomials or investigated as an initial value problem with additional time-dependent basis functions that are linear solutions of the problem. Previous work has demonstrated that for inviscid shear flows, the initial value calculation leads to the “discrete" eigenmodes with time-dependence $\exp[i\omega t]$ described in the previous section, and an additional set of “continuum" modes[@Case]. These continuum modes either oscillate with frequency $k$ or decay algebraically. For the present calculation we only consider perturbations that can be expressed as linear combinations of the two discrete eigenmodes $\phi_1$ and $\phi_2$, representing a truncation of the complete system. If we are able to demonstrate a significant impact from $\phi_2$, that suffices to demonstrate the importance of stable modes, relative to existing models that only consider the unstable mode. By focusing on perturbations that are linear combinations of $\phi_1$ and $\phi_2$ (i.e. limiting ourselves to the subspace spanned by $\phi_1$ and $\phi_2$), the vector representation and invertible linear transformation between the state of the system and the eigenmode amplitudes of Ref. [@Terry2006] can be recovered. Consequently, the governing Eq.  can be manipulated to derive nonlinear equations that describe the evolution of the eigenmode amplitudes and their interactions. The method relies on the jump conditions given in Eq. . Since the jump conditions for one eigenmode differ from those for the other eigenmode, one can form an invertible map between the discontinuity of $d\phi/dz$ at each interface and the amplitude of each eigenmode. Additionally, because there are two jump conditions that will serve as our dynamical quantities, only the two eigenmodes of the previous section are needed to construct an invertible map between eigenmodes and dynamical quantities. To derive equations describing the nonlinear interaction between the eigenmodes, we start by deriving nonlinear jump conditions. First, let $\hat{\phi}(k,z,t) = \mathcal{F}[\Phi(x,z,t)]$ be the Fourier transformed stream function, and assume $$\label{phihat combo} \hat{\phi}(k,z,t) = \beta_1(k,t)\phi_1(k,z) + \beta_2(k,t)\phi_2(k,z).$$ The nonlinear jump conditions are obtained by performing the same steps that led to Eq.  without dropping nonlinear terms (and explicitly taking the Fourier transform rather than assuming normal modes). Taking the Fourier transform and integrating from $\pm 1 - \epsilon$ to $\pm 1 + \epsilon$ with $\epsilon \to 0$ yields $$\label{deltahat} \frac{\partial}{\partial t}\hat{\Delta}_{\pm} \pm ik\hat{\Delta}_{\pm} \pm ik\hat{\phi}(k,\pm 1) + \lim\limits_{\epsilon\to 0}ik\int \limits_{-\infty}^{\infty}\frac{dk'}{2\pi}\left[\frac{d}{dz}\hat{\phi}(k',z)\frac{d}{dz}\hat{\phi}(k'',z)\right]_{\pm 1-\epsilon}^{\pm 1+\epsilon} = 0,$$ where $k'' \equiv k-k'$, while $$\begin{aligned} \hat{\Delta}_{\pm}(k,t) &\equiv \lim\limits_{\epsilon\to 0}\left[ \frac{d}{dz}\hat{\phi}(k,\pm 1 + \epsilon,t) - \frac{d}{dz}\hat{\phi}(k,\pm 1 - \epsilon,t)\right]\\ &= \beta_1(k,t)\Delta_{\pm 1}(k) + \beta_2(k,t)\Delta_{\pm 2}(k)\end{aligned}$$ and $$\Delta_{\pm j}(k) \equiv \lim\limits_{\epsilon\to 0}\left[ \frac{d}{dz}\phi_j(k,\pm 1 + \epsilon) - \frac{d}{dz}\phi_j(k,\pm 1 - \epsilon)\right]$$ are the discontinuities in $d\hat{\phi}/dz$ and $d\phi_j/dz$ at $z = \pm 1$. With $\hat{\phi}$ given by Eq.  and $\phi_j$ given by Eq. , one can show that $$\phi_j(k,1) = \frac{-\Delta_{+ j}}{2|k|} - \frac{\Delta_{- j}}{2|k|e^{2|k|}},$$ and $$\phi_j(k,-1) = \frac{-\Delta_{+ j}}{2|k|e^{2|k|}} - \frac{\Delta_{- j}}{2|k|}.$$ The $\hat{\phi}(k,\pm 1)$ term in Eq.  can then be written in terms of $\hat{\Delta}_{\pm}$ to yield $$\label{deltahat corrected} \frac{\partial}{\partial t}\begin{pmatrix}\hat{\Delta}_+\\\hat{\Delta}_-\end{pmatrix} = \mathcal{D}\begin{pmatrix}\hat{\Delta}_+\\\hat{\Delta}_-\end{pmatrix} + \begin{pmatrix}N_+\\N_-\end{pmatrix},$$ with $$\label{Dmatrix} \mathcal{D} = ik\begin{pmatrix} \frac{1}{2|k|} - 1 & \frac{e^{-2|k|}}{2|k|}\\ \frac{-e^{-2|k|}}{2|k|} & -\frac{1}{2|k|} + 1\\ \end{pmatrix},$$ and $N_{\pm}$ representing the nonlinearities in Eq. . Note that taking $N_{\pm} \to 0$ and $\partial / \partial t \to i\omega$ reduces this to the linear system solved in the previous section. We now have all of the necessary tools to treat this system in a manner similar to the previously-mentioned calculations[@Terry2006; @Makwana]. Using our definitions for $\hat{\Delta}_{\pm}$ and $\Delta_{\pm j}$, the $z$-derivative of Eq.  evaluated between $z = \pm 1 + \epsilon$ and $z = \pm 1 - \epsilon$ with $\epsilon \to 0$ is $$\label{Mdef} \begin{pmatrix}\hat{\Delta}_+\\\hat{\Delta}_-\end{pmatrix} = \mathbf{M}\begin{pmatrix}\beta_1\\\beta_2\end{pmatrix},$$ where $$\label{Mmatrix} \mathbf{M} = \begin{pmatrix} \Delta_{+ 1} & \Delta_{+ 2}\\ \Delta_{- 1} & \Delta_{- 2}\\ \end{pmatrix} = -2|k|e^{|k|}\begin{pmatrix} 1 & 1\\ b_1 & b_2\\ \end{pmatrix},$$ and $b_j$ is given in Eqn. . Equation is equivalent to Eq. : for this calculation, the dynamical quantities that we use to specify the state of the system are $\hat{\Delta}_{\pm}$, and their eigenmode structure is given by the columns of the matrix $\mathbf{M}$. The governing nonlinear PDE, Eq.  has been rewritten as a system of nonlinear ODEs, Eq. . The linearized system of ODEs (Eq.  with $N_{\pm} \to 0$) can be diagonalized: substituting $\hat{\Delta}_{\pm}$ for $\beta_j$ via Eq.  and multiplying by $\mathbf{M}^{-1}$ on the left gives $$\label{Leigenmode} \begin{pmatrix}\dot{\beta}_1\\\dot{\beta}_2\end{pmatrix} = \mathbf{M}^{-1}\mathcal{D}\mathbf{M}\begin{pmatrix}\beta_1\\\beta_2\end{pmatrix},$$ where the matrix $\mathbf{M}^{-1}\mathcal{D}\mathbf{M}$ is diagonal with entries $i\omega_j$. The nonlinear interactions between the eigenmodes can now be investigated. Applying the steps that led to Eq.  to the full, nonlinear Eq.  yields $$\label{NLeigenmode} \begin{pmatrix}\dot{\beta}_1\\\dot{\beta}_2\end{pmatrix} = \mathbf{M}^{-1}\mathcal{D}\mathbf{M}\begin{pmatrix}\beta_1\\\beta_2\end{pmatrix} + \mathbf{M}^{-1}\begin{pmatrix}N_+\\N_-\end{pmatrix},$$ where, again, $N_{\pm}$ are the nonlinearities in Eq. . Using Eq.  and the forms for $\phi_j$ given by Eq. , $N_{\pm}$ can be written in terms of products of the form $\beta_i \beta_j$ with $i,j$ each taking values $1,2$. Equation then becomes $$\label{unstablemode} \begin{split} \dot{\beta}_1(k) = i\omega_1(k)\beta_1(k) + \int \limits_{-\infty}^{\infty}\frac{dk'}{2\pi}\bigg[ &C_1(k,k') \beta_1(k')\beta_1(k'') + C_2(k,k') \beta_1(k')\beta_2(k'')\\ + &C_3(k,k') \beta_1(k'')\beta_2(k') + C_4(k,k')\beta_2(k')\beta_2(k'')\bigg], \end{split}$$ $$\label{stablemode} \begin{split} \dot{\beta}_2(k) = i\omega_2(k)\beta_2(k) + \int \limits_{-\infty}^{\infty}\frac{dk'}{2\pi}\bigg[ &D_1(k,k')\beta_1(k')\beta_1(k'') + D_2(k,k')\beta_1(k')\beta_2(k'')\\ + &D_3(k,k')\beta_1(k'')\beta_2(k') + D_4(k,k')\beta_2(k')\beta_2(k'')\bigg]. \end{split}$$ The coefficients $C_j,D_j$ arise from writing the nonlinearities $N_{\pm}$ in the basis of the linear eigenmodes, so their functional forms include information about both the linear properties of the system and the nonlinearities $N_{\pm}$. The exact expressions for $C_j, D_j$ are given in the Appendix, where it is shown that $C_2(k,k') = C_3(k,k-k')$, so that the $C_3$ integral is equal to the $C_2$ integral. Equations and are equivalent to Eq. , but they directly show how $\beta_1$ and $\beta_2$ interact. An analogy can be made here to the use of Els[ä]{}sser variables in incompressible, homogeneous MHD turbulence, which are a familiar example of an eigenmode decomposition that makes explicit the nonlinear interaction of the linear eigenmodes. The linearized equations have as solutions counterpropagating, noninteracting waves of the form $\mathbf{z}^{\pm} = \mathbf{v} \pm \mathbf{b}/(4\pi\rho_0)^{1/2}$. Expressing the nonlinear equations in terms of $\mathbf{z}^{\pm}$, the nonlinearity in the equation for $\partial \mathbf{z}^{\pm}/\partial t$ is $\mathbf{z}^{\mp}\cdot\nabla\mathbf{z}^{\pm}$, which describes the nonlinear interactions between linearly noninteracting modes. In the present calculation, the linearly noninteracting $\phi_1, \phi_2$ are comparable to $\mathbf{z}^{\pm}$, and the terms proportional to $\beta_1(k')\beta_2(k'')$ and $\beta_1(k'')\beta_2(k')$ are comparable to $\mathbf{z}^{\mp}\cdot\nabla\mathbf{z}^{\pm}$. However, unlike the $\mathbf{z}^{\pm}$ equations, the $\dot{\beta}_j$ equations include other nonlinear terms that are proportional to $\beta_1(k')\beta_1(k'')$ and $\beta_2(k')\beta_2(k'')$. If all of the nonlinearities are zero except for the $C_1$ term, then the evolution of $\beta_1(k)$ is just a combination of its linear drive $i\omega_1(k)$ and three-wave interactions with $\beta_1(k')$ and $\beta_1(k-k')$, allowing $\phi_1$ to saturate through a Kolmogorov-like cascade to smaller scales. This is effectively the assumption of standard quasilinear calculations of momentum transport, where only $\phi_1,\omega_1$ are considered. Figure \[CDplot\] shows some of the nonlinear coupling coefficients plotted over a range of wavenumbers. Since $D_1, C_2$, and $C_3$ are not identically zero, there is some interaction between the eigenmodes. Systems where such interactions have been identified are all gyroradius-scale, quasihomogeneous systems driven by drift-wave instabilities[@Terry2006; @Makwana]. Equations and represent a demonstration that these interactions occur for larger-scale, inhomogeneous plasmas and neutral fluids. ![Three of the eight nonlinear coupling coefficients in Eqs.  and , $C_1, C_2$, and $D_1$, evaluated over the most relevant scales. Color indicates absolute values of the coefficients. The coefficients are all roughly the same magnitude, indicating significant coupling between stable and unstable eigenmodes.[]{data-label="CDplot"}](Fig3.eps){width="16cm"} The Threshold Parameter ======================= By comparing the nonlinearities that transfer energy to stable modes with those that cause the Kolmogorov-like cascade of energy to small scales, one can investigate how important stable modes are in instability saturation. A quantity known as the threshold parameter $P_t$ has been used to evaluate the relative importance of the stable eigenmodes in situations where instability saturation is described by eigenmode-projected ODEs. The threshold parameter $P_t$ estimates the relative importance in saturation of the nonlinearities responsible for energy transfer to the stable mode versus the nonlinearity of the forward cascade[@Terry2006]. If $P_t$ is small compared to unity, it indicates that the instability saturates via a Kolmogorov-like transfer of energy to smaller scales, and only the term in Eq.  corresponding to the most unstable eigenmode needs to be included to accurately describe the system. If $P_t \gtrsim 0.3$, it was found that the transfer of energy from the unstable mode to other modes at similar scales is an important mechanism in saturation. In that case, additional terms in Eq.  must therefore be included[@Makwana]. The quantity $P_t$ is the ratio of the $C_1\beta_1\beta_2$ and $C_2\beta_1\beta_1$ terms in Eq.  and therefore includes information about both linear and nonlinear properties of the system. In previous work[@Terry2006; @Makwana], simplifying assumptions – such as treating growth rates $\gamma_j = -\mathrm{Im}(\omega_j)$ as independent of wavenumber – allowed the threshold parameter to be written as $$\label{Ptold} P_t = \frac{2 D_1 C_2}{C_1^2(2-\gamma_2/\gamma_1)}$$ for $\gamma_2<0$. This form of $P_t$ is useful because it illustrates how $P_t$ depends on different parameters of the system: the size of $P_t$ relative to unity is determined by the ratios $D_1C_2/C_1^2$ and $\gamma_1/\gamma_2$. When the former is small, stable modes are only weakly coupled to unstable modes and have little impact on saturation dynamics. When the latter is small, stable modes decay too quickly to achieve significant amplitude by the time the instability saturates unless $D_1C_2/C_1^2 \gg 1$ and compensates. Previous work evaluated this form of $P_t$ in several systems and found that whenever $P_t$ is at least a few tenths, energy transfer to stable modes becomes comparable to the energy injection rate of the instability[@Makwana]. Note that in the system considered here $|\gamma_1/\gamma_2| = 1$, and numerically evaluating $C_j,D_j$ shows that $D_1$ and $C_2$ are of the same order as or even larger than $C_1$ \[see Fig. \[CDplot\]\]. These features alone yield $P_t \approx 0.7$, which implies stable modes are important for KH saturation. Here we extend previous analyses of $P_t$ by including the full wavenumber dependence of $\gamma_j, C_j$, and $D_j$. Consider the evolution of the system from a small initial amplitude $\beta_i$. When amplitudes are small every nonlinear term is negligible, so the dynamics are linear with $\beta_2$ decaying and $\beta_1$ growing exponentially at every wavenumber. Eventually couplings in $\int (dk'/2\pi) D_1(k,k')\beta_1(k')\beta_1(k-k')$ dominate in Eq. . This occurs in the linear phase, before saturation, because nonlinearities dominate the decaying linear response of $\beta_2$ long before matching the growing linear response of $\beta_1$. Thus, Eq.  can be approximated as $$\label{parametricstable} \dot{\beta}_2(k) = i\omega_2(k)\beta_2(k) + \int \limits_{-\infty}^{\infty}\frac{dk'}{2\pi}D_1(k,k')\beta_1(k')\beta_1(k'').$$ Note that for these times $\beta_2 \ll \beta_1$ therefore the $D_1$ terms are the largest of the $D_j$ terms. Since the $C_j$ nonlinearities have not reached the amplitudes of the growing linear terms, $\beta_1$ can be approximated as $\beta_i \exp[i \omega_1 t]$. These approximations are referred to as the parametric instability approximations[@Terry2006]. Then Eq.  is solved by $$\label{parametricsolution} \beta_2(k,t) = \int \limits_{-\infty}^{\infty}\frac{dk'}{2\pi} \frac{D_1(k,k')\beta_i^2}{i\left(-\omega_2(k)+\omega_1(k')+\omega_1(k-k')\right)} \left[ e^{i(\omega_1(k')+\omega_1(k-k'))t} - e^{i\omega_2(k)t} \right] + \beta_i e^{i \omega_2(k) t}.$$ In assessing $P_t$ the above integral is only taken over unstable wavenumbers, as they drive $\beta_2$ more strongly than marginally stable modes. ![Nonlinear terms in Eq.  at saturation for $k=0.4$ and $\beta_i=0.01,$ with $k'$ and $k-k'$ ranging from $-0.6$ to $0.6$. The $C_1$ term is responsible for the Kolmogorov-like saturation of the instability by energy transfer to unstable modes at smaller wavelengths. The $C_2$ term represents the previously-neglected coupling between unstable modes at $k$ and $k'$ with stable modes at $k''$. The threshold parameter is evaluated by dividing the peak value of the $C_2$ term by the peak value of the $C_1$ term. Here we find $P_t \approx 6$, indicating that stable modes are important in KH saturation.[]{data-label="PTscan"}](Fig4.eps){width="16cm"} To evaluate $P_t$, the ratio of the largest $\beta_1\beta_2$ term and the largest $\beta_1\beta_1$ term in Eq.  is taken at the time of saturation $t_s$: $$\label{Ptnew} P_t = \left[\frac{\max|2C_2\beta_1(k')\beta_2(k'')|}{\max|C_1\beta_1(k')\beta_1(k'')|}\right]_{t=t_s},$$ where $t_s$ is defined as the time at which one of the nonlinearities in Eq.  reaches the same amplitude as the linear term. Figure \[PTscan\] shows the size of these terms at saturation for $k=0.4$ with an initial amplitude of $\beta_i = 0.01$. We choose $k=0.4$ because it is the most unstable wavenumber and is therefore the wavenumber of the most dominant unstable mode leading into saturation. From Fig. \[PTscan\], it is inferred that $P_t \approx 6$, indicating that even before the nominal saturation time energy transfer to stable modes has become as important to the saturation of the unstable mode at $k=0.4$ as the Kolmogorov-like transfer to unstable modes at other scales. In previous calculations of $P_t$, the parameter was independent of the initial amplitude $\beta_i$ (which is assumed to be the same for each $k$). However, in the above evaluation of $P_t$, we do find that it depends on $\beta_i$; for instance, reducing $\beta_i$ to $0.001$ yields $P_t \approx 15$. In previous calculations, the lack of dependence of $P_t$ on $\beta_i$ is an artifact of treating growth rates as independent of wavenumber[@Terry2006]. In considering Eq.  for the most unstable wavenumber, both $\beta_1(k')$ and $\beta_1(k'')$ were assumed to grow at the same rate as the most unstable mode, when in fact three-wave interactions require $k \neq k'$. When including wavenumber dependence, these nonlinear terms will necessarily grow at less than twice the peak growth rate. On the other hand, stable modes near $k=0$ can be driven by $D_1\beta_1\beta_1$ terms where one of the driving modes is near $k=0.4$ and the other is near $k=-0.4$. Thus, our inclusion of the wavenumber dependence of $\omega_j$ causes $\beta_2$ to grow large enough that Eq.  becomes invalid before saturation time. This makes the precise value of $P_t$ less meaningful, as the stable modes have grown so large that the approximations made in obtaining $P_t$ are invalid. However, the size of $\beta_2$ relative to $\beta_1$ and the comparable amplitudes of $C_2$ and $C_1$ imply $P_t \gtrsim 1$, and therefore $P_t \gtrsim 0.3$ is still well satisfied. The above nonlinear analysis demonstrates that energy transfer to stable modes is significant relative to energy transfer to smaller scales, modifying the usual understanding of instability saturation by a cascade to small scales. The analysis employs approximations, hence it is instructive to consider a second, complementary form of approximate nonlinear analysis based on a three-wavenumber truncation of Eqs.  and . Such a calculation complements the $P_t$ analysis because it makes different assumptions. The $P_t$ analysis makes parametric instability approximations when considering the evolution of $\beta_2$ (c.f. Eq. ), but samples a broad continuum of wavenumbers. On the other hand, a three-wavenumber truncation makes no assumptions about the evolution of the modes, and instead limits the system to only three wavenumbers that are evolved according to Eqs.  and . ![Time evolution of $|\beta_j(k,t)|$ for a three-wavenumber truncation with $k=0.3, k'=0.9$, and $k-k'=-0.6$. As expected from the $P_t$ analysis, the stable mode decays linearly, then is nonlinearly pumped to an amplitude that is comparable to the unstable mode.[]{data-label="ThreeModePlot"}](Fig5.eps){width="16cm"} The result of a three-wavenumber truncation is plotted in Fig. \[ThreeModePlot\], showing the time evolution of $\beta_j(k,t)$ obtained by solving Eqs.  and numerically with only interactions between $k=0.3$, $k'=0.9$, and $k-k'=-0.6$ considered. The linear growth phase of $\beta_1$ is clearly seen, as is the linear decay and nonlinear driving of $\beta_2$. The linear growth phase for $\beta_1$ ends with both eigenmodes reaching comparable amplitudes, consistent with the $P_t$ analysis. Once the stable mode reaches a value that is comparable to the unstable mode there is continuous exchange of energy between the two modes. The saturation levels slowly grow as $t \to \infty$. That can be understood as a consequence of the inviscid dynamics in a three-mode system, in that previous work has demonstrated that a necessary condition for bounded solutions to three-mode truncations is that the sum of the growth rates is negative[@Terry1982]. Without viscosity, the present system does not meet the necessary condition. Note that the time scale for nonlinear energy exchange is very short compared to the time scale of the saturation level increase, strongly suggesting that the nonlinearities of Eqs.  and conserve energy. This calculation demonstrates that the system can saturate by energy transfer to stable modes, and shows that the assumptions made regarding the growth of $\beta_1$ and $\beta_2$ in the $P_t$ analysis are reasonable. As an illustration of the effect of finite $\beta_2$ on the fluctuating flow, Fig. \[summed\_mode\_plot\] shows the flows arising from linear combinations of $\beta_1$ and $\beta_2$ with the weight of $\beta_2$ varied. The flow arising purely from the unstable mode is strikingly different from the flow that combines $\beta_1$ and $\beta_2$ with equal weights. Regions of hyperbolic flow appear to be less likely for the equally weighted combination, suggesting that secondary structure generation and cascading may be weakened when the stable mode is excited. This will be the subject of future research.\ ![Examples of superpositions of stable and unstable modes at $k=0.4$ plotted over one wavelength in $x$ and from $z=-2$ to $z=2$ (cf. Fig. \[eigenmode\_plot\]). In the right column, the unstable and stable modes have an equal contribution to the overall flow. In the top and bottom rows, the relative phase between the two modes is $+\pi$ and $-\pi$, respectively.[]{data-label="summed_mode_plot"}](Fig6.eps){width="16cm"} Momentum Transport ================== Reynolds stresses and the associated momentum transport due to unstable modes tend to broaden the original flow profile. Here we show that stable modes have the potential to reduce the broadening of the profile. The transport of momentum in the $x$ direction across the interface at $z=1$ is found by integrating the $x$-component of the divergence of the stress tensor $\tau_{ij} = \langle v_{1i} v_{1j} \rangle$ across the interface. Integrating $d\tau_{xz}/dz$ across the interface gives $$S = -\lim\limits_{\epsilon \to 0} \int \limits_{1-\epsilon}^{1+\epsilon}dz\langle v_{1x}v_{1z} \rangle = -\lim\limits_{\epsilon \to 0} \int \limits_{1-\epsilon}^{1+\epsilon}dz\frac{d}{dz}\langle \frac{d\Phi}{dz}\frac{\partial \Phi}{\partial x}\rangle$$ where $\langle \rangle$ denotes averaging in $x$, while $\mathbf{v}_1$ is the perturbed velocity. Taking $\Phi = \mathcal{F}^{-1}[\hat{\phi}]$ with $\hat{\phi} = \beta_1\phi_1 + \beta_2\phi_2$ gives $$\label{shear} \begin{split} S = \int \limits_{-\infty}^{\infty}\frac{dk}{2\pi}4k^2e^{2|k|} \bigg[ &\mathrm{Im}(\omega_1^*)|\beta_1|^2 + \mathrm{Im}(\omega_2^*)|\beta_2|^2\\ + &\mathrm{Im}\left[(\omega_2^*+k)\beta_1\beta_2^*\right] + \mathrm{Im}\left[(\omega_1^*+k)\beta_2\beta_1^*\right]\bigg]. \end{split}$$ When the stable modes are ignored, only the first term contributes to $S$. The coefficient $4k^2e^{2|k|}$ is positive, and Eq.  shows that $\mathrm{Im}(\omega^*_1) \leq 0$ and $\mathrm{Im}(\omega^*_2) = -\mathrm{Im}(\omega^*_1)$, indicating that the transport due to unstable modes alone is negative, and the second term acts against the first to reduce $|S|$. Clearly the amplitude of $\beta_2(k)$ relative to $\beta_1(k)$ has a significant impact on the momentum transport in this system. The relative phase between $\beta_2(k)$ and $\beta_1(k)$ determines the contribution of the last two terms. If $|\beta_2(k)| = |\beta_1(k)|$, then the first two terms cancel and the transport is entirely determined by the last two terms. Analysis of other systems shows there are situations where eigenmode cross correlations significantly affect transport[@Baver; @Terry2009]. To determine the actual properties of $S$, it is necessary to solve Eqs.  and for $\beta_j(k)$ and integrate Eq. , either analytically or numerically. This is beyond the scope of the present paper, but will be considered in the future. In lieu of such solutions, we construct an estimate of the ratio $|\beta_2(k)|/|\beta_1(k)|$ using the threshold parameter. In the previous section the threshold parameter was defined as the ratio of the maximum amplitudes of the $C_2$ terms and the $C_1$ terms in Eq.  at the onset of saturation. An estimate of $|\beta_2(k)|/|\beta_1(k)|$ in terms of $P_t$ is obtained by taking $$P_t \sim \frac{|2C_2\beta_1(k')\beta_2(k'')|}{|C_1\beta_1(k')\beta_1(k'')|} = 2\left|\frac{C_2}{C_1}\right|\frac{|\beta_2(k'')|}{|\beta_1(k'')|} \sim 2\left|\frac{\beta_2(k)}{\beta_1(k)}\right|.$$ While the threshold parameter estimates the relative amplitudes of the modes, it does not capture information about their cross-phase. Taking $\beta_2 = \beta_1 \exp[i\theta_{12}] P_t/2$ allows $S$ to be rewritten as $$\label{shearapprox} S = \int \limits_{-\infty}^{\infty}\frac{dk}{2\pi}4k^2e^{2|k|}|\beta_1|^2\left\{ \mathrm{Im}(\omega_1^*)\left(1 - \frac{P_t^2}{4}\right) + \frac{P_t}{2}\mathrm{Im}\left[\omega_1^*(2i\sin(\theta_{12}))\right]\right\}.$$ Due to the form of $\omega_1$ \[see Fig. \[dispersion\_plot\]\], the first term is only nonzero for $|k| \lesssim 0.64$, and the second term is only nonzero for $|k| \gtrsim 0.64$. It is clear that $P_t \sim 1$ reduces the magnitude of the first term, while the contribution of the second term to $S$ depends significantly on the cross-phase $\theta_{12}$ between the eigenmodes. Having shown that momentum transport can be affected by stable mode activity, we next summarize the main findings of this paper. Conclusion ========== Shear-flow instabilities are widely studied due to their potential to drive turbulence in systems where the turbulent transport of momentum, particles, and heat are of interest. While the linear regime of these instabilities are generally well-understood, saturation and the resulting nonlinear flows are difficult to model. We have presented a nonlinear analysis of an unstable shear layer with piecewise-linear shear flow, showing that the complex conjugate stable linear eigenmode is excited nonlinearly and strongly affects saturation. This result is significant because it represents the first demonstration that nonlinear excitation of linearly stable modes is an important aspect of saturation in global-scale unstable plasma and hydrodynamic systems. Previous studies were limited to quasihomogeneous systems on gyroradius scales[@Terry2006; @Makwana]. A critical aspect of this work is the development of a mapping technique that allows analytical saturation analyses derived for spatially homogeneous systems to be applied to the strongly inhomogeneous situation of shear flow instability. Assuming the flow is a linear combination of the linear eigenmodes allows the global state of the system to be described in terms of its behavior at the edges of the shear layer (as is also done to determine the dispersion relation). The nonlinearity, originally written in terms of flow components and their spatial derivatives, is then written in terms of the eigenmodes to demonstrate that unstable modes nonlinearly pump stable modes. This allows the eigenfunctions of this system to be treated similarly to the eigenvectors of previous systems. Using a parameter that quantifies the threshold for a stable mode to impact saturation, we have estimated the impact of stable modes on instability saturation and found it to be significant. Analysis of the flow associated with stable modes indicates that, at the predicted saturation levels, the fluctuating flow undergoes significant topological changes relative to flows characterized by the unstable mode alone. Such changes may affect the propensity for the turbulent flow structure to generate secondary structures through transient amplification and other processes. Because the system described here is inviscid, this work indicates that stable modes have the potential to modify the evolution of instabilities even when they are not subject to dissipation. Finally, we consider the contribution of stable modes to momentum transport and give an estimate in terms of the threshold parameter, demonstrating that stable modes can significantly reduce the broadening of the shear layer, thereby counteracting the effect of the unstable modes. One may similarly expect that stable modes can affect other transport channels such as matter entrainment and heat transport. This line of inquiry will be left for future investigations. The authors would like to thank F. Waleffe for valuable discussions and insights. Partial support for this work was provided by the Wisconsin Alumni Research Foundation and the U S Department of Energy, Office of Science, Fusion Energy Sciences, under award No. DE-FG02-89ER53291. Coupling Coefficients ===================== In Eqs.  and , the nonlinear coupling coefficients $C_j,D_j$, which are obtained by expressing the nonlinearities of Eq.  in terms of the eigenmode amplitudes $\beta_j$, are as follows: $$\begin{split} C_1 &= \alpha \left[ ( b_2b_1' + b_1'' ) e^{2|k''|} + ( b_2b_1'' + b_1' ) e^{2|k'|} \right]\\ C_2 &= \alpha \left[ ( b_2b_1' + b_2'' ) e^{2|k''|} + ( b_2b_2'' + b_1' ) e^{2|k'|} \right]\\ C_3 &= \alpha \left[ ( b_2b_2' + b_1'' ) e^{2|k''|} + ( b_2b_1'' + b_2' ) e^{2|k'|} \right]\\ C_4 &= \alpha \left[ ( b_2b_2' + b_2'' ) e^{2|k''|} + ( b_2b_2'' + b_2' ) e^{2|k'|} \right]\\ D_1 &= -\alpha \left[ ( b_1b_1' + b_1'' ) e^{2|k''|} + ( b_1b_1'' + b_1' ) e^{2|k'|} \right]\\ D_2 &= -\alpha \left[ ( b_1b_1' + b_2'' ) e^{2|k''|} + ( b_1b_2'' + b_1' ) e^{2|k'|} \right]\\ D_3 &= -\alpha \left[ ( b_1b_2' + b_1'' ) e^{2|k''|} + ( b_1b_1'' + b_2' ) e^{2|k'|} \right]\\ D_4 &= -\alpha \left[ ( b_1b_2' + b_2'' ) e^{2|k''|} + ( b_1b_2'' + b_2' ) e^{2|k'|} \right], \end{split}$$ where $$\alpha = \frac{ik|k'||k''|e^{-|k|-|k'|-|k''|}}{2|k|(b_1-b_2)},$$ with $b_j' \equiv b_j(k')$ and $b_j'' \equiv b_j(k'')$. For convenience, the definition of $b_j(k)$ is repeated here: $$b_j = e^{2|k|}\frac{2|k|(\omega_j+k)-k}{k}.$$ Notice that $\alpha(k,k') = \alpha(k,k-k')$ and $C_3(k,k') = C_2(k,k-k')$. Thus, changing the integration variable to $k'' = k-k'$ in the $C_3$ integral yields $$\begin{split} \int_{-\infty}^{\infty}\frac{dk'}{2\pi}C_3(k,k')\beta_1(k'')\beta_2(k') &= \int_{-\infty}^{\infty}\frac{dk'}{2\pi}C_2(k,k'')\beta_1(k'')\beta_2(k')\\ &= \int_{-\infty}^{\infty}\frac{dk''}{2\pi}C_2(k,k'')\beta_1(k'')\beta_2(k-k'')\\ &= \int_{-\infty}^{\infty}\frac{dk'}{2\pi}C_2(k,k')\beta_1(k')\beta_2(k-k'), \end{split}$$ so the $C_3$ and $C_2$ integrals in Eq.  are identical. [9]{} K.A. Browning, Quart. J. Roy. Met. Soc. **97**, 283 (1971). A.M. Dimits, T.J. Williams, J.A. Byers, and B.I. Cohen, Phys. Rev. Lett. **77**, 71 (1996). B.N. Rogers, W. Dorland, and M. Kotschenreuther, Phys. Rev. Lett. **85**, 5336 (2000). E.B. Churchwell, Astrophys. J. **479**, L59 (1997). 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--- author: - Chen Sun - Austin Myers - Carl Vondrick - Kevin Murphy - Cordelia Schmid bibliography: - 'egbib.bib' title: 'VideoBERT: A Joint Model for Video and Language Representation Learning' ---

--- abstract: 'We prove new bilinear estimates for the $X^{s, b}_{\pm}({\mathbb{R}}^2)$ spaces which are optimal up to endpoints. These estimates are often used in the theory of nonlinear Dirac equations on ${\mathbb{R}}^{1+1}$. The proof of the bilinear estimates follows from a dyadic decomposition in the spirit of Tao [@Tao2001] and D’Ancona, Foschi, and Selberg [@D''Ancona2010]. As an application, by using the $I$-method of Colliander, Keel, Staffilani, Takaoka, and Tao, we extend the work of Tesfahun [@Tesfahun2009] on global existence below the charge class for the Dirac-Klein-Gordon equation on ${\mathbb{R}}^{1+1}$.' author: - | Timothy Candy\ *D M, U E\ E EH9 3JE, U K\ E: T.L.C@* title: 'Bilinear Estimates and Applications to Global Well-Posedness for the Dirac-Klein-Gordon equation on ${\mathbb{R}}^{1+1}$' --- Introduction ============ We consider the problem of proving bilinear estimates in the Bourgain-Klainerman-Machedon type spaces $X^{s, b}_{\pm}$ on ${\mathbb{R}}^2$, where we define the spaces $X^{s, b}_\pm$ via the norm $$\| \psi \|_{X^{s, b}_\pm} = \big\| {\langle}\tau \pm \xi{\rangle}^b {\langle}\xi {\rangle}^s \widetilde{\psi}(\tau, \xi) \|_{L^2_{\tau, \xi}({\mathbb{R}}^2)}$$ with ${\langle}\cdot {\rangle}= \sqrt{ 1+ |\cdot|^2}$. These spaces have been used in the low regularity theory of various nonlinear Dirac equations in one space dimension, [@Machihara2005; @Selberg2010b], as well as the Dirac-Klein-Gordon (DKG) system [@Pecher2006; @Selberg2008]. Though recently, product Sobolev spaces based on the null coordinates $x\pm t$ have also proved useful [@Candy2010; @Machihara2010]. In applications of the $X^{s, b}_\pm$ spaces to low regularity well-posedness, we often require product estimates of the form $$\label{standard form} \| u v\|_{X^{ -s_1, -b_1}_{\pm_1}} {\lesssim}\| u \|_{X^{s_2, b_2}_{\pm_2}} \| v \|_{X^{s_3, b_3}_{\pm_3}}$$ where $s_j, b_j \in {\mathbb{R}}$ and $\pm_j$ are independent choices of $\pm$. A number of estimates of this form, for specific values of $s_j$ and $b_j$, have appeared previously in the literature [@Machihara2005; @Selberg2008; @Selberg2010b]. The case where $\pm_1 = \pm_2 = \pm_3$ is not particularly interesting, as a simple change of variables reduces (\[standard form\]) to two applications of the 1-dimensional Sobolev product estimate $$\| f g\|_{H^{-s_1}({\mathbb{R}})} {\lesssim}\| f\|_{H^{s_2}({\mathbb{R}})} \| g\|_{H^{s_3}({\mathbb{R}})}.$$ Thus leading to the conditions[^1] $$\label{product sobolev cond 1} b_j + b_k>0, \qquad b_1 + b_2 + b_3>\frac{1}{2}$$ and $$\label{product sobolev cond 2} s_j + s_k>0, \qquad s_1 + s_2 + s_3>\frac{1}{2}$$ where $j \neq k$. On the other hand, if we have $\pm_1=\pm_2=\pm$ and $\pm_3 = \mp$, then we can make significant improvements over (\[product sobolev cond 2\]). This observation allows one to exploit the null structure that is often found in nonlinear hyperbolic systems in one dimension, see for instance [@Selberg2010b].\ To state our first result we use the following conventions. For a set of real numbers $\{a_1, a_2, a_3\}$, we let $a_{max}=\max_{i} a_i$, $a_{min}=\min_i a_i$, and use $a_{med}$ to denote the median. If $a \in {\mathbb{R}}$ then we define $$a_+ = \begin{cases} a \qquad &a>0 \\ 0 &a{\leqslant}0. \end{cases}$$ We state our product estimate in the dual form. \[thm main X\^sb esimate\] Let $s_j$, $b_j \in {\mathbb{R}}$, $j=1, 2, 3$ satisfy $$b_1+ b_2 + b_3 >\frac{1}{2},\qquad b_j + b_k >0, \qquad (j\neq k) \label{thm main X^sb esitimate - cond on b}$$ and for $k \in \{ 1, 2\}$ $$\label{thm main X^sb esitimate - cond on s} \begin{split} s_1 + s_2 &{\geqslant}0, \\ s_k + s_3 &> -b_{min},\\ s_k +s_3&> \frac{1}{2} - b_1 - b_2 - b_3, \\ s_1 + s_2 + s_3 &> \frac{1}{2} - b_3,\\ s_1 + s_2 + s_3 &> \Big(\frac{1}{2} - b_{max}\Big)_+ + \Big(\frac{1}{2} - b_{med}\Big)_+ - b_{min}. \end{split}$$ Then $$\label{thm main X^sb est - main trilinear est} \Big|\int_{{\mathbb{R}}^2} \Pi_{j=1}^3 \psi_j(t, x) dx dt\Big| {\lesssim}\| \psi_1\|_{X^{s_1, b_1}_\pm} \| \psi_2 \|_{X^{s_2, b_2}_\pm} \| \psi_3 \|_{X^{s_3, b_3}_\mp}.$$ Moreover the conditions (\[thm main X\^sb esitimate - cond on b\]) and (\[thm main X\^sb esitimate - cond on s\]) are sharp up to equality. There are cases where we can allow equality in (\[thm main X\^sb esitimate - cond on b\]) or (\[thm main X\^sb esitimate - cond on s\]), for instance the case $$s_1=s_2=s_3=0, \qquad b_1=0,\qquad b_2=b_3=\frac{1}{2}+\epsilon$$ holds [@Selberg2008 Corollary 1]. We have not attempted to list or prove the endpoint cases here, as this would significantly complicate the statement of Theorem \[thm main X\^sb esimate\]. Additionally, Theorem \[thm main X\^sb esimate\] is sufficient for our intended application to global well-posedness for the Dirac-Klein-Gordon equation. Define the Wave-Sobolev spaces $H^{s, b}$ by using the norm $$\| \psi\|_{H^{s, b}} = \big\| {\langle}|\tau| - |\xi| {\rangle}^b {\langle}\xi {\rangle}^s \widetilde{\psi}(\tau, \xi) \|_{L^2_{\tau, \xi}({\mathbb{R}}^2)}.$$ Then as a simple corollary to Theorem \[thm main X\^sb esimate\] we can replace one of the $X^{s, b}_{\pm}$ norms on the righthand side of (\[thm main X\^sb est - main trilinear est\]) with a $H^{s, b}$ norm. \[cor wave-sobolev and X\^sb estimate\] Let $r, s_1, s_2, b_j \in {\mathbb{R}}$, $j=1, 2, 3$ satisfy $$b_1+ b_2 + b_3>\frac{1}{2}, \qquad b_j + b_k>0, \qquad (j \neq k)$$ and for $k \in \{ 1, 2\}$ $$\begin{aligned} s_k +r &{\geqslant}0, \\ s_k + r &>-b_{min} \\ s_1 + s_2 &> -b_{min},\\ s_1 + s_2 &> \frac{1}{2} - b_1 - b_2 - b_3, \\ s_1 + s_2 + r &> \frac{1}{2} - b_k ,\\ s_1 + s_2 + r&> \Big(\frac{1}{2} - b_{max}\Big)_+ + \Big(\frac{1}{2} - b_{med}\Big)_+ - b_{min}. \end{aligned}$$ Then $$\Big|\int_{{\mathbb{R}}^2} \Pi_{j=1}^3 \psi_j(t, x) dx dt\Big| {\lesssim}\| \psi_1 \|_{X^{s_1, b_1}_+} \| \psi_2 \|_{X^{s_2, b_2}_-}\| \psi_3\|_{H^{r, b_3}}.$$ We decompose $\psi_3$ into the regions $\{ (\tau, \xi) \in {\mathbb{R}}^{1+1} \,| \, \pm \tau \xi {\geqslant}0 \}$ and observe that on the first region ${\langle}|\tau| - |\xi| {\rangle}= {\langle}\tau - \xi {\rangle}$ while in the second region $ {\langle}|\tau| - |\xi| {\rangle}= {\langle}\tau + \xi {\rangle}$. The corollary now follows from two applications of Theorem \[thm main X\^sb esimate\]. This result should be compared to the similar estimates contained in [@Selberg2008] and [@Tesfahun2009]. Also we note that the decomposition used in the proof of Corollary \[cor wave-sobolev and X\^sb estimate\] can be used to give bilinear estimates in the Wave-Sobolev spaces $H^{r, b}$, thus giving an alternative (though closely related) proof of Theorem 7.1 in [@D'Ancona2010b] (up to endpoints). The second main result contained in this article concerns the global existence problem for the DKG equation on ${\mathbb{R}}^{1+1}$. The DKG equation can be written as $$\label{DKG general form} \begin{split} \big( \gamma^0 {\partial}_t + \gamma^1 {\partial}_x\big) \psi &= -i M \psi + i \phi \psi \\ \big( - \Box + m^2\big)\phi &= {\langle}\gamma^0 \psi, \psi {\rangle}_{{\mathbb{C}}^2} \end{split}$$ with initial data $$\label{DKG general initial data} \psi(0) = \psi_0 \in H^s, \qquad \phi(0) = \phi_0 \in H^r, \qquad {\partial}_t \phi(0) = \phi_1 \in H^{r-1}$$ for some values of $s, r \in {\mathbb{R}}$. The d’Alembertian is defined by $\Box= - {\partial}_t^2 + {\partial}_x^2$ and we take the standard representation of the Dirac matrices $$\gamma^0 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \gamma^{1} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.$$ The Dirac spinor $\psi \in {\mathbb{C}}^2$, and the real-valued scalar field $\phi \in {\mathbb{R}}$, are functions of $(t, x) \in {\mathbb{R}}^{1+1}$. The notation ${\langle}\cdot, \cdot{\rangle}_{{\mathbb{C}}^2}$ refers to the standard inner product on ${\mathbb{C}}^2$, and $m, M \in {\mathbb{R}}$ are constants. There are two main features of the DKG equation (\[DKG general form\]) which we wish to highlight here. The first feature concerns the conservation of charge which can be stated as follows: if $(\psi, \phi)$ is a smooth solution to (\[DKG general form\]) with sufficient decay at infinity, then for all times $t \in {\mathbb{R}}$ we have $$\label{conservation of charge} \| \psi(t) \|_{L^2} = \| \psi(0) \|_{L^2}.$$ The conservation of charge is crucial in controlling the global behaviour of the solution $(\psi, \phi)$. The second feature we would like to note is that the nonlinearity in the DKG equation has null structure. Roughly speaking, this refers to the fact that the nonlinear terms in (\[DKG general form\]) behave significantly better than generic products. The null structure is a crucial component in the low regularity existence theory for the DKG equation and has been used by a number of authors [@Bournaveas2006; @Fang2004a; @Machihara2007b; @Pecher2006; @Selberg2008]. The observation that null structure can be used to improve local existence results for nonlinear wave equations is due to Klainerman and Machedon in [@Klainerman1993].\ The question of local well-posedness (LWP) for the DKG equation was first considered by Chadam [@Chadam1973]. Subsequently, much progress has been made by numerous authors [@Bournaveas2006; @Fang2004a; @Machihara2007b; @Pecher2006; @Selberg2008]. The best result to date is due to Machihara, Nakanishi, and Tsugawa [@Machihara2010] where it was shown that (\[DKG general form\]) with initial data (\[DKG general initial data\]) is locally well-posed provided $$s>-\frac{1}{2}, \qquad |s| {\leqslant}r {\leqslant}s+1.$$ Moreover, this region is essentially sharp, except possibly at the endpoint $s= - \frac{1}{2}$. More precisely, outside this region the solution map is either ill-posed, or fails to be twice differentiable, see [@Machihara2010] for a more precise statement.\ In the current article we are interested in the minimum regularity required on the initial data (\[DKG general initial data\]) to ensure that the corresponding local in time solution $(\psi, \phi)$ to (\[DKG general form\]) can be extended globally in time. Global well-posedness (GWP) in the high regularity case $s=r=1$ was first proven by Chadam [@Chadam1973], this was then progressively lowered to $s{\geqslant}0$ by a number of authors [@Bournaveas2000; @Bournaveas2006a; @Chadam1973; @Fang2004a; @Pecher2006] by exploiting the conservation of charge (\[conservation of charge\]) together with the local well-posedness theory. The first result below the charge class was due to Selberg [@Selberg2007] where it was shown that the DKG equation is GWP in the region[^2] $$- \frac{1}{8}<s<0, \qquad -s+\sqrt{s^2 - s} < r{\leqslant}s+1.$$ Note that when $s<0$, the conservation of charge cannot be used directly since $\psi \not \in L^2$, thus the problem of global existence is significantly more difficult. Instead Selberg made use of the Fourier truncation method of Bourgain [@Bourgain1998], which allows one to take initial data just below a conserved quantity. There is a difficulty in directly applying this method to the DKG equation however, as there is no conservation law for the scalar $\phi$. Instead, one needs to exploit the fact the nonlinearity for $\phi$ depends only on the spinor $\psi$. Thus, as we have control over $\psi$ via the conservation of charge, we should be able to estimate the growth of $\phi$. This strategy was implemented by Selberg via an induction argument involving the cascade of free waves. Currently, the best result for GWP for the DKG equation is due to Tesfahun [@Tesfahun2009] where the GWP region of Selberg was extended to $$- \frac{1}{8}<s<0, \qquad s+\sqrt{s^2 - s} < r{\leqslant}s+1.$$ The improvement comes from applying the $I$-method of Colliander, Keel, Staffilani, Takaoka, and Tao, see for instance [@Colliander2002] for an introduction to the $I$-method. In the current article, we prove the following. \[thm DKG in intro\] The DKG equation (\[DKG general form\]) is globally well-posed for initial data $ \psi_0 \in H^s$, $(\phi_0, \phi_1) \in H^r\times H^{r-1}$ provided $$-\frac{1}{6}<s<0, \qquad s-\frac{1}{4} + \sqrt{\Big(s-\frac{1}{4}\Big)^2 - s}<r {\leqslant}s+1.$$ (-2.5,-2.5)(2.5,2.5) (0.66,1.33)(2.5,2.25)(2.5,-1.75)(2,-2)(1.58,-1.6)(0.66,-1.33)(0.66,1.33) (-2.5,-2)(2.5,-2) (2,-2.5)(2,2.5) (-2,0) (-1.97,0.03)(2.5,2.25) (-1.97,-0.03)(2,-2) (2,-2)(2.5,-1.75) (1.9,2)(2,2) (1.9,0)(2,0) (1.9,-1.33)(2,-1.33) (2.1,1.75)[$1$]{} (2.15,-0.22)[[$\frac{1}{2}$]{}]{} (2.15,-1.55)[$\frac{1}{6}$]{} (-2,-2)(-2,-1.9) (0.66,-2)(0.66,-1.9) (-2.1,-2.5)[[$-\frac{1}{2}$]{}]{} (0.55,-2.5)[$-\frac{1}{6}$]{} (2.2,2.3)[$r$]{} (2.5,-2.3)[$s$]{} (0.66,-1.33)(0.66,1.33) (0.66,-1.33)(1.58,-1.6)(2,-2) The proof of Theorem \[thm DKG in intro\] follows the argument used in [@Tesfahun2009] together with the bilinear estimates in Theorem \[thm main X\^sb esimate\]. More precisely, we use the $I$-method together with the induction on free waves approach of Selberg. The main idea, following the usual $I$-method, is to define a mild smoothing operator $I$ such that, firstly, for some large constant $N$, we have the estimate $$\label{intro explanation of Imethod} \| If \|_{L^2({\mathbb{R}})} {\lesssim}N^{-s} \| f \|_{H^s({\mathbb{R}})} {\lesssim}N^{-s} \| I f \|_{L^2}.$$ Secondly, we require $I$ to be the identity on low frequencies. We then try to estimate the growth of $\| I \psi(t) \|_{L^2}$ in terms of $t$. It turns out that despite the fact that $I\psi$ no longer solves the DKG equation, there is sufficient cancelation of frequencies to ensure that the charge $\| I\psi(t) \|_{L^2_x}$ is almost conserved. This almost conservation property follows from the usual proof of the conservation of charge, together with a number of applications of Theorem \[thm main X\^sb esimate\]. Thus we can estimate the growth of $\| \psi(t) \|_{H^s}$ from (\[intro explanation of Imethod\]). The induction on free waves approach of Selberg then allows us to control the scalar field $\phi$ and completes the proof of Theorem \[thm DKG in intro\].\ We now give a brief outline of this article. In Section \[sec - linear est\], we recall some properties of the $X^{s, b}$ and $H^{r, b}$ spaces which we require in the proof of Theorem \[thm DKG in intro\]. The proof of Theorem \[thm DKG in intro\] is contained in Section \[sec gwp for DKG\]. In Section \[sec bilinear est\] we prove that the conditions in Theorem \[thm main X\^sb esimate\] are sufficient for the estimate (\[thm main X\^sb est - main trilinear est\]). Finally, the counter examples showing that Theorem \[thm main X\^sb esimate\] is sharp up to equality are contained in Section \[sec counter examples\].\ **Notation**: The Fourier transform on ${\mathbb{R}}$ of a function $f \in L^1({\mathbb{R}})$ is denoted by $\widehat{f}(\xi) = \int_{\mathbb{R}}f(x) e^{-i x\xi} dx$. We use the notation $\widetilde{f}(\tau, \xi)$ for the space-time Fourier transform of a function $f(t, x)$ on ${\mathbb{R}}^{1+1}$. We write $a {\lesssim}b$ if there is some constant $C$, independent of the variables under consideration, such that $a {\leqslant}C b$. If we wish to make explicit that the constant $C$ depends on $\delta$ we write $a {\lesssim}_\delta b$. Occasionally we write $a \ll b$ if $C<1$. We use $a \approx b$ to denote the inequalities $a {\lesssim}b$ and $b {\lesssim}a$. All sums such as $\sum_{N} f(N)$ are over dyadic numbers $N \in 2^{{\mathbb{N}}}$. Given dyadic variables $N_1, N_2, N_3 \in 2^{{\mathbb{N}}}$, we use the short hand $$\sum_{N_{max} \approx N_{med}} = \sum_{N_{max} \in 2^{{\mathbb{N}}}} \sum_{\substack{ \\ N_{med} \in 2^{{\mathbb{N}}} \\ N_{med} \approx N_{max} } } \sum_{\substack{ N_{min} \in 2^{{\mathbb{N}}} \\ N_{min} {\lesssim}N_{med} } }.$$ We let ${\mathbbold{1}}_\Omega$ denote the characteristic function of the set $\Omega$, we occasionally abuse notation and write $ {\mathbbold{1}}_{|x| \approx N} $ instead of ${\mathbbold{1}}_{\{ |x| \approx N\}}$. The standard Sobolev space $H^s$ is defined as the completion of $C^\infty_0$ using the norm $$\| f \|_{H^s} = \| {\langle}\xi {\rangle}^s \widehat{f} \|_{L^2}.$$ If $u$ is a function of $(t, x) \in {\mathbb{R}}^{1+1}$ we use the notation $$\| u[t] \|_{H^s} = \| u(t) \|_{H^s} + \| {\partial}_t u(t) \|_{H^{s-1}}.$$ To handle solutions to the wave equation, we make use of the Banach space $\mathcal{H}^{r, b}$ defined via the norm $$\| {\varphi}\|_{\mathcal{H}^{r, b}} = \| {\varphi}\|_{H^{r, b}} + \| {\partial}_t {\varphi}\|_{H^{r-1, b}}.$$ The proof of Theorem \[thm DKG in intro\] requires the use of the local in time versions of the $X^{s, b}_\pm$ and $\mathcal{H}^{r, b}$ spaces. Let $S_{\Delta T} = [0, \Delta T] \times {\mathbb{R}}$. We define $X^{s, b}_\pm(S_{\Delta T})$ by restricting elements of $X^{s,b}_\pm$ to $S_{\Delta T}$. More precisely, $$X^{s, b}_\pm(S_{\Delta T})= X^{s, b}_\pm/ \{ f \in X^{s, b}_\pm \, |\, f|_{S_{\Delta T}} = 0\}.$$ The local in time space $X^{s, b}_\pm(S_{\Delta T})$ is a Banach space with norm $$\| {\varphi}\|_{X^{s,b}_\pm(S_{\Delta T})} = \inf_{ u = {\varphi}\text{ on } S_{\Delta T}} \| u \|_{X^{s, b}_\pm}.$$ If $b>\frac{1}{2}$, then we have the continuous embedding $X^{s, b}_\pm(S_{\Delta T}) \subset C\big([0, \Delta T], H^s\big)$. We define the Banach spaces $\mathcal{H}^{r, b}(S_{\Delta T})$ similarly and note that, if $b>\frac{1}{2}$, then we have the continuous embedding $\mathcal{H}^{r, b}(S_{\Delta T}) \subset C\big( [0, \Delta T], H^r \big) \cap C^1\big( [0, \Delta T], H^{r-1} \big)$. Linear Estimates {#sec - linear est} ================ Here we briefly recall some of the important properties of the $X^{s, b}_\pm$ and $\mathcal{H}^{r, b}$ spaces which we make use of in the proof of Theorem \[thm DKG in intro\], for more details we refer the reader to [@D'Ancona2007b] and [@Tao2006b]. We start by recalling some properties of the localised spaces $X^{s, b}_\pm(S_{\Delta T})$. \[lem time dilation for Xsb\] Let $s \in {\mathbb{R}}$, $0<\Delta T<1$, and $\nu \in C^\infty_0({\mathbb{R}})$. If $- \frac{1}{2} < b_1 {\leqslant}b_2 < \frac{1}{2}$ then $$\Big\| \nu\Big( \frac{t}{\Delta T} \Big) u(t, x) \Big\|_{X^{s, b_1}_\pm} {\lesssim}\Delta T^{b_2 - b_1 } \| u \|_{X^{s, b_2}_\pm}.$$ Consequently, we have $ \| u \|_{X^{s, b_1}_\pm(S_{\Delta T})} {\lesssim}\Delta T^{b_2 - b_1} \| u \|_{X^{s, b_2}_\pm(S_{\Delta T})}$. Moreover if $-\frac{1}{2} <b < \frac{1}{2}$ then $$\| {\mathbbold{1}}_{[0, \Delta T]}(t) u \|_{X^{s, b}_\pm} {\lesssim}\| u \|_{X^{s, b}_\pm(S_{\Delta T})}$$ with constant independent of $\Delta T$. The first conclusion is well known and can be found in, for instance, [@Tao2006b]. The second conclusion is perhaps not as well known and for the convenience of the reader we include the proof here. The definition of $X^{s, b}_\pm(S_{\Delta T})$ together with a change of variables on the frequency side shows that is suffices to prove $$\| {\mathbbold{1}}_{[0, \Delta T]}(t) f \|_{H^{b}} {\lesssim}\| f \|_{H^{b} }.$$ By duality we may assume that $0<b<\frac{1}{2}$. Then by a well-known characterisation of the Sobolev spaces $H^s$, (see for instance [@Adams2003]) we have $$\begin{aligned} \| {\mathbbold{1}}_{[0, \Delta T]} f \|_{H^b}^2 &\approx \| {\mathbbold{1}}_{[0, \Delta T]} f \|_{L^2}^2 + \int_{{\mathbb{R}}^2} \frac{ | {\mathbbold{1}}_{[0, \Delta T]}(t) f(t) - {\mathbbold{1}}_{[0, \Delta T]}(t') f(t') |^2}{|t - t'|^{1+ 2b}} dt dt'\\ &{\lesssim}\| f\|_{L^2}^2 + \int_0^{\Delta T} \int_0^{\Delta T} \frac{ |f(t) - f(t') |^2}{|t - t'|^{1+ 2b} } dt dt' + 2 \int_0^{\Delta T} \int_{t' \not \in [0, \Delta T]} \frac{|f(t)|^2}{ |t - t'|^{1+2b}} dt' dt\\ &{\lesssim}\| f\|_{H^b}^2 + 2 \int_0^{\Delta T} \int_{t' \not \in [0, \Delta T]} \frac{|f(t)|^2}{ |t - t'|^{1+2b}} dt' dt. \end{aligned}$$ To complete the proof we use Hardy’s inequality (see for instance [@Tao2006b Lemma A.2]) together with the assumption $0<b<\frac{1}{2}$ to deduce that $$\begin{aligned} \int_0^{\Delta T} \int_{t' \not \in [0, \Delta T]} \frac{|f(t) |^2}{ |t - t'|^{1+2b}} dt' dt &{\lesssim}\int_0^{\Delta T} |f(t)|^2 \Big( \frac{1}{|t|^{2b}} + \frac{1}{|t - \Delta T|^{2b}} \Big) dt\\ &{\lesssim}\Big\| \frac{f(t)}{|t|^{b}} \Big\|^2_{L^2} + \Big\| \frac{f(t)}{|t-\Delta T|^{b}} \Big\|^2_{L^2}\\ &{\lesssim}\| f\|_{H^b}^2. \end{aligned}$$ To control the solution to the Dirac equation we make use of the energy estimate for the $X^{s, b}_\pm$ spaces. \[lem energy est for Xsb\] Let $s \in {\mathbb{R}}$, $b>\frac{1}{2}$, and $0<\Delta T<1$. Suppose $f \in H^s$, $F \in X^{s, b-1}_{\pm}(S_{\Delta T})$, and let $u$ be the solution to $$\begin{split} {\partial}_t u \pm {\partial}_x u &= F \\ u(0)&=f. \end{split}$$ Then $u \in X^{s, b}_\pm(S_{\Delta T})$ and we have the estimate $$\| u \|_{X^{s, b}_\pm(S_{\Delta T})} {\lesssim}\| f\|_{H^s} + \| F \|_{X^{s, b-1}_\pm(S_{\Delta T})}.$$ We also require the $H^{r, b}$ versions of the above results. \[lem time dilation for Hrb\] Let $r \in {\mathbb{R}}$, $0<\Delta T<1$, and $\nu \in C^\infty_0({\mathbb{R}})$. Then if $- \frac{1}{2} < b_1 {\leqslant}b_2 < \frac{1}{2}$ we have $$\Big\| \nu\Big( \frac{t}{\Delta T} \Big) u(t, x) \Big\|_{H^{r, b_1}} {\lesssim}\Delta T^{b_2 - b_1 } \| u \|_{H^{r, b_2}}.$$ Consequently, we have $ \| u \|_{H^{r, b_1}(S_{\Delta T})} {\lesssim}\Delta T^{b_2 - b_1} \| u \|_{H^{r, b_2}(S_{\Delta T})}$. \[lem energy est for Hrb\] Let $r \in {\mathbb{R}}$, $b>\frac{1}{2}$, $0<\Delta T<1$, and $m\in {\mathbb{R}}$. Suppose $f \in H^r$, $g \in H^{r-1}$, and $F \in H^{r-1, b-1}(S_{\Delta T})$ and let $u$ be the solution to $$\begin{split} \Box u &= m^2 u + F \\ u(0)&=f, \qquad {\partial}_t u(0) = g. \end{split}$$ Then $u \in \mathcal{H}^{r, b}(S_{\Delta T})$ and we have the estimate $$\| u \|_{\mathcal{H}^{r, b}(S_{\Delta T})} {\lesssim}\| f\|_{H^r} + \| g\|_{H^{r-1}} + \| F \|_{H^{r-1, b-1}(S_{\Delta T})}.$$ See [@Tesfahun2009]. Global Well-Posedness for the Dirac-Klein-Gordon Equation {#sec gwp for DKG} ========================================================= We are now ready to consider the proof of global well-posedness for the DKG equation. To uncover the null structure for the DKG equation, we let $\psi =( \psi_+, \psi_-)^T$. Then the DKG equation (\[DKG general form\]) can be written as $$\label{DKG equation} \begin{split} {\partial}_t \psi_{\pm} \pm {\partial}_x \psi_{\pm} &= -i M\psi_{\mp} + i \phi \psi_{\mp} \\ \Box \phi &= m^2 \phi - 2 \Re \big(\psi_+ \overline{\psi}_- \big) \end{split}$$ with initial data $$\label{DKG data} \psi_{\pm}(0) = f_{\pm} \in H^s, \qquad\phi(0) = \phi_0 \in H^r, \qquad {\partial}_t \phi(0) = \phi_1\in H^{r-1}.$$ Note that the right hand side of (\[DKG equation\]) has the bilinear product $\psi_+ \overline{\psi}_-$, which, as we have seen in Theorem \[thm main X\^sb esimate\], behaves significantly better than the corresponding product with $++$. The $+-$ structure can also be seen in the term $\phi \psi_\pm$ via a duality argument [@Selberg2008]. These are the key observations used in the local well-posedness theory for the DKG equation. To prove the global well-posedness result of Theorem \[thm DKG in intro\], by the local well-posedness result in [@Selberg2008], it suffices to prove that the data norms $\| \psi_\pm(T) \|_{H^s}$, $\| u[T] \|_{H^r}$ remain finite for all large times $ 0<T<\infty$. To this end, we make use of the $I$-method together with ideas from [@Selberg2007] and [@Tesfahun2009]. Let $\rho_0 \in C^{\infty}$ be even, decreasing, and satisfy $$\rho_0(\xi) = \begin{cases} 1 \qquad &|\xi|<1\\ |\xi|^s &|\xi|>2. \end{cases}$$ Let $\rho(\xi) = \rho_0\Big(\frac{|\xi|}{N}\Big)$ and define the $I$ operator by $\widehat{I\psi}(\xi) = \rho(\xi) \widehat{\psi}(\xi).$ We have the following straightforward estimates. Firstly, since $s<0$, we have for any $\sigma \in {\mathbb{R}}$, $$\label{Imethod implies control of H^s norm} \| f\|_{H^\sigma} {\lesssim}\| I f\|_{H^{\sigma - s}} {\lesssim}N^{-s} \| f\|_{H^\sigma}.$$ In particular, by taking $\sigma=0$, we observe that to obtain control over $\| \psi(t) \|_{H^s_x}$, it suffices to estimate $\| I\psi(t)\|_{L^2_x}$. Secondly, if ${\text{supp }\,}\widehat{g}\subset \{ |\xi| \gtrsim N \}$, $s<0$, and $s_1<s_2$, then we can trade regularity for decay in terms of $N$, $$\label{Imethod can trade regularity for powers of N} \| g \|_{H^{s_1}} {\lesssim}N^{s_1 - s_2} \| g \|_{H^{s_2}} \approx N^{s_1 - s_2 + s} \| Ig \|_{H^{s_2 - s}}.$$ Thirdly, we note that the $I$ operator is the identity on low frequencies, so if ${\text{supp }\,}\widehat{f} \subset \{ |\xi| < N\}$ then $ If = f$. Finally, if $f$ is real-valued, then $If$ is also real-valued since $\rho$ was assumed to be even. The $I$-method proceeds as follows. Assume we have a local solution $$\psi_{\pm} \in C\big([0, \Delta T], H^s\big), \qquad \phi \in C\big([0, \Delta T], H^r\big) \cap C^1\big( [0, \Delta T], H^{r-1}\big)$$ to (\[DKG equation\]), (\[DKG data\]). Note that from (\[Imethod implies control of H\^s norm\]) we have $I\psi(t) \in L^2_x$. We would like to use the conservation of charge to control $\| I\psi(t)\|_{L^2_x}$. However $I\psi$ is no longer a solution to (\[DKG equation\]) and so we can not expect $\| I \psi(t) \|_{L^2_x}$ to be conserved. Despite this, if we follow the proof of conservation of charge, then $$\begin{aligned} {\partial}_t \int_{{\mathbb{R}}} |I \psi_+(t)|^2 + | I \psi_- (t)|^2 dx &=2\Re\bigg( \int_{\mathbb{R}}\overline{I\psi}_+ {\partial}_t I \psi_+ + \overline{I\psi}_- {\partial}_t I \psi_- dx \bigg)\notag \\ &=2\Re\bigg( \int_{\mathbb{R}}\overline{I\psi}_+ \big( - {\partial}_x I \psi_+ - i M I \psi_- + i I(\phi \psi_-) \big) \notag \\ &\qquad \qquad \qquad \qquad + \overline{I\psi}_- \big( {\partial}_x I \psi_- - i M I \psi_+ + i I(\phi \psi_+) \big) dx \bigg)\notag\\ &=2\Re\bigg( i \int_{\mathbb{R}}\overline{I\psi}_+ I(\phi \psi_-) + \overline{I\psi}_- {\partial}_t I(\phi \psi_+) dx \bigg). \label{derivation of almost conservation law} \end{aligned}$$ Now as $\phi$ is real-valued, $I^2\phi$ is also real-valued and hence $$2\Re\Big( i I^2 \phi \big( \overline{I \psi}_+ I \psi_- + \overline{I\psi}_- I \psi_+\big) \Big)=0.$$ Subtracting this term from (\[derivation of almost conservation law\]) and using the fundamental theorem of Calculus then gives $$\begin{aligned} \sup_{t' \in [0, \Delta T] } \big( \| I \psi_+(t') \|_{L^2_x}^2 + \| I \psi_-(t') &\|_{L^2_x}^2 \big) {\leqslant}\| f_+ \|_{L^2}^2 + \| f_- \|_{L^2}^2 \notag\\ & + 2 \sum_{\pm} \sup_{t' \in [0, \Delta T]} \Big| \int_0^{t'} \int_{\mathbb{R}}\big( I(\phi \psi_\pm) - I^2 \phi I \psi_\pm\big) \overline{I \psi}_\mp dx dt\Big|.\label{growth of Ipsi} \end{aligned}$$ Thus provided we can show the last term in (\[growth of Ipsi\]) is small, we can deduce that over a small time $[0, \Delta T]$, $\| I \psi_\pm(t)\|_{L^2}$ does not grow to large. The first step in this direction is the following. \[lem smoothing\] Let $\frac{-1}{4}<s<0$ and $-s< r{\leqslant}1+2s$. Assume $b=\frac{1}{2}+\epsilon$ with $\epsilon>0$ sufficiently small. Then for any $\Delta T \ll 1$, $N \gg 1$ we have $$\begin{aligned} \sup_{t' \in [0, \Delta T]} \Big| \int_0^{t'} \int_{\mathbb{R}}\big(I(\phi &u) - I^2 \phi I u\big)\overline{I v} \,dx dt \Big| \notag\\ &{\lesssim}\Delta T^{\frac{1}{2} - 2\epsilon} N^{2s - r + 2\epsilon} \| I^2 \phi \|_{H^{r-2s, b}(S_{\Delta T})} \| Iu \|_{X^{0, b}_\pm (S_{\Delta T})} \| I v \|_{X^{0, b}_{\mp}(S_{\Delta T})}\label{lem smoothing - main eqn}\end{aligned}$$ where $S_{\Delta T} = [0, \Delta T] \times {\mathbb{R}}$. See Subsection \[subsec - proof of lem smoothing\] below. The use of $I^2\phi$ instead of just $\phi$ or $I \phi$ on the right hand side of (\[lem smoothing - main eqn\]) may require some explanation. Roughly speaking, the larger the negative exponent on $N$ in (\[lem smoothing - main eqn\]), the better the eventual GWP result will be. Moreover, an examination of the proof of Lemma \[lem smoothing\] shows that the exponent on $N$ depends entirely on the number of derivatives on $\phi$. In other words, we could replace the term $N^{2s - r} \| I^2\phi \|_{H^{r-2s, b}}$ with $N^{ks -r} \| I^k \phi \|_{H^{r-ks, b}}$ for any $k \in {\mathbb{N}}$ (provided $r-ks {\leqslant}1$). However, the size of $\phi$ with respect to $N$ ends up being of the order $N^{-2s}$. This follows by observing that schematically $\phi$ is a solution to $\Box \phi = \psi^2$, and by (\[Imethod implies control of H\^s norm\]), the low frequency component of $\psi^2$ is essentially of size $N^{-2s}$. Thus it is natural to take $I^2 \phi$, which via (\[Imethod implies control of H\^s norm\]), also has size roughly $N^{-2s}$. The powers of $\Delta T$ and $N$ on the right hand side of (\[lem smoothing - main eqn\]) are essentially sharp if we are working in the spaces $X^{s, b}_\pm$, $H^{s, b}$. This follows from the counter examples in Section \[sec counter examples\] together with a scaling argument. Lemma \[lem smoothing\] allows us to estimate the growth of $\| I \psi_\pm(t)\|_{L^2}$ on $[0, \Delta T]$, provided that we can control the size of the norms $\| I \psi_\pm\|_{X^{0, b}_\pm(S_{\Delta T} )}$ and $\| I^2 \phi \|_{H^{r-2s, b}(S_{\Delta T})}$. This control is provided by a modification of the usual local well-posedness theory. \[lem mod lwp\] Let $\frac{-1}{6} < s<0$, $-s<r {\leqslant}\frac{1}{2} + 2s$, and $b= \frac{1}{2} + \epsilon$ with $\epsilon>0$ sufficiently small. Assume $f_\pm \in H^s$ and $\phi[0] \in H^r \times H^{r-1}$. Choose $\Delta T \ll 1$ and $N \gg 1$ such that $$\label{lem mod lwp - required est for wave data} \Big( \Delta T^{\frac{1}{2} + r-2s - 3 \epsilon} + N^{-r+2s + 2\epsilon}\Big) \| I^2\phi[0] \|_{H^{r-2s}} \ll 1$$ and $$\label{lem mod lwp - required est for spinor data} \Big( \Delta T^{1 - \epsilon} + N^{ - \frac{1}{2}+ 2\epsilon}\Big) \Big( \| If_+ \|_{L^2} + \| If_-\|_{L^2}\Big)^2 \ll 1.$$ Then the Dirac-Klein-Gordon equation (\[DKG equation\]) with initial data (\[DKG data\]) is locally well-posed on the domain $S_{\Delta T}=[0, \Delta T]\times {\mathbb{R}}$. Moreover, the solution $(\psi, \phi)$ satisfies $$\| I \psi_+ \|_{X^{0, b}_+(S_{\Delta T})} + \| I \psi_- \|_{X^{0, b}_-(S_{\Delta T})} {\lesssim}\| If_+\|_{L^2} + \| If_-\|_{L^2}$$ and $$\| I^2 \phi \|_{\mathcal{H}^{r-2s, b}(S_{\Delta T}) } {\lesssim}\| I^2 \phi[0]\|_{H^{r-2s}} + \big( \| If_+\|_{L^2} + \| If_-\|_{L^2}\big)^2.$$ See Subsection \[subsec - proof of lem mod lwp\] below. Note that since $\| I^2\phi[0]\|_{H^{r-2s}} {\lesssim}N^{-2s}$, by choosing $N$ sufficiently large and $\Delta T$ sufficiently small, we can ensure that the inequality (\[lem mod lwp - required est for wave data\]) is satisfied. A similar comment applies to (\[lem mod lwp - required est for spinor data\]). The reason that we can extend the work of Tesfahun [@Tesfahun2009] is due to the conclusions in Lemma \[lem smoothing\] and Lemma \[lem mod lwp\]. In more detail, Lemma \[lem smoothing\] improves [@Tesfahun2009 Lemma 8] by adding a power of $\Delta T$ on the right hand side of (\[lem smoothing - main eqn\]). Since $\Delta T$ will be taken small, this is a significant gain. Similarly, Lemma \[lem mod lwp\] extends [@Tesfahun2009 Theorem 8] by having a larger exponent on $\Delta T$ in (\[lem mod lwp - required est for wave data\]). As a consequence, we can take $\Delta T$ larger, which improves the eventual GWP result. The point here is that the larger $\Delta T$ becomes, the fewer time steps of length $\Delta T$ are required to reach a large time $T$. We now follow the argument used in [@Tesfahun2009] and sketch the proof of Theorem \[thm DKG in intro\]. The persistence of regularity result in [@Selberg2008] shows that it suffices to prove GWP in the case $$\label{thm main X^sb esimate - proof - conditions on s, r} -\frac{1}{6}<s<0, \qquad s-\frac{1}{4} + \sqrt{\Big(s-\frac{1}{4}\Big)^2 - s}<r < \frac{1}{2} + 2s.$$ Note that this region is non-empty as the intersection of the curves $s-\frac{1}{4} + \sqrt{\Big(s-\frac{1}{4}\Big)^2 - s}$ and $ \frac{1}{2} + 2s$ occurs at $s=-\frac{1}{6}$. Choose some large time $T>0$ and assume $\epsilon>0$ is small. Let $N$ be some large fixed constant to be chosen later depending on the initial data $\|\psi(0)\|_{H^s}$ and $\|\phi[0]\|_{H^r}$, as well as the various constants appearing in Lemma \[lem smoothing\] and Lemma \[lem mod lwp\]. Take $\Delta T = N^{ \frac{4 s - 2\epsilon }{ 1 + 2r-4s - 6\epsilon}}$. If $N$ is sufficiently large then from (\[Imethod implies control of H\^s norm\]) $$\begin{aligned} \Big( \Delta T^{\frac{1}{2} + r-2s - 3 \epsilon} + N^{-r+2s + 2\epsilon}\Big) \| I^2\phi[0] \|_{H^{r-2s}} &\ll 1\\ \Big( \Delta T^{1 - \epsilon} + N^{ - \frac{1}{2} + 2 \epsilon}\Big) \Big( \| If_+ \|_{L^2} + \| If_-\|_{L^2}\Big)^2 &\ll 1. \end{aligned}$$ Therefore by Lemma \[lem mod lwp\] we get a solution $(\psi, \phi)$ to (\[DKG equation\]) on $[0, \Delta T]$. We would now like to repeat this argument $\frac{T}{\Delta T}$ times to advance to the time $T$. The only obstruction is the possible growth of the norms $\| I \psi_\pm(t) \|_{L^2}$ and $\| I^2\phi[t]\|_{H^{r-2s}}$. Our aim is to use Lemma (\[lem smoothing\]) to show that $\| I \psi_\pm(t) \|_{L^2}$ is “almost conserved” and consequently obtain large time control over the norm $\| I \psi_\pm (t) \|_{L^2}$. This is accomplished by using an induction argument as follows. Assume $n{\lesssim}\frac{T}{\Delta T}$ and suppose we have a solution $(\psi, \phi)$ on $[0, n\Delta T]$ with the bounds $$\label{proof of gwp - induc assump for psi} \sup_{t \in [0, n \Delta T]} \Big( \| I \psi_+(t) \|_{L^2_x}^2 + \| I \psi_- (t) \|_{L^2_x}^2\Big) {\leqslant}2 \| If_+ \|^2_{L^2_x} + 2\| If_-\|^2_{L^2_x}$$ and $$\label{proof of gwp - induc assump for phi} \sup_{t \in [0, n \Delta T]} \| I^2 \phi[t] \|_{H^{r-2s}_x} {\leqslant}C^* \Big( \| I^2 \phi[0] \|_{H^{r-2s}_x} + \big( \| If_+ \|_{L^2_x} + \| If_-\|_{L^2_x}\big)^2 \Big)$$ where the constant $C^*$ is some large constant independent of $N$, $\Delta T$, and $n$. If $N$ is sufficiently large, depending on $C^*$ and the initial data $\| f_\pm \|_{H^s}$, $\| \phi[0]\|_{H^r}$, then we can apply Lemma \[lem mod lwp\] with initial data $\psi(n\Delta T)$, $\big(\phi(n\Delta T), {\partial}_t \phi(n\Delta T) \big)$, and extend the solution to $[0, (n+1)\Delta T]$. Suppose we could show that the bounds (\[proof of gwp - induc assump for psi\]) and (\[proof of gwp - induc assump for phi\]) on $[0, n\Delta T]$ implied that they also hold on the larger interval $[0, (n+1) \Delta T]$ with the same constant $C^*$. Then by induction we would have (\[proof of gwp - induc assump for psi\]) and (\[proof of gwp - induc assump for phi\]) on $[0, T]$. Since $T$ was arbitrary, Theorem \[thm DKG in intro\] would follow. Thus it suffices to verify the estimates (\[proof of gwp - induc assump for psi\]) and (\[proof of gwp - induc assump for phi\]) on the interval $[0, (n+1) \Delta T]$. We break this into two parts, proving the bound on $\| I \psi_\pm (t ) \|_{L^2}$, and then estimating $\| I^2 \phi[t] \|_{H^{r-2s}}$.\ **Bound on the Spinor $\psi_\pm$.** Let $$\Gamma(z) = \sup_{t \in [0, z]} \Big( \| I \psi_+(t) \|_{L^2_x}^2 + \| I \psi_- (t) \|^2_{L^2_x}\Big) .$$ Note that the bounds (\[proof of gwp - induc assump for psi\]) and (\[proof of gwp - induc assump for phi\]) imply that $$\label{proof of gwp - rough bounds} \begin{split} \Gamma(n \Delta T) &{\leqslant}A N^{-2s} \\ \sup_{t \in [0, n \Delta T]} \| I^2 \phi[t] \|_{H^{r-2s}_x} &{\leqslant}B N^{-2s} \end{split}$$ where $A$ and $B$ depend on the initial data, the constant $C^*$, and $T$, but are independent of $n$, $N$, and $\Delta T$. If we now combine Lemma \[lem smoothing\], Lemma \[lem mod lwp\] together with (\[growth of Ipsi\]) we obtain the following control on the growth of $\Gamma(t)$. \[cor almost conservation law\] Let $\frac{-1}{6} < s <0$ and $ -s< r {\leqslant}\frac{1}{2} + 2s$ and $b=\frac{1}{2} + \epsilon$ with $\epsilon>0$ sufficiently small. Suppose $$\Delta T = N^{\frac{4 s - 2 \epsilon}{ 1 + 2r-4s - 6\epsilon} }$$ and we have the bounds (\[proof of gwp - rough bounds\]). Then provided $N$ is sufficiently large, $$\Gamma(\Delta T) {\leqslant}\Gamma(0) + C \Delta T^{\frac{1}{2} - 2\epsilon} N^{ - r + 2 \epsilon} \big( A +B\big) \Gamma(0).$$ By Lemma \[lem smoothing\], Lemma \[lem mod lwp\], and (\[growth of Ipsi\]) it suffices to show that $$\Delta T^{\frac{1}{2} + r-2s - 3 \epsilon } N^{-2s} B + N^{-r + 2\epsilon} B \ll 1$$ and $$\Delta T^{1-\epsilon} N^{-2s} A + N^{2\epsilon- \frac{1}{2} - 2s} B \ll 1.$$ However these inequalities follow provided $\Delta T = N^{\frac{4 s - 2 \epsilon}{ 1 + 2r-4s - 6\epsilon} }$ and we choose $N$ sufficiently large. We can now iterate the previous corollary to get control over $\Gamma(t)$ at time $(n+1)\Delta T$ $$\Gamma\big((n+1) \Delta T\big) {\leqslant}\Gamma(0) + C n \Delta T^{\frac{1}{2} - 2\epsilon} N^{-r + 2\epsilon} (A + B) \Gamma(0).$$ Since the number of steps $n{\lesssim}\frac{T}{\Delta T}$ we get $$\Gamma\big((n+1) \Delta T\big){\leqslant}\Gamma(0) + C T \Delta T^{-\frac{1}{2} - 2\epsilon} N^{-r + 2 \epsilon} (A + B) \Gamma(0).$$ We want to make the coefficient of the second term small. Thus we need to ensure that, using the requirement on $\Delta T$ in Corollary \[cor almost conservation law\], $$\label{almost conservation law deduction} C T \Delta T^{-\frac{1}{2} - 2\epsilon} N^{-r + 2 \epsilon} (A + B) \approx N^{ \frac{ -(1+4\epsilon)(2s - \epsilon) }{1 + 2r - 4s - 6\epsilon} -r + 2\epsilon} \ll 1.$$ By choosing $N$ large, and $\epsilon>0$ sufficiently small, we see that (\[almost conservation law deduction\]) will follow provided $ -2s- r \big( 1 + 2r-4s\big) <0$. Rearranging, we get the quadratic polynomial $2 r^2 +(1 - 4s) r + 2s >0$ and so we need $$s - \frac{1}{4} + \sqrt{ \Big(s - \frac{1}{4}\Big)^2 - s} <r .$$ Therefore, provided we choose $N$ large enough, depending on $T$, $A$, and $B$, we get $$\Gamma\big((n+1) \Delta T\big) {\leqslant}2\Gamma(0)$$ as required.\ **Bound on $\phi$.** Recall that our goal was to show that, if the bounds (\[proof of gwp - induc assump for psi\]) and (\[proof of gwp - induc assump for phi\]) hold for $t \in [0, n \Delta T]$, then in fact they also held on the larger domain $[0, (n+1)\Delta T]$ (with the same constants). The bound for $\| I \psi_\pm\|_{L^2}$ was obtained above. Thus it remains to bound $\| I^2\phi[t] \|_{H^{r-2s}}$ on the interval $[0, (n+1)\Delta T]$. The argument that gives the required bound makes use of an idea due to Selberg in [@Selberg2007] on induction of free waves. The idea is to break $\phi$ into a sum of homogeneous waves, together with an inhomogeneous term and then use an induction argument to estimate the contribution that each of these homogeneous waves makes to the size of $\| I^2\phi[t] \|_{H^{r-2s}}$. We note that this idea was also used in [@Tesfahun2009].\ We begin by observing that the induction assumptions (\[proof of gwp - induc assump for psi\]) and (\[proof of gwp - induc assump for phi\]) together with Lemma \[lem mod lwp\] give for every $0 {\leqslant}j {\leqslant}n$ $$\label{bound on phi eqn1} \| I \psi_+ \|_{X^{0, b}_+(S_j)} + \| I \psi_- \|_{X^{0, b}_-(S_j)} {\leqslant}C_1 \Big( \| I f_+ \|_{L^2_x} + \| If_-\|_{L^2_x}\Big)$$ where $S_j=[j \Delta T, (j+1)\Delta T]$ and the constant $C_1$ is independent of $C^*$, $j$, $n$, $N$, and $\Delta T$. Suppose we could show that (\[bound on phi eqn1\]) implies that $$\label{bound on phi eqn2} \sup_{ t \in [n \Delta T, (n+1)\Delta T]} \| I^2 \phi[t] \|_{H^{r-2s}} {\leqslant}C_2 \Big( \| I^2 \phi[0] \|_{H^{r-2s}_x} + \Big( \| I f_+ \|_{L^2_x} + \| I f_- \|_{L^2_x}\Big)^2\Big).$$ Then by taking $C^*=C_2$ we see that the bound (\[proof of gwp - induc assump for phi\]) holds for $t \in [0, (n+1) \Delta T]$. Thus by induction, together with the fact that the constants in (\[proof of gwp - induc assump for psi\]) and (\[proof of gwp - induc assump for phi\]) are independent of $n$, we would obtain control over the solution on $[0, T]$ and Theorem \[thm DKG in intro\] would follow. We now show that (\[bound on phi eqn1\]) implies (\[bound on phi eqn2\]). We make use of the following result which is a variant of a corresponding result in [@Tesfahun2009]. \[lem control of inhomogeneous term\] Let $m \in {\mathbb{R}}$, $0<\Delta T<1$, $\frac{-1}{4}<s<0$, $0<r<\frac{1}{2} + 2s$, and $b>\frac{1}{2}$. Assume $u \in X^{s, b}_+(S_{\Delta T})$ and $v \in X^{s, b}_-(S_{\Delta T})$. Then there exists a unique solution $\Phi \in H^{r, b}(S_{\Delta T})$ to $$\begin{aligned} \Box \Phi &= \Re( u v) + m^2 \Phi \\ \Phi(0)&= {\partial}_t \Phi(0) = 0. \end{aligned}$$ on $S_{\Delta T}=[0, \Delta T]\times {\mathbb{R}}$. Moreover we have $$\label{lem control of inhomogeneous term - estimate} \sup_{ t \in [0, \Delta T]} \| I^2 \Phi[t] \|_{H^{r-2s}_x} {\lesssim}\big( \Delta T + N^{ -\frac{1}{2} + 2\epsilon}\big) \| I u\|_{X^{0, b}_+(S_{\Delta T})} \| Iv \|_{X^{0, b}_-(S_{\Delta T})}.$$ The existence/uniqueness claim follows from Lemma \[lem energy est for Hrb\] together with an application of Theorem \[thm main X\^sb esimate\]. To prove (\[lem control of inhomogeneous term - estimate\]) we write $\Phi = \Phi_1 + \Phi_2$ where $$\begin{aligned} \Box \Phi_1 &= \Re( u_{low} v_{low}) + m^2 \Phi_1\\ \Phi_1(0)&= 0, \qquad {\partial}_t \Phi_1(0) = 0. \end{aligned}$$ and $\widehat{u_{low}} = {\mathbbold{1}}_{|\xi|< \frac{N}{2}} \widehat{u}$, $\widehat{v_{low}} = {\mathbbold{1}}_{|\xi|< \frac{N}{2}} \widehat{v}$. The standard representation of solutions to the Klein-Gordon equation, together with the Sobolev product law and the observation that $I^2(u_{low} v_{low}) = u_{low} v_{low}$, gives $$\begin{aligned} \sup_{t \in [0, \Delta T]} \| I^2 \Phi_1[t] \|_{H^{r-2s}_x} &{\lesssim}\int_0^{\Delta T} \| u_{low}(t) v_{low}(t) \|_{H^{r-2s -1}_x} dt\\ &{\lesssim}\int_0^{\Delta T} \| u_{low}(t)\|_{L^2_x} \|v_{low}(t) \|_{L^2_x} dt \\ &{\lesssim}\Delta T \| I u \|_{X^{0, b}_+(S_{\Delta T}) } \| I v \|_{X^{0, b}_-(S_{\Delta T})}. \end{aligned}$$ To bound the remaining term, $\Phi_2$, we note that by the energy estimate for $H^{s, b}$ spaces in Lemma \[lem energy est for Hrb\], $$\begin{aligned} \sup_{t \in [0, \Delta T]} \| I^2 \Phi_2[t] \|_{H^{r-2s}_x} &{\lesssim}\| I^2 \Phi_2 \|_{\mathcal{H}^{r - 2s, b}(S_{\Delta T})} \notag\\ &{\lesssim}\| I^2( u v - u_{low} v_{low}) \|_{H^{r - 2s -1, b-1}(S_{\Delta T})} \notag\\ &{\lesssim}\| u_{low} v_{hi} \|_{H^{-\frac{1}{2}, b-1}(S_{\Delta T})} + \| u_{hi} v_{low} \|_{H^{-\frac{1}{2}, b-1}(S_{\Delta T})} + \| u_{hi} v_{hi} \|_{H^{-\frac{1}{2}, b-1}(S_{\Delta T})} \label{lem control of inhomogeneous term - decomp into terms} \end{aligned}$$ where $u_{hi} = u - u_{low}$ is the high frequency component of $u$, $v_{hi}$ is defined similarly, and we used the assumption $r < \frac{1}{2} + 2s$. By Corollary \[cor wave-sobolev and X\^sb estimate\] we have the estimate $$\label{lem control of inhomogeneous term - application of corollary} \|\psi_1 \psi_2 \|_{H^{ - \frac{1}{2}, b-1}} {\lesssim}\| \psi_1 \|_{X^{-\frac{1}{2}- s_1 + 2\epsilon, b}_+} \| \psi_2 \|_{X^{s_1, b}_-}$$ for $\frac{-1}{2} < s_1{\leqslant}0$. To control the first term in (\[lem control of inhomogeneous term - decomp into terms\]) we use (\[lem control of inhomogeneous term - application of corollary\]) with $s_1 = -\frac{1}{2} + 2\epsilon$ together with (\[Imethod can trade regularity for powers of N\]) to obtain $$\begin{aligned} \| u_{low} v_{hi} \|_{H^{-\frac{1}{2}, b-1}(S_{\Delta T})} &{\lesssim}\| u_{low} \|_{X^{0, b}_+(S_{\Delta T})} \| v_{hi} \|_{X^{-\frac{1}{2} + 2 \epsilon, b}_-(S_{\Delta T})}\\ &{\lesssim}N^{-\frac{1}{2} + 2\epsilon} \| I u \|_{X^{0, b}_+(S_{\Delta T})} \| Iv \|_{X^{0, b}_-(S_{\Delta T})} \end{aligned}$$ A similar application of (\[lem control of inhomogeneous term - application of corollary\]) allows us to estimate the second term in (\[lem control of inhomogeneous term - decomp into terms\]). Finally, for the last term in (\[lem control of inhomogeneous term - decomp into terms\]) we use (\[Imethod can trade regularity for powers of N\]) and (\[lem control of inhomogeneous term - application of corollary\]) with $s_1= s$ to deduce that $$\begin{aligned} \| u_{hi} v_{hi} \|_{H^{-\frac{1}{2}, b}(S_{\Delta T})} &{\lesssim}\| u_{hi} \|_{X^{-\frac{1}{2} -s + 2\epsilon, b}_+(S_{\Delta T})} \| v_{hi} \|_{X^{s, b}_-(S_{\Delta T})}\\ &{\lesssim}N^{-\frac{1}{2} + 2\epsilon} \| I u \|_{X^{0, b}_+(S_{\Delta T})} \| Iv \|_{X^{0, b}_-(S_{\Delta T})} \end{aligned}$$ where we needed $-\frac{1}{2} - s + 2\epsilon{\leqslant}s$ which holds provided $s> - \frac{1}{4}$ and $\epsilon$ sufficiently small. We now have the necessary results to control the growth of $\| I^2\phi[t] \|_{H^{r-2s}}$. Let $0{\leqslant}j{\leqslant}n$ and define $\phi_j^{(0)}$ to be the solution to $$\begin{split} \Box \phi_j^{(0)} &=m^2 \phi_j^{(0)}\\ \phi_j^{(0)}(j\Delta T) &= \phi(j\Delta T), \qquad {\partial}_t \phi_j^{(0)}(j\Delta T) = {\partial}_t \phi(j\Delta T). \end{split}$$ Let $\Phi_j = \phi - \phi_j^{(0)}$ be the inhomogeneous component of $\phi$. The inequality (\[bound on phi eqn1\]) together with Lemma \[lem control of inhomogeneous term\] and the assumption $\Delta T = N^{\frac{4 s - 2 \epsilon}{1 + 2r - 4s - 6 \epsilon} } $, shows that for every $0{\leqslant}j {\leqslant}n$ $$\label{control of inhomogeneous term} \sup_{ t \in [j\Delta T, (j+1) \Delta T]} \| I^2 \Phi_j[t] \|_{H^{r-2s}_x} {\lesssim}\Delta T \big( \| I f_+ \|_{L^2_x} + \| If_-\|_{L^2_x} \big)^2.$$ We now claim that for $1{\leqslant}j {\leqslant}n$ we have the estimate $$\label{control of KG component - estimate for free waves} \sup_{t\in [0, (n+1) \Delta T]} \| I^2 \phi^{(0)}_j[t] \|_{H_x^{r-2s}} {\leqslant}\sup_{ t \in [0, (n+1) \Delta T]} \| I^2\phi^{(0)}_{j-1}[t] \|_{H^{r-2s}_x} + C \Delta T\big( \| If_+ \|_{L_x^2} + \| I f_- \|_{L^2_x}\big)^2.$$ Assume for the moment that (\[control of KG component - estimate for free waves\]) holds. Then after $n$ applications of (\[control of KG component - estimate for free waves\]), together with the standard energy inequality for the homogeneous wave equation, we obtain $$\begin{aligned} \sup_{t\in [0, (n+1) \Delta T]} \| I^2 \phi^{(0)}_n[t] \|_{H^{r-2s}_x} &{\leqslant}\sup_{t \in [0, (n+1) \Delta T]} \| I^2 \phi^{(0)}_0[t] \|_{H^{r-2s}_x} + C n \Delta T \big( \| If_+ \|_{L^2_x} + \| I f_- \|_{L^2_x} \big)^2\notag \\ &{\lesssim}\| I^2\phi[0] \|_{H_x^{r-2s}} + C n \Delta T \big( \| If_+ \|_{L^2_x} + \| I f_- \|_{L^2_x} \big)^2.\label{control of KG component - estimate for free waves2} \end{aligned}$$ If we now combine (\[control of inhomogeneous term\]) and (\[control of KG component - estimate for free waves2\]) we see that since $n {\lesssim}\frac{T}{\Delta T}$ $$\begin{aligned} \sup_{ t \in [n\Delta T, (n+1) \Delta T]} \| I^2\phi[t] \|_{H^{r-2s}_x} &{\leqslant}\sup_{ t \in [n\Delta T, (n+1) \Delta T]} \| I^2\phi^{(0)}_{n}[t] \|_{H^{r-2s}_x} + \sup_{ t \in [n\Delta T, (n+1) \Delta T]} \| I^2\Phi_n[t] \|_{H^{r-2s}_x} \\ &{\lesssim}\| I^2 \phi[0] \|_{H^{r-2s}_x} + (n+1) \Delta T \Big( \| I f_+ \|_{L^2_x} + \| I f_- \|_{L^2_x}\Big)^2 \\ &{\lesssim}\| I^2 \phi[0] \|_{H^{r-2s}_x} + \Big( \| I f_+ \|_{L^2_x} + \| I f_- \|_{L^2_x}\Big)^2 \end{aligned}$$ where the implied constant is independent of $N$, $C^*$, and $\Delta T$. Thus we obtain (\[bound on phi eqn2\]) as required. It only remains to prove (\[control of KG component - estimate for free waves\]). We begin by observing that $$\big(\phi^{(0)}_{j} - \phi^{(0)}_{j-1} \big) (j \Delta T) = \phi( j\Delta T) - \phi^{(0)}_{j-1}( j\Delta T) = \Phi_{j-1}( j \Delta T).$$ Hence the difference $\phi^{(0)}_{j} - \phi^{(0)}_{j-1}$ satisfies the equation $$\begin{aligned} \Box( \phi^{(0)}_{j} - \phi^{(0)}_{j-1} ) &= m^2( \phi^{(0)}_{j} - \phi^{(0)}_{j-1}) \\ \big(\phi^{(0)}_{j} - \phi^{(0)}_{j-1}\big)(j\Delta T) &= \Phi_{j-1}(j\Delta T), \\ {\partial}_t \big(\phi^{(0)}_{j} - \phi^{(0)}_{j-1}\big)(j\Delta T) &= {\partial}_t \Phi_{j-1}(j\Delta T). \end{aligned}$$ Therefore $$\begin{aligned} \sup_{ t \in [0, (n+1)\Delta T]} \| I^2\phi^{(0)}_j[t] \|_{H^{r-2s}_x} &{\leqslant}\sup_{ t\in [0, (n+1) \Delta T]} \| I^2 \phi^{(0)}_{j-1}[t] \|_{H^{r-2s}_x } + \sup_{ t\in [0, (n+1) \Delta T]} \| I^2 \big(\phi^{(0)}_{j} - \phi^{(0)}_{j-1}\big)[t] \|_{H^{r-2s}_x } \\ &{\leqslant}\sup_{ t\in [0, (n+1) \Delta T]} \| I^2 \phi^{(0)}_{j-1}[t] \|_{H^{r-2s}_x } + C \| \Phi_{j-1}[j\Delta T] \|_{H^{r-2s}_x} \end{aligned}$$ and so (\[control of KG component - estimate for free waves\]) follows from (\[control of inhomogeneous term\]). Consequently, we deduce that the induction assumptions (\[proof of gwp - induc assump for psi\]) and (\[proof of gwp - induc assump for phi\]) hold on the larger interval $[0, (n+1) \Delta T]$ and hence Theorem \[thm DKG in intro\] follows. Proof of Lemma \[lem smoothing\] {#subsec - proof of lem smoothing} -------------------------------- Let $Q(f, g) = I(fg) - I^2 f I g$. Note that $$\widehat{Q(f, g)}(\xi) = \int_{\mathbb{R}}\big(\rho(\xi ) - \rho(\xi - \eta)^2 \rho(\eta) \big) \widehat{f}(\xi - \eta) \widehat{g}(\eta) d\eta .$$ An application of Cauchy-Schwarz together with Lemma \[lem time dilation for Xsb\] gives $$\begin{aligned} \Big| \int_0^{t'} \int_{\mathbb{R}}\big(I(\phi u) - I^2 \phi I u\big)\overline{I v} \,dx dt \Big| &{\lesssim}\| {\mathbbold{1}}_{[0, t']} Q(\phi, u) \|_{X^{0, -\frac{1}{2} + \epsilon}_\mp} \| I v \|_{X^{0, \frac{1}{2} - \epsilon}_\mp([0, t']\times {\mathbb{R}})}\\ &{\lesssim}\| Q(\phi, u) \|_{X^{0, -\frac{1}{2} + \epsilon}_\mp([0, t']\times {\mathbb{R}})} \| I v \|_{X^{0, \frac{1}{2} - \epsilon}_\mp([0, t']\times {\mathbb{R}})}\\ &{\lesssim}\|Q(\phi, u) \|_{X^{0, -\frac{1}{2} + \epsilon}_\mp(S_{\Delta T})} \| I v \|_{X^{0, b}_\mp(S_{\Delta T})}. \end{aligned}$$ Thus, by the definition of $X^{0, b}_\pm(S_{\Delta T})$, it suffices to prove that $$\label{appendix lem smoothing - main eqn} \| Q(\phi, u) \|_{X^{0, -\frac{1}{2} +\epsilon}_\mp(S_{\Delta T})} {\lesssim}\Delta T^{\frac{1}{2} - 2\epsilon} N^{2s - r + 2\epsilon} \| I^2 \phi \|_{H^{r-2s, b}} \| I u \|_{X^{0, b}_\pm}.$$ where we may assume that $\phi$ and $u$ are supported in $[-\Delta T, 2 \Delta T]\times {\mathbb{R}}$. Note that since the $I$ operator only acts on the spatial variable $x$, $I^2\phi$ and $I u$ are also supported in $[-\Delta T, 2 \Delta T]\times {\mathbb{R}}$. Write $\phi = \phi_{low} + \phi_{hi}$ and $u = u_{low} + u_{hi}$ where, as in the proof of Lemma \[lem control of inhomogeneous term\], we define $\widetilde{\phi}_{low} = {\mathbbold{1}}_{|\xi|{\leqslant}\frac{N}{2}}\widetilde{\phi}$, and $u_{low}$ is defined similarly. We consider each of the possible interactions separately.\ **$\bullet$ Case 1 ( $low$-$low$).** In this case we simply note that $Q(\phi, u) =0$ and hence (\[appendix lem smoothing - main eqn\]) holds trivially.\ **$\bullet$ Case 2 ( $low$-$hi$).** We need to use the smoothing property of the bilinear form $Q(\phi, u)$ to transfer a derivative from $\phi_{low}$ to $u_{hi}$. More precisely, suppose $|\xi - \eta|<\frac{N}{2}$ and $|\eta|>\frac{N}{2}$. Then since $\rho'(z) {\lesssim}N^{-s} |z|^{s-1}$ for $|z| {\geqslant}\frac{N}{2}$ we have $$\begin{aligned} |\rho(\xi) - \rho(\xi - \eta)^2 \rho(\eta)| &= |\rho(\xi) - \rho(\eta)| \\ &{\lesssim}N^{-s} |\eta|^{s-1} |\xi - \eta| \\ &\approx \rho(\eta) \frac{|\xi - \eta|}{|\eta|} {\lesssim}\rho(\eta) \frac{ |\xi - \eta|^{r-2s}}{|\eta|^{r-2s}}\end{aligned}$$ provided $r-2s < 1$. Hence $$|\widetilde{Q(\phi_{low}, u_{hi})}(\tau, \xi)| {\lesssim}\int_{{\mathbb{R}}^2} |\xi-\eta|^{r-2s}|\widetilde{\phi}_{low}(\tau - \lambda, \xi - \eta) |\eta|^{-r+2s} \rho(\eta) |\widetilde{u}_{hi}(\lambda, \eta)| d \lambda d \eta.$$ Thus we can move the derivative $|\nabla|^{r-2s}$ from $u_{hi}$ to $\phi_{low}$, where we let $\widehat{(|\nabla|^s f)}(\xi) = |\xi|^s \widehat{f}(\xi)$. This is the essential step which allows us to prove (\[appendix lem smoothing - main eqn\]) in the $low$-$hi$ case. We now apply (\[Imethod can trade regularity for powers of N\]) and Theorem \[thm main X\^sb esimate\] with $s_1 = s_2 = 0$, $s_3 = 2\epsilon$, $b_1 = \frac{1}{2} -\epsilon$, $b_2=0$, and $b_3 = b$ to obtain $$\begin{aligned} \| Q(\phi_{low}, u_{hi} )\|_{X^{0, -\frac{1}{2} + \epsilon}_\mp(S_{\Delta T})} &{\lesssim}\big\| |\nabla|^{r-2s} \phi_{low} |\nabla|^{-r+2s} Iu_{hi} \big\|_{X^{0, -\frac{1}{2} + \epsilon}_{\mp}}\notag\\ &{\lesssim}\big\| |\nabla|^{r-2s} \phi_{low} \big\|_{L^2_{t, x} } \big\| |\nabla|^{-r + 2s } I u_{hi} \big\|_{X^{2\epsilon, b}_{\pm}}\notag\\ &{\lesssim}\Delta T^{\frac{1}{2}} N^{-r +2s + 2\epsilon} \| I^2 \phi \|_{L^\infty_tH_x^{r-2s}} \| I u\|_{X^{0, b}_{\pm}} \\ &{\lesssim}\Delta T^{\frac{1}{2}} N^{-r +2s + 2\epsilon} \| I^2 \phi \|_{H^{r-2s, b}} \| I u\|_{X^{0, b}_{\pm}} \end{aligned}$$ where we used the assumption ${\text{supp }\,}\phi \subset \{ [-\Delta T, 2 \Delta T] \times {\mathbb{R}}\}$.\ **$\bullet$ Case 3 ( $hi$-$low$).** In this case we do not have to transfer any regularity and we simply use the estimate $ \rho(\xi) - \rho(\xi - \eta)^2 \rho(\eta) {\lesssim}1$. Then (\[Imethod can trade regularity for powers of N\]) together with an identical application of Theorem \[thm main X\^sb esimate\] to the $low$-$hi$ case gives $$\begin{aligned} \| Q(\phi_{hi}, u_{low} )\|_{X^{0, -\frac{1}{2} + \epsilon}_\mp(S_{\Delta T})} &{\lesssim}\big\| \phi_{hi} u_{low} \big\|_{X^{0, -\frac{1}{2} + \epsilon}_{\mp}}\\ &{\lesssim}\big\| \phi_{hi} \big\|_{L^2_{t, x} } \big\| u_{low} \big\|_{X^{2\epsilon, b}_{\pm}}\\ &{\lesssim}\Delta T^{\frac{1}{2}} N^{2s -r + 2\epsilon} \| I^2 \phi \|_{L^\infty_tH_x^{r-2s}} \| Iu \|_{X^{0, b}_{\pm}} \\ &{\lesssim}\Delta T^{\frac{1}{2}} N^{2s -r + 2\epsilon} \| I^2 \phi \|_{H^{r-2s, b}} \| Iu \|_{X^{0, b}_{\pm}} \end{aligned}$$ where as before, we used the assumption ${\text{supp }\,}\phi \subset \{ [-\Delta T, 2 \Delta T] \times {\mathbb{R}}\}$.\ **$\bullet$ Case 4 ( $hi$-$hi$).** This is the most difficult case and we need to make full use of the generality of Theorem \[thm main X\^sb esimate\] to obtain the term $\Delta T^{\frac{1}{2} - \epsilon}$. We decompose $\phi_{hi} = \phi_{hi}^+ + \phi_{hi}^-$ where $$\widetilde{\phi}_{hi}^+ = {\mathbbold{1}}_{\{ \tau\xi <0\}} \widetilde{\phi}_{hi}$$ is the restriction of $\widetilde{\phi}_{hi}$ to the second and fourth quadrants of ${\mathbb{R}}^{1+1}$. Note that $\| \phi^\pm \|_{X^{s, b}_\pm} {\lesssim}\| \phi \|_{H^{s, b}}$. Assume that we have $\pm =+$, $\mp=-$ in (\[appendix lem smoothing - main eqn\]), it will be clear that the proof will also apply to the $\pm=-$, $\mp =+$ case.\ **$\bullet$ Case 4a ($hi$-$hi$ $+$).** As in $hi$-$low$ case we start by discarding the smoothing multiplier $Q$. We now apply Theorem \[thm main X\^sb esimate\] with $s_1=-s + 2\epsilon$, $s_2 = s$, $s_3=0$, $b_1 = b_2 = \frac{1}{4}$, and $b_3 = \frac{1}{2} - \epsilon$ to obtain $$\begin{aligned} \| Q( \phi_{hi}^+, u_{hi}) \|_{X^{0, -\frac{1}{2} + \epsilon}_-(S_{\Delta T})} &{\lesssim}\| \phi^+_{hi} u_{hi} \|_{X^{0, -\frac{1}{2} + \epsilon}_-} \\ &{\lesssim}\| \phi^+_{hi} \|_{X^{-s + 2\epsilon, \frac{1}{4}}_+} \| u_{hi} \|_{X^{s, \frac{1}{4}}_+} \\ &{\lesssim}N^{2s-r + 2\epsilon} \| I^2 \phi \|_{H^{r-2s, \frac{1}{4} }} \| I u \|_{X^{0, \frac{1}{4}}_+}\\ &{\lesssim}\Delta T^{\frac{1}{2} - \epsilon} N^{2s -r + 2\epsilon} \| I^2 \phi \|_{H^{r-2s, b}} \| I u \|_{X^{0, b}_+} \end{aligned}$$ where we needed $-s<r$, $\epsilon>0$ sufficiently small, and in the final line we used the assumption that $\phi$, $u$, are compactly supported in the interval $[-\Delta T, 2\Delta T]$ together with Lemma \[lem time dilation for Xsb\] and Lemma \[lem time dilation for Hrb\].\ **$\bullet$ Case 4b ( $hi$-$hi$ $-$).** Here we first apply Lemma \[lem time dilation for Xsb\], discard the multiplier $Q$, and then apply Theorem \[thm main X\^sb esimate\] with $s_1 =0$, $s_2=-s + \epsilon$, $s_3=s$, $b_1=b_2=\frac{1}{4}$, and $b_3=\frac{1}{2} + \epsilon$ to obtain $$\begin{aligned} \| Q( \phi_{hi}^-, u_{hi}) \|_{X^{0, -\frac{1}{2} + \epsilon}_-(S_{\Delta T})} &{\lesssim}\Delta T^{\frac{1}{4} - \epsilon}\| \phi^-_{hi} u_{hi} \|_{X^{0, -\frac{1}{4} }_-} \\ &{\lesssim}\Delta T^{\frac{1}{4} - \epsilon}\| \phi^-_{hi} \|_{X^{-s + \epsilon, \frac{1}{4}}_-} \| u_{hi} \|_{X^{s, b}_+} \\ &{\lesssim}\Delta T^{\frac{1}{4} - \epsilon}N^{2s-r + \epsilon} \| I^2 \phi \|_{H^{r-2s, \frac{1}{4} }} \| I u \|_{X^{0, b}_+}\\ &{\lesssim}\Delta T^{\frac{1}{2} - 2\epsilon} N^{2s -r + \epsilon} \| I^2 \phi \|_{H^{r-2s, b}} \| I u \|_{X^{0, b}_+} \end{aligned}$$ where, as previously, we used the assumption on the support of $\phi$ in the last line. Proof of Lemma \[lem mod lwp\] {#subsec - proof of lem mod lwp} ------------------------------ Lemma \[lem mod lwp\] follows by a standard fixed point argument using Lemma \[lem energy est for Xsb\], Lemma \[lem energy est for Hrb\], and the estimates $$\label{appendix mod lwp - dirac est} \| I ( uv) \|_{X^{0, b-1}_{\pm}(S_{\Delta T})} {\lesssim}\Big( \Delta T^{\frac{1}{2} + r - 2s - 3\epsilon} + N^{-r + 2 s + 2\epsilon}\Big) \| I^2 u \|_{H^{r-2s, b}(S_{\Delta T})} \| I v\|_{X^{0, b}_{\mp}(S_{\Delta T})}$$ and $$\label{appendix mod lwp - wave est} \| I^2 ( uv) \|_{H^{r-2s -1, b-1}(S_{\Delta T})} {\lesssim}\Big( \Delta T^{1 -\epsilon} + N^{-\frac{1}{2} + 2\epsilon}\Big) \| I u \|_{X_+^{0, b}(S_{\Delta T})} \| I v\|_{X^{0, b}_{-}(S_{\Delta T})}.$$ See for instance [@Tesfahun2009].\ We start by proving (\[appendix mod lwp - dirac est\]). As in the proof of Lemma \[lem smoothing\], we decompose $u= u_{low} + u_{hi}$ and $v=v_{low} + v_{hi}$.\ **$\bullet$ Case 1 ( $low$-$low$).** We split $u_{low} = u_{low}^+ + u_{low}^{-}$ where we use the same notation as in Subsection \[subsec - proof of lem smoothing\], Case 4. Observe that an application of Theorem \[thm main X\^sb esimate\] gives $$\label{lem mod lwp proof - low low case eqn 1} \int_{ {\mathbb{R}}^{2}} \Pi_{j=1}^3 \psi_j dx dt {\lesssim}\| \psi_1 \|_{X^{0, \epsilon}_\pm} \| \psi_2 \|_{X^{r-2s, \frac{1}{2} - r + 2s + \frac{\epsilon}{2} }_\pm} \| \psi_3 \|_{X^{0, \frac{1}{2}- \epsilon}_{\mp}}$$ provided that $0<r - 2s< \frac{1}{2}$ and $\epsilon>0$ is sufficiently small. Hence, using Lemma \[lem time dilation for Xsb\] together with two applications of (\[lem mod lwp proof - low low case eqn 1\]) we see that $$\begin{aligned} \| I(u_{low} v_{low}) \|_{X^{0, b-1}_\pm(S_{\Delta T})} &{\lesssim}\Delta T^{\frac{1}{2} - 2 \epsilon} \| u_{low}^\pm v_{low} \|_{X^{0, - \epsilon}_\pm(S_{\Delta T})} + \| u_{low}^{\mp} v_{low} \|_{X^{0, b-1}_{\pm}(S_{\Delta T})} \\ &{\lesssim}\Delta T^{\frac{1}{2} - 2 \epsilon} \| u_{low}^{\pm} \|_{X^{r-2s, \frac{1}{2} - r + 2s + \frac{\epsilon}{2}}_{\pm}(S_{\Delta T})} \| v_{low} \|_{X^{0, \frac{1}{2} + \epsilon}_{\mp}(S_{\Delta T})} \\ &\qquad \qquad \qquad + \| u_{low}^{\mp} \|_{X^{r-2s, \frac{1}{2} - r + 2s + \frac{\epsilon}{2}}_{\mp}(S_{\Delta T})} \| v_{low} \|_{X^{0, \epsilon}_{\mp}(S_{\Delta T})}\\ &{\lesssim}\Delta T^{\frac{1}{2} + r - 2s - 3\epsilon} \| I^2 u \|_{H^{r-2s, b}(S_{\Delta T})} \| I v \|_{X^{0, b}_\pm(S_{\Delta T})}. \end{aligned}$$ **$\bullet$ Case 2 ( $low$-$hi$).** Note that Corollary \[cor wave-sobolev and X\^sb estimate\] implies that $$\label{lem mod lwp proof - low hi case eqn3} \| \psi {\varphi}\|_{X^{0, b-1}_{\pm}} {\lesssim}\| \psi \|_{H^{s_1, b}} \| \psi \|_{X^{s_2, b}_{\mp}}$$ provided $$s_1>0, \qquad s_2> - \frac{1}{2} + \epsilon, \qquad s_1 + s_2>\epsilon.$$ We now apply (\[lem mod lwp proof - low hi case eqn3\]) with $s_1 = r-2s$, $s_2 = 2s - r + 2 \epsilon$ to get $$\begin{aligned} \| I( u_{low} v_{hi} ) \|_{X^{0, b-1}_{\pm}(S_{\Delta T})} &{\lesssim}\| u_{low} \|_{H^{r-2s, b}(S_{\Delta T})} \| v_{hi } \|_{X^{2s - r + 2\epsilon, b}_{\mp}(S_{\Delta T})} \\ &{\lesssim}N^{2s - r + 2\epsilon} \| I^2 u \|_{H^{r-2s, b}(S_{\Delta T})} \| I v \|_{X^{0, b}_{\mp}(S_{\Delta T})}. \end{aligned}$$ **$\bullet$ Case 3 ( $hi$-$low$).** An application of (\[lem mod lwp proof - low hi case eqn3\]) with $s_1 = 2\epsilon$, $s_2 = 0$ gives $$\begin{aligned} \| I( u_{hi} v_{low} ) \|_{X^{0, b-1}_{\pm}(S_{\Delta T})} &{\lesssim}\| u_{hi} \|_{H^{2\epsilon, b}(S_{\Delta T})} \| v_{low } \|_{X^{0, b}_{\mp}(S_{\Delta T})} \\ &{\lesssim}N^{2s - r + 2\epsilon} \| I^2 u \|_{H^{r-2s, b}(S_{\Delta T})} \| I v \|_{X^{0, b}_{\mp}(S_{\Delta T})}. \end{aligned}$$ **$\bullet$ Case 4 ( $hi$-$hi$).** We apply (\[lem mod lwp proof - low hi case eqn3\]) with $s_1=r$, $s_2=- r + 2\epsilon$ and observe that $$\begin{aligned} \| I( u_{hi} v_{hi} ) \|_{X^{0, b-1}_{\pm}(S_{\Delta T})} &{\lesssim}\| u_{hi} \|_{H^{r, b}(S_{\Delta T})} \| v_{hi } \|_{X^{ - r + 2\epsilon, b}_{\mp}(S_{\Delta T})} \\ &{\lesssim}N^{2s - r + 2\epsilon} \| I^2 u \|_{H^{r-2s, b}(S_{\Delta T})} \| I v \|_{X^{0, b}_{\mp}(S_{\Delta T})} \end{aligned}$$ where we used the assumption $r>-s$ together with (\[Imethod can trade regularity for powers of N\]).\ We now prove prove (\[appendix mod lwp - wave est\]). We again break $u = u_{low} + u_{hi}$ and $v = v_{low} + v_{hi}$ and consider each of the possible interactions separately.\ **$\bullet$ Case 1 ( $low$-$low$).** Corollary \[cor wave-sobolev and X\^sb estimate\] together with the assumption $r - 2s <\frac{1}{2}$ gives $$\begin{aligned} \| I^2(u_{low} v_{low} ) \|_{H^{r-2s -1, b-1}(S_{\Delta T})} &{\lesssim}\| u_{low} v_{low} \|_{H^{-\frac{1}{2} , b-1}(S_{\Delta T})} \\ &{\lesssim}\| u_{low} \|_{X^{0, \epsilon}_+(S_{\Delta T})} \| v_{low} \|_{X^{0, \epsilon}_-(S_{\Delta T})} \\ &{\lesssim}\Delta T^{1 - 2\epsilon} \| I u \|_{X^{0, b}_+(S_{\Delta T})} \| I v\|_{X^{0, b}_{-}(S_{\Delta T})}. \end{aligned}$$ **$\bullet$ Case 2 ( $low$-$hi$).** For the remaining cases we will use the estimate $$\label{lem mod lwp proof - low hi case eqn4} \| \psi {\varphi}\|_{H^{-\frac{1}{2}, b-1}} {\lesssim}\| \psi\|_{X^{s_1, b}_+} \| {\varphi}\|_{X^{s_2, b}_-}$$ which follows from Corollary \[cor wave-sobolev and X\^sb estimate\] provided $$s_1>-\frac{1}{2},\qquad s_2>-\frac{1}{2}, \qquad s_1 + s_2> -\frac{1}{2} + \epsilon.$$ The $low$-$hi$ case now follows by taking $s_1=0$, $s_2 = - \frac{1}{2} + 2\epsilon$ and observing that $$\begin{aligned} \| I^2 ( u_{low} v_{hi} ) \|_{H^{r-2s -1, b-1}(S_{\Delta T})} &{\lesssim}\| u_{low} v_{hi} \|_{H^{-\frac{1}{2} , b-1}(S_{\Delta T})} \\ &{\lesssim}\| u_{low} \|_{X_+^{0, b}(S_{\Delta T})} \| v_{hi} \|_{X^{-\frac{1}{2} + 2\epsilon, b}_{-}(S_{\Delta T})}\\ &{\lesssim}N^{-\frac{1}{2} + 2\epsilon} \| I u \|_{X_+^{0, b}(S_{\Delta T})} \| I v\|_{X^{0, b}_{-}(S_{\Delta T})}. \end{aligned}$$ **$\bullet$ Case 3 ( $hi$-$low$).** Follows by taking $s_1 = -\frac{1}{2} + 2\epsilon$, $s_2= 0$ in (\[lem mod lwp proof - low hi case eqn4\]) and using an identical argument to the previous case.\ **$\bullet$ Case 4 ( $hi$-$hi$).** As before, we use (\[lem mod lwp proof - low hi case eqn4\]) with $s_1 = - \frac{1}{2} +2\epsilon - s$ and $s_2 = s$ and apply a similar argument to the above cases. Bilinear Estimates {#sec bilinear est} ================== In this section we prove Theorem \[thm main X\^sb esimate\]. To help simplify the proof, we start by introducing some notation. Let $m : {\mathbb{R}}^3\times{\mathbb{R}}^3 \rightarrow {\mathbb{C}}$ and consider the inequality $$\label{general multiplier estimate} \Big| \int_{\Gamma} m(\tau, \xi) \Pi_{j=1}^3 f_j(\tau_j, \xi_j) d\sigma(\tau, \xi)\Big| {\lesssim}\Pi_{j=1}^3 \| f_j \|_{L^2_{\tau, \xi}}$$ where $\tau, \xi \in {\mathbb{R}}^3$, $\Gamma = \{ \xi_1 + \xi_2 + \xi_3 =0, \,\,\,\, \tau_1 + \tau_2 +\tau_3 =0\}$, and $d\sigma$ is the surface measure on the hypersurface $\Gamma$. Without loss of generality, we may assume $f_j{\geqslant}0$ as we are using $L^2$ norms on the right hand side of (\[general multiplier estimate\]). Note that the $X^{s, b}$ estimate contained in Theorem \[thm main X\^sb esimate\] can be written in the form (\[general multiplier estimate\]) after applying Plancherel and relabeling. Following Tao in [@Tao2001], for a multiplier $m$, we use the notation $\| m \|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}$ to denote the optimal constant in (\[general multiplier estimate\]). This norm $\| \cdot\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}$ was studied in detail in [@Tao2001]. We recall the following elementary properties. Firstly, if $m_1 {\leqslant}m_2$ then it is easy to see that $\| m_1 \|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} {\leqslant}\| m_2 \|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}$. Secondly, via Cauchy-Schwarz, for $j, k \in \{1, 2, 3\}$, $j \neq k$, we have the characteristic function estimate $$\label{characteristic function est} \| {\mathbbold{1}}_A(\tau_j, \xi_j) {\mathbbold{1}}_B(\tau_k, \xi_k) \|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} {\lesssim}\sup_{(\tau, \xi) \in {\mathbb{R}}^2} \big| \{ (\lambda, \eta) \in A \,\,:\,\,(\tau - \tau, \xi - \xi) \in B\, \}\big|^{\frac{1}{2}}$$ where $|\Omega|$ denotes the measure of the set $\Omega \subset {\mathbb{R}}^2$. We refer the reader to [@Tao2001] for a proof as well a number of other properties of the norm $\| \cdot \|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}$. Let $$\lambda_1 = \tau_1 \pm \xi_1, \qquad \lambda_2 = \tau_2 \pm \xi_2, \qquad \lambda_3 = \tau_3 \mp \xi_3.$$ Note that if $(\tau, \xi ) \in \Gamma$, then $$\label{lambda sum} \lambda_1 + \lambda_2 + \lambda_3 = \pm2 \xi_3.$$ Let $N_j, L_j \in 2^{{\mathbb{N}}}$, $j = 1, 2, 3$, be dyadic numbers. Our aim is to decompose the $\xi_j$ and $\lambda_j$ variables dyadically, and reduce the problem of estimating $\| m \|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}$ to trying to bound the frequency localised version $$\Big\| m(\tau, \xi) \Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j| \approx N_j, \,|\lambda_j|\approx L_j\}} \Big\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}$$ together with computing a dyadic summation. Note that if we restrict $|\xi_j| \approx N_j$, then since $\xi_1+ \xi_2 + \xi_3 =0$ we must have $N_{max} \approx N_{med}$ where, as in the introduction, $N_{max} = \max\{ N_1, N_2, N_3\}$, $N_{med}$ and $N_{min}$ are defined similarly. Similarly, if $|\lambda_j| \approx L_j$, then (\[lambda sum\]) implies that $L_{max} \approx \max\{ L_{med}, N_3\}$. Hence $$1 \approx \sum_{N_{max} \approx N_{med} } \sum_{L_{max} \approx \max\{ N_3, L_{med} \} }\Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j| \approx N_j, \,|\lambda_j|\approx L_j\}}.$$ Combining these observations with results from [@Tao2001] leads to the following. \[lem reduction to dyadic pieces\] $$\| m \|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} {\lesssim}\sup_{N} \sum_{N_{max} \approx N_{med} \approx N} \sum_{L_{max} \approx \max\{ N_3, L_{med} \} } \Big\| m(\tau, \xi) \Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j| \approx N_j, \,|\lambda_j|\approx L_j\}} \Big\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}.$$ The inequality follows from the triangle inequality together with [@Tao2001 Lemma 3.11]. Alternatively, we can just compute by hand. For ease of notation, let $a_{N_1} = \| f_1 {\mathbbold{1}}_{|\xi_1| \approx N_1} \|_{L^2}$, $b_{N_2} = \| f_2 {\mathbbold{1}}_{|\xi_2| \approx N_2} \|_{L^2}$, $c_{N_3} = \| f_3 {\mathbbold{1}}_{|\xi_3| \approx N_3} \|_{L^2}$, and $A_{N_1, N_2, N_3} = \big\| m(\tau, \xi) \Pi_{j=1}^3 {\mathbbold{1}}_{|\xi_j| \approx N_j} \big\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}$. Then since $\xi_j$ lie on the surface $\Gamma$, we have $\xi_1 + \xi_2 + \xi_3 = 0$ and so $$\begin{aligned} \int_{\Gamma} m(\tau, \xi) \Pi_{j=1}^3 f_j(\tau_j, \xi_j) d\sigma(\tau, \xi) &= \sum_{N_{max} \approx N_{med} } \sum_{N_{min} {\leqslant}N_{med}} \int_{\Gamma} m(\tau, \xi) \Pi_{j=1}^3 f_j(\tau_j, \xi_j) {\mathbbold{1}}_{|\xi_j| \approx N_j} d\sigma(\tau, \xi) \\ &{\leqslant}\sum_{N_{max} \approx N_{med}} \sum_{N_{min} {\lesssim}N_{med}} a_{N_1} b_{N_2} c_{N_3} A_{N_1, N_2, N_3}. \end{aligned}$$ Without loss of generality we may assume that $N_1 {\geqslant}N_2 {\geqslant}N_3$ and so $N_1 \approx N_2$. For simplicity we also assume that $N_1=N_2$ as the general case $N_1 \approx N_2$ is essentially the same. Then $$\begin{aligned} \int_{\Gamma} m(\tau, \xi) \Pi_{j=1}^3 f_j(\tau_j, \xi_j) d\sigma(\tau, \xi) &{\leqslant}\sum_{N_1} a_{N_1} b_{N_1} \sum_{N_3 {\leqslant}N_1} c_{N_3} A_{N_1, N_1, N_3} \\ &{\lesssim}\Big( \sup_{N_3} c_{N_3}\Big) \Big( \sup_{N_1} \sum_{N_3{\leqslant}N_1} A_{N_1, N_1, N_3} \Big) \sum_{N_1} a_{N_1} b_{N_1} \\ &{\lesssim}\Big( \sup_{N_1} \sum_{N_3{\leqslant}N_1} A_{N_1, N_1, N_3} \Big) \Pi_{j=1}^3 \| f_j \|_{L^2}.\end{aligned}$$ Thus we have $$\| m \|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} {\lesssim}\sup_{N} \sum_{N_{max} \approx N_{med} \approx N} \sum_{N_{min}{\leqslant}N_{med}} \Big\| m(\tau, \xi) \Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j| \approx N_j \}} \Big\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}.$$ To decompose the $\lambda_j$ variables follows an similar argument. We omit the details. We now come to the proof of Theorem \[thm main X\^sb esimate\]. To begin with, by taking the Fourier transform and relabeling, the required estimate (\[thm main X\^sb est - main trilinear est\]) is equivalent to showing $$\label{main est fourier side} \Big| \int_{\Gamma} \mathfrak{m}(\tau, \xi) \Pi_{j=1}^3 f_j(\tau_j, \xi_j) d\sigma(\tau, \xi)\Big| {\lesssim}\Pi_{j=1}^3 \| f_j \|_{L^2_{\tau, \xi}}$$ where $$\mathfrak{m}(\tau, \xi) = \frac{ {\langle}\xi_1 {\rangle}^{-s_1} {\langle}\xi_2 {\rangle}^{-s_2} {\langle}\xi_3 {\rangle}^{-s_3}}{{\langle}\tau_1 \pm \xi_1 {\rangle}^{b_1} {\langle}\tau_2 \pm \xi_2 {\rangle}^{b_2} {\langle}\tau_3 \mp \xi_3 {\rangle}^{b_3} }.$$ Note that Theorem \[thm main X\^sb esimate\] follows from the estimate $\| \mathfrak{m} \|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} < \infty$. Now since $$\big\| \mathfrak{m} \,\Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j| \approx N_j, \,|\lambda_j|\approx L_j\}} \big\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} \approx \big\|\Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j| \approx N_j, \,|\lambda_j|\approx L_j\}} \big\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} \Pi_{j=1}^3 N_j^{-s_j} L_j^{-b_j},$$ an application of Lemma \[lem reduction to dyadic pieces\] shows that is suffices to estimate, for every $N \in 2^{{\mathbb{N}}}$, $$\begin{aligned} \label{main dyadic sum} \sum_{N_{max} \approx N_{med} \approx N} N_1^{-s_1} N_2^{-s_2} N_3^{-s_3} \sum_{L_{max} \approx \max\{ L_{med}, N_3\} } L_1^{-b_1} L_2^{-b_2} L_3^{-b_3} \big\| \Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j|\approx N_j, \, |\lambda_j| \approx L_j \}}\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}. \end{aligned}$$ The first step to estimate this sum is the following estimate on the size of the frequency localised multiplier. \[lem dyadic multiplier estimate\] $$\big\| \Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j|\approx N_j, \, |\lambda_j| \approx L_j \}}\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} {\lesssim}\min\Big\{ N_{min}^{\frac{1}{2}} L_{min}^{\frac{1}{2}}, \,L_1^{\frac{1}{2}} L_3^{\frac{1}{2}}, \,L_2^{\frac{1}{2}}L_3^{\frac{1}{2}}\Big\}$$ Let $I= \big\| \Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j|\approx N_j, \, |\lambda_j| \approx L_j \}}\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} $. If we let $A= {\mathbbold{1}}_{|\lambda_j|\approx L_j, \,\, |\xi_j |\approx N_j}$ and $B = {\mathbbold{1}}_{|\lambda_k| \approx L_k, \, |\xi_k| \approx N_k}$ in (\[characteristic function est\]), then an application of Fubini gives $$\begin{aligned} I &{\lesssim}\big\| {\mathbbold{1}}_{\{|\xi_j|\approx N_j, \, |\lambda_j| \approx L_j \}} {\mathbbold{1}}_{\{|\xi_k|\approx N_k, \, |\lambda_k| \approx L_k \}} \|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} \\ &{\lesssim}\sup_{\lambda, \xi \in {\mathbb{R}}} \big| \big\{ |\lambda_j| \approx L_j \,\, : \,\, |\lambda - \lambda_j| \approx L_k \big\} \big|^{\frac{1}{2}} \big| \big\{ |\xi_j| \approx N_j \,\, : \,\, |\xi - \xi_j| \approx N_k \, \big\} \big|^{\frac{1}{2}}\\ &{\lesssim}\min\{ L_j^{\frac{1}{2}}, L_k^{\frac{1}{2}}\} \min\{ N_j^{\frac{1}{2}}, N_k^{\frac{1}{2}} \} \end{aligned}$$ and hence $I {\lesssim}L_{min}^{\frac{1}{2}} N_{min}^{\frac{1}{2}}$. On the other hand, another application of (\[characteristic function est\]) together with a change of variables gives $$\begin{aligned} I {\lesssim}\big\| {\mathbbold{1}}_{\{|\lambda_1| \approx L_1 \}} {\mathbbold{1}}_{\{ |\lambda_3| \approx L_3 \}} \|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} &{\lesssim}\sup_{\tau, \xi \in {\mathbb{R}}} \big| \big\{ |\tau_1 \pm \xi_1| \approx L_1\,\, : \, |\tau \mp\xi - (\tau_1 \mp \xi_1) | \approx L_3 \big\} \big|^{\frac{1}{2}}\\ &{\lesssim}L_1^{\frac{1}{2}} L_3^{\frac{1}{2}}. \end{aligned}$$ A similar argument gives $I {\lesssim}L_2^{\frac{1}{2}} L_3^{\frac{1}{2}}$ and hence lemma follows. We are now ready to preform the computations needed to estimate the dyadic summation (\[main dyadic sum\]). We split this into two parts, by computing the inner summation and then the outer summation. We note the following estimate $$\sum_{a{\leqslant}N {\leqslant}b} N^{\delta} \approx \begin{cases} a^{\delta} \qquad \qquad&\delta<0 \\ \log(b) &\delta=0\\ b^{\delta} &\delta>0 \end{cases}$$ which we use repeatedly. Moreover, we have $\log( r ) {\lesssim}r^\epsilon$ for any $\epsilon>0$ and $r{\geqslant}1$. \[lem inner summation\] Let $b_j + b_k>0$ and $b_1 + b_2 + b_3>\frac{1}{2}$. Then for any sufficiently small $\epsilon>0$ $$\begin{aligned} \sum_{L_{max} \approx \max\{ L_{med}, N_3\}}& L^{-b_1}_1 L^{-b_2}_2 L^{-b_3}_3 \big\| \Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j|\approx N_j, \, |\lambda_j| \approx L_j \}}\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}\\ &{\lesssim}N_3^\epsilon \,\Big( N_3^{\frac{1}{2} - b_1 - b_2 - b_3} N_{min}^\frac{1}{2} + N_{3}^{-b_3} N_{min}^{\frac{1}{2}} + N_3^{-b_{min}} N_{min}^{(\frac{1}{2} - b_{max})_+ +\, (\frac{1}{2} - b_{med})_+}\Big). \end{aligned}$$ We split into the cases $L_{med} {\leqslant}N_3 $ and $L_{med}{\geqslant}N_3$.\ **$\bullet$ Case 1 ($L_{med} {\leqslant}N_3$).** Since the the righthand side of Lemma \[lem dyadic multiplier estimate\] does not behave symmetrically with respect to the sizes of the $L_j$, we need to decompose further into $L_{max} =L_3$ and $L_{max} \neq L_3$.\ **$\bullet$ Case 1a ($L_{med} {\leqslant}N_3$ and $L_{max} \neq L_3$).** We have by Lemma \[lem dyadic multiplier estimate\] $$\big\| \Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j|\approx N_j, \, |\lambda_j| \approx L_j \}}\big\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} {\lesssim}L_{min}^{\frac{1}{2}} \min\{ N_{min}^{\frac{1}{2}}, L_{med}^{\frac{1}{2}}\}.$$ Since the righthand side is symmetric under permutations of $\{1, 2, 3\}$, we may assume $L_1 {\geqslant}L_2 {\geqslant}L_3$. Then for any $\epsilon>0$ $$\begin{aligned} \sum_{L_{max} \approx N_3 \gtrsim L_{med} } L^{-b_1}_1 L^{-b_2}_2 L^{-b_3}_3 \big\| \Pi_{j=1}^3 &{\mathbbold{1}}_{\{|\xi_j|\approx N_j, \, |\lambda_j| \approx L_j \}}\big\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} \notag \\ &{\lesssim}N^{-b_1}_3 \sum_{L_2 {\leqslant}N_3} L^{-b_2}_2 \min\{ N_{min}^{\frac{1}{2}}, L_2^{\frac{1}{2}}\} \sum_{L_3 {\leqslant}L_2} L_3^{\frac{1}{2}-b_3}\notag\\ &{\lesssim}N^{-b_1}_3 \sum_{L_2 {\leqslant}N_3} L^{(\frac{1}{2}- b_3)_+ - b_2}_2 \log(L_2) \min\{ N_{min}^{\frac{1}{2}}, L_2^{\frac{1}{2}}\} \notag\\ &{\lesssim}N^{-b_1 + \frac{\epsilon}{2} }_3 \sum_{L_2 {\leqslant}N_{min}} L_2^{(\frac{1}{2} - b_3)_+ \,+\, \frac{1}{2} - b_2} \notag\\ &\qquad\qquad + N_{min}^{\frac{1}{2}} N^{-b_1 + \frac{\epsilon}{2} }_3 \sum_{N_{min} {\leqslant}L_2 {\leqslant}N_3} L^{(\frac{1}{2} - b_3)_+ - b_2}_2 \label{lem dyadic multiplier estimate - eqn 4} \end{aligned}$$ Now for the first sum in (\[lem dyadic multiplier estimate - eqn 4\]) we have $$\begin{aligned} N^{-b_1}_3 \sum_{L_2 {\leqslant}N_{min}} L_2^{(\frac{1}{2} - b_3)_+ \,+\, \frac{1}{2} - b_2} &{\lesssim}N_{min}^{ \big((\frac{1}{2} - b_3)_+ + \frac{1}{2} - b_2\big)_+ } N_3^{ - b_1} \log(N_{min})\\ &{\lesssim}N_{min}^{ (\frac{1}{2} - b_{max})_+ +( \frac{1}{2} - b_{med} )_+ } N_3^{ - b_{min} + \frac{\epsilon}{2} }. \end{aligned}$$ For the second sum we first consider the case $(\frac{1}{2} - b_3)_+ - b_2>0$. Then $$\begin{aligned} N_{min}^{\frac{1}{2}} N^{-b_1}_3 \sum_{N_{min} {\leqslant}L_2 {\leqslant}N_3} L^{(\frac{1}{2} - b_3)_+ - b_2}_2 &{\lesssim}N^{\frac{1}{2}}_{min} N_3^{(\frac{1}{2} - b_3)_+ -b_1 - b_2 } \\ &{\lesssim}N^{\frac{1}{2}}_{min} N_3^{(\frac{1}{2} - b_{max})_+ - b_{med} - b_{min}} \end{aligned}$$ On the other hand if $(\frac{1}{2} - b_3)_+ - b_2{\leqslant}0$ we get $$\begin{aligned} N_{min}^{\frac{1}{2}} N^{-b_1}_3 \sum_{N_{min} {\leqslant}L_2 {\leqslant}N_3} L^{(\frac{1}{2} - b_3)_+ - b_2}_2 &{\lesssim}N^{\frac{1}{2} - b_2 + (\frac{1}{2}- b_3)_+}_{min} N_3^{-b_1} \log(N_3) \\ &{\lesssim}N^{(\frac{1}{2} - b_{max})_+ + (\frac{1}{2} - b_{med})_+ }_{min} N^{-b_{min}+\frac{\epsilon}{2}}_3. \end{aligned}$$ Together with (\[lem dyadic multiplier estimate - eqn 4\]) this then gives $$\begin{aligned} \sum_{L_{max} \approx N_3 \gtrsim L_{med} } L^{-b_1}_1 L^{-b_2}_2 &L^{-b_3}_3 \big\| \Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j|\approx N_j, \, |\lambda_j| \approx L_j \}}\big\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} \\ &{\lesssim}N_3^\epsilon \Big( N_3^{(\frac{1}{2} - b_{max})_+ - b_{med} - b_{min}}N^{\frac{1}{2}}_{min} + N_3^{-b_{min}} N_{min}^{(\frac{1}{2} - b_{max})_+ + (\frac{1}{2} - b_{med})_+}\Big) \\ &{\lesssim}N_3^\epsilon \Big( N_3^{\frac{1}{2} - b_1 - b_2 - b_3}N^{\frac{1}{2}}_{min} + N_3^{-b_{min}} N_{min}^{(\frac{1}{2} - b_{max})_+ + (\frac{1}{2} - b_{med})_+}\Big) \end{aligned}$$ where we used the inequality $$\label{lem dyadic multiplier estimate - eqn 1} N^{\frac{1}{2}}_{min} N^{(\frac{1}{2} - b_{max})_+ - b_{med} - b_{min}}_3 {\leqslant}N^{\frac{1}{2}}_{min} N_3^{\frac{1}{2} - b_1 - b_2 - b_3} + N^{(\frac{1}{2} - b_{max})_+ + (\frac{1}{2} - b_{med})_+}_{min} N_3^{-b_{min}} .$$ which is trivial if $b_{max}<\frac{1}{2}$. On the other hand, if $b_{max} {\geqslant}\frac{1}{2}$, then (\[lem dyadic multiplier estimate - eqn 1\]) follows by noting that since $b_j + b_k > 0$ we have $b_{med} > 0$ and so $$N_{min}^{\frac{1}{2} } N_{3}^{-b_{med} - b_{min}} {\leqslant}N^{\frac{1}{2} - b_{med}}_{min} N_3^{-b_{min}} {\leqslant}N_{min}^{(\frac{1}{2} - b_{med})_+} N_3^{-b_{min}}$$ as required.\ **$\bullet$ Case 1b ($L_{med} {\leqslant}N_3$ and $L_{max} = L_3$).** Lemma \[lem dyadic multiplier estimate\] together with the assumption $L_{max} = L_3$ gives $$\big\| \Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j|\approx N_j, \, |\lambda_j| \approx L_j \}}\big\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} {\lesssim}L_{min}^{\frac{1}{2}} N_{min}^{\frac{1}{2}}.$$ Suppose $L_1 {\leqslant}L_2$. Then $$\begin{aligned} \sum_{L_{max} \approx N_3 \gtrsim L_{med} } L^{-b_1}_1 L^{-b_2}_2 L^{-b_3}_3 \big\| \Pi_{j=1}^3 &{\mathbbold{1}}_{\{|\xi_j|\approx N_j, \, |\lambda_j| \approx L_j \}}\big\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}\label{lem dyadic multiplier estimate - eqn 2} \\ &{\lesssim}N_{min}^{\frac{1}{2}} N_3^{-b_3} \sum_{L_2 {\leqslant}N_3 } L_2^{ - b_2} \sum_{L_1{\leqslant}L_2} L_1^{\frac{1}{2} - b_1} \notag \\ \notag &{\lesssim}N_{min}^{\frac{1}{2}} N_3^{-b_3} \sum_{L_2 {\leqslant}N_3 } L_2^{ (\frac{1}{2}-b_1)_+ - b_2} \log(L_2) \\ \notag &{\lesssim}N_{min}^{\frac{1}{2}} N_3^{((\frac{1}{2} - b_1)_+ - b_2)_+ - b_3 + \epsilon} \end{aligned}$$ for any $\epsilon>0$. If we have $$\label{lem dyadic multiplier estimate - eqn 3} N_{min}^{\frac{1}{2}} N_3^{((\frac{1}{2} - b_1)_+ - b_2)_+ - b_3}{\leqslant}N_3^{\frac{1}{2} - b_1 - b_2 - b_3} N_{min}^\frac{1}{2} + N_{3}^{-b_3} N_{min}^{\frac{1}{2}} + N_3^{-b_{min}} N_{min}^{(\frac{1}{2} - b_{max})_+ + (\frac{1}{2} - b_{med})_+}$$ then we get $$(\ref{lem dyadic multiplier estimate - eqn 2}) {\lesssim}N_3^\epsilon \Big( N_3^{\frac{1}{2} - b_1 - b_2 - b_3} N_{min}^\frac{1}{2} + N_{3}^{-b_3} N_{min}^{\frac{1}{2}} + N_3^{-b_{min}} N_{min}^{(\frac{1}{2} - b_{max})_+ + (\frac{1}{2} - b_{med})_+}\Big)$$ as required. The case $L_1 {\geqslant}L_2$ follows an identical argument and so it remains to show (\[lem dyadic multiplier estimate - eqn 3\]). To this end note that if $(\frac{1}{2} - b_1)_+ - b_2 <0$ then we simply have $$N_{min}^{\frac{1}{2}} N_3^{((\frac{1}{2} - b_1)_+ - b_2)_+ - b_3} = N_{min}^{\frac{1}{2}} N_3^{- b_3}.$$ On the other hand, if $(\frac{1}{2} - b_1 )_+ - b_2 {\geqslant}0$, then by using (\[lem dyadic multiplier estimate - eqn 1\]) we have $$\begin{aligned} N_{min}^{\frac{1}{2}} N_3^{((\frac{1}{2} - b_1)_+ - b_2)_+ - b_3}&=N^{\frac{1}{2}}_{min} N_3^{(\frac{1}{2} - b_1)_+ - b_2 - b_3}\\ &{\leqslant}N^{\frac{1}{2}}_{min} N_3^{(\frac{1}{2} - b_{max})_+ - b_{med} - b_{min}}\\ &{\leqslant}N^{\frac{1}{2}}_{min} N_3^{\frac{1}{2} - b_1 - b_2 - b_3} + N^{(\frac{1}{2} - b_{max})_+ + (\frac{1}{2} - b_{med})_+}_{min} N_3^{-b_{min}} \end{aligned}$$ and so we obtain (\[lem dyadic multiplier estimate - eqn 3\]).\ **$\bullet$ Case 2 ($ L_{med} {\geqslant}N_3$).** In this case we have $L_{max} \approx L_{med}$ and by Lemma \[lem dyadic multiplier estimate\] $$\big\| \Pi_{j=1}^3 {\mathbbold{1}}_{\{|\xi_j|\approx N_j, \, |\lambda_j| \approx L_j \}}\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]} {\lesssim}N^{\frac{1}{2}}_{min} L^{\frac{1}{2}}_{min} .$$ Suppose $L_1 {\geqslant}L_2 {\geqslant}L_3$. Then $$\begin{aligned} \sum_{L_{max} \approx L_{med} \gtrsim N_3} L^{-b_1}_1 L^{-b_2}_2 L^{-b_3}_3 N^{\frac{1}{2}}_{min} L^{\frac{1}{2}}_{min} &\approx N_{min}^{\frac{1}{2}} \sum_{L_2 \gtrsim N_3} L^{-b_1 -b_2}_2 \sum_{L_3 {\leqslant}L_2} L^{\frac{1}{2} - b_3}_3 \\ &{\lesssim}N^{\frac{1}{2}}_{min} \sum_{L_2 \gtrsim N_3} L^{(\frac{1}{2} - b_3)_+ \, -b_1 -b_2}_2 \log(L_2) \\ &{\lesssim}N^{\frac{1}{2}}_{min} N^{ (\frac{1}{2} - b_3)_+ - b_1 - b_2 + \epsilon}_3\\ &{\lesssim}N^{\frac{1}{2}}_{min} N^{ (\frac{1}{2} - b_{max})_+ - b_{med} - b_{min} + \epsilon}_3 \end{aligned}$$ provided $b_1 + b_2 + b_3 > \frac{1}{2}$, $b_j + b_k > 0$, and we choose $\epsilon>0$ sufficiently small. Since this argument also holds for all other size combinations of the $L_j$, we get from (\[lem dyadic multiplier estimate - eqn 1\]) $$\begin{aligned} \sum_{L_{max} \approx L_{med} \gtrsim N_3} L^{-b_1}_1 L^{-b_2}_2 L^{-b_3}_3 \big\| \Pi_{j=1}^3 &{\mathbbold{1}}_{\{|\xi_j|\approx N_j, \, |\lambda_j| \approx L_j \}}\big\|_{[3, {\mathbb{R}}\times {\mathbb{R}}]}\\ &{\lesssim}N_3^\epsilon\Big( N_3^{\frac{1}{2} - b_1 - b_2 - b_3} N^{\frac{1}{2}}_{min} + N_3^{-b_{min}} N^{(\frac{1}{2} - b_{max})_+ + (\frac{1}{2} - b_{med})_+}_{min}\Big) \end{aligned}$$ and so lemma follows. We now come to the proof of Theorem \[thm main X\^sb esimate\]. By Lemma \[lem reduction to dyadic pieces\] and Lemma \[lem inner summation\] it suffices to estimate the sum $$\sup_{N} \sum_{N_{max} \approx N_{med} \approx N} \Big(\Pi_{j=1}^3 N_{j}^{-s_j}\Big) N_{min}^{\alpha} N_3^{-\beta}$$ for the pairs $$(\alpha, \beta) \in \Bigg\{ \Bigg(\frac{1}{2}, \,\,\,b_1 + b_2 + b_3 - \frac{1}{2} - \epsilon\Bigg),\,\,\, \Bigg( \frac{1}{2}, \,\,\,b_3 - \epsilon \Bigg), \,\,\,\Bigg( \Big(\frac{1}{2} - b_{max}\Big)_+ + \Big( \frac{1}{2} - b_{med}\Big)_+,\,\,\, b_{min} - \epsilon \Bigg) \Bigg\}$$ where $\epsilon>0$ may be taken arbitrarily small. Let $s_1' = s_1$, $s_2' = s_2$, and $s_3' = s_3 + \beta$. Then we have to show $$\sup_{N} \sum_{N_{max} \approx N_{med} \approx N} \Big(\Pi_{j=1}^3 N_{j}^{-s_j'}\Big) N_{min}^{\alpha} < \infty.$$ Since this summation is symmetric with respect to the $N_j$, we may assume $ N_1 {\leqslant}N_2 {\leqslant}N_3$. Then $$\begin{aligned} \sum_{N_{max} \approx N_{med} \approx N} \Big(\Pi_{j=1}^3 N_{j}^{-s_j'}\Big) N_{min}^{\alpha} &{\lesssim}N^{-s_2' - s_3'} \sum_{N_1 {\leqslant}N} N_1^{ -s_1' + \alpha} <\infty \end{aligned}$$ provided $s_j' + s_k' {\geqslant}0$ and $s_1' + s_2' + s_3' > \alpha$. These conditions hold by the assumptions in Theorem \[thm main X\^sb esimate\] provided we choose $\epsilon$ sufficiently small. Counter Examples {#sec counter examples} ================ Here we prove that the conditions in Theorem \[thm main X\^sb esimate\] are sharp up to equality. \[prop - counter examples\] Assume the estimate (\[thm main X\^sb est - main trilinear est\]) holds. Then we must have $$b_j + b_k \geqslant 0, \qquad b_1+ b_2 + b_3 \geqslant \frac{1}{2} \label{prop - counter examples - b cond}$$ and for $k \in \{ 1, 2\}$ $$\begin{aligned} \label{prop - counter examples - s cond 1} s_1 + s_2 &\geqslant 0, \\ s_k + s_3 &\geqslant -b_{min},\label{prop - counter examples - s cond 2}\\ s_k +s_3&\geqslant \frac{1}{2} - b_1 - b_2 - b_3, \label{prop - counter examples - s cond 3}\\ s_1 + s_2 + s_3 &\geqslant \frac{1}{2} - b_3,\label{prop - counter examples - s cond 4}\\ s_1 + s_2 + s_3 &\geqslant \Big(\frac{1}{2} - b_{max}\Big)_+ + \Big(\frac{1}{2} - b_{med}\Big)_+ - b_{min}. \label{prop - counter examples - s cond 5}\end{aligned}$$ We note that in some regions the $\pm$ structure in (\[thm main X\^sb esimate\]) is redundant and so the counter examples for the Wave-Sobolev spaces used in [@D'Ancona2010] and [@Selberg2008] would apply. In fact, the counterexamples in [@D'Ancona2010] already essentially show that we must have (\[prop - counter examples - b cond\]), (\[prop - counter examples - s cond 1\]), and (\[prop - counter examples - s cond 5\]). On the other hand, the conditions (\[prop - counter examples - s cond 2\] - \[prop - counter examples - s cond 4\]) reflect the $\pm$ structure and thus cannot be deduced from [@D'Ancona2010]. It suffices to find necessary conditions for the estimate (\[main est fourier side\]). Moreover we may assume $\pm = +$ since the case $\pm = - $ follows by a reflection in the $\tau_j$ variables. Let $\lambda \gg 1$ be some large parameter. The main idea is as follows. Assume we have sets $A, B, C \subset {\mathbb{R}}^{1+1}$ with $$\label{prop - counter examples - set cond 1} |A|\approx \lambda^{d_1}, \qquad |B| \approx \lambda^{d_2}, \qquad |C|\approx \lambda^{d_3}.$$ Moreover, suppose that if $(\tau_2, \xi_2 ) \in B$ and $ (\tau_3, \xi_3) \in C$, then $$\label{prop - counter examples - set cond 2} - ( \tau_2 + \tau_3, \xi_2 + \xi_3) \in A$$ and $$\label{prop - counter examples - set cond 3} \frac{ {\langle}\xi_2 + \xi_3 {\rangle}^{-s_1} {\langle}\xi_2 {\rangle}^{-s_2} {\langle}\xi_3 {\rangle}^{-s_3} }{ {\langle}\tau_2 +\tau_3 + \xi_2 + \xi_3 {\rangle}^{b_1} {\langle}\tau_2 + \xi_2 {\rangle}^{b_2}{\langle}\tau_3 - \xi_3 {\rangle}^{b_3}} \approx \lambda^{ - \delta}.$$ Let $f_1 = {\mathbbold{1}}_A$, $f_2 = {\mathbbold{1}}_B$, $f_3 = {\mathbbold{1}}_C$. Then using the conditions (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) we have $$\begin{aligned} \int_{\Gamma} \mathfrak{m}(\tau, \xi) \Pi_{j=1}^3 f_j(\tau_j, \xi_j) d\sigma(\tau, \xi) &\gtrsim \lambda^{-\delta} \int_B \int_C d\tau_3d\xi_3d\tau_2d\xi_2 \\ &\approx \lambda^{d_2 + d_3 - \delta}. \end{aligned}$$ Therefore, assuming that the inequality (\[main est fourier side\]) holds, we must have $$\lambda^{d_2 + d_3 - \delta} {\lesssim}|A|^{\frac{1}{2}} |B|^{\frac{1}{2}} |C|^{\frac{1}{2}} \approx \lambda^{\frac{d_1 + d_2 + d_3}{2}}.$$ By choosing $\lambda$ large, we then derive the necessary condition $$\label{prop - counter examples - necessary cond} \delta + \frac{ d_1 - d_2 - d_3}{2} {\geqslant}0.$$ Thus it will suffice to find sets $A$, $B$, and $C$ satisfying the conditions (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) with particular values of $\delta$, $d_1$, $d_2$, and $d_3$.\ **$\bullet$ Necessity of (\[prop - counter examples - b cond\]).** We first show that $b_j + b_k {\geqslant}0$. Since the estimate (\[main est fourier side\]) is symmetric in $b_1$, $b_2$, it suffices to consider the pairs $(j, k) \in \{(1, 2), \, (1, 3)\}$. For the first pair, we choose $$B= \{|\tau +\lambda|{\leqslant}1, \,\,\,|\xi|{\leqslant}1 \}, \qquad C = \{|\tau | {\leqslant}1, \,\,\, |\xi|{\leqslant}1 \}, \qquad A = \{|\tau-\lambda|{\leqslant}2, \,\,\, |\xi|{\leqslant}2\} .$$ Then the conditions (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) hold with $d_1 = d_2 = d_3 = 0$ and $\delta = b_1 + b_2$ and so from (\[prop - counter examples - necessary cond\]) we obtain the necessary condition $b_1 + b_2 {\geqslant}0.$ On the other hand, for the pair $(1, 3)$ we choose $$B= \{|\tau |{\leqslant}1,\,\,\,|\xi|{\leqslant}1\}, \qquad C = \{|\tau + \lambda| {\leqslant}1, \,\,\,\,|\xi|{\leqslant}1 \}, \qquad A = \{|\tau-\lambda|{\leqslant}2, \,\,\, |\xi|{\leqslant}2 \} .$$ Then as in the previous case, the conditions (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) hold with $d_1 = d_2 = d_3 = 0$ and $\delta = b_1 + b_3$ and so from (\[prop - counter examples - necessary cond\]) we obtain the necessary condition $b_1 + b_3 {\geqslant}0.$ To show the second condition in (\[prop - counter examples - b cond\]) is also necessary, we take $$B= \{ |\tau - 2\lambda|{\leqslant}\lambda, \,\,\, |\xi|{\leqslant}1\}, \qquad C = \{ |\tau - 2\lambda| {\leqslant}\lambda, \,\,\,\, |\xi|{\leqslant}1\}, \qquad A = \{ |\tau+ 4\lambda|{\leqslant}2\lambda, \,\,\,|\xi|{\leqslant}2 \} .$$ Then (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) hold with $d_1=d_2=d_3 =1$ and $\delta = b_1 +b_2 + b_3$ which leads to the condition $ b_1 + b_2 +b_3 {\geqslant}\frac{1}{2}$.\ **$\bullet$ Necessity of (\[prop - counter examples - s cond 1\]).** Let $$B= \{ |\tau - \lambda|{\leqslant}1, \,\,\, |\xi + \lambda|{\leqslant}1\}, \qquad C = \{ |\tau | {\leqslant}1, \,\,\,\, |\xi|{\leqslant}1\}, \qquad A = \{ |\tau + \lambda|{\leqslant}2, \,\,\,|\xi - \lambda|{\leqslant}2 \} .$$ Then (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) hold with $d_1=d_2=d_3 =0$ and $\delta =s_1 + s_2$ and so we must have (\[prop - counter examples - s cond 1\]).\ **$\bullet$ Necessity of (\[prop - counter examples - s cond 2\]).** By symmetry we may assume $k=1$. Suppose $b_{min} = b_1$ and choose $$B= \{ |\tau |{\leqslant}1, \,\,\, |\xi|{\leqslant}1\}, \qquad C = \{ |\tau - \lambda | {\leqslant}1, \,\,\,\, |\xi - \lambda|{\leqslant}1\}, \qquad A = \{ |\tau + \lambda|{\leqslant}2, \,\,\,|\xi+\lambda|{\leqslant}2 \} .$$ Then (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) hold with $d_1=d_2=d_3 =0$ and $\delta =s_1 + s_3 + b_1$ and so we must have $ s_1 + s_3 + b_1 {\geqslant}0.$ On the other hand, if $b_{min} = b_2$ we let $$B= \{ |\tau + 2\lambda|{\leqslant}1, \,\,\, |\xi |{\leqslant}1\}, \qquad C = \{ |\tau - \lambda | {\leqslant}1, \,\,\,\, |\xi - \lambda|{\leqslant}1\}, \qquad A = \{ |\tau -\lambda |{\leqslant}2, \,\,\,|\xi + \lambda|{\leqslant}2 \} .$$ Then (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) hold with $d_1=d_2=d_3 =0$ and $\delta =s_1 + s_3 + b_2$ and so we obtain the condition $ s_1 + s_3 + b_2 {\geqslant}0.$ The final case, $b_{min} = b_3$, follows by taking $$B= \{ |\tau |{\leqslant}1, \,\,\, |\xi |{\leqslant}1\}, \qquad C= \{ |\tau - \lambda | {\leqslant}1, \,\,\,\, |\xi + \lambda|{\leqslant}1\}, \qquad A = \{ |\tau + \lambda|{\leqslant}2, \,\,\,|\xi-\lambda|{\leqslant}2 \} .$$ Again the conditions (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) hold with $d_1=d_2=d_3 =0$ and $\delta =s_1 + s_3 + b_3$. Hence (\[prop - counter examples - s cond 2\]) is necessary.\ **$\bullet$ Necessity of (\[prop - counter examples - s cond 3\]).** As in the previous case, by symmetry, we may assume $k=1$. Let $$B= \Big\{ |\tau - \lambda | {\leqslant}\frac{\lambda}{4} , \,\,\,\, |\xi|{\leqslant}1\Big\}, \qquad C =\Big\{ |\tau |{\leqslant}\frac{\lambda}{4} , \,\,\,|\xi -\lambda|{\leqslant}\frac{\lambda}{4} \Big\}, \qquad A = \Big\{ |\tau + \lambda|{\leqslant}\frac{\lambda}{2}, \,\,\, |\xi + \lambda|{\leqslant}\frac{\lambda}{2} \Big\} .$$ Then (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) hold with $d_1=d_3=2$, $d_2=1$, and $\delta =s_1 + s_3 + b_1 + b_2 + b_3$. Thus we obtain the necessary condition (\[prop - counter examples - s cond 3\]).\ **$\bullet$ Necessity of (\[prop - counter examples - s cond 4\]).** In this case we choose $$B= \Big\{ |\tau + \xi|{\leqslant}1, \,\,\, |\xi - \lambda|{\leqslant}\frac{\lambda}{4} \Big\}, \qquad C = \Big\{ |\tau + \xi |{\leqslant}1, \,\,\,|\xi -\lambda|{\leqslant}\frac{\lambda}{4} \Big\}, \qquad A=\Big\{ |\tau + \xi|{\leqslant}2, \,\,\, |\xi + 2 \lambda|{\leqslant}\frac{\lambda}{2} \Big\}.$$ Then a simple computation shows that (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) hold with $d_1=d_2=d_3=1$, and $\delta =s_1 + s_2 + s_3 + b_3$. So we see that (\[prop - counter examples - s cond 4\]) is necessary.\ **$\bullet$ Necessity of (\[prop - counter examples - s cond 5\]).** We break this into the 3 conditions $$\label{prop - counter examples - split into 3 cond} s_1 + s_2 + s_3 {\geqslant}1 - b_1 - b_2 - b_3, \qquad s_1 + s_2 + s_3{\geqslant}\frac{1}{2} - b_j - b_k, \qquad s_1 + s_2 + s_3 {\geqslant}- b_{min}.$$ For the first inequality, we take $$B= \Big\{ |\tau|{\leqslant}\frac{\lambda}{4}, \,\,\, |\xi - \lambda|{\leqslant}\frac{\lambda}{4} \Big\}, \qquad C = \Big\{ |\tau |{\leqslant}\frac{\lambda}{4}, \,\,\,|\xi - \lambda|{\leqslant}\frac{\lambda}{4} \Big\}, \qquad A=\Big\{ |\tau|{\leqslant}\frac{\lambda}{2}, \,\,\, |\xi + 2\lambda|{\leqslant}\frac{\lambda}{2} \Big\}.$$ Then we have (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) with $d_1=d_2=d_3=2$, and $\delta =s_1 + s_2 + s_3 +b_1 + b_2+ b_3$. Therefore we must have $ s_1+s_2+s_3 {\geqslant}1 - b_1 - b_2 - b_3.$ We now consider the second inequality in (\[prop - counter examples - split into 3 cond\]). By symmetry, it suffices to consider $(j, k) \in \{ (1, 2), (1, 3)\}$. Let $$B= \Big\{ |\tau + \xi - \lambda|{\leqslant}\frac{\lambda}{4}, \,\,\, |\xi - \lambda|{\leqslant}\frac{\lambda}{4} \Big\}, \,\,\,\, C = \Big\{ |\tau - \xi |{\leqslant}1, \,\,\,|\xi - \lambda|{\leqslant}\frac{\lambda}{4} \Big\}, \,\,\,\, A=\Big\{ |\tau + \xi + 3\lambda|{\leqslant}\lambda, \,\,\, |\xi + 2\lambda|{\leqslant}\frac{\lambda}{2} \Big\}.$$ Then (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) hold with $d_1=d_2=2$, $d_3=1$, and $\delta =s_1 + s_2 + s_3 +b_1 + b_2$. Therefore we must have $ s_1+s_2+s_3 >\frac{1}{2} - b_1 - b_2.$ On the other hand, for the case $(j, k) = (1, 3)$, we take $$B= \Big\{ |\tau + \xi |{\leqslant}1, \,\,\, |\xi - \lambda|{\leqslant}\frac{\lambda}{4} \Big\}, \qquad C = \Big\{ |\tau |{\leqslant}\frac{\lambda}{4}, \,\,\,|\xi - \lambda|{\leqslant}\frac{\lambda}{4} \Big\}, \qquad A=\Big\{ |\tau + \xi + \lambda|{\leqslant}\frac{3\lambda}{4}, \,\,\, |\xi + 2\lambda|{\leqslant}\frac{\lambda}{2} \Big\}.$$ A simple computation shows that (\[prop - counter examples - set cond 1\] - \[prop - counter examples - set cond 3\]) are satisfied with $d_1=d_3=2$, $d_2=1$, and $\delta =s_1 + s_2 + s_3 +b_1 + b_3$. Finally, the third condition in (\[prop - counter examples - split into 3 cond\]) follows from the conditions (\[prop - counter examples - s cond 1\]) and (\[prop - counter examples - s cond 2\]). [10]{} R. Adams and J. Fournier, *Sobolev spaces*, $2^{nd}$ ed., Academic Press, 2003. 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--- abstract: 'We derive a free boson representation of the Yangian double $DY_\hbar(sl_N)$ with arbitrary level $k$ using the observation that there is a correspondence between the $q$-affine algebra and Yangian double associated with the same Cartan matrix. Vertex operator and screening currents are not obtained in the same way.' author: - | Bo-Yu Hou Liu Zhao\ Institute of Modern Physics, Northwest University, Xian 710069, China\ Xiang-Mao Ding\ Institute of Theoretical Physics, Academy of China, Beijing 100080, China date: January 1997 title: | **$q$-affine-Yangian double correspondence\ and free boson representation of\ Yangian double with arbitrary level** --- -1cm -2cm = 15 pt Introduction ============ $q$-algebra and Yangian were proposed by Drinfeld as generalizations of classical Lie algebras with nontrivial Hopf algebra structures [@D1; @D2; @D3; @Dr:new]. Following the Faddeev-Reshetikhin-Takhtajan formalism [@FRT], both kinds of algebras can be considered as associative algebras defined through the Yang-Baxter relation (i.e. $RLL$-relations) with the structure constants determined by the solutions of the quantum Yang-Baxter equation (QYBE). $q$-affine algebra [@Dr:new] and Yangian double [@KT; @K; @KL] with center are respectively affine extensions of $q$-algebra and Yangian. $q$-affine algebra corresponds to the trigonometric solution of QYBE and Yangian corresponds to the rational one–if one considers the Reshetikhin-Semenov-Tian-Shansky realization [@RS] which is the affine analog of Faddeev-Reshetikhin-Takhtajan formalism–and they both were proved to have important applications in certain physical problems, especially in describing the dynamical symmetries and calculating the correlation functions and/or form factors of some two-dimensional exactly solvable lattice statistical model and (1+1)-dimensional completely integrable quantum field theories [@BL; @LS; @S2] In such applications, the infinite-dimensional representations of $q$-affine algebra and Yangian double are often required, especially the representations with higher ($k>1$) level. In practice, realization of complicated algebra in terms of a relatively simple one is proved to be quite effective and useful. In this aspect, the Heisenberg algebra (or free boson) representation has become a common method for obtaining representations of ($q$-)affine algebras. For examples, the free boson representations of $U_q(\widehat{sl_2})$ with an arbitrary level have been obtained in Refs.[@Srs; @M1; @M2; @kim; @abg]. Free boson representation of $U_q(\widehat{sl_N})$ with level 1 was constructed in [@FJ]. Free boson representations of $U_q(\widehat{sl_3})$ and $U_q(\widehat{sl_N})$ with arbitrary level were constructed in [@sl3] and [@sln] respectively. For the Yangian doubles, the free field representation of $DY_\hbar(sl_2)$ with level $k$ was constructed in [@konno]. The level 1 free boson representation of $DY_\hbar(sl_N)$ was given in [@iohara]. But free field representations for Yangian doubles of higher rank and with arbitrary level are still unknown. In this paper we shall address the problem of free field representation of the Yangian double $DY_\hbar(sl_N)$ with arbitrary level $k$. For this purpose we largely rely on the result of [@sln] on free field representation of $U_q(\widehat{sl_N})$ with arbitrary level and observe that there is a simple correspondence between the $q$-affine algebra $U_q(\widehat{sl_N})$ and the Yangian double $DY_\hbar(sl_N)$. This correspondence makes our derivation of free field representation for $DY_\hbar(sl_N)$ greatly simplified. However, we have been unable to obtain the vertex operators and screening currents for $DY_\hbar(sl_N)$ following the same spirit. $q$-affine and Yangian double correspondence ============================================ In this section we first establish the correspondence between the $q$-affine algebra $U_q(\widehat{sl_N})$ and the Yangian double $DY_\hbar(sl_N)$. For this and the subsequent purposes we use the Drinfeld current realizations for both algebras. Other realizations such as the Reshetikhin-Semenov-Tian-Shansky realization of them can be found in [@RS] and [@iohara] (which are actually the quotient algebras of $U_q(\widehat{gl_N})$ and $DY_\hbar(gl_N)$ respectively with respect to a Heisenberg subalgebra), and the equivalence (algebra isomorphism) to Drinfeld realizations were given in [@DF] and [@iohara] respectively. We remark that the Ding-Frenkel isomorphism only provides an algebra isomorphism but not a Hopf algebra isomorphism for $q$-affine algebras at least in the $\widehat{gl_2}$ case [@BSnew]. Whether this is also the case for Yangian doubles is an interesting open problem. Drinfeld currents realization of $U_q(\widehat{sl_N})$ ------------------------------------------------------ $U_q(\widehat{sl_N})$ is an associative algebra generated by the Drinfeld generators $E^{\pm,i}_n~(n\in {\Bbb Z})$, $H^i_n~(n\in {\Bbb Z})~ (i=1,~2,~...,~N-1)$ and the center $\gamma$. Let $$K_i = \mbox{exp}\left((q-q^{-1}) \frac{1}{2} H^i_0 \right),$$ then we can write the Drinfeld currents in the form of formal power series of the complex parameter $z$ with coefficients given by the above generators, $$\begin{aligned} & & H^i(z) = \sum_{n\in {\Bbb Z}}H^i_n z^{-n-1},~~ E^{\pm,i}(z) = \sum_{n \in {\Bbb Z}} E^{\pm,i}_n z^{-n-1},\nonumber\\ & & \psi^i_\pm(z) = \sum_{n \in {\Bbb Z}} \psi^i_{\pm,n} z^{-n} \equiv K_i^{\pm 1} \mbox{exp} \left( \pm (q-q^{-1}) \sum_{\pm n > 0} H^i_n z^{-n} \right). \nonumber\end{aligned}$$ The generating relations for $U_q(\widehat{sl_N})$ in terms of these currents can be written as follows [@sln], $$\begin{aligned} & & [ \psi^i_\pm(z),~\psi^j_\pm(w) ] =0, \label{1} \\ & & (z-q^{a_{ij}} \gamma^{-1} w) (z-q^{-a_{ij}} \gamma w) \psi^i_+(z) \psi^j_-(w) \nonumber\\ & &~~~~= (z-q^{a_{ij}} \gamma w) (z-q^{-a_{ij}} \gamma^{-1} w) \psi^i_-(w) \psi^j_+(z), \label{2}\\ & & (z-q^{\pm a_{ij}} \gamma^{\mp \frac{1}{2}} w) \psi^i_+(z) E^{\pm,j} (w) = (q^{\pm a_{ij}} z- \gamma^{\mp \frac{1}{2}} w) E^{\pm,j} (w) \psi^i_+(z), \label{3} \\ & & (z-q^{ \pm a_{ij}} \gamma^{\mp \frac{1}{2}} w) E^{\pm,j} (z) \psi^i_-(w) = (q^{ \pm a_{ij}} z- \gamma^{\mp \frac{1}{2}} w) \psi^i_-(w) E^{\pm,j} (z), \label{4} \\ & & [ E^{+,i}(z),~E^{-,j}(w) ] = \frac{\delta^{ij}}{(q-q^{-1})zw} \left( \delta(z^{-1}w \gamma) \psi^i_+( \gamma^{\frac{1}{2}}w) - \delta(z^{-1}w \gamma^{-1}) \psi^i_-( \gamma^{- \frac{1}{2}}w) \right), \label{5} \\ & & (z- q^{\pm a_{ij}} w ) E^{\pm,i}(z)E^{\pm,j}(w) = (q^{\pm a_{ij}} z - w ) E^{\pm,j}(w)E^{\pm,i}(z), \label{6} \\ & & E^{\pm,i}(z)E^{\pm,j}(w) = E^{\pm,j}(w)E^{\pm,i}(z) ~~\mbox{for}~a_{ij}=0, \label{7} \\ & & E^{\pm,i}(z_1) E^{\pm,i}(z_2) E^{\pm,j}(w) -(q+q^{-1}) E^{\pm,i}(z_1) E^{\pm,j}(w) E^{\pm,i}(z_2) \nonumber\\ & &~~~~~+ E^{\pm,j}(w) E^{\pm,i}(z_1) E^{\pm,i}(z_2) + (\mbox{replacement:}~z_1 \leftrightarrow z_2) = 0 ~\mbox{for}~a_{ij} = -1, \label{8}\end{aligned}$$ where $a_{ij}$ are elements of the Cartan matrix of the type $A_{N-1}$ and $$\begin{aligned} \delta(x)= \sum_{n\in {\Bbb Z}} x^n.\end{aligned}$$ In this paper we only consider $q$-affine algebra and Yangian double as associative algebras and do not care about the Hopf algebra aspect. The Yangian double $DY_\hbar(sl_N)$ ----------------------------------- As an associative algebra, the Yangian double $DY_\hbar(sl_N)$ is generated by the Drinfeld generators $\{h_{il},~e^{\pm}_{il} | i=1,~2,~...,~N-1;~l \in {\Bbb Z}_{ \geq 0} \}$ and the center $c$. In terms of the formal power series (Drinfeld currents) $$\begin{aligned} & & H^{+}_i(u) = 1 + \hbar \sum_{l \geq 0} h_{il} u^{-l-1},~ H^{-}_i(u) = 1 - \hbar \sum_{l < 0} h_{il} u^{-l-1}, \\ & & E^{\pm}(u) = \sum_{l \in {\Bbb Z}} e^{\pm}_{il} u^{-l-1}\end{aligned}$$ we can write the generating relations for $DY_\hbar(sl_N)$ as follows [@iohara], $$\begin{aligned} & & [ H_i^\pm(u),~H_j^\pm(v) ] =0, \label{y1}\\ & & (u_\mp-v_\pm + B_{ij} \hbar) (u_\pm-v_\mp - B_{ij} \hbar) H_i^+(u) H_j^-(v) \nonumber\\ & &~~~~= (u_\mp-v_\pm - B_{ij} \hbar) (u_\pm-v_\mp + B_{ij} \hbar) H_j^-(v) H_i^+(u), \label{y2} \\ & & (u_\pm-v \mp B_{ij} \hbar) H_i^+(u) E^{\pm}_j(v) = (u_\pm-v \pm B_{ij} \hbar) E^{\pm}_j(v) H_i^+(u), \label{y3} \\ & & (u_\mp-v \mp B_{ij} \hbar) H_i^-(u) E^{\pm}_j(v) = (u_\mp-v \pm B_{ij} \hbar) E^{\pm}_j(v) H_i^-(u), \label{y4} \\ & & (u-v \mp B_{ij} \hbar) E^{\pm}_i(u) E^{\pm}_j(v) = (u-v \pm B_{ij} \hbar) E^{\pm}_j(v) E^{\pm}_i(u), \label{y5} \\ & & [ E^+_i(u),~E^-_j(v) ] = \frac{1}{\hbar} \delta_{ij} \left( \delta( u_- - v_+) H^+_i(v_+) - \delta( u_+ - v_-) H^-_i(v_-) \right), \label{y6} \\ & & E^{\pm}_i(u_1) E^{\pm}_i(u_2) E^{\pm}_j(v) -2 E^{\pm}_i(u_1) E^{\pm}_j(v) E^{\pm}_i(u_2) \nonumber\\ & &~~~~~+ E^{\pm}_j(v) E^{\pm}_i(u_1) E^{\pm}_i(u_2) + (\mbox{replacement:}~u_1 \leftrightarrow u_2) = 0 ~\mbox{for}~|i-j|=1, \label{y7}\\ & & E^{\pm}_i(u) E^{\pm}_j(v) = E^{\pm}_j(v) E^{\pm}_i(u) ~~\mbox{for}~|i-j|>1, \label{y8}\end{aligned}$$ where $$u_\pm = u \pm \frac{1}{4} \hbar c$$ and $$B_{ij} = \frac{1}{2} a_{ij}.$$ $q$-affine-Yangian double correspondence ---------------------------------------- Our central goal is to establish a free boson representation of the Yangian double $DY_\hbar(sl_N)$. For this we would like to use the known results [@sln] for the $q$-affine algebra $U_q(\widehat{sl_N})$ by establishing a correspondence principle between these two algebras. Such a correspondence principle has been expected for some time and was “quite mysterious” as stated in Ref.[@iohara]. For the present authors, however, such a correspondence is rather obvious by making use of the Drinfeld current realizations for both $U_q(\widehat{sl_N})$ and $DY_\hbar(sl_N)$. For other realizations no such an obvious observation could be obtained. We give the following \[ob1\] ($q$-affine-Yangian double correspondence). The following gives a simple correspondence between $U_q(\widehat{sl_N})$ and $DY_\hbar(sl_N)$ as associative algebras $$\begin{aligned} & & q \rightarrow \mbox{e}^{\frac{\hbar}{2}},~~~ \gamma \rightarrow \mbox{e}^{\frac{\hbar c}{2}},\\ & & z \rightarrow \mbox{e}^{u} ,\\ & & \psi^i_{\pm}(z) \rightarrow H^{\pm}_i(u),\\ & & z E^{\pm,i}(z) \rightarrow E^{\pm}_i( u)\end{aligned}$$ in the limit $\hbar \rightarrow 0,~u \rightarrow 0$ up to the linear approximation in $\hbar$ and $u$. We remark that the above observation only gives a rule for obtaining equations (\[y1\]-\[y8\]) from (\[1\]-\[8\]) and does not imply any more fundamental Hopf algebraic or algebraic relations. Free boson representation of $DY_\hbar(sl_N)$ with arbitrary level ================================================================== In this section we shall consider our central problem–the establishment of a free boson representation of $DY_\hbar(sl_N)$ with arbitrary level. For $N=2$ this problem has already been solved in Ref. [@konno]. For generic $N$, the desired expressions are rather complicated and our construction depend largely on the observation \[ob1\] and the result of [@sln]. One crucial difference of our construction from the one in [@sln] is that, in our case, the Yangian double $DY_\hbar(sl_N)$ should be realized through [*ordinary*]{} Heisenberg algebras (i.e. [*without*]{} deformation), whereas in Ref.[@sln], $U_q(\widehat{sl_N})$ was realized via a set of $q$-deformed Heisenberg algebras. Therefore our observation \[ob1\] has to be used in somewhat a nontrivial way (for example, the vertex operators and screening currents cannot be obtained using our correspondence principles). Free bosons and Fock space -------------------------- We introduce the following set of $N^2-1$ Heisenberg algebras with generators $a^i_n~(1 \leq i \leq N-1),~b^{ij}_n~\mbox{and}~c^{ij}_n~ (1 \leq i < j \leq N)$ with $ n \in {\Bbb Z} - \{ 0 \}$ and $p_{a^i},~q_{a^i}~(1 \leq i \leq N-1),~p_{b^{ij}},~q_{b^{ij}},~ p_{c^{ij}},~q_{c^{ij}}~(1 \leq i < j \leq N)$, $$\begin{aligned} \begin{array}{ll} $$[ a^i_n,~a^j_m ] = (k+g) B_{ij} n \delta_{n+m,0},$$ & $$[ p_{a^i},~q_{a^j} ] = (k+g) B_{ij},$$ \cr $$[ b^{ij}_n,~b^{i'j'}_m ] = - n \delta^{i,i'} \delta^{i,j'} \delta_{n+m,0},$$ & $$[ p_{b^{ij}},~q_{b^{ij}} ] = -\delta^{i,i'} \delta^{i,j'},$$ \cr $$[ c^{ij}_n,~c^{i'j'}_m ] = n \delta^{i,i'} \delta^{i,j'} \delta_{n+m,0},$$ & $$[ p_{c^{ij}},~q_{c^{ij}} ] = \delta^{i,i'} \delta^{i,j'},$$ \end{array}\end{aligned}$$ where $g=N$ is the dual Coexter number for the Cartan matrix of type $A_{N-1}$. The Fock space corresponding to the above Heisenberg algebras can be specified as follows. Let $| 0 \rangle$ be the vacuum state defined by $$\begin{aligned} & & a^i_n| 0 \rangle = b^{ij}_n| 0 \rangle = c^{ij}_n| 0 \rangle =0~ ( n >0),\\ & & p_{a^i}| 0 \rangle = p_{b^{ij}}| 0 \rangle = p_{c^{ij}}| 0 \rangle =0.\end{aligned}$$ Define $$\begin{aligned} & & | l_a,l_b,l_c \rangle = \\ & &~~~~ \mbox{exp} \left( \sum_{i,j=1}^{N-1} \sum_{n>0} l_{a^i} \frac{1}{k+g} (B^{-1})^{ji} a^j_n - \sum_{1 \leq i < j \leq N} l_{b^{ij}} q_{b^{ij}} + \sum_{1 \leq i < j \leq N} l_{c^{ij}} q_{c^{ij}} \right) | 0 \rangle,\end{aligned}$$ it can be shown that the following equations hold, $$\begin{aligned} & & a^i_n| l_a,l_b,l_c \rangle = b^{ij}_n| l_a,l_b,l_c \rangle = c^{ij}_n| l_a,l_b,l_c \rangle =0 ~( n > 0),\\ & & p_{a^i} | l_a,l_b,l_c \rangle = l_{a^i} | l_a,l_b,l_c \rangle,\\ & & p_{b^{ij}} | l_a,l_b,l_c \rangle = l_{b^{ij}} | l_a,l_b,l_c \rangle,\\ & & p_{c^{ij}} | l_a,l_b,l_c \rangle = l_{c^{ij}} | l_a,l_b,l_c \rangle.\end{aligned}$$ The Fock space ${\cal F}(l_a,l_b,l_c)$ is then generated by the actions of the negative modes of $a^i,~b^{ij},~c^{ij}$. We shall see later that this Fock space actually forms a (Wakimoto-like [@wakimoto; @fff]) module for the Yangian double $DY_\hbar(sl_N)$ with level $k$. For $X= a^i,~b^{ij},~c^{ij}$, let us now define $$\begin{aligned} & & X(u;A,B)=\sum_{n>0} \frac{X_{-n}}{n} (u+A \hbar)^n - \sum_{n>0} \frac{X_{n}}{n} (u+B \hbar)^{-n} + \mbox{log}(u+B \hbar) p_X +q_X, \\ & & X_+(u;B)= - \sum_{n>0} \frac{X_{n}}{n} (u+B \hbar)^{-n} + \mbox{log}(u+B \hbar) p_X,\\ & & X_-(u;A)=\sum_{n>0} \frac{X_{-n}}{n} (u+A \hbar)^n +q_X,\\ & & X(u;A) = X(u;A,A),~~X(u)=X(u,0).\end{aligned}$$ Then we have $$: \mbox{exp} \left( X(u;A,B) \right) : = \mbox{exp} \left( X_-(u;A) \right) \mbox{exp} \left( X_+(u;B) \right).$$ Following the standard quantum field theory we have $$X^{\alpha}(u;A,B) X^{\beta}(v;C,D) = \langle X^{\alpha}(u;A,B) X^{\beta}(v;C,D) \rangle + :X^{\alpha}(u;A,B) X^{\beta}(v;C,D):, \label{contract}$$ where [^1] $$\begin{aligned} & & \langle a^{i}(u;A,B) a^{j}(v;C,D) \rangle =(k+g) B_{ij} \mbox{log}(u-v+(B-C) \hbar),\\ & & \langle b^{ij}(u;A,B) b^{i'j'}(v;C,D) \rangle =- \delta^{ii'} \delta^{jj'} \mbox{log}(u-v+(B-C) \hbar),\\ & & \langle c^{ij}(u;A,B) c^{i'j'}(v;C,D) \rangle = \delta^{ii'} \delta^{jj'} \mbox{log}(u-v+(B-C) \hbar),\end{aligned}$$ and all other contractions vanish. From eq.(\[contract\]) it is easy to calculate that $$\begin{aligned} & & :\mbox{exp} \left( X^{\alpha}(u;A,B) \right): :\mbox{exp} \left( X^{\beta}(v;C,D) \right) : \nonumber \\ & &~~~~= \mbox{exp} \left( \langle X^{\alpha}(u;A,B) X^{\beta}(v;C,D) \rangle \right) :\mbox{exp} \left( X^{\alpha}(u;A,B) \right) \mbox{exp} \left( X^{\beta}(v;C,D) \right) :. \label{expt}\end{aligned}$$ It should be noticed that equations (\[contract\]) and (\[expt\]) hold unchanged if we change everywhere $X^{\alpha}(u;A,B) \rightarrow X^{\alpha}_+(u;B)$ and $X^{\beta}(v;C,D) \rightarrow X^{\beta}_-(v;C)$. For later use let us introduce some more definitions. For $X=b^{ij},~c^{ij}$, define $$\begin{aligned} \hat{X}_\pm(u) = \mp \left( X_\pm(u;-\frac{1}{2}) - X_\pm(u; \frac{1}{2}) \right).\end{aligned}$$ For the bosonic fields $a^{i}(u;A,B)$, define $$\begin{aligned} & & \hat{a}^i_+(u) = a^i_+(u;0) - a^i_+(u; k+g), \label{ap}\\ & & \hat{a}^i_-(u) = \frac{1}{k+g} \sum_{j,l=1}^{N-1} (B^{-1})^{jl} \left(a^j_-(u; B_{ij}) - a^j_-(u; -B_{ij}) \right). \label{an}\end{aligned}$$ It is easy to obtain the following operator product expansion (OPE) relations, $$\begin{aligned} & & \mbox{exp} \left( \hat{a}^{i}_+(u) \right) \mbox{exp} \left( \hat{a}^{j}_-(v) \right) \nonumber\\ & &~~~~= \frac{(u-v-B_{ij} \hbar) (u-v+(k+g+B_{ij}) \hbar)} {(u-v+B_{ij} \hbar) (u-v+(k+g-B_{ij}) \hbar)} \mbox{exp} \left( \hat{a}^{j}_-(v) \right) \mbox{exp} \left( \hat{a}^{i}_+(u) \right), \label{aapn}\\ & & \mbox{exp} \left( \hat{b}^{ij}_+(u) \right) \mbox{exp} \left( \hat{b}^{i'j'}_-(v) \right) \nonumber\\ & &~~~~= \left( \frac{(u-v)^2} {(u-v - \hbar) (u-v+\hbar)} \right)^{\delta^{ii'} \delta^{jj'}} \mbox{exp} \left( \hat{b}^{i'j'}_-(v) \right) \mbox{exp} \left( \hat{b}^{ij}_+(u) \right), \nonumber\\ & & \mbox{exp} \left( \hat{c}^{ij}_+(u) \right) \mbox{exp} \left( \hat{c}^{i'j'}_-(v) \right) \nonumber \\ & &~~~~= \left( \frac{(u-v - \hbar) (u-v+\hbar)}{(u-v)^2} \right)^{\delta^{ii'} \delta^{jj'}} \mbox{exp} \left( \hat{c}^{i'j'}_-(v) \right) \mbox{exp} \left( \hat{c}^{ij}_+(u) \right). \nonumber\end{aligned}$$ Moreover, we have the following relations, $$\begin{aligned} & & \mbox{exp} \left( \hat{b}^{ij}_+(u) \right) : \mbox{exp} \left( \hat{b}^{i'j'}(v) \right) : \\ & &~~~~= \left( \frac{u-v-\frac{1}{2} \hbar} {u-v +\frac{1}{2} \hbar} \right)^{\delta^{ii'} \delta^{jj'}} : \mbox{exp} \left( \hat{b}^{i'j'}(v) \right): \mbox{exp} \left( \hat{b}^{ij}_+(u) \right), \\ & & \mbox{exp} \left( \hat{c}^{ij}_+(u) \right) : \mbox{exp} \left( \hat{c}^{i'j'}(v) \right) : \\ & &~~~~= \left( \frac{u-v+\frac{1}{2} \hbar} {u-v -\frac{1}{2} \hbar} \right)^{\delta^{ii'} \delta^{jj'}} : \mbox{exp} \left( \hat{c}^{i'j'}(v) \right): \mbox{exp} \left( \hat{c}^{ij}_+(u) \right). \\\end{aligned}$$ To specify the correspondence of our notations and that of Ref. [@sln] for $q$-bosons, we give the second observation \[ob2\] The expressions $\hat{a}^{i}_\pm(u)$, $\hat{b}^{ij}_\pm(u)$ and $\hat{c}^{ij}_\pm(u)$ correspond to the fields $a^i_{\pm}(q^{\pm \frac{k+g}{2}}z)$, $b^{ij}_\pm(z)$ and $c^{ij}_\pm(z)$ of Ref. [@sln] respectively. \[rem1\] Notice that in Ref. [@sln], the explicit expressions for $a^i_{\pm}(q^{\pm \frac{k+g}{2}}z)$ are symmetric with respect to $+ \leftrightarrow -$, but this is not the case for $\hat{a}^{i}_\pm(u)$. The partial reason for this difference is that, for $q$-affine algebras, the Drinfeld currents $\psi^i_\pm(z)$ are defined in an symmetric way in $H^i_n$, whilst for Yangian doubles the currents $H^{\pm}_i$ are defined asymmetrically. In the next subsection, we shall see that, despite the difference stated in Remark \[rem1\], the above observations are rather useful to guess the bosonic expressions for the Drinfeld currents of $DY_\hbar(sl_N)$. Free boson representation of $DY_\hbar(sl_N)$ with level $k$ ------------------------------------------------------------ Let us define $$\begin{aligned} & & H^{\pm}_i(u) = : \mbox{exp} \left\{ \sum_{l=1}^i \hat{b}^{l,i+1}_{\pm} ( u \pm \frac{1}{2} (\frac{k}{2} + l -1) \hbar ) - \sum_{l=1}^{i-1} \hat{b}^{l,i}_{\pm} ( u \pm \frac{1}{2} (\frac{k}{2} + l) \hbar ) \right. \nonumber\\ & & ~~~~ + \hat{a}^i_\pm (u \mp \frac{1}{4} k \hbar ) \nonumber \\ & &~~~~ + \left. \sum_{l=i+1}^N \hat{b}^{il}_{\pm} ( u \pm \frac{1}{2} (\frac{k}{2} + l) \hbar ) - \sum_{l=i+2}^N \hat{b}^{i+1,l}_{\pm} ( u \pm \frac{1}{2} (\frac{k}{2} + l -1) \hbar ) \right\}:~, \label{hpn}\end{aligned}$$ $$\begin{aligned} & & E^+_i(u) = - \frac{1}{\hbar} \sum_{m=1}^i :\mbox{exp} \left\{(b+c)^{mi} ( u + \frac{1}{2} (m-1) \hbar ) \right\} \nonumber\\ & & ~~~~\times \left[ \mbox{exp} \left( \hat{b}^{m,i+1}_+ ( u + \frac{1}{2} (m-1) \hbar ) -(b+c)^{m,i+1} ( u + \frac{1}{2} m \hbar ) \right) \right. \nonumber \\ & &~~~~~~~~ - \left. \mbox{exp} \left( \hat{b}^{m,i+1}_- ( u + \frac{1}{2} (m-1) \hbar ) -(b+c)^{m,i+1} ( u + \frac{1}{2} (m-2) \hbar ) \right) \right] \nonumber\\ & &~~~~\times \mbox{exp} \left\{ \sum_{l=1}^{m-1} \left[ \hat{b}^{l,i+1}_+ ( u + \frac{1}{2} (l-1) \hbar ) -\hat{b}^{li}_+ ( u + \frac{1}{2} l \hbar ) \right] \right\} :~, \label{ep}\end{aligned}$$ $$\begin{aligned} & & E^-_i(u) = - \frac{1}{\hbar} \left\{ \sum_{m=1}^{i-1} :\mbox{exp} \left( (b+c)^{m,i+1} ( u - \frac{1}{2} (k+m) \hbar ) \right) \right. \nonumber\\ & &~~~~\times \left[ \mbox{exp} \left( -\hat{b}^{mi}_- ( u - \frac{1}{2} (k+m) \hbar ) -(b+c)^{mi} ( u - \frac{1}{2} (k+m-1) \hbar ) \right) \right. \nonumber\\ & &~~~~~~~~ - \left. \mbox{exp} \left( -\hat{b}^{mi}_+ ( u - \frac{1}{2} (k+m) \hbar ) -(b+c)^{mi} ( u - \frac{1}{2} (k+m+1) \hbar ) \right) \right] \nonumber \\ & &~~~~\times \mbox{exp} \left( \sum_{l=m+1}^i \hat{b}^{l,i+1}_{-} ( u - \frac{1}{2} (k+l-1) \hbar ) - \sum_{l=m+1}^{i-1} \hat{b}^{li}_{-} ( u- \frac{1}{2} (k+l) \hbar) \right. \nonumber\\ & &~~~~~~~~ + \left.\hat{a}^i_- (u) + \sum_{l=i+1}^N \hat{b}^{il}_{-} ( u - \frac{1}{2} (k+l) \hbar) - \sum_{l=i+2}^N \hat{b}^{i+1,l}_{\pm} ( u- \frac{1}{2} (k + l -1) \hbar ) \right): \nonumber\\ & & ~~~~+ :\mbox{exp} \left( (b+c)^{i,i+1} ( u - \frac{1}{2} (k+i) \hbar ) \right) \nonumber\\ & & ~~~~~~~~\times \mbox{exp} \left( \hat{a}^i_- (u) + \sum_{l=i+1}^N \hat{b}^{il}_{-} ( u - \frac{1}{2} (k+l) \hbar ) - \sum_{l=i+2}^{N} \hat{b}^{i+1,l}_{-} ( u- \frac{1}{2} (k+l-1) \hbar) \right): \nonumber\\ & & ~~~~- :\mbox{exp} \left( (b+c)^{i,i+1} ( u + \frac{1}{2} (k+i) \hbar ) \right) \nonumber\\ & & ~~~~~~~~\times \mbox{exp} \left( \hat{a}^i_+ (u) + \sum_{l=i+1}^N \hat{b}^{il}_{+} ( u + \frac{1}{2} (k+l) \hbar ) - \sum_{l=i+2}^{N} \hat{b}^{i+1,l}_{-} ( u+ \frac{1}{2} (k+l-1) \hbar) \right): \nonumber\\ & &~~~~- \sum_{m=i+2}^N :\mbox{exp} \left( (b+c)^{im} ( u + \frac{1}{2} (k+m-1) \hbar ) \right) \nonumber\\ & & ~~~~~~~~\times \left[ \mbox{exp} \left( \hat{b}^{i+1,m}_+ ( u + \frac{1}{2} (k+m-1) \hbar ) -(b+c)^{i+1,m} ( u + \frac{1}{2} (k+m) \hbar ) \right) \right. \nonumber\\ & &~~~~~~~~~~~~ - \left. \mbox{exp} \left( \hat{b}^{i+1,m}_- ( u + \frac{1}{2} (k+m-1) \hbar ) -(b+c)^{i+1,m} ( u + \frac{1}{2} (k+m-2) \hbar ) \right) \right] \nonumber\\ & &~~~~~~~~\times \left. \mbox{exp} \left( \sum_{l=m}^{N} \left[ \hat{b}^{il}_+ ( u + \frac{1}{2} (k+l) \hbar ) -\hat{b}^{i+1,l}_+ ( u + \frac{1}{2} (k+l-1) \hbar ) \right] \right) : \right\}~. \label{en}\end{aligned}$$ The following proposition is the main result of this paper: \[prop1\] The fields $H^{\pm}_i(u)$, $E^{\pm}_i(u)$ defined in equations (\[hpn\]), (\[ep\]) and (\[en\]) are well-defined on the Fock space ${\cal F}(l_a,l_b,l_c)$ and satisfy equations (\[y1\]-\[y4\]) with $c=k$ and $$\begin{aligned} & & E^{\pm}_i(u) E^{\pm}_j(v) \simeq E^{\pm}_j(v) E^{\pm}_i(u) \sim reg. ~~\mbox{for}~B_{ij}=0,\\ & & (u-v \mp B_{ij} \hbar) E^{\pm}_i(u) E^{\pm}_j(v) \simeq (u-v \pm B_{ij} \hbar) E^{\pm}_j(v) E^{\pm}_i(u) \sim reg. ~~\mbox{for}~B_{ij} \neq 0,\\ & & E^{+}_i(u) E^{-}_j(v) - E^{-}_j(v) E^{+}_i(u) \\ & &~~~~ \sim reg. + \frac{1}{\hbar} \left( \delta( u_- - v_+) H^+_i(v_+) - \delta( u_+ - v_-) H^-_i(v_-) \right),\end{aligned}$$ where $reg.$ means some regular expressions and $\simeq$ and $\sim$ imply “equals up to” such expressions. [*Proof*]{}: The proposition follow by straightforward but tedious calculations. Actually the calculations are step by step analogous to that of Ref. [@sln] for $q$-affine case. So we omit all such calculations and only refer to [@sln] and remind the readers of our correspondence rules (Observations 1 and 2). \[rem2\] In proving Proposition \[prop1\], only the OPE relation (\[aapn\]) for $\hat{a}^i_\pm$ is used and the exact expressions for the fields $\hat{a}^i_\pm$ are not important. Actually, there are infinite many choices for $\hat{a}^i_\pm$ which satisfy the relation (\[aapn\]). For example, the following is another example which differs from the original definitions (\[ap\]-\[an\]), $$\begin{aligned} & & \hat{a}^i_+(u) = \frac{1}{k+g} \sum_{j,l=1}^{N-1} (B^{-1})^{lj} a^j_+(u;\frac{k+g}{2} - B_{ij}) - a^j_+(u;\frac{k+g}{2} + B_{ij}), \label{ap2}\\ & & \hat{a}^i_-(u) = a^i_-(u;-\frac{k+g}{2}) - a^i_+(u;\frac{k+g}{2}). \label{an2}\end{aligned}$$ However, no matter which choice we use, we cannot make the definition of $\hat{a}^i_+(u)$ and $\hat{a}^i_-(u)$ symmetric, i.e. no violation of Remark \[rem1\] could occur. \[rem3\] While $N=2$, equations (\[hpn\]-\[en\]) become $$\begin{aligned} & & H^{+}(u) = : \mbox{exp}\left( \hat{b}_+(u+\frac{1}{4}k\hbar) + \hat{b}_+(u+\frac{1}{2}(\frac{k}{2}+2) \hbar) + \hat{a}_+(u-\frac{1}{4}k\hbar) \right):~, \label{hppp}\\ & & H^{-}(u) = : \mbox{exp}\left( \hat{b}_-(u-\frac{1}{4}k\hbar) + \hat{b}_-(u-\frac{1}{2}(\frac{k}{2}+2) \hbar) + \hat{a}_-(u+\frac{1}{4}k\hbar) \right):~,\label{hnnn}\\ & & E^+(u)= -\frac{1}{\hbar} : \left[ \mbox{exp}\left( \hat{b}_+(u) -(b+c)(u+\frac{\hbar}{2}) \right) - \mbox{exp}\left( \hat{b}_-(u) -(b+c)(u-\frac{\hbar}{2}) \right) \right]:~, \label{eppp}\\ & & E^-(u)= \frac{1}{\hbar} : \left[ \mbox{exp}\left( (b+c)(u+\frac{1}{2}(k+1)\hbar ) \right) \mbox{exp}\left( \hat{a}_+(u) + \hat{b}_+(u+ \frac{1}{2}(k+2)\hbar) \right) \right. \nonumber \\ & &~~~~\left. - \mbox{exp}\left( (b+c)(u-\frac{1}{2}(k+1)\hbar ) \right) \mbox{exp}\left( \hat{a}_-(u) + \hat{b}_-(u- \frac{1}{2}(k+2)\hbar) \right) \right]: \label{ennn}\end{aligned}$$ which is different from the result of Ref. [@konno] for $DY_\hbar(sl_2)_k$. The reason for this difference is that, first, the Yangian double $DY_\hbar(sl_2)_k$ of Ref. [@konno] is realized in an asymmetric way which differs from the symmetric one which we are using by a shift of parameter [@KL]; Second, as we have remarked in Remarks \[rem1\] and \[rem2\], for the same realization of Yangian double, there still exist infinite many choices for the bosonization formulas. Therefore the difference of our result (\[hppp\]-\[ennn\]) from that of Ref. [@konno] is reasonable. Problems in obtaining Vertex operators and screening currents ============================================================= After successfully obtained the bosonization formulas for the Yangian double $DY_\hbar(sl_N)$ using the correspondence rules (Observations \[ob1\] and \[ob2\]), one naturally expects that the Vertex operators and screening currents for $DY_\hbar(sl_N)$ could also be obtained in the same way. In this section we briefly give why this is difficult. Following Observations \[ob1\] and \[ob2\] and Ref.[@sln], we expects that the screening currents for $DY_\hbar(sl_N)$ might be written in the following form, $$\begin{aligned} S^i(u) = :\mbox{exp}\left( X^i [a](u) \right): \tilde{S}^i(u),\end{aligned}$$ where $\tilde{S}^i(u)$ is nothing but $E^+_{N-i}(u)$ with the replacement $\hat{b}^{ij}_\pm \rightarrow - \hat{b}^{N+1-j,N+1-i}_\mp$, $(b+c)^{ij} \rightarrow (b+c)^{N+1-j,N+1-i}$, and $X^i [a](u)$ is some field depending only on $a^i$ but not on $b^{ij}$ and $c^{ij}$. In the $q$-affine case, $X^i [a](u)$ is just the field $-\left( \frac{1}{k+g} a^i \right)(z; \frac{k+g}{2})$ [@sln]. At present, we expect that $:\mbox{exp}\left( X^i [a](u) \right):$ have the following OPE relations with $\mbox{exp}\left( \hat{a}^i_+(u) \right)$ and $\mbox{exp}\left( \hat{a}^i_-(u) \right)$ (see equations (C.17), (C.18) of Ref. [@sln]) $$\begin{aligned} & & \mbox{exp} ( \hat{a}^i_+(u) ) :\mbox{exp} ( X^j [a](v) ): \nonumber \\ & &~~~~~~= \frac{u-v+(\frac{k+g}{2}-B_{ij})\hbar} {u-v+(\frac{k+g}{2}+B_{ij})\hbar} :\mbox{exp} ( X^j [a](v) ): \mbox{exp} ( \hat{a}^i_+(u) ), \label {1a}\\ & & :\mbox{exp} ( X^i [a](u) ): \mbox{exp} ( \hat{a}^j_-(v) ) \nonumber \\ & &~~~~~~= \frac{u-v+(\frac{k+g}{2}-B_{ij})\hbar} {u-v+(\frac{k+g}{2}+B_{ij})\hbar} \mbox{exp} ( \hat{a}^j_-(v) ) :\mbox{exp} ( X^i [a](u) ):~. \label{2a}\end{aligned}$$ Notice that in the rational factors of equations (\[1a\]) and (\[2a\]), the $\frac{k+g}{2}$ appear with the same sign in both the numerators and the denominators. This fact makes it difficult to obtain an explicit expression for $X^i [a](u)$. For examples, if we adopt the definitions (\[ap\]) and (\[an\]) of $\hat{a}^i_\pm(u)$, then the “positive frequency part” of $X^i [a](u)$ can be easily seen to be equal to $\hat{a}^i(u+\frac{k+g}{2})$, but the “negative frequency part could not be written in a simple form (and it is not known whether it is possible to write down such an expression). If we adopt the definitions (\[ap2\]) and (\[an2\]) instead of (\[ap\]) and (\[an\]), then the negative frequency part of $X^i [a](u)$ can be obtained easily but the positive frequency part is unknown. That is why we could not obtain a simple analogy of screening currents for Yangian double and $q$-affine algebras. Due to similar reasons the vertex operators for $DY_\hbar(sl_N)$ is also not obtained from that of $U_q(\widehat{sl_N})$. Discussions =========== In this paper we established the free boson representation of the Yangian double $DY_\hbar(sl_N)$ with arbitrary level $k$. Our construction is based on the crucial correspondence Observations \[ob1\] and \[ob2\]. Such representations of the Yangian double $DY_\hbar(sl_N)$ are expected to be useful in calculating the correlation functions of various quantum integrable systems in (1+1)-spacetime dimensions, e.g. the spin Calogero-Sutherland model [@spinCS], quantum nonlinear Schrodinger equation [@nonlinear] and some field theoretic models such as Thirring model, Gross-Neveu model with $U(N)$ gauge symmetries etc. Our representation of $DY_\hbar(sl_N)$ may also be used to analyze the behavior of Yangian double at the critical level $k=-g$, a very fascinating area of great interest of study [@FR]. Besides what have been solved in this paper, the unsolved problem of the construction of vertex operators and screening currents are also of great interest. Especially if we know these quantities we could have been able to calculate the cohomology of the action of our bosonization formulas on the Fock spaces ${\cal F}(l_a,~l_b,~l_c)$. We hope these problems could be solved in the future. [99]{} Abada,A., Bougourzi, A.H., El Gradechi, M.A.: A deformation of the Wakimoto construction, [*Preprint [CRM-1829]{}*]{} (1992). Awata, H., Odake, S., Shiraishi, J.: [*Lett. Math. Phys.*]{} [**30**]{} (1994), 207. Awata, H., Odake, S., Shiraishi, J.: [*Commun. Math. Phys.*]{} [ **162**]{} (1994), 61. Bernard, D., LeClair, A.: [*Nucl. Phys.*]{} [**B399**]{} (1993), 709. Bougourzi, A.H., Sebbar, A., A Hopf algebra isomorphism between two realizations of the quantum affine algebra $U_q(\widehat{gl_2})$, [*Preprint [q-alg/9701004]{}, [ITP-SB-97-01]{}*]{} (1997). Ding, J., and Frenkel, I.B.: [*Commun. Math. Phys.*]{} [**156**]{} (1993), 277. Drinfeld, V.G.: [*Soviet Math. Dokl.*]{} [**283**]{} (1985), 1060. Drinfeld, V.G.: [*Soviet Math. Dokl.*]{} [**32**]{} (1988), 212. Drinfeld, V.G.: In [*Proceedings of the International Congress of Mathematicians*]{}, p798, Berkeley, (1987). Drinfeld, V.G.: [*Sov. Math. Dokl.*]{} [**36**]{} (1987), 212. Faddeev, L.D., Reshetikhin, N.Yu., Takhtajan, L.A.: [*Advanced Series in Mathematical Physics*]{}, [**Vol.10**]{}, Singapore, World Scientific (1989), p299. Feigin, B.L., Frenkel, E.: [*Russ. Math. Surv. [**43**]{}*]{} (1989), 221. Frenkel, E., Reshetikhin, N.: [*Commun. Math. Phys. [**178**]{}*]{} (1996), 237. Frenkel, I.B., Jing, N.: [*Proc. Natl. Acad. Sci. USA [**85**]{}*]{} (1988), 9373. Iohara, K.: Bosonic representations of Yangian double $DY_\hbar(g)$ with $g=gl_N,~sl_N$, [*Preprint [q-alg/9603033]{}*]{} (1996). Iohara, K., Konno, M.: [*Lett. Math. Phys. [**37**]{}*]{} (1996), 319. Khoroshkin, S., Tolstoy, V.: [*Lett. Math. Phys. [**36**]{}*]{} (1996), 373. Khoroshkin, S.: Central Extension of the Yangian Double. In Collection SMF, Colloque “Septièmes Rencontres du Contact Franco-Belge en Algèbre", June 1995, Reins; [*Preprint [q-alg/9602031]{}*]{}. Khoroshkin, S., Lebedev, D.: Intertwining operators for the central extension of the Yangian double, [*Preprint [q-alg/9602030]{}*]{} (1996). Kimura, K.: On free field representation of the quantum affine algebra $U_q(\widehat{sl_2})$, [*Preprint [RIMS-910]{}*]{} (1992). Kojima, T., Korepin, V.E.: Determinant representations for dynamical correlation functions of the quantum nonlinear Schrodinger equation, [*Preprint [RIMS-1115]{}*]{} (1996). Konno, H.: Free field representation of level-$k$ Yangian double $DY(sl_2)_k$ and deformation of Wakimoto Modules, [*Preprint [YITP-96-10]{}*]{} (1996). LeClair, A., Smirnov, F.: [*Int. J. Mod. Phys. [**A7**]{}*]{} (1992), 2997. Matsuo, A.: [*Phys. Lett. [**B308**]{}*]{} (1993), 260. Matsuo, A.: [*Commun. Math. Phys. [**160**]{}*]{} (1994), 33. Reshetikhin, N. Yu., Semenov-Tyan-Shansky, M. A.: [*Let. Math. Phys. [**19**]{}*]{} (1990), 133. Shiraishi, J.: [*Phys. Lett. [**A171**]{}*]{} (1992), 243. Smirnov, F.A.: [*Int. J. Mod. Phys. [**A7 suppl. 1B**]{}*]{} (1992), 813, 839. Takimura, K., Uglov, D.: The orthogonal eigenbasis and norms of eigenvectors in the spin Calogero-Sutherland model, [*Preprint [RIMS-1114]{}*]{} (1996). Wakimoto, M.: [*Commun. Math. Phys. [**104**]{}*]{} (1986), 605. [^1]: Here and below, all OPE relations should be understood to hold in the analytic continuation sense.

--- abstract: 'We carry out a comprehensive study of the dynamics of large-scale perturbations in quintessence scenarios. We model the contents of the Universe by a perfect fluid with equation of state ${w_\mathrm{f}}$ and a scalar field $Q$ with potential $V(Q)$. We are able to reduce the perturbation equations to a system of four first-order equations. During each of the five main regimes of quintessence field behaviour, these equations have constant coefficients, enabling analytic solution of the perturbation evolution by eigenvector decomposition. We determine these solutions and discuss their main properties.' author: - 'Michaël Malquarti and Andrew R. Liddle' title: 'Evolution of large-scale perturbations in quintessence models' --- Introduction ============ Recent observations seem to indicate that the Universe is undergoing a period of accelerated expansion [@acc]. Whereas cosmologists initially introduced a cosmological constant in order to explain this, a range of different models have emerged, amongst which quintessence has been particularly prominent in the literature [@RP; @qui]. It is defined as a scalar field rolling down its potential and presently dominating the dynamics of the Universe. An important class of quintessence models are known as tracking models [@RP; @qui], where the late-time evolution of the field has an attractor behaviour rendering its evolution fairly independent of initial conditions. In contrast to a cosmological constant, which is by definition perfectly homogeneous, the quintessence field can, and indeed must, have perturbations. The evolution of perturbations in quintessence models have been studied by many authors [@RP; @quintperts; @BMR; @perts2]. In this paper we carry out an exhaustive and elegant analysis of those in the large-scale approximation. We model the contents of the Universe by a perfect fluid with equation of state ${w_\mathrm{f}}$ and a scalar field $Q$ with potential $V(Q)$. We assume a flat Universe throughout. Background evolution ==================== Before studying the perturbations, we recall some results for the homogeneous background evolution. The geometry of the Universe is described by a flat Robertson–Walker metric $${\mathrm{d}}s^2= - {\mathrm{d}}t^2 + a^2(t){\mathrm{d}}\mathbf{x}^2{\,\mathrm{.}}$$ The Einstein equations $$\begin{aligned} H^2\,=\,\Big(\frac{\dot{a}}{a}\Big)^2 &=&\frac{8\pi}{3{m_\mathrm{Pl}}^2}{\left(}{\rho_\mathrm{f}}+{\rho_Q}{\right)}\,,\\ 2\dot{H}+3H^2\,=\,2\frac{\ddot{a}}{a}+\Big(\frac{\dot{a}}{a}\Big)^2 &=&-\frac{8\pi}{{m_\mathrm{Pl}}^2}{\left(}p_\mathrm{f}+p_Q{\right)}\end{aligned}$$ relate the matter components to the geometry. The indices “f” and “$Q$” always refer to the perfect fluid and the quintessence field respectively, and dots are time derivatives. We will use a prime to denote a derivative with respect to $N\equiv\log(a/a_0)$. The evolution of the fluid is straightforward, with its energy density scaling as $a^{-3(1+{w_\mathrm{f}})}$, where ${w_\mathrm{f}}$ is the ratio of pressure to energy density of the fluid. The quintessence field follows the Euler–Lagrange equation $$\ddot{Q}=-3H\dot{Q}-\frac{{\mathrm{d}}V}{{\mathrm{d}}Q}$$ and its equation of state is $${w_Q}\equiv\frac{p_Q}{\rho_Q}=\frac{\dot{Q}^2/2-V(Q)}{\dot{Q}^2/2+V(Q)} {\,\mathrm{.}}$$ Depending on the precise model and on the choice of initial conditions, the quintessence dynamics can feature up to five main regimes, which were classified by Brax et al. [@BMR] and which appear in a sequential order. During the first three the quintessence field is sub-dominant.[^1] The “kinetic” regime is characterized by the domination of the kinetic energy which scales as $a^{-6}$. In the “transition” and “potential” regimes the potential energy dominates and the energy density remains constant. The sound speed of the quintessence field is defined by $$\label{csdef} {c_{\mathrm{s}Q}^2}\equiv\frac{\dot{p}_Q}{\dot{\rho}_Q} ={w_Q}-\frac{{w_Q}'}{3(1+{w_Q})} =1+\frac{2}{3}\frac{{\mathrm{d}}V/{\mathrm{d}}Q}{H\dot{Q}}{\,\mathrm{,}}$$ and is equal to $1$ or $-2-{w_\mathrm{f}}$ respectively during those regimes. During the “tracker” regime the quintessence field approximately mimics the behaviour of the fluid, and usually its energy density is still sub-dominant when tracking begins. If ${w_Q}={w_\mathrm{f}}$ there is perfect tracking, even if the quintessence field is not sub-dominant. Finally the field enters its “domination” phase, during which ${w_Q}$ tends to $-1$ in most cases. Four useful parameters can be defined to describe these five regimes. They are the quintessence density parameter ${\Omega_Q}$, the equation of state conveniently parametrized as ${\gamma_Q}\equiv 1+{w_Q}$, the speed of sound ${c_{\mathrm{s}Q}^2}$ as defined in Eq. (\[csdef\]), and one further parameter relating to the speed of sound defined as $${\theta_Q}\equiv\frac{\left({c_{\mathrm{s}Q}^2}\right)^\prime}{1-{c_{\mathrm{s}Q}^2}} =-3(1+{c_{\mathrm{s}Q}^2})-\frac{{\mathrm{d}}}{{\mathrm{d}}N} \log{\left(}Q' {\mathrm{d}}V/{\mathrm{d}}Q {\right)}{\,\mathrm{.}}$$ In order to simplify the notation, we define the vector ${\mathbf{x}}\equiv ({\Omega_Q},{\gamma_Q},{c_{\mathrm{s}Q}^2},{\theta_Q})$. In [Fig. \[regimes\]]{}, we show an example of the evolution of those parameters. We take the realistic case ${w_\mathrm{f}}=1/3$ during radiation domination and ${w_\mathrm{f}}=0$ during matter domination, and we use an inverse power-law quintessence potential. We clearly see the five different regimes, and also the transition between radiation domination and matter domination which in this case occurs during the tracking regime. In [Table \[val\_reg\]]{} we give the values of the parameters in the general case for each regime. ----------------- -------------- ----------------------- --------------------- -------------------------- Kinetic 0 2 1 $-3(3+{w_\mathrm{f}})/2$ Transition 0 0 1 $-3(3+{w_\mathrm{f}})/2$ Potential 0 0 $-2-{w_\mathrm{f}}$ 0 Usual Tracker 0 ${\gamma_Q}$ ${w_Q}$ 0 Perfect Tracker ${\Omega_Q}$ ${\gamma_\mathrm{f}}$ ${w_Q}$ 0 Domination 1 ${\gamma_Q}$ ${w_Q}$ 0 ----------------- -------------- ----------------------- --------------------- -------------------------- : Values of the four parameters ${\Omega_Q}$, ${\gamma_Q}$, ${c_{\mathrm{s}Q}^2}$, and ${\theta_Q}$ during the different regimes.[]{data-label="val_reg"} Perturbation evolution ====================== In order to describe the perturbations we choose the conformal Newtonian gauge [@MFB]. As long as there is no anisotropic stress, the perturbed metric is $${\mathrm{d}}s^2=-(1+2\Phi){\mathrm{d}}t^2+a^2(t)(1-2\Phi){\mathrm{d}}\mathbf{x}^2{\,\mathrm{,}}$$ where in this case $\Phi$ is equal to the gauge-invariant potential defined as in Ref. [@MFB]. We work in Fourier space and compute the first-order perturbed Einstein equations $$\begin{aligned} -3H(H\Phi+\dot{\Phi})-\frac{k^2}{a^2}\Phi&=&4\pi G({\delta\rho_\mathrm{f}}+{\delta\rho_Q}){\,\mathrm{,}}\\ \ddot{\Phi}+4H\dot{\Phi}+(2\dot{H}+3H^2)\Phi&=&4\pi G({\delta p_\mathrm{f}}+{\delta p_Q}){\,\mathrm{,}}\end{aligned}$$ where the perturbed fluid pressure ${\delta p_\mathrm{f}}={w_\mathrm{f}}{\delta\rho_\mathrm{f}}$ and the perturbations in the quintessence energy density and pressure are given by $$\begin{aligned} {\delta\rho_Q}&=&\dot{Q}\delta\dot{Q}-\dot{Q}^2\Phi+\frac{{\mathrm{d}}V}{{\mathrm{d}}Q} {\delta Q}{\,\mathrm{,}}\\ {\delta p_Q}&=&\dot{Q}\delta\dot{Q}-\dot{Q}^2\Phi-\frac{{\mathrm{d}}V}{{\mathrm{d}}Q} {\delta Q}{\,\mathrm{.}}\end{aligned}$$ The perturbed Euler–Lagrange equation and fluid conservation equation lead to $$\begin{aligned} &\delta\ddot{Q}+3H\delta\dot{Q}+\frac{k^2}{a^2}{\delta Q}+\frac{{\mathrm{d}}^2 V}{{\mathrm{d}}Q^2}{\delta Q}=4\dot{Q}\dot{\Phi}-2\frac{{\mathrm{d}}V}{{\mathrm{d}}Q}\Phi{\,\mathrm{,}}\\ &\dot{{\delta_\mathrm{f}}}-3(1+{w_\mathrm{f}})\dot{\Phi}=-(1+{w_\mathrm{f}})\frac{k}{a}\mathcal{V}_\mathrm{f}{\,\mathrm{,}}\end{aligned}$$ where ${\delta_\mathrm{f}}\equiv{\delta\rho_\mathrm{f}}/{\rho_\mathrm{f}}$ and $\mathcal{V}_\mathrm{f}$ gives the fluid velocity. From now on we use $N$ as a time variable and study the evolution of $\Phi$, ${\delta_\mathrm{f}}$, ${\delta_Q}\equiv{\delta\rho_Q}/{\rho_Q}$ and ${\delta_P}\equiv{\delta p_Q}/{\rho_Q}$ in the long-wavelength limit ($k/aH\ll1$). We define the vector ${\mathbf{y}}\equiv(\Phi,{\delta_\mathrm{f}},{\delta_Q},{\delta_P})^T$, and considerable algebra leads to the expression $$\label{lin_sys} {\mathbf{y}}'={\mathcal{F}}({\gamma_\mathrm{f}},{\mathbf{x}})\times{\mathbf{y}}{\,\mathrm{,}}$$ where the matrix ${\mathcal{F}}({\gamma_\mathrm{f}},{\mathbf{x}})$ is given by $$\label{Fmat} {\mathcal{F}}({\gamma_\mathrm{f}},{\mathbf{x}})={\left(}\begin{array}{cccc} -1 & -{\Omega_\mathrm{f}}/2 & -{\Omega_Q}/2 & 0 \\ -3{\gamma_\mathrm{f}}& -3{\gamma_\mathrm{f}}{\Omega_\mathrm{f}}/2 & -3{\gamma_\mathrm{f}}{\Omega_Q}/2 & 0 \\ -3{\gamma_Q}& -3{\gamma_Q}{\Omega_\mathrm{f}}/2 & -3{\gamma_Q}{\Omega_Q}/2+3{w_Q}& -3 \\ -3{\gamma_Q}{c_{\mathrm{s}Q}^2}& -3{\gamma_Q}{\Omega_\mathrm{f}}/2 & 3{\gamma_\mathrm{f}}{\Omega_\mathrm{f}}/2+{\theta_Q}& -3{\gamma_Q}{\Omega_Q}/2+3{w_Q}-3{\gamma_\mathrm{f}}{\Omega_\mathrm{f}}/2-{\theta_Q}-3{c_{\mathrm{s}Q}^2}\end{array}{\right)}{\,\mathrm{.}}$$ In order to display the above expression in compact form we used the variables ${\Omega_\mathrm{f}}=1-{\Omega_Q}$ and , but the matrix depends only on the five independent parameters ${\gamma_\mathrm{f}}$, ${\Omega_Q}$, ${\gamma_Q}$, ${c_{\mathrm{s}Q}^2}$ and ${\theta_Q}$. Recall that ${\gamma_\mathrm{f}}$ and ${\gamma_Q}$ take on values between $0$ and $2$. The general evolution is of course very complicated, but as seen in [Fig. \[regimes\]]{}, during each of the five regimes described above the coefficients within the matrix take on constant values, and this allows us to study the main features of the perturbation evolution. We are interested in the eigenvalues $n_i$ of the matrix ${\mathcal{F}}$ and their corresponding eigenvectors ${\bf y}_i$. The solution then takes the form $${\bf y} = \sum_{i=1}^4 A_i \, {\bf y}_i \exp(n_iN) \,,$$ where the $A_i$ are constants given by the initial conditions. The adiabatic case ------------------ Before considering the general case, we restrict ourselves to adiabatic perturbations. For perturbations to be adiabatic, they must share a common perturbation according to the prescription $$\frac{{\delta\rho_\mathrm{f}}}{\dot{\rho}_\mathrm{f}} = \frac{{\delta\rho_Q}}{\dot{\rho}_Q} = \frac{{\delta p_Q}}{\dot{p}_Q} {\,\mathrm{,}}$$ which ensures that all matter perturbations vanish on uniform-density hypersurfaces. Note that the quintessence pressure perturbation, as well as its density perturbation, must satisfy the adiabatic condition (for the perfect fluid adiabaticity of its pressure perturbation is automatically guaranteed by its equation of state). These conditions can be rewritten as $$\label{adiabatic} \frac{{\delta_\mathrm{f}}}{{\gamma_\mathrm{f}}} = \frac{{\delta_Q}}{{\gamma_Q}} = \frac{{\delta_P}}{{c_{\mathrm{s}Q}^2}{\gamma_Q}} {\,\mathrm{.}}$$ It is well known that initially adiabatic perturbations remain purely adiabatic, and indeed it is not difficult to check that these conditions are conserved through evolution by our equations. We can therefore reduce the dynamical system in the adiabatic case to a system of two first-order equations. We define the vector ${\mathbf{z}}\equiv(\Phi,{\delta_\mathrm{f}})^T$, and using Eqs. (\[lin\_sys\]) and (\[adiabatic\]) we find $${\mathbf{z}}'={\mathcal{G}}({\gamma_\mathrm{f}},{\Omega_Q},{\gamma_Q})\times{\mathbf{z}}{\,\mathrm{,}}$$ where the matrix ${\mathcal{G}}({\gamma_\mathrm{f}},{\Omega_Q},{\gamma_Q})$ is given by $${\mathcal{G}}({\gamma_\mathrm{f}},{\Omega_Q},{\gamma_Q})={\left(}\begin{array}{cc} -1 & -{\gamma_\mathrm{tot}}/2{\gamma_\mathrm{f}}\\ -3{\gamma_\mathrm{f}}& -3{\gamma_\mathrm{tot}}/2 \end{array}{\right)}{\,\mathrm{,}}$$ where ${\gamma_\mathrm{tot}}={\gamma_\mathrm{f}}{\Omega_\mathrm{f}}+{\gamma_Q}{\Omega_Q}$. The eigenvalues and eigenvectors are given in the upper part of [Table \[results\]]{}. We see, as is well known, that there are always a constant and a decaying adiabatic mode, the former giving the late-time solution $\Phi = -{\gamma_\mathrm{tot}}{\delta_\mathrm{f}}/2{\gamma_\mathrm{f}}= -\delta_{{\rm tot}}/2$, where $\delta_{{\rm tot}}\equiv \delta\rho_{{\rm tot}}/\rho_{{\rm tot}}$. [lcl]{} &&\ \ $n_1=0$ && ${\mathbf{z}}_1=(-{\gamma_\mathrm{tot}}/2,{\gamma_\mathrm{f}})$\ $n_2=-1-3{\gamma_\mathrm{tot}}/2$ && ${\mathbf{z}}_2=(1/3,{\gamma_\mathrm{f}})$\ &&\ &&\ \ $n_1=0$ && ${\mathbf{y}}_1=(-{\gamma_\mathrm{f}}/2,{\gamma_\mathrm{f}},2,2)$\ $n_2=-1-3{\gamma_\mathrm{f}}/2$ && ${\mathbf{y}}_2=(1/3,{\gamma_\mathrm{f}},2,2)$\ $n_3=+6$ && ${\mathbf{y}}_3=(0,0,1,-1)$\ $n_4=0$ && ${\mathbf{y}}_4=(0,0,1,1)$\ &&\ \ $n_1=0$ && ${\mathbf{y}}_1=(-1/2,1,0,0)$\ $n_2=-1-3{\gamma_\mathrm{f}}/2$ && ${\mathbf{y}}_2=(1/3,{\gamma_\mathrm{f}},0,0)$\ $n_3=0$ && ${\mathbf{y}}_3=(0,0,1,-1)$\ $n_4=-6$ && ${\mathbf{y}}_4=(0,0,1,1)$\ &&\ \ $n_1=0$ && ${\mathbf{y}}_1=(-1/2,1,0,0)$\ $n_2=-1-3{\gamma_\mathrm{f}}/2$ && ${\mathbf{y}}_2=(1/3,{\gamma_\mathrm{f}},0,0)$\ $n_3=0$ && ${\mathbf{y}}_3=(0,0,1,-1)$\ $n_4=-3+3{\gamma_\mathrm{f}}/2$ && ${\mathbf{y}}_4=(0,0,-2,{\gamma_\mathrm{f}})$\ &&\ \ $n_1=0$ && ${\mathbf{y}}_1=(-{\gamma_\mathrm{f}}/2,{\gamma_\mathrm{f}},{\gamma_Q},{w_Q}{\gamma_Q})$\ $n_2=-1-3{\gamma_\mathrm{f}}/2$ && ${\mathbf{y}}_2=(1/3,{\gamma_\mathrm{f}},{\gamma_Q},{w_Q}{\gamma_Q})$\ $n_3={n_\mathrm{ut1}}+{n_\mathrm{ut3}}$ && ${\mathbf{y}}_3=(0,0,{n_\mathrm{ut2}}+{n_\mathrm{ut3}},3{\gamma_\mathrm{f}}/2)$\ $n_4={n_\mathrm{ut1}}-{n_\mathrm{ut3}}$ && ${\mathbf{y}}_4=(0,0,{n_\mathrm{ut2}}-{n_\mathrm{ut3}},3{\gamma_\mathrm{f}}/2)$\ &&\ \ $n_1=0$ && ${\mathbf{y}}_1=(-{\gamma_\mathrm{f}}/2,{\gamma_\mathrm{f}},{\gamma_\mathrm{f}},{w_\mathrm{f}}{\gamma_\mathrm{f}})$\ $n_2=-1-3{\gamma_\mathrm{f}}/2$ && ${\mathbf{y}}_2=(1/3,{\gamma_\mathrm{f}},{\gamma_\mathrm{f}},{w_\mathrm{f}}{\gamma_\mathrm{f}})$\ $n_3={n_\mathrm{pt1}}+{n_\mathrm{pt2}}$ && ${\mathbf{y}}_3=\ldots$\ $n_4={n_\mathrm{pt1}}-{n_\mathrm{pt2}}$ && ${\mathbf{y}}_4=\ldots$\ &&\ \ $n_1=0$ && ${\mathbf{y}}_1=(-{\gamma_Q}/2,{\gamma_\mathrm{f}},{\gamma_Q},{w_Q}{\gamma_Q})$\ $n_2=-1-3{\gamma_Q}/2$ && ${\mathbf{y}}_2=(1/3,{\gamma_\mathrm{f}},{\gamma_Q},{w_Q}{\gamma_Q})$\ $n_3=0$ && ${\mathbf{y}}_3=(0,1,0,0)$\ $n_4=-3+3{\gamma_Q}/2$ && ${\mathbf{y}}_4=(1/3,{\gamma_\mathrm{f}},1/3-{w_Q},-{\gamma_Q}/3)$\ &&\ The general case ---------------- We now return to the full set of perturbation equations Eq. (\[lin\_sys\]), continuing to consider the regimes in each of which the coefficients of the matrix ${\mathcal{F}}$ remain constant. We summarize our main results in the lower part of [Table \[results\]]{}. For the perfect tracker regime the eigenvectors ${\mathbf{y}}_3$ and ${\mathbf{y}}_4$ have not been given, since they are long and complicated formulas which are anyway not very relevant. In order to simplify some expressions we have used the variables $$\begin{aligned} {n_\mathrm{ut1}}& = & -\frac{3}{4}({\gamma_\mathrm{f}}-2{w_Q}){\,\mathrm{,}}\label{nua}\\ {n_\mathrm{ut2}}& = & \frac{3}{4}({\gamma_\mathrm{f}}+2{w_Q}){\,\mathrm{,}}\\ {n_\mathrm{ut3}}& = & \frac{3}{4}\sqrt{(2{w_Q}+{\gamma_\mathrm{f}})^2-8{\gamma_\mathrm{f}}}{\,\mathrm{,}}\\ {n_\mathrm{pt1}}& = & -\frac{3}{4}{\left(}2-{\gamma_\mathrm{f}}{\right)}{\,\mathrm{,}}\\ {n_\mathrm{pt2}}& = & \frac{3}{4}\sqrt{(2-{\gamma_\mathrm{f}})(2-{\gamma_\mathrm{f}}-8{\gamma_\mathrm{f}}{\Omega_\mathrm{f}})}{\,\mathrm{.}}\label{npb}\end{aligned}$$ For each regime the adiabatic modes can easily be identified as the first two entries in the table. Now let us analyze the two other, non-adiabatic, modes. In the kinetic case there are a growing mode, for which ${\delta_P}=-{\delta_Q}$ and thus $\delta\dot{Q}=0$ (since $\Phi = 0$), and another constant mode corresponding to $\delta Q = 0$. During the transition and potential regimes the former growing mode becomes constant and for each regime the fourth mode is decaying. Therefore, before entering the tracker the quintessence field may feature large non-adiabatic perturbations. As long as the Universe is dominated by the fluid, they are isocurvature perturbations. In the usual tracker case the last two eigenvalues may have an imaginary part, leading to oscillations, and their real part can either be negative or positive according to the value of ${\gamma_\mathrm{f}}$ and ${\gamma_Q}$. The regimes of the eigenvalues are shown in [Fig. \[eigenvalues\]]{}. However, since the quintessence field has to dominate at the present epoch, ${\rho_Q}$ must decrease more slowly than ${\rho_\mathrm{f}}$, and hence ${\gamma_Q}<{\gamma_\mathrm{f}}$. As one can see in [Fig. \[eigenvalues\]]{}, this implies ${\mathrm{Re}}(n_{3,4})<0$ . In the perfect tracker case one easily sees that the last two modes decay, possibly oscillating as they do. As long as all the other modes are decaying, during the tracker regime the constant adiabatic mode ${\mathbf{y}}_1$ is an attractor. As a result, a long tracker period implies the suppression of all non-adiabatic modes [@BMR; @perts2]. Moreover, in the case of a sub-dominant tracker the late-time evolution of the perturbations is even independent of the quintessence field initial conditions. Finally, during the domination regime, which is reached after the present epoch, the non-adiabatic modes are constant and decaying. Conclusions =========== We have derived four first-order equations to describe large-scale perturbations in quintessence scenarios. During each of the five main regimes of quintessence behaviour, these equations have constant coefficients, enabling analytic solution of the perturbations by eigenvector decomposition. We have seen that during the kinetic regime there is a growing isocurvature mode which then remains constant until tracking begins. However, if the quintessence field undergoes a long period of tracking, there remain only adiabatic perturbations which, in case of a sub-dominant tracker, are independent of its initial conditions. A low initial quintessence energy density [@ML], or a long kinetic period, may prevent the non-adiabatic modes disappearing completely. It is possible in principle to carry out the same analysis without using the large-scale approximation, although the equations may then be too complicated to be useful. M.M. was supported by the Fondation Barbour, the Fondation Wilsdorf and the Janggen-Pöhn-Stiftung, and A.R.L. in part by the Leverhulme Trust. We thank Pier-Stefano Corasaniti for useful discussions. A. G. Riess et al., Astron. J. **116**, 1009 (1998), `astro-ph/9805201`; P. Garnavich et al., Ap. J. **509**, 74 (1998), `astro-ph/9806396`; S. Perlmutter et al., Ap. J. **517**, 565 (1998), `astro-ph/9812133`; G. Efstathiou et al., Mon. Not. Roy. Ast. Soc. **330**, L29 (2002), astro-ph/0109152. B. Ratra and P. J. E. Peebles, Phys. Rev. D**37**, 3406 (1988). C. Wetterich, Nucl. Phys. B**302**, 668 (1988); E. J. Copeland, A. R. Liddle, and D. Wands, Ann. N. Y. Acad. Sci. [**688**]{}, 647 (1993); R. R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev. Lett. **80**, 1582 (1998), `astro-ph/9708069`; P. G. Ferreira and M. Joyce, Phys. Rev. D**58**, 023503 (1998), `astro-ph/9711102`; E. J. Copeland, A. R. Liddle, and D. Wands, Phys. Rev. D[**57**]{}, 4686 (1998), [gr-qc/9711068]{}; I. Zlatev, L. Wang, and P. J. Steinhardt, Phys. Rev. Lett. **82**, 896 (1999), `astro-ph/9807002`; A. R. Liddle and R. J. Scherrer, Phys. Rev. D[**59**]{}, 023509 (1999), [astro-ph/9809272]{}; V. Sahni and A. Starobinsky, Int. J. Mod. Phys. **D9**, 373 (2000), `astro-ph/9904398`. R. R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev. Lett. [**80**]{}, 1582 (1998), `astro-ph/9708069`; P. T. P. Viana and A. R. Liddle, Phys. Rev. D**57**, 674 (1998), `astro-ph/9708247`; F. Perrotta and C. Baccigalupi, Phys. Rev. D**59**, 123508 (1999), `astro-ph/9811156`. P. Brax, J. Martin, and A. Riazuelo, Phys. Rev. D**62**, 103505 (2000), `astro-ph/0005428`. L. R. Abramo and F. Finelli, Phys. Rev. D**64**, 083513 (2001), `astro-ph/0101014`; M. Kawasaki, T. Moroi, and T. Takahashi, Phys. Lett. B [ **533**]{}, 294 (2002), `astro-ph/0108081`; R. Dave, R. R. Caldwell, and Paul J. Steinhardt, Phys. Rev. D**66**, 023516 (2002), `astro-ph/0206372`. V. Mukhanov, H. Feldman, and R. Brandenberger, Phys. Rep. **215**, 203 (1992). M. Malquarti and A. R. Liddle, Phys. Rev. D**66**, 023524 (2002), `astro-ph/0203232`. [^1]: Actually, one can consider initial conditions with domination of the quintessence field, but they usually lead to an “overshoot” of the required present energy density, and are therefore not interesting.

--- abstract: | We derive explicit lower and upper bounds for the probability generating functional of a stationary locally stable Gibbs point process, which can be applied to summary statistics like the $F$ function. For pairwise interaction processes we obtain further estimates for the $G$ and $K$ functions, the intensity and higher order correlation functions. The proof of the main result is based on Stein’s method for Poisson point process approximation. [**Keywords:**]{} Gibbs process, probability generating functional, intensity, correlation function, $F$,$G$ and $K$ function, Poisson saddlepoint approximation, Stein’s method. [**AMS 2010 Subject Classification:**]{} Primary 60G55; secondary 62M30. author: - | Kaspar Stucki[^1] [^2] and Dominic Schuhmacher [^3]\ University of Bern date: 15 October 2012 title: 'Bounds for the probability generating functional of a Gibbs point process[^4]' --- Introduction {#sec:introduction} ============ Gibbs processes are very popular point process models which are extensively used both in spatial statistics and in statistical physics, see e.g. [@moellerwaage04; @ruelle69]. Especially the pairwise interaction processes allow a simple yet flexible modelling of point interactions. However, a major drawback of Gibbs processes is that in general there are no analytic formulas available for their intensities or higher order correlation functions. In a recent couple of articles [@baddeleynair12; @bn12] Baddeley and Nair proposed an approximation method that is fast to compute and accurate as verified by Monte Carlo methods. There are however no theoretical results in this respect and hence no guarantees for accuracy in most concrete models nor quantifications of the approximation error. The aim of the present paper is to derive rigorous lower and upper bounds for correlation functions and related quantities. These allow us to narrow down the true values quite precisely if the Gibbs process is not too far away from a Poisson process. Figure \[fig:Strauss50\] shows our bounds on the intensity for a two-dimensional Strauss process in dependence of its interaction parameter $\gamma$. The pluses are estimates of the true intensity obtained as averages over the numbers of points in $[0,1]^2$ of 10,000 Strauss processes simulated by dominated coupling from the past. The point processes were simulated on a larger window in order avoid noticeable edge effects. All simulations and numerical computations in this paper were performed in the R language [@r12] using the contributed package `spatstat` [@spatstat12]. (0,0) rectangle (254.39,231.26); ( 0.00, 0.00) rectangle (254.39,231.26); ( 50.64, 28.91) – (232.66, 28.91); ( 50.64, 28.91) – ( 50.64, 22.91); ( 87.05, 28.91) – ( 87.05, 22.91); (123.45, 28.91) – (123.45, 22.91); (159.85, 28.91) – (159.85, 22.91); (196.25, 28.91) – (196.25, 22.91); (232.66, 28.91) – (232.66, 22.91); at ( 50.64, 15.71) [0.0]{}; at ( 87.05, 15.71) [0.2]{}; at (123.45, 15.71) [0.4]{}; at (159.85, 15.71) [0.6]{}; at (196.25, 15.71) [0.8]{}; at (232.66, 15.71) [1.0]{}; ( 43.36, 35.87) – ( 43.36,209.85); ( 43.36, 35.87) – ( 37.36, 35.87); ( 43.36, 64.86) – ( 37.36, 64.86); ( 43.36, 93.86) – ( 37.36, 93.86); ( 43.36,122.86) – ( 37.36,122.86); ( 43.36,151.86) – ( 37.36,151.86); ( 43.36,180.85) – ( 37.36,180.85); ( 43.36,209.85) – ( 37.36,209.85); at ( 34.96, 33.46) [20]{}; at ( 34.96, 62.45) [25]{}; at ( 34.96, 91.45) [30]{}; at ( 34.96,120.45) [35]{}; at ( 34.96,149.45) [40]{}; at ( 34.96,178.44) [45]{}; at ( 34.96,207.44) [50]{}; ( 43.36, 28.91) rectangle (239.94,216.81); ( 50.64,138.76) – ( 52.46,139.20) – ( 54.28,139.65) – ( 56.10,140.09) – ( 57.92,140.54) – ( 59.74,141.00) – ( 61.56,141.46) – ( 63.38,141.92) – ( 65.20,142.38) – ( 67.02,142.85) – ( 68.84,143.33) – ( 70.66,143.80) – ( 72.48,144.28) – ( 74.30,144.77) – ( 76.12,145.26) – ( 77.94,145.75) – ( 79.76,146.25) – ( 81.58,146.75) – ( 83.40,147.25) – ( 85.23,147.76) – ( 87.05,148.28) – ( 88.87,148.79) – ( 90.69,149.32) – ( 92.51,149.84) – ( 94.33,150.37) – ( 96.15,150.91) – ( 97.97,151.45) – ( 99.79,152.00) – (101.61,152.55) – (103.43,153.10) – (105.25,153.66) – (107.07,154.23) – (108.89,154.80) – (110.71,155.37) – (112.53,155.95) – (114.35,156.54) – (116.17,157.13) – (117.99,157.72) – (119.81,158.32) – (121.63,158.93) – (123.45,159.54) – (125.27,160.16) – (127.09,160.78) – (128.91,161.41) – (130.73,162.04) – (132.55,162.68) – (134.37,163.33) – (136.19,163.98) – (138.01,164.64) – (139.83,165.31) – (141.65,165.98) – (143.47,166.65) – (145.29,167.34) – (147.11,168.03) – (148.93,168.72) – (150.75,169.43) – (152.57,170.14) – (154.39,170.86) – (156.21,171.58) – (158.03,172.31) – (159.85,173.05) – (161.67,173.79) – (163.49,174.55) – (165.31,175.31) – (167.13,176.08) – (168.95,176.85) – (170.77,177.63) – (172.59,178.43) – (174.41,179.22) – (176.23,180.03) – (178.05,180.85) – (179.87,181.67) – (181.69,182.50) – (183.51,183.35) – (185.33,184.20) – (187.15,185.05) – (188.97,185.92) – (190.79,186.80) – (192.61,187.69) – (194.43,188.58) – (196.25,189.49) – (198.07,190.40) – (199.89,191.33) – (201.71,192.26) – (203.53,193.21) – (205.35,194.16) – (207.17,195.13) – (208.99,196.10) – (210.81,197.09) – (212.63,198.09) – (214.45,199.10) – (216.27,200.12) – (218.09,201.15) – (219.91,202.20) – (221.74,203.25) – (223.56,204.32) – (225.38,205.40) – (227.20,206.49) – (229.02,207.60) – (230.84,208.72) – (232.66,209.85) – (232.66,209.85) – (230.84,208.72) – (229.02,207.59) – (227.20,206.47) – (225.38,205.37) – (223.56,204.27) – (221.74,203.18) – (219.91,202.09) – (218.09,201.02) – (216.27,199.95) – (214.45,198.89) – (212.63,197.84) – (210.81,196.80) – (208.99,195.77) – (207.17,194.74) – (205.35,193.72) – (203.53,192.71) – (201.71,191.70) – (199.89,190.71) – (198.07,189.72) – (196.25,188.73) – (194.43,187.76) – (192.61,186.79) – (190.79,185.83) – (188.97,184.88) – (187.15,183.93) – (185.33,182.99) – (183.51,182.05) – (181.69,181.13) – (179.87,180.20) – (178.05,179.29) – (176.23,178.38) – (174.41,177.48) – (172.59,176.58) – (170.77,175.69) – (168.95,174.81) – (167.13,173.93) – (165.31,173.06) – (163.49,172.20) – (161.67,171.34) – (159.85,170.49) – (158.03,169.64) – (156.21,168.80) – (154.39,167.96) – (152.57,167.13) – (150.75,166.30) – (148.93,165.48) – (147.11,164.67) – (145.29,163.86) – (143.47,163.06) – (141.65,162.26) – (139.83,161.47) – (138.01,160.68) – (136.19,159.90) – (134.37,159.12) – (132.55,158.35) – (130.73,157.58) – (128.91,156.82) – (127.09,156.06) – (125.27,155.30) – (123.45,154.56) – (121.63,153.81) – (119.81,153.07) – (117.99,152.34) – (116.17,151.61) – (114.35,150.89) – (112.53,150.16) – (110.71,149.45) – (108.89,148.74) – (107.07,148.03) – (105.25,147.33) – (103.43,146.63) – (101.61,145.93) – ( 99.79,145.24) – ( 97.97,144.56) – ( 96.15,143.88) – ( 94.33,143.20) – ( 92.51,142.53) – ( 90.69,141.86) – ( 88.87,141.19) – ( 87.05,140.53) – ( 85.23,139.87) – ( 83.40,139.22) – ( 81.58,138.57) – ( 79.76,137.92) – ( 77.94,137.28) – ( 76.12,136.64) – ( 74.30,136.01) – ( 72.48,135.38) – ( 70.66,134.75) – ( 68.84,134.13) – ( 67.02,133.51) – ( 65.20,132.89) – ( 63.38,132.28) – ( 61.56,131.67) – ( 59.74,131.06) – ( 57.92,130.46) – ( 56.10,129.86) – ( 54.28,129.27) – ( 52.46,128.68) – ( 50.64,128.09) – cycle; ( 50.64,138.76) – ( 52.46,139.20) – ( 54.28,139.65) – ( 56.10,140.09) – ( 57.92,140.54) – ( 59.74,141.00) – ( 61.56,141.46) – ( 63.38,141.92) – ( 65.20,142.38) – ( 67.02,142.85) – ( 68.84,143.33) – ( 70.66,143.80) – ( 72.48,144.28) – ( 74.30,144.77) – ( 76.12,145.26) – ( 77.94,145.75) – ( 79.76,146.25) – ( 81.58,146.75) – ( 83.40,147.25) – ( 85.23,147.76) – ( 87.05,148.28) – ( 88.87,148.79) – ( 90.69,149.32) – ( 92.51,149.84) – ( 94.33,150.37) – ( 96.15,150.91) – ( 97.97,151.45) – ( 99.79,152.00) – (101.61,152.55) – (103.43,153.10) – (105.25,153.66) – (107.07,154.23) – (108.89,154.80) – (110.71,155.37) – (112.53,155.95) – (114.35,156.54) – (116.17,157.13) – (117.99,157.72) – (119.81,158.32) – (121.63,158.93) – (123.45,159.54) – (125.27,160.16) – (127.09,160.78) – (128.91,161.41) – (130.73,162.04) – (132.55,162.68) – (134.37,163.33) – (136.19,163.98) – (138.01,164.64) – (139.83,165.31) – (141.65,165.98) – (143.47,166.65) – (145.29,167.34) – (147.11,168.03) – (148.93,168.72) – (150.75,169.43) – (152.57,170.14) – (154.39,170.86) – (156.21,171.58) – (158.03,172.31) – (159.85,173.05) – (161.67,173.79) – (163.49,174.55) – (165.31,175.31) – (167.13,176.08) – (168.95,176.85) – (170.77,177.63) – (172.59,178.43) – (174.41,179.22) – (176.23,180.03) – (178.05,180.85) – (179.87,181.67) – (181.69,182.50) – (183.51,183.35) – (185.33,184.20) – (187.15,185.05) – (188.97,185.92) – (190.79,186.80) – (192.61,187.69) – (194.43,188.58) – (196.25,189.49) – (198.07,190.40) – (199.89,191.33) – (201.71,192.26) – (203.53,193.21) – (205.35,194.16) – (207.17,195.13) – (208.99,196.10) – (210.81,197.09) – (212.63,198.09) – (214.45,199.10) – (216.27,200.12) – (218.09,201.15) – (219.91,202.20) – (221.74,203.25) – (223.56,204.32) – (225.38,205.40) – (227.20,206.49) – (229.02,207.60) – (230.84,208.72) – (232.66,209.85); ( 50.64,128.09) – ( 52.46,128.68) – ( 54.28,129.27) – ( 56.10,129.86) – ( 57.92,130.46) – ( 59.74,131.06) – ( 61.56,131.67) – ( 63.38,132.28) – ( 65.20,132.89) – ( 67.02,133.51) – ( 68.84,134.13) – ( 70.66,134.75) – ( 72.48,135.38) – ( 74.30,136.01) – ( 76.12,136.64) – ( 77.94,137.28) – ( 79.76,137.92) – ( 81.58,138.57) – ( 83.40,139.22) – ( 85.23,139.87) – ( 87.05,140.53) – ( 88.87,141.19) – ( 90.69,141.86) – ( 92.51,142.53) – ( 94.33,143.20) – ( 96.15,143.88) – ( 97.97,144.56) – ( 99.79,145.24) – (101.61,145.93) – (103.43,146.63) – (105.25,147.33) – (107.07,148.03) – (108.89,148.74) – (110.71,149.45) – (112.53,150.16) – (114.35,150.89) – (116.17,151.61) – (117.99,152.34) – (119.81,153.07) – (121.63,153.81) – (123.45,154.56) – (125.27,155.30) – (127.09,156.06) – (128.91,156.82) – (130.73,157.58) – (132.55,158.35) – (134.37,159.12) – (136.19,159.90) – (138.01,160.68) – (139.83,161.47) – (141.65,162.26) – (143.47,163.06) – (145.29,163.86) – (147.11,164.67) – (148.93,165.48) – (150.75,166.30) – (152.57,167.13) – (154.39,167.96) – (156.21,168.80) – (158.03,169.64) – (159.85,170.49) – (161.67,171.34) – (163.49,172.20) – (165.31,173.06) – (167.13,173.93) – (168.95,174.81) – (170.77,175.69) – (172.59,176.58) – (174.41,177.48) – (176.23,178.38) – (178.05,179.29) – (179.87,180.20) – (181.69,181.13) – (183.51,182.05) – (185.33,182.99) – (187.15,183.93) – (188.97,184.88) – (190.79,185.83) – (192.61,186.79) – (194.43,187.76) – (196.25,188.73) – (198.07,189.72) – (199.89,190.71) – (201.71,191.70) – (203.53,192.71) – (205.35,193.72) – (207.17,194.74) – (208.99,195.77) – (210.81,196.80) – (212.63,197.84) – (214.45,198.89) – (216.27,199.95) – (218.09,201.02) – (219.91,202.09) – (221.74,203.18) – (223.56,204.27) – (225.38,205.37) – (227.20,206.47) – (229.02,207.59) – (230.84,208.72) – (232.66,209.85); at ( 50.64,129.88) [+]{}; at ( 59.74,132.55) [+]{}; at ( 68.84,136.21) [+]{}; at ( 77.94,139.52) [+]{}; at ( 87.05,142.40) [+]{}; at ( 96.15,146.00) [+]{}; at (105.25,148.24) [+]{}; at (114.35,152.28) [+]{}; at (123.45,155.59) [+]{}; at (132.55,159.26) [+]{}; at (141.65,162.66) [+]{}; at (150.75,166.11) [+]{}; at (159.85,170.42) [+]{}; at (168.95,174.24) [+]{}; at (178.05,178.79) [+]{}; at (187.15,182.46) [+]{}; at (196.25,187.50) [+]{}; at (205.35,192.28) [+]{}; at (214.45,196.71) [+]{}; at (223.56,203.13) [+]{}; at (232.66,208.13) [+]{}; ( 0.00, 0.00) rectangle (254.39,231.26); ( 43.36, 28.91) – (239.94, 28.91) – (239.94,216.81) – ( 43.36,216.81) – ( 43.36, 28.91); at (141.65, 2.51) [$\lambda$]{}; at ( 16.96,122.86) [$\gamma$]{}; Our main result, Theorem \[thm:bounds\], more generally gives bounds on the probability generating functional of a Gibbs process. Let $\Xi$ be an arbitrary point process on ${\mathbb{R}}^d$. The *probability generating functional (p.g.fl.)* $\Psi_{\Xi}$ is defined as $$\label{eq:pgfl} \Psi_\Xi(g)={{\mathbb E}}\Bigl(\prod_{y\in \Xi}g(y)\Bigr)$$ for any measurable function $g\colon {\mathbb{R}}^d\to [0,1]$ for which $1-g$ has bounded support, see e.g. [@dvj08 p.59] for details. Many statistics of point processes, such as the *empty space function* ($F$ function), contain expectations as in . For pairwise interaction processes the situation is even better. By the Georgii–Nguyen–Zessin equation  the *nearest neighbour function* ($G$ function), Ripley’s $K$ function and the correlation functions of all orders can be rewritten using the p.g.fl. The idea for proving Theorem \[thm:bounds\] is to replace the Gibbs process $\Xi$ in by a suitable Poisson process and bound the error using Stein’s method. The rest of the paper is organised as follows. In Section \[sec:pre\] we introduce some notation and state the main result. In Section \[sec:bounds-int\] we provide bounds on the intensity, and in Section \[sec:s-stat\] bounds on other summary statistics are derived. Section \[sec:proofs\] contains the proof of the main result. Preliminaries and main result {#sec:pre} ============================= Let $({\mathfrak{N}},{\mathcal{N}})$ denote the space of locally finite point measures on ${\mathbb{R}}^d$ equipped with the $\sigma$-algebra generated by the evaluation maps $[{\mathfrak{N}}\ni\xi\mapsto \xi(A) \in {\mathbb{Z}_{+}}]$ for bounded Borel sets $A\in {\mathbb{R}}^d$. A point process is just a ${\mathfrak{N}}$-valued random element. We assume the point processes to be *simple*, i.e. do not allow multi-points. Thus we can use set notation, i.e. $x\in \xi$ means that the point $x$ lies in the support of the measure $\xi$. In spatial statistics point processes are usually defined on a bounded window ${\mathcal{W}}\subset {\mathbb{R}}^d$. Let ${\mathfrak{N}}\vert_{\mathcal{W}}$ denote the restriction of the ${\mathfrak{N}}$ to ${\mathcal{W}}$. A point process $\Xi$ on ${\mathcal{W}}$ is called a *Gibbs process* if it has a hereditary density $u$ with respect to the distribution of the Poisson process with unit intensity. Hereditarity means that $u(\xi)>0$ implies $u(\eta)>0$ for all subconfigurations $\eta\subset \xi$. By hereditarity we can define the *conditional intensity* as $$\label{eq:cond-int} \lambda(x\mid \xi)=\frac{u(\xi\cup\{x\})}{u(\xi)},$$ where $0/0=0$. Roughly speaking, the conditional intensity is the infinitesimal probability that $\Xi$ has a point at $x$, given that $\Xi$ coincides with the configuration $\xi$ everywhere else. Furthermore $\lambda(\cdot{\, \vert \,}\cdot)$ uniquely characterises the distribution of $\Xi$, since by one can recursively recover an unnormalised density. It is well-known that the conditional intensity is the $dx \otimes {\mathscr{L}}({\Xi})$-almost everywhere unique product measurable function that satisfies the *Georgii–Nguyen–Zessin equation* $$\label{eq:gnz} {\mathbb{E}}\biggl( \int_{{\mathcal{W}}} h(x, {\Xi}\setminus\{x\}) \; {\Xi}({\mathrm{d}}x) \biggr) = \int_{{\mathcal{W}}} {\mathbb{E}}\bigl( h(x, {\Xi}) \lambda(x {\, \vert \,}{\Xi}) \bigr) \; dx$$ for every measurable $h \colon {\mathcal{W}}\times {\mathfrak{N}}\vert_{\mathcal{W}}\to {\mathbb{R}_{+}}$. So far $\lambda(\cdot{\, \vert \,}\cdot)$ is only a function on ${\mathcal{W}}\times {\mathfrak{N}}\vert_{\mathcal{W}}$, but in many cases there exists a natural extension to the whole space, which we shall also denote by $\lambda(\cdot{\, \vert \,}\cdot)$. One way to generalise Gibbs processes to the whole space ${\mathbb{R}}^d$ is then by the so-called *integral characterisation*. A point process $\Xi$ on ${\mathbb{R}}^d$ is a Gibbs process corresponding to the conditional intensity $\lambda(\cdot{\, \vert \,}\cdot)$ if it satisfies with ${\mathcal{W}}$ replaced by ${\mathbb{R}}^d$ for all measurable $h \colon {\mathbb{R}}^d\times {\mathfrak{N}}\to {\mathbb{R}_{+}}$; see [@moellerwaage04 p.95], or [@nguyenzessin79] for a more rigorous presentation. Unlike in the case of a bounded domain, $\Xi$ may not be uniquely determined by . For the rest of this paper we will only deal with the conditional intensity, i.e. if we say that a result holds for a Gibbs process with conditional intensity $\lambda(\cdot{\, \vert \,}\cdot)$, we mean that it holds for *all* processes corresponding to this conditional intensity. A Gibbs process $\Xi$ is said to be a *pairwise interaction process* if its conditional intensity is of the form $$\label{eq:pip} \lambda(x{\, \vert \,}\xi)=\beta \prod_{y \in \xi}\varphi (x,y)$$ for a constant $\beta>0$ and a symmetric *interaction function* $\varphi$. We denote the distribution of $\Xi$ by $\operatorname{PIP}(\beta,\varphi)$. The process $\Xi$ is called *inhibitory* if $\varphi \le 1$ and it is said to have a *finite interaction range* if $1-\varphi$ is compactly supported. $\Xi$ is *stationary* if $\varphi(x,y)$ depends only on the difference $x-y$; we then write $\varphi(x,y)=\varphi(x-y)$. If in addition the interaction function is rotation invariant, i.e. $\varphi(x)=\varphi({\lVert x \rVert})$, then $\Xi$ is called *isotropic*. For conditions on $\varphi$ ensuring the existence of $\Xi$ we refer the reader to [@ruelle69]. If $\Xi$ is a general point process, its *expectation measure or first order moment measure* ${{\mathbb E}}\Xi$ on ${\mathbb{R}}^d$ is simpy given by $({{\mathbb E}}\Xi)(A) = {{\mathbb E}}(\Xi(A))$ for every Borel set $A \subset {\mathbb{R}}^d$. For $k \in {\mathbb{N}}$ the *$k$-th order factorial moment measure* of $\Xi$ is the expectation measure of the factorial product measure $$\Xi^{[k]} = \sum_{\substack{X_1,\ldots,X_k \in \Xi \\[2pt] \text{pairwise different}}} \delta_{(X_1,\ldots,X_k)}$$ on $({\mathbb{R}}^d)^k$. Any moment measure is said to *exist* if it is locally finite. The *intensity (function)* $\lambda(x)$ of a Gibbs process $\Xi$ is the density of the first moment measure of $\Xi$ with respect to Lebesgue measure, provided the first moment measure exists. For a bounded $A\subset {\mathbb{R}}^d$, Equation  yields $${{\mathbb E}}\Xi(A)=\int_A {{\mathbb E}}\big(\lambda(x{\, \vert \,}\Xi)\big)\; dx,$$ hence the existence of the intensity and the form $\lambda(x)={{\mathbb E}}(\lambda(x{\, \vert \,}\Xi))$. For stationary processes the intensity is obviously constant and we just write $\lambda$. For a stationary pairwise interaction process we get $$\label{eq:lambda-pip} \lambda={{\mathbb E}}\big(\lambda(0{\, \vert \,}\Xi)\big)=\beta {\hspace*{1.5pt}}{{\mathbb E}}\Bigl(\prod_{y\in \Xi}\varphi(y)\Bigr)=\beta {\hspace*{1.5pt}}\Psi_\Xi( \varphi).$$ In a similar manner it is possible to obtain the densities of the higher order factorial moment measures, the so-called *correlation functions*; see [@mw07; @mase90]. For a stationary process $\Xi\sim\operatorname{PIP}(\beta,\varphi)$ the $k$-th correlation function is given by $$\begin{aligned} \lambda_k(x_1,\dots,x_k)&=\beta^k\Bigl(\prod_{1\le i<j\le k}\varphi(x_i-x_j)\Bigr){\hspace*{1.5pt}}{{\mathbb E}}\Bigl(\prod_{y\in \Xi}\varphi(y-x_1)\cdots\varphi(y-x_k)\Bigr) \nonumber \\ &=\beta^k\Bigl(\prod_{1\le i<j\le k}\varphi(x_i-x_j)\Bigr){\hspace*{1.5pt}}\Psi_\Xi\big(\varphi(\cdot-x_1)\cdots\varphi(\cdot-x_k)\big). \label{eq:corr-fun}\end{aligned}$$ A frequently used function in spatial statistics is the *pair correlation function* which is defined as $$\label{eq:pcf} \rho(x,y)=\frac{\lambda_2(x,y)}{\lambda(x)\lambda(y)}.$$ In the stationary isotropic case this simplifies to $\rho(s)=\lambda_2(x,y)/\lambda^2$, where $s=\|x-y\|$. For our results we need a stability condition for the Gibbs processes. A Gibbs process $\Xi$ is called *locally stable* if there exists a non-negative function $c^*$ such that $\int_{\mathcal{W}}c^*(x)\,dx < \infty$ for all bounded domains ${\mathcal{W}}\subset {\mathbb{R}}^d$ and the conditional intensity satisfies $$\label{eq:loc-s} \lambda(x{\, \vert \,}\xi)\le c^*(x),$$ for all $\xi \in {\mathfrak{N}}$. For the rest of the paper we restrict ourselves to stationary point processes on ${\mathbb{R}}^d$ and require $c^{*}$ to be a constant. The following is the key theorem for obtaining the results in Sections \[sec:bounds-int\] and \[sec:s-stat\]. Its proof is the subject of Section \[sec:proofs\]. \[thm:bounds\] Let $\Xi$ be a stationary locally stable Gibbs process with intensity $\lambda$ and local stability constant $c^*$, and let $g\colon {\mathbb{R}}^d\to [0,1]$ be a function for which $1-g$ has bounded support. Then $$\label{eq:bounds} 1-\lambda G \le {{\mathbb E}}\Bigl(\prod_{y\in \Xi}g(y)\Bigr) \le 1-\frac{\lambda}{c^*}\big(1-e^{-c^*G}\big),$$ where $G=\int_{{\mathbb{R}}^d} 1-g(x)\, dx$. Bounds on the intensity {#sec:bounds-int} ======================= For the intensity of a inhibitory pairwise interaction process we immediately obtain from Theorem \[thm:bounds\] the following result. \[thm:lambdaPIP\] Let $\Xi\sim \operatorname{PIP}(\beta,\varphi)$ be inhibitory with finite interaction range. Then $$\label{eq:lambdaPIP} \frac{\beta}{1+\beta G}\le \lambda \le \frac{\beta}{2-e^{-\beta G}},$$ where $G=\int_{{\mathbb{R}}^d}1-\varphi(x)\,dx$. Recall from that $\lambda=\beta {\hspace*{1.5pt}}{{\mathbb E}}\prod_{y\in \Xi}\varphi(y)$ and use $c^*=\beta$. Theorem \[thm:bounds\] then yields $$1-\lambda G \le \frac{\lambda}{\beta}\le 1-\frac{\lambda}{\beta}\big(1-e^{-\beta G}\big)$$ which can be rearranged as . The lower bound of can also be found in [@ruelle69 p.96] with the restriction $\beta G <e^{-1}$, whereas our inequality holds for all values of $\beta$ and $G$. Let $\Xi$ be a Strauss process, i.e. $$\varphi(x) =\begin{cases} \gamma \quad &\text{if} \quad \|x\| \le r\\ 1 \quad &\text{if} \quad \|x\|>r \end{cases}$$ for some parameters $r>0$ and $0\le \gamma \le 1$. Then $G=(1-\gamma)\alpha_dr^d$, where $\alpha_d$ denotes the volume of the unit ball. Figure \[fig:Strauss50\] shows that for a reasonable choice of the parameters $(\beta,r,\gamma)$ the bounds on $\lambda$ are quite good. The maximal relative error between the bounds and the simulated values is about 3.5%. 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The first one is the *Poisson-saddlepoint* approximation proposed in [@baddeleynair12]. The authors replaced in the Gibbs process $\Xi\sim\operatorname{PIP}(\beta,\varphi)$ by a Poisson process ${\mathrm{H}}_{\lambda_{PS}}$ with intensity $\lambda_{PS}$ such that the following equality holds $$\label{eq:ps} \lambda_{PS}={{\mathbb E}}\lambda(0{\, \vert \,}{\mathrm{H}}_{\lambda_{PS}})=\beta {\hspace*{1.5pt}}{{\mathbb E}}\Bigl(\prod_{y\in {\mathrm{H}}_{\lambda_{PS}}}\varphi(y)\Bigr).$$ Solving this equation yields $$\label{eq:lambda-PS} \lambda_{PS}=\frac{W(\beta G)}{G},$$ where $W$ is *Lambert’s W function*, the inverse of $x\mapsto xe^x$, and $G=\int_{{\mathbb{R}}^d}1-\varphi(x)\,dx$ as above. The second method is the *mean-field* approximation that was also described in [@baddeleynair12] and is given by $$\label{eq:lambda-MF} \lambda_{MF}=\frac{W(\beta \Gamma)}{\Gamma},$$ where $\Gamma=-\int_{{\mathbb{R}}^d}\log(\varphi(x))\,dx$. Figure \[fig:Strauss100\] shows the two approximations and our bounds from Inequality  for two-dimensional Strauss processes. In [@baddeleynair12] it is shown that under the conditions of Theorem \[thm:lambdaPIP\] we have $\lambda\ge \lambda_{MF}$. The authors also conjectured, based on simulations for Strauss processes, that $\lambda_{PS}$ is an upper bound for $\lambda$. However, the next example indicates that this is not generally true. (0,0) rectangle (420.48,183.96); ( 14.45, 0.00) rectangle (210.24,169.51); ( 33.87, 6.28) – (190.82, 6.28) – (190.82,163.23) – ( 33.87,163.23) – ( 33.87, 6.28); at ( 50.75, 87.67) [+]{}; at (156.13,153.79) [+]{}; at ( 97.62,127.92) [+]{}; at (171.27, 93.02) [+]{}; at ( 92.11, 69.67) [+]{}; at (168.68, 53.00) [+]{}; at (132.32, 37.20) [+]{}; at (159.91,151.13) [+]{}; at ( 99.79, 31.61) [+]{}; at ( 83.57, 39.26) [+]{}; at (131.29, 96.82) [+]{}; at ( 61.11, 54.40) [+]{}; at ( 53.80, 13.67) [+]{}; at (152.29,108.78) [+]{}; at (136.19, 18.46) [+]{}; at ( 39.94,135.09) [+]{}; at (133.97, 96.63) [+]{}; at ( 50.53,149.63) [+]{}; at (178.92,124.98) [+]{}; at (189.54,101.33) [+]{}; at (141.71, 74.55) [+]{}; at (164.45, 56.19) [+]{}; at ( 71.32, 88.91) [+]{}; at ( 69.81, 5.09) [+]{}; at ( 65.70, 5.85) [+]{}; at ( 59.88,100.79) [+]{}; at ( 61.97, 22.05) [+]{}; at ( 44.82, 27.44) [+]{}; at (136.64, 92.70) [+]{}; at (164.58, 6.95) [+]{}; at (186.23, 82.58) [+]{}; at ( 96.54, 84.64) [+]{}; at (157.23,150.24) [+]{}; 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at (129.63, 81.42) [+]{}; at ( 74.98, 51.37) [+]{}; at ( 39.08,138.38) [+]{}; at ( 59.97, 59.06) [+]{}; at (166.05,125.54) [+]{}; at (114.01,126.36) [+]{}; at (154.36, 37.60) [+]{}; at (121.62, 7.52) [+]{}; at (179.37,111.66) [+]{}; at (156.71,154.74) [+]{}; at ( 99.36,153.68) [+]{}; at (172.37, 93.10) [+]{}; at (186.39, 26.92) [+]{}; at ( 99.64,123.40) [+]{}; at (140.14,137.30) [+]{}; at ( 80.72,123.56) [+]{}; at (187.24, 83.14) [+]{}; at (178.62, 41.95) [+]{}; at (158.98,152.42) [+]{}; at (127.15, 36.05) [+]{}; at ( 85.88,141.03) [+]{}; at ( 93.15, 87.78) [+]{}; at (143.36,149.33) [+]{}; at (107.35, 78.49) [+]{}; at ( 46.96, 32.81) [+]{}; at (152.88, 20.40) [+]{}; at ( 79.01, 71.66) [+]{}; at (103.82, 19.50) [+]{}; at ( 67.72,151.07) [+]{}; at ( 97.68, 86.64) [+]{}; at ( 69.23,151.90) [+]{}; at (165.28, 54.35) [+]{}; at ( 42.76,140.92) [+]{}; at (156.32,106.44) [+]{}; at ( 64.42, 75.90) [+]{}; at ( 50.80, 89.57) [+]{}; at ( 68.28,117.61) [+]{}; at (116.56,140.42) [+]{}; at (131.32,125.99) [+]{}; 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at (160.92,154.37) [+]{}; at ( 65.13,120.71) [+]{}; at ( 64.27,113.98) [+]{}; at ( 68.48, 5.76) [+]{}; at ( 46.35,114.05) [+]{}; at ( 50.90,153.66) [+]{}; at ( 89.25, 8.34) [+]{}; at (156.52, 91.94) [+]{}; at ( 68.05,153.05) [+]{}; at ( 65.69, 57.66) [+]{}; at (107.20, 77.45) [+]{}; at ( 97.26, 34.11) [+]{}; at (142.55, 74.72) [+]{}; at (170.40, 76.61) [+]{}; at ( 92.78, 70.48) [+]{}; at (170.69, 20.60) [+]{}; at (173.79,109.87) [+]{}; at ( 39.11, 68.66) [+]{}; at ( 80.70, 64.14) [+]{}; at ( 92.51,156.61) [+]{}; at (147.91, 74.27) [+]{}; at ( 46.35, 53.70) [+]{}; at ( 77.78, 75.04) [+]{}; at ( 38.59,137.31) [+]{}; at (154.31,130.32) [+]{}; at ( 69.86,137.48) [+]{}; at (129.75,150.57) [+]{}; at ( 99.55,108.64) [+]{}; at ( 41.18, 65.21) [+]{}; at (103.85, 57.12) [+]{}; at (104.37, 57.42) [+]{}; at (176.42,154.14) [+]{}; at ( 85.58, 21.87) [+]{}; at (131.23,144.97) [+]{}; at (112.62, 39.22) [+]{}; at (139.98,136.64) [+]{}; at (133.54, 39.36) [+]{}; at (155.77,106.80) [+]{}; at (155.54,153.58) [+]{}; at (150.79, 53.54) [+]{}; at ( 68.67, 21.51) [+]{}; at (170.08, 53.73) [+]{}; at ( 49.86,154.74) [+]{}; at (127.51,146.60) [+]{}; at (189.46,148.70) [+]{}; at ( 92.70, 89.18) [+]{}; at (128.15, 81.68) [+]{}; at (105.75,108.61) [+]{}; at (174.45, 18.53) [+]{}; at (111.42,141.54) [+]{}; at (169.20, 53.46) [+]{}; at ( 70.15, 93.39) [+]{}; at ( 98.76, 68.88) [+]{}; at ( 81.36,123.69) [+]{}; at ( 80.76,124.38) [+]{}; at ( 85.36,106.37) [+]{}; at ( 61.85, 53.08) [+]{}; at (181.92, 59.66) [+]{}; at (135.29, 20.40) [+]{}; at ( 58.84,100.26) [+]{}; at ( 74.00, 47.92) [+]{}; at (148.86, 19.34) [+]{}; at ( 84.70,103.89) [+]{}; at ( 35.71,156.91) [+]{}; at ( 39.08, 66.75) [+]{}; at ( 44.37,136.32) [+]{}; at (186.47, 26.34) [+]{}; at ( 65.03, 39.66) [+]{}; at (139.13,111.56) [+]{}; at ( 48.12, 90.02) [+]{}; at ( 96.56, 37.59) [+]{}; at ( 74.41, 49.81) [+]{}; at (148.48, 75.45) [+]{}; at (132.95, 35.81) [+]{}; at (132.43, 95.36) [+]{}; at ( 79.38, 79.14) [+]{}; at (110.68, 38.34) [+]{}; at (185.28, 30.00) [+]{}; 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at (154.63, 86.72) [+]{}; at ( 83.66, 37.00) [+]{}; at (163.75, 52.56) [+]{}; at (112.31,146.76) [+]{}; at (128.46, 68.00) [+]{}; at ( 50.54, 87.60) [+]{}; at (161.01,154.84) [+]{}; at (123.22,108.47) [+]{}; at (170.72,137.74) [+]{}; at ( 71.95, 88.08) [+]{}; at (121.12, 53.30) [+]{}; at ( 65.39, 20.61) [+]{}; at (155.52, 18.50) [+]{}; at (186.27, 28.36) [+]{}; at (108.50, 57.82) [+]{}; at (127.92,129.56) [+]{}; at ( 64.56, 39.56) [+]{}; at (165.65,140.29) [+]{}; at (181.29, 44.76) [+]{}; at ( 91.54, 48.40) [+]{}; at (163.82,123.79) [+]{}; at (160.57,153.00) [+]{}; at ( 42.90, 27.88) [+]{}; at (169.74, 72.11) [+]{}; at (106.49, 93.70) [+]{}; at ( 58.98, 78.27) [+]{}; at (107.66, 57.11) [+]{}; at (114.41,125.31) [+]{}; at (153.06, 89.01) [+]{}; at ( 83.86, 20.82) [+]{}; at ( 63.18, 24.06) [+]{}; at (156.22, 92.75) [+]{}; at ( 87.45, 50.62) [+]{}; at (112.22, 37.04) [+]{}; at (187.41,143.43) [+]{}; at ( 81.76,119.79) [+]{}; at (109.40, 41.70) [+]{}; at (128.46,126.91) [+]{}; at ( 70.41, 93.50) [+]{}; at (167.77, 53.31) [+]{}; at (186.01, 29.08) [+]{}; at ( 62.84,151.78) [+]{}; at ( 45.00, 26.41) [+]{}; at (117.09, 18.40) [+]{}; at (134.36, 91.75) [+]{}; at ( 39.20, 71.14) [+]{}; at (153.11, 22.14) [+]{}; at ( 69.73, 94.23) [+]{}; at (170.32, 33.55) [+]{}; at (150.36, 59.59) [+]{}; at (166.10,141.41) [+]{}; at ( 69.76,132.24) [+]{}; at (106.23,111.83) [+]{}; at (116.37, 23.00) [+]{}; at ( 50.80, 15.88) [+]{}; at (108.24, 60.85) [+]{}; at ( 81.84, 40.01) [+]{}; at ( 69.31,136.06) [+]{}; at ( 67.59, 22.99) [+]{}; at (105.22, 59.00) [+]{}; at ( 51.78,117.67) [+]{}; at ( 34.77,124.58) [+]{}; at ( 48.34, 84.34) [+]{}; at (141.81,118.71) [+]{}; at (119.43,156.26) [+]{}; at (173.60,153.44) [+]{}; at ( 64.60, 76.80) [+]{}; at ( 68.47,134.19) [+]{}; at ( 41.63, 67.97) [+]{}; at (156.11, 85.39) [+]{}; at (151.04, 5.87) [+]{}; at ( 64.66,116.91) [+]{}; at (132.63, 22.99) [+]{}; at (153.02,128.92) [+]{}; at ( 93.70, 67.25) [+]{}; at (110.13, 58.47) [+]{}; at (168.74, 69.22) [+]{}; at (161.75,156.47) [+]{}; at (155.12,108.05) [+]{}; at ( 66.56,118.62) [+]{}; at (143.39,153.86) [+]{}; at (184.97, 76.16) [+]{}; at (117.78,104.80) [+]{}; at (144.46, 75.06) [+]{}; at (151.92, 56.19) [+]{}; at (141.52,149.64) [+]{}; at (184.13,126.53) [+]{}; at (107.90, 61.14) [+]{}; at ( 60.34, 71.14) [+]{}; at (106.96, 93.92) [+]{}; at (173.57, 90.59) [+]{}; at ( 97.89, 37.46) [+]{}; at (133.45, 38.37) [+]{}; at (102.04, 16.66) [+]{}; at ( 63.67, 36.74) [+]{}; at ( 45.33, 85.69) [+]{}; at (111.33, 39.84) [+]{}; at ( 86.11,120.52) [+]{}; at (115.63,128.22) [+]{}; at (151.69,108.67) [+]{}; at (136.42, 96.42) [+]{}; at ( 45.52, 48.27) [+]{}; at ( 89.71,137.73) [+]{}; at (154.14,130.07) [+]{}; at (156.36, 92.83) [+]{}; (224.69, 0.00) rectangle (420.48,169.51); (244.11, 6.28) – (401.06, 6.28) – (401.06,163.23) – (244.11,163.23) – (244.11, 6.28); at (386.19,111.68) [+]{}; at (385.36,158.59) [+]{}; at (326.38, 67.61) [+]{}; at (308.94,126.08) [+]{}; at (381.76,118.94) [+]{}; at (360.81, 65.79) [+]{}; at (365.27,108.59) [+]{}; 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at (280.09,118.17) [+]{}; at (255.24, 76.57) [+]{}; at (343.20,149.16) [+]{}; at (386.09, 19.65) [+]{}; at (340.72, 65.53) [+]{}; at (314.60, 73.94) [+]{}; at (265.44, 79.02) [+]{}; at (298.52,112.78) [+]{}; at (390.83,124.37) [+]{}; at (368.88, 94.29) [+]{}; at (349.08,132.10) [+]{}; at (330.32, 28.70) [+]{}; at (365.09, 41.00) [+]{}; at (349.93, 89.65) [+]{}; at (359.82, 57.14) [+]{}; at (355.48, 75.20) [+]{}; at (284.06,152.63) [+]{}; at (326.42,112.67) [+]{}; at (306.58, 72.06) [+]{}; at (388.70, 68.73) [+]{}; at (400.07,106.14) [+]{}; at (279.74,105.44) [+]{}; at (349.45,100.81) [+]{}; at (262.41, 98.93) [+]{}; at (298.35, 39.62) [+]{}; \[ex:ringprozess\] Consider the process $\Xi\sim \operatorname{PIP}(\beta,\varphi)$ with the interaction function $$\varphi(x) =\begin{cases} 1 \quad &\text{if} \quad \|x\| \le r\\ 0 \quad &\text{if} \quad r<\|x\| \le R\\ 1 \quad &\text{if} \quad \|x\| > R \end{cases}$$ for constants $0\le r\le R $. We refer to this as a *hard annulus process*. It is a special case of a so-called *multiscale Strauss process*, see [@moellerwaage04 Ex. 6.2]. Let $d=2$, $\beta=3000$, $r=0.05$ and $R=\sqrt{2}r$. Then $$\lambda_{MF}=0,\quad \frac{\beta}{1+\beta G}=122.1, \quad \lambda_{PS}=295.2\quad \text{and}\quad \frac{\beta}{2-e^{-\beta G}}=1500.$$ An estimate of the intensity based on $300$ simulations gave $\hat{\lambda}=493.8 > \lambda_{PS}$. For comparison we also estimated the intensity of a Strauss hard core process ($\gamma=0$) with the same $\beta$ and $G$ and obtained $\hat{\lambda}=193.3$. Figure \[fig:ringprozess\] shows that although the two processes have the same $\beta$ and $G$, their realisations look quite different. All simulations were performed by long runs ($10^7$ steps) of Markov Chain Monte Carlo. We were not able to prove that $\lambda>\lambda_{PS}$ in this case, but bring forward the following heuristic argument for the observed phenomenon. The simulations shows that for large $\beta$ the points tend to cluster on “island” of radius $\le r/2$ which are separated by a distance $\ge R$. Since the points within each island do not interact, we expect the intensity to grow linearly in $\beta$ for large $\beta$. However $\lambda_{PS}$ only grows logarithmically for large $\beta$, so that at some point the intensity will overtake. Even if $\lambda_{PS}$ may not serve as a bound on $\lambda$ it remains useful as an approximation. Empirically its values stay relatively close to the simulated values, whereas the difference of our upper and lower bounds in increases for large $\beta G$. The following result in connection with Theorem \[thm:lambdaPIP\] gives an upper bound on the error in Poisson saddlepoint approximation. \[lemma:PS\] Under the conditions of Theorem \[thm:lambdaPIP\] we have $$\label{eq:bounds-lambda-PS} \frac{\beta}{1+\beta G}\le \lambda_{PS} \le \frac{\beta}{2-e^{-\beta G}}.$$ Since $\lambda_{PS}=W(\beta G)/G$, it suffices to show the following two inequalities: $$\label{eq:W} \frac{x}{1+x}\le W(x) \quad \text{and} \quad W(x) \le \frac{x}{2-e^{-x}}$$ for all $x\ge0$. The first one follows from $x/(1+x)\le \log(1+x)$, see [@as64 Eq. 4.1.33], by transforming it to $$\frac{x}{1+x} \exp \bigl( \frac{x}{1+x} \bigr) \leq x$$ and applying the increasing function $W$ on both sides. For the second inequality note that $$\log(2-e^{-x})\le \frac{x}{2-e^{-x}}.$$ This holds because we have equality for $x=0$ and it is straightforward to see that the derivative of the left hand side is less than or equal to the derivative of the right hand side for all $x\ge 0$. A similar transformation as above and applying $W$ on both sides again gives the second inequality in . Summary statistics {#sec:s-stat} ================== For a point process $\Xi$ the *empty space function* or *$F$ function* is defined as the cumulative distribution function of the distance from the origin to the nearest point in $\Xi$, i.e. $$\begin{aligned} F(t)&={{\mathbb P}}(\exists y\in \Xi \colon \|y\| \le t)=1-{{\mathbb P}}(\Xi({\mathbb{B}}(0,t))=0)\\ &=1-{{\mathbb E}}\Bigl(\prod_{y\in\Xi}{\mathbbm{1}}\{y\notin {\mathbb{B}}(0,t)\}\Bigr)=1-\Psi_\Xi({\mathbbm{1}}\{\cdot \notin {\mathbb{B}}(0,t)\}),\end{aligned}$$ where ${\mathbb{B}}(x,t)$ denotes the closed ball centred at $x\in{\mathbb{R}}^d$ with radius $t\ge0$. Thus for a locally stable process $\Xi$ with constant $c^*$ we obtain from Theorem \[thm:bounds\] $$\label{eq:boundsF} \frac{\lambda}{c^*}\big(1-\exp(-c^*\alpha_dt^d)\big)\le F(t) \le \lambda \alpha_d t^d.$$ Note that for a Poisson process with intensity $\lambda$ we may choose $c^*=\lambda$, in which case the lower bound in is exact. A minor drawback of the bounds in is that the intensity is in general not known and has to be estimated as well, e.g. by the methods of Section \[sec:bounds-int\]. The *nearest neighbour function* or *$G$ function* is defined as the cumulative distribution function of the distance from a typical point of $\Xi$ (in the sense of the Palm distribution) to its nearest neighbour. For pairwise interaction processes $\Xi\sim\operatorname{PIP}(\beta,\varphi)$ the $G$ function is computed in [@mase90 Sec. 5] as $$\label{eq:G} G(t)=1-\frac{\beta}{\lambda}{{\mathbb E}}\Big(\prod_{y\in \Xi} {\mathbbm{1}}\{y\notin {\mathbb{B}}(0,t)\}\varphi(y)\Big)=1-\frac{\beta}{\lambda}\Psi_\Xi\big({\mathbbm{1}}\{\cdot \notin {\mathbb{B}}(0,t)\}\varphi(\cdot)\big).$$ Thus if $\Xi$ is inhibitory and has finite interaction range, setting $c^*=\beta$ in Theorem \[thm:bounds\] yields $$\label{eq:boundsG} 2-\frac{\beta}{\lambda}-\exp(-\beta \tilde{G}_t)\le G(t) \le 1-\frac{\beta}{\lambda}+\beta\tilde{G}_t,$$ where $\tilde{G}_t=\int_{{\mathbb{R}}^d}1-\varphi(x){\mathbbm{1}}\{\|x\|>t\}\,dx$. The left panel of Figure \[fig:StraussK\] shows these bounds for the hard annulus process from Example \[ex:ringprozess\] with parameters $\beta=70$, $r=0.025$ and $R=0.035$. Let us furthermore assume that $\Xi$ is isotropic. Then the *$K$ function* is defined as $$K(t)=\alpha_dd\int_0^ts^{d-1}\rho(s)\;ds,$$ where $\rho$ is the pair correlation function. By , and Theorem \[thm:bounds\] we obtain bounds on $\rho$ as $$\label{eq:boundspcf} \varphi(x)\biggl(\frac{\beta^2}{\lambda^2}-\frac{\beta^2\tilde{G}_x}{\lambda}\biggr)\le \rho(\|x\|) \le \varphi(x)\biggl(\frac{\beta^2}{\lambda^2}-\frac{\beta}{\lambda}\big(1-\exp(-\beta\tilde{G}_x)\big)\biggr),$$ where $\tilde{G}_x=\int_{{\mathbb{R}}^d}1-\varphi(y)\varphi(y-x)\;dy$, and (in most cases numeric) integration of yields bounds on the $K$ function. \[ex:StraussK\] Let $\Xi$ be a Strauss process in two dimensions. Then $$\tilde{G}_x=2\pi r^2(1-\gamma)-2 r^2 (1-\gamma)^2\Biggr(\mathrm{arccos}\biggl(\frac{\|x\|}{2r}\biggr)-\frac{\|x\|}{2r}\sqrt{1-\Bigl(\frac{\|x\|}{2r}\Bigr)^2}\Biggr);$$ see also [@bn12]. Since we do not know the true intensity of the Strauss process, we plug in the bounds of into to obtain bounds on the $K$ function. This procedure causes twice an error and therefore the estimates on $K$ are good only for smaller values of $\beta G$. The right panel of Figure \[fig:StraussK\] shows these estimates for $\beta=40$, $r=0.05$ and $\gamma=0$. 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35.09) – (310.09, 35.09) – (310.49, 35.09) – (310.88, 35.09) – (311.27, 35.09) – (311.66, 35.09) – (312.06, 35.09) – (312.45, 35.09) – (312.84, 35.09) – (313.23, 35.09) – (313.63, 35.09) – (314.02, 35.09) – (314.41, 35.09) – (314.80, 35.09) – (315.20, 35.09) – (315.59, 35.09) – (315.98, 35.09) – (316.37, 35.09) – (316.77, 35.09) – (317.16, 35.09) – (317.55, 35.09) – (317.94, 35.09) – (318.34, 35.09) – (318.73, 35.09) – (319.12, 35.09) – (319.51, 35.09) – (319.91, 35.09) – (320.30, 35.09) – (320.69, 35.09) – (321.08, 35.09) – (321.48, 35.19) – (321.87, 35.38) – (322.26, 35.59) – (322.65, 35.77) – (323.05, 35.91) – (323.44, 36.08) – (323.83, 36.27) – (324.22, 36.45) – (324.62, 36.60) – (325.01, 36.79) – (325.40, 36.96) – (325.79, 37.15) – (326.19, 37.35) – (326.58, 37.52) – (326.97, 37.73) – (327.36, 37.93) – (327.76, 38.19) – (328.15, 38.32) – (328.54, 38.53) – (328.93, 38.79) – (329.33, 38.99) – (329.72, 39.22) – (330.11, 39.39) – (330.50, 39.62) – (330.90, 39.82) – (331.29, 40.03) – 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53.92) – (353.66, 54.23) – (354.05, 54.47) – (354.44, 54.75) – (354.84, 55.00) – (355.23, 55.28) – (355.62, 55.57) – (356.01, 55.87) – (356.41, 56.21) – (356.80, 56.50) – (357.19, 56.78) – (357.58, 57.09) – (357.98, 57.43) – (358.37, 57.80) – (358.76, 58.05) – (359.15, 58.34) – (359.55, 58.68) – (359.94, 58.96) – (360.33, 59.28) – (360.72, 59.58) – (361.12, 59.82) – (361.51, 60.15) – (361.90, 60.47) – (362.29, 60.81) – (362.69, 61.09) – (363.08, 61.43) – (363.47, 61.74) – (363.86, 62.04) – (364.26, 62.36) – (364.65, 62.65) – (365.04, 62.99) – (365.43, 63.34) – (365.83, 63.76) – (366.22, 64.12) – (366.61, 64.42) – (367.00, 64.75) – (367.40, 65.10) – (367.79, 65.44) – (368.18, 65.77) – (368.57, 66.09) – (368.97, 66.43) – (369.36, 66.77) – (369.75, 67.13) – (370.14, 67.53) – (370.54, 67.84) – (370.93, 68.17) – (371.32, 68.57) – (371.71, 68.91) – (372.11, 69.31) – (372.50, 69.64) – (372.89, 70.03) – (373.28, 70.36) – (373.68, 70.71) – (374.07, 71.11) – (374.46, 71.44) – (374.85, 71.79) – 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95.29) – (397.22, 95.78) – (397.62, 96.24) – (398.01, 96.76) – (398.40, 97.23) – (398.79, 97.68) – (399.19, 98.23) – (399.58, 98.71) – (399.97, 99.19) – (400.36, 99.66) – (400.76,100.20) – (401.15,100.65) – (401.54,101.18) – (401.93,101.66) – (402.33,102.07) – (402.72,102.58) – (403.11,103.00) – (403.50,103.55) – (403.90,104.09) – (404.29,104.53) – (404.68,105.04) – (405.07,105.52) – (405.47,105.97) – (405.86,106.47) – (406.25,106.90) – (406.64,107.35) – (407.04,107.83) – (407.43,108.37) – (407.82,108.88) – (408.21,109.36) – (408.61,109.86) – (409.00,110.38) – (409.39,110.90) – (409.78,111.39) – (410.18,111.94) – (410.57,112.42) – (410.96,112.90) – (411.35,113.47) – (411.75,113.91) – (412.14,114.39) – (412.53,114.90) – (412.92,115.42) – (413.32,115.99) – (413.71,116.47) – (414.10,116.93) – (414.49,117.39) – (414.89,117.93) – (415.28,118.42) – (415.67,118.98) – (416.06,119.57) – (416.46,120.07) – (416.85,120.57) – (417.24,121.11) – (417.63,121.63) – (418.03,122.16) – (418.42,122.69) – (418.81,123.25) – (419.20,123.81) – (419.60,124.36) – (419.99,124.90) – (420.38,125.42) – (420.77,125.97) – (421.17,126.49) – (421.56,127.04) – (421.95,127.63) – (422.34,128.21) – (422.74,128.79) – (423.13,129.29) – (423.52,129.85) – (423.91,130.41) – (424.31,130.99) – (424.70,131.50) – (425.09,132.06) – (425.48,132.63) – (425.88,133.21) – (426.27,133.75) – (426.66,134.33) – (427.05,134.98) – (427.45,135.58) – (427.84,136.13) – (428.23,136.69) – (428.62,137.26) – (429.02,137.80) – (429.41,138.38) – (429.80,138.95) – (430.19,139.57) – (430.59,140.16) – (430.98,140.73) – (431.37,141.28) – (431.76,141.82) – (432.16,142.42) – (432.55,143.03) – (432.94,143.60) – (433.33,144.24) – (433.73,144.82) – (434.12,145.36) – (434.51,145.99) – (434.90,146.63) – (435.30,147.24) – (435.69,147.85) – (436.08,148.49) – (436.47,149.13) – (436.87,149.79) – (437.26,150.42) – (437.65,151.05) – (438.04,151.68) – (438.44,152.33) – (438.83,153.01) – (439.22,153.63) – (439.61,154.30) – (440.01,154.97) – (440.40,155.58) – (440.79,156.21) – (441.18,156.82) – (441.58,157.44) – (441.97,158.12) – (442.36,158.78) – (442.75,159.33) – (443.15,159.96) – (443.54,160.59) – (443.93,161.22) – (444.32,161.95) – (444.71,162.58) – (445.11,163.21) – (445.50,163.76) – (445.89,164.45) – (446.28,165.04) – (446.68,165.63) – (447.07,166.25) – (447.46,166.92) – (447.85,167.50) – (448.25,168.17) – (448.64,168.77) – (449.03,169.43) – (449.42,170.03) – (449.82,170.72) – (450.21,171.42) – (450.60,172.08) – (450.99,172.72) – (451.39,173.33) – (451.78,173.96) – (452.17,174.63) – (452.56,175.35) – (452.96,176.02) – (453.35,176.66) – (453.74,177.28) – (454.13,178.06) – (454.53,178.72) – (454.92,179.41) – (455.31,180.07) – (455.70,180.64) – (456.10,181.36) – (456.49,182.04) – (456.88,182.70) – (457.27,183.34) – (457.67,184.07) – (458.06,184.77) – (458.45,185.40) – (458.84,186.13) – (459.24,186.81) – (459.63,187.55) – (460.02,188.25) – (460.41,189.05) – (460.81,189.72) – (461.20,190.40) – (461.59,191.15) – (461.98,191.88) – (462.38,192.66) – (462.53,192.89); (281.05, 35.09) – (281.44, 35.09) – (281.83, 35.09) – (282.23, 35.10) – (282.62, 35.10) – (283.01, 35.11) – (283.40, 35.12) – (283.80, 35.13) – (284.19, 35.14) – (284.58, 35.15) – (284.97, 35.17) – (285.37, 35.18) – (285.76, 35.20) – (286.15, 35.22) – (286.54, 35.24) – (286.94, 35.26) – (287.33, 35.29) – (287.72, 35.31) – (288.11, 35.34) – (288.51, 35.37) – (288.90, 35.40) – (289.29, 35.43) – (289.68, 35.46) – (290.08, 35.50) – (290.47, 35.53) – (290.86, 35.57) – (291.25, 35.61) – (291.65, 35.65) – (292.04, 35.69) – (292.43, 35.74) – (292.82, 35.78) – (293.22, 35.83) – (293.61, 35.88) – (294.00, 35.93) – (294.39, 35.98) – (294.79, 36.03) – (295.18, 36.09) – (295.57, 36.15) – (295.96, 36.20) – (296.36, 36.26) – (296.75, 36.32) – (297.14, 36.39) – (297.53, 36.45) – (297.93, 36.52) – (298.32, 36.58) – (298.71, 36.65) – (299.10, 36.72) – (299.50, 36.79) – (299.89, 36.87) – (300.28, 36.94) – (300.67, 37.02) – (301.07, 37.10) – (301.46, 37.18) – (301.85, 37.26) – (302.24, 37.34) – (302.64, 37.42) – (303.03, 37.51) – (303.42, 37.60) – (303.81, 37.69) – (304.21, 37.78) – (304.60, 37.87) – (304.99, 37.96) – (305.38, 38.06) – (305.78, 38.15) – (306.17, 38.25) – (306.56, 38.35) – (306.95, 38.45) – (307.35, 38.56) – (307.74, 38.66) – (308.13, 38.77) – (308.52, 38.87) – (308.92, 38.98) – (309.31, 39.09) – (309.70, 39.20) – (310.09, 39.32) – (310.49, 39.43) – (310.88, 39.55) – (311.27, 39.67) – (311.66, 39.79) – (312.06, 39.91) – (312.45, 40.03) – (312.84, 40.16) – (313.23, 40.28) – (313.63, 40.41) – (314.02, 40.54) – (314.41, 40.67) – (314.80, 40.80) – (315.20, 40.93) – (315.59, 41.07) – (315.98, 41.21) – (316.37, 41.34) – (316.77, 41.48) – (317.16, 41.63) – (317.55, 41.77) – (317.94, 41.91) – (318.34, 42.06) – (318.73, 42.21) – (319.12, 42.36) – (319.51, 42.51) – (319.91, 42.66) – (320.30, 42.81) – (320.69, 42.97) – (321.08, 43.12) – (321.48, 43.28) – (321.87, 43.44) – (322.26, 43.60) – (322.65, 43.77) – (323.05, 43.93) – 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(453.74,184.61) – (454.13,185.29) – (454.53,185.98) – (454.92,186.66) – (455.31,187.34) – (455.70,188.03) – (456.10,188.72) – (456.49,189.41) – (456.88,190.10) – (457.27,190.79) – (457.67,191.49) – (458.06,192.18) – (458.45,192.88) – (458.84,193.58) – (459.24,194.28) – (459.63,194.98) – (460.02,195.69) – (460.41,196.39) – (460.81,197.10) – (461.20,197.81) – (461.59,198.52) – (461.98,199.23) – (462.38,199.94) – (462.53,200.22); ( 0.00, 0.00) rectangle (462.53,210.24); at (361.35, 2.51) [$t$]{}; at (248.23,112.35) [$K(t)$]{}; (274.63, 28.91) – (448.07, 28.91) – (448.07,195.79) – (274.63,195.79) – (274.63, 28.91); Proof of Theorem \[thm:bounds\] {#sec:proofs} =============================== The main strategy of the proof is to replace in the process $\Xi$ by a Poisson process ${\mathrm{H}}$, and then use Stein’s method to bound the error $$\label{eq:error} {{\mathbb E}}\Big( \prod_{y\in \Xi} g(y) \Big)-{{\mathbb E}}\Big( \prod_{y\in {\mathrm{H}}} g(y) \Big).$$ In the context of Poisson process approximation, Stein’s method for bounding expressions of the form ${\lvert {{\mathbb E}}f(\Xi)-{{\mathbb E}}f({\mathrm{H}}) \rvert}$ uniformly in $f$ from a class of functions ${\mathcal{F}}$ was first introduced in [@bb92]. It has undergone numerous developments in the subsequent years, including a recent generalisation to Gibbs process approximation [@ss12]. In what follows we briefly touch on the key points of Stein’s method needed for our proof, referring to the nice expository chapter of [@xia05] for details. Assume that $g \colon {\mathbb{R}}^d \to [0,1]$ is such that $A = \operatorname{supp}(1-g)$ is compact. Consider the function $f \colon {\mathfrak{N}}\to [0,1]$, $$\label{eq:f} f(\xi)=f(\xi \vert_A)=\prod_{y\in\xi}g(y).$$ For any $\xi \in {\mathfrak{N}}\vert_A$ let $Z_{\xi} = \{Z_\xi(t)\}_{t\ge0}$ be a spatial immigration-death process on $A$ with immigration rate $\nu>0$ and unit per-capita death rate, started in the configuration $\xi$. This is a pure-jump Markov process on ${\mathfrak{N}}\vert_A$ that holds any state $\eta \in {\mathfrak{N}}\vert_A$ for an exponentially distributed time with mean $1/(\nu {\lvert A \rvert} + \eta(A))$, where ${\lvert A \rvert}$ denotes the Lebesgue measure of $A$; then a uniformly distributed point in $A$ is added with probability $\nu {\lvert A \rvert}/(\nu {\lvert A \rvert} + \eta(A))$, or a uniformly distributed point in $\eta$ is deleted with probability $\eta(A)/(\nu {\lvert A \rvert} + \eta(A))$. The process $Z_{\xi}$ has the Poisson process distribution on $A$ with intensity $\nu$ as its stationary distribution. If the process is started at the empty configuration $\emptyset$, then $Z_{\emptyset}(t)$ is a Poisson process with intensity $\nu (1-e^{-t})$ for every $t \geq 0$. Let $\{E_x\}_{x\in \xi}$ be i.i.d. standard exponentially distributed random variables and introduce the death process $D_\xi(t)=\sum_{x\in\xi}{\mathbbm{1}}\{E_x>t\}\delta_x$. Constructing $Z_{\emptyset}$ and $D_{\xi}$ independently on the same probability space, $Z_\xi$ can be represented as $Z_\xi(t) {\text{\raisebox{0pt}[0pt][0pt]{${}\stackrel{\mathscr{D}}{=}{}$}}}Z_\emptyset(t)+D_\xi(t)$ for every $t \geq 0$; see [@xia05 Thm. 3.5] Let ${\mathrm{H}}_{\nu}$ be a Poisson process with intensity $\nu$ and define $h_f \colon {\mathfrak{N}}\to {\mathbb{R}}$ as $$\label{eq:stein-sol} h_f(\xi)=h_f(\xi \vert_A) = -\int_0^\infty \big[{{\mathbb E}}\bigl(f(Z_{\xi\vert_A}(t))\bigr)-{{\mathbb E}}(f({\mathrm{H}}_\nu))\big]\;dt.$$ By [@xia05 Lem. 5.2] $h_f$ is well-defined and satisfies $$\begin{aligned} {{\mathbb E}}f(\Xi)-{{\mathbb E}}f({\mathrm{H}}_\nu) &= {{\mathbb E}}\int_{A} \bigl[ h_f(\Xi+\delta_x)-h_f(\Xi) \bigr] {\hspace*{1.5pt}}\nu \; dx+ {{\mathbb E}}\int_{A} \bigl[ h_f(\Xi-\delta_x)-h_f(\Xi) \bigr] \; \Xi(d x) \nonumber \\ &= {{\mathbb E}}\int_{A}\big[h_f(\Xi+\delta_x)-h_f(\Xi)\big]\big( \nu-\lambda(x{\, \vert \,}\Xi)\big)\; dx \nonumber \\ &= {{\mathbb E}}\int_{{\mathbb{R}}^d}\big[h_f(\Xi+\delta_x)-h_f(\Xi)\big]\big( \nu-\lambda(x{\, \vert \,}\Xi)\big)\; dx, \label{eq:SteinGNZ}\end{aligned}$$ where we applied the Georgii–Nguyen–Zessin equation on ${\mathbb{R}}^d$ to the function $\bigl[(x,\xi) \mapsto {\mathbbm{1}}_A(x) \bigl( h_f(\xi)-h_f(\xi+\delta_x) \bigr)\bigr]$ for obtaining the second equality. Equation  is our starting point for further considerations. \[prop:1\] Let $\Xi$ be a stationary Gibbs process with intensity $\lambda$ and conditional intensity $\lambda(\cdot{\, \vert \,}\cdot)$. Let $g\colon {{\mathbb{R}}^d} \to [0,1]$ be a function such that $1-g$ has bounded support. Then for all $\nu>0$ $$\label{eq:prop1} {{\mathbb E}}\Big(\prod_{y\in\Xi}g(y)\Big)=1-\frac{\lambda}{\nu} \big(1-e^{-\nu G}\big)+I_\nu(g),$$ where $$\label{eq:I} I_\nu(g)=e^{-\nu G}{\hspace*{1.5pt}}{{\mathbb E}}\biggl( \int_0^1e^{\nu Gs}\Big(1-\prod_{y\in \Xi}\big(1-s(1-g(y))\big)\Big)\;ds \hspace*{0.4em} \cdot \, \int_{{\mathbb{R}}^d}(1-g(x))(\lambda(x{\, \vert \,}\Xi)-\nu)\;dx\biggr).$$ It is well known that for the Poisson process ${\mathrm{H}}_{\nu}$ we have $$\label{eq:Pois-pgfl} {{\mathbb E}}\Big(\prod_{y\in {\mathrm{H}}_\nu}g(y)\Big)=\exp\bigg(-\nu\int_{{\mathbb{R}}^d}1-g(x)\; dx\bigg)=e^{-\nu G};$$ see e.g. [@dvj08 Eq. 9.4.17]. We then follow the main proof strategy laid out above in order to re-express $${{\mathbb E}}\Big(\prod_{y\in \Xi}g(y)\Big)-e^{-\nu G}.$$ Starting from Equation , we may use the decomposition $Z_{\xi+\delta_x} {\text{\raisebox{0pt}[0pt][0pt]{${}\stackrel{\mathscr{D}}{=}{}$}}}Z_\xi+D_{\delta_x}$ with independent $Z_\xi$ and $D_{\delta_x}$ to see that for any $\xi \in {\mathfrak{N}}\vert_A$ $$\begin{aligned} h_f(\xi+\delta_x)-h_f(\xi)&=\int_0^\infty {{\mathbb E}}f(Z_\xi(t))-{{\mathbb E}}f(Z_{\xi+\delta_x}(t))\;dt \\ &=\int_0^\infty {{\mathbb E}}f(Z_\xi(t))-{{\mathbb E}}f(Z_{\xi}(t)+D_{\delta_x}(t))\;dt \\ &=(1-g(x))\int_0^\infty {{\mathbb E}}f(Z_\xi(t)) {\hspace*{1.5pt}}{{\mathbb P}}(D_{\delta_x}(t) \neq \emptyset)\;dt \\ &=(1-g(x))\int_0^\infty {{\mathbb E}}f(Z_\xi(t)) {\hspace*{1.5pt}}e^{-t}\;dt \\ &=(1-g(x))\int_0^\infty {{\mathbb E}}\big(f(Z_\emptyset(t))+(f(Z_\xi(t)) - f(Z_\emptyset(t))\big) {\hspace*{1.5pt}}e^{-t}\;dt.\end{aligned}$$ By further decomposing $Z_\xi {\text{\raisebox{0pt}[0pt][0pt]{${}\stackrel{\mathscr{D}}{=}{}$}}}Z_\emptyset+D_\xi$ with independent $Z_{\emptyset}$ and $D_\xi$, we obtain $$\begin{aligned} {{\mathbb E}}\big(f(Z_\xi(t)) - f(Z_\emptyset(t))\big)&={{\mathbb E}}\Big(\prod_{y\in Z_\emptyset(t)}g(y)\Big){\hspace*{1.5pt}}{{\mathbb E}}\Big(\prod_{y\in D_\xi(t)}g(y)-1\Big)\\ &=\exp\big(-\nu(1-e^{-t})G\big)\Big(\prod_{y\in \xi}\bigl(1-e^{-t}(1-g(y))\bigr)-1 \Big),\end{aligned}$$ where for the first expectation we used and that $Z_\emptyset(t)$ is a Poisson process with intensity $\nu(1-e^{-t})$; for the second expectation note that each point of $\xi$ survives independently up to time $t$ with probability $e^{-t}$. Thus in total $$\begin{split} h_f(&\xi+\delta_x)-h_f(\xi)\\[1mm] &=(1-g(x))e^{-\nu G}\int_0^\infty\Bigl[ e^{\nu Ge^{-t}} + e^{\nu Ge^{-t}}\Bigl(\prod_{y\in \xi}\bigl(1-e^{-t}(1-g(y))\bigr) -1\Bigr)\Bigr]e^{-t}\;dt \\ &= (1-g(x))\frac{1-e^{-\nu G}}{\nu G} + (1-g(x)) e^{-\nu G}\int_0^1 e^{\nu Gs}\Bigl(\prod_{y\in \xi}\bigl(1-s(1-g(y))\bigr) -1 \Bigr)\;ds \end{split}$$ by the substitution $s=e^{-t}$. Plugging this into Equation  and using ${{\mathbb E}}\lambda(x{\, \vert \,}\Xi)=\lambda$ finally yields $${{\mathbb E}}\Big(\prod_{y\in \Xi}g(y)\Big)-e^{-\nu G}=\frac{\nu-\lambda}{\nu}\big(1-e^{-\nu G}\big)+I_\nu(g).$$ \[prop:bounds-I\] Let $\Xi$ be a stationary locally stable Gibbs process with constant $c^*$. Then for all $0<\nu< c^*$ $$\underline{I_{\nu}}(g)\le I_\nu(g)\le \overline{I_\nu}(g),$$ where $$\begin{aligned} \underline{I_{\nu}}(g)&= -\frac{1}{c^*-\nu}\big(c^*(1-e^{-\nu G})- \nu(1-e^{-c^*G})\big) \le 0,\label{eq:boundsI1}\\ \overline{I_\nu}(g)&= \frac{1}{\nu}\big(c^*(1-e^{-\nu G})- \nu(1-e^{-c^*G})\big)\ge 0. \label{eq:boundsI2}\end{aligned}$$ Furthermore $I_\nu(g) \le 0$ for all $\nu \ge c^*$. Since $-\nu \le \lambda(x{\, \vert \,}\Xi)-\nu\le c^*-\nu$ a.s., we get for $\nu<c^*$ the upper bound $$\label{eq:proof-b-I} I_\nu(g)\le (c^*-\nu)e^{-\nu G}G \int_0^1e^{\nu Gs}\Big(1-{{\mathbb E}}\prod_{y\in \Xi}\big(1-s(1-g(y))\big)\Big)\;ds ,$$ and a similar lower bound, where $c^*-\nu$ is replaced by $-\nu$. Because $A=\operatorname{supp}(1-g)$ is bounded, $\Xi$ can be replaced by $\Xi\vert_A$ in . It is a known fact that every locally stable Gibbs process on a bounded domain can be obtained as a dependent random thinning of a Poisson process; see [@km00 Remark 3.4]. In particular, there exists a Poisson process ${\mathrm{H}}_{c^*}$ such that $\Xi\vert_A\subset {\mathrm{H}}_{c^*}$ a.s. Since $(1-s(1-g(y))\le 1$ for all $s\in [0,1]$ and for all $y\in {\mathbb{R}}^d$, we obtain $$1-{{\mathbb E}}\prod_{y\in \Xi}\bigl(1-s(1-g(y))\bigr) \le 1-{{\mathbb E}}\prod_{y\in {\mathrm{H}}_{c^*}}\bigl(1-s(1-g(y))\bigr) = 1-e^{-sc^*G},$$ where the last equality follows by . Integrating and rearranging the terms yields the formulas and . If $\nu \ge c^*$, $I_\nu(g)$ is obviously non-positive. *Remainder of the proof of Theorem \[thm:bounds\].* Propositions \[prop:1\] and \[prop:bounds-I\] yield the upper bounds $$\begin{aligned} {{\mathbb E}}\Big(\prod_{y\in \Xi}g(y)\Big)&\le 1-\frac{\lambda}{\nu}\big(1-e^{-\nu G}\big)+\frac{c^*}{\nu}\big(1-e^{-\nu G}\big)-\big(1-e^{-c^* G}\big)\\ &=(c^*-\lambda)G\frac{1-e^{-\nu G}}{\nu G}+e^{-c^*G}\end{aligned}$$ for $0<\nu<c^*$ and $${{\mathbb E}}\Big(\prod_{y\in \Xi}g(y)\Big)\le 1-\lambda G\frac{1-e^{-\nu G}}{\nu G}$$ for $\nu \ge c^*$. Since the function $[x \mapsto (1-\exp(-x))/x]$ is monotonically decreasing for $x\ge 0$, we obtain the minimal upper bound for $\nu = c^{*}$, as $${{\mathbb E}}\Big(\prod_{y\in \Xi}g(y)\Big)\le 1-\frac{\lambda}{c^*} \big(1-e^{-c^* G}\big).$$ For the lower bound recall the Weierstrass product inequality, which states $$\prod_{i=1}^n(1-a_i)\ge 1-\sum_{i=1}^na_i$$ for $0\le a_1,\dots,a_n\le 1$. Then, noting that the products below contain only finitely many factors $\neq 1$ by the boundedness of $\operatorname{supp}(1-g)$, we have $$\begin{aligned} {{\mathbb E}}\Big(\prod_{y\in \Xi}g(y)\Big)&={{\mathbb E}}\Big(\prod_{y\in \Xi}\big(1-(1-g(y))\big)\Big)\\ &\ge 1-{{\mathbb E}}\sum_{y\in \Xi}(1-g(y))\\ &=1-{{\mathbb E}}\int_{{\mathbb{R}}^d}1-g(x)\; \Xi(dx)\\ &=1-\lambda\int_{{\mathbb{R}}^d}1-g(x)\;dx =1-\lambda G\end{aligned}$$ by Campell’s formula; see [@dvj08 Section $9.5$]. An alternative proof for the lower bound can be obtained by using Propositions \[prop:1\] and \[prop:bounds-I\] in the analogous way as for the upper bound. [10]{} Milton Abramowitz and Irene A. Stegun. , volume 55 of [*National Bureau of Standards Applied Mathematics Series*]{}. 1964. Adrian Baddeley and Gopalan Nair. Approximating the moments of a spatial point process. , 1:18–30, 2012. Adrian Baddeley and Gopalan Nair. Fast approximation of the intensity of [G]{}ibbs point processes. , 6:1155–1169, 2012. Adrian Baddeley and Rolf Turner. Spatstat: an [R]{} package for analyzing spatial point patterns. , 12(6):1–42, 2005. Andrew D. Barbour and Timothy C. Brown. Stein’s method and point process approximation. , 43(1):9–31, 1992. Daryl J. Daley and David Vere-Jones. . Probability and its Applications (New York). Springer, New York, second edition, 2008. Wilfrid S. Kendall and Jesper M[ø]{}ller. Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. , 32(3):844–865, 2000. Shigeru Mase. Mean characteristics of [G]{}ibbsian point processes. , 42(2):203–220, 1990. Jesper M[ø]{}ller and Rasmus P. Waagepetersen. , volume 100 of [*Monographs on Statistics and Applied Probability*]{}. Chapman & Hall/CRC, Boca Raton, FL, 2004. Jesper M[ø]{}ller and Rasmus P. Waagepetersen. Modern statistics for spatial point processes. , 34(4):643–684, 2007. Xuan-Xanh Nguyen and Hans Zessin. Integral and differential characterizations of the [G]{}ibbs process. , 88:105–115, 1979. . . R Foundation for Statistical Computing, Vienna, Austria, 2012. David Ruelle. . W. A. Benjamin, Inc., New York–Amsterdam, 1969. Dominic Schuhmacher and Kaspar Stucki. On bounds for [G]{}ibbs point process approximation. , 2012. Available at http://arxiv.org/abs/1207.3096. Aihua Xia. Stein’s method and [P]{}oisson process approximation. In [*An introduction to Stein’s method*]{}, volume 4 of [*Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap.*]{}, pages 115–181. Singapore Univ. Press, Singapore, 2005. [^1]: Institute of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland. [^2]: Email address: kaspar.stucki@stat.unibe.ch [^3]: Email address: schuhmacher@stat.unibe.ch [^4]: Supported by Swiss National Science Foundation Grant 200021-137527.

--- abstract: | This paper is the first one in a series investigating the properties of the S stars belonging to the Henize sample (205 S stars with $\delta<-25^\circ$ and $R<10.5$) in order to derive the respective properties (like galactic distribution and relative frequencies) of intrinsic (i.e. genuine asymptotic giant branch) S stars and extrinsic (i.e. post mass-transfer binary) S stars. High-resolution (R=30000 to 60000) spectra covering the range $\lambda\lambda4230-4270$Å  have been obtained for 76 S stars, 8 M stars and 2 symbiotic stars. The $\lambda4262$Å  and $\lambda4238$Å  blends involving a line were analysed separately and yield consistent conclusions regarding the presence or absence of technetium. Only one ‘transition’ case (Hen 140 = HD 120179, a star where only weak lines of technetium are detectable) is found in our sample. A resolution greater than R $=30\,000$ is clearly required in order to derive unambiguous conclusions concerning the presence or absence of technetium. The Tc/no Tc dichotomy will be correlated with radial velocity and photometric data in a forthcoming paper. author: - 'S. Van Eck' - 'A. Jorissen [^1]' date: 'Received date; accepted date' subtitle: 'I. The technetium dichotomy[^2] ' title: The Henize sample of S stars --- \#1\#2[BD[$#1$]{}$^\circ$[\#2]{}]{} \#1\#2[[$#1$]{}$^\circ$[\#2]{}]{} Introduction ============ S stars have been identified as a class of peculiar red giants by Merrill ([@Merrill22]). Basically, the S stars emerge as a sequence parallel to the M stars as far as temperature is concerned, but with enhanced abundances of s-process elements. The chief observational difference between M and S spectra is the presence of ZrO bands in the latter. The s-process overabundances in S stars are explained in the framework of the Thermally Pulsing Asymptotic Giant Branch (TPAGB) evolution. Thermal instabilities (called [*thermal pulses*]{}) that affect the helium-burning shell of these stars have two important consequences: they provide the proper environment for the nucleosynthesis of s-process elements, and they trigger an envelope response (the [*third dredge-up*]{}) that allows the s-process elements and carbon to be brought to the surface \[see e.g. Mowlavi ([@Mowlavi]) for a review\]. Technetium is an s-process element with no stable isotope that was first identified in the spectra of some M and S stars by Merrill ([@Merrill52]). $^{99}$Tc, with a laboratory half-life of $t_{1/2} = 2.13\times 10^5$ yr, is the only technetium isotope produced by the s-process. The high temperatures encountered during thermal pulses strongly shorten the effective half-life of $^{99}$Tc ($t_{1/2}\sim 1$ yr at $\sim 3\times 10^8$ K, Cosner et al. [@Cosner]) but the large neutron densities at these high temperatures more than compensate the reduction of $^{99}$Tc life-time (Mathews et al. [@Mathews]) and enable a substantial technetium production. Third dredge-up episodes then carry technetium to the envelope, where it decays steadily at its terrestrial rate of $t_{1/2}=2.13 \times 10^5$ yr. Starting from an abundance corresponding to the maximum observed one, technetium should remain detectable during $1.0-1.5\times 10^{6}$ yr (Smith & Lambert [@Smith]). If the dredge-up of heavy elements occurs after each thermal pulse (occurring every $\sim 1-3\times 10^{5}$ yr), virtually all s-process enriched TPAGB stars should exhibit technetium lines. However, Straniero et al. ([@Straniero]) advocated that the s-process nucleosynthesis mainly occurs during the interpulse. When technetium is engulfed in the subsequent thermal pulse, it will decay at a fast rate because of the high temperature, and will not be replenished if there is no neutron source operating within the pulse itself. The conclusion that s-process enriched TPAGB stars should necessarily exhibit technetium would then be challenged. Nevertheless, all the S stars identified as TPAGB stars by Van Eck et al. ([@Van; @Eck98]) thanks to the HIPPARCOS parallaxes turned out to be Tc-rich. A survey of technetium in a large sample of S stars like the Henize sample may be expected to provide further constraints on the s-process environment in AGB stars (e.g. interpulse s-process versus thermal-pulse s-process, thermal-pulse duration and temperature versus $^{99}$Tc half-life). Not all S stars exhibit Tc lines though (Little-Marenin & Little [@Little-Marenin], Little et al. [@Little]), but technetium-poor stars (also called [*extrinsic*]{}, as opposed to technetium-rich, [*intrinsic*]{} S stars) are currently believed to emerge from a totally different evolutionary history: because they are members of binary systems (Brown et al. [@Brown], Jorissen et al. [@Jorissen93], Johnson et al. [@Johnson], Jorissen et al. [@Jorissen]), they rather owe their chemical peculiarities to the accretion of s-process-rich matter from their companion (formerly a TPAGB star, now an undetected white dwarf). They are technetium-poor, because enough time has elapsed for the technetium to decay since the mass transfer event. The $^{99}$Tc half-life is indeed much shorter than any stellar evolutionary timescale (but the TPAGB). Such a polluted giant star will be classified either as a G or K giant with enhanced heavy elements (i.e. as a barium star) or, if it has cooled enough for the ZrO molecular bands to appear, as a technetium-poor S star. Besides technetium detection, several spectroscopic criteria (of various efficiencies) aiming at distinguishing extrinsic from intrinsic S stars have been mentioned in the literature \[e.g. oxygen isotopic ratio (Smith & Lambert [@Smith90]), presence of the $\lambda 10830$ line (Brown et al. [@Brown]), zirconium isotopic ratio (Busso et al. [@Busso])\], but technetium detection appears to be, by far, the most secure and tractable way to unmask extrinsic S stars. This unmasking operation is crucial when deriving fundamental stellar quantities such as the third dredge-up luminosity threshold (commonly measured as the minimum luminosity of S stars). Evolutionary timescales of TPAGB stars can be strongly in error if the considered star samples are polluted by non-AGB, mass-transfer S stars. We therefore decided to study a large and properly defined sample of S stars in order to disentangle the two sub-families and to study their respective characteristics. The Henize sample of S stars (Henize [@Henize], as listed in Stephenson [@Stephenson]) consists of 205 S stars south of $\delta=-25^\circ$ and brighter than $R=10.5$. Radial velocity data, low- and high-resolution spectroscopy, as well as Geneva photometry have been collected over several years. The present paper deals with high-resolution technetium spectra for 72 Henize stars. Some additional K, M and symbiotic stars data are also presented. Results concerning binarity and photometry, as well as the global analysis of the different data sets, are postponed to a forthcoming paper. Observations and reduction ========================== Instrumental set-up {#Sect:setting} ------------------- The high-resolution spectra used in the present study were obtained during several runs (1991-1998) at the European Southern Observatory, with the Coudé Echelle Spectrometer (CES) fed by the 1.4m Coudé Auxiliary Telescope (CAT). The 1991-1993 runs were performed with the short camera (f/1.8) and CCD \#9 (RCA SID 503 thinned, backside illuminated, $1024 \times 640$ pixels of $15~\mu$m), whereas the long camera (f/4.7) and CCD \#38 (Loral/Lesser thinned, backside illuminated, UV flooded, $2688 \times 512$ pixels of $15~\mu$m) were used during the 1997-1998 runs. Details on these configurations can be found in Lindgren & Gilliotte ([@Lindgren]) and Kaper & Pasquini ([@Kaper]). The resolution ranges from 0.14Å  (R=30000) to 0.07Å  (R=60000) for a central wavelength of 4250Å. The spectra approximately cover the wavelength range $\lambda\lambda$ 4230-4270Å. Stellar samples --------------- The observed stars are a subset of the sample of 205 S stars collected by Henize ([@Henize]) from his objective-prism survey (with a dispersion of 450Å/mm at H$\alpha$) of ZrO stars south of $\delta = -25^{\circ}$ and brighter than $R=10.5$. Given the limitations on the CAT pointing and on the detectors sensitivity, only stars with $\delta > -75^{\circ}$, $V<11$ and $B-V<2$ (translating into 70 objects) could be observed in a reasonable amount of time, i.e. less than 1h30 per star. A few bright redder stars could also be observed (but see the discussion on SC stars in Sect. \[Sect:SC\]). A sample of bright M stars with an excess at $60\mu$m \[indicative of a possibly detached dust shell; see Zijlstra et al. ([@Zijlstra])\], as well as the two symbiotic stars RW Hya and SY Mus, and some radial-velocity standards have also been observed. Three non-Henize S stars from our radial-velocity monitoring (Udry et al. [@Udry]) have been included as well. The log of the observations, including the instrumental setting, is given in Table \[Tab1\] and, for Henize stars, in Table \[Tab2\]. Data reduction and S/N ratio ---------------------------- The CCD frames were corrected for the electronic offset (bias) and for the relative pixel-to-pixel response variations (flat-field). Wavelength calibration was performed from thorium lamp spectra taken several times per night. An optimal extraction of the spectra was performed according to the method of Horne ([@Horne]). The whole reduction sequence was performed within the ‘long’ context of the MIDAS software package. The signal-to-noise (S/N) ratio was estimated for each spectrum in the following way: three S/N values were computed for the three best exposed CCD lines (along the dispersion axis), in the neighborhood of the spectral region of interest (either 4262Å  or 4238Å). These three S/N values were then combined according to Eq. 17 of Newberry ([@Newberry]). When the exposure time on a given star has been split in two (in order to reduce cosmics detrimental effect), the final S/N ratio was computed using Eq. 18 of Newberry ([@Newberry]). The degradation of the S/N ratio due to flat-field correction has not been taken into account, since flat-fields have little degrading effect for the low S/N values under consideration. The S/N ratio values are listed in Tables \[Tab1\] and  \[Tab2\] for each target star. Because of the CCD spectral response, the S/N ratio near 4238Å  is systematically lower than the one near 4262Å. Analysis ======== Fit of the technetium blends {#Sect:Analysis} ---------------------------- The three strong resonance lines of are located at 4238Å, 4262Å  and 4297Å, with intensity ratios of 3:4:5. All three lines are severely blended (Little-Marenin & Little 1979, their Table III). With the adopted instrumental configurations, a single exposure spans 35 to 50Å; it is thus possible to observe simultaneously the 4238Å  and 4262Å  lines. In this analysis we follow the guidelines provided in the landmark paper of Smith & Lambert ([@Smith]) and therefore concentrate on the most useful 4262Å  line, while the 4238Å  line is used as an independent confirmation. Fig. \[Fig:figTc62\] shows examples of spectra in the 4262Å  region for an M3-4 giant (HD 73341) and for seven S stars (four being technetium-poor: Hen 3, 187, 31, 7 and three technetium-rich: Hen 140, 39 and 202 = $\pi^1$ Gru). It can be seen that the Tc $\lambda 4262.270$Å  line is blended with two features; the bluest includes primarily (4262.050Å) and (4262.087Å), and the reddest (4262.373Å) (see Fig. \[Fig:figTc62\]). A weaker contribution of at 4262.228Å, almost on the top of the line, may also be present. These composite features are much weaker than the resonance line at its maximum strength; moreover, the - blend and the line are 0.18Å  apart. Therefore the shape and location of the - (-) blend (hereafter called $X_{4262}$ feature) clearly depends on whether it contains the technetium line or not. Quantitatively, the minimum of a gaussian fitted to a Tc-containing $X_{4262}$ feature is shifted redward by $\sim0.14$Å  with respect to the minimum of a gaussian fitted to a no-Tc $X_{4262}$ feature; such a shift is easily detectable on our spectra (compare Hen 3 or Hen 7 with Hen 39 on Fig. \[Fig:figTc62\]). In practice, each spectrum has been rebinned to zero-redshift in the following way: 10 nearby ($\le 5$Å  on either side) apparently unblended stellar features with unambiguous identification, are adopted as wavelength standards. Gaussian profiles are fitted to these lines and provide a mean redshift. The wavelength of the $X_{4262}$ feature is then computed as the minimum of a gaussian centered on the $X_{4262}$ feature of the redshift-corrected spectrum. Typical uncertainties on the $X_{4262}$ wavelength amount to 0.013Å  for technetium-poor stars and 0.020Å  for technetium-rich stars (as derived from the standard deviation on the mean redshift). The same method is applied to the 4238Å  technetium line, where the CH- blend has been taken as the $X_{4238}$ feature (Fig. \[Fig:figTc38\]). Typical uncertainties on the $X_{4238}$ wavelength are slightly larger (0.023Å  for Tc-rich S stars and 0.025Å  for Tc-poor S stars) because of the lower S/N ratio and the stronger blending at 4238Å. Results are listed in Tables \[Tab1\] and  \[Tab2\]; the $X_{4238}$ and $X_{4262}$ features always yield consistent results regarding the absence or presence of technetium, except for Hen 140 (=HD 120179). This star is indeed unique in having very weak technetium features (see Figs. \[Fig:figTc62\] and \[Fig:figTc38\]). A second spectrum, taken 3.5 years later, is almost identical to the one displayed in Fig. \[Fig:figTc62\] and \[Fig:figTc38\]. The blind application of gaussian fitting to the $X_{4238}$ feature of Hen 140 yields a central wavelength that would qualify it as Tc-poor; however, the extreme weakness of the pseudo-emission separating the LaII-CH blend from the SmII line as seen in Hen 140 (Fig. \[Fig:figTc38\]) is unusual for Tc-poor stars, and suggests the presence of a weak technetium line, as confirmed from the appearance of the $X_{4262}$ feature. We therefore believe that Hen 140 is the unique example in our sample of an S star with very weak Tc lines. In all the other cases, technetium (non-) detection relies on the location of the minimum of a gaussian fitted to the $X_{4262}$ (or $X_{4238}$) blend. Are there, with this method, risks (i) to misclassify as Tc-rich a truly Tc-poor star, and (ii) to misclassify as Tc-poor a truly Tc-rich star? We show in the remaining of this section that both risks are most probably non-existent in the present study. Error (i) could, in principle, affect very luminous Tc-poor stars, because their large macroturbulence would broaden their $X_{4262}$ feature, which could then possibly mimick a Tc-rich feature. In order to test this hypothesis, gaussian filters of different widths have been applied to Tc-poor spectra, so as to make their line widths comparable to those of the stars classified as Tc-rich. This simulation clearly shows that even the largest macroturbulence value observed in our sample (T Ceti) is not large enough to make truly Tc-poor stars appear as Tc-rich from the broadening of their $X_{4262}$ feature. That conclusion is even more stringent when considering the $X_{4238}$ feature. However, this risk cannot be excluded for very luminous stars (class I or II) if observed at lower resolution (R $< 30\,000$). Error (ii) could, in principle, occur for stars displaying a ‘weak technetium line’ (weaker than the Tc lines of Hen 140 discussed above) with an intensity not large enough to shift the $X_{4262}$ blend redward from the Tc-poor wavelength. In fact, some stars in our sample exhibit an ‘ambiguous’ $X_{4262}$ blend, in the sense that the pseudo-emission located between the - lines and the line becomes very weak or even disappears, mimicking a ‘weak technetium line’ (a typical example is Hen 7 on Fig. \[Fig:figTc62\]). Such a star is classified by our method as technetium-poor, for the minimum of the $X_{4262}$ blend remains unchanged with respect to the no-Tc cases. In fact, all intermediates exist between the ‘unambiguous’ $X_{4262}$ Tc-poor blends (with a clear central pseudo-emission, see HD 73341 and Hen 3 on Fig. \[Fig:figTc62\]) and the ‘ambiguous’ $X_{4262}$ blends (where this pseudo-emission is absent, as in Hen 7); two typical transition cases are plotted on Fig. \[Fig:figTc62\] (Hen 187 and Hen 31). These ‘ambiguous’ spectra were taken during different observing runs; the shape of the $X_{4262}$ blend is independent of the resolution and of the S/N ratio of the spectra. Do these ‘ambiguous’ spectra correspond to stars with a weak technetium line, intermediate between the clear Tc-poor and Tc-rich cases? In fact these ‘ambiguous’ spectra are clearly different from the spectum of the weakly Tc-rich star Hen 140, for [*their $X_{4238}$ feature is identical to the $X_{4238}$ feature of the unambiguous Tc-poor stars*]{} (Fig. \[Fig:figTc38\]), which clearly indicates that technetium is absent in these stars. It may therefore be concluded that our method of gaussian fit to the $X_{4238}$ and $X_{4262}$ features is able to properly separate technetium-rich from technetium-poor S stars. What then causes the variety of $X_{4262}$ features observed in Fig. \[Fig:figTc62\] for technetium-poor S stars? The spectral sequence going from Hen 3 to Hen 7 on Fig. \[Fig:figTc62\] is not a temperature sequence. The temperature of the stars of our sample have been derived from the $V-K$ color index using the Ridgway et al. ([@Ridgway]) calibration, the $K$ magnitudes from Catchpole et al. ([@Catchpole79]) and our Geneva photometry. Although the bulk of technetium-rich S stars are clearly cooler than technetium-poor S stars (see also Van Eck et al. [@Van; @Eck98]), there is no sign whatsoever of a possible correlation between the shape of the $X_{4262}$ blend of technetium-poor stars and their temperature. MOOG (Sneden [@Sneden]) synthetic spectra (for stars with T$_{eff}\sim 3400-3800$K as derived from their $V-K$ index) indicate that neither gravity nor metallicity can significantly modify the $X_{4262}$ blend. A closer inspection of the spectral sequence of Fig. \[Fig:figTc62\] (from HD 73341 to Hen 7) reveals that several lines become stronger as the central pseudo-emission of the $X_{4262}$ blend weakens. The major contributors to these features, identified with the help of synthetic spectra, are indicated on the top of Fig. \[Fig:figTc62\]. It is noteworthy that [*all these elements are s-process elements*]{}. The sequence of spectra (drawn with a thin line) in Fig. \[Fig:figTc62\] is thus, from top to bottom, a sequence of increasing s-process line strengths (s-process lines being weak, as expected, in the M star HD 73341). The line which progressively blends the $X_{4262}$ feature of technetium-poor stars is thus probably an s-process line as well. Since it cannot be technetium (see above), a good candidate is the 4262.228Å  line of , or perhaps the wing of the line at 4262.087Å. It is not surprising to find a wide range of s-process enhancements among technetium-poor S stars, since these stars have accreted their s-process-enriched matter from a companion star. Hence the level of chemical peculiarities is not linked to the evolutionary status of the star, but rather depends upon the amount of s-process accreted matter (see Jorissen et al. [@Jorissen] for a detailed discussion). These s-process lines are more difficult to see in the technetium-rich S stars, probably because in these cooler and more luminous stars, lines are broader (because of a larger macroturbulence) and the molecular blanketing is more severe. Misclassified and SC stars {#Sect:SC} -------------------------- The method outlined in Sect. \[Sect:Analysis\] cannot be applied to four stars of our sample which exhibit peculiar spectra (Hen 22, 135, 154 and 198). In order to check the assignment of the Henize stars to spectral type S, low-resolution spectra ($\Delta \lambda \sim 0.3$ nm, $4400$Å$<\lambda< 8200$Å) have been obtained for all stars from Hen 3 to Hen 165 at ESO on the 1.52m telescope equipped with the Boller & Chivens spectrograph and grating \#23 (Van Eck et al. [@Van; @Eck]). Two misclassified stars have been uncovered: Hen 22 and Hen 154 show no sign of ZrO bands whatsoever in their spectra. Besides, Hen 22 is classified as ‘S:’ by Henize. Both stars cannot be dwarfs because their NaD and MgH $\lambda$4780Å  features are too weak. Their prominent $\lambda$ 4455Å  line and their weak CN $\lambda$7895Å  band point towards them being giant stars rather than supergiants. Type Ia supergiants can certainly be ruled out because their absolute magnitudes ($M_{\rm v}=-7.8$ and -7.5 for G8Ia and K3-5I respectively, Landolt-Börnstein [@Landolt]) would result in much too large heights above the galactic plane (7.5 kpc for Hen 154 and 13.2 kpc for Hen 22). Hen 154 is probably a late G giant ($\sim$G8), and Hen 22 a mid-K giant ($\sim$K3-5). These assignments are compatible with the Geneva photometry available for these two stars. As far as Hen 135 and Hen 198 are concerned, Fig. \[Fig:figTc62except\] shows that the spectra of these stars are very different from those of other S stars of the Henize sample. Many spectral features adopted as wavelength standards, as well as the technetium blend, are difficult or even impossible to identify in the spectra of Hen 135 and Hen 198. In fact, we show below that these two stars are the only two SC stars in the subsample of Henize S stars observed with the CAT[^3]. Hen 135 ($V\sim 7$) and Hen 198 ($V\sim 7-10$) were indeed the only very red stars ($B-V>2$) which were bright enough to allow spectra to be taken in the violet. SC stars are known to have very peculiar spectra. Their spectrum is filled with strong atomic lines and almost no molecular bands in the optical, a consequence of their C/O ratio being very close to unity (Scalo [@Scalo]). Catchpole & Feast ([@Catchpole71]) define SC stars from the following three criteria: (i) extremely strong Na D lines, (ii) drop in the continuum intensity shortward of 4500Å, and (iii) bands of ZrO and CN simultaneously present (though quite weak), as well as general resemblance of the spectrum (i.e. regarding ‘the absolute and relative strength of metal lines’) with that of UY Cen. Hen 135 (=UY Cen) is thus the prototype SC star. Our two spectra of that star (taken in March 1993 and January 1997, see Fig. \[Fig:figTc62except\]) are quite different; in particular the shape of the $X_{4262}$ feature has changed noticeably. Therefore it is hazardous to infer the technetium content of UY Cen from these data alone without the help of appropriate model atmospheres and synthetic spectra, which is beyond the scope of this paper. Hen 198 (=RZ Sgr) has an Se-type spectrum; Stephenson ([@Stephenson]) quotes the HD catalogue noting that ‘the spectrum is similar to class N, but does not belong to that class’. It is probably associated with a reflection nebula (Whitelock [@Whitelock94]). RZ Sgr is a large-amplitude ($\sim2.5$ mag) SRb-type variable ($P=203.6$ d). Its H$\alpha$ emission, as well as the TiO and ZrO band strengths, are variable. Catchpole & Feast ([@Catchpole76]) also note that the Zr:Ti ratio of RZ Sgr is unusually high for an S star, and rather close to the one of N-type carbon stars. Although RZ Sgr has not been classified as an SC star, it shares many common features with that family. Indeed, it reasonably meets the three criteria mentioned above for SC stars:\ (i) Reid and Mould ([@Reid]) measured the strength of the Na D lines for several S, SC and C stars, including RZ Sgr. A spectrophotometric index of 1.07 is found for RZ Sgr, much larger than typical values for S stars (0.22 for and 0.28 for NQ Pup), but comparable to values obtained for SC stars (0.55 for LMC 441, 1.73 for R CMi, 2.57 for VX Aql). Thus RZ Sgr has abnormally strong Na D lines with respect to other S stars. \(ii) The ultraviolet flux deficiency of SC stars is clearly apparent from photometric data in the Geneva system. Indeed, the mean wavelengths of the $B$ and $V$ filters are $\lambda_0(B)$ = 4227Å  and $\lambda_0(V)$ = 5488Å  (Rufener & Nicolet [@Rufener]); therefore the $B-V$ index is highly sensitive to the ultraviolet flux deficiency of SC stars occurring for $\lambda < 4500$Å. SC stars have $B-V>2$, whereas the bulk of S stars have $B-V<2$. In that respect again, RZ Sgr ($2.0<B-V<3.0$) is typical of SC stars. \(iii) ZrO is present (although weak) in RZ Sgr; we found no information about the possible presence of CN bands. Infrared CO bands are stronger in RZ Sgr than in many other S and SC stars (Whitelock et al. [@Whitelock85]), probably locking a great quantity of carbon. The IRAS colours of RZ Sgr also share many similarities with SC stars: it is located in a region of the ($K-[12]$,$[25]-[60]$) color-color diagram (‘region E’ as defined by Jorissen & Knapp [@JorissenKnapp]) containing mainly SC stars with large 60$\mu$m excess and often resolved shells (see also Young et al. [@Young]). All these arguments therefore indicate that RZ Sgr is closely related to the SC family. As pointed out for UY Cen, the 4262Å  and 4238Å  lines of are very difficult to analyse in SC stars. An assignment of these two stars to either the Tc-rich or Tc-poor group has therefore not been attempted here. Abia & Wallerstein ([@Abia]) nevertheless suggest that SC stars are Tc-rich, based on a quantitative analysis. Discussion {#Sect:RESULTS} ========== The technetium dichotomy ------------------------ The lower part of Fig. \[Fig:Tc\] shows the frequency histogram of the wavelength of the $X_{4262}$ spectral feature for stars of Tables \[Tab1\] and  \[Tab2\]. The stars of our sample clearly segregate in two groups. The average wavelength of the bluer group is 4262.093Å; this group thus corresponds to Tc-poor S stars. The average wavelength of the redder group is 4262.235Å, thus revealing the contribution of the 4262.270Å  line to the - blend. The standard deviation on the $X_{4262}$ wavelengths is 0.012Å  for Tc-poor stars and 0.017Å  for Tc-rich stars. These values are in good agreement with the estimated errors on the $X_{4262}$ wavelength (0.013Å  for Tc-poor and 0.020Å  for Tc-rich stars, Sect. \[Sect:Analysis\]). The two groups are clearly separated by a 0.08Å  gap, [*with no intermediate cases*]{}. Therefore, in order to distinguish Tc-poor from Tc-rich stars (on our spectra of resolution in the range 30000-60000), a delimiting wavelength of 4262.16Å  may be safely adopted. A similar conclusion holds for the $X_{4238}$ feature (Fig. \[Fig:Tc3862\]), where a boundary wavelength of 4238.29Å  unambiguously separates the two kinds of S stars. Fig. \[Fig:Tc3862\] further shows that the diagnostics provided by the $X_{4238}$ and $X_{4262}$ features are consistent with each other. For comparison purpose the frequency histogram of the wavelength of the $X_{4262}$ spectral feature as obtained by Smith & Lambert ([@Smith]) is plotted in the upper part of Fig. \[Fig:Tc\], for their sample of MS and S stars (their Table 2). The segregation into Tc-poor and Tc-rich S stars (with 4 stars falling on their boundary wavelength at 4262.14Å) is not as clean as with our higher resolution spectra. The small number of ‘transition stars’ in our sample (i.e. stars with weak Tc lines, the only case being Hen 140) is noteworthy. This result may provide constraints on the evolution with time of the technetium abundance along the TPAGB (Smith & Lambert [@Smith]; Busso et al. [@Busso]) and clearly deserves further studies. For example, it would be of interest to investigate whether the small number of S stars with weak technetium lines found in our sample implies that the very first objects to dredge-up heavy elements on the TPAGB are not S stars but rather M stars. Indeed, the Tc detection threshold might not coincide with the ZrO detection threshold, but be slightly lower (i.e. [*stars would appear Tc-rich before being ZrO-rich*]{}); this would explain the puzzling Tc-rich M stars discovered by Little-Marenin & Little ([@Little-Marenin]) and Little et al. ([@Little]). It is moreover necessary to disentangle abundance effects from atmospheric effects on the technetium line strength. Although a more detailed study is deferred to a forthcoming paper, it may already be mentioned at this point that the Tc/no-Tc dichotomy reported in this paper is not due to technetium being entirely ionized in the warmer S stars. Indeed, the /( + ) ratio is still $\sim40\%$ in the warmest S stars (T$_{eff}$=3800 K) while it amounts to $\sim70\%$ at T$_{eff}$=3500 K and to $\sim95\%$ at T$_{eff}$=3000 K, according to the Saha ionization equilibrium formula (with representative electron densities taken from model atmospheres). M stars with 60$\mu$m excess and symbiotic stars ------------------------------------------------ None of the four M stars with 60$\mu$m excess taken from the sample of Zijlstra et al. ([@Zijlstra]) show technetium. This observation clearly indicates that these stars, which are surrounded by cool dust dating back to a former episode of strong mass loss, do not currently experience heavy elements synthesis followed by third dredge-ups. The same conclusion holds true for the two observed symbiotic stars (SY Mus and RW Hya). Conclusion ========== High-resolution spectra have been obtained and analysed to infer the technetium content of 76 S, 8 M and 2 symbiotic stars. The presence or absence of technetium was deduced from the shape of two blends involving technetium at 4238Å  and 4262Å  (more precisely: from the wavelength of their minimum). However this method does not apply to SC stars. Two misclassified S stars (Hen 22 and Hen 154) have emerged. The technetium (non-)detection at 4238Å  is consistent with the result at 4262Å. Only one ‘transition’ case (Hen 140 = HD 120179, a star where only weak lines of technetium are detectable) is found in our sample. A resolution in excess of $30\,000$ is definitely required to provide unambiguous conclusions regarding presence or absence of technetium. For example, at 4262Å, an s-process line (possibly ) is suspected to sometimes mimick a weak technetium line (although the 4238Å  feature clearly shows that technetium is absent). The shape of the $\lambda$4262Å  feature varies from one Tc-poor star to another, depending on the s-process overabundance level, which is in turn a function of the amount of accreted matter by these binary S stars. Among the 70 analysed Henize S stars, 41 turn out to be technetium-poor and 29 technetium-rich. That fraction may not be used, however, to infer the relative frequencies of intrinsic and extrinsic S stars, since the subsample of Henize S stars observed with the CAT is biased towards the brightest and bluest stars. The frequency of extrinsic/intrinsic S stars will be derived from the whole data set in a forthcoming paper. 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--- abstract: 'Some studies suggested that a correlation between locations of BL Lacertae objects (BL Lacs) and the arrival directions of the ultrahigh energy cosmic rays (UHECRs) exists. Especially by assuming the primary particles charged $+1$ and using a galactic magnetic field (GMF) model to calculate the deflections of the UHECRs, the significance of correlation is improved. We construct a new GMF model by incorporating all progresses in the GMF measurements in recent years. Based on a thorough study of the deflections of the UHECRs measured by the AGASA experiment, we study the GFM model dependence of the correlation between the UHECRs and the selected BL Lacs using the new model together with others. It turns out that only specific one of those GMF models makes the correlation significant, even if neither GMF models themselves nor deflections of the UHECRs are not significantly different. It indicates that the significance of the correlation, calculated using a method suggested in those studies, is intensively depending on the GMF model. Great improvement in statistics may help to suppress the sensitivity to the GMF models.' address: - 'Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, 10039, P. R. China' - 'Department of Physics, University of Utah, Salt Lake City, UT 84112, USA' - 'Department of Physics, Yunnan University, Kunming, 650091, P. R. China ' - 'Yunnan Agricultural University, Kunming, 650201, P. R. China ' - 'National Astronomical Observatories/Yunnan Observatory, Chinese Academy of Sciences, P.O. Box 110, Kunming, 650011, P. R. China' author: - Zhen Cao - Ben Zhong Dai - Jian Ping Yang - Li Zhang title: 'The correlations between BL Lacs and ultra-high-energy cosmic rays deflected by using different GMF models ' --- ultrahigh energy cosmic rays ,BL Lacertae objects ,magnetic field 98.70.Sa ,98.35.Eg ,98.54.Cm Introduction ============ Search for sources of ultrahigh energy cosmic rays (UHECRs) is very important and many researches have been done recently, including searches for correlations between the UHECRs measured by AGASA and HiRes experiments and BL Lacertae objects (BL Lacs) [@BL1; @BL:GMF; @BL2]. Identifying the sources will provide us direct information about the acceleration mechanisms of the UHECRs that is essential to understand phenomena such as Greisen-Zatsepin-Kuzmin (GZK) ¡®cutoff¡¯[@Greisen; @Zatsepin]. The distribution of arrival directions of UHECRs, however, is remarkably isotropic. There is no correlation between astronomical objects and cosmic rays being confirmed. Arrival directions should not point back to sources unless the UHECRs are neutral, e.g. $\gamma$’s or neutrinos. The charged cosmic ray primaries will be deflected while propagate through the galactic and intergalactic magnetic fields (GMF and IGMF). UHECRs above $4\times10^{19}$ eV might be considered for small-scale anisotropy searching. Deflections of them would be so small that the uncertainty of the deflection estimation would not be harmful. If there was any correlations between the UHECRs and any known objects, the correlation should not be strongly dependent on specific GMF models. However, this has yet to be seriously tested. This paper is devoted to test the model dependence. At first, a new GMF model is constructed by incorporating all recent progresses in GMF measurements. Secondly, the deflections of UHECRs using different GMF models, including the models used for the AGASA data analysis, are compared. Finally, the correlations between the UHECRs and the selected BL Lacs are investigated to check the influences from the different GMF models. Active Galactic Nuclei(AGNs) have been considered as UHECR sources by authors [@Berezinsky90; @Rachen; @Berezinsky:2002nc]. The AGASA data set [@takeda] has exhibited clustering in the experimental resolution. Some authors have suggested that the clusters may be due to point sources (although no cluster has been confirmed by the HiRes experiment[@Abbasi1; @Abbasi2] ). It has been argued that those clusters might be aligned with BL Lacs [@BL1; @BL:GMF; @BL2]. In those papers, significant correlations are expressed no matter what primary particles of those UHECRs are assumed, neutral or charged. In case of the proton primary is assumed, deflections of the protons in the GMF are estimated using a specific model. Even more significant correlation is claimed with the proton primary assumption. BL Lacs are blazars (AGNs with relativistic jets directed along the line of sight) characterized in particular by absence of emission lines. This indicates low ambient matter, therefore it is a favorable condition for accelerating particles to ultra high energies. Studies have suggested that the acceleration of particles in jets can be explained using pinch mechanism and the maximum energy of particles can be greater than $10^{20 }$eV[@Veniamin]. The AGASA results seem to enhance the hypothesis. However, the correlation between the UHECRs and the BL Lacs must be concretely confirmed. If the UHECRs are not neutral, the GMF model dependence needs to be investigated thoroughly before drawing any conclusion on this mechanism. Deflections of charged UHECRs in the intergalactic magnetic fields are assumed random and unpredictable due to lack of knowledge about the magnetic fields. The deflections in the GMF are better understood because the knowledge about the GMF is greatly enhanced in the last ten years using rotation measures (RMs) of radio polarization from pulsars and extragalactic radio sources. The RM data reveals many new features of the GMF, such as a central rotating bar in the galactic plane, a dynamo structure in the galactic halo, a magnetic dipole at the galactic center (G.C.) and so on. The GMF model could be improved by taking all of the features into account. There are many GMF models available in the market depending on implementations of those observational facts. This offers an opportunity for investigating the model dependence of the correlation between the BL Lacs and the UHECRs. The GMF model is improved by incorporating the latest updates of the RM measurements in this paper, then the GMF model dependence is studied using the AGASA data and the new model and other available GMF models. This study is essential for further investigation using HiRes data. This paper is organized as following. Improvements of the GMF model and differences between models are described in Sect. \[sect:GMF\]. Deflections of the UHECRs using different GMF models are analyzed in Sect. \[sect:DF\]. Correlations between the BL Lacs and the UHECRs are re-estimated for the GMF model dependence testing in Sect.\[sect:CA\]. Conclusions are drawn based on the comparisons in Sect. \[sect:conclusions\]. Galactic Magnetic Field {#sect:GMF} ======================= The Galactic Magnetic Field (GMF) is composed of regular field and turbulent field. The regular field keeps no change in time and distributes close to matters in our galaxy. The turbulent field that is due to localized activities of objects in our galaxy. Recent studies [@han04paper] has discovered that the turbulent fields exist on all scales from a few pc’s to the whole galaxy. Strengths of the turbulent fields can be up to twice of the regular field at the same place. Directions of the turbulent fields are isotropic. To model the turbulent fields, their directions are randomly chosen and their strengths are randomly sampled from a half to twice of the regular field strength at the same place. The regular field includes three components according to their sources and regions where they are distributed. The main component is located inside the disk of our galaxy where a majority of the galactic matter is distributed in a spiral structure. The magnetic field is also distributed in the same 2-dimensional spiral structure. Outside the galactic plane, the filed strengths decrease with the distance from the plane as the galactic matter distribution does. Directions of the fields are parallel to the disk. In the galactic halo, a pair of toric structures of the GMF are found. The toruses are located in regions about 4kpc from the disk in both upper(north) and lower(south) hemispheres. Directions of the toric fields are also parallel to the disk, rotating in a clockwise direction in the upper hemisphere but reversed in the lower. Moreover, evidences indicate that there is a dipole field at the center of our galaxy. All three components are described in details in the following subsections. The 2-dimensional magnetic field component in the galactic disk {#mf} ---------------------------------------------------------------- The field is distributed in a spiral structure as the baryonic matter does in the area beyond 4 kpc from the G.C. The model of the magnetic field component in the galactic disk is described by the following parameters : - Distance from the Sun to the Galactic center, $R = 8.5$ kpc, - Local field strength, $B_0 = 1.4 \,\mu {\rm G}$[@han04paper; @Beck00], - Pitch angle $p= - 8.2^\circ$[@Beck00; @Han94; @Han01], - Distance from the sun to the first field reversal $d$=-0.2kpc [@Han94; @Han01],. The field strength at a point ($r,\theta$) in the galactic disk is $$B(r,\theta) = B(r)\, \, \cos\left(\theta - \beta \ln\left(\frac{r}{R}\right) + \phi \right) \,\,\,\,\,(r> 4kpc), \label{Bd1}$$ where $\beta \equiv 1/\tan(p)$, the constant phase $\phi$ is given by $$\phi = \beta \ln\left(1 + \frac{d}{R}\right) - \frac{\pi}{2} \, ,$$ and $B(r)$ describes the change of the strength with distance from the center of our galaxy. The strengths of the fields exponentially fall off with the distance from the G.C. and reads as $$B(r) = \frac{B_0\:exp(- \frac{(r-R)}{r_b})}{\cos (\phi )} \,\,\,\,\,(r>4kpc), \label{Bd3}$$ where the scale radius $ r_b=7 $ kpc [@han-lecture]. The direction of the field at $(r,\theta)$ is determined by the pitch angle and described by the two components $$B_\theta = B(r,\theta)\, \cos(p), \qquad B_r = B(r,\theta)\, \sin(p).$$ Recently, there is a clear evidence from GLIMPSE (Galactic Legacy Mid-Plane Survey Extraordinaire) about a bar-like structure, consisting of relatively old and red stars, in the central area of our galaxy. The bar is about 27,000 light years ($\sim$8kpc) long, longer than previously believed. This survey also shows that the bar is oriented at 45$^\circ$ respect to a line connecting the sun and the center of the galaxy[@Benjamin]. This bar-shaped structure has been built in the new GMF model for $r<4kpc$. A Gaussian is used for the description, i.e. $$B(r,\theta) = 4 \,\cdot exp( -\frac{r^2}{4}(sin\theta-cos\theta)^2) \,\,\,\,\,(r< 4kpc),$$ where the field strength is 4 $\mu$G in the bar area that is about 1kpc wide. The magnetic fields outside the galactic disk ---------------------------------------------- There are two components in the region of $z\ne 0$. The first part is the extension of the spiral and bar-shaped field. Outside the galactic disk, the strengths of the fields decreases following a Gaussian function of $z$, i.e. $$B(r,\theta,z) = B(r,\theta)\,exp( -\frac{1}{2}(\frac{z}{h})^2),$$ Where the Gaussian width $h$=0.6 kpc[@Han-priv] being consistent with the matter distribution. Directions of the fields are the same as the 2-dimensional fields at $z=0$ in both the upper and lower hemispheres. This is a natural description. The magnetic fields in the halo near the galactic disk are dominated by the extensions of the spiral field. In regions further away from the disk, the magnetic fields are found to have a dynamo-like structure which is described better by a toric field model[@Han01]. The magnetic fields are distributed in two toruses located in the upper and lower hemispheres, respectively. The fields of the two toruses are in opposite directions. The centers of the toruses are about $\pm$1.22 kpc from the disk and are parallel to the disk. There are also evidences showing that the strengths of the fields are about 1$\mu {\rm G}$ around z=1.5 kpc[@Han99]. The toric fields can be modelled by $$B(r,\theta,z)=B_t exp( -\frac{1}{2}(\frac{r}{r_t})^2)[exp(-\frac{1}{2}(\frac{z-1.22}{h_t})^2)-exp(-\frac{1}{2}(\frac{z+1.22}{h_t})^2)]$$ where $r_t$ is 8.5 kpc, $h_t$ is 1.22 kpc and $B_t$ is 1.85 $\mu$ G. dipole magnetic fields located at the galactic center ------------------------------------------------------ The local magnetic fields in the vicinity of the solar system have been found to have a small vertical component about 0.3 $\mu {\rm G}$ and pointing from the South to the North [@Han94]. This observational fact can be interpreted as that there exists a dipole at the center of our galaxy and points to the north. The dipole fields are $$\label{r31} B_x = -3 \mu {_{\rm G}} \sin(\theta) \cos(\theta) \cos(\phi)/r^3$$ $$\label{r32} B_y = -3 \mu {_{\rm G}} \sin(\theta) \cos(\theta) \sin(\phi)/r^3$$ $$\label{r33} B_z = \mu {_{\rm G}} (1-3\cos^2(\theta))/r^3$$ where the constant $\mu {_{\rm G}}$ must be 184.2 $\mu {\rm G} (kpc)^3$, and $r$, $\theta$, $\phi$ are spherical coordinates at the center of galaxy and $\phi=0$ is the direction of the sun. This is a very strong dipole moment determined only by a small value measured at 8.5 kpc away from the dipole. It has to be tested with great care. There is not enough experimental data to provide sufficient constraint on the model beyond 20kpc from the G.C., therefore the new model is only valid in a range of 20 kpc from the center of our galaxy in all directions. In order to compare the new model with others, two models that are used in Ref. [@BL:GMF] (model-T) and Ref. [@HAJIME; @Prouza; @Jaime; @Hiroyuki] (model-M) are chosen. Those models are widely used in the correlation analyses between the BL Lacs and the UHECRs. At first, the fields in the galactic disk are compared. In Fig. \[GMF\_r\], the field strength is plotted against the distance from the G.C. along the direction $\phi=0$. The model-T has very similar behavior compared with the new model beyond 4 kpc, including the strengths, reversals and the phases of the fields. This similarity indicates that the spiral structure and the reversals of the fields are essentially same in those two models. The model-M shows a slightly different behavior. The reversals occur at different locations and more importantly the field strengths remain quite strong at large distances, e.g. near 20kpc from the G.C., where the field strengths are reduced to very weak in the other two models. The rather large and flat field strengths between 16 kpc and 20 kpc may have strong consequences in deflections of the UHECRs. The other major difference is that the latest progress about the bar-shaped baryonic matter distribution near the G.C. has been incorporated in the new model. It may cause significant differences in the deflections of UHECRs passing by the G.C. It is noticed that the strengths have discontinuities at $z=0$ in the descriptions of the halo fields in both the model-T and model-M. Those discontinuities occur also at all reversals of the fields in the galactic disk. Those discontinuities are not natural and cause the deflections of the UHECRs in different ranges to cancel each other. In the new model the strength has a smooth transition cross the disk at $z=0$ everywhere in our galaxy. In Fig. \[GMF\_z\], the GMF strengths are plotted as functions of distances from the disk in the vicinity of the sun. It is important to note that the halo fields have similar behaviors beyond 4 kpc from the galactic disk in the new model and the model-T, however, the model-M assumes that the fields are quite strong in a larger region of the halo. The strengths of the fields are much stronger than other models beyond 3kpc from the disk. Those features cause much stronger deflections for most UHECRs that come from high galactic latitudes. ![ The change of the galactic magnetic field strength as a function of the distance from the G.C., r, in the direction of the sun, $\phi=0$. []{data-label="GMF_r"}](gmf_r.eps){width="9cm"} ![ The change of the galactic magnetic field strength as a function of the distance from the galactic disk, z, in the vicinity of the sun.[]{data-label="GMF_z"}](gmf_z.eps){width="9cm"} Analysis of deflections using three GMF models {#sect:DF} ============================================== A deflection angle is defined as the difference between the observed UHECR direction and the primary direction of UHECR i.e. the direction outside our galaxy which is calculated by using time reversing symmetry by assigning a charge to the cosmic ray particle, e.g. assuming it is a proton. The 57 UHECR events observed by AGASA experiment ($E>4 \times 10 ^{19} eV$) adopted from Ref. [@takeda] are used as examples for studying the deflections of the UHECRs in the GMF and model dependence in this paper. Using the new GMF model, most of AGASA events are found to be bent less than $10^\circ$ with a peak at $3^\circ$, only one event is bent about $13^\circ$( Fig. \[our\_defl\] ). With or without the turbulent components of the GMF, the deflection angles of the cosmic ray samples are compared in the same figure. It is shown that turbulent fields systematically shift the deflection angle by about $0.1^\circ$. It is so small compared with the effect of different GMF models that the turbulent component is negligible for the UHECR samples. The average bending angle is 3.16$^\circ$. The dipole field at the galactic center is so strong that any particles passing by the G.C. could be bent severely. In Fig. \[four\_defl\], the deflections are compared with or without the dipole fields. The most deflected cosmic ray is from the direction with the galactic longitude and latitude of (l,b)=(22.8$^\circ$,15.7$^\circ$) and is bent by 13.6$^\circ$ using the new model. The bending is mainly caused by the dipole field, since the trajectory is close to the G.C., i.e. the angle between the trajectory and a connection from the G.C. to the sun is about 27$^\circ$. Once the dipole field is turned off in the deflection calculation, this event is bent with a smaller angle, while the most deflected cosmic ray is replaced by another from (l,b)=(154.5$^\circ$,15.6$^\circ$) and is bent by 9.53$^\circ$. According to this analysis, one should cautiously treat those events that their trajectories are close to the G.C. with models that the dipole fields are included. To avoid a great uncertainty associated with the dipole component, those events that pass by the G.C. with close distances should be cut. For comparison, the model-T and model-M are used to the same data set for deflection estimation. Deflections of all events are confined to $7^\circ$ using model-T (dotted dash line in Fig. \[four\_defl\]). The average bending angle is about 3.3$^\circ$ which is not significantly different from the new model. Within the statistic fluctuations, one might draw a conclusion that the deflection estimations are essentially same by using those two models. However, using the model-T, the bending angles are populated around about $3^\circ\sim 4^\circ$, which differ from the new model. Using the new model, most of cosmic rays are bent with angles less than 4$^\circ$. 5 cosmic ray events are bent more than 7$^\circ$, which are caused by the dipole component. There is no event being bent more than 7$^\circ$ if the dipole component is turned off in the new model. Deflections are scattered in a wide range using model-M. Many cosmic rays are bent more than $10^\circ$ (dotted line in Fig. \[four\_defl\]). Statistically, this model has different behavior from the new model and the model-T. The wide distribution is mainly caused by the halo field. From Fig. \[GMF\_z\], one can see that the halo field of model-M has a very large tail extending 8kpc away from the disk. On the other hand, both new model and the model-T have similar behaviors beyond 4kpc from the disk, so that the comic rays have the similar deflections even if the halo fields are very different near the disk. This indicates that the fields in the halo, for instance beyond 4kpc, are crucial for UHECR deflections. It is expected that the GMF measurements in the high galactic latitude region will be improved in the future. They are essential for understanding the cosmic ray deflections. ![ The distribution of deflection angles of 57 AGASA events using the new GMF model. The thick solid histogram represent the case that the turbulent component is turned off, while the thin histogram represent that the turbulent component of the GMF is considered. []{data-label="our_defl"}](our_defl.eps){width="9cm"} ![ The distribution of deflection angles of the 57 AGASA events different GMF models. Thick solid line is for the new model, thin solid line is for the new model without the dipole component, dotted dash line is for the model-T and dot line is for the model-M. []{data-label="four_defl"}](four_defl.eps){width="9cm"} The correlation between UHECRs and BL Lacs {#sect:CA} ========================================== Using the chance probability function $p(\delta)$ introduced in Refs. [@BL1; @BL:GMF; @BL2; @BL3; @BL4], the correlations between the UHECR events and sources were quantitatively estimated for given angular interval $\delta$. The probability is calculated by counting how often the numbers of MC events and source pairs are equal or greater than numbers of real events and source pairs, i.e. $p(\delta)=\frac{ number\; of \; trial \; sets\; with \;N_{mc} \geq N_{real}}{total \;number\; of\; trial \;sets\; }$, where the $N_{real}$ and the $N_{mc}$ are numbers of pairs of real cosmic rays matching with sources and simulated events matching with sources, respectively. A pair is defined as an event falls in a circle with an angular radius of $\delta$ centered any source in a selected samples. The smaller is this probability, the more significant the correlation is. The Monte-Carlo events are generated in the horizon reference frame with a geometrical acceptance $dn \propto \cos \theta_z\sin\theta_z d\theta_z$, where $\theta_z$ is zenith angle and the coordinates are transformed into equatorial frame assuming random arrival time. All events are generated with zenith angles $\theta_z < 45^\circ$, same as the real event samples are selected[@Uchihori]. The deflections of both data and MC samples are calculated by assuming pure proton primary. If the function $p(\delta)$ exists a minimum nearby the experimental resolution(about $2.5^\circ$ for the AGASA experiment), it indicates a correlation between the observed UHECR’s and the sources. The value of the $p(\delta)$ is an estimation of the chance probability. Ten thousand sets of the MC events are generated for each case. From the data of QSO catalog [@veron], 178 BL Lacs are selected according to a criterion of apparent magnitude less than $ 18$ Ref.[@BL:GMF]. The focus of this paper is the GMF model dependence of the correlation between selected BL Lacs and the 57 AGASA events with deflections. First of all, the result in Ref.[@BL:GMF] is reproduced using the model-T as the dash-dotted line in Fig. \[our\_gmf\]. It clearly shows a minimum around $\delta =2.7^{\circ}$. Based on this, the authors claimed a significant correlation between the UHECRs and the selected BL Lac samples. However, the same chance probability calculated using the new GMF model and model-M do not show any correlation between the AGASA cosmic ray events and the BL Lac samples as solid and dotted lines in the same figure. This shows that the GMF model dependence is nontrivial. ![ $P(\delta)$ for the set of 178 BL Lacs (${\rm mag} < 18)$ and AGASA 57 events ($E>4\times10^{19}$ eV). The new GMF model (solid), model-T (dash-dotted) and model-M (dotted) are used for the cosmic ray deflection calculation. []{data-label="our_gmf"}](three_gmf.eps){width="9cm"} As described in Sec.\[sect:GMF\], the models themselves are very similar between the new model and the model-T, except for the fields both in the galactic disk and halo is improved based on more modern observations in the new model. The artificial discontinuities at $z=0$ are replaced by a smooth transition between the upper and lower hemispheres and the singularity at $r=0$ is overcome by introducing the bar-structure in the central area in the new model. Those small improvements are not expected to significantly change the deflections of UHECRs. This has been shown in the comparison in Sec.\[sect:DF\]. Although the deflections are not exactly same, within statistical fluctuation, the deflection angle distributions from those two models agree well with each other. This also shows that the deflection of UHECRs are correctly treated which maintains the consistency. It is noted that the GMF model-T does not include the dipole field component. It is also aware of that the dipole field component might be determined with a great deal of uncertainty. Therefore, the deflection behaviors are studied using the new model without the dipole component. From the Fig.\[four\_defl\], it is found that the distribution of the deflection angles is even more close to that from the model-T. Only for those the deflection angles are less than 1$^\circ$, the two models have different behaviors. Without the dipole component, a similar analysis of the correlation between the UHECRs and the BL Lacs is still not significant. To understand the discrepancy between results of the correlation analyses from different models, the numbers of the event-source matching pairs around the AGASA resolution $2.5^\circ$ are listed for different GMF models in Table \[tab:probability\]. The differences between the new model and the model-T are small. However, the chance probability jumps 2 orders of magnitudes from 10$^{-4}$ to 10$^{-2}$. This indicates that the function $p(\delta)$ is too sensitive. A usual problem of such a sensitive method is poor stability or robustness. It may be very useful for a big sample of UHECRs where more matching pairs are expected. It is noticed that the numbers of pairs are not affected by the dipole field component. Conclusions {#sect:conclusions} =========== Using AGASA 57 UHECR ($E>4 \times 10 ^{19} eV$) and the new GMF model, model-T and model-M, the GMF model dependence of the correlation analysis between the BL Lacs and the UHECRs are tested in this paper. To calculate the deflection of the UHECRs , charge $Q=+1$ is assigned to all events and deflections by IGMF are assumed negligible. Using the model-M, the deflections are found significantly larger than other two models. The reason is that the magnetic fields in the galactic halo extend much further from the galactic disk than the other models. The deflection behaviors of the UHECRs are very similar using the new GMF model and the model-T. Within the statistical fluctuation, the two models agree with each other in terms of the deflection angle distributions. However, the numbers of event-BL Lac matching pairs by using those two models are different, i.e. slightly larger than one Poissonian standard deviation. This difference causes a discrepancy of about 2 orders of magnitudes in the chance probability function between the two models. Both the new GMF model and the model-M does not support correlations between the AGASA UHECRs and the selected BL Lacs. This indicates that the correlation analysis method might be too sensitive to the model, especially at the stage where there is not enough statistics for the UHECR samples. To draw a conclusion on the correlation between the UHECRs and the BL Lacs, there needs to be many more UHECR event samples. There have been lots of UHECR events collected by the HiRes Experiment and Auger Experiment recently. To complete this study, however, more constraints on the GMF models based on further observations are crucial according to the discussion in this paper. angle new model model-T model-M ------------- ----------- --------- --------- -- $2.3^\circ$ 12 12 6 $2.4^\circ$ 12 15 6 $2.5^\circ$ 12 15 8 $2.6^\circ$ 12 16 9 $2.7^\circ$ 12 18 9 $2.8^\circ$ 12 19 11 Acknowledgments {#acknowledgments .unnumbered} =============== [=400 We are grateful to Dr. J. L.Han for his valuable comments and discussions with us. This work is supported by NSFC under contract 10445001, Knowledge Innovation fund (U-526) of IHEP, China and by Hundred Talents & Outstanding Young Scientists Abroad Program of CAS, China(U-610). ]{} [00]{} P. G. Tinyakov and I. I. Tkachev, JETP Lett.  [**74**]{}, 445 (2001) \[Pisma Zh. Eksp. Teor. Fiz., 499 (2001)\]. P. G. Tinyakov and I. I. Tkachev, Astropart. Phys.  [**18**]{}, 165 (2002). D. S. Gorbunov [*et al.*]{}, JETP Lett. [**80**]{}, 145 (2004) \[arXiv:astro-ph 0406654\]. K. 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[Ground States of Quantum Electrodynamics with Cutoffs ]{}\ $\;$\ [Toshimitsu Takaesu ]{}\ $\;$\ *Faculty of Science and Technology, Gunma University,\ Gunma, 371-8510, Japan* > **Abstract** In this paper, we investigate a system of quantum electrodynamics with cutoffs. The total Hamiltonian is defined on a tensor product of a fermion Fock space and a boson Fock. It is shown that, under spatially localized conditions and momentum regularity conditions, the total Hamiltonian has a ground state for all values of coupling constants. In particular, its multiplicity is finite.\ > $\; $\ > [MSC 2010 : 47A10, 81Q10. $\; $\ > key words : Fock spaces, Spectral analysis, Quantum Electrodynamics]{}. Introduction ============= $\;$ This articles is concerned with a system of quantum electrodynamics with cutoffs. In quantum field theory, the interactions of charged particles and photons are described by quantum electrodynamics. We consider the system of a massive Dirac field coupled to a radiation field. The radiation filed is quantized in the Coulomb gauge. In this system, the process of electron-positron pair production and annihilation occurs. We mathematically investigate the spectrum of the total Hamiltonian for the system. The Hilbert space for the system is defined by a tensor product of a fermion Fock space and boson Fock space, which is called a boson-fermion Fock space. The total Hamiltonian is given by $$\begin{aligned} {H_{\textrm{QED}}}= H_{{ \textrm{D} }}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}}}& + {\kappa_{\textrm{I}}}\sum_{j=1}^{3} {\int_{\mathbf{R}^{3}}}\chi_{{\textrm{I}}} ({\ensuremath{\mathbf{x}}}) ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}}) }{\otimes}A_j ({\ensuremath{\mathbf{x}}} ) ) d{\ensuremath{\mathbf{x}}} \notag \\ & \; \; + {\kappa_{\textrm{II}}}\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|} \left( {\psi^{\dagger} (\mathbf{x}) }{ \psi ({\ensuremath{\mathbf{x}}}) }{\psi^{\dagger} (\mathbf{y}) }{ \psi ({\ensuremath{\mathbf{y}}}) }{\otimes}{{\small \text{1}}\hspace{-0.32em}1}\right) d {\ensuremath{\mathbf{x}}} \, d {\ensuremath{\mathbf{y}}} \notag \end{aligned}$$ on the Hilbert space. Here $H_{{ \textrm{D} }}$ and $H_{{\textrm{rad}}} $ denote the energy Hamiltonians of the Dirac field and radiation field, respectively, $\psi ({\ensuremath{\mathbf{x}}})$ the Dirac field operator, ${\ensuremath{\mathbf{A}}}({\ensuremath{\mathbf{x}}})=(A_j({\ensuremath{\mathbf{x}}}))_{j=1}^3$ the radiation field operator, ${\ensuremath{\mathbf{\alpha}}}=(\alpha_j)_{j=1}^3$ $4\times 4$ Dirac matrices, and ${\chi_{\textrm{I}} (\mathbf{x}) }$ and ${\chi_{\textrm{II}} (\mathbf{x}) }$ the spatial cutoffs. The constants ${\kappa_{\textrm{I}}}\in {\ensuremath{\mathbf{R}}}$ and ${\kappa_{\textrm{II}}}\in {\ensuremath{\mathbf{R}}}$ are called coupling constants. Ultraviolet cutoffs are imposed on ${ \psi ({\ensuremath{\mathbf{x}}}) }$ and ${\ensuremath{\mathbf{A}}}({\ensuremath{\mathbf{x}}})$, respectively.\ By making use of the spacial cutoffs and ultraviolet cutoffs, ${H_{\textrm{QED}}}$ is self-adjoint operator on the Hilbert space, and the spectrum of ${H_{\textrm{QED}}}$ is bounded from below. The main interest in this paper is the lower bound of the spectrum of ${H_{\textrm{QED}}}$. If the infimum of the spectrum of a self-adjoint operator is eigenvalue, the eigenvector is called ground state. The infimum of the spectrum of $H_{0}= H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}}}$ is eigenvalue, but it is embedded in continuous spectrum. This is because the radiation field is a massless field. It is not clear that ${H_{\textrm{QED}}}$ has a ground state since the embedded eigenvalue is not stable when interactions are turned on.\ The ground state of ${H_{\textrm{QED}}}$ for sufficiently small values of coupling constants was proven in [@Ta09]. The aim of this paper is to prove that ${H_{\textrm{QED}}}$ has a ground state for all values of coupling constants. In particular, its multiplicity is finite. For the ground states of other QED models, Dimassi-Guillot [@DiGu03] and Barbaroux-Dimassi-Guillot [@BDG04] investigated the system of the Dirac field in external potential coupled to the radiation field. They proved the existence of the ground state of the total Hamiltonian with generalized interactions for sufficiently small values of coupling constants. As far as we know, the existence of the ground states for the systems of a fermionic field coupled to a massless bosonic field, which include QED models, has not been proven for all values of coupling constants until now.\ To prove the existence of the ground state of ${H_{\textrm{QED}}}$ for all values of coupling constants, we apply the methods for systems of particles coupled bosonic fields. The spectral analysis and scattering theory for these systems, which include the non-relativistic QED models, have been progressed since the middle of ’90s. The existence of the ground states was established by Arai-Hirokawa [@AH97], Bach-Fr[ö]{}hlich-Sigal [@BFS98; @BFS99], G[é]{}rard [@Ge00], Griesemer-Lieb-Loss [@GLL01], Lieb-Loss [@LiLo03], Spohn [@Sp98] and many researchers. The strategy is as follows. $\; $\ **\[1st Step\]** We introduce approximating Hamiltonians $H_{m}$, $m>0$. Physically, $m>0$ denotes the artificial mass of photon, and we call $H_{m}$ a massive Hamiltonian. To prove the existence of ground states of $H_{m}$, we use partition of unity on Fock space, which was developed by Dereziński-Gérard [@DeGe99]. We especially need the partitions of unity for both Dirac field and radiation field. By the partitions of unity and the Weyl sequence method, we prove that a positive spectral gap above the infimum of the spectrum exists for all values of coupling constants. From this, the existence of the ground states of $H_m$ for all values of coupling constants follows. $\; $\ **\[2nd Step\]** Let $\Psi_m $ be the ground state of $H_{m}$, $m>0$. Without loss of generality, we may assume that the $\Psi_m$ is normalized. Then, there exists a subsequence of $\{ \Psi_{m_j} \}_{j=1}^\infty$ with $m_{j+1} < m_{j}$, $j\in {\ensuremath{\mathbf{N}}}$, such that the weak limit of $ \{ \Psi_{m_j}\}_{j=1}^{\infty}$ exists. The key point is to show that the the weak limit is non-zero vector. To prove this, we consider a combined method of Gerard [@Ge00] and Griesemer-Lieb-Loss in [@GLL01]. We use the electron positron derivative bounds and photon derivative bounds. To derive these bounds, the argument of the spatially localization is needed. For the spatially localized conditions, we suppose $$\int_{{\mathbf{R}^{3} }} |{\ensuremath{\mathbf{x}}}| \, | \chi_{{\textrm{I}}}({\ensuremath{\mathbf{x}}}) | dx\, < \, \infty , \qquad \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{| \chi_{{\textrm{II}}}({\ensuremath{\mathbf{x}}}) \chi_{{\textrm{II}}}({\ensuremath{\mathbf{y}}}) |}{|{\ensuremath{\mathbf{x}}}- {\ensuremath{\mathbf{y}}}|} |{\ensuremath{\mathbf{x}}}| {d \mathbf{x} }{d \mathbf{y} }< \infty.$$ I addition, We imposed momentum regularity conditions on the Dirac field and radiation field, which include the infrared regularity condition $$\int_{{\mathbf{R}^{3} }} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |^2}{| {\ensuremath{\mathbf{k}}} |^{5} } {d \mathbf{k} }< \infty .$$ $\;$\ $\; $\ We briefly review the results for the systems of fermionic fields coupled bosonic fields. For QED models, the Gell-Mann - Low formula of ${H_{\textrm{QED}}}$ was obtained by Futakuchi-Usui [@FuUs14]. For the Yukawa model, which is the system for a massive Dirac field interacting with a massive Klein-Gordon field, the existence of the ground state was proven in [@Ta11]. The spectaral analysis for the the weak interaction models has been analyzed, and refer to Barbaroux-Faupin-Guillot [@BFG14], Guillot [@Gu15] and the reference therein. $\;$\ This paper is organized as follows. In section 2, full Fock spaces, fermion Fock spaces and boson Fock spaces are introduced, and Dirac field operators and radiation field operators are defined on a Fermion Fock space and boson Fock space, respectively. The total Hamiltonian is defined on a boson-fermion Fock space and the main theorem is stated. In Section 3, partitions of unity for the Dirac field and radiation field are investigated. Then the existence of the ground state of $H_{m}$ is proven. In section 4, the derivative bounds for electrons-positrons and photons are derived. In Section 5, we give the proof of the main theorem.\ Notations and Main Results ========================== Fock Spaces ----------- **(i) Full Fock Space**\ The full Fock space over a complex Hilbert space ${\ensuremath{\mathscr{Z}}}$ is defined by $ {\ensuremath{\mathscr{F}}}( {\ensuremath{\mathscr{Z}}})= \oplus_{n=0}^\infty ( {\otimes^{n}}{\ensuremath{\mathscr{Z}}})$ where ${\otimes^{n}}{\ensuremath{\mathscr{Z}}}$ is the $n$ fold tensor product of $Z$. The Fock vacuum is defined by $\Omega = \{1, 0,0, \cdots \} \in {\ensuremath{\mathscr{F}}}( {\ensuremath{\mathscr{Z}}})$. Let $ {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Z}}})$ be the set which consists of all linear operators on ${\ensuremath{\mathscr{Z}}}$. The functor of $Q\in {\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{Z}}})$ is defined by $\Gamma (Q) = \oplus_{n=0}^{\infty} \left( {\otimes^{n}}Q \right)$ and the second quantization of $T\in{\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{Z}}})$ is given by $ {\ensuremath{d\Gamma({T}) }} = \oplus_{n=0}^{\infty} \tilde{T}^{(n)} $ with $ \tilde{T}^{(n)} =\sum\limits_{j=1}^{n} ( ({\otimes}^{j-1} {{\small \text{1}}\hspace{-0.32em}1}) {\otimes}T {\otimes}( {\otimes}^{n-j} {{\small \text{1}}\hspace{-0.32em}1}) )$. The number operator is defined by $N= {\ensuremath{d\Gamma({{{\small \text{1}}\hspace{-0.32em}1}}) }} $.\ $\;$\ **(ii) Fermion Fock Space**\ The fermion Fock space over a complex Hilbert space ${\ensuremath{\mathscr{X}}}$ is defined by $ {\mathscr{F}_{\textrm{f}} }( {\ensuremath{\mathscr{X}}})= \oplus_{n=0}^\infty ( {\otimes^{n}_{\textrm{a}}}{\ensuremath{\mathscr{X}}})$ where $ {\otimes^{n}_{\textrm{a}}}{\ensuremath{\mathscr{X}}}$ denotes the $n$-fold anti-symmetric tensor product of ${\ensuremath{\mathscr{X}}}$. The Fock vacuum is defined by $\Omega_{{\textrm{f}}} = \{1, 0, 0, \cdots \} \in {\ensuremath{\mathscr{F}}}( {\ensuremath{\mathscr{X}}})$. Let $T_{{\textrm{f}}}$ and $Q_{{\textrm{f}}}$ be linear operators on ${\ensuremath{\mathscr{X}}}$. We set ${\ensuremath{d\Gamma_{\textrm{f}}({T_{{\textrm{f}}}}) }}= {\ensuremath{d\Gamma({T_{{\textrm{f}}}}) }}_{\upharpoonright {\mathscr{F}_{\textrm{f}} }( {\ensuremath{\mathscr{X}}})}$ and $\Gamma_{{\textrm{f}}}( Q_{{\textrm{f}}})= \Gamma (Q_{{\textrm{f}}})_{\upharpoonright {\mathscr{F}_{\textrm{f}} }( {\ensuremath{\mathscr{X}}})}$ where $ X_{\upharpoonright {\ensuremath{\mathscr{M}}}} $ is the restriction of the operator $X$ to the subspace ${\ensuremath{\mathscr{M}}} $. The number operator is defined by $N_{{\textrm{f}}}= {\ensuremath{d\Gamma_{\textrm{f}}({{{\small \text{1}}\hspace{-0.32em}1}}) }} $. The creation operator $C^{\, \dagger}(f)$, $f \in {\ensuremath{\mathscr{X}}}$, is defined by $(C^{\, \dagger}(f) \Psi )^{(n)} = \sqrt{n} \, U_{\textrm{a}}^{ n}(f {\otimes}\Psi^{(n-1)})$, $n \geq 1$, and $(C^{\, \dagger }(f) \Psi )^{(0)} = 0 $ where $ U_{\textrm{a}}^{ n} $ is the projection from $ {\otimes^{n}}{\ensuremath{\mathscr{X}}} $ to ${\otimes^{n}_{\textrm{a}}}{\ensuremath{\mathscr{X}}} $. The annihilation operator $C(f)$ is defined by $C(f)=(C^{\, \dagger}(f))^{\ast}$ where $X^{\ast} $ denotes the adjoint of the operator $X$. For each subspace ${\ensuremath{\mathscr{M}}} \subset {\ensuremath{\mathscr{X}}}$, the finite particle space $ {\ensuremath{\mathscr{F}}}_{{\textrm{f}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{M}}}) $ is defined by the linear hull of $ \Omega_{{\textrm{f}}} $ and $C^{\, \dagger}(f_{1}) , \cdots C^{\, \dagger}(f_{n}) \Omega_{{\textrm{f}}} $, $j=1 , \cdots , n$, $n \in {\ensuremath{\mathbf{N}}}$. The creation and annihilation operators satisfy the canonical anti-commutation relations on ${\mathscr{F}_{\textrm{f}} }({\ensuremath{\mathscr{X}}})$ : $$\qquad \{ C(f) , \, C^{ \, \dagger}(f') \} = (f , f' ) , \quad \{ C^{ \, \dagger }(f) , C^{ \, \dagger }(f') \} = \{ C(f) , C (f') \} =0 , \quad f, f' \in {\ensuremath{\mathscr{X}}},$$ where $\{ X ,Y \} = XY +YX $. $\;$\ **(ii) Boson Fock Space**\ The boson Fock space over a complex Hilbert space ${\ensuremath{\mathscr{Y}}}$ is defined by $ {\mathscr{F}_{\textrm{b}} }( {\ensuremath{\mathscr{Y}}})= \oplus_{n=0}^\infty ( {\otimes^{n}_{\textrm{s}}}{\ensuremath{\mathscr{Y}}})$ where $ {\otimes^{n}_{\textrm{s}}}{\ensuremath{\mathscr{Y}}}$ denotes the $ n$-fold symmetric tensor product of ${\ensuremath{\mathscr{Y}}}$. The Fock vacuum is given by $\Omega_{{\textrm{b}}} = \{1, 0, 0, \cdots \} \in {\ensuremath{\mathscr{F}}}( {\ensuremath{\mathscr{Y}}})$ Let $T_{{\textrm{b}}}$ and $Q_{{\textrm{b}}}$ be linear operators on ${\ensuremath{\mathscr{Y}}}$. Then we define ${\ensuremath{d\Gamma_{\textrm{b}}({T_{{\textrm{b}}}}) }}= {\ensuremath{d\Gamma({T_{{\textrm{b}}} }) }}_{\upharpoonright {\mathscr{F}_{\textrm{b}} }( {\ensuremath{\mathscr{Y}}})}$ and $\Gamma_{{\textrm{b}}}( Q)= \Gamma (Q_{{\textrm{b}}})_{\upharpoonright {\mathscr{F}_{\textrm{b}} }( {\ensuremath{\mathscr{Y}}})}$. The number operator is defined by $N_{{\textrm{f}}}= {\ensuremath{d\Gamma_{\textrm{f}}({{{\small \text{1}}\hspace{-0.32em}1}}) }} $. The creation operator $A^{\dagger}(g)$, $g \in {\ensuremath{\mathscr{Y}}}$, is defined by $(A^{\dagger}(g) \Phi )^{(n)} = \sqrt{n} \, U_{\textrm{s}}^{ n}(f {\otimes}\Phi^{(n-1)})$, $n \geq 1$, and $(A^{\dagger}(g) \Phi )^{(0)} = 0 $ where $ U_{\textrm{s}}^{ n} $ is the projection from $ {\otimes^{n}}{\ensuremath{\mathscr{Y}}} $ to $ {\otimes^{n}_{\textrm{s}}}{\ensuremath{\mathscr{Y}}} $. The annihilation operator $A(f)$ is defined by $A(g)= ( A^{\dagger}(g) )^{\ast} $. The finite particle space $ {\ensuremath{\mathscr{F}}}_{{\textrm{b}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{N}}}) $ on the subspace ${\ensuremath{\mathscr{N}}} \subset {\ensuremath{\mathscr{Y}}}$ defined by the linear hull of $ \Omega_{{\textrm{f}}} $ and $A^{\, \dagger}(g_{1}) , \cdots A^{\, \dagger}(g_{n}) \Omega_{{\textrm{f}}} $, $j=1 , \cdots , n$, $n \in {\ensuremath{\mathbf{N}}}$. The creation and annihilation operators satisfy the canonical commutation relations on $ {\ensuremath{\mathscr{F}}}_{{\textrm{b}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{N}}}) $: $$\qquad [ A(g),A^{ \dagger }(g ') ] = (g , g' ) , \quad [ A^{ \dagger }(g) , A^{ \dagger }(g') ] = [ A(g) , A (g') ] =0 , \quad g, g' \in {\ensuremath{\mathscr{Y}}} ,$$ where $[X ,Y]=XY-YX $.\ Dirac field ----------- Let ${\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} = {\mathscr{F}_{\textrm{f}} }(L^2( {\mathbf{R}^{3} };{\ensuremath{\mathbf{C}}}^4))$. The energy Hamiltonian of the Dirac field is defined by $$H_{{ \textrm{D} }} = {\ensuremath{d\Gamma_{\textrm{f}}({\omega_{\,M} }) }}$$ where $\omega_M ({\ensuremath{\mathbf{p}}}) = \sqrt{| {\ensuremath{\mathbf{p}}}|^2 + M^2 }$, ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$, and $M>0$. Let $C^{\, \dagger } (^{t}(f_{1} , \cdots , f_{4}))$, $f_{l} \in L^2( {\mathbf{R}^{3} })$, $l=1, \cdots ,4$, be the creation operator on ${\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}}$. For each $f \in L^2( {\mathbf{R}^{3} })$, we set $$\begin{aligned} &b_{1/2}^{\dagger}(f) = C^{\, \dagger}({}^{t} (f,0,0,0) ), \quad \quad b_{-1/2}^{\dagger}(f) = C^{\, \dagger}({}^{t}(0,f,0,0) ), \\ &d_{1/2}^{\, \dagger}(f) = C^{\, \dagger}({}^{t}(0,0,f,0) ), \quad \quad d_{-1/2}^{\, \dagger}(f) = C^{\, \dagger}({}^{t}(0,0,0,f) ) . \end{aligned}$$ We define $b_s ( f)$ and $d_{s} ( g)$ by the conjugate of $b_s^{\dagger} ( f)$ and $d_{s}^{ \, \dagger } ( g)$, respectively. Then the canonical anti-commutation relations $$\begin{aligned} & \{ b_s (f) ,b_{s'}^{\dagger} (g ) \} = \{ d_s (f) ,d_{s'}^{\, \dagger } (g ) \}= \delta_{s, s'} (f,g ), \quad \label{ACR1} \\ & \{ b_s (f) ,b_{s'} (g) \} =\{ d_s (f) ,d_{s'} (g) \}= \{ b_s (g) ,d_{s'}^{\, \dagger} (g) \} = 0 , \quad \label{ACR2}\end{aligned}$$ are satisfied and it holds that $$\|b_s (f) \| = \|b_s^{\dagger }(f) \| = \| f \| , \qquad \|d_s(g) \| = \|d_s^{\, \dagger }(g) \|= \| g \| . \label{bdNorm}$$ Let $h_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}})={\ensuremath{\mathbf{\alpha}}} \cdot {\ensuremath{\mathbf{p}}}+ M\beta $ be the Fourier transformed Dirac operator with $4 \times 4$ Dirac matrices $\boldsymbol{\alpha} = (\alpha^j)_{j=1}^3$ and $\beta $. Let ${\ensuremath{\mathbf{S}}}({\ensuremath{\mathbf{p}}})= {\ensuremath{\mathbf{S}}} \cdot {\ensuremath{\mathbf{p}}}$, ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$, where ${\ensuremath{\mathbf{S}}}= -\frac{i}{4} \boldsymbol{\alpha} \wedge \boldsymbol{\alpha} $ is the spin angular momentum. The spinors $ u_{s}({\ensuremath{\mathbf{p}}} ) = (u_{s}^{l} ({\ensuremath{\mathbf{p}}}) )_{l=1}^{4} \; $ and $v_{s}({\ensuremath{\mathbf{p}}} ) = (v_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) )_{l=1}^{4} \; $ are function which satisfy the following : $$\begin{aligned} \textbf{(D.1)}\; \; \; &h_{D} ({\ensuremath{\mathbf{p}}}) u_{s} ({\ensuremath{\mathbf{p}}}) = E_{M} ({\ensuremath{\mathbf{p}}}) u_{s} ({\ensuremath{\mathbf{p}}}), \quad h_{D} ({\ensuremath{\mathbf{p}}}) v_{s} ({\ensuremath{\mathbf{p}}}) = -E_{M} ({\ensuremath{\mathbf{p}}}) v_{s} ({\ensuremath{\mathbf{p}}}), \\ \textbf{(D.2)} \; \; \; & S({\ensuremath{\mathbf{p}}} ) u_{s} ({\ensuremath{\mathbf{p}}}) = s | {\ensuremath{\mathbf{p}}} | u_{s} ({\ensuremath{\mathbf{p}}}), \quad S({\ensuremath{\mathbf{p}}} ) v_{s} ({\ensuremath{\mathbf{p}}}) = s | {\ensuremath{\mathbf{p}}} | v_{s} ({\ensuremath{\mathbf{p}}}) , \\ \textbf{(D.3)} \; \; \; & \sum_{l=1}^4 u_{s}^{\, l} ({\ensuremath{\mathbf{p}}} )^{\ast} u_{s'}^{\,l} ({\ensuremath{\mathbf{p}}}' ) = \sum_{l=1}^4 v_{s}^{\, l} ({\ensuremath{\mathbf{p}}} )^{\ast} v_{s'}^{\, l} ({\ensuremath{\mathbf{p}}}' ) =\delta_{s,s'} ,\quad \sum_{l=1}^4 u_{s}^{\, l} ({\ensuremath{\mathbf{p}}} )^{\ast} v_{s'}^{\, l} ({\ensuremath{\mathbf{p}}}' ) = 0 . \end{aligned}$$ \[exa1\] We review the example of spinors in the standard representation (see [@Tha] ; Section 1). The Pauli matrices are defined by $ \sigma_{1} = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) $, $ \sigma_{2} = \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right) $ and $ \sigma_{3} = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) $. Then, the Dirac marices are $ \alpha = \left( \begin{array}{cc} O & \sigma \\ \sigma & O \end{array} \right) $, $ \beta = \left( \begin{array}{cc} {{\small \text{1}}\hspace{-0.32em}1}& O \\ O & -{{\small \text{1}}\hspace{-0.32em}1}\end{array} \right) $, and the spin angular momentum is ${\ensuremath{\mathbf{S}}} = \frac{1}{2} \left( \begin{array}{cc} \ \sigma & O \\ O & \sigma \end{array} \right) $. Let $ O_{\textrm{SR}} = \{ {\ensuremath{\mathbf{p}}} = (p^{1} , p^{2}, p^{3} ) \in {\mathbf{R}^{3} }\; \left| \frac{}{} \right. |{\ensuremath{\mathbf{p}}}| -p^3 = 0 \} $. We see that the Lebesgue measure of $O_{\textrm{SR} }$ is zero. We set $$\eta_{+} ({\ensuremath{\mathbf{p}}} ) = \left\{ \begin{array}{c} \frac{1}{\sqrt{ 2 | {\ensuremath{\mathbf{p}}} | ( | {\ensuremath{\mathbf{p}}} | -p^{3} ) } } \begin{pmatrix} p^{1} - i p^{2} \\ | {\ensuremath{\mathbf{p}}} | - p^{3} \end{pmatrix} , \; {\ensuremath{\mathbf{p}}} \notin O_{\textrm{SR} } , \\ \qquad \qquad \begin{pmatrix} 1 \\ 0 \end{pmatrix} , \quad \quad \qquad {\ensuremath{\mathbf{p}}} \in O_{\textrm{SR} } , \end{array} \right. \; \; \eta_{-} ({\ensuremath{\mathbf{p}}} ) = \left\{ \begin{array}{c} \frac{1}{\sqrt{ 2 | {\ensuremath{\mathbf{p}}} | ( | {\ensuremath{\mathbf{p}}} | - p^{3} ) } } \begin{pmatrix} p^{3} - | {\ensuremath{\mathbf{p}}} | \\ p^{1} + i p^{2} \end{pmatrix} , \, {\ensuremath{\mathbf{p}}} \notin O_{\textrm{SR} } , \\ \qquad \quad \begin{pmatrix} 0 \\ 1 \end{pmatrix} , \quad \qquad \qquad {\ensuremath{\mathbf{p}}} \in O_{\textrm{SR} } . \end{array} \right.$$ Let $$u_{\pm1/2} ({\ensuremath{\mathbf{p}}}) = \begin{pmatrix} \lambda_{+} ({\ensuremath{\mathbf{p}}} ) \eta_{\pm} ({\ensuremath{\mathbf{p}}}) \\ \pm \lambda_{-} ({\ensuremath{\mathbf{p}}} ) \eta_{\pm} ({\ensuremath{\mathbf{p}}}) \end{pmatrix} , \qquad v_{\pm 1/2} ({\ensuremath{\mathbf{p}}}) = \begin{pmatrix} \mp \lambda_{-} ({\ensuremath{\mathbf{p}}} ) \phi_{\pm} ({\ensuremath{\mathbf{p}}}) \\ \pm \lambda_{+} ({\ensuremath{\mathbf{p}}} ) \eta_{\pm} ({\ensuremath{\mathbf{p}}}) \end{pmatrix} ,$$ with $ \lambda_{\pm} ({\ensuremath{\mathbf{p}}}) = \frac{1}{\sqrt{2}} \sqrt{ 1 \pm M \, E_{M} ({\ensuremath{\mathbf{p}}})^{-1 }}$. Here note that $ u_{s} $ and $ v_{s} $ satisfy $ u_{s} , v_s \in \oplus^4 ( C^{1} ( {\mathbf{R}^{3} }\backslash O_{\textrm{SR}} ) ) $.\ $\;$ $\;$\ The Dirac field operator $ \psi({\ensuremath{\mathbf{x}}}) = {}^{t} (\psi_{1}({\ensuremath{\mathbf{x}}}) , \cdots , \psi_{4}({\ensuremath{\mathbf{x}}}))$ is defined by $$\qquad \qquad \quad \psi_{l}({\ensuremath{\mathbf{x}}}) = \sum_{s=\pm 1/2} \left( b_s ( f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ) + d^{\, \dagger}_s ( g_{s , {\ensuremath{\mathbf{x}}}}^l) \frac{}{} \right) , \quad \qquad l=1, \cdots , 4 ,$$ where $f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}})= f_{s }^{\, l} ({\ensuremath{\mathbf{p}}})e^{-{\ensuremath{\mathbf{p}}}\cdot {\ensuremath{\mathbf{x}}} }$ with $f_{s }^{\, l} ({\ensuremath{\mathbf{p}}})= \frac{1}{\sqrt{ (2 \pi )^3 }} \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) u_s^{l}({\ensuremath{\mathbf{p}}})$ and $g_{s , {\ensuremath{\mathbf{x}}}}^l ({\ensuremath{\mathbf{p}}})= g_{s }^l ({\ensuremath{\mathbf{p}}})e^{-{\ensuremath{\mathbf{p}}}\cdot {\ensuremath{\mathbf{x}}} }$ with $g_{s }^l ({\ensuremath{\mathbf{p}}})= \frac{1}{\sqrt{ (2 \pi )^3 }}\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) \tilde{v}_s^{l}({\ensuremath{\mathbf{p}}})$ and $\tilde{v}_s^{\, l}({\ensuremath{\mathbf{p}}})= v_{s}^{\, l} (-{\ensuremath{\mathbf{p}}})$. Here $\chi_{{ \textrm{D} }} $ satisfy the following condition.\ > **(A.1 ; Ultraviolet Cutoff for Dirac Field)** $$\int_{{\mathbf{R}^{3} }} | \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) |^2 d {\ensuremath{\mathbf{p}}} > < \infty .$$ Then it holds that $$\qquad \|\psi_l ({\ensuremath{\mathbf{x}}}) \| \leq c_{\, { \textrm{D} }}^{ \, l } , \label{psiBound}$$ where $ c_{\, { \textrm{D} }}^{\, l } = \frac{1}{\sqrt{(2\pi)^3}} \sum\limits_{s= \pm 1/2} \left( \| \chi_{{ \textrm{D} }} \, u_{s}^{\,l} \| + \| \chi_{{ \textrm{D} }} \, \tilde{v}_{s}^{\, l} \| \right)$, $l=1 , \cdots , 4.$\ Radiation Field in the Coulomb Gauge ------------------------------------ Let ${\mathscr{F}_{\textrm{rad}}}= {\mathscr{F}_{\textrm{b}} }( \oplus_{r= 1,2} L^2( {\mathbf{R}^{3} }))$. The free Hamiltonian is defined by $${H_{\textrm{rad}}}= {\ensuremath{d\Gamma_{\textrm{b}}({\omega }) }}$$ where $\omega ({\ensuremath{\mathbf{k}}}) = |{\ensuremath{\mathbf{k}}}|$, ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }$. Let $A^{\ast }(h_1 , h_2 )$, $h_{r} \in L^2( {\mathbf{R}^{3} })$, $r=1,2$, be the creation operators on ${\mathscr{F}_{\textrm{rad}}}$. Let $$\qquad a^{\dagger }_{1} (h ) = A (( h, 0)) , \quad a^{\dagger }_{2} (h )= A (( 0, h )) , \qquad h \in L^2( {\mathbf{R}^{3} }) ,$$ and $a_{r}(h')=(a^{\dagger} (h'))^{\ast}$, $h' \in L^2( {\mathbf{R}^{3} }) $, $r=1,2$. The creation operators and annihilation operators satisfy the canonical commutation relations $$\begin{aligned} & [a_r (h) ,a_{r'}^{\dagger} (h') ] = \delta_{r, r'} (h, \, h' ), \; \; \label{radCCR1} \\ & [a_r (h) ,a_{r'} ( h' ) ] = [a_r^{\dagger } (h' ) ,a_{r'}^{\dagger} (h') ] = 0 , \label{radCCR2}\end{aligned}$$ on ${\mathscr{F}_{\textrm{rad}}}^{\, {\textrm{fin}}}({\ensuremath{\mathscr{M}}})$ where ${\ensuremath{\mathscr{M}}} $ is a subspace of $\oplus_{r= 1,2} L^2( {\mathbf{R}^{3} }) $. For all $h \in {\ensuremath{\mathscr{D}}}(\omega^{-1/2}) $, it follows that $$\quad \| a_r (h ) ( {H_{\textrm{rad}}}+1 )^{-1/2} \| \leq \| \frac{h}{\sqrt{\omega }} \| , \quad \| a^{\dagger }_r (h ) ( {H_{\textrm{rad}}}+1 )^{-1/2} \| \leq \| \frac{h}{\sqrt{\omega }} \| + \| h \| . \label{radfiedBound}$$ The polarization vectors ${\ensuremath{\mathbf{e}}}_{r} ({\ensuremath{\mathbf{k}}}) = (e_r^{j} ({\ensuremath{\mathbf{k}}}))$, $r=1,2$, satisfy the following relations. $$\textbf{(R.1)} \; \qquad {\ensuremath{\mathbf{e}}}_{r} ({\ensuremath{\mathbf{k}}}) \cdot {\ensuremath{\mathbf{e}}}_{r'} ({\ensuremath{\mathbf{k}}}) =0 , \quad {\ensuremath{\mathbf{e}}}_{r} ({\ensuremath{\mathbf{k}}})\cdot {\ensuremath{\mathbf{k}}} = 0 , \quad \quad {\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }\backslash \{ {\ensuremath{\mathbf{0}}} \} . \notag$$ $\;$ \[exa2\] We check the example of the polarization vectors. For all ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }\backslash \{ {\ensuremath{\mathbf{0}}} \} $, we set $$\qquad {\ensuremath{\mathbf{e}}}_1({\ensuremath{\mathbf{k}}}) = \frac{1}{\sqrt{(k^1)^2 + (k^2) ^2}} \left( \begin{array}{c} - k^{2} \\ k^{1} \\ 0 \end{array} \right) , \; \; \; {\ensuremath{\mathbf{e}}}_2({\ensuremath{\mathbf{k}}}) =\frac{1}{|{\ensuremath{\mathbf{k}}}|\sqrt{k_1^2 +k_2^2}} \left( \begin{array}{c} k^{1} k^3 \\ k^{2} k^3 \\ - (k^1)^2 - (k^2)^2 \end{array} \right) . \quad$$ Then **(R.1)** is satisfied. Here it is noted that ${\ensuremath{\mathbf{e}}}_{r} \in \oplus^3 \, ( C^{1}({\mathbf{R}^{3} }\backslash \{ {\ensuremath{\mathbf{0}}} \} ))$, $r=1,2$. $\;$\ The radiation field operator ${\ensuremath{\mathbf{A}}}({\ensuremath{\mathbf{x}}})= ( A_j ({\ensuremath{\mathbf{x}}}))_{j=1}^3$ is defined by $$A_j({\ensuremath{\mathbf{x}}}) = \sum_{r=1,2} \left( a_r (h_{r , {\ensuremath{\mathbf{x}}}}^j) + a^{\dagger}_r (h_{r , {\ensuremath{\mathbf{x}}}}^j) \frac{}{} \right)$$ where $h_{r , {\ensuremath{\mathbf{x}}}}^j ({\ensuremath{\mathbf{k}}})= h_{r }^j ({\ensuremath{\mathbf{k}}})e^{-{\ensuremath{\mathbf{k}}}\cdot {\ensuremath{\mathbf{x}}} }$ with $h_{r }^j ({\ensuremath{\mathbf{k}}})= \frac{1}{\sqrt{(2 \pi )^3}} \frac{\chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) e_r^{j}({\ensuremath{\mathbf{k}}})}{\sqrt{2 \omega({\ensuremath{\mathbf{k}}})}}$, and $\chi_{{\textrm{rad}}}$ satisfy the following condition.\ > **(A.2 : Ultraviolet Cutoff for Radiation Field)** $$\qquad > \int_{{\mathbf{R}^{3} }} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |^2}{ \omega({\ensuremath{\mathbf{k}}})^{l}} {d \mathbf{k} }< \infty , \quad l= 1,2 .$$ Then $$\|A_j ({\ensuremath{\mathbf{x}}}) ( {H_{\textrm{rad}}}+1 )^{-1/2} \| \leq c^{\, j}_{{\textrm{rad}}}$$ where $c^{\,j}_{{\textrm{rad}}}= \frac{1}{\sqrt{(2\pi )^3}}\sum\limits_{r=1,2}\left( \sqrt{2} \| \frac{\chi_{{\textrm{rad}}} e_r^j }{ \omega} \| + \| \frac{\chi_{{\textrm{rad}}} e_r^j }{\sqrt{2 \omega}} \| \right) $.\ Total Hamiltonian and Main Theorem ---------------------------------- We define the system of the Dirac field interacting with the radiation field. The Hilbert space for the system is defined by ${\mathscr{F}_{\textrm{QED}}}= {\mathscr{F}_{\textrm{Dirac}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$. The free Hamiltonian is defined by $$H_0= H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}}}\notag$$ on the domain ${\ensuremath{\mathscr{D}}}(H_{0}) = {\ensuremath{\mathscr{D}}}( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \cap {\ensuremath{\mathscr{D}}}({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}}})$. To define the interactions, we introduce spatial cutoff $\chi_{{\textrm{I}}}$ and $\chi_{{\textrm{II}}}$, which satisfy the condition below.\ > **(A.3 : Spatial Cutoff )** $$\int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| {d \mathbf{x} }< \infty , \qquad \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} > \frac{|{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }|}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|} {d \mathbf{x} }{d \mathbf{y} }< \infty .$$ First we define a functional on $ {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \times {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} $ by $$\qquad \qquad \ell_{{\textrm{I}}}(\Phi , \Psi ) = \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left( \Phi , ({\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}}) }{\otimes}A_j ({\ensuremath{\mathbf{x}}}) ) \Psi \right) , \; \; \Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} , \, \Psi \in {\ensuremath{\mathscr{D}}}(H_0 ) ,$$ where $ {\psi^{\dagger} (\mathbf{x}) }= ( \psi_{1}({\ensuremath{\mathbf{x}}})^{\ast} , \cdots , \psi_{4}({\ensuremath{\mathbf{x}}})^{\ast} ) $. We see that $$| \ell_{{\textrm{I}}}(\Phi , \Psi ) | \leq \left( \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| {d \mathbf{x} }\right) \, \sum\limits_{j=1}^3 \, \sum_{l,l'=1}^4 |\alpha_{l,l'}^j | c_{\, { \textrm{D} }}^{\, l } c_{\, { \textrm{D} }}^{ \, l' } c_{{\textrm{rad}}}^{\, j } \, \| \Phi \| \, \| {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}({H_{\textrm{rad}}}+1)^{1/2} \Psi \| . \notag$$ By the Riesz representation theorem, we can define the operator ${H_{\textrm{I}}}$ which satisfy $ (\Phi , {H_{\textrm{I}}}\Psi ) = \ell_{{\textrm{I}}}(\Phi , \Psi ) $ and $$\| {H_{\textrm{I}}}\Psi \| \leq c_{\, {\textrm{I}}} \, \| {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}({H_{\textrm{rad}}}+1)^{1/2} \Psi \| , \label{HIbound}$$ where $ c_{\, {\textrm{I}}} = \| \chi_{{\textrm{I}}} \|_{L^1} \sum\limits_{j=1}^3 \, \sum\limits_{l,l'=1}^4 |\alpha_{l,l'}^j | c_{\, { \textrm{D} }}^{\, l } c_{\, { \textrm{D} }}^{ \, l' } c_{{\textrm{rad}}}^{\, j }$. By the spectral decomposition theorem, it is proven that for all $\epsilon > 0$, $$\|{H_{\textrm{I}}}\Psi \| \leq c_{{\textrm{I}}} \epsilon \|H_{0} \Psi \| +c_{\textrm{I}}\left( \frac{1 }{2 \epsilon} +1 \right) \| \Psi \| . \label{HIbound'}$$ Next we define a functional on $ {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} $ by $$\ell_{{\textrm{II}}}(\Phi , \Psi ) = \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|} \left( \Phi , \left( {\psi^{\dagger} (\mathbf{x}) }{ \psi ({\ensuremath{\mathbf{x}}}) }{\psi^{\dagger} (\mathbf{y}) }{ \psi ({\ensuremath{\mathbf{y}}}) }{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} \right) \Psi \right) d {\ensuremath{\mathbf{x}}} \, d {\ensuremath{\mathbf{y}}} , \; \; \Phi , \, \Psi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} . \\$$ We see that $$|\ell_{{\textrm{II}}}(\Phi , \Psi ) | \leq \left( \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \left| \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{ |{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \right| {d \mathbf{x} }{d \mathbf{y} }\right) \, \sum_{l , l' =1}^4 (c_{\, { \textrm{D} }}^{ \, l} c_{\, { \textrm{D} }}^{\, l'})^2 \| \Phi \| \, \| \Psi \| . \notag$$ Then, by the Riesz representation theorem, we can define an operator ${H_{\textrm{II}}}$ satisfying $ (\Phi , {H_{\textrm{II}}}\Psi ) = \ell_{{\textrm{II}}}(\Phi , \Psi )$ and $$\|{H_{\textrm{II}}}\| \leq c_{{\textrm{II}}} , \label{HIIbound}$$ where $ c_{{\textrm{II}}} = \left\| \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{ |{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \right\|_{L^1} \, \sum\limits_{l , l' =1}^4 (c_{\, { \textrm{D} }}^{ \, l} c_{\, { \textrm{D} }}^{\, l'})^2$. By (\[HIbound’\]) and (\[HIIbound\]), it holds that $$\| ({\kappa_{\textrm{I}}}{H_{\textrm{I}}}+ {\kappa_{\textrm{II}}}{H_{\textrm{II}}}) \Psi \| \leq c_{{\textrm{I}}} {\kappa_{\textrm{I}}}\epsilon \| H_0 \Psi \| + \left( c_{{\textrm{I}}} {\kappa_{\textrm{I}}}\left( \frac{1 }{2 \epsilon} +1 \right) + c_{{\textrm{II}}} {\kappa_{\textrm{II}}}\right) \| \Psi \| . \notag$$ Then the Kato-Rellich theorem yields that that ${H_{\textrm{QED}}}$ is self-adjoint on ${\ensuremath{\mathscr{D}}}(H_{0})$ and essentially self-djoint on any core of $H_{0}$. Hence, in particular, ${H_{\textrm{QED}}}$ is essentially self-adjoin on $${\ensuremath{\mathscr{D}}}_{0} = {\mathscr{F}_{\textrm{Dir}}}^{\, {\textrm{fin}}}{{\ensuremath{\mathscr{D}}}(\omega_{\, M})} \hat{{\otimes}}{\mathscr{F}_{\textrm{rad}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{D}}}(\omega ))$$ where $\hat{{\otimes}}$ denotes the algebraic tensor product. $\; $\ To prove the existence of the ground state of ${H_{\textrm{QED}}}$, we suppose additional conditions below. > **(A.4 : Spatial Localization)** $$\int_{{\mathbf{R}^{3} }} |{\ensuremath{\mathbf{x}}}|| {\chi_{\textrm{I}} (\mathbf{x}) }| {d \mathbf{x} }< \infty , \quad \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} > \frac{|{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }|}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|}|{\ensuremath{\mathbf{x}}}| {d \mathbf{x} }{d \mathbf{y} }< \infty .$$ > **(A.5 : Momentum Regularity Condition for Dirac Field)**\ > There exists a subset $O_{{ \textrm{D} }} \subset {\mathbf{R}^{3} }$ with Lebesgue measure zero such that $u_{s}, v_{s} \in \oplus^4 \, ( C^1 ({\mathbf{R}^{3} }\backslash O_{{ \textrm{D} }})) $, $s= \pm 1/2$. $\chi_{{ \textrm{D} }} \in C^{1}({\mathbf{R}^{3} }) $, and it satisfies that $$\int_{{\mathbf{R}^{3} }} | \partial_{p^{\nu}}\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) |^2 d {\ensuremath{\mathbf{p}}} > < \infty , > \; \int_{{\mathbf{R}^{3} }} | \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p }}}) \partial_{p^{\nu }} u_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) |^2 d {\ensuremath{\mathbf{p}}} > < \infty , \; \int_{{\mathbf{R}^{3} }} | \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p }}}) \partial_{p^{\nu }} v_{s}^{\, l} (-{\ensuremath{\mathbf{p}}}) |^2 d {\ensuremath{\mathbf{p}}} > < \infty ,$$ for all $\nu =1, 2,3$, $ l = 1 , \cdots ,4$, $ s= \pm 1/2 $. > **(A.6 : Momentum Regularity Condition for Radiation Field )**\ > There exists a subset $O_{{\textrm{rad}}} \subset {\mathbf{R}^{3} }$ with Lebesgue measure zero such that ${\ensuremath{\mathbf{e}}}_{r} \in \oplus^3 \, ( C^{1}({\mathbf{R}^{3} }\backslash O_{{\textrm{rad}}} ))$, $r=1,2$, where $O_{{\textrm{rad}}} $. $\chi_{{\textrm{rad}}} \in C^{1}({\mathbf{R}^{3} })$ and it satisfies that $$\int_{{\mathbf{R}^{3} }} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |^2}{ | {\ensuremath{\mathbf{k}}}|^5} {d \mathbf{k} }< \infty , \quad > \int_{{\mathbf{R}^{3} }} \frac{ | \partial_{k^\nu }\chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |^2}{ | {\ensuremath{\mathbf{k}}} |^3} {d \mathbf{k} }< \infty , \quad \int_{{\mathbf{R}^{3} }} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) \partial_{k^\nu } e_{r}^{j} ({\ensuremath{\mathbf{k}}}) |^2}{ |{\ensuremath{\mathbf{k}}}|^3} {d \mathbf{k} }< \infty ,$$ for all $\nu =1, 2, 3$, $j=1, 2, 3$, $r=1,2$. \[exa3\] Examples of $O_{{ \textrm{D} }}$ and $O_{{\textrm{rad}}}$ in **(A.5)** and **(A.6)** are as follows. In the case of the standard representation, $O_{{ \textrm{D} }} = O_{\textrm{SR}}$ where $ O_{\textrm{SR}} $ is defined in Remark \[exa1\]. For the polarization vectors considered in Remark \[exa2\], $O_{{\textrm{rad}}} = \{ {\ensuremath{\mathbf{0}}} \}$. $\; $\ The main theorem in this paper is as follows. \[Main-Theorem\] **(Existence of a Ground State)**\ Suppose **(A.1)** - **(A.6)**. Then $ {H_{\textrm{QED}}}$ has a ground state for all values of coupling constants. In particular, its multiplicity is finite. Ground States of Massive case ============================= In this section, we consider a massive Hamiltonian defined by $$H_m = H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}, \, m}}+ {\kappa_{\textrm{I}}}{H_{\textrm{I}}}+ {\kappa_{\textrm{II}}}{H_{\textrm{II}}}, \notag$$ where ${H_{\textrm{rad}, \, m}}= {\ensuremath{d\Gamma_{\textrm{b}}({\omega_{m}}) }}$ with $\omega_m ({\ensuremath{\mathbf{k}}})=\sqrt{{\ensuremath{\mathbf{k}}}^2 + m^2} $, $m>0$.\ Fock Spaces on Direct Sum of Hilbert Spaces ------------------------------------------- $\, $ We review basic properties of Fock spaces on direct sum of Hilbert spaces. These are useful for constructing partitions of unity on Fock spaces (see, Dereziński-Gérard [@DeGe99]). $\, $\ **(i) Full Fock Space on** ${\ensuremath{\mathscr{Z}}} \oplus {\ensuremath{\mathscr{Z}}}$\ Let $ Z = \left[ \begin{array}{c} Z_0 \\ Z_\infty \end{array} \right] $, $ Z_0 , Z_\infty \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Z}}}) $, where ${\ensuremath{\mathscr{Z}}}$ is a complex Hilbert space. We consider $ Z = \left[ \begin{array}{c} Z_0 \\ Z_\infty \end{array} \right] $ is an operator $ {\ensuremath{\mathscr{Z}}} \to {\ensuremath{\mathscr{Z}}} \oplus {\ensuremath{\mathscr{Z}}}$ which acts for $$\qquad \qquad Z h = \left[ \begin{array}{c} Z_0 \, h \\ Z_\infty \, h \end{array} \right] , \quad \; h \in {\ensuremath{\mathscr{D}}} (Z_{0}) \cap {\ensuremath{\mathscr{D}}} (Z_{\infty}) .$$ Let $J = \left[ \begin{array}{c} J_0 \, \\ J_\infty \, \end{array} \right]$, $ J_0 , J_\infty \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Z}}}) $ and $ B = \left[ \begin{array}{c} B_0 \, \\ B_\infty \, \end{array} \right] $, $ B_0 , B_\infty \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Z}}}) $. We define $d \Gamma ( J , B) : {\ensuremath{\mathscr{F}}}({\ensuremath{\mathscr{Z}}}) \to {\ensuremath{\mathscr{F}}} ({\ensuremath{\mathscr{Z}}}) \oplus {\ensuremath{\mathscr{F}}}({\ensuremath{\mathscr{Z}}})$ by $$d \Gamma (J , B ) = \oplus_{n=0}^{\infty} \left( \sum\limits_{j=1}^n ( {\otimes}^{j-1} J ) {\otimes}B {\otimes}( {\otimes}^{n-j} J ) \right) .$$ If $B_{0}$ and $B_{\infty}$ are bounded, and $J_{0}^{\ast}J_0 + J_{\infty}^{\ast}J_{\infty} \leq 1 $, it holds that $$\| d \Gamma ( J, B ) (N +1)^{-1} \| \leq \sqrt{ \|B_{0} \|^2 + \| B_{\infty} \|^2 } . \label{3.1.1}$$ Let $T \in {\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{Z}}})$. Then it holds that $$\Gamma (J ) {\ensuremath{d\Gamma({T}) }} = d \Gamma \left( \left[ \begin{array}{cc} T & 0 \\ 0 & T \end{array} \right] \right) \Gamma ( J ) + d \Gamma ( J ,\tilde{\text{ad}}_{T}( J ) ) , \label{3.1.2}$$ where $ \tilde{\text{ad}}_{T}(J) : {\ensuremath{\mathscr{Z}}} \to {\ensuremath{\mathscr{Z}}} \oplus {\ensuremath{\mathscr{Z}}}$ is defined by $$\qquad \qquad \quad \tilde{\text{ad}}_{T}(J) h=\left[ \begin{array}{c} [T ,J_0 ] h \\ {[} T , J_\infty {]} h \end{array} \right] , \quad h \in {\ensuremath{\mathscr{D}}}([T ,J_0 ]) \cap {\ensuremath{\mathscr{D}}}([T ,J_\infty ] ) .$$ $\;$\ **(ii) Fermion Fock Space on** ${\ensuremath{\mathscr{X}}} \oplus {\ensuremath{\mathscr{X}}}$\ Let ${\ensuremath{\mathscr{X}}}$ be a complex Hilbert space. Let $ J_{{\textrm{f}}} = \left[ \begin{array}{c} J_{{\textrm{f}}}^{\, 0 } \, \\ J_{{\textrm{f}}}^{\, \infty} \, \end{array} \right] $, $ J_{{\textrm{f}}}^{\, 0 } , J_{{\textrm{f}}}^{\, \infty} \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{X}}}) $ and $ B_{{\textrm{f}}} = \left[ \begin{array}{c} B_{{\textrm{f}}}^{\, 0 } \, \\ B_{{\textrm{f}}}^{\, \infty}\, \end{array} \right] $, $ B_{{\textrm{f}}}^{\, 0 }, B^{\, \infty}_{{\textrm{f}}} \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{X}}}) $. We set ${\ensuremath{d\Gamma_{\textrm{f}}({ J_{{\textrm{f}}} , B_{{\textrm{f}}} }) }} = {\ensuremath{d\Gamma({ J_{{\textrm{f}}} , B_{{\textrm{f}}} }) }}_{{\upharpoonright}{\mathscr{F}_{\textrm{f}} }({\ensuremath{\mathscr{X}}})} $. Suppose that $B_{{\textrm{f}}}^{\, 0 } $ and $B^{\, \infty}_{{\textrm{f}}} $ are bounded, and $ (J_{{\textrm{f}}}^{\, 0 })^\ast J_{{\textrm{f}}}^{\, 0 } + (J_{{\textrm{f}}}^{\, \infty})^{\ast} J_{{\textrm{f}}}^{\, \infty} \leq 1 $. By (\[3.1.1\]), it holds that $$\| d {\Gamma_{\textrm{f}}}( J_{{\textrm{f}}} , B_{{\textrm{f}}} ) (N_{{\textrm{f}}} +1)^{-1} \| \leq \sqrt{ \| B_{{\textrm{f}}}^{\, 0 } \|^2 + \| B_{{\textrm{f}}}^{\, \infty} \|^2 } . \label{3.1.3}$$ Let $T_{{\textrm{f}}} \in {\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{X}}})$. From (\[3.1.2\]), it holds that $${\Gamma_{\textrm{f}}}( J_{{\textrm{f}}} ) {\ensuremath{d\Gamma_{\textrm{f}}({T_{{\textrm{f}}} }) }} = d {\Gamma_{\textrm{f}}}\left( \left[ \begin{array}{cc} T_{{\textrm{f}}} & 0 \\ 0 & T_{{\textrm{f}}} \end{array} \right] \right) {\Gamma_{\textrm{f}}}( J_{{\textrm{f}}} ) + d {\Gamma_{\textrm{f}}}( J_{{\textrm{f}}} ,\tilde{\text{ad}}_{T_{{\textrm{f}}} }( J_{{\textrm{f}}} ) ). \label{3.1.4}$$ Let $ C(f)$ and $C^{ \, \dagger} (f)$, $f \in {\ensuremath{\mathscr{X}}}$, be the annihilation and creation operators on ${\mathscr{F}_{\textrm{f}} }({\ensuremath{\mathscr{X}}})$, respectively. Then it follows that $$\begin{aligned} & \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) C (f) = C \left( \left[ \begin{array}{c} f \\ 0 \end{array} \right] \right) \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) + \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) \, C \left( (1-( J_{{\textrm{f}}}^{\, 0 })^{\ast})f \frac{}{}\right) , \label{3.1.5} \\ & \Gamma ( J_{{\textrm{f}}} ) C^{\, \dagger} (f) = C^{\, \dagger} \left( \left[ \begin{array}{c} f \\ 0 \end{array} \right] \right) \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) + C^{\, \dagger} \left( \left[ \begin{array}{c} J_{{\textrm{f}}}^{\, 0 } -1 \\ J_{{\textrm{f}}}^{\, \infty} \end{array} \right] f \right) \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) . \label{3.1.6}\end{aligned}$$ $\;$\ $\; $\ **(iii) Boson Fock Space on** ${\ensuremath{\mathscr{Y}}} \oplus {\ensuremath{\mathscr{Y}}}$\ Let ${\ensuremath{\mathscr{Y}}}$ be a complex Hilbert space. Let $ J_{{\textrm{b}}} = \left[ \begin{array}{c} J_{{\textrm{b}}}^{\, 0 } \, \\ J_{{\textrm{b}}}^{\, \infty} \, \end{array} \right] $, $ J_{{\textrm{b}}}^{\, 0 } , J_{{\textrm{b}}}^{\, \infty} \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Y}}}) $ and $ B_{{\textrm{b}}} = \left[ \begin{array}{c} B_{{\textrm{b}}}^{\, 0 } \, \\ B_{{\textrm{b}}}^{\, \infty} \, \end{array} \right] $, $ B_{{\textrm{b}}}^{\, 0 }, B_{{\textrm{b}}}^{\, \infty} \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Y}}}) $. We define ${\ensuremath{d\Gamma_{\textrm{b}}({ J_{{\textrm{b}}} , B_{{\textrm{b}}} }) }} = {\ensuremath{d\Gamma({ J_{{\textrm{b}}} , B_{{\textrm{b}}} }) }}_{{\upharpoonright}{\mathscr{F}_{\textrm{b}} }({\ensuremath{\mathscr{Y}}})} $. Assume that $B_{{\textrm{b}}}^{\, 0 } $ and $ B_{{\textrm{b}}}^{\, \infty} $ are bounded, and $ (J_{{\textrm{b}}}^{\, 0 })^\ast J_{{\textrm{b}}}^{\, 0 } + (J_{{\textrm{b}}}^{\, \infty})^{\ast} J_{{\textrm{b}}}^{\, \infty} \leq 1 $. By (\[3.1.1\]), it follows that $$\| d {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} , B_{{\textrm{b}}} ) (N_{{\textrm{b}}} +1)^{-1} \| \leq \sqrt{ \| B_{{\textrm{b}}}^{\, 0 } \|^2 + \| B_{{\textrm{b}}}^{\, \infty} \|^2 } . \label{3.1.7}$$ Let $ T_{{\textrm{b}}} \in {\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{Y}}})$. Then (\[3.1.2\]) yields that $${\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) {\ensuremath{d\Gamma_{\textrm{b}}({T_{{\textrm{b}}} }) }} = d {\Gamma_{\textrm{b}}}\left( \left[ \begin{array}{cc} T_{{\textrm{b}}} & 0 \\ 0 & T_{{\textrm{b}}} \end{array} \right] \right) {\Gamma_{\textrm{b}}}( J_{{\textrm{f}}} ) + d {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ,\tilde{\text{ad}}_{T_{{\textrm{b}}} }( J_{{\textrm{b}}} ) ). \label{3.1.8}$$ Let $ A(g)$ and $A^{\dagger} (g)$, $g \in {\ensuremath{\mathscr{Y}}}$, be the annihilation and creation operators on ${\mathscr{F}_{\textrm{b}} }({\ensuremath{\mathscr{Y}}})$, respectively. Then it follows that $$\begin{aligned} & \Gamma_{{\textrm{b}}} ( J_{{\textrm{b}}} ) A (g) = A \left( \left[ \begin{array}{c} g \\ 0 \end{array} \right] \right) {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) + {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) \, A \left( (1-( J_{{\textrm{b}}}^{\, 0 } )^{\ast}) g \frac{}{}\right) , \label{3.1.9} \\ & {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) A^{\dagger } (g) = A^{\dagger} \left( \left[ \begin{array}{c} g \\ 0 \end{array} \right] \right) \Gamma_{{\textrm{b}}} ( J_{{\textrm{b}}} ) + A^{\dagger } \left( \left[ \begin{array}{c} J_{{\textrm{b}}}^{\, 0 } -1 \\ J_{{\textrm{b}}}^{\, \infty } \end{array} \right] g \right) {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) . \label{3.1.10}\end{aligned}$$ Partition of Unity for the Dirac Field -------------------------------------- We construct a partition of unity for the Dirac field. For general properties of partition of unity for fermionic fields, refer to Ammari [@Am04]. $\;$\ Let $$\quad c_{ \tau , s} (f) = \left\{ \begin{array}{c} b_{s}(f), \; \; \tau = + , \\ d_{s}(f) , \; \;\tau = - . \end{array} \right.$$ Let $U_{{\textrm{f}}}: {\mathscr{F}_{\textrm{f}} }\left( L^2( {\mathbf{R}^{3} }_{\bf{p}} ; {\ensuremath{\mathbf{C}}}^4 ) \oplus L^2( {\mathbf{R}^{3} }_{\bf{p}} ; {\ensuremath{\mathbf{C}}}^4 ) \right) \to {\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}} {\otimes}{\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}} $ be an isometric operator which satisfy $U_{{\textrm{f}}} \, \Omega_{{ \textrm{D} }} =\Omega_{{ \textrm{D} }} {\otimes}\Omega_{{ \textrm{D} }} $ and $$\begin{aligned} &U_{{\textrm{f}}} \; c^{\dagger }_{\tau_1, s_1} \left(\left[ \begin{array}{c} f_1 \\ g_1 \end{array} \right]\right) \cdots c^{\dagger }_{\tau_1 , s_1} \left(\left[ \begin{array}{c} f_1 \\ g_n \end{array} \right]\right) \Omega_{{ \textrm{D} }} \notag \\ & = \left( c^{\dagger}_{\tau_1 , s_1} (f_1) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ (-1)^{N_{{ \textrm{D} }}} {\otimes}c^{\dagger}_{\tau_1 , s_1}(g_1) \frac{}{}\right) \cdots \left( c^{\dagger }_{\tau_n, s_n} (f_n) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ (-1)^{N_{{ \textrm{D} }}} {\otimes}c^{\dagger}_{\tau_n , r_n}(g_n) \frac{}{} \right) \Omega_{{ \textrm{D} }} {\otimes}\Omega_{{ \textrm{D} }} . \notag\end{aligned}$$ Here note that $(-1)^{N_{{ \textrm{D} }}}\Psi = (-1)^n \Psi $ for the vector of the form $ \Psi = c^\dagger_{\tau_1 , s_1}(f_1) \cdots c^\dagger_{\tau_n, s_n} (f_n) \Omega_{{ \textrm{D} }} $, $f_{j} \in L^2 ({\mathbf{R}^{3} }) $, $j= 1, \cdots ,n$, $n \in {\ensuremath{\mathbf{N}}}$. Let $j_{0} , j_{\infty} \in C^{\, \infty} ({\ensuremath{\mathbf{R}}})$. We assume that $j_0 \geq 0 $, $j_{\infty} \geq 0$, $j_{0} ({\ensuremath{\mathbf{x}}})^2 + j_{\infty} ({\ensuremath{\mathbf{x}}})^2 =1$, $j_{0}({\ensuremath{\mathbf{x}}})=1 $ for $|{\ensuremath{\mathbf{x}}}| \leq 1 $ and $j_{0}({\ensuremath{\mathbf{x}}})=0 $ for $|{\ensuremath{\mathbf{x}}}| \geq 2 $. Let $j_{{\textrm{f}}, R } = \left[ \begin{array}{c} j_{{\textrm{f}}, R }^{\, 0} \\ j_{{\textrm{f}}, R }^{\, \infty} \end{array} \right]$ where $ j_{{\textrm{f}}, R }^{\, 0}= j_{0} (\frac{-i \, {\ensuremath{\mathbf{\nabla_{{\ensuremath{\mathbf{p}}}}}}}}{R})$ and $ j_{{\textrm{f}}, R }^{\, \infty}= j_{\infty} (\frac{-i \, {\ensuremath{\mathbf{ \nabla_{{\ensuremath{\mathbf{p}}}}}}}}{R}) $ with $\nabla_{{\ensuremath{\mathbf{p}}}}= (\partial_{p^1}, \partial_{p^2},\partial_{p^3} ) $.\ $\;$\ Let $X_{{\textrm{f}}, R} : {\mathscr{F}_{\textrm{Dir}}}\to {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} $ defined by $$\quad X_{{\textrm{f}}, R} = \, U_{{\textrm{f}}} \, \Gamma_{{\textrm{f}}}(j_{{\textrm{f}}, R}) . \notag$$ From (\[3.1.4\])-(\[3.1.6\]), it holds that $$\begin{aligned} & X_{{\textrm{f}}, R} \, H_{{ \textrm{D} }} = \left( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}H_{{ \textrm{D} }} \frac{}{} \right) X_{{\textrm{f}}, R} \, + U_{{\textrm{f}}} \, d \Gamma_{{\textrm{f}}} (j_{{\textrm{f}}, R} , \tilde{\text{ad}}_{\omega_{M} }(j_{{\textrm{f}},R})) , \label{3.2.1} \\ & X_{{\textrm{f}}, R} \, c_{\tau , s}(f) = (c_{\tau, s } (f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \, + X_{{\textrm{f}}, R} \, c_{\tau , s}((1-j_{{\textrm{f}}, R}^{\, 0} ) f) , \label{3.2.2} \\ & X_{{\textrm{f}}, R} \, c^{\dagger}_{\tau , s }(f) = ( c^\ast_{\tau , s } (f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} + \left( c^{\dagger}_{\tau , s } ((j_{{\textrm{f}}, R}^{\, 0} -1 )f ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ (-1)^{N_{{ \textrm{D} }}} {\otimes}c^{\dagger }_{\tau , s }(j_{{\textrm{f}}, R}^{\, \infty} f ) \right) X_{{\textrm{f}}, R} . \label{3.2.3}\end{aligned}$$ \[lemma32a\] Assume **(A.1)**. Then, $$\begin{aligned} & \textbf{(i)} \left\| \left( X_{{\textrm{f}}, R} \, H_{{ \textrm{D} }} - ( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}H_{{ \textrm{D} }} ) \right) X_{{\textrm{f}}, R} ( N_{{ \textrm{D} }} + 1 )^{-1} \right\| \leq \frac{c_{ \, {\textrm{f}}}}{R} , \\ & \textbf{(ii)} \left\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}}) - ( \psi_{l}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right\| \leq \delta^{1,l}_{ \, {\textrm{f}},R } ({\ensuremath{\mathbf{x}}}) ,\quad l= 1, \cdots 4, \\ &\textbf{(iii)} \left\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} - ( \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right\| \leq \delta^{2 , l}_{ \, {\textrm{f}},R } ({\ensuremath{\mathbf{x}}}) , \quad l= 1, \cdots 4 . \end{aligned}$$ Here $c_{ \, {\textrm{f}}} \geq 0$ is a constant, and $\delta^{i, l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) \geq0 $, $l=1 ,\cdots ,4$, $i= 1, 2$, are error terms which satisfy $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^{ i , l }_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) | < \infty$ and $\lim\limits_{R \to \infty }\delta^{i,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) =0 $ for all ${\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }$. $\;$\ **(Proof)** **(i)** By (\[3.2.1\]), we have $$\left\| \left( X_{{\textrm{f}}, R} H_{{ \textrm{D} }} - ( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}H_{{ \textrm{D} }} ) X_{{\textrm{f}}, R} \right) (N_{{ \textrm{D} }} +1)^{-1} \right\| \leq \| d \Gamma_{{\textrm{f}}} ( j_{{\textrm{f}}, R} , \tilde{\text{ad}}_{\omega_{\, M}}(j_{{\textrm{f}},R})) (N_{{ \textrm{D} }} +1)^{-1} \| , \notag$$ and (\[3.1.3\]) yields that $$\| d \Gamma_{{\textrm{f}}} ( j_{{\textrm{f}}, R} , \tilde{\text{ad}}_{\omega_{\, M}}(j_{{\textrm{f}},R})) (N_{{ \textrm{D} }} +1)^{-1} \| \leq \sqrt{ \| [\omega_{\, M},j^{\, 0}_{{\textrm{f}}, R} ] \|^2_{B(L^2({\mathbf{R}^{3} }))} + \| [\omega_{\, M},j^{\, \infty}_{{\textrm{f}}, R} ] \|^2_{B(L^2({\mathbf{R}^{3} }) )} } \notag$$ By pseudo-differential calculus (e.g., [@FGS02] ; Appendix A, [@Hida11] ; Section IV), it follows that $\;$ $\| [\omega_{M},j^{\, \sharp}_{{\textrm{f}}, R} ] \|_{B(L^2({\mathbf{R}^{3} }) )} \leq$ $ \frac{ c_{\sharp }}{R}$, $\sharp = 0, \infty $, where $c_{\sharp} \geq 0$ are constants. Thus **(i)** is proven.\ **(ii)** By the definition of $\psi_{l} ({\ensuremath{\mathbf{x}}})= \sum\limits_{s= \pm 1/2}(b_{s} (f_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) + d^{\, \dagger }_{s} (g_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) )$, we have from (\[3.2.2\]) and (\[3.2.3\]) that $$\begin{aligned} & X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}}) - ( \psi_{l}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \notag \\ & =\sum_{s= \pm 1/2} \left( X_{{\textrm{f}}, R} \, b_{s} ((1-j_{{\textrm{f}}, R}^{\, 0}) f_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) + \left( d^{\, \dagger }_{s} ((j_{{\textrm{f}}, R}^{\, 0}-1) g_{s,{\ensuremath{\mathbf{x}}}}^{\, l}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ (-1)^{N_{{ \textrm{D} }}} {\otimes}d_{s}^{\, \dagger } (j_{{\textrm{f}}, R}^{\, \infty} g_{s,{\ensuremath{\mathbf{x}}}}^{l\, } )\right) X_{{\textrm{f}}, R} \right) . \end{aligned}$$ Then we have $$\begin{aligned} & \left\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}}) - ( \psi_{l}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right\| \notag \\ & \leq \sum_{s= \pm 1/2 } \left( \| b_{s} ((1-j_{{\textrm{f}}, R}^{\, 0}) f_{s ,{\ensuremath{\mathbf{x}}}}^{\, l}) \| + \| ( d^{\, \dagger}_{s} ((j_{{\textrm{f}}, R}^{\, 0}-1) g_{s ,{\ensuremath{\mathbf{x}}}}^l){\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \| + \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}d_{s}^{\, \dagger } (j_{{\textrm{f}}, R}^{\, \infty} g_{s,{\ensuremath{\mathbf{x}}}}^l ) X_{{\textrm{f}}, R} \| \right) \notag \\ & \leq \sum_{s= \pm 1/2} \left( \| ((1-j_{{\textrm{f}}, R}^{\, 0}) f_{s ,{\ensuremath{\mathbf{x}}}}^{\, l}) \| + \| ((j_{{\textrm{f}}, R}^{\, 0}-1) g_{s,{\ensuremath{\mathbf{x}}}}^l \| + \| j_{{\textrm{f}}, R}^{\, \infty} g_{s,{\ensuremath{\mathbf{x}}}}^l \| \frac{}{} \right) .\end{aligned}$$ Let $\delta^{1,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) = \sum\limits_{s= \pm 1/2 } \left( \| ((1-j_{{\textrm{f}}, R}^{\, 0}) f_{s,{\ensuremath{\mathbf{x}}}}^{\, l}) \| + \| ((1-j_{{\textrm{f}}, R}^{\, 0}) g_{s,{\ensuremath{\mathbf{x}}}}^l) \| + \| j_{{\textrm{f}}, R}^{\, \infty} g_{s,{\ensuremath{\mathbf{x}}}}^l \| \frac{}{} \right) $. We see that $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^{1, l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}})| \leq \sum\limits_{s= \pm 1/2 } \left( \|f_{s}^{\,l} \| + 2 \|g_{s}^l \|\right) $ and $\lim\limits_{R \to \infty} \delta^{1,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) =0$ for all ${\ensuremath{\mathbf{x}}} \in {\ensuremath{\mathbf{R}}}$. Hence **(ii)** follows.\ $\; $\ **(iii)** From the definition of $\psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast}= \sum\limits_{s= \pm 1/2}(b^{\dagger }_{s} (f_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) + d_{s} (g_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) )$, (\[3.2.2\]) and (\[3.2.3\]) yield that $$\begin{aligned} & X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} - ( \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \notag \\ & = \sum_{s= \pm 1/2} \left( \left( b^{\dagger }_{s} ((j_{{\textrm{f}}, R}^{\, 0}-1) f_{s,{\ensuremath{\mathbf{x}}}}^{\, l}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ (-1)^{N_{{ \textrm{D} }}} {\otimes}b_{s}^{\dagger } (j_{{\textrm{f}}, R}^{\, \infty} f_{s,{\ensuremath{\mathbf{x}}}}^{\, l } ) \right) X_{{\textrm{f}}, R} + X_{{\textrm{f}}, R} \, d_{s} ((1-j_{{\textrm{f}}, R}^{\, 0}) g_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) \right) . \end{aligned}$$ Then it follows that $$\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} - ( \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \| \leq \delta^{2,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) , \notag$$ where $\delta^{2,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) = \sum\limits_{s= \pm 1/2 } \left( \| ((j_{{\textrm{f}}, R}^{\, 0} -1) f_{s,{\ensuremath{\mathbf{x}}}}^l) \| + \| j_{{\textrm{f}}, R}^{\, \infty} f_{s,{\ensuremath{\mathbf{x}}}}^l \| + \| ((1-j_{{\textrm{f}}, R}^{\, 0}) g_{s,{\ensuremath{\mathbf{x}}}}^{\, l}) \| \, \right) $. It is seen that $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^{2, l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}})| \leq \sum\limits_{s= \pm 1/2 } \left( 2 \|f_{s}^{\,l} \| + \|g_{s}^l \|\right) $ and $\lim\limits_{R \to \infty} \delta^{2,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) =0$ for all ${\ensuremath{\mathbf{x}}} \in {\ensuremath{\mathbf{R}}}$. Thus we obtain **(iii)**. $\blacksquare $\ \[coro32a\] Assume **(A.1)**. Then, for all $l ,l' = 1, \cdots 4$, $$\begin{aligned} \textbf{(i)} \, & \left\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{x}}}) - ( \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right\| \leq \delta^{3, l,l'}_{ \, {\textrm{f}},R } ({\ensuremath{\mathbf{x}}}) , \\ \textbf{(ii)} & \left\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l}({\ensuremath{\mathbf{x}}}) \psi_{l'}({\ensuremath{\mathbf{y}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{y}}}) - ( \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l}({\ensuremath{\mathbf{x}}}) \psi_{l'}({\ensuremath{\mathbf{y}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right\| \leq \delta^{ \, 4, l, l' }_{ \, {\textrm{f}},R } ({\ensuremath{\mathbf{x}}}, {\ensuremath{\mathbf{y}}}) .\notag\end{aligned}$$ Here $\delta^{ 3 , l, l' }_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) \geq 0 $ satisfies $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^{ 3, l,l'}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) | < \infty$ and $\lim\limits_{R \to \infty }\delta^{ 3, l ,l'}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) =0 $ for all ${\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }$, and $\delta^{ 4 , l, l' }_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}} ,{\ensuremath{\mathbf{y}}} ) \geq0 $ satisfies $ \sup\limits_{ ({\ensuremath{\mathbf{x}}} ,{\ensuremath{\mathbf{y}}}) \in {\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} | \delta^{ \, 4, l,l'}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}} ,{\ensuremath{\mathbf{y}}}) | < \infty$ and $\lim\limits_{R \to \infty }\delta^{ \, 4, l ,l'}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}}) =0 $ for all $ {\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}} \in {\mathbf{R}^{3} }$. **(Proof)** **(i)** By Lemma \[lemma32a\] **(ii)** and **(iii)**, it is seen that $$\begin{aligned} & \left\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{x}}}) - ( \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right\| \notag \\ &\leq \left\| \left( X_{{\textrm{f}}, R} \, \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} - \left( ( \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right) \right) \psi_{l'}({\ensuremath{\mathbf{x}}}) \right\| \notag \\ & \qquad \qquad \qquad + \left\| (\psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \left( X_{{\textrm{f}}, R} \, \psi_{l'}({\ensuremath{\mathbf{x}}}) - ( \psi_{l'}({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right)\right\| \notag \\ & \leq \delta_{{\textrm{f}}, R}^{2,l} \|\psi_{l'}({\ensuremath{\mathbf{x}}}) \| + \delta_{{\textrm{f}}, R}^{1,l'} \| \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \| . \notag \end{aligned}$$ Note that $ \| \psi_{\, l'}({\ensuremath{\mathbf{x}}}) \| \leq c_{\, { \textrm{D} }}^{\, l'}$ and $ \| \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \| \leq c_{\, { \textrm{D} }}^{l} $. Hence **(i)** is obtained. Similarly, we can prove **(ii)** by using **(i)**. $\blacksquare$\ Partition of Unity for Radiation Field --------------------------------------- Let $U_{{\textrm{b}}}: {\ensuremath{\mathscr{F}}}_{{\textrm{b}}}( L^2( {\mathbf{R}^{3} }_{{\ensuremath{\mathbf{k}}}} \times \{ 1,2 \} ) \oplus L^2( {\mathbf{R}^{3} }_{{\ensuremath{\mathbf{k}}}} \times \{ 1,2 \} ) ) \to {\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} {\otimes}{\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} $ an isometric operator satisfying $U_{{\textrm{b}}} \, {\Omega_{\textrm{rad}}}= {\Omega_{\textrm{rad}}}{\otimes}{\Omega_{\textrm{rad}}}$ and $$\begin{aligned} &U_{{\textrm{b}}} \, a_{r_1}^{\dagger } \left(\left[ \begin{array}{c} f_1 \\ g_1 \end{array} \right]\right) \cdots a_{r_1}^{\dagger } \left(\left[ \begin{array}{c} f_1 \\ g_n \end{array} \right]\right) {\Omega_{\textrm{rad}}}\\ = & \, \left( a^{\dagger }_{r_1} (f_1) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a^{\dagger }_{r_1}(g_1) \frac{}{}\right) \cdots \left( a^{\dagger}_{r_n} (f_n) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a^{\dagger}_{r_n}(g_n) \frac{}{} \right) {\Omega_{\textrm{rad}}}{\otimes}{\Omega_{\textrm{rad}}}. \end{aligned}$$ Let $j_{0} , j_{\, \infty} \in C^{\infty} ({\ensuremath{\mathbf{R}}})$. We suppose that $j_0 \geq 0$, $j_{\infty} \geq0 $, $j_{0}^2 + j_{\infty}^2 =1$, $j_{0}({\ensuremath{\mathbf{y}}})=1 $ if $|{\ensuremath{\mathbf{y}}}| \leq 1 $ and $j_{0}({\ensuremath{\mathbf{y}}})=0 $ if $|{\ensuremath{\mathbf{y}}}| \geq 2 $. We set $j_{{\textrm{b}}, R } =\left[ \begin{array}{c} j_{{\textrm{b}}, R }^{\, 0} \\ j_{{\textrm{b}}, R }^{\, \infty} \end{array} \right] $ where $ j_{{\textrm{b}}, R }^{\, 0}= j_{0} (\frac{-i {\ensuremath{\mathbf{\nabla_{{\ensuremath{\mathbf{k}}}}}}}}{R})$ and $ j_{{\textrm{b}}, R }^{\, \infty}= j_{\infty} (\frac{-i {\ensuremath{\mathbf{ \nabla_{{\ensuremath{\mathbf{k}}}}}}}}{R}) $ with $\nabla_{{\ensuremath{\mathbf{k}}}}= (\partial_{k^1}, \partial_{k^2},\partial_{k^3} ) $.\ $\;$\ Let $Y_{{\textrm{b}}, R} : {\mathscr{F}_{\textrm{rad}}}\to {\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} {\otimes}{\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} $ defined by $$Y_{{\textrm{b}}, R} = U_{{\textrm{b}}} \, \Gamma_{{\textrm{b}}}(j_{{\textrm{b}}, R}) . \notag$$ From (\[3.1.8\]) - (\[3.1.10\]), it follows that $$\begin{aligned} & Y_{{\textrm{b}}, R} \, {H_{\textrm{rad}, \, m}}= \left( {H_{\textrm{rad}, \, m}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}, \, m}}\frac{}{} \right) Y_{{\textrm{b}}, R} - U_{{\textrm{b}}} \, d \Gamma_{{\textrm{b}}} ( j_{{\textrm{b}}, R} , \tilde{\text{ad}}_{\omega_{m} }(j_{{\textrm{b}},R})) , \label{3.3.1} \\ & Y_{{\textrm{b}}, R} \, a_{r}(h) = (a_r (h) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R} + Y_{{\textrm{b}}, R} \, a_{r}((1-j_{{\textrm{b}}, R}^{\, 0} )h) , \label{3.3.2} \\ &Y_{{\textrm{b}}, R} \, a^{\dagger}_{r}(h) = (a^\ast_r (h) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R} + \left(a^{\dagger}_r ((j_{{\textrm{b}}, R}^{\, 0}-1) h ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}^{\dagger }(j_{{\textrm{b}}, R}^{\, \infty} h) \right) Y_{{\textrm{b}}, R} . \label{3.3.3}\end{aligned}$$ \[lemma33a\] Assume **(A.2)**. Then $$\begin{aligned} & \textbf{(i)} \left\| \left( Y_{{\textrm{b}}, R} \, {H_{\textrm{rad}, \, m}}- ( {H_{\textrm{rad}, \, m}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}, \, m}}) \, Y_{{\textrm{b}}, R} \frac{}{} \right) (N_{{\textrm{rad}}} +1)^{-1} \right\| \leq \frac{c_{ \, {\textrm{b}}}}{R} , \\ & \textbf{(ii)} \left\| \left( Y_{{\textrm{b}}, R} \, A_{j}({\ensuremath{\mathbf{x}}}) - ( A_{j}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R} \frac{}{} \right) (N_{{\textrm{rad}}}+1)^{-1/2} \right\| \leq \delta^j_{ \, {\textrm{b}},R } ({\ensuremath{\mathbf{x}}}) .\end{aligned}$$ Here $c_{ \, {\textrm{b}}} \geq 0$ is a constant and $\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) \geq0 $, $j=1, 2,3$, are error terms which satisfy $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) | < \infty$ and $\lim\limits_{R \to \infty }\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) =0 $ for all $ {\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }$. **(Proof)** **(i)** It is proven in a similar way to Lemma \[lemma32a\] **(i)**.\ **(ii)** By the definition of $ A_{j}({\ensuremath{\mathbf{x}}}) = \sum\limits_{r=1,2} \left( a_{r}(h^j_{r , {\ensuremath{\mathbf{x}}}}) + a^{\dagger}_{r}(h^j_{r , {\ensuremath{\mathbf{x}}}}) \right)$, it follows from (\[3.3.2\]) and (\[3.3.3\]) that $$\begin{aligned} & Y_{{\textrm{b}}, R} \, A_{j}({\ensuremath{\mathbf{x}}}) - ( A_{j}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R} \notag \\ &= \sum_{r=1,2} \left( Y_{{\textrm{b}}, R} \, a_{r}((1-j_{{\textrm{b}}, R}^{\, 0} )h_{r, {\ensuremath{\mathbf{x}}}}^j) + \left(a^{\dagger}_r ((j_{{\textrm{b}}, R}^{\, 0}-1) h_{r, {\ensuremath{\mathbf{x}}}}^j ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}^{\dagger }(j_{{\textrm{b}}, R}^{\, \infty} h_{r, {\ensuremath{\mathbf{x}}}}^j ) \right) Y_{{\textrm{b}}, R} \right) . \notag \end{aligned}$$ Since $\|a_{r}(h) (N_{{\textrm{rad}}} + 1)^{-1/2} \| \leq \| h \|$ and $\|a^{\dagger }_{r}(h) (N_{{\textrm{rad}}} + 1)^{-1/2} \| \leq 2 \| h \|$, we have $$\begin{aligned} & \left\| \left( Y_{{\textrm{b}}, R} \, A_{j}({\ensuremath{\mathbf{x}}}) - ( A_{j}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R} \right) (N_{{\textrm{rad}}} +1)^{-1/2} \right\| \notag \\ & \leq \sum_{r=1,2} \left( \| ( a_{r} (1-j_{{\textrm{b}}, R}^{\, 0}) h_{r,{\ensuremath{\mathbf{x}}}}^j) (N_{{\textrm{rad}}} +1)^{-1/2} \| \right. \notag \\ & \qquad \qquad \left. + \| (a^{\dagger}_{r} ((j_{{\textrm{b}}, R}^{\, 0}-1) h_{r,{\ensuremath{\mathbf{x}}}}^j)(N_{{\textrm{rad}}} +1)^{-1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) ( (N_{{\textrm{rad}}} +1)^{1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})Y_{{\textrm{b}}, R} (N_{{\textrm{rad}}} +1)^{-1/2} \| \right. \notag \\ &\qquad \qquad \qquad \left. + \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}^{\dagger} (j_{{\textrm{b}}, R}^{\, \infty} h_{r,{\ensuremath{\mathbf{x}}}}^j ) (N_{{\textrm{rad}}} +1)^{-1/2} ) ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}(N_{{\textrm{rad}}} +1)^{1/2} ) Y_{{\textrm{b}}, R} (N_{{\textrm{rad}}} +1)^{-1/2} \| \frac{}{} \right) \notag \\ & \leq \sum\limits_{r=1,2} \left( \| (1-j_{{\textrm{b}}, R}^{\, 0}) h_{r,{\ensuremath{\mathbf{x}}}}^j \| +2 \| (j_{{\textrm{b}}, R}^{\, 0}-1)h_{r,{\ensuremath{\mathbf{x}}}}^j \| + 2 \| j_{{\textrm{b}}, R}^{\, \infty} h_{r,{\ensuremath{\mathbf{x}}}}^j \| \frac{}{} \right) . \notag \end{aligned}$$ Let $\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) = \sum\limits_{r=1,2} \left( 3 \| ((1-j_{{\textrm{b}}, R}^{\, 0}) h_{r,{\ensuremath{\mathbf{x}}}}^j) \| + 2 \| j_{{\textrm{b}}, R}^{\, \infty} h_{r,{\ensuremath{\mathbf{x}}}}^j \| \frac{}{} \right) $, $j=1,2 ,3$. We see that $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} |\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}})| \leq 5 \left( \|h_{1}^j \| + \|h_{2}^j \|\right) $ and $\lim\limits_{R \to \infty}\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) =0$ for all ${\ensuremath{\mathbf{x}}} \in {\ensuremath{\mathbf{R}}}$. Thus we obtain the proof. $\blacksquare$\ Existence of Ground State of $H_{m}$ ------------------------------------ We recall that the massive Hamiltonian is defined by $$H_{m} = H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}, \, m}}+ {\kappa_{\textrm{I}}}{H_{\textrm{I}}}+ {\kappa_{\textrm{II}}}{H_{\textrm{II}}}. \notag$$ Throughout this subsection, we do not omit the subscripts of the identities ${{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}$ and ${{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}$.\ $\;$\ Since $\frac{1}{\omega_m({\ensuremath{\mathbf{k}}})^{\lambda}} \leq \frac{1}{\omega ({\ensuremath{\mathbf{k}}})^{\lambda}} $, $\lambda>0$, it holds that $$\|A_j ({\ensuremath{\mathbf{x}}}) ( {H_{\textrm{rad}, \, m}}+1 )^{-1/2} \| \leq \sum_{r=1,2} \left( 2 \| \frac{ \chi_{{\textrm{rad}}} e_{r}^{\, j}}{ \omega_{m}} \| + \| \frac{ \chi_{{\textrm{rad}}} e_{r}^{\, j}}{ \sqrt{\omega_m}} \| \right) \leq c_{{\textrm{rad}}}^{\, j} . \label{9/1.1}$$ Then, we have $$\| {H_{\textrm{I}}}\Psi \| \leq c_{\,{\textrm{I}}} \, \| {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}+1)^{1/2} \Psi \| , \quad \label{HImbound}$$ and it holds that for all $\epsilon > 0$, $$\|{H_{\textrm{I}}}\Psi \| \leq c_{{\textrm{I}}} \epsilon \|H_{0 , m} \Psi \| + c_{\textrm{I}}\left( \frac{1 }{2 \epsilon} +1 \right) \| \Psi \| . . \label{HImbound'}$$ From (\[HImbound’\]) and $\| H_{{\textrm{II}}}\| < \infty $, it is proven that $H_{m}$ is self-adjoint and essentially self adjoint on any core of $H_{0,m}$.\ \[Massive-Case\] **(Existence of a Ground State of $H_m $)**\ Suppose **(A.1)** - **(A.3)**. Let $m <M$. Then $H_m $ has purely discrete spectrum in $[ E_{0} (H_m ) , E_{0} (H_m ) + m )$. In particular, $ {H_{m}}$ has a ground state. $\; $\ To prove Theorem \[Massive-Case\], we need some preparations. We define $ \tilde{X}_{{\textrm{f}}, R } : {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \to {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ by $$\tilde{X}_{{\textrm{f}}, R } = X_{{\textrm{f}}, R} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} .$$ We introduce Hamiltonian $\tilde{H}_{m} : {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}\to {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ defined by $$\tilde{H}_{m} = \tilde{H}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}\tilde{H}_{{\textrm{rad}}} + {\kappa_{\textrm{I}}}\tilde{H}_{{\textrm{I}}} + {\kappa_{\textrm{II}}}\tilde{H}_{{\textrm{II}}} , \notag$$ where $\tilde{H}_{{ \textrm{D} }} = H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} $, $ \tilde{H}_{{\textrm{rad}}} = {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}, \, m}}$ and $$\begin{aligned} &\tilde{H}_{{\textrm{I}}} = \sum_{j=1}^{3} {\int_{\mathbf{R}^{3}}}\chi_{{\textrm{I}}} ({\ensuremath{\mathbf{x}}}) (\tilde{\psi}^{\dagger}({\ensuremath{\mathbf{x}}}) \tilde{\alpha}^j \tilde{\psi}({\ensuremath{\mathbf{x}}} ) {\otimes}A_j ({\ensuremath{\mathbf{x}}} )) d{\ensuremath{\mathbf{x}}} , \\ & \tilde{H}_{{\textrm{II}}} = \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|} \left( \tilde{\psi}^{\dagger } ({\ensuremath{\mathbf{x}}}) \tilde{\psi} ({\ensuremath{\mathbf{x}}}) \tilde{\psi}^{\dagger} ({\ensuremath{\mathbf{y}}}) \tilde{\psi} ({\ensuremath{\mathbf{y}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} \right) d {\ensuremath{\mathbf{x}}} \, d {\ensuremath{\mathbf{y}}}. \end{aligned}$$ with $\tilde{\psi}({\ensuremath{\mathbf{x}}})=\psi({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}$ and $\tilde{\alpha}^{j} = \alpha^j {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}$, $j=1, \cdots 3$. $\; $\ \[9/9.a\] Assume **(A.1)** - **(A.3)**. Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_{m})$. Then, it holds that $$\begin{aligned} & \textbf{(i)} \; \; \left\| \left( \tilde{X}_{{\textrm{f}}, R } ( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) - ( \tilde{H}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right\| \notag \\ & \quad \; \; \; \; \leq \frac{ c_{ \, {\textrm{f}}}}{R} \, \left( \left\| ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \right\| + \left\| \Psi \right\| \right) , \\ & \textbf{(ii)} \; \, \, \left\| \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}- \tilde{{H_{\textrm{I}}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right\| \leq \delta_{\, {\textrm{f}}, {\textrm{I}}} (R ) \, \left( \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi \| + \left\| \Psi \right\| \right), \\ & \textbf{(iii)} \; \left\| \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}- \tilde{{H_{\textrm{II}}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right\| \leq \delta_{\, {\textrm{f}}, {\textrm{II}}} (R) \, \| \Psi \| .\end{aligned}$$ Here $c_{{\textrm{f}}} \geq 0$ is the constant in Lemma \[lemma32a\]**(i)**, and $\delta_{\, {\textrm{f}}, {\textrm{I}}}(R ) \geq 0 $ and $\delta_{\, {\textrm{f}}, {\textrm{II}}} (R) \geq 0 $ are error terms satisfying that $\lim\limits_{R \to \infty }\delta_{\, {\textrm{f}}, {\textrm{I}}} (R) =0 $ and $\lim\limits_{R \to \infty }\delta_{\, {\textrm{f}}, {\textrm{II}}} (R) =0 $, respectively. **(Proof)**\ **(i)** It directly follows from Lemma \[lemma32a\] (**i**).\ **(ii)** Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m ) $ and $\tilde{\Phi} \in {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Phi} \| =1 $. Then, $$\begin{aligned} & \left( \tilde{\Phi} , \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}\, - \tilde{H}_{{\textrm{I}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right) \notag \\ & = \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left( \Phi , \left( \left( X_{{\textrm{f}}, R} {\psi^{\dagger} (\mathbf{x}) }({\ensuremath{\mathbf{x}}}) \alpha^j \psi ({\ensuremath{\mathbf{x}}}) - \tilde{\psi}^{\dagger} ({\ensuremath{\mathbf{x}}}) \tilde{\alpha}^j \tilde{\psi} ({\ensuremath{\mathbf{x}}}) X_{{\textrm{f}}, R} \right) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) \right) \Psi \right) {d \mathbf{x} }. \notag \end{aligned}$$ Then we have $$\begin{aligned} & \left| \left( \tilde{\Phi} , \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}\, - \tilde{H}_{{\textrm{I}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right) \right| \notag \\ & \leq \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} \left| {\chi_{\textrm{I}} (\mathbf{x}) }\right| \, \left\| X_{{\textrm{f}}, R} \psi^{\dagger} ({\ensuremath{\mathbf{x}}}) \alpha^j \psi ({\ensuremath{\mathbf{x}}}) - \tilde{\psi}^{\dagger} ({\ensuremath{\mathbf{x}}}) \tilde{\alpha}^j \tilde{\psi} ({\ensuremath{\mathbf{x}}}) X_{{\textrm{f}}, R} \right\| \, \left\| \left( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) \right) \Psi \right\| {d \mathbf{x} }\notag \\ & \leq \sum_{j=1}^3 \sum_{l,l'=1}^4 |\alpha_{l , l'}^j | \int_{{\mathbf{R}^{3} }} \left| {\chi_{\textrm{I}} (\mathbf{x}) }\right| \, \| \left( X_{{\textrm{f}}, R} \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'} ({\ensuremath{\mathbf{x}}}) - \tilde{\psi}_{l} ({\ensuremath{\mathbf{x}}})^{\ast} \tilde{\psi}_{l'} ({\ensuremath{\mathbf{x}}}) X_{{\textrm{f}}, R} \right) \| \left\| \left( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) \right) \Psi \right\| {d \mathbf{x} }. \notag\end{aligned}$$ By Corollary \[coro32a\] **(i)**, we have $ \| \left( X_{{\textrm{f}}, R} \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'} ({\ensuremath{\mathbf{x}}}) - \tilde{\psi}_{l} ({\ensuremath{\mathbf{x}}})^{\ast} \tilde{\psi}_{l'} ({\ensuremath{\mathbf{x}}}) X_{{\textrm{f}}, R} \right) \| \leq \delta_{\, {\textrm{f}}, R}^{3, l, l'} ({\ensuremath{\mathbf{x}}}) $. We also see that $ \| A_{j} ({\ensuremath{\mathbf{x}}}) (N_{{\textrm{rad}}} +1)^{-1/2}\| \leq 3 \sum\limits_{r=1,2}\| h_{r}^j\| $. Then it follows that $$\begin{aligned} & \left| \left( \tilde{\Phi} , \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}\, - \tilde{H}_{{\textrm{I}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right) \right| \notag \\ & \quad \qquad \leq \sum_{r=1,2} \sum_{j=1}^3 \sum_{l,l'=1}^4 |\alpha_{l , l'}^j | \| h_{r}^j \| \left( \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \delta_{\, {\textrm{f}}, R}^{3, l, l'} ({\ensuremath{\mathbf{x}}}) {d \mathbf{x} }\frac{}{} \right) \, \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}} +1 )^{1/2}) \Psi \| . \label{9/9.2}\end{aligned}$$ Since (\[9/9.2\]) holds for all $\tilde{\Phi} \in {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}} {\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Phi} \| =1 $, it follows that $$\left\| \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}\, - \tilde{H}_{{\textrm{I}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right\| \leq \delta_{\, {\textrm{f}}, {\textrm{I}}} (R) \, \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}} +1 )^{1/2}) \Psi \| ,$$ where $\delta_{\, {\textrm{f}}, {\textrm{I}}} (R) = 3 \sum\limits_{r=1,2} \sum\limits_{j=1}^3 \sum\limits_{l,l'=1}^4 \, |\alpha_{l , l'}^j | \, \| h_{r}^j \| \, \| \chi_{{\textrm{I}}} \, \delta_{\, {\textrm{f}}, R}^{3, l, l'} \|_{L^1} $. We see that $ \lim\limits_{R \to \infty} \delta_{\, {\textrm{f}},{\textrm{I}}} (R) =0 $, and hence **(ii)** follows.\ **(iii)** Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m)$. We set $Q_{ l }({\ensuremath{\mathbf{x}}})= \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l} ({\ensuremath{\mathbf{x}}}) $. Then for all $\tilde{\Phi} \in {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Phi} \| =1 $, $$\begin{aligned} & \left( \tilde{\Phi} , \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}\, - \tilde{H}_{{\textrm{II}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right) \notag \\ & = \sum_{l, l'=1}^4\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \left( \tilde{\Phi} , \left( ( X_{{\textrm{f}}, R} Q_{ l }({\ensuremath{\mathbf{x}}}) Q_{l' } ({\ensuremath{\mathbf{y}}} ) - (( {Q}_{ l }({\ensuremath{\mathbf{x}}}) {Q}_{l'} ({\ensuremath{\mathbf{y}}} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} ) X_{{\textrm{f}}, R} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} \right) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }. \notag\end{aligned}$$ Then we have $$\begin{aligned} & \left| \left( \tilde{\Phi} , \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}\, - \tilde{H}_{{\textrm{II}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right) \right| \notag \\ & \leq \sum_{l, l'=1}^4\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{ | {\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }| }{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \left\| \left( X_{{\textrm{f}}, R} Q_{ l }({\ensuremath{\mathbf{x}}}) Q_{l' } ({\ensuremath{\mathbf{y}}} ) - ( {Q}_{ l }({\ensuremath{\mathbf{x}}}) {Q}_{l' } ({\ensuremath{\mathbf{y}}} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} ) X_{{\textrm{f}}, R} \right) \Psi \right\| {d \mathbf{x} }{d \mathbf{y} }. \notag \end{aligned}$$ From Corollary \[coro32a\] **(ii)**, it holds that $\left\| X_{{\textrm{f}}, R} Q_{ l }({\ensuremath{\mathbf{x}}}) Q_{l' } ({\ensuremath{\mathbf{y}}} ) - ({Q}_{ l }({\ensuremath{\mathbf{x}}}) {Q}_{l' } ({\ensuremath{\mathbf{y}}} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} ) X_{{\textrm{f}}, R} \right\| \leq \delta_{{\textrm{f}}, R}^{4, l, l'} ({\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}})$. Then we have $$\left| \left( \tilde{\Phi} , \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}\, - \tilde{H}_{{\textrm{II}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right) \right| \leq \left( \sum_{l, l'=1}^4\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{ | {\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }| }{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \delta_{{\textrm{f}}, R}^{4, l, l'} ({\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}}) {d \mathbf{x} }{d \mathbf{y} }\right) \, \left\| \Psi \right\| . \notag$$ This implies that $$\left\| \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}\, - \tilde{H}_{{\textrm{II}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right\| \leq \delta_{\, {\textrm{f}}, {\textrm{II}}} (R) \, \left\| \Psi \right\| , \notag$$ where $ \delta_{ \, {\textrm{f}}, {\textrm{II}}} (R) = \sum\limits_{l, l'=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{ | {\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }|}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \delta_{\, {\textrm{f}}, R}^{4, l, l'} ({\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}}) {d \mathbf{x} }{d \mathbf{y} }$. We see that $\lim\limits_{R \to \infty } \delta_{\, {\textrm{f}}, {\textrm{II}}} (R) =0$, and thus the proof is obtained. $\blacksquare$\ $\;$\ We define $\tilde{Y}_{{\textrm{b}}, R } : {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \to {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ by $$\tilde{Y}_{{\textrm{b}}, R } = {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}Y_{{\textrm{b}}, R} .$$ \[9/9.b\] Assume **(A.1)** - **(A.3)**. Then it holds that for all $\Psi \in {\ensuremath{\mathscr{D}}}(H_m )$, $$\begin{aligned} & \textbf{(i)} \left\| \left( \tilde{Y}_{{\textrm{b}}, R } ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}, \, m}}) - ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}, \, m}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{QED} }} {\otimes}{H_{\textrm{rad}, \, m}}) \tilde{Y}_{{\textrm{b}}, R } \right) \Psi \right\| \notag \\ & \qquad \qquad \leq \frac{ c_{ \, {\textrm{b}}}}{R} \, \left( \left\| ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}} ) \Psi \right\| + \left\| \Psi \right\| \right) , \\ & \textbf{(ii)} \left\| \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}- ( {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{{\textrm{b}}, R } \right) \Psi \right\| \leq \delta_{ \, {\textrm{b}}, {\textrm{I}}} (R ) \left( \| ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi \| + \| \Psi \| \right) ,\end{aligned}$$ where $c_{ \, {\textrm{b}}} \geq 0 $ and $\delta_{ \, {\textrm{b}},{\textrm{I}}} (R) \geq 0 $ satisfying $\lim\limits_{R \to \infty } \delta_{ \, {\textrm{b}}, {\textrm{I}}} (R) =0 $. **(Proof)** **(i)** It immediately follows from Lemma \[lemma33a\] **(i)**.\ **(ii)** Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m )$ and $\tilde{\Xi} \in {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Xi} \| =1$. We see that $$\begin{aligned} & \left( \tilde{\Xi} , \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}- ( {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{{\textrm{rad}}, R } \right) \Psi \right) \notag \\ & = \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left( \tilde{\Xi} , \left( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}}) \right) {\otimes}\left( Y_{{\textrm{b}}, R } A_{j} ({\ensuremath{\mathbf{x}}}) \, - ( A_{j} ({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) Y_{{\textrm{b}}, R } ) \right) \Psi \right) {d \mathbf{x} }, \notag .\end{aligned}$$ Then, $$\begin{aligned} & \left| \left( \tilde{\Xi} , \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}- ( {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{{\textrm{b}}, R } \right) \Psi \right) \right| \notag \\ & \leq \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \, \left\| ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}}) {\otimes}\left( Y_{{\textrm{b}}, R } A_{j} \, - ( A_{j} ({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) Y_{{\textrm{b}}, R } ) \right) \Psi \right\| {d \mathbf{x} }\notag \\ & \leq \left( \sum_{j=1}^3 \sum_{l,l'=1 }^4 |\alpha^j_{l,l'}| c_{\, { \textrm{D} }}^{\, l} c_{\, { \textrm{D} }}^{\, l'} \right) \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \left\| \left( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}\left( Y_{{\textrm{b}}, R } A_{j} \, - ( A_{j} ({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) Y_{{\textrm{b}}, R } ) \right) \right) \Psi \right\| {d \mathbf{x} }. \notag \end{aligned}$$ From Lemma \[lemma33a\] **(ii)**, it holds that $$\left\| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}\left( Y_{{\textrm{b}}, R } A_{j} \, - ( A_{j} ({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) Y_{{\textrm{b}}, R } ) \right) \Psi \right\| \leq \delta_{\, {\textrm{b}},R}^j ({\ensuremath{\mathbf{x}}}) \|({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}}+1 )^{1/2}) \Psi \| , \notag$$ where $ \delta_{{\textrm{b}}, R}^j ({\ensuremath{\mathbf{x}}}) \geq 0 $ is the error term, and hence, $$\left| \left( \tilde{\Xi} , \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}- ( {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{{\textrm{b}}, R } \right) \Psi \right) \right| \leq \delta_{\, {\textrm{b}}, {\textrm{I}}} (R) \, \|({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}}+1 )^{1/2}) \Psi \| , \label{9/9.3}$$ where $\delta_{\, {\textrm{b}},{\textrm{I}}} (R) = \sum\limits_{l,l'=1 }^4 |\alpha^j_{l,l'}| c_{\, { \textrm{D} }}^{\, l} c_{\, { \textrm{D} }}^{\, l'} \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \delta^j_{\, {\textrm{b}}} ({\ensuremath{\mathbf{x}}}) dx $. Since (\[9/9.3\]) holds for all $\tilde{\Xi} \in {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Xi} \| =1$, we have $$\left\| \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}- ( {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{{\textrm{rad}}, R } \right) \Psi \right\| \leq \delta_{\, {\textrm{b}}, {\textrm{I}}} (R) \, \|({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}}+1 )^{1/2}) \Psi \| . \notag$$ Since $\lim\limits_{R \to \infty }\delta_{\, {\textrm{b}}, {\textrm{I}}}(R)=0$, the proof is obtained. $\blacksquare$\ $\; $\ Here we introduce a new norm defined by $$\qquad \| \Psi \|_{ \lambda , \, \lambda ' } \; = \; \|( N_{{ \textrm{D} }}^{\, \lambda /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \| + \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{ \, \lambda ' /2} )\Psi \| + \| \Psi \| , \quad \Psi \in {\ensuremath{\mathscr{D}}} (N_{{ \textrm{D} }}^{\lambda /2 } {\otimes}N_{{\textrm{rad}}}^{ \lambda ' /2} ) .$$ $\;$\ From Proposition \[9/9.a\] and Proposition \[9/9.b\], the next corollary follows. $\;$\ \[9/9.c\] Assume **(A.1)** - **(A.3)**. Then for all $\Psi \in {\ensuremath{\mathscr{D}}}(H_m)$, $$\begin{aligned} &\textbf{(i)} \; \left\| \left( \tilde{X}_{{\textrm{f}}, R } H_{m} - ( \tilde{H}_{m} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{X}_{{\textrm{f}}, R} \right) \Psi \right\| \leq \delta_{ \, {\textrm{f}}} (R) \| \Psi \|_{2,1 } , \\ &\textbf{(ii)} \; \left\| \left( \tilde{Y}_{{\textrm{b}}, R } H_{m} - ( H_{m} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} {\otimes}{H_{\textrm{rad}, \, m}}) \tilde{Y}_{{\textrm{b}}, R } \right) \Psi \right\| \leq \delta_{ \,{\textrm{b}}} (R) \| \Psi \|_{0, 2} \, .\end{aligned}$$ Here $ \delta_{ \, {\textrm{f}}} (R) \geq 0 $ and $ \delta_{ \, {\textrm{b}}} (R) \geq 0 $ are error terms which satisfy that $ \lim\limits_{R \to \infty} \delta_{ \, {\textrm{f}}} (R)=0 $ and $ \lim\limits_{R \to \infty} \delta_{ \, {\textrm{b}}} (R) =0 $, respectively.\ \[9/9.d\] \[LformboundHm\] Assume **(A.1)** - **(A.3)**. Let $q_{\, {\textrm{f}}, R} = (j^{\, 0}_{\, {\textrm{f}}, R})^2$ and $q_{\, {\textrm{b}}, R} = (j^{\, 0}_{\, {\textrm{b}}, R})^2$. Then, for all $\Psi \in {\ensuremath{\mathscr{ D}}} (H_m ) $ with $\| \Psi \| =1$, $$\begin{aligned} (\Psi, H_m \Psi ) \geq & E_{0} (H_m) + \, m \, + \, (M-m) \left( \Psi, ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}\Gamma_{{\textrm{b}}} (q_{\, {\textrm{b}}, R}) ) \Psi \right) \\ & - M \left( \Psi , \left( \Gamma_{{\textrm{f}}} (q_{\, {\textrm{f}}, R}) {\otimes}\Gamma_{{\textrm{b}}} (q_{\, {\textrm{b}}, R}) \right) \Psi \right) + \left( \delta_{\, {\textrm{f}}} (R ) \| \Psi \|_{2,1 } + \delta_{\, {\textrm{b}}} (R ) \| \Psi \|_{0,2 } \right) . \end{aligned}$$ **(Proof)** Let $\Psi \in {\ensuremath{\mathscr{ D}}} (H_m ) $ with $\| \Psi \| =1$. By Lemma Corollary \[9/9.c\] **(ii)**, $$\begin{aligned} (\Psi , H_m \Psi ) & = \left( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} \tilde{Y}_{\, {\textrm{b}},R} H_m \Psi \right) \notag \\ & \geq ( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} ( H_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}) \tilde{Y}_{\, {\textrm{b}},R} \Psi ) + ( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} {\otimes}{H_{\textrm{rad}, \, m}}) \tilde{Y}_{\, {\textrm{b}},R} \Psi ) - \delta_{\, {\textrm{b}}} (R) \| \Psi \|_{0,2} . \notag \end{aligned}$$ We see that $ {H_{\textrm{rad}, \, m}}\geq m ( {{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} - {P_{\textrm{rad}}}) $ with $ {P_{\textrm{rad}}}= E_{N_{{\textrm{rad}}}}(\{ 0\})$ where $E_{X}(J)$ denotes the spectral projection on a Borel set $J \in {\ensuremath{\mathscr{B}}}({\ensuremath{\mathbf{R}}})$ for a self-adjoint operator $X$. Then $$\begin{aligned} (\Psi , H_m \Psi ) & \geq ( \Psi , \tilde{Y}_{\, {\textrm{rad}},R}^{\ast} ( H_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{\, {\textrm{rad}},R} \Psi ) + m \notag \\ & \qquad \qquad \quad - m ( \Psi , \tilde{Y}_{\, {\textrm{rad}},R}^\ast ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} {\otimes}{P_{\textrm{rad}}}) \tilde{Y}_{\, {\textrm{rad}},R} \Psi ) - \delta_{\, {\textrm{b}}} (R) \| \Psi \|_{0,2 } \notag \\ & \geq ( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} ( H_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{\, {\textrm{b}},R} \Psi ) + m-m (\Psi , ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{\Gamma_{\textrm{b}}}( q_{\, {\textrm{b}}, R}) \Psi ) - \delta_{\, {\textrm{b}}} (R) \| \Psi \|_{0,2 } . \label{9/10.1}\end{aligned}$$ Here we used $ {Y}_{\, {\textrm{b}},R}^{\ast} ( {{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} {\otimes}{P_{\textrm{rad}}}) {Y}_{\, {\textrm{b}},R} = {\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) $ in the last line. We evaluate the first term in the right hand side of (\[9/10.1\]). Let $\tilde{\tilde{X}}_{{\textrm{f}},R}= \tilde{X}_{\, {\textrm{f}},R} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}$. By Corollary \[9/9.c\] **(i)**, $$\begin{aligned} & ( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} ( H_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{\, {\textrm{b}},R} \Psi ) \notag \\ & = \left( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} ( ( \tilde{X}_{\, {\textrm{f}},R}^{\ast} \tilde{X}_{\, {\textrm{f}},R} H_m) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{\, {\textrm{b}},R} \Psi ) \right) \notag \\ &\geq \left( \Psi , \tilde{Y}_{\, {\textrm{rad}},R}^{\ast} \tilde{\tilde{X}}^{\ast}_{\, {\textrm{f}},R} ( \tilde{H}_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}) \tilde{\tilde{X}}_{\, {\textrm{f}},R} \tilde{Y}_{\, {\textrm{b}},R} \Psi \right) \notag \\ & \qquad \qquad \qquad \quad + \left( \Psi_n , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} \tilde{\tilde{X}}^{\ast}_{\, {\textrm{f}},R} ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{\tilde{X}}_{\, {\textrm{b}},R} \tilde{Y}_{ {\textrm{b}},R} \Psi \right) - \delta_{\, {\textrm{f}}} (R) \| \tilde{Y}_{ {\textrm{b}},R} \Psi \|_{2,1}^{\sim }, \label{9/9.3}\end{aligned}$$ where we set $$\| \tilde{ \Phi} \|_{ \lambda , \lambda ' }^{\sim} \; = \; \|( N_{{ \textrm{D} }}^{\lambda /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{ \Phi} \| + \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{\lambda ' /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{ \Phi} \| + \| \tilde{ \Phi} \| ,$$ for $ \tilde{ \Phi} \in {\ensuremath{\mathscr{D}}} ( N_{{ \textrm{D} }}^{\lambda /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}) \cap {\ensuremath{\mathscr{D}}}({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{\lambda ' /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) $. We see that $$\begin{aligned} \| \tilde{Y}_{ {\textrm{b}},R} \Psi \|_{ 2,1 }^{\sim} & = \| \tilde{Y}_{ {\textrm{b}},R} ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \| + \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{ {\textrm{b}},R} \Psi \| + \| \tilde{Y}_{ {\textrm{b}},R} \Psi \| \\ & \leq \| \tilde{Y}_{ {\textrm{b}},R} ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \| + \|\tilde{Y}_{ {\textrm{b}},R} ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi \|+ \| \tilde{Y}_{ {\textrm{b}},R} \Psi \| = \| \Psi \|_{2,1} ,\end{aligned}$$ and $ H_{{ \textrm{D} }} \geq M ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}- P_{{ \textrm{D} }}) $ with $ P_{{ \textrm{D} }} = E_{N_{{ \textrm{D} }}}(\{ 0 \})$. Then we have $$\begin{aligned} (\ref{9/9.3})& \geq E_{0} (\tilde{H}_m ) + M - M \left( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} \tilde{\tilde{X}}_{\, {\textrm{f}},R}^{\ast} ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{P_{\Omega_{\textrm{D}}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{\tilde{X}}_{\, {\textrm{f}},R} \tilde{Y}_{\, {\textrm{b}},R} \Psi \right) - \delta_{\, {\textrm{f}}} (R) \| \Psi \|_{ 2,1} \notag \\ & \geq E_{0} (H_m ) + M - M ( \Psi , ( {\Gamma_{\textrm{f}}}( q_{ \, {\textrm{f}}, R} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi ) - \delta_{\, {\textrm{f}}, m } (R) \| \Psi \|_{2,1} . \notag\end{aligned}$$ Here we used $ E_{0} (\tilde{H}_m)= E_{0} (H_m)$ and $ {X}_{\, {\textrm{f}},R}^{\ast} ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}P_{{ \textrm{D} }} ) {X}_{\, {\textrm{f}},R} = {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}}, R} ) $ in the last line. Thus we have $$\begin{aligned} (\Psi, H_m \Psi ) & \geq E_{0} (H_m ) + m + M - M \left( \Psi , ( {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \right) \notag \\ & \qquad \qquad -m \left( \Psi , \left( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right) \right) \Psi \right) -\delta_{\, {\textrm{b}}} (R) \| \Psi \|_{0,2} - \delta_{\, {\textrm{f}}} (R) \| \Psi \|_{2,1} .\end{aligned}$$ Note that $$\begin{aligned} {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} & \geq {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}+ ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} - {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) ) {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right) \notag \\ & = {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}+ {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right) - {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right) . \notag\end{aligned}$$ Then we have $$\begin{aligned} (\Psi, H_m \Psi ) & \geq E_{0} (H_m) + \, m \, + \, (M-m) \left( \Psi, ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right) ) \Psi \right) \\ & \quad- M \left( \Psi , ({\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right) ) \Psi \right) -\delta_{\, {\textrm{b}}} (R) \| \Psi \|_{0,2} - \delta_{\, {\textrm{f}}} (R) \| \Psi \|_{2,1} .\end{aligned}$$ Thus the proof is obtained. $\blacksquare$\ \[9/9.e\] Assume **(A.1)** - **(A.3)**. Then for all $ 0< \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $$\qquad \qquad \| H_{0, m} \Psi \| \leq L_{\epsilon} \| H_{m} \Psi \| + R_{\epsilon} \| \Psi \| , \qquad \Psi \in {\ensuremath{\mathscr{D}}}(H_m ) , \notag$$ where $L_{\epsilon} = \frac{1}{1- c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|\epsilon} $ and $R_{\epsilon}= \frac{1}{1- c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|\epsilon}\left( c_{\textrm{I}}|{\kappa_{\textrm{I}}}| \, ( \frac{1 }{2 \epsilon} +1 ) + |{\kappa_{\textrm{II}}}| \, \| {H_{\textrm{II}}}\| \right)$. **(Proof)** Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m)$. Since $H_{0,m} = H_m - {\kappa_{\textrm{I}}}{H_{\textrm{I}}}- {\kappa_{\textrm{II}}}{H_{\textrm{II}}}$, we see that $$\|H_{0,m} \Psi \| \leq \| H_m \Psi \| + \left| {\kappa_{\textrm{I}}}\right| \, \left| {H_{\textrm{I}}}\Psi \right\| + \left| {\kappa_{\textrm{II}}}\right| \, \, \| {H_{\textrm{II}}}\| \, \| \Psi \| . \notag$$ From (\[HImbound’\]), it holds that $ \| {H_{\textrm{I}}}\Psi \| \leq c_{{\textrm{I}}} \epsilon \| H_{0 , m } \Psi \| + c_{{\textrm{I}}} ( \frac{1 }{2 \epsilon} +1 ) \| \Psi \| $ for all $\epsilon > 0$. Hence $$(1- c_{\, {\textrm{I}}} |{\kappa_{\textrm{I}}}| \epsilon )\| H_{0,m} \Psi \| \leq \| H_m \Psi \| + \left( c_{\textrm{I}}|{\kappa_{\textrm{I}}}| \, \left( \frac{1 }{2 \epsilon} +1 \right) + | {\kappa_{\textrm{II}}}| \, \| {H_{\textrm{II}}}\| \, \right) \| \Psi \| .$$ Taking $\epsilon >0$ such that $ \epsilon < \frac{1}{c_{\, {\textrm{I}}}|{\kappa_{\textrm{I}}}|}$, we obtain the proof. $\blacksquare $ $\;$\ Since $\| N_{{ \textrm{D} }} \Psi \| \leq \frac{1}{M} \| H_{{ \textrm{D} }} \Psi \|$, $\Psi \in {\ensuremath{\mathscr{D}}}(H_{{ \textrm{D} }})$, and $\| N_{{\textrm{rad}}} \Phi \| \leq \frac{1}{m} \| H_{{\textrm{rad}}} \Phi \|$, $\Phi \in {\ensuremath{\mathscr{D}}}(H_{{\textrm{rad}}})$, the next corollary follows from Lemma \[9/9.e\].\ \[9/9.f\] Assume **(A.1)** - **(A.3)**. Then for all $ 0< \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$ and $ \Psi \in {\ensuremath{\mathscr{D}}}(H_m )$, $$\begin{aligned} & \textbf{(i)} \; \| (N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \| \leq \frac{L_{\epsilon}}{M} \|H_{m} \Psi \| + \frac{R_\epsilon}{M} \| \Psi \| , \\ & \textbf{(ii)} \;\| ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}} ) \Psi \| \leq \frac{L_{\epsilon }}{m} \|H_{m} \Psi \| + \frac{R_\epsilon}{m} \| \Psi \| . \end{aligned}$$ $\;$\ [**(Proof of Theorem \[Massive-Case\] )**]{}\ It is enough to show that $ \sigma_{{ \textrm{ess} }} (H_m ) \subset [E_{0} (H_m ) + m , \infty )$. Let $\lambda \in \sigma_{{ \textrm{ess} }} (H_m )$. Then by the Weyl’s theorem, there exists a sequence $\{ \Psi_n \}_{n=1}^{\infty}$ of ${\ensuremath{\mathscr{D}}}(H_m )$ such that (i) $\| \Psi_n \| =1$, $n \in {\ensuremath{\mathbf{N}}}$, (ii) s-$\lim\limits_{ n \to \infty} (H_m -\lambda ) \Psi_n =0 $, and (iii) w-$\lim\limits_{n \to \infty} \Psi_n = 0$. Since $ | \lambda- (\Psi_n , H_m \Psi_n ) | \leq | ( \Psi_n , ( H_m- \lambda )\Psi_n | \leq \| ( H_m- \lambda )\Psi_n \| $, it holds that $ \lambda = \lim\limits_{n \to \infty} (\Psi_n , H_m \Psi_n ) $. Here we show that $$\lim\limits_{n \to \infty} (\Psi_n , H_m \Psi_n ) \geq E_{0} ({H_{m}}) + m , \notag$$ and then, the proof is obtained. Let $m \leq M$. From Lemma \[LformboundHm\], $$\begin{aligned} ( \Psi_{n} ,H_m \Psi_{n} ) \geq & E_{0} (H_m) + m - M (\Psi_n , ({\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) ,\Psi_n ) \notag \\ & \qquad \qquad \qquad \quad \;\;- \delta_{\, {\textrm{f}}} (R) \| \Psi_n \|_{2,1} -\delta_{\, {\textrm{b}}, m } (R) \| \Psi_n \|_{0,2 }. \notag \end{aligned}$$ Since s-$\lim\limits_{ n \to \infty} (H_m -\lambda ) \Psi_n =0 $, we can set $$E_{m} = \sup_{n \in {\ensuremath{\mathbf{N}}}} \| H_m \Psi_n \| < \infty .$$ Let $0 \leq \lambda \leq 2 $ and $0 \leq \lambda' \leq 2$. From Corollary \[9/9.f\], it is seen that for all $ 0< \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $$\begin{aligned} \| \Psi_{n} \|_{\lambda ,\lambda ' } & = \|( N_{{ \textrm{D} }}^{\lambda /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi_n \| + \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{\lambda ' } ) \Psi_n \| + \| \Psi_n \| \notag \\ & \leq \|( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi_n \| + \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}} ) \Psi_n \| + 3 \| \Psi_n \| \notag \\ & \leq (\frac{1}{M} + \frac{1}{m} ) \left( L_{\epsilon } \|H_{m} \Psi_n \| + 2R_{\epsilon} \right) +3\| \Psi_n \| \notag \\ & \leq E_{m} L_{\epsilon} (\frac{1}{M} + \frac{1}{m} ) + 2 (\frac{1}{M} + \frac{1}{m} )R_{\epsilon} +3 . \notag\end{aligned}$$ Then we have $$( \Psi_{n} ,H_m \Psi_{n} ) \geq E_{0} (H_m) + m - M (\Psi_n , ({\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) ,\Psi_n ) - \delta_{m, \epsilon} (R) , \label{9/9.6}$$ where $\delta_{m , \epsilon} (R) = c_{\, m , \epsilon} ( \delta_{\, {\textrm{b}}} (R) + \delta_{ \, {\textrm{f}}} (R) ) $ with $c_{\, m , \epsilon } = E_{m} L_{\epsilon} (\frac{1}{M} + \frac{1}{m} ) + 2 (\frac{1}{M} + \frac{1}{m} )R_{\epsilon} +3$. It is seen that $$\begin{aligned} &\left| ( \Psi_n , ( {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) \Psi_n ) \right| \notag \\ & \qquad \qquad \leq \| (H_{0,m} +1)^{1/2} \Psi_n \| \, \| (H_{0,m} +1)^{-1/2} ({\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) \Psi_n \| . \label{9/16.1}\end{aligned}$$ From Lemma \[9/9.e\], we see that $$\| (H_{0,m} +1)^{1/2} \Psi_n \| \leq \| H_{0,m} \Psi_n \| + \|\Psi_n \| \leq L_\epsilon \| H_{m} \Psi_n \| + ( R_\epsilon +1) \|\Psi_n \| = E_{0} (H_m) L_{\epsilon} + R_\epsilon +1 ,$$ and hence, $$\sup\limits_{n \in {\ensuremath{\mathbf{N}}}} \| (H_{0,m} +1)^{1/2} \Psi_n \| \leq E_{m} L_\epsilon + R_\epsilon +1 . \label{9/16.2}$$ It holds that $$\begin{aligned} (H_{0,m} +1)^{-1/2} ( {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) = & (H_{0,m} +1)^{-1/2} ( (H_{{ \textrm{D} }} + 1)^{1/2} {\otimes}({H_{\textrm{rad}, \, m}}+1)^{1/2} ) \notag \\ & \; \times ( (H_{{ \textrm{D} }} + 1)^{-1/2} {\Gamma_{\textrm{f}}}(q_{ \, {\textrm{f}},R} ) ) {\otimes}( ({H_{\textrm{rad}, \, m}}+1)^{-1/2} {\Gamma_{\textrm{b}}}(q_{ \, {\textrm{b}},R} ) ) ) , \notag\end{aligned}$$ and hence, $(H_{0,m} +1)^{-1/2} ( {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) )$ is compact, since $\| (H_{0,m} +1)^{-1/2} ( (H_{{ \textrm{D} }} + 1)^{1/2} {\otimes}({H_{\textrm{rad}, \, m}}+1)^{1/2} )\| \leq 1$ and $ \left( (H_{{ \textrm{D} }} + 1)^{-1/2}{\Gamma_{\textrm{f}}}(q_{ \, {\textrm{f}},R}) \right) {\otimes}\left( ({H_{\textrm{rad}, \, m}}+1)^{-1/2} {\Gamma_{\textrm{b}}}(q_{ \, {\textrm{b}}, R}) \right)$ is compact. Therefore it holds that $$\lim_{n \to \infty} \left\| (H_{0,m} +1)^{-1/2} ( {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) \Psi_n \right\|=0. \label{9/16.3}$$ From (\[9/16.1\]) - (\[9/16.3\]) we have $ \lim\limits_{n \to \infty}\left| \left( \Psi_n , ({\Gamma_{\textrm{f}}}(q_{ \, {\textrm{f}},R}) {\otimes}{\Gamma_{\textrm{b}}}(q_{ \, {\textrm{b}},R}) ) ,\Psi_n \right) \right| =0 $. Then by taking the limit of (\[9/9.6\]) as $R \to \infty$, we have $ \lim\limits_{n \to \infty} (\Psi_n , H_m \Psi_n ) \geq E_{0}(H_m ) + m $. $\blacksquare$\ Derivative Bounds ================= From Theorem \[Massive-Case\], $H_{m}$ has the ground state. Let $\Psi_{m}$ be the normalized ground state of $H_{m}$, i.e. $$\qquad \qquad H_{m} \Psi_m = E_{0}(H_m ) \Psi_m , \quad \| \Psi_m \| = 1. \notag$$ Electron-Positron Derivative Bounds ----------------------------------- We introduce the distribution kernel of the annihilation operator for the Dirac field. For all $ \Psi = \left\{ \Psi^{(n)} = {}^{t}\left( \Psi^{(n)}_1 , \cdots , \Psi^{(n)}_4 \right) \right\}_{n=0}^{\infty} \in {\ensuremath{\mathscr{D}}} ( H_{{ \textrm{D} }} )$, we set $$\qquad C_{l}({\ensuremath{\mathbf{p}}})\Psi^{(n , \nu )}( {\ensuremath{\mathbf{p}}}_{1} , \cdots , {\ensuremath{\mathbf{p}}}_{n} ) = \delta_{\, l , \nu } \sqrt{n+1} \Psi^{(n+1 , \nu )}( {\ensuremath{\mathbf{p}}} , {\ensuremath{\mathbf{p}}}_{1} , \cdots , {\ensuremath{\mathbf{p}}}_{n} ) . \quad l=1 ,\cdots 4 .$$ Let $$b_{1/2} ({\ensuremath{\mathbf{p}}}) = C_{1} ({\ensuremath{\mathbf{p}}}), \; \; b_{-1/2} ({\ensuremath{\mathbf{p}}}) = C_{2} ({\ensuremath{\mathbf{p}}}) , \; \; d_{1/2} ({\ensuremath{\mathbf{p}}}) = C_{3} ({\ensuremath{\mathbf{p}}}), \; \; d_{-1/2} ({\ensuremath{\mathbf{p}}}) = C_{4} ({\ensuremath{\mathbf{p}}}) .$$ $\;$\ Then it follows that for all $\Phi \in {\mathscr{F}_{\textrm{Dirac}}}$ and $ \Psi \in {\ensuremath{\mathscr{D}}}( H_{{ \textrm{D} }} )$, $$\begin{aligned} \qquad \quad & ( \Phi , b_{s} (f) \Psi ) \, = \, \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} ( \Phi , b_{s} ({\ensuremath{\mathbf{p}}}) \Psi ) d {\ensuremath{\mathbf{p}}} , \quad \quad \; f \in {\ensuremath{\mathscr{D}}}(\omega_{\, M}) , \\ & ( \Phi , d_{s} (g) \Psi ) \, = \, \int_{{\mathbf{R}^{3} }} g({\ensuremath{\mathbf{p}}})^{\ast} ( \Phi , d_{s} ({\ensuremath{\mathbf{p}}}) \Psi ) d {\ensuremath{\mathbf{p}}} , \quad \quad \; g \in {\ensuremath{\mathscr{D}}}(\omega_{\, M}) . \end{aligned}$$ The number operator for electrons and positrons are defined by $$N_{{ \textrm{D} }}^{+} = d {\Gamma_{\textrm{f}}}\left( \left( \begin{array}{cc} {{\small \text{1}}\hspace{-0.32em}1}& O \\ O& O \end{array} \right) \right) , \qquad N_{{ \textrm{D} }}^{-} = d {\Gamma_{\textrm{f}}}\left( \left( \begin{array}{cc} O & O \\ O& {{\small \text{1}}\hspace{-0.32em}1}\end{array} \right) \right) ,$$ respectively. It holds that for all $\Phi , \Psi \in {\ensuremath{\mathscr{D}}}( H_{{ \textrm{D} }} )$, $$\begin{aligned} &(\Phi , N_{{ \textrm{D} }}^{+} \Psi ) = \sum_{\pm1/2}\int_{{\mathbf{R}^{3} }}( b_{s}({\ensuremath{\mathbf{p}}}) \Phi , b_{s}({\ensuremath{\mathbf{p}}}) \Psi) d {\ensuremath{\mathbf{p}}} , \notag \\ &(\Phi , N_{{ \textrm{D} }}^{-} \Psi ) = \sum_{\pm1/2} \int_{{\mathbf{R}^{3} }}( d_{s}({\ensuremath{\mathbf{p}}}) \Phi , d_{s}({\ensuremath{\mathbf{p}}}) \Psi) d {\ensuremath{\mathbf{p}}} \notag .\end{aligned}$$ $\; $\ By the canonical anti-commutation relation, it is proven in ([@Ta09] ; Section III) that $$\begin{aligned} &[ {\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}}) }, b_{s}(f ) ] = - \sum_{l,l'=1}^4 \alpha_{l,l'}^{j} ( f , f_{s,{\ensuremath{\mathbf{x}}}}^{l} ) \; \psi_{l'}({\ensuremath{\mathbf{x}}}) , \label{ori1} \\ &[ {\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}}) }, d_{s}(g ) ] = \sum_{l,l'=1}^4 \alpha_{l,l'}^{j} ( g ,g_{s ,{\ensuremath{\mathbf{x}}}}^{l'} ) \; \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} , \label{ori2} \end{aligned}$$ and for $\rho({\ensuremath{\mathbf{x}}})={\psi^{\dagger} (\mathbf{x}) }\psi ({\ensuremath{\mathbf{x}}})$, $$\begin{aligned} &[ \rho({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}), \; b_{s}(f ) ] = -\sum_{l=1}^4 \left( ( f , f_{s, {\ensuremath{\mathbf{y}}}}^{l} ) \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) + ( f , f_{s, {\ensuremath{\mathbf{x}}}}^{l} ) \; \psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) \right) , \label{ori3} \\ &[ \rho ({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}), \; d_{s}( g ) ] = \sum_{l=1}^4 \left( ( g , g_{s, {\ensuremath{\mathbf{y}}}}^{l} ) \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) ^{\ast} + (g , g_{s, {\ensuremath{\mathbf{x}}}}^{l} ) \; \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \rho ({\ensuremath{\mathbf{y}}}) \right) . \label{ori4}\end{aligned}$$ $\; $\ Let $X$ and $Y$ be operators on a Hilbert space. The weak commutator is defined by $$[X, \,Y ]^0 (\Phi , \Psi ) = (X^{\ast}\Phi , Y \Psi ) - (Y^{\ast}\Phi , X \Psi ) ,$$ where $ \Psi \in {\ensuremath{\mathscr{D}}}(X) \cap {\ensuremath{\mathscr{D}}} ( Y) $ and $ \Phi \in {\ensuremath{\mathscr{D}}}(X^{\ast}) \cap {\ensuremath{\mathscr{D}}} ( Y^{\ast}) $.\ \[9/12.a\] Assume **(A.1)** - **(A.3)**. Then it holds that for all $f \in L^2 ({\mathbf{R}^{3} })$, $$\begin{aligned} \textbf{(i)} & \; \; [{H_{\textrm{I}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi ) \, = \, \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , K_{s}^{+} ({\ensuremath{\mathbf{p}}}) \Psi \right) d {\ensuremath{\mathbf{p}}} , \quad \Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} ,\; \Psi \in {\ensuremath{\mathscr{D}}}(H_m ) , \\ \textbf{(ii)} & \; \; [{H_{\textrm{II}}}, b_{s}(f){\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi) \, = \, \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , (S_s^{\, +} ({\ensuremath{\mathbf{p}}}) + T_s^{+} ({\ensuremath{\mathbf{p}}}) ) \Psi \right) d {\ensuremath{\mathbf{p}}} , \quad \Phi , \Psi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} . \end{aligned}$$ Here $ K_s^{+}({\ensuremath{\mathbf{p}}})$, $S_s^{\, +} ({\ensuremath{\mathbf{p}}}) $ and $T_s^{+} ({\ensuremath{\mathbf{p}}}) $ are operators which satisfy $$\begin{aligned} & (\Phi , K_s^{+}({\ensuremath{\mathbf{p}}}) \Psi ) = -\sum_{j=1}^3 \sum_{l,l'=1}^4 \alpha^{j}_{l,l'}\, \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) \left( \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi \right) {d \mathbf{x} }, \notag \\ & (\Phi , S^{+}_{s}({\ensuremath{\mathbf{p}}}) \Psi ) = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} f_{s , {\ensuremath{\mathbf{y}}} }^{\, l}({\ensuremath{\mathbf{p}}}) \left( \Phi , \, ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }, \notag \\ &(\Phi , T^{+}_{s}({\ensuremath{\mathbf{p}}}) \Psi ) = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}}) \left( \Phi , \, (\psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }. \notag\end{aligned}$$ **(Proof)**\ **(i)** Let $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ and $ \Psi \in {\ensuremath{\mathscr{D}}}(H_m )$. By (\[ori1\]), we have $$\begin{aligned} [{H_{\textrm{I}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi) & =\sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }[ {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}} ) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) , b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 ( \Phi , \Psi ) {d \mathbf{x} }\notag \\ & = \sum_{j=1}^3\int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left( \Phi , \left( [ {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}} ), b_{s}(f) ] {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) \right) \Psi \right) {d \mathbf{x} }\notag \\ & = - \sum_{ j=1}^3 \sum_{ l, l'=1}^4 \alpha_{l , l'}^j \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\, (f, f_{s, {\ensuremath{\mathbf{x}}}}^{\, l}) \left( \Phi , ( \psi_{l'} ({\ensuremath{\mathbf{x}}} ) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}})) \Psi \right) {d \mathbf{x} }. \notag \end{aligned}$$ Let $\ell_{s, {\ensuremath{\mathbf{p}}}} : {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \times {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \to {\ensuremath{\mathbf{C}}} $ be a functional defined by $$\ell_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ') = - \sum_{j=1}^3 \sum_{l, l'=1}^4 \alpha_{l , l'}^j \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\, f_{s, {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) \left( \Phi ' , ( \psi_{l'} ({\ensuremath{\mathbf{x}}} ) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}})) \Psi '\right) {d \mathbf{x} },$$ for $ \Phi ' \in {\mathscr{F}_{\textrm{QED}}}, \Psi ' \in {\ensuremath{\mathscr{D}}}(H_{0, m} )$. We see that $$\ell_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ' ) \leq c_{ \, {\textrm{I}}, \, s, {\ensuremath{\mathbf{p}}}} \| \Phi ' \| \, \| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}( {H_{\textrm{rad}, \, m}}+1)^{1/2}) \Psi ' \| ,$$ where $c_{\, {\textrm{I}}, \, s, {\ensuremath{\mathbf{p}}}} = \sum\limits_{j=1}^3 \sum\limits_{l, l'=1}^4 | \alpha_{l , l'}^j| \, \| \chi_{\, {\textrm{I}}} \|_{L^1} \, |f_{s}^{\, l}({\ensuremath{\mathbf{p}}})|c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j} $. Then from the Riesz Representation theorem, we can define an operator $ K_{s}^{\, +} ({\ensuremath{\mathbf{p}}})$ which satisfy $\ell_{s, {\ensuremath{\mathbf{p}}}} (\Phi ', \Psi ' ) = (\Phi ' , K_{s}^{\, +} ({\ensuremath{\mathbf{p}}}) \Psi' ) $. Then it holds that $$[{H_{\textrm{I}}}, b_{s}(f) ]^0 (\Phi , \Psi) = \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \ell_{s, {\ensuremath{\mathbf{p}}}} (\Phi , \Psi ) d {\ensuremath{\mathbf{p}}} = \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , K_{s}^{+} ({\ensuremath{\mathbf{p}}}) \Psi \right) d {\ensuremath{\mathbf{p}}} .$$ **(ii)** From (\[ori3\]), we see that for all $\Phi , \Psi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$, $$\begin{aligned} [{H_{\textrm{II}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi) & = \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } [ \rho ({\ensuremath{\mathbf{x}}} ) \rho ({\ensuremath{\mathbf{y}}} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi ,\Psi ) {d \mathbf{x} }{d \mathbf{y} }\notag \\ & =\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } \left( \Phi , \left( [ \rho ({\ensuremath{\mathbf{x}}} ) \rho ({\ensuremath{\mathbf{y}}} ) , b_{s}(f) ] {\otimes}{{\small \text{1}}\hspace{-0.32em}1}\right) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }\notag \\ & = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } \left\{ ( f , f_{s, {\ensuremath{\mathbf{y}}}}^{\, l} ) ( \Phi , ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l} ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi ) \right. \notag \\ & \qquad \qquad \qquad \qquad \left. + ( f , f_{s, {\ensuremath{\mathbf{x}}}}^{\, l} ) (\Phi , \, ( \psi_{l} ({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi ) \right\} {d \mathbf{x} }{d \mathbf{y} }. \notag \end{aligned}$$ We set functionals $q_{s, {\ensuremath{\mathbf{p}}}} $ and $r_{s,{\ensuremath{\mathbf{p}}}} $ on ${\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \times {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ by $$\begin{aligned} &\quad q_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ' ) = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } f_{s, {\ensuremath{\mathbf{y}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) ( \Phi ' , ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l} ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi ' ) {d \mathbf{x} }{d \mathbf{y} }, \quad \Phi ', \, \Psi ' \in {\mathscr{F}_{\textrm{QED}}}, \\ &\quad r_{s, {\ensuremath{\mathbf{p}}}} (\Phi '', \Psi '') = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } f_{s, {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) ( \Phi '' , (\psi_{l} ({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi '' ) {d \mathbf{x} }{d \mathbf{y} }, \quad \Phi '' , \, \Psi '' \in {\mathscr{F}_{\textrm{QED}}}.\end{aligned}$$ We see that $$\begin{aligned} & q_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ' ) \leq c_{ \, {\textrm{II}}, \, s, {\ensuremath{\mathbf{p}}}} \| \Phi ' \| \, \| \Psi '\| , \notag \\ & r_{s, {\ensuremath{\mathbf{p}}}} (\Phi '' , \Psi '' ) \leq c_{ \, {\textrm{II}}, \, s, {\ensuremath{\mathbf{p}}}} \| \Phi '' \| \, \| \Psi '' \| ,\end{aligned}$$ where $c_{ \, {\textrm{II}}, \, s, {\ensuremath{\mathbf{p}}}} = \sum\limits_{l, l'=1}^4 \left\| \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}- {\ensuremath{\mathbf{y}}}| } \right\|_{L^1} \, |f_{s}^{\, l}({\ensuremath{\mathbf{p}}})|(c_{\, { \textrm{D} }}^{\, l'})^2 c_{\, { \textrm{D} }}^{\, l} $. Then from Riesz Representation theorem, we can define operators $ S_s^{+} ({\ensuremath{\mathbf{p}}})$ and $ T_s^{+} ({\ensuremath{\mathbf{p}}})$ such that $q_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ' ) = (\Phi ', S_{s}^{+} ({\ensuremath{\mathbf{p}}}) \Psi ' ) $ and $r_{{\ensuremath{\mathbf{p}}}} (\Phi '' , \Psi '' ) = (\Phi '' , T_s^{+} ({\ensuremath{\mathbf{p}}}) \Psi '' ) $, respectively. Then it holds that $$\begin{aligned} [{H_{\textrm{II}}}, b_{s}(f) ]^0 (\Phi , \Psi) & = \int_{{\mathbf{R}^{3} }} \overline{f({\ensuremath{\mathbf{p}}})} \left( q_{s, {\ensuremath{\mathbf{p}}}} (\Phi , \Psi ) + r_{ s, {\ensuremath{\mathbf{p}}}} (\Phi , \Psi ) \right) d {\ensuremath{\mathbf{p}}} \\ &= \int_{{\mathbf{R}^{3} }} \overline{f({\ensuremath{\mathbf{p}}})} \left( \Phi , (S_s^{+} ({\ensuremath{\mathbf{p}}}) + T_s^{+} ({\ensuremath{\mathbf{p}}}) ) \Psi \right) d {\ensuremath{\mathbf{p}}}.\end{aligned}$$ Thus proof is obtained. $\blacksquare $ $\; $\ In a similar way to Lemma \[9/12.a\], the following lemma is also proven.\ \[9/12.b\] Assume **(A.1)** - **(A.3)**. Then it holds that for all $g \in L^2 ({\mathbf{R}^{3} })$, $$\begin{aligned} \textbf{(i)} \; & \; \; [{H_{\textrm{I}}}, d_{s}(g) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi ) \, = \, \int_{{\mathbf{R}^{3} }} g({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , K_{s}^{\,-} ({\ensuremath{\mathbf{p}}}) \Psi \right) d {\ensuremath{\mathbf{p}}} , \qquad \Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} ,\Psi \in {\ensuremath{\mathscr{D}}}(H_m ) , \\ \textbf{(ii)} & \; \; [{H_{\textrm{II}}}, d_{s}(g){\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi) \, = \, \int_{{\mathbf{R}^{3} }} g({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , (S_s^{\, -} ({\ensuremath{\mathbf{p}}}) + T_s^{-} ({\ensuremath{\mathbf{p}}}) ) \Psi \right) d {\ensuremath{\mathbf{p}}} , \qquad \Phi , \Psi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} . \end{aligned}$$ Here $ K_s^{\, -}({\ensuremath{\mathbf{p}}})$, $S_s^{\, -} ({\ensuremath{\mathbf{p}}}) $ and $T_s^{-} ({\ensuremath{\mathbf{p}}}) $ are operators which satisfy $$\begin{aligned} & (\Phi , K_s^{\, -}({\ensuremath{\mathbf{p}}}) \Psi ) = \sum_{j=1}^3 \sum_{l,l'=1}^4 \alpha^{j}_{l,l'}\, \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }g_{s , {\ensuremath{\mathbf{x}}}}^{\, l'} ({\ensuremath{\mathbf{p}}}) \left( \Phi , \, (\psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi \right) {d \mathbf{x} }, \notag \\ & (\Phi , S^{-}_{s}({\ensuremath{\mathbf{p}}}) \Psi ) = \sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} g_{s , {\ensuremath{\mathbf{y}}} }^{\, l}({\ensuremath{\mathbf{p}}}) \left( \Phi , \, ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }, \notag \\ &(\Phi , T^{-}_{-}({\ensuremath{\mathbf{p}}}) \Psi ) = \sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} g_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}}) \left( \Phi , \, (\psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }, \notag\end{aligned}$$ $\; $ \[9/12.c\] Assume **(A.1)** - **(A.5)**. Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m )$. Then, $K_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi$, $S_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi$ and $T_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi$, $s= \pm 1/2$, are strongly differentiable for all ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }\backslash O_{{ \textrm{D} }}$. **(Proof)** We show that $K_{s}^{+}({\ensuremath{\mathbf{p}}}) \Psi$ is strongly differentiable. Let $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ with $ \| \Phi \| =1$. From **(A.4)**, $K_{s}^{+}({\ensuremath{\mathbf{p}}}) \Psi$ is weakly differentiable for all ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }\backslash O_{{ \textrm{D} }}$, and we have $$\partial_{p^{\nu}} (\Phi , K_{s}^{+ }({\ensuremath{\mathbf{p}}}) \Psi ) = -\sum_{j=1}^3 \sum_{l,l'=1}^4 \alpha^{j}_{l,l'}\, \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\partial_{p^\nu}f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) \left( \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi \right) {d \mathbf{x} },$$ and $ |\partial_{p^{\nu}} (\Phi , K_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi )| \leq \sum\limits_{j=1}^3 \sum\limits_{l , l'=1}^4 | \alpha^{j}_{l, l'} | c_{{ \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j} \left( \int_{{\mathbf{R}^{3} }}| \partial_{p^\nu}f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) | {d \mathbf{x} }\, \right) \| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}, \, m}}^{1/2} ) \Psi \| $. Then the Riesz representation theorem shows that there exists a vector $\Xi_{\Psi} ({\ensuremath{\mathbf{p}}}) \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ such that $( \Phi , \Xi_{\Psi} ({\ensuremath{\mathbf{p}}}) )= \partial_{p^{\nu}} (\Phi , K_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi )$. Let ${\ensuremath{\mathbf{e}}}_{\nu} = (\delta_{\nu , j})_{j=1}^3 $. It is seen that $$\begin{aligned} &(\Phi , \frac{K_{s}^{+ }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - K_{s}^{+ }({\ensuremath{\mathbf{p}}} )}{\epsilon} \Psi) - ( \Phi , \Xi ({\ensuremath{\mathbf{p}}}) ) \notag \\ & = -\sum_{j=1}^3 \sum_{l , l'=1}^4 \alpha^{j}_{l, l'} \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left( \frac{f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - f_{s , {\ensuremath{\mathbf{x}}}}^{\,l } ({\ensuremath{\mathbf{p}}})}{\epsilon} - \partial_{p^{\nu}} f_{s , {\ensuremath{\mathbf{x}}}}^{l } ({\ensuremath{\mathbf{p}}})\right) (\Phi , ( \psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) ) \Psi ) {d \mathbf{x} }, \notag \end{aligned}$$ and hence, $$\begin{aligned} &| (\Phi , ( \frac{K_{s}^{+ }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - K_{s}^{+ }({\ensuremath{\mathbf{p}}} )}{\epsilon} \Psi - \Xi ({\ensuremath{\mathbf{p}}})) | \notag \\ & \leq \sum_{j=1}^3 \sum_{l , l'=1}^4 | \alpha^{j}_{l, l'} | c_{{ \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j'} \left( \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \left| \frac{f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - f_{s , {\ensuremath{\mathbf{x}}}}^{\, l } ({\ensuremath{\mathbf{p}}}) }{\epsilon} - \partial_{p^{\nu}} f_{s , {\ensuremath{\mathbf{x}}}}^{\, l } ({\ensuremath{\mathbf{p}}})\right| {d \mathbf{x} }\right) \| \Psi \| . \label{9/13.1}\end{aligned}$$ Since (\[9/13.1\]) holds for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ with $ \| \Phi \| =1$, we have $$\begin{aligned} & \left\| \frac{K_{s}^{+ }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - K_{s}^{+ }({\ensuremath{\mathbf{p}}} )}{\epsilon} \Psi - \Xi ({\ensuremath{\mathbf{p}}}) \right\| \notag \\ & \leq \sum_{j=1}^3 \sum_{l , l'=1}^4 | \alpha^{j}_{l, l'} | c_{{ \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j'} \left( \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \left| \frac{f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - f_{s , {\ensuremath{\mathbf{x}}}}^{\, l } ({\ensuremath{\mathbf{p}}}) }{\epsilon} - \partial_{p^{\nu}} f_{s , {\ensuremath{\mathbf{x}}}}^{\, l } ({\ensuremath{\mathbf{p}}})\right| {d \mathbf{x} }\right) \| \Psi \| \to 0 , \notag\end{aligned}$$ as $\epsilon \to 0$. Thus $K_{s}^{+ }({\ensuremath{\mathbf{p}}} ) \Psi $ is strongly differentiable. Similarly, it is proven that $ K_{s}^{\, -}({\ensuremath{\mathbf{p}}} ) \Psi$, $ S_{s}^{\pm }({\ensuremath{\mathbf{p}}} ) \Psi$ and $ T_{s}^{\pm }({\ensuremath{\mathbf{p}}} ) \Psi$ are strongly differentiable for all ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }\backslash O_{{ \textrm{D} }}$. $\blacksquare $\ \[9/12.d\] For all $\Phi , \Psi \in {\ensuremath{\mathscr{D}}}(H_{{ \textrm{D} }})$, it holds that $$\begin{aligned} &\textbf{(i)} \; \; [H_{{ \textrm{D} }} , b_{s}(f) ]^0 (\Phi , \Psi )=- \left( \Phi , b_{s}( \omega_{\, M} f )\Psi \right) , \\ & \textbf{(ii)} \; \; [H_{{ \textrm{D} }} , d_{s}(f) ]^0 (\Phi , \Psi )=- \left( \Phi , d_{s}( \omega_{\, M} f )\Psi \right) .\end{aligned}$$ **(Proof)** It holds that for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{D}}} ( \omega_{\, M} )$, $$[H_{{ \textrm{D} }}, b^{\dagger}_{s}(f) ]\Phi = b^{\dagger}_{s}( \omega_{\, M} f) \Phi .$$ Let $\Psi \in {\ensuremath{\mathscr{D}}} (H_m )$. Then $$(H_{{ \textrm{D} }} \Phi, b_{s}(f) \Psi ) - ( b_{s}(f) \Phi , H_{{ \textrm{D} }} \Psi ) = ( [ b^{\dagger}_{s}(f) , H_{{ \textrm{D} }} ] \Phi , \Psi ) = (- b^{\dagger}_{s}( \omega_{\, M} f) \Phi , \Psi) ,$$ and hence, $$(H_{{ \textrm{D} }} \Phi, b_{s}(f) \Psi ) - ( b_{s}(f) \Phi , H_{{ \textrm{D} }} \Psi ) = -( \Phi , b_{s}( \omega_{\, M} f) \Psi) . \label{9/11.1}$$ Since $ {\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{D}}} ( \omega_{\, M} ))$ is a core of $H_{{ \textrm{D} }}$ and $ b_{s}(f)$ is bounded, (\[9/11.1\]) holds for all $\Phi \in {\ensuremath{\mathscr{D}}}(H_{{ \textrm{D} }})$. Hence **(i)** follows. Similarly, we can also prove **(ii)**. $\blacksquare $\ \[EP-Pullth\] **(Electron-Positron Pull-Through Formula)**\ Assume **(A.1)** - **(A.3)**. Then that $$\begin{aligned} & \textbf{(i)} \; \; \; (b_s ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m = (H_m- E_0 (H_m) + \omega_{M}({\ensuremath{\mathbf{p}}}))^{-1} \left( {\kappa_{\textrm{I}}}K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + {\kappa_{\textrm{II}}}( S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m , \\ & \textbf{(ii)} \; \; (d_s ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m = (H_m- E_0 (H_m) + \omega_{M}({\ensuremath{\mathbf{p}}}))^{-1} \left( {\kappa_{\textrm{I}}}K^{\,-}_{s} ({\ensuremath{\mathbf{p}}} ) + {\kappa_{\textrm{II}}}( S^{\,-}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,-}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m , \end{aligned}$$ for almost everywhere ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$. **(Proof)** Let $\Phi, \in {\ensuremath{\mathscr{D}}} (H_m )$. By Lemma \[9/12.d\] **(i)**, we have $$\begin{aligned} & [{H_{m}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) \notag \\ &= - \left( \Phi , ( b_{s}( \omega_M f ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) + {\kappa_{\textrm{I}}}[{H_{\textrm{I}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) + {\kappa_{\textrm{II}}}[{H_{\textrm{II}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) . \notag\end{aligned}$$ On the other hand, $H_m \Psi_m=E_{0}(H_m) \Psi_m$ yields that $$[{H_{m}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) = \left( ( {H_{m}}- E_{0}({H_{m}}) ) \Phi , ( b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) . \notag$$ Then, we have $$\begin{aligned} & ( ({H_{m}}- E_{0}({H_{m}})) \Phi , ( b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m ) + ( \Phi , ( b_{s}( \omega_M f ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m ) \notag \\ &\qquad \qquad \qquad = {\kappa_{\textrm{I}}}[{H_{\textrm{I}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) + {\kappa_{\textrm{II}}}[{H_{\textrm{II}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) . \notag\end{aligned}$$ By Lemma \[9/12.a\], it follows that $$\begin{aligned} & \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( ({H_{m}}- E_{0}({H_{m}}) + \omega_M ({\ensuremath{\mathbf{p}}})) \Phi , ( b_{s}({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \frac{}{} \right) d {\ensuremath{\mathbf{p}}} \notag \\ &\quad \qquad \quad = \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , \left( {\kappa_{\textrm{I}}}K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + {\kappa_{\textrm{II}}}( S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m \right) d {\ensuremath{\mathbf{p}}}. \label{9/12.3}\end{aligned}$$ Since (\[9/12.3\]) holds for all $f \in L^2 ({{\mathbf{R}^{3} }} )$, it follows that $$( ({H_{m}}- E_{0}({H_{m}}) + \omega_M ({\ensuremath{\mathbf{p}}})) \Phi , ( b_{s}({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m ) =( \Phi , \left( {\kappa_{\textrm{I}}}K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + {\kappa_{\textrm{II}}}( S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m ) , \notag$$ for almost everywhere ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$. This implies that $ ( b_{s}({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \in {\ensuremath{\mathscr{D}}} ({H_{m}}) $ and $$( {H_{m}}- E_{0}({H_{m}}) + \omega_M ({\ensuremath{\mathbf{p}}})) ( b_{s}({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m = \left( {\kappa_{\textrm{I}}}K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + {\kappa_{\textrm{II}}}( S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m . \label{9/12.7}$$ From (\[9/12.7\]), we obtain **(i)**. Similarly, **(ii)** is also proven. $\blacksquare $\ \[EP-DB\] (**Electron-Positron Derivative Bounds**)\ Assume **(A.1)** - **(A.5)**. Then, it holds that for all ${\ensuremath{\mathbf{p}}} \in {\ensuremath{\mathbf{R}}}^3 \backslash O_{{ \textrm{D} }}$ and $0 < \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $$\begin{aligned} &\textbf{(i)} \; \; \left\| \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \right\| \leq \left( ( L_{\epsilon} E_{0}(H_{m} ) + R_\epsilon + 1 \,) |{\kappa_{\textrm{I}}}|+ 2|{\kappa_{\textrm{II}}}| \frac{}{} \right) F_{s , +}^{\nu}({\ensuremath{\mathbf{p}}}) , \\ &\textbf{(ii)} \; \; \left\| \partial_{p^{\nu}} (d_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \right\| \leq \left( ( L_{\epsilon} E_{0}(H_{m} ) + R_\epsilon + 1 \,) |{\kappa_{\textrm{I}}}|+ 2|{\kappa_{\textrm{II}}}| \frac{}{} \right) F_{s , -}^{\nu}({\ensuremath{\mathbf{p}}}) . \end{aligned}$$ Here $ F_{s ,\pm }^{\nu} $ are functions satisfying $ F_{s , \pm }^{\nu} \in L^{2} ({\mathbf{R}^{3} }) $, $s= \pm 1/2$, $\nu=1 , \cdots , 3$. **(Proof)** Let $R_{m,M}({\ensuremath{\mathbf{p}}}) =(H_m- E_0 (H_m) + \omega_{\, M}({\ensuremath{\mathbf{p}}}))^{-1} $. From Proposition \[EP-Pullth\] it holds that for all $\Phi \in {\mathscr{F}_{\textrm{QED}}}$ with $ \| \Phi \|=1$, $$\begin{aligned} ( \Phi , \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} ) & = {\kappa_{\textrm{I}}}\left( \Phi, \partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) + {\kappa_{\textrm{II}}}\left( \Phi, \partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) \notag \\ & \qquad \qquad + {\kappa_{\textrm{II}}}\left( \Phi, \partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) . \label{9/13.3} \end{aligned}$$ Here we evaluate the three terms in the right-hand side of (\[9/13.3\]) as follows.\ (First term) We see that $$\begin{aligned} & \left( \Phi,\partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) \notag \\ & =- \sum_{j=1}^{3} \sum_{l,l'=1}^{4} \alpha^{j}_{l,l'}\, \partial_{p^{\nu}} \left( f_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\right) , \notag \\ & = -\sum_{j=1}^{3} \sum_{l,l'=1}^{4} \alpha^{j}_{l,l'}\, \left\{ ( \partial_{p^{\nu}} f_{s}^{\, l} ({\ensuremath{\mathbf{p}}})) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\right. , \notag \\ & \qquad \qquad \qquad -i f_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }x^{\nu} \, e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\notag \\ & \qquad \qquad \qquad \left. - \frac{ f_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) p^{\nu} }{ \omega_{\, M}({\ensuremath{\mathbf{p}}})} \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}})^2 \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\right\} \notag .\end{aligned}$$ Since $ \| R_{m,M}({\ensuremath{\mathbf{p}}})\| \leq \frac{1}{\omega_{\, M}({\ensuremath{\mathbf{p}}})} \leq \frac{1}{M}$ and $\|\Phi \|=1$, we have [$$\begin{aligned} & \left| \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\right| \leq \frac{ c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j}}{ M} \| \chi_{{\textrm{I}}} \|_{L^1} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}+1)^{1 /2} ) \Psi_{m} \|, \\ & \left| \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }x^{\nu} \, e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\right| \leq \frac{ c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j}}{ M} \, \| |{\ensuremath{\mathbf{x}}}| \chi_{{\textrm{I}}} \|_{L^1} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}+1)^{1 /2} ) \Psi_{m} \|, \\ & \left| \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}})^2 \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\right| \leq \frac{c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j}}{ M^2} \, \| \chi_{{\textrm{I}}} \|_{L^1} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}+1)^{1 /2} ) \Psi_{m} \| .\end{aligned}$$ ]{} It is seen that $ \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}^{1/2} +1)^{1/2}) \Psi_m \| \leq \| H_{0,m} \Psi_{m} \| + \| \Psi_m\| = \| H_{0,m} \Psi_{m} \| +1 $, and hence, $$\begin{aligned} & \left| \partial_{p^{\nu}} \left( \Phi, R_{m,M}({\ensuremath{\mathbf{p}}}) K_{s}^{\, +} ({\ensuremath{\mathbf{p}}})\Psi_m \right) \right| \notag \\ & \leq \| (1+ |{\ensuremath{\mathbf{x}}}|) \chi_{{\textrm{I}}} \|_{L^1} \sum_{j=1}^{3} \sum_{l,l'=1}^{4} c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j} \left( \frac{| \partial_{p^{\nu}}f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) |}{ M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{M^2} \right) \, \left( \| H_{0,m} \Psi_m \| +1 \frac{}{} \right) . \label{9/13.I} \end{aligned}$$ (Second term) It is seen that $$\begin{aligned} & \left( \Phi, \partial_{p^{\nu}}R_{m,M}({\ensuremath{\mathbf{p}}}) S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) \notag \\ & =- \sum_{l=1}^4 \partial_{p^{\nu}} \left( f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\right) \notag \\ & =- \sum_{l=1}^4 \left\{ ( \partial_{p^{\nu}} f_{s }^{\, l}({\ensuremath{\mathbf{p}}})) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\right. \notag \\ &\qquad \quad - i f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} y^{\nu} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\notag \\ & \qquad \quad \left. - \frac{f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) p^{\nu} }{ \omega_{\, M}({\ensuremath{\mathbf{p}}})} \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}})^2 \Phi , \, ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\right) . \label{9/13.9}\end{aligned}$$ By evaluating the right-hand side of (\[9/13.9\]), we have $$\left| \partial_{p^{\nu}} \left( \Phi, R_{m,M}({\ensuremath{\mathbf{p}}}) S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) \right| \leq \| (1+ |{\ensuremath{\mathbf{x}}}|) \chi_{{\textrm{I}}} \|_{L^1} \sum_{l,l'=1}^{4} ( c_{\, { \textrm{D} }}^{\, l'})^2 c_{\, { \textrm{D} }}^{\, l}\left( \frac{| \partial_{p^{\nu}}f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) |}{ M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{ M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{M^2} \right) . \label{9/13.II}$$ (Third term) We see that $$\begin{aligned} & \left( \Phi, \partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) \notag \\ & =-\sum_{l=1}^4 \partial_{p^{\nu}} \left( f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, ( \psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\right) \notag \\ & =- \sum_{l=1}^4 \left\{ ( \partial_{p^{\nu}} f_{s }^{\, l}({\ensuremath{\mathbf{p}}})) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\right. \notag \\ &\qquad \quad - i f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} y^{\nu} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\notag \\ & \qquad \quad \left. - \frac{f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) p^{\nu} }{ \omega_{\, M}({\ensuremath{\mathbf{p}}})} \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}})^2 \Phi , \, (\psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\right) . \label{9/13.10}\end{aligned}$$ We estimate the right-hand side of the absolute value of (\[9/13.10\]), and then, $$\left| \partial_{p^{\nu}} \left( \Phi, R_{m,M}({\ensuremath{\mathbf{p}}}) T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) \right| \leq \| (1+ |{\ensuremath{\mathbf{x}}}|) \chi_{{\textrm{I}}} \|_{L^1} \sum_{l,l'=1}^{4}c_{\, { \textrm{D} }}^{\, l} ( c_{\, { \textrm{D} }}^{\, l'})^2 \left( \frac{| \partial_{p^{\nu}}f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) |}{M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{ M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{M^2} \right) . \label{9/13.III}$$ From (\[9/13.I\]), (\[9/13.II\]) and (\[9/13.III\]), we have $$\begin{aligned} & \left| ( \Phi , \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} ) \right| \notag \\ & \leq \sum_{l=1}^4 c_{+}^{\,l} \left( \frac{| \partial_{p^{\nu}}f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) |}{ M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{ M^2} \right) \left( |\kappa_{\, {\textrm{I}}}| \, \| H_{0,m} \Psi_m \| + |{\kappa_{\textrm{I}}}| + 2 |\kappa_{\, {\textrm{II}}}| \frac{}{} \right) , \notag \end{aligned}$$ where $c_{+}^{\,l} = \| (1+ |{\ensuremath{\mathbf{x}}}|) \chi_{{\textrm{I}}} \|_{L^1} \, \times \max \left\{ \sum\limits_{j=1}^{3}\sum\limits_{l'=1}^{4} |\alpha^j_{l,l'}| c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j} , \; \sum\limits_{l'=1}^{4} ( c_{\, { \textrm{D} }}^{\, l'})^2 c_{\, { \textrm{D} }}^{\, l} \right\}$. By the definition of $f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) = \frac{\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) \, u_{s}^{\,l}({\ensuremath{\mathbf{p}}})}{ \sqrt{(2 \pi )^3 } }$, we have $$\left| ( \Phi , \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} ) \right| \leq F_{s ,+}^{\nu}({\ensuremath{\mathbf{p}}}) \, \left( |\kappa_{\, {\textrm{I}}}| \, \| H_{0,m} \Psi_m \| + |{\kappa_{\textrm{I}}}|+ 2 |\kappa_{\, {\textrm{II}}}| \frac{}{} \right) , \label{9/13.11}$$ where $$F_{s ,+}^{\nu}({\ensuremath{\mathbf{p}}}) = \frac{1}{\sqrt{(2 \pi )^3}} \sum_{l=1}^4 c_{+}^{\, l} \left( \frac{| \partial_{p^{\nu}} \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) \, |}{ M} + \frac{ | \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) \partial_{p^{\nu}} u_{s}^{\,l}({\ensuremath{\mathbf{p}}}) | \, }{M} + \frac{ |\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) | \, }{ M} + \frac{ |\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) | \, }{ M^2} \right) .$$ We see that (\[9/13.11\]) holds for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ with $\| \Phi \| =1$, and this implies that $$\left\| \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} ) \right\| \leq F_{s ,+}^{\nu}({\ensuremath{\mathbf{p}}}) \, \left( |\kappa_{\, {\textrm{I}}}| \| H_{0,m} \Psi_m \| +| {\kappa_{\textrm{I}}}| + 2 |\kappa_{\, {\textrm{II}}}| \frac{}{} \right) . \notag$$ From Lemma \[9/9.e\], it holds that for all $0 < \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $$\| H_{0,m} \Psi_{m} \| \leq \, L_{\epsilon} \| H_{m} \Psi_m \| + R_{\epsilon} \| \Psi_m \| = \, L_{\epsilon} E_{0}(H_m) + R_{\epsilon} .$$ Thus **(i)** is obtained. Similarly, **(ii)** is also proven in a same way as **(i)**. $\blacksquare $\ Photon Derivative Bound ----------------------- In a similar to the Dirac field, we introduce the distribution kernel of the annihilation operator for the radiation field. For all $ \Psi = \left\{ \Psi^{(n)} = \left( \Psi^{(n)}_1 , \Psi^{(n)}_2 \right) \right\}_{n=0}^{\infty} \in {\ensuremath{\mathscr{D}}} ( H_{{\textrm{rad}}, m } )$, we define $a_{r}({\ensuremath{\mathbf{k}}})$, by $$a_{r}({\ensuremath{\mathbf{k}}})\Psi^{(n)}_{\varrho} ( {\ensuremath{\mathbf{k}}}_{1} , \cdots , {\ensuremath{\mathbf{k}}}_{n} ) = \delta_{\, r , \varrho } \sqrt{n+1} \Psi^{(n+1 )}_{\varrho }( {\ensuremath{\mathbf{k}}} , {\ensuremath{\mathbf{k}}}_{1} , \cdots , {\ensuremath{\mathbf{k}}}_{n} ) , \qquad \varrho = 1,2.$$ It holds that $$\qquad \qquad (\Phi , a_{r}(h) \Psi )= \int_{{\mathbf{R}^{3} }}h({\ensuremath{\mathbf{k}}})^{\ast}(\Phi , a_{r}({\ensuremath{\mathbf{k}}}) \Psi) d {\ensuremath{\mathbf{k}}}, \quad \Phi \in {\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} , \; \Psi \in {\ensuremath{\mathscr{D}}}(H_{{\textrm{rad}}, m}) .$$ \[9/13.c\]Assume (**A.2**). Then for all $\Phi , \Psi \in {\ensuremath{\mathscr{D}}}(H_{{\textrm{rad}},m })$, $$\begin{aligned} & \textbf{(i)} \; \; [H_{{\textrm{rad}}, m } \, , a_{r} (h)]^0 (\Phi , \Psi )= \left( \Phi , a_{r}( \omega_m h )\Psi \right) , \\ &\textbf{(ii)} \; \; [A_{j}({\ensuremath{\mathbf{x}}} ) , a_{r}(h) ]^0 (\Phi , \Psi )= - (h, h_{r , {\ensuremath{\mathbf{x}}}}^j )\left( \Phi , \Psi \right) . \end{aligned}$$ **(Proof)** It holds that for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{\textrm{rad}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{D}}} ( \omega_{\, m} ))$, $$\begin{aligned} & [ H_{{\textrm{rad}}, m}, a^{\dagger}_{r}(h) ]\Phi = - a^{\dagger}_{r}(\omega_{\,m }h)\Phi , \label{9/14.1} \\ &[ A_{j} ({\ensuremath{\mathbf{x}}}), a^{\dagger}_{r}(h) ]\Phi = ( h_{r , {\ensuremath{\mathbf{x}}}}^j ,h ) \Phi .\label{9/14.2}\end{aligned}$$ In a similar way to Lemma \[9/12.d\], we can prove **(i)** by (\[9/14.1\]) and **(ii)** by (\[9/14.2\]). $\blacksquare $\ \[9/13.d\] Assume **(A.1)** - **(A.3)**. Then\ **(i)** it holds that for all $\Phi , \Psi \in {\ensuremath{\mathscr{D}}}(H_m )$, $$[{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ( h )]^0 (\Phi , \Psi) \, = \, \int_{{\mathbf{R}^{3} }} h({\ensuremath{\mathbf{k}}})^{\ast} \left( \Phi , Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi \right) d {\ensuremath{\mathbf{k}}} . \notag$$ Here $ Q_r ({\ensuremath{\mathbf{k}}})$ is an operator which satisfy $$(\Phi , Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi ) = - \sum_{j=1}^3 \, \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }h_{r, {\ensuremath{\mathbf{x}}}}^{\, j} ({\ensuremath{\mathbf{k}}}) \left( \Phi , \, ({\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}}) }{\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi \right) {d \mathbf{x} }$$ with $ \| Q_{r} ({\ensuremath{\mathbf{k}}}) \| \leq \|\chi_{{\textrm{I}}} \|_{L^1} \sum\limits_{j=1}^3 \sum\limits_{l,l'=1}^4 |h_{r}^j ({\ensuremath{\mathbf{k}}})| \, |\alpha^{j}_{l,l'}| \, |c_{\, { \textrm{D} }}^{\, l}| \, |c_{\, { \textrm{D} }}^{\, l'}|$.\ **(ii)** Additionally assume **(A.4)** and **(A.6)**. Then, $Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi$ is strongly differential for all ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }\backslash O_{{\textrm{rad}}}$. **(Proof)** **(i)** Let $\Phi \in {\ensuremath{\mathscr{D}}}(H_m)$ From Lemma \[9/13.c\], $$\begin{aligned} [{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h ) ]^0 (\Phi , \Psi ) &= \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }[( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}}) {\otimes}A_{j}({\ensuremath{\mathbf{x}}}), {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h)]^0( \Phi , \Psi ) {d \mathbf{x} }\notag \\ &= \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }[ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}A_{j}({\ensuremath{\mathbf{x}}}) , {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h) ]^0 ( \Phi ,( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi ) {d \mathbf{x} }\notag \\ & =- \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }( h, h_{r ,{\ensuremath{\mathbf{x}}}}^j )( \Phi ,( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi ) {d \mathbf{x} }.\end{aligned}$$ We define $\ell_{r, {\ensuremath{\mathbf{k}}}}:{\mathscr{F}_{\textrm{QED}}}{\otimes}{\mathscr{F}_{\textrm{QED}}}\to {\ensuremath{\mathbf{C}}}$ by $$\ell_{r, {\ensuremath{\mathbf{k}}}} (\Phi ' , \Psi ') = - \sum_{j=1}^3 h_{r}^j ({\ensuremath{\mathbf{k}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}} \, ( ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Phi ', \Psi ' ) {d \mathbf{x} }$$ We see that $|\ell_{r, {\ensuremath{\mathbf{k}}}} (\Phi ' , \Psi ' ) | \leq \|\chi_{{\textrm{I}}} \|_{L^1} \sum\limits_{j=1}^3 \sum\limits_{l,l'=1}^4 |h_{r}^j ({\ensuremath{\mathbf{k}}})| \, |\alpha^{j}_{l,l'}| \, |c_{\, { \textrm{D} }}^{\, l}| \, |c_{\, { \textrm{D} }}^{\, l'}|\, \| \Phi ' \| \, \| \Psi' \| $. By Riesz representation theorem, we can define an operator $Q_{r} ({\ensuremath{\mathbf{k}}})$ such that $\ell_{r, {\ensuremath{\mathbf{k}}}} (\Phi ' , \Psi ') = (\Phi ', Q_{r}({\ensuremath{\mathbf{k}}}) \Psi ')$. Then we have $$[{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(f) ]^0 (\Phi , \Psi ) = \int_{{\mathbf{R}^{3} }} h ({\ensuremath{\mathbf{k}}})^{\ast} \ell_{r, {\ensuremath{\mathbf{k}}}} (\Phi , \Psi ) {d \mathbf{k} }= \int_{{\mathbf{R}^{3} }} h ({\ensuremath{\mathbf{k}}})^{\ast} \left( \Phi ,Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi \right) {d \mathbf{k} }. \notag$$ Then **(i)** is obtained.\ **(ii)** The strong differentiability of $ Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi $ is proven by **(A.4)** and **(A.6)** in a similar way to Lemma \[9/12.c\], and the proof is omitted. $\blacksquare $.\ \[PF-P\] ()\ Assume **(A.1)** - **(A.3)**. Then it holds that for almost everywhere ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }$, $$({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}})) \Psi_m ={\kappa_{\textrm{I}}}(H_m- E_0 (H_m) + \omega_{m}({\ensuremath{\mathbf{k}}}))^{-1} Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi_m .$$ **(Proof)** Let $\Phi \in {\ensuremath{\mathscr{D}}}(H_m) $. By Lemma \[9/13.c\] **(i)**, $$[{H_{m}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h ) ]^0 (\Phi , \Psi_m )= - \left( \Phi , ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ( \omega_m h)) \Psi_m \right) + {\kappa_{\textrm{I}}}\, [{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h) ]^0 (\Phi , \Psi_m ) . \notag$$ It also holds that $$[{H_{m}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h) ]^0 (\Phi , \Psi_m ) = \left( (H_{m}-E_{0}({H_{m}}) ) \Phi , ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h)) \Psi_m \right) . \notag$$ Then we have $$\left( (H_{m}-E_{0}({H_{m}}) ) \Phi , ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h)) \Psi_m \right) + \left( \Phi, ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(\omega_m h)) \Psi_m \right) = {\kappa_{\textrm{I}}}[{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h )]( \Phi, \Psi_{m} ). \notag$$ By Lemma \[9/13.d\], $$\int_{{\mathbf{R}^{3} }} h ({\ensuremath{\mathbf{k}}})^{\ast} \left( (H_{m}-E_{0}({H_{m}})+\omega_{m}({\ensuremath{\mathbf{k}}}) ) \Phi , ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \right) {d \mathbf{k} }= {\kappa_{\textrm{I}}}\int_{{\mathbf{R}^{3} }} h ({\ensuremath{\mathbf{k}}})^{\ast} \left( \Phi ,Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi_m \right) {d \mathbf{k} }. \label{9/13.14}$$ Note that (\[9/13.14\]) holds for all $h \in L^2 ({\mathbf{R}^{3} })$. Then we have $$\left( (H_{m}-E_{0}({H_{m}})+\omega_{m}({\ensuremath{\mathbf{k}}}) ) \Phi , ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \right) = \left( \Phi ,\, {\kappa_{\textrm{I}}}Q_r ({\ensuremath{\mathbf{k}}}) \Psi_m \right) , \label{9/13.15}$$ for almost everywhere $ {\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }$. In addition, (\[9/13.15\]) yields that $ ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \in {\ensuremath{\mathscr{D}}} (H_m )$ and $$(H_{m}-E_{0}({H_{m}})+\omega_{m}({\ensuremath{\mathbf{k}}}) ) ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m = {\kappa_{\textrm{I}}}Q_r ({\ensuremath{\mathbf{k}}}) \Psi_m . \notag$$ Thus the proof is obtained. $\blacksquare $ \[P-DB\] (**Photon Derivative Bounds**)\ Assume **(A.1)**-**(A.4)** and **(A.6)**. Then it holds that for all ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }\backslash O_{{\textrm{rad}}}$, $$\left\| \partial_{k^\nu} ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}})) \Psi_m \right\| \leq |{\kappa_{\textrm{I}}}| F_{r}^{\, \nu} ({\ensuremath{\mathbf{k}}}) \notag$$ where $ F_{r}^{\, \nu} $ is a function which satisfy $ F_{r}^{\, \nu} \in L^2 ({\mathbf{R}^{3} }) $. **(Proof)** $\; $\ Let $ R_{m}({\ensuremath{\mathbf{k}}}) = (H_m- E_0 (H_m) + \omega_{\, m}({\ensuremath{\mathbf{k}}}))^{-1} $. From Proposition \[PF-P\], it holds that ${{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}}) \Psi_m = R_{m}({\ensuremath{\mathbf{k}}}) Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi_m $. Then for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$, $$\begin{aligned} & = ( \Phi , \partial_{k^\nu } ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}})) \Psi_m ) \notag \\ & =- {\kappa_{\textrm{I}}}\sum_{j=1}^3 \, \partial_{k^{\nu}} \left( h_{r}^{\, j} ({\ensuremath{\mathbf{k}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m}({\ensuremath{\mathbf{k}}}) \Phi , \, ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) {d \mathbf{x} }\right) \notag \\ & =- {\kappa_{\textrm{I}}}\sum_{j=1}^3 \, \left\{ ( \partial_{k^{\nu}} h_{r}^{\, j} ({\ensuremath{\mathbf{k}}})) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m}({\ensuremath{\mathbf{k}}}) \Phi , \, ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) {d \mathbf{x} }\right. \notag \\ & \qquad \qquad \qquad -i h_{r}^{\, j} ({\ensuremath{\mathbf{k}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }x^{\nu} \, e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m}({\ensuremath{\mathbf{k}}}) \Phi , \, ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) {d \mathbf{x} }\notag \\ & \qquad \qquad \qquad \left. - \frac{ h_{r}^{\, j} ({\ensuremath{\mathbf{k}}}) k^{\nu} }{\omega_{\, m}({\ensuremath{\mathbf{k}}})} \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m}({\ensuremath{\mathbf{k}}})^2 \Phi , \, ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) {d \mathbf{x} }\right\} . \label{9/13.16}\end{aligned}$$ By estimating the absolute value of the right-hand side of (\[9/13.16\]), we have $$\begin{aligned} & \left| \partial_{k^{\nu}} \left( \Phi, R_{m}({\ensuremath{\mathbf{k}}})Q ({\ensuremath{\mathbf{k}}})\Psi_m \right) \right| \notag \\ & \leq \| (1+ |{\ensuremath{\mathbf{x}}}| ) \chi_{{\textrm{I}}} \|_{L^1}|{\kappa_{\textrm{I}}}| \sum_{j=1}^3 \sum_{l,l'=1}^4 \, | \alpha^j_{l,l'} | \, | c_{\, { \textrm{D} }}^{\,l}| \, | c_{\, { \textrm{D} }}^{\,l'}| \, \left( \frac{| \partial_{k^{\nu}} h_{r }^{\, j}({\ensuremath{\mathbf{k}}}) |}{\omega_{\, m}({\ensuremath{\mathbf{k}}})} + \frac{ | h_{r }^{\, j}({\ensuremath{\mathbf{k}}}) | \, }{\omega_{\, m}({\ensuremath{\mathbf{k}}})} + \frac{ | h_{r }^{\, j}({\ensuremath{\mathbf{k}}}) | \, }{\omega_{\, m}({\ensuremath{\mathbf{k}}})^2} \right) . \notag \end{aligned}$$ From the definition of $h_{r}^{\, j}({\ensuremath{\mathbf{k}}})=\frac{\chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) e_{r}^{\,j}({\ensuremath{\mathbf{k}}}))}{\sqrt{2 (2 \pi )^3 \omega ({\ensuremath{\mathbf{k}}}) }}$, we have $$\partial_{k^\nu} h_{r}^{\, j}({\ensuremath{\mathbf{k}}}) = \frac{1}{\sqrt{2 (2 \pi )^3 }} \left( \frac{ ( \partial_{k^\nu} \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) ) e_{r}^{\,j}({\ensuremath{\mathbf{k}}}) }{\omega({\ensuremath{\mathbf{k}}})^{1/2}} + \frac{ \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) \partial_{k^\nu} e_{r}^{\,j}({\ensuremath{\mathbf{k}}})}{\omega ({\ensuremath{\mathbf{k}}})^{1/2}} - \frac{1}{2} \frac{ \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) k^{\nu}}{ \omega ({\ensuremath{\mathbf{k}}})^{5/2}} \right) . \notag$$ Hence, it holds that $$\left| ( \Phi , \partial_{k^\nu } ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}})) \Psi_{m} ) \right| \leq | {\kappa_{\textrm{I}}}| \, F_{r}^{\nu}({\ensuremath{\mathbf{k}}}) , \label{9/13.17}$$ where $$\begin{aligned} F_{r}^{\nu}({\ensuremath{\mathbf{k}}}) =\frac{\| (1+ |{\ensuremath{\mathbf{x}}}| ) \chi_{{\textrm{I}}} \|_{L^1}}{\sqrt{2 (2 \pi )^3 }} & \sum_{j=1}^3 \sum_{l,l'=1}^4 \, | \left\{ \alpha^j_{l,l'} | \, | c_{\, { \textrm{D} }}^{\,l}| \, | c_{\, { \textrm{D} }}^{\,l'}| \, \frac{}{} \right.\notag \\ & \times \left. \left( \frac{| \partial_{k^\nu} \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |+| \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) \partial_{k^\nu} e_{r}^{\,j}({\ensuremath{\mathbf{k}}})| + | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) | }{ \omega ({\ensuremath{\mathbf{k}}})^{3/2}} + \frac{3}{2} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |}{ \omega ({\ensuremath{\mathbf{k}}})^{5/2}} \right) \right\} . \notag \end{aligned}$$ Since (\[9/13.17\]) holds for all $\Phi \in {\mathscr{F}_{\textrm{QED}}}$, we have $$\| \partial_{k^{\nu}} ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_{m} \| \leq | {\kappa_{\textrm{I}}}| F_{r}^{\nu}({\ensuremath{\mathbf{k}}}) .$$ The condition **(A.6)** yields that $ F_{r}^{\nu} \in L^2 ({\mathbf{R}^{3} }) $, and hence the proof is obtained. $\blacksquare $ Proof of Theorem \[Main-Theorem\] ================================== Let $\{ \Psi_m\}_{m>0}$ be the sequence of the normalized ground state of $H_m$, $m>0$. Then there exists a subsequence of $\{ \Psi_{m_{j}} \}_{j=1}^{\infty}$ with $m_{j+1} < m_{j} $, $j \in {\ensuremath{\mathbf{N}}}$, such that the weak limit $\Psi_{0} := $w-$\lim\limits_{j \to \infty } \Psi_{m_{j}} $ exists. \[9/16.a\] Suppose **(A.1)** - **(A.3)**. Then,\ $\qquad $ **(i)** ${\ensuremath{\mathscr{D}}}_{0}$ is a common core of $H_{{ \textrm{QED} }}$ and $H_{m}$, $m>0$, and $H_{m}$ strongly converges to $H_{{ \textrm{QED} }}$ on ${\ensuremath{\mathscr{D}}}_{0}$\ $\qquad $ **(ii)** $ \lim\limits_{m \to \infty} E_{0}(H_m) = E_{0}(H_{{ \textrm{QED} }} ) $. **(i)** Since ${\ensuremath{\mathscr{D}}}_{0}$ is a core of $H_{0,m}$, ${\ensuremath{\mathscr{D}}}_{0}$ is also a core of $H_{m}$. It is directly proven that $ \lim\limits_{m \to 0} H_{m} \Psi = H_{{ \textrm{QED} }} \Psi $ for all $\Psi \in {\ensuremath{\mathscr{D}}}_0$.\ **(ii)** We see that $(\Psi , H_{m} \Psi ) \geq ( \Psi, H_{{ \textrm{QED} }} \Psi ) \geq {E_{0}(H_{{ \textrm{QED} }})}$, for all $\Psi \in {\ensuremath{\mathscr{D}}}_{0}$. Hence $\inf\limits_{m>0}E_{0}(H_{m}) \geq E_{0}(H_m )$. From **(i)**, it follows that $H_{m}$ converges to $H_{{ \textrm{QED} }}$ as $m \to 0$ in the strong resolvent sense, and this yields that $\limsup\limits_{m \to 0}E_{0}(H_{m}) \leq E_{0}({H_{\textrm{QED}}})$. Hence **(ii)** follows. $\blacksquare $\ $\;$\ From Lemma \[9/16.a\] **(ii)**, we can set $$E_{\infty} = \sup\limits_{j\in {\ensuremath{\mathbf{N}}}} |E_{0}(H_{m_j })| \; < \infty . \\$$ \[9/14.a\] **(Number Operator Bounds)**\ Suppose **(A.1)** - **(A.6)**. Then, for all $ 0 < \epsilon < \frac{1}{c_{\textrm{I}}| {\kappa_{\textrm{I}}}|} ,$ $$\begin{aligned} \quad {\ensuremath{\mathbf{(i)}}} \; \; & \sup_{j\in {\ensuremath{\mathbf{N}}} } \| (N_{{ \textrm{D} }}^{1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m_j } \| \leq \left( \frac{L_{\epsilon}}{M}E_{\infty} + \frac{R_{\epsilon} }{M}\right)^{1/2} , \notag \\ {\ensuremath{\mathbf{(ii)}}} \; \;& \sup_{j\in {\ensuremath{\mathbf{N}}} } \| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_{m_j } \| \leq c_0 |{\kappa_{\textrm{I}}}| \, \left\| \frac{ \chi_{{\textrm{rad}}}}{\omega^{3/2}} \right\| , \notag \end{aligned}$$ where $c_{0} = \sqrt{\frac{11}{2(2\pi)^3}} \sum\limits_{j=1}^3 \sum\limits_{l,l'=1}^4 |\alpha^{j}_{l,l'}| c_{{ \textrm{D} }}^{\,l} c_{{ \textrm{D} }}^{\,l'} $. **(Proof)** **(i)** We see that $ \|( N_{{ \textrm{D} }}^{1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \|^2 = (\Psi_m , ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} )\leq \|( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \|$, and Corollary \[9/9.f\] yields that for all $0 < \epsilon < \frac{1}{c_{\textrm{I}}| {\kappa_{\textrm{I}}}| }$, $$\|( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \| \leq \frac{L_{\epsilon}}{M} \| H_m \Psi_m \| + \frac{ R_{\epsilon} }{M}\|\Psi_m \| = \frac{L_{\epsilon}}{M}E_{0}(H_m ) + \frac{R_{\epsilon}}{M} . \notag$$ Hence **(i)** follows.\ **(ii)** From the photon pull-through formula in Proposition \[PF-P\], it follows that $$\begin{aligned} (\Psi_m , ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}) \Psi_m ) & = \sum_{r=1,2} \int_{{\mathbf{R}^{3} }} \| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \|^2 d {\ensuremath{\mathbf{k}}} \notag \\ & = |{\kappa_{\textrm{I}}}|^2 \sum_{r=1,2} \int_{{\mathbf{R}^{3} }}\| (H_m -E_{0}(H_m ) + \omega_{m}({\ensuremath{\mathbf{k}}})) Q_{r}({\ensuremath{\mathbf{k}}}) \Psi_m \|^2 d {\ensuremath{\mathbf{k}}} \notag \\ & \leq |{\kappa_{\textrm{I}}}|^2 \frac{11}{{2(2\pi)^3}} \sum_{r=1,2} \sum_{j=1}^3 \sum_{l,l'=1}^4 |\alpha^{j}_{l,l'}|^2 ( c_{{ \textrm{D} }}^{\,l} c_{{ \textrm{D} }}^{\,l'} )^2 \left( \int_{{\mathbf{R}^{3} }} \frac{| \chi_{{\textrm{rad}}}|^2}{|{\ensuremath{\mathbf{k}}}|^3} d {\ensuremath{\mathbf{k}}} \right) . \label{9/15.1}\end{aligned}$$ From (\[9/15.1\]), we obtain **(ii)**. $\blacksquare $\ \[9/15.a\] Assume **(A.1)**-**(A.6)**. Let $ F \in C_0^{\, \infty} ({\ensuremath{\mathbf{R}}}^3)$ which satisfy $0 \leq F \leq 1$ and $F({\ensuremath{\mathbf{x}}})=1$ for $|{\ensuremath{\mathbf{x}}}| \leq 1$, and set $ F_{ R}({\ensuremath{\mathbf{x}}}) = F (\frac{ {\ensuremath{\mathbf{x}}} }{R} ) $. Let $\hat{{\ensuremath{\mathbf{p}}}} = - i \nabla_{{\ensuremath{\mathbf{p}}}} $ and $\hat{{\ensuremath{\mathbf{k}}}} = - i \nabla_{{\ensuremath{\mathbf{k}}}} $. Then for all $0 < \epsilon < \frac{1}{c_{{\textrm{I}}} {\kappa_{\textrm{I}}}}$, $R \geq 1$ and $R' \geq 1$, $$\begin{aligned} &\textbf{(i)} \; \; \, \sup_{j\in {\ensuremath{\mathbf{N}}} } \| ( {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{f}}}( F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m_j } \|) \leq \frac{c_{1,\epsilon }}{\sqrt{R}} , \\ &\textbf{(ii)} \; \; \, \sup_{ j\in {\ensuremath{\mathbf{N}}}} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}( {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) )) \Psi_{m_j } \| ) \leq \frac{c_2}{\sqrt{R'}} , \notag \end{aligned}$$ where $$c_{1, \epsilon } = \left( 4 \frac{L_{\epsilon} E_{\infty} + R_{\epsilon}}{M}\right)^{1/4} \left( \left( \frac{L_{\epsilon} E_{\infty} + R_{\epsilon}}{M}\right)^{1/2} + ( L_{\epsilon} E_{\infty} + R_{\epsilon} +1 ) |{\kappa_{\textrm{I}}}| + 2|{\kappa_{\textrm{II}}}| \sum_{s=\pm 1/2} \sum_{\nu =1}^3 \sum_{\tau= \pm } \|F_{s ,\tau}^{\nu} \| \right)^{1/2} \notag$$ and $$c_2 = |{\kappa_{\textrm{I}}}|^{1/2} \left( c_0 \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}} \right\| \,\right)^{1/2} \, \left( c_0 \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}} \right\| + \, \sum_{r=1,2} \sum_{\nu =1}^3 \| F_{r}^{\nu} \|_{L^2}\right)^{1/2} . \notag$$ **(Proof)** It follows that $ ({{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{f}}}(F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )^2 \leq {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{f}}}(F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) \leq {\ensuremath{d\Gamma_{\textrm{f}}({1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}})}) }} $, and then, $$\begin{aligned} \| ( ( {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{f}}}(F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \|^2 &\leq ( \Psi_m ,\left( {d \Gamma_{\textrm{f}}}( 1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}\right) \Psi_{m} ) \notag \\ & = \sum_{s= \pm 1/2} \left( \int_{{\mathbf{R}^{3} }} \left( (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_{m} , (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) d {\ensuremath{\mathbf{p}}} \right. \notag \\ & \qquad \quad \left. + \int_{{\mathbf{R}^{3} }} \left( (d_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_{m} , (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(d_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m } \right) d {\ensuremath{\mathbf{p}}} \right) . \label{9/15.2}\end{aligned}$$ We evaluate the two terms in the right-hand side of (\[9/15.2\]). The first term is estimated as $$\begin{aligned} &\left| \int_{{\mathbf{R}^{3} }} \left( (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_{m} , (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) d {\ensuremath{\mathbf{p}}} \right| \notag \\ & \leq \left( \int_{{\mathbf{R}^{3} }} \left\| (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \right)^{1/2} \times \left( \int_{{\mathbf{R}^{3} }} \left\| (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \right)^{1/2} \notag \\ & = \| ( N_{{ \textrm{D} }}^{+} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| \times \left( \int_{{\mathbf{R}^{3} }} \left\| (1-F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \right)^{1/2}. \notag\end{aligned}$$ It is seen that $$\begin{aligned} & \int_{{\mathbf{R}^{3} }} \| (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \|^2 d {\ensuremath{\mathbf{p}}} \notag \\ & \leq 4 \int_{{\mathbf{R}^{3} }} \| (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) \, \frac{1}{1+ \hat{{\ensuremath{\mathbf{p}}}}^2} (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \|^2 d {\ensuremath{\mathbf{p}}} \notag \\ & \qquad \qquad \qquad \qquad + 4 \sum_{\nu=1}^3 \int_{{\mathbf{R}^{3} }} \| (1-F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) \, \frac{(\hat{p}^\nu )^2}{1+ \hat{{\ensuremath{\mathbf{p}}}}^2} \, (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \|^2 d {\ensuremath{\mathbf{p}}} . \notag\end{aligned}$$ Note that for all ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$, $$\sup_{{\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }} \left| \left( 1- F_{ R} ({\ensuremath{\mathbf{p}}})\right) \, \frac{1}{ 1+ {\ensuremath{\mathbf{p}}}^2} \right| \leq \frac{1}{R^2} \; , \quad \; \; \; \; \sup_{{\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }} \left| \left( 1- F_{ R} ({\ensuremath{\mathbf{p}}})\right) \, \frac{p^{\nu}}{ 1+ {\ensuremath{\mathbf{p}}}^2} \right| \leq \frac{1}{R} .$$ Then by the electron derivative bounds in Theorem \[EP-DB\] **(i)** and the spectral decomposition theorem, we have $$\begin{aligned} & \int_{{\mathbf{R}^{3} }} \left\| (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}))(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \ \notag \\ & \leq \frac{4}{R^4} \int_{{\mathbf{R}^{3} }} \left\| (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} + \frac{4}{R^2} \sum_{\nu =1}^3 \int_{{\mathbf{R}^{3} }} \left\| \partial_{p^{\nu}}(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \notag \\ & \leq \frac{4}{R^4} \| ( N_{{ \textrm{D} }}^{+} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \|^2 + \frac{ c_m( \epsilon )^2}{R^2} \sum_{\nu =1}^3 \int_{{\mathbf{R}^{3} }}|F_{s, +}^{\nu}({\ensuremath{\mathbf{p}}})|^2 d {\ensuremath{\mathbf{p}}} , \notag \end{aligned}$$ where $c_m ( \epsilon ) = 2 ( L_{\epsilon} E_{0} (H_m ) + R_{\epsilon} +1) |{\kappa_{\textrm{I}}}| + 4|{\kappa_{\textrm{II}}}|$. Therefore, $$\begin{aligned} & \left| \int_{{\mathbf{R}^{3} }} (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_m , (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m ) d {\ensuremath{\mathbf{p}}} \right| \notag \\ & \leq \| ( N_{{ \textrm{D} }}^{+} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| \times \left( \frac{2}{R^2} \| ( N_{{ \textrm{D} }}^+ {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| + \frac{c_m ( \epsilon )}{R} \sum_{\nu =1}^3 \|F_{s ,+}^{\nu} \|_{L^2} \right) . \label{9/15.3}\end{aligned}$$ In a same way as the first term, we can estimate the second term in the right-hand side of (\[9/15.2\]) by the positron derivative bounds in Theorem \[EP-DB\] **(ii)**, and then, $$\begin{aligned} & \left| \int_{{\mathbf{R}^{3} }} (d_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_m , (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(d_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m ) d {\ensuremath{\mathbf{p}}} \right| \notag \\ & \leq \| ( N_{{ \textrm{D} }}^{-} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| \times \left( \frac{2}{R^2} \| ( N_{{ \textrm{D} }}^- {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| + \frac{c_m ( \epsilon )}{R} \sum_{\nu =1}^3 \|F_{s , -}^{\nu} \|_{L^2} \right) . \label{9/15.4}\end{aligned}$$ From (\[9/15.3\]) and (\[9/15.4\]), we have for all $R>1$,\ $$\begin{aligned} & \| ( ( {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{f}}}(F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \|^2 \notag \\ &\leq \frac{1}{R}\sum_{\tau = \pm} \| ( N_{{ \textrm{D} }}^{\tau} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| \left( 2 \| ( N_{{ \textrm{D} }}^{\tau} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| + c_m ( \epsilon ) \sum_{s=\pm 1/2} \sum_{\nu =1}^3 \|F_{s ,\tau}^{\nu} \|_{L^2} \right) . \label{9/15.7}\end{aligned}$$ From Lemma \[9/14.a\] **(i)**, $$\sup_{j \in {\ensuremath{\mathbf{N}}}} \| ( N_{{ \textrm{D} }}^{\pm} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_{m_j} \| \leq \sup_{j \in {\ensuremath{\mathbf{N}}}} \| ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_{m_j} \| \leq \left( \frac{L_{\epsilon}}{M} E_{\infty} +\frac{R_{\epsilon}}{M} \right)^{1/2} ,$$ and we see that $$\sup_{j \in {\ensuremath{\mathbf{N}}}} c_{m_{j}} ( \epsilon ) = \sup_{j \in {\ensuremath{\mathbf{N}}}} \left(2 ( L_{\epsilon} E_{0}(H_{m_j}) + R_{\epsilon} +1 ) |{\kappa_{\textrm{I}}}| + 4|{\kappa_{\textrm{II}}}| \right) \; \leq \; 2 ( L_{\epsilon} E_{\infty} + R_{\epsilon} +1 ) |{\kappa_{\textrm{I}}}| + 4|{\kappa_{\textrm{II}}}|.$$ Hence **(i)** follows.\ **(ii)** In a similar way to the proof of **(i)**, it follows that $ ({{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})))^2 \leq {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{b}}}(F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})) \leq {\ensuremath{d\Gamma_{\textrm{b}}({1-F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) }) }} $, and hence, $$\begin{aligned} \| ( {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{b}}}( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})) ) \Psi_{m} \|^2 &\leq ( \Psi_m ,\left( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{d \Gamma_{\textrm{b}}}( 1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) \right) \Psi_{m} ) \notag \\ & = \sum_{r= 1,2} \int_{{\mathbf{R}^{3} }} \left( ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) )\Psi_{m} , (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right) d {\ensuremath{\mathbf{k}}} . \label{9/15.5}\end{aligned}$$ We see that $$\begin{aligned} &\left| \int_{{\mathbf{R}^{3} }} \left( ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) )\Psi_{m} , (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right) d {\ensuremath{\mathbf{k}}} \right| \notag \\ & \leq \left( \int_{{\mathbf{R}^{3} }} \left\| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \right)^{1/2} \times \left( \int_{{\mathbf{R}^{3} }} \left\| (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \right)^{1/2} \notag \\ & = \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \| \times \left( \int_{{\mathbf{R}^{3} }} \left\| (1-F_{{\textrm{b}}, R'} )( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right\|^2 d {\ensuremath{\mathbf{k}}} \right)^{1/2} . \notag\end{aligned}$$ By the photon derivative bounds in Theorem \[P-DB\] and the spectral decomposition theorem, $$\begin{aligned} & \int_{{\mathbf{R}^{3} }} \| (1-F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) )( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \|^2 d {\ensuremath{\mathbf{k}}} \notag \\ & \leq 4 \int_{{\mathbf{R}^{3} }} \| (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) \, \frac{1}{1+ \hat{{\ensuremath{\mathbf{k}}}}^2} ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \|^2 d {\ensuremath{\mathbf{k}}} \notag \\ & \qquad \qquad + 4 \sum_{\nu =1}^3 \int_{{\mathbf{R}^{3} }} \| (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) \, \frac{(\hat{k}^\nu )^2}{1+ \hat{{\ensuremath{\mathbf{k}}}}^2} \, ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \|^2 d {\ensuremath{\mathbf{k}}} \notag \\ & \leq \frac{4}{R'^4} \int_{{\mathbf{R}^{3} }} \left\| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right\|^2 d {\ensuremath{\mathbf{k}}} + \frac{4}{R'^2} \sum_{\nu=1}^3\int_{{\mathbf{R}^{3} }} \left\| \partial_{k^{\nu}}( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right\|^2 d {\ensuremath{\mathbf{k}}} \notag \\ & \leq \frac{4}{R'^4} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \|^2 + \frac{ 4 |{\kappa_{\textrm{I}}}|^2}{R'^2}\sum_{\nu =1}^3 \int_{{\mathbf{R}^{3} }}|F_{r}^{\nu}({\ensuremath{\mathbf{k}}})|^2 d {\ensuremath{\mathbf{k}}} . \notag \end{aligned}$$ Then we have $$\begin{aligned} & \left| \int_{{\mathbf{R}^{3} }} \left( ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) )\Psi_{m} , (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right) d {\ensuremath{\mathbf{k}}} \right| \notag \\ & \leq \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \| \times \left( \frac{4}{{R'}^4} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \|^2 + \frac{ 4|{\kappa_{\textrm{I}}}|^2 }{{R'}^2} \sum_{\nu =1}^3 \| F_{r}^{\nu} \|_{L^2}^2 \right)^{1/2} \notag , \end{aligned}$$ and hence, for all $R'>1$, $$\begin{aligned} & \| ( ( {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{b}}}( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )) ) \Psi_{m} \|^2 \notag \\ &\leq \frac{2}{R'} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \| \times \left( \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \| +|{\kappa_{\textrm{I}}}|\, \sum_{r=1,2} \sum_{\nu =1}^3 \| F_{r}^{\nu} \|_{L^2}\right) .\end{aligned}$$ From Lemma \[9/14.a\] (**ii**), it holds that $ \sup\limits_{j \in {\ensuremath{\mathbf{N}}}} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_{m_j} \| < c_{0} |{\kappa_{\textrm{I}}}| \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}} , \right\| $. Therefore the proof is obtained. $\blacksquare$\ $\;$\ [**(Proof of Theorem \[Main-Theorem\])**]{}\ From Proposition \[9/16.a\] and a general theorem ([@AH97] ; Lemmma 4.9), it is enough to show that w-$ \lim\limits_{j \to \infty} \Psi_{m_j} \neq 0$. We see that $$\begin{aligned} {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} & = ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} -{\Gamma_{\textrm{f}}}(F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) ) {\otimes}( {{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} - {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) )) \notag \\ & \quad + {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) ) {\otimes}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) ) E_{N_{{\textrm{rad}}}}([0, n] )) + {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) ) E_{N_{{\textrm{rad}}}}([ n+1, \infty) )) . \notag \end{aligned}$$ Then by Proposition \[9/15.a\], we have for all $0 < \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $R>1$ and $R'> 1 $, $$\begin{aligned} & \left\| \left( {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) ) E_{N_{{\textrm{rad}}}}([0, n ] ) )\right) \Psi_{m_j} \right\| \notag \\ & \geq 1- \left( \frac{}{} \| (( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} -{\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ))) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi_{m_j} \| \right. \notag \\ & \qquad \qquad \qquad \left. + \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}( {{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} - {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) ) ) \Psi_{m_j} \| + \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}E_{N_{{\textrm{rad}}}}([n+1, \infty) )) \Psi_{m_j} \| \frac{}{} \right) . \notag \\ & \geq 1- \left( \frac{c_{1, \epsilon }}{\sqrt{R}} + \frac{c_2}{\sqrt{R'}} + \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}E_{N_{{\textrm{rad}}}}([n+1, \infty) )) \Psi_{m_j} \| \right) , \notag\end{aligned}$$ It is seen that $$\sqrt{n+1 }\| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}E_{N_{{\textrm{rad}}}}([n+1, \infty) )) \Psi_{m_j} \| \leq \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_{m_j} \| \leq c_{0} |{\kappa_{\textrm{I}}}| \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}}\right\| .$$ Then from Lemma \[9/14.a\] **(ii)**, we have $$\sup\limits_{j \geq 1}\| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}E_{N_{{\textrm{rad}}}}([n+1, \infty) )) \Psi_{m_j} \| \leq \frac{c_3 }{(n+1)^{1/2}}. \notag$$ where $c_{3}= c_{0} |{\kappa_{\textrm{I}}}| \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}}\right\| $. Then it follows that $$\| {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{m_j} \| \geq 1 - \left( \frac{c_{1 , \epsilon}}{R} + \frac{c_2}{R'} + \frac{c_3}{(n+1)^{1/2}} \right) . \label{9/15.9}$$ We also see that $$\begin{aligned} & \| {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )) {\otimes}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )) E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{m_j} \|^2 \notag \\ & = ((H_0 +1) E_{N_{{\textrm{rad}}}}([0, n ] ) \Psi_{m_j} ,(H_0 +1)^{-1} ( {\Gamma_{\textrm{f}}}(F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )^2) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )^2 ) E_{N_{{\textrm{rad}}}}([0, n ] )) ) \Psi_{m_j} ) \notag \\ & \leq \| (H_0 +1)\Psi_{m_j} \| \, \| (H_0 +1)^{-1} \, ( {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )^2 ) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )^2 ) E_{N_{{\textrm{rad}}}}([0, n ] ))) \, \Psi_{m_j} \| . \notag\end{aligned}$$ We see that $\| (H_0 +1)\Psi_{m_j} \| \leq \| H_0 \Psi_{m_j} \| +1 \leq \| H_{0,m_{j}} \Psi_{m_j} \| +1 $ and Lemma \[9/9.e\] yields that $$\| H_{0,m_{j}} \Psi_{m_j} \| \leq \frac{L_{\epsilon } }{M} \| H_{m_j} \Psi_{m_j} \| + \frac{R_{\epsilon }}{M} \leq \frac{L_{\epsilon}}{M} E_{\infty} + \frac{R_{\epsilon}}{M} .$$ Then we have $$\begin{aligned} &\| {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )) {\otimes}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )) E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{m_j} \| \notag \\ & \qquad \qquad \qquad \leq c_{4 ,\epsilon}\| (H_0 +1)^{-1} \, \left( ( {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) ) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )^2 ) E_{N_{{\textrm{rad}}}}([0, n ] ))) \right) \, \Psi_{m_j} \|^{1/2}, \label{9/15.10}\end{aligned}$$ where $c_{4 , \epsilon }= (\frac{L_{\epsilon}E_{\infty} + R_{\epsilon} }{M} +1)^{1/2}$. From (\[9/15.9\]) and (\[9/15.10\]) $$\|(H_0 +1)^{-1}{\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )^2) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})^2 E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{m_j} \| \geq \frac{1}{c_{4, \epsilon }}^2 \left( 1 - \left( \frac{c_{1,\epsilon}}{\sqrt{R}} + \frac{c_2}{\sqrt{R'}} + \frac{c_3}{(n+1)^{1/2}} \right) \right)^2 \notag$$ Since $(H_0 +1)^{-1} \left( {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )^2) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})^2) E_{N_{{\textrm{rad}}}}([0, n ] ))) \right) $ is compact, we have $$\|(H_0 +1)^{-1}{\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}})^2 ) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})^2 ) E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{0} \| \geq \frac{1}{c_{4, \epsilon }^2}\left( 1 - \left( \frac{c_{1,\epsilon}}{\sqrt{R}} + \frac{c_2}{\sqrt{R'}} + \frac{c_3}{(n+1)^{1/2}} \right) \right)^2 , \label{9/15.20}$$ where we set $\Psi_{0} =$ w-$ \lim\limits_{j \to \infty} \Psi_{m_j} $. Then for sufficiently large $R>0$, $R'>0 $ and $n >0$, the right-hand side of (\[9/15.20\]) is greater than zero, and hence $ \Psi_{0} \ne 0 $.\ $\;$\ **(Multiplicity)** Assume dim ker $(H -E_{0}(H_{{ \textrm{QED} }})) = \infty $. Let $ \Psi_{l} $, $l \in {\ensuremath{\mathbf{N}}}$, be the ground states. Let ${\ensuremath{\mathscr{M}}} $ be the closure of the linear hull of $\{ \Psi_l \}_{l=0}^{\infty}$. Then ${\ensuremath{\mathscr{H}}}_{{ \textrm{QED} }}$ is decomposed as ${\ensuremath{\mathscr{H}}}_{{ \textrm{QED} }} = {\ensuremath{\mathscr{M}}} \oplus {\ensuremath{\mathscr{M}}}^{\bot} $. Let $\{ \Phi_l \}_{l=0}^{\infty}$ be a complete orthogonal system of ${\ensuremath{\mathscr{M}}}^{\bot}$. We can set a complete orthonormal system $\{ \Xi_l \}_{l=0}^{\infty}$ of ${\ensuremath{\mathscr{H}}}_{{ \textrm{QED} }}$ by $ \Xi_{2 l-1 } = \Psi_{l} $ and $ \Xi_{2 l} = \Phi_{l} $ for all $l \in {\ensuremath{\mathbf{N}}}$. Since $\{ \Xi_l \}_{l=0}^{\infty}$ is a complete orthonormal system , w-$\lim\limits_{l \to \infty}\Xi_l =0$. On the other hand, $\Xi_{2 l-1 }$ is ground state for all $l \in {\ensuremath{\mathbf{N}}}$, and hence $H_{{ \textrm{QED} }}\Xi_{2 l-1 }=E_{0}({H_{\textrm{QED}}}) \Xi_{2 l-1 } $. In a same argument of the proof of the existence of the ground state, we have $$\|(H_0 +1)^{-1}{\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}})^2 ) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})^2 ) E_{N_{{\textrm{rad}}}}([0, n ] ))\Xi_{2l-1} \| \geq \frac{1}{\tilde{c}_{4, \epsilon }^2}\left( 1 - \left( \frac{ \tilde{c}_{1,\epsilon}}{\sqrt{R}} + \frac{c_2}{\sqrt{R'}} + \frac{c_3}{(n+1)^{1/2}} \right) \right)^2 , \label{9/15.15}$$ where $\tilde{c}_{1, \epsilon} $ and $\tilde{c}_{4, \epsilon} $ are the constants $ c_{1 , \epsilon}$ and $ c_{1 , \epsilon}$ replacing $E_{\infty}$ with $E_{0}({H_{\textrm{QED}}})$. Then by taking sufficiently large $R>0$, $R'>0 $ and $n >0$, we have w-$\lim\limits_{l \to \infty}\Xi_{2l-1} \ne 0$, but this is contradict to w-$\lim\limits_{l \to \infty}\Xi_l = 0$. Hence dim ker $(H_{{ \textrm{QED} }} -E_{0}(H_{{ \textrm{QED} }})) < \infty $. $\blacksquare $\ $\;$\ $\;$\ [**\[Concluding remarks\]**]{}\ **(1) The case of Massless Dirac field**\ It is not realistic model, but we can consider the system of a massless Dirac field coupled to the radiation field. In such a case, by replacing (**A.5**) with similar conditions to (**A.6**), we can also prove the existence of the ground state in a same ways as ${H_{\textrm{QED}}}$. $\; $\ **(2) Infrared divergent problem**\ For some systems of particles coupled to massless Bose fields, the existence of the ground states without infrared regularity conditions was obtained (refer to e.g., Bach-Fröhlich-Sigal [@BFS99], Griesemer-Lieb-Loss [@GLL01] and Hasler-Herbst [@HaHe11]), and non-existence of the ground states for other other systems was also investigated (see e.g., Arai-Hirokawa-Hiroshima [@AHH99]). To prove the existence or non-existence of the ground state of ${H_{\textrm{QED}}}$ without infrared regularity conditions is left for future study. [99]{}

--- abstract: 'A relatively simple model problem where a single electron moves in two relativistically-strong obliquely intersecting plane wave-packets is studied using a number of different numerical solvers. It is shown that, in general, even the most advanced solvers are unable to obtain converged solutions for more than about 100 fs in contrast to the single plane-wave problem, and that some basic metrics of the orbit show enormous sensitivity to the initial conditions. At a bare minimum this indicates an unusual degree of non-linearity, and may well indicate that the dynamics of this system are chaotic.' author: - 'A.P.L.Robinson' - 'K.Tangtartharakul' - 'K.Weichman' - 'A.V.Arefiev' title: Extreme Nonlinear Dynamics in Vacuum Laser Acceleration with a Crossed Beam Configuration --- Introduction ============ Since the development of Chirped Pulse Amplification lasers [@danson_vulcan_2004; @hernandez-gomez_vulcan_2010; @tang_optical_2008; @hooker_improving_2011], the field of ultra-intense laser-matter interactions has grown considerably. Initially this technology allowed the development of TW-scale lasers that breached the 10$^{18}$Wcm$^{-2}\mu$m$^2$, but subsequent progress has lead to the construction of 10PW scale systems [@hernandez-gomez_vulcan_2010], with 100PW systems under development. The field now spans a large number of sub-topics including laser wakefield acceleration of electrons [@mangles_monoenergetic_2004], laser-driven ion acceleration [@gaillard_increased_2011], laser- driven x-ray [@kneip_observation_2008] and neutron sources, advanced inertial fusion concepts such as Fast Ignition [@m.tabak_review_2005], studies of both Warm Dense Matter and Hot Dense Matter [@d.j.hoarty_observations_2013], radiation reaction studies, and even probing QED physics [@heinzl_observation_2006; @heinzl_exploring_2009]. It is likely that the latter topics in that list will become more dominant as multi-PW facilities become fully operational in the following years. Numerical simulation codes, particularly Particle-in-Cell (PIC) codes [@birdsall_plasma_2018; @pukhov_strong_2003], have been instrumental in driving the field forward, both in terms of interpreting experiments and in making predictions that have motivated crucial experimental work. Perhaps the best known example of PIC’s predictive capabilities is that of Pukhov and Meyer-ter-Vehn’s prediction [@pukhov_laser_2002] of the ‘bubble regime’ of laser wakefield acceleration, which was later validated by three different research groups [@mangles_monoenergetic_2004]. The PIC algorithm is itself dependent on a number of algorithms, some of which were developed separately, such as the Yee FDTD method [@kane_yee_numerical_1966] for numerical electromagnetics. Importantly this includes a ‘particle-pusher’ algorithm which advances the macroparticles position and momentum. The quality and capability of any individual PIC code will depend on the set of algorithms chosen for these different components. A common choice for the particle-pusher is the Boris method [@birdsall_plasma_2018]. The Boris method is a second order accurate leapfrog-type method that is centred in time. It is a method that has enjoyed considerable success, and which has been employed in a number of different PIC codes including EPOCH [@arber_contemporary_2015]. Developing higher order versions of the Boris method is a non-trivial proposition, and it is has been questioned whether or this endeavour would actually yield any serious benefits to laser-plasma or accelerator science [@higuera_structure-preserving_2017; @londrillo_2010]. In the past few years however it has been recognized that the Boris method has at least one serious defect, namely that constant motion is not maintained in the case of uniform crossed ${\bf E} \ne 0$ and ${\bf B} \ne 0$ fields (for the choice of particle velocity for which a force-free scenario is obtained). This was first identified by Vay [@vay_simulation_2008], who proposed a variation on the Boris pusher that resolved this issue. Later Higuera and Cary [@higuera_structure-preserving_2017] proposed an algorithm that both solved the issue of the ${\bf E} \times {\bf B}$ velocity and which also preserved phase-space volume (unlike Vay’s method). Alongside these developments, Arefiev also showed that considerable care needs to be taken in setting the time step when integrating the orbits of an electron in a relativistic laser pulse. Altogether these developments underline how the particle-pusher problem needs careful study to ensure that particle-pusher algorithms can be trusted when employed to study the strongly relativistic and highly complex configurations encountered in ultra-intense laser-matter problems. Despite these developments the methods of Vay and Higuerra-Cary are still only second order accurate methods. For problems where the overall behaviour of the system is quite ’regular’ this means that they will be quite adequate in the majority of cases. What has not been given so much consideration is whether the dynamics can always be assumed to be sufficiently ‘regular’. Some researchers have pointed out that some laser interaction problems will have a ‘stochastic’ nature [@z.-m.sheng_stochastic_2002; @sheng_efficient_2004; @meyer-ter-vehn_electron_1999], this terminology appears to actually mean that the dynamics are [*chaotic*]{}[@strogatz_nonlinear_2015; @ott_coping_1994; @handfinch]. If the Lyapunov time is larger than the time-scale of interest then this is not a problem for numerical simulation. However if the Lyapunov time becomes much shorter than the time-scale of interest then the ability to predict future states of the system will be highly limited even with very sophisticated numerical solvers. In this paper we present a relatively simple test problem for a single electron : two plane EM Gaussian wave-packets that cross at an oblique angle and which are $\pi$ out of phase. The electron is initially at rest and which sits ‘off-axis’ by a fraction of the vacuum wavelength. To the best of the authors’ knowledge this problem does not have an analytic solution. We have studied the ability of a number of leapfrog pushers, RK4 method, and more sophisticated adaptive algorithms to solve the electron orbits in this problem. We have found that, in general, all of these solvers are only able to obtain converged orbits for a fraction ($<$20%) of the total problem duration (100-200 fs out of 1 ps). Complete converged orbits are only obtained in a few cases, and usually only the RK4 method (or better) is able to do this. A survey of the sensitivity to initial conditions was carried out, and it was found that there are regions of parameter space which exhibit extreme sensitivity to initial conditions. This indicates that this problem, however simple it may seem, in fact is [*chaotic*]{} in nature, as expected given earlier studies [@z.-m.sheng_stochastic_2002; @sheng_efficient_2004; @meyer-ter-vehn_electron_1999], however in this case it would appear that the chaotic dynamics is severly problematic for numerical integration. We suggest that this may have important ramifications for both Vacuum Laser Acceleration (VLA) [@hartemann_chirped-pulse_1999; @thevenet_vacuum_2016; @plettner_proof--principle_2005; @troha_vacuum_1999; @robinson_interaction_2018] and Direct Laser Acceleration (DLA) [@pukhov_particle_1999; @naseri_channeling_2012; @arefiev_beyond_2016; @robinson_breaking_2017; @arefiev_spontaneous_2016; @huang_characteristics_2016; @zhang_synergistic_2015; @willingale_surface_2013; @robinson_generating_2013]. Description of Model Problem {#model} ============================ We consider a problem where two relativisitically-strong plane EM wave-packets intersect obliquely. We want to study the relativistic motion of an electron that is initially at rest. This can be described by the following formulae for the electric fields of the incident waves: $$\begin{aligned} &&{\bf E} = {\bf E}_1 + {\bf E}_2, \\ &&{\bf E}_1 = E\cos\psi_1f_{env,1}\left[ -\sin(\theta_{cb}/2),\cos(\theta_{cb}/2),0\right], \\ &&{\bf E}_2 = E\cos\psi_2f_{env,2}\left[ \sin(\theta_{cb}/2),\cos(\theta_{cb}/2),0\right],\end{aligned}$$ ![Schematic of the simulation set-up showing key parameters.[]{data-label="sketch"}](sketch5.png){width="\columnwidth"} where $\psi_1 = {\bf k_1.r} - \omega_L{t} + \phi_1$, $\psi_2 = {\bf k_2.r} - \omega_L{t}+\phi_2$, ${\bf k}_1 = [\cos(\theta_{cb}/2),\sin(\theta_{cb}/2),0]$, ${\bf k}_2 = [\cos(\theta_{cb}/2),-\sin(\theta_{cb}/2),0]$. For the envelope functions, we use $f_{env} = \exp(-(\psi/k_L + 5c\tau_L)^2/(2c\tau_L))$. There are corresponding magnetic fields in the z-direction. This corresponds to two intersecting plane wave-packets that are aligned obliquely to the $x$-axis with the E-field polarized in the $xy$-plane in each case. The angle between the wavevectors of the two wave-packets is $\theta_{cb}$. For our baseline problem we consider the case where $E = 5\omega_Lm_ec/e$ (i.e. each plane wave-packet has $a_0 =$ 5), $\theta_{cb}= $40$^\circ$, $\lambda_L =$1 $\mu$m, and $\tau_L = $20 fs. The two wavepackets are $\pi$ out of phase, i.e. $\phi_1 =$ 0, $\phi_2 = \pi$. The electron is initially at rest at $x =$0,$z = $0, and $y = y_0$. A schematic of the problem is shown in fig.\[sketch\]. Since this problem is quite close to that considered previously [@z.-m.sheng_stochastic_2002; @sheng_efficient_2004; @meyer-ter-vehn_electron_1999], we should expect that chaotic dynamics are likely to be encountered. A very significant difference with earlier studies is that the value of the normalized vector potential in this case is significantly larger ($a >$ 5 here). However since Mendonca’s [@mendonca_1983] criterion is $a_1a_2 >$1/16 we expect that chaotic dynamics will only be more prevalent in this problem. Analysis with Standard Algorithms {#standard} ================================= In the first part of our study we have used the following solvers : (i) the Boris pusher [@birdsall_plasma_2018], (ii) the Vay pusher[@vay_simulation_2008], (iii) the Higuera-Cary pusher[@higuera_structure-preserving_2017], and (iv) the 4th order Runge-Kutta (RK4) algorithm [@numrecipes], to study this problem. Note that (i)–(iii) are formally 2nd order algorithms (although they differ in their treatment of the ${\bf E} \times {\bf B}$ velocity) , and only (iv) is formally 4th order. These were applied to study the baseline case (case 1). We shall not re-state the details of these here, and we refer the reader to the given references for further details. We have tested and checked our implementations, in particular by testing that they reproduce the motion in a single plane wave-packet. The baseline numerical integration is carried out over 18000 time steps with $\Delta{t} = $0.05 fs. To examine convergence, the time step is multiplied by a factor $1/M$, and the total number of time steps by $M$ in order to keep the total duration of the integration constant. In general, we regard two trajectories as being converged if the variables in question are within 5% of one another. All of these solvers reproduce the analytic prediction for the single plane-wave problem with $M$ = 1 and the solutions of each solver are practically identical. For each solver we obtained solutions of $M =$1,2,4, and 8. The results for the Boris pusher, in terms of $p_y$ are shown in fig.\[fig:figure1\]. By following sequence of cases, we can see that the solution is not converging. ![\[fig:figure1\]The results from the Boris Pusher for the baseline case. Value of $M$ for each line is shown in the legend. Solution shows no sign of convergence with increasing $M$.](figure1.pdf){width="\columnwidth"} The behaviour of the Boris pusher is in sharp contrast with the RK4 algorithm. The results of the RK4 algorithm, also in terms of $p_y$, are shown in fig.\[fig:figure2\]. Here the four curves almost perfectly overlap, showing clearly that there has been very good convergence, and that it has happened very rapidly. ![\[fig:figure2\]The results from the RK4 algorithm for the baseline case. Value of $M$ for each line is shown in the legend. All four curves overlap almost perfectly, indicating extremely rapid convergence.](figure4.pdf){width="\columnwidth"} The behaviour of both the Vay and Higuera-Cary pushers are shown in fig.s \[fig:figure3\] and \[fig:figure4\]. By comparing fig.s \[fig:figure3\] and \[fig:figure4\] to fig. \[fig:figure2\] we can see that, when $M =$8, both the Vay and the Higuera-Cary pushers come very close to the solution obtained by the RK4 algorithm. This should lead to confidence in the solution obtained by the RK4 solver. It is clearly good that both the Vay and Higuera-Cary pushers are able to eventually reach this solution, however the rate of convergence is rather slow, and it requires that one adopts a very small time ($M =$8) time step. In figure \[fig:figure5\] we directly compare the Vay, Higuera-Cary, and RK4 solutions for $M =$8. As can be seen they all lie extremely close to one another, showing that the Vay and Higuera-Cary solvers are able to approach the RK4 solution, whereas the Boris solver cannot for $M \le 8$. ![\[fig:figure3\]The results from the Vay Pusher for the baseline case. Value of $M$ for each line is shown in the legend. ](figure2.pdf){width="\columnwidth"} ![\[fig:figure4\]The results from the Higuera-Cary pusher for the baseline case. Value of $M$ for each line is shown in the legend. ](figure3.pdf){width="\columnwidth"} ![\[fig:figure5\]Comparison of the solutions from the Vay, Higuera-Cary, and RK4 pushers for $M =$8, showing that, in the $M =$8 case, convergence is obtained. ](figure5.pdf){width="\columnwidth"} In the second part of our study we extended this to multiple cases to see if these findings reflected a general trend. As is evident from fig.s \[fig:figure1\]–\[fig:figure4\], even when convergence is not obtained over the entire 900 fs, convergence in fact can occur over a time period that is a fraction of the total duration of the problem. When extending the study we instead looked at the fraction of the problem duration over which convergence was obtained (instead of whether or not [*total*]{} convergence was obtained). The results are summarized in table \[table1\], which shows the convergence obtained for each case as a percentage of the total problem duration (900 fs), and for each solver tried. The special cases of the convergence obtained by the Vay and Higuera-Cary pushers in the baseline case are noted by an asterisk. Case Boris Vay Hig.-Cary RK4 ------------------------------------------------------------- ------- ------- ----------- ------- 1.$a_0 =$5, $y_0 = \lambda/4$, $\theta_{cb} = $ 40$^\circ$ 14.7 100\* 100\* 100 2.$a_0 =$5, $y_0 = \lambda/2$, $\theta_{cb} = $ 40$^\circ$ 11.1 11.0 9.1 12.1 3.$a_0 =$10, $y_0 = \lambda/2$, $\theta_{cb} = $ 40$^\circ$ 12.8 13.3 12.9 100.0 4.$a_0 =$10, $y_0 = \lambda/4$, $\theta_{cb} = $ 40$^\circ$ 10.2 10.2 11.3 17.8 5.$a_0 =$5, $y_0 = \lambda/8$, $\theta_{cb} = $ 40$^\circ$ 10.9 11.6 11.6 17.6 6.$a_0 =$10, $y_0 = \lambda/8$, $\theta_{cb}= $ 40$^\circ$ 10.8 11.9 11.9 17.1 7.$a_0 =$5, $y_0 = \lambda/4$, $\theta_{cb} = $ 60$^\circ$ 11.6 11.9 11.7 14.0 8.$a_0 =$5, $y_0 = \lambda/4$, $\theta_{cb}= $ 80$^\circ$ 9.3 9.3 9.8 16.6 9.$a_0 =$5, $y_0 = \lambda/4$, $\theta_{cb}= $ 20$^\circ$ 14.8 14.8 14.8 41.3 10.$a_0 =$5, $y_0 = \lambda/4$, $\theta_{cb} = $ 10$^\circ$ 48.7 48.3 49.0 62.5 : \[table1\]Summary of results for different cases. Shown in the percentage of the total problem duration for which a given pusher is able to obtain convergence for $M \le$8. The special cases of the Vay and Higuera-Cary pushers in the baseline case are noted by an asterisk. From Table \[table1\] we find that the baseline case unfortunately represents a rather optimistic one from the point of view of numerically solving this problem. In general we found that even the RK4 pusher was unable to produce converged solutions for more than 18% of the problem duration. Converged solutions over the full duration were only obtained by the RK4 solver in a couple of cases. Also as the approach angle, $\theta_{cb}$, becomes very small, it is much easier to obtain convergence. All the leapfrog solvers perform less well than the RK4 pusher. The differences between the three are usually rather small (again suggesting that the baseline case, happens to be a special case). It therefore appears that, in general, the enhanced leapfrog solvers are not substantially better at the crossed beam problem than the Boris pusher. We have also examined the effect that the different solvers have on distributions arising from an ensemble of different initial conditions. This was done for 10000 different particles initialized at rest with the initial $x$-position spanning -0.5 to +0.5$\lambda_L$ ($y_{init} = \lambda_L/4$). The problem was run up to 450 fs with $M = $1. Otherwise the problem corresponds to the baseline case. We compared the distributions that arose from using the Boris and the RK4 solvers, which are shown in fig.s \[fig:figure9\] and \[fig:figure10\] respectively. ![\[fig:figure9\]Distribution at 450 fs of ensemble calculation (see text) for the case of the Boris solver. ](crossed_borisphase.jpeg){width="\columnwidth"} ![\[fig:figure10\]Distribution at 450 fs of ensemble calculation (see text) for the case of the RK4 solver. ](crossed_rk4phase.jpeg){width="\columnwidth"} In the case of the RK4 solver we see that there is a very strong spike at high energy, denoted as ’B’ in fig. \[fig:figure10\]. This feature is absent in fig. \[fig:figure9\], and instead we see a different feature denoated as ’A’ in this figure. Given that the strongest accumulations of particles are completely different for different solvers applied to the same ensemble/problem, we can conclude that the issues observed with single trajectories will lead to significant differences in particles distributions as well. It therefore appears to be the case that the crossed beam problem presented here represents a far harder test than the single plane wave of single electron trajectory calculation. To the best of the author’s knowledge this is currently the hardest test case, at least specifically for laser-plasma studies, as the conventional particle pushers tested here are known to capable of producing fully converged solutions (for $M \le$ 8) over the full duration. This is certainly the case for the single plane wave problem. More importantly the results presented in Table \[table1\] already indicate the most likely reason as to why this problem is so challenging : namely that the dynamics has become chaotic. We see that, in the general case, a converged solution can only be obtained for a short period of time. We also see that there are strange isolated cases where a full converged solution can be obtained. The observation of these features motivated a more detailed study of the problem. Parameter Scans with Advanced Algorithms {#advanced} ======================================== In the second phase of this study, another class of solvers was used, namely the [MATLAB]{} suite of ODE solvers. In broad terms, applying these solvers to the problem lead to results similar to those presented in Sec. \[standard\], with convergence only obtained over a limited period of time and for a small angle between the beams. Out of the entire suite, ODE113 performed the best. This solver is a variable-step, variable-order (VSVO) Adams-Bashforth-Moulton Predictor-Corrector solver of order 1 to 13. It was found that convergence was reliably obtained when the angle between the beams was limited to no more than $\theta_{cb} =$ 30$^\circ$. We have cross-checked the results obtained with ODE113 against the RK4 algorithm, and found good agreement between the two. In order to study the sensitivity to the initial conditions, parameters scans were then carried out by varying $\theta_{cb}$, $\phi_1$, and $\phi_2$. For each set of initial conditions a calculation was run up to 200 fs. Two outputs were recorded : (a) the ratio of the final displacement in $y$ to that in $x$ ($r_y/r_x$), and (b) the time at which the maximum $\gamma$ occurred ($\tau_{\gamma,{max}}$, normalized to the laser period). Two types of scans were carried out, [*coarse*]{} and [*fine*]{}. For the coarse scans, 100 points were used for each parameter over a large range : $\pm \pi$ for phases and 10–30$^\circ$ for $\theta_{cb}$. For the fine scans, a fraction of each range was used and 200 points were then used for each parameter. In all other respects, the calculations are the same as the baseline calculation described in Sec. \[model\]. By moving from the analysis of Sec. \[standard\] where we looked at 10 cases to these parameter scans where we look at 10000-40000 cases per scan, we can obtain a much clearer idea of how sensitive this problem can be to the initial conditions. In Fig.s \[fig:figure6\] and \[fig:figure7\] we show the results from a coarse parameter scan of $\theta_{cb}$ and $\phi_2$ with $\phi_1$ held fixed at 0$^{\circ}$. The sub-figures show plots of fine parameter scans in the regions indicated. ![\[fig:figure6\]Results from parameter scan over $\phi_2$ and $\theta_{cb}$ in terms of $r_y/r_x$. Main plot is a coarse scan, with fine scans as sub-plots a and b. ](AngleDeltaPhi_Slope_All.png){width="\columnwidth"} ![\[fig:figure7\]Results from parameter scan over $\phi_2$ and $\theta_{cb}$ in terms of $\tau_{\gamma,{max}}$. Main plot is a coarse scan, with fine scans as sub-plots a and b. ](AngleDeltaPhi_TimeMaxGamma_All.png){width="\columnwidth"} What we observe from these extensive parameters scans is that the parameter space appears to consist of two types of regions. There are regions where the results of the calculations vary (relatively) slowly and smoothly as the initial conditions are changed. Examples of these are shown in the ‘a’ sub-figures in both fig. \[fig:figure6\] and fig. \[fig:figure7\]. There are also regions where small variations of the initial conditions leads to gross changes in the results including rapid changes in sign. Examples of these sub-regions are shown in the ‘b’ sub-figures in both fig. \[fig:figure6\] and fig. \[fig:figure7\]. As we were observing strong point-to-point changes in fig. \[fig:figure7\](b) along $\theta_{cb} = $ 28$^\circ$ we repeated this set of calculations at twice the resolution in $\phi_2$ (i.e. now with 400 points in $\phi_2$ across the ‘fine’ range). The results are shown in fig. \[fig:figure8\]. It can be seen that there is no improvement in terms of being able to ‘resolve’ the detail in this region. ![\[fig:figure8\]Line-out of Fig.\[fig:figure7\](b) along $\theta_{cb} =$ 28$^\circ$](LineOut_TimeGammaMax2.png){width="\columnwidth"} We can summarize the results from this second phase of the study as follows : (i) we have done an extensive parameter scan of the initial conditions / problem parameters using an advanced ODE solver, (ii) this reveals regions in parameter space that are very sensitive to the initial conditions / problem parameters, (iii) we are not able to ‘resolve’ this sensitivity by successively refining the set of points over which we scan. These observations suggest that we are actually looking at a system that exhibits chaotic dynamics, as we expected from earlier studies. Conclusions =========== In this paper we have examined an apparently simple model problem in relativistic single electron motion relevant to ultra-intense laser-plasma interactions, involving two obliquely intersecting plane wave-packets. The findings for this model problem, which are presented herein can be summarized as follows: 1. [Under a wide range of conditions converged solutions cannot be obtained for a 1 ps period using a wide range of different solvers including the Boris method, 4th order Runge-Kutta, and the [MATLAB]{} suite of ODE solvers.]{} 2. [Converged solutions appear to occur in isolated ranges of problem parameters.]{} 3. [Converged solutions can, in general, only be obtained over quite short durations, especially compared to benchmarks such as the single plane-wave problem where this is not an issue.]{} 4. [When extensive parameter scans are carried out across initial conditions / problem parameters, it is found that regions in parameter space exist where there is a very high degree of sensitivity to these initial conditions (or problem parameters).]{} 5. [Progressively increasing the resolution of these sensitive regions does not lead to any improved resolution of the highly sensitive region.]{} Our findings have, in the authors’ view, two main consequences. Firstly, great care needs to be taken when using PIC codes to study laser-plasma interactions. Prior to this study it was generally assumed that algorithms such as the Boris pusher would produce reasonably accurate results irrespective of the field configuration under consideration. In light of this study, we no longer think this can be assumed. We suggest that PIC simulations are accompanied by complementary studies of the single particle motion to ensure that converged orbits can be obtained. Secondly, these findings suggest that the root cause of both the issues of convergence and the sensitivity to initial conditions is at the very least indicative of extreme nonlinearity, but it quite strongly suggests that the dynamics of this problem are [*chaotic*]{}. This is entirely consistent with earlier studies [@z.-m.sheng_stochastic_2002; @sheng_efficient_2004; @meyer-ter-vehn_electron_1999], however these results now indicate that it is quite easy for the Lyapunov time to become sufficiently short that numerical integration is inhibited. This would explain the very limited ability of nearly all methods to obtain converged solutions, and it also explains the very high sensitivity to initial conditions. We do not claim to provide any rigorous proof that the dynamics of this system are chaotic, only to submit the results of numerical calculations that show that this might be the case, and that further investigation should be carried out. We do however draw the attention of the reader to earlier studies where such detailed analysis was carried out [@mendonca_1983]. If this simple model problem is indeed shown to have chaotic dynamics then this could have quite profound implications for the field of ultra-intense laser-plasma interactions, as it would then imply that a number of laser-target configurations where there are interfering laser fields would have the potential for chaotic dynamics. Acknowledgements {#acknowledgements .unnumbered} ================ The work of K.T. and A.V.A was supported by the National Science Foundation (PHY 1632777). K.W. was supported in part by the DOE Office of Science under Grant No. DE-SC0018312 and in part by the DOE Computational Science Graduate Fellowship under Grant No. DE-FG02-97ER25308.

--- abstract: | We consider Ricci flow invariant cones $\mathcal{C}$ in the space of curvature operators lying between nonnegative Ricci curvature and nonnegative curvature operator. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to Ricci flow has its curvature operator which satsisfies $\operatorname{R}+{\varepsilon}\operatorname{I}\in\mathcal{C}$ at the initial time, then it satisfies $\operatorname{R}+K{\varepsilon}\operatorname{I}\in\mathcal{C}$ on some time interval depending only on the scalar curvature control. This allows us to link Gromov-Hausdorff convergence and Ricci flow convergence when the limit is smooth and $\operatorname{R}+\operatorname{I}\in\mathcal{C}$ along the sequence of initial conditions. Another application is a stability result for manifolds whose curvature operator is almost in $\mathcal{C}$. Finally, we study the case where $\mathcal{C}$ is contained in the cone of operators whose sectional curvature is nonnegative. This allow us to weaken the assumptions of the previously mentioned applications. In particular, we construct a Ricci flow for a class of (not too) singular Alexandrov spaces. author: - Thomas Richard title: 'Lower bounds on Ricci flow invariant curvatures and geometric applications.' --- Introduction and statement of the results ========================================= In the study of the Ricci flow, various nonnegative curvature conditions have been shown to be preserved, and the discovery of new invariant conditions has often given rise to new geometric applications. One of the most famous occurence of this fact is the discovery by Brendle and Schoen in [@MR2449060] and independently by Nguyen in [@MR2587576] of the preservation of nonnegative isotropic curvature, which plays a crucial role in the proof by Brendle and Schoen of the differentiable sphere theorem in [@MR2449060]. Once one has understood the behaviour of the Ricci flow assuming the nonnegativity of a certain curvature, it is natural to ask if something can be done under arbitrary lower bounds on this given curvature. Such a work has been done for Ricci curvature in dimension 3 by Simon in [@MR2526789] and [@MS2009]. An important feature of this work is that, in order to control lower bounds on the Ricci curvature along the flow, one has to impose further geometric conditions on the initial manifold. In Simon’s work, a non-collapsing assumption is required. Our estimate will rely on an a priori bound on the scalar curvature. In order to state the results of this paper, we need some terminology. A nonnegativity condition on the curvature is given by a closed convex cone $\mathcal{C}$ in the space of algebraic curvature operators $S^2_B\Lambda^2\mathbb{R}^n$ such that : - The identity operator $\operatorname{I}:\Lambda^2\mathbb{R}^n\to\Lambda^2\mathbb{R}^n$ lies in the interior of $\mathcal{C}$. - $\mathcal{C}$ is invariant under the action of $O(n,\mathbb{R})$ on $S^2_B\Lambda^2\mathbb{R}^n$ given by : $${\left\langle g.\operatorname{R}(x\wedge y),z\wedge t \right\rangle}={\left\langle \operatorname{R}(gx\wedge gy),gz\wedge gt \right\rangle}.$$ Recalls and references on algebraic curvtaure operators are included in Section \[sec:prel-about-algebr\]. Given a nonnegativity condition $\mathcal{C}$ and a Riemannian manifold $(M,g)$, we can canonically embed $\mathcal{C}$ in $S^2_B\Lambda^2T_mM$ for each $m\in M$, thanks to the $O(n,\mathbb{R})$ invariance of $\mathcal{C}$. We say that $(M,g)$ has $\mathcal{C}$-nonnegative curvature (or $\operatorname{R}\geq_\mathcal{C}0$) if, for each $x\in M$ the curvature operator of $(M,g)$ at $x$ belongs to $\mathcal{C}$. Classical condtions of nonnegative curvature operator, nonnegative sectional curvature, nonnegative Ricci curvature or nonnegative scalar curvature fit in this framework. Similarly we say that $(M,g)$ has $\mathcal{C}$-curvature bounded from below by $-k\operatorname{I}$ (or $\operatorname{R}\geq_\mathcal{C} -k\operatorname{I}$) for some $k\in\mathbb{R}$ if for each $x\in M$ the curvature operator $\operatorname{R}$ at $x$ is such that $\operatorname{R}+k\operatorname{I}\in\mathcal{C}$. We now define a class of nonnegativity condition which behaves well with Ricci flow. A nonnegativity condition $\mathcal{C}$ is said to be (Ricci Flow) invariant if $\mathcal{C}$ is preserved by Hamilton’s ODE $\dot{\operatorname{R}}=2Q(\operatorname{R})$. Namely, if $\operatorname{R}(t)$ is a solution to Hamilton’s ODE on some time interval $[0,T)$ such that $\operatorname{R}(0)\in\mathcal{C}$, then $\operatorname{R}(t)\in\mathcal{C}$ for all $t\in[0,T)$. Details and references about Hamilton’s ODE are given in Section \[sec:prel-about-algebr\]. Hamilton’s maximum principle for tensors ([@MR862046 Theorem 4.3]) implies that such a cone is preserved by Ricci flow in the sense that, if $(M,g_0)$ is a compact Riemannian manifold such that $\operatorname{R}\geq_\mathcal{C} 0$, then the Ricci flow $(M,g(t))$ such that $g(0)=g_0$ satisfies $\operatorname{R}(g(t))\geq_\mathcal{C} 0$ as long as it exists. We are now ready to state our result. It roughly says the following. We consider a manifold whose $\mathcal{C}$-curvature is bounded from below, where $\mathcal{C}$ is an invariant condition between nonnegative Ricci curvature and nonnegative curvature operator. We furthermore assume that an a priori estimate on the blow up rate of the scalar curvature of the Ricci flow as $t$ goes to zero is true. Then the $\mathcal{C}$-curvature can be bounded from below on a small time interval. \[lboundthm\] For any dimension $n\in\mathbb{N}$, any $A\in(0,\frac{1}{4})$ and any $B>0$, one can find $T=T(n,A,B)$ and $K=K(n,A,B)$ such that if $\mathcal{C}\subset S^2_B\Lambda^2\mathbb{R}^n$ is a closed convex cone which satisfies : 1. $\mathcal{C}$ is an invariant nonnegativity condition, 2. $\mathcal{C}$ contains the cone of nonnegative curvature operators, 3. $\mathcal{C}$ is contained in the cone of curvature operators whose Ricci curvature is nonnegative, and $(M^n,g(t))_{t\in [0,T')}$ is a Ricci flow on a smooth compact manifold satisfying : 1. $\operatorname{R}(g(0))\geq_{\mathcal{C}}-{\varepsilon}\operatorname{I}$ at $t=0$ for some ${\varepsilon}\in [0,1]$, 2. $|\operatorname{Scal}(g(t))|\leq A/t+B$ for $t$ in $(0,T')$, we have : $$\operatorname{R}(g(t))\geq_{\mathcal{C}}-K{\varepsilon}\operatorname{I}$$ for all $t$ in $[0,T')\cap [0,T)$. During the redaction of this article, the author has been informed that a similar estimate was also known by Miles Simon and Arthur Schlichting. \[excones\] Known examples of cones which satisfy the assumptions of the theorem include : - the cone $\mathcal{C}_{CO}$ of nonnegative curvature operators, - the cone $\mathcal{C}_{2CO}$ of 2-nonnegative curvature operators, - the cone $\mathcal{C}_{IC1}$ of curvature operators which have nonnegative isotropic when extended by $0$ to $\Lambda^2\mathbb{R}^{n+1}$, - the cone $\mathcal{C}_{IC2}$ of curvature operators which have nonnegative isotropic when extended by $0$ to $\Lambda^2\mathbb{R}^{n+2}$. All these conditions have been extensively studied ([@MR2415394],[@MR2449060],[@MR2386107],[@MR2462114]) and compact manifolds with $\mathcal{C}$-nonnegative curvature have been classified when $\mathcal{C}$ is one of these four cones. An exposition of the relations between these condtions and how nonnegativity of these curvatures affect the topology of the underlying manifold can be found in the Brendle’s book [@MR2583938] together with precise definitions and additional references. It should also be noted that Wilking has given a unified proof of the preservation of these conditions (along with others) in [@2010arXiv1011.3561W]. Some continuous families of such cones have also been constructed in [@MR2415394] and [@2011arXiv1105.5311G]. It should be noted that in dimension greater or equal to $4$, nonnegative Ricci curvature is not preserved, see [@MR2736347]. \[remwilk\] If $\mathcal{C}$ satisfies the assumptions of the theorem and moreover is a Wilking cone (see [@2010arXiv1011.3561W]), it follows from the work of Gururaja, Maity and Seshadri in [@2011arXiv1101.5884G] that $\mathcal{C}$ is included in $\mathcal{C}_{IC1}$. [**From now on any curvature condition $\mathcal{C}$ is supposed to satisfy the assumptions of Theorem \[lboundthm\].**]{} The estimate of Theorem \[lboundthm\] allows us to adapt the methods of [@MR2526789] and [@MS2009] to some higher dimensional situations. In the first two applications, the estimate on the scalar curvature which is required to apply Theorem \[lboundthm\] will be obtained by Perelman’s pseudolocality theorem, first stated in [@PerelEnt] but omitting the crucial assumption of completeness as pointed by Topping, see [@2011arXiv1106.2493G Theorem A.3], complete statement and proofs can be found in [@KleiLott; @MR2779131]. Our first application is to show that, if the $\mathcal{C}$-curvature is bounded from below along a sequence of compact $n$-dimensional smooth manifolds which Gromov-Hausdorff converges (we will write GH-converges in the sequel) to a compact $n$-dimensional smooth manifold, then, up to a subsequence, the associated Ricci flows converge to a Ricci flow of the limit manifold (where the initial condition is to be understood in a weak sense). Here convergence is smooth convergence of the Ricci flows up to diffeomorphisms, as in [@HamComp]. More precisely, we prove the following theorem, which is an higher dimensional analogue of [@MR2526789 Theorem 9.2], where such a theorem has been proved in dimension 3 under lower bounds on the Ricci curvature and without assuming smoothness and compactness of the limit : \[GHrfthm\] Let $(M_k,g_k)$ be a sequence of compact $n$-manifolds which satisfies $R\geq_\mathcal{C}-\operatorname{I}$ and which GH-converges to a compact smooth $n$-manifold $(M,g)$. Let $(M_k,g_k(t))_{t\in[0,T_k)}$ be the maximal solution of the Ricci flow satisfying $g_k(0)=g_k$. Then : 1. there is a positive constant $T$ such that each Ricci flow $(M_k,g_k(t))$ is defined at least on $[0,T)$ and the sequence of Ricci flows $(M_k,g_k(t))_{t\in(0,T)}$ is precompact in the sense of Cheeger-Gromov-Hamilton. 2. any Cheeger-Gromov-Hamilton limit $(\tilde{M},\tilde{g}(t))_{t\in(0,T)}$ of a convergent subsequence of $(M_k,g_k(t))_{t\in(0,T)}$ is such that $\tilde{M}$ is homeomorphic to $M$ and the distance functions $d_{\tilde{g}(t)}$ uniformly converge as $t$ goes to $0$ to some distance $\tilde{d}$ which is isometric to the distance $d_g$. In particular the $M_k$’s are homeomorphic to $M$ for $k$ large enough. Along the proof ot Theorem \[GHrfthm\], we will see that the precompactness of the sequence of flows $(M_k,g_k(t))_{t\in(0,T)}$ still holds when one replaces $\mathcal{C}$-curvature bounded from below by Ricci curvature bounded from below (see Lemma \[GHvsRF\]). However, our method of proof requires the lower bound on the $\mathcal{C}$-curvature to control the initial condition of the limit flow. In the conclusions of the theorem, the fact that the $M_k$’s are homeomorphic to $M$ for $k$ large enough can be seen using Cheeger and Colding’s work on manifolds with Ricci curvature bounded from below (see [@CheeCol Theorem A.1.12]). Additionaly, Cheeger and Colding’s result allow to strengthen the conclusion from homeomorphism to diffeomorphism. However, our proof is independent of this work. Another application is a result about manifolds whith almost nonnegative $\mathcal{C}$-curvature, in the spirit of [@MR2526789 Theorem 1.7] : \[almostNNCC\] For any $i>0$ and $D>0$, for any $n\in\mathbb{N}$, there is an $\varepsilon>0$ such that any manifold $(M,g)$ satisfying : 1. $\operatorname{inj}(g)\geq i$ 2. $\operatorname{diam}(M,g)\leq D$ 3. $\operatorname{R}\geq_{\mathcal{C}}-\varepsilon \operatorname{I}$ admits a metric whose curvature is $\mathcal{C}$-nonnegative. Using the classifcation results of Brendle [@MR2583938 Theorem 9.33], Micallef and Wang [@MR1253619 Theorem 3.1] and remark \[remwilk\], if we moreover assume that $\mathcal{C}$ is a Wilking cone, we have that the universal cover of $M$ is diffeomorphic to a product $\mathbb{R}^k\times N_1\times\dots\times N_l$ where each $N_i$ is one of the following : - a standard sphere $\mathbb{S}^n$ with $n\geq 2$, - a compact symmetric space. We then impose stronger requirements on the cone $\mathcal{C}$. We assume that $\mathcal{C}$ is included in the cone of curvature operators whose sectional curvature is nonnegative. The cones which satisfy this assumptions in the list of Example \[excones\] are $\mathcal{C}_{CO}$ and $\mathcal{C}_{IC2}$. This allows us to weaken the hypothesis of our results. In this context, it turns out that a convenient assumption that can be used to fulfill the hypothesis of Theorem \[lboundthm\] is that balls have almost euclidean volume. This is proved in Lemma \[lemalmosteuc\], and was inspired to the author by the recent work of Cabezas-Rivas and Wilking [@2011arXiv1107.0606C]. For instance, Theorem \[GHrfthm\] becomes : \[GHRFnnsec\] Let $\mathcal{C}$ be a cone satifying the hypothesis of Theorem \[lboundthm\] and which is contained in the cone of curvature operator whose sectional curvature is nonnegative. For any $n\in\mathbb{N}$, there exist $\kappa>0$, $T>0$ and $\delta>0$ such that if $(X,d)$ is a metric space which is a Gromov-Hausdorff limit of a sequence of compact manifolds $(M_i^n,g_i)$ such that : - $\operatorname{R}(g_i)\geq_{\mathcal{C}} -\kappa\operatorname{I}$, - for any $x\in M_i^n$, $\operatorname{vol}_{g_i}(B_{g_i}(x,1))\geq (1-\delta)\omega_n$, where $\omega_n$ is the volume of the unit ball in $\mathbb{R}^n$, then one can find a Ricci flow $(M,g(t))$ defined on $(0,T)$ with bounded curvature on each time slice such that $M$ is homeomorphic to $X$ and the distance $d_g(t)$ converge uniformly on any compact of $M$ to a distance $\tilde{d}$ such that $(M,\tilde{d})$ is isometric to $(X,d)$. The fact that $X$ is a manifold is a direct consequence of Perelman’s stability theorem (see [@MR2408265]), but we will not use this result in the proof. The metric space $(X,d)$ in our result is an Alexandrov space with curvature bounded form below and can have cone-like singularities, but the almost euclidean volume condition forbids too sharp cone angles. Similarly, we get a stronger analogue of Theorem \[almostNNCC\] : \[almostnnccnnsc\] Let $\mathcal{C}$ be a cone satifying the hypothesis of Theorem \[lboundthm\] and which is contained in the cone of curvature operator whose sectional curvature is nonnegative. For any $n\in\mathbb{N}$, there exists $\delta>0$ such that for any $D>0$, one can find ${\varepsilon}>0$ such that if $(M^n,g)$ is a compact Riemannian manifold such that : - $\operatorname{R}(g)\geq_\mathcal{C}-{\varepsilon}\operatorname{I}$, - $\forall x\in M,\ \operatorname{vol}_g(B_g(x,1))\geq(1-\delta)\omega_n$, - $\operatorname{diam}(M,g)\leq D$, then $M$ admits a metric with $\mathcal{C}$-nonnegative curvature. The article is organised as follows : in Section \[sec:prel-about-algebr\] we set up the notations and give some background about the evolution equation of the curvature operator along the Ricci flow that will be used in the proof of Theorem \[lboundthm\]. In Section \[sec:proofthm\], we give the proof of Theorem \[lboundthm\]. The applications are discussed in Section \[sec:applications\]. Section \[sec:manif-with-mathc\] is devoted to the applications in the case where $\mathcal{C}$-nonnegative curvature implies nonneagtive sectional curvature. Acknowledgements {#acknowledgements .unnumbered} ---------------- The author is grateful to Gilles Carron and Harish Seshadri for helpful discussions during the elaboration of this paper. The author also thanks his supervisor Gérard Besson for his interest and support. Preliminaries about algebraic curvature operators. {#sec:prel-about-algebr} ================================================== In this section, we set up the notations that will be used in this paper. Our conventions follow closely those of Böhm and Wilking in [@MR2415394]. We will denote by $S^2\Lambda^2\mathbb{R}^n$ the vector space of symmetric operators on $\Lambda^2\mathbb{R}^n$ equiped with the standard inner product. $S^2_B\Lambda^2\mathbb{R}^n$ is the vector space of operators in $S^2\Lambda^2\mathbb{R}^n$ which in addition satisfy the first Bianchi identity. It is called the space of algebraic curvature operators on $\mathbb{R}^n$. As a norm on this space we use the classical Frobenius norm $\|\operatorname{R}\|^2=\operatorname{trace}(\operatorname{R}^2)$. Similar constructions hold on the tangent bundle of a Riemannian manifold $(M,g)$ and give rise to the bundles $S^2\Lambda^2TM$ and $S^2_B\Lambda^2TM$. The curvature tensor of a manifold $(M,g)$ will always be viewed as a section of the bundle of curvture operators, $S^2_B\Lambda^2TM$. We follow the convention of [@MR2415394] for the curvature operator, namely, the curvature operator of a round sphere of radius 1 is the identity. We will use $\operatorname{R}$, $\operatorname{Ric}$ and $\operatorname{Scal}$ to denote the curvature operator, Ricci curvature and scalar curvature. When considering a Ricci flow $(M,g(t))$, we will often not specify the dependence on $t$ of these various curvature when no confusion is possible. We will write $\operatorname{I}$ for the identity operator of $S^2_B\Lambda^2\mathbb{R}^n$ and $\operatorname{id}$ for the identity of $\mathbb{R}^n$. Hamilton defined a bilinear map : $$\begin{aligned} \#:S^2\Lambda^2\mathbb{R}^n\times S^2\Lambda^2\mathbb{R}^n &\to S^2\Lambda^2\mathbb{R}^n\\ (\operatorname{R},\operatorname{L})&\mapsto \operatorname{R}\#\operatorname{L}\end{aligned}$$ whose expression can be found in [@MR862046] or [@MR2415394]. If $g(t)$ is a family of metric on $M$ evolving along the Ricci flow, Hamilton showed in [@MR862046] that in appropriate coordinates the curvature operator $\operatorname{R}_{g(t)}$ satisfy the following evolution equation : $${\frac{\partial \operatorname{R}}{\partial t}}=\Delta \operatorname{R}+ 2(\operatorname{R}^2+\operatorname{R}^\#),$$ where $\Delta$ is the connection laplacian and $\operatorname{R}^\#=\operatorname{R}\#\operatorname{R}$. Removing the laplacian in this evolution equation, we obtain Hamilton’s ODE : $$\dot{\operatorname{R}}=2(\operatorname{R}^2+\operatorname{R}^\#)=2Q(\operatorname{R}).$$ We will need the following algebraic fact about the $\#$ operator, which was proved by Böhm and Wilking [@MR2415394 Lemma 2.1] : \[BWid\] $\operatorname{R}+\operatorname{R}\#\operatorname{I}=\operatorname{Ric}\wedge\operatorname{id}$ Here $\operatorname{Ric}\wedge\operatorname{id}$ is the curvature operator defined by, for any $u$ and $v$ in $\mathbb{R}^n$ : $$\operatorname{Ric}\wedge\operatorname{id}(u\wedge v)=\frac{1}{2}(\operatorname{Ric}(u)\wedge v+u\wedge\operatorname{Ric}(v)),$$ where $\operatorname{Ric}$ is viewed as an operator on $\mathbb{R}^n$. In particular, if $(\lambda_i)_{1\leq i\leq n}$ are the eigenvalues of $\operatorname{Ric}$ then the eigenvalues of $\operatorname{Ric}\wedge\operatorname{id}$ are $(\frac{\lambda_i+\lambda_j}{2})_{1\leq i<j\leq n}$. Proof of Theorem \[lboundthm\]. {#sec:proofthm} =============================== According to our hypothesis, if we define a new section of the bundle of curvature operators $\operatorname{L}$ by : $$\operatorname{L}=\operatorname{R}+{\varepsilon}({\varphi}(t)+t\alpha\operatorname{Scal})\operatorname{I},$$ it is enough to find a positive smooth function ${\varphi}$, a constant $\alpha$ and a time $T>0$, all depending only on $A$ and $B$ such that $\operatorname{L}\in\mathcal{C}$ for $t\in [0,T)$. The fact that $t\operatorname{Scal}$ and ${\varphi}$ are unformly bounded on $[0,T]$ will then give the required bound. To ensure that $\operatorname{L}\in\mathcal{C}$ at time $0$, we impose that ${\varphi}(0)=1$. Since such lower bounds are likely to get worse with time, we will assume that ${\varphi}'\geq 0$. To prove that $\operatorname{L}$ remains in $\mathcal{C}$, we will apply Hamilton’s maximum principle for tensors [@MR862046], or more precisely a variant of it called maximum principle with avoidance set proved by Chow and Lu in [@MR2042930 Theorem 4]. This variant allows us to use our a priori estimate on the scalar curvature (which is not implied by the ODE) in the study of the ODE associated to the PDE satisfied by $\operatorname{L}$. We will impose conditions on ${\varphi}$ and $\alpha$ during the proof and verify that these conditions can be fullfilled at the end of the proof. We first compute the evolution of $\operatorname{L}$ : $$\begin{aligned} {\frac{\partial \operatorname{L}}{\partial t}} = & \Delta \operatorname{R}+2Q(\operatorname{R}) +{\varepsilon}(\varphi'+\alpha \operatorname{Scal}+t\alpha(\Delta \operatorname{Scal}+2|\operatorname{Ric}|^2))\operatorname{I}\\ = & \Delta \operatorname{L}+2Q(\operatorname{R}) +{\varepsilon}(\varphi' +\alpha \operatorname{Scal}+2t\alpha |\operatorname{Ric}|^2)\operatorname{I}\\ = & \Delta \operatorname{L}+ 2N(\operatorname{L}).\end{aligned}$$ We now have to show that $\mathcal{C}$ is preserved by the differential equation $\dot{\operatorname{L}}=2N(\operatorname{L})$. That is, given $\operatorname{L}\in\partial\mathcal{C}$, we need to show that $N(\operatorname{L})\in\mathcal{C}$. Since $\mathcal{C}$ is preserved by Hamilton’s ODE, we know that $Q(\operatorname{L})\in\mathcal{C}$ and we just need to show (since $\mathcal{C}$ is convex) that $D(\operatorname{L})=N(\operatorname{L})-Q(\operatorname{L})\in\mathcal{C}$. This idea comes from the work of Böhm and Wilking in [@MR2415394]. We will in fact prove that $D(\operatorname{L})$ is a nonnegative curvature operator, which will be enough since $\mathcal{C}$ contains the cone of nonnegative curvature operator. Using Böhm and Wilking identity (proposition \[BWid\]), we have : $$\begin{aligned} Q(\operatorname{L})= & Q(\operatorname{R})+2{\varepsilon}({\varphi}+t\alpha\operatorname{Scal})(\operatorname{R}+\operatorname{R}\# \operatorname{I}) +{\varepsilon}^2({\varphi}+t\alpha\operatorname{Scal})^2 Q(\operatorname{I})\\ = & Q(\operatorname{R}) +2{\varepsilon}({\varphi}+t\alpha\operatorname{Scal})(\operatorname{Ric}\wedge \operatorname{id})+(n-1) {\varepsilon}^2({\varphi}+t\alpha\operatorname{Scal})^2 \operatorname{I}.\end{aligned}$$ We then compute $D(\operatorname{L})$ : $$\begin{aligned} D(\operatorname{L})= & N(\operatorname{L})-Q(\operatorname{L})\\ = & \frac{{\varepsilon}}{2}(\varphi' +\alpha \operatorname{Scal}+2t\alpha |\operatorname{Ric}|^2)\operatorname{I}\\ & -2{\varepsilon}({\varphi}+t\alpha\operatorname{Scal})(\operatorname{Ric}\wedge\operatorname{id})-(n-1) {\varepsilon}^2({\varphi}+t\alpha\operatorname{Scal})^2 \operatorname{I}.\end{aligned}$$ In order to estimate the $2\operatorname{Ric}\wedge \operatorname{id}$ term, we use that $\operatorname{L}\in\mathcal{C}$ has nonnegative Ricci curvature, which gives that $\operatorname{Ric}\geq -(n-1){\varepsilon}({\varphi}+t\alpha\operatorname{Scal})\operatorname{id}$ as symmetric operators. Since $\operatorname{trace}(\operatorname{Ric})=\operatorname{Scal}$, we have : $$-(n-1){\varepsilon}({\varphi}+t\alpha\operatorname{Scal})\operatorname{id}\leq\operatorname{Ric}\leq (\operatorname{Scal}+(n-1)^2{\varepsilon}({\varphi}+t\alpha\operatorname{Scal}))\operatorname{id}.$$ This implies : $$2\operatorname{Ric}\wedge \operatorname{id}\leq(\operatorname{Scal}+(n-1)^2{\varepsilon}({\varphi}+t\alpha\operatorname{Scal}))\operatorname{I}.$$ We now assume that : $${\varphi}+t\alpha\operatorname{Scal}\geq 0\qquad\text{condition (C1)}.$$ This allows us to estimate $D(\operatorname{L})$ : $$\begin{aligned} D(\operatorname{L})\geq &\ \frac{{\varepsilon}}{2}(\varphi' +\alpha \operatorname{Scal})\operatorname{I}\\ & -{\varepsilon}({\varphi}+t\alpha\operatorname{Scal})(\operatorname{Scal}+(n-1)^2{\varepsilon}({\varphi}+t\alpha\operatorname{Scal}))\operatorname{I}\\ & -(n-1){\varepsilon}^2({\varphi}+t\alpha\operatorname{Scal})^2 \operatorname{I}.\end{aligned}$$ We rearrange the terms in the following way[^1] : $$\begin{aligned} D(\operatorname{L})\geq&\ \frac{{\varepsilon}}{2} {\varphi}'\\ & +{\varepsilon}\operatorname{Scal}((\frac{1}{2}-t\operatorname{Scal})\alpha-{\varphi})\\ & -(2n-1)(n-1){\varepsilon}^2({\varphi}+t\alpha\operatorname{Scal})^2\end{aligned}$$ We now assume that : $$0\leq (\frac{1}{2}-t\operatorname{Scal})\alpha-{\varphi}\leq 1\qquad\text{condition (C2)},$$ Since $\operatorname{Scal}\geq -{\varepsilon}n(n-1)$ at $t=0$, it remains so as long as the solution exists. Therefore we have : $$\begin{aligned} D(\operatorname{L})\geq&\ \frac{{\varepsilon}}{2}{\varphi}'\\ & -{\varepsilon}^2 n(n-1)\\ & -(2n-1)(n-1){\varepsilon}^2({\varphi}+t\alpha\operatorname{Scal})^2,\end{aligned}$$ and since ${\varepsilon}\in [0,1]$ : $$\begin{aligned} \frac{1}{{\varepsilon}^2}D(\operatorname{L})\geq&\ \frac{{\varphi}'}{2}\\ & -n(n-1)\\ & -(2n-1)(n-1)({\varphi}+t\alpha\operatorname{Scal})^2.\end{aligned}$$ We now use that $|t\operatorname{Scal}|\leq A+Bt$ to get : $$\begin{aligned} \frac{1}{{\varepsilon}^2}D(\operatorname{L})\geq&\ \frac{{\varphi}'}{2}\\ & -n(n-1)\\ & -(n-1)(2n-1)({\varphi}+\alpha(A+Bt))^2.\end{aligned}$$ To ensure that $D(\operatorname{L})$ is a nonnegative operator, it is then enough to show that : $$\frac{{\varphi}'}{2} -n(n-1) -(n-1)(2n-1)({\varphi}+\alpha(A+Bt))^2\geq 0\qquad\text{condition (C3)}.$$ We now have to find ${\varphi}$, $\alpha$ and $T$ such that conditions (C1), (C2) and (C3) are satisfied on $[0,T]$. Using again that $-n(n-1)t\leq t\operatorname{Scal}\leq A+Bt$, we have that conditions (C1) and (C2) are implied by the following inequalities which involves only $A$, $B$ and the dimension $n$ : $$\begin{aligned} \left . \begin{array}{l} (\frac{1}{2}-(A+Bt))\alpha-{\varphi}\geq 0\\ (\frac{1}{2}+tn(n-1))\alpha-{\varphi}\leq 1\\ {\varphi}-n(n-1)t\alpha\geq 0 \end{array} \right\} \qquad\text{condition (C4)}\end{aligned}$$ Looking at conditions (C4) at $t=0$, we see that it is fulfilled if $\alpha$ belongs to $[\frac{2}{1-2A},4]$. We now impose that $A<\frac{1}{4}$. Let $\alpha\in (\frac{2}{1-2A},4)$, and ${\varphi}(t)=1+\beta t$. Conditition (C4) is then satisfied at time $0$ with strict inequalities. We now choose $\beta$ big enough such that condition (C3) is fulfilled with a strict inequality. By continuity of $\varphi$, these conditions are still fulfilled for $t$ in some small time interval $[0,T)$. Our choices of ${\varphi}$, $\alpha$ and $T$ depend only on $A$, $B$ and $n$, the theorem is then proved. First applications. {#sec:applications} =================== Gromov-Hausdorff converging sequences whose $\mathcal{C}$-curvature is bounded from below. {#sec:grom-hausd-conv} ------------------------------------------------------------------------------------------ In this section, we prove Theorem \[GHrfthm\]. We first state a lemma which is of independent interest, the idea of using pseudolocality and convergence of the isoperimetric profiles in the proof of the following lemma was suggested to the author by Gilles Carron : \[GHvsRF\] Let $(M_k,g_k)_{k\in\mathbb{N}}$ be a sequence of smooth compact $n$-dimensional Riemannian manifold which satisfies $\operatorname{Ric}(g_k)\geq -(n-1)g_k$ and which GH-converges to a smooth compact $n$-dimensional Riemannian manifold $(M,g)$. Then for every every $A>0$, there exist $k_0\in \mathbb{N}$, $B>0$ and $T>0$ such that, for any $k\geq k_0$ the Ricci flows $(M_k,g_k(t))$ whith initial condition $(M_k,g_k)$ exist at least on $[0,T)$ and satisfy : 1. $\|\operatorname{R}(g_k(t))\|\leq A/t+B$ for all $t\in(0,T)$, 2. $\operatorname{vol}(B_{g_k(t)}(x,\sqrt{t}))\geq ct^{n/2}$ for all $t\in (0,T)$ and $x\in M_i$. In particular, the Ricci flows $(M_k,g_k(t))_{t\in(0,T)}$ form a precompact sequence in the sense of Cheeger Gromov and Hamilton. We want to apply Perelman’s pseudolocality ([@PerelEnt Section 10], [@KleiLott Theorem 30.1, Corollary 35.1]) to get the two estimates of the lemma. The precompactness statement then follows from Hamilton’s compactness theorem [@HamComp]. Let $A>0$ be fixed. We already know that for any $x\in M_k$, $\operatorname{Scal}_{g_k}(x)\geq -n(n-1)$. Thus we just need to find some $r_0\in\left (0,(n(n-1))^{-1/2}\right ]$ such that any smooth domain $\Omega$ contained in a ball of radius $r_0$ in $M_k$ for $k$ large enough satisfies the almost Euclidean isoperimetric estimate : $$|\partial\Omega|^{\frac{n}{n-1}}\geq (1-\delta)\gamma_n |\Omega|\label{eq:1}$$ where $\gamma_n$ is the euclidean isoperimetric constant and $\delta$ is given by the pseudolocality theorem. To obtain this estimate, we will consider the isoperimetric profiles of the $(M_k,g_k)$’s, that will be denoted by $h_k(\beta)$. Since $(M,g)$ is smooth, by a result of Bérard and Meyer [@MR690651 Appendice C], its isoperimetric profile $h(\beta)$ is equivalent to the euclidean one as $\beta$ goes to zero. Thus we can find, for any given $\varepsilon>0$, some $\rho>0$ such that : $$\beta<\rho \Rightarrow\ h(\beta)\geq (1-\varepsilon) \frac{\gamma_n}{\operatorname{vol}(M,g)^{\frac{1}{n}}} \beta^{\frac{n-1}{n}}$$ We then use a result from Bayle thesis [@Bayle] : under non collapsing GH-convergence to a smooth manifold with Ricci curvature bounded from below, the ratio of the isoperimetric profiles $h_k/h$ is going to $1$ uniformly on $(0,1)$. Then, for $i$ large enough : $$h_k\geq (1-\varepsilon)h.$$ Let $\Omega\subset M_i$ be a smooth domain whose volume is less than $\rho \operatorname{vol}(M_k,g_k)$. We then have : $$\begin{aligned} |\partial\Omega| & \geq \operatorname{vol}(M,g)\times h_k\left(\frac{|\Omega|}{\operatorname{vol}(M,g)}\right)\\ & \geq \operatorname{vol}(M,g)\times (1-\varepsilon)h \left(\frac{|\Omega|}{\operatorname{vol}(M,g)}\right)\\ & \geq \operatorname{vol}(M,g)\times (1-\varepsilon)^2 \frac{\gamma_n}{\operatorname{vol}(M,g)^{\frac{1}{n}}} \left(\frac{|\Omega|}{\operatorname{vol}(M,g)}\right)^{\frac{n-1}{n}}\\ & = (1-\varepsilon)^2 \gamma_n|\Omega|^{\frac{n-1}{n}} \end{aligned}$$ If we take ${\varepsilon}$ small enough, we get estimate for domains of volume less then $\rho \operatorname{vol}(M_k,g_k)$. Now, using Colding’s theorem on the continuity of volume [@ColVol], for $k$ large enough, $\operatorname{vol}(M_k,g_k)\geq V/2$ where $V$ is the volume of $(M,g)$. In particular, our almost Euclidean isoperimetric inequality is valid for domains of volume less than $\rho V/2$. Since the Ricci curvature is bounded from below, Bishop Gromov inequality gives us that : $$\operatorname{vol}(B_{g_k}(x,r))\leq V_{-1}(r)$$ where $V_{-1}(r)$ is the volume a radius $r$ ball in the $n$-dimensional hyperbolic space. This shows that our isoperimetric inequality is valid for domains included in balls of radius less than $r_0$ where $r_0$ is such that $V_{-1}(r_0)=\rho V/2$. Finally, pseudolocality applies and we get the required bounds. We now proove Theorem \[GHrfthm\]. We now consider a sequence $(M_k^n,g_k)$ of smooth compact manifolds whose $\mathcal{C}$-curvature is bounded from below by $-\operatorname{I}$ and which in addition satisfy the assumptions of Lemma \[GHvsRF\]. Thanks to the previous lemma, we can find $i_0\in\mathbb{N}$, $T>0$ and a constant $B$ such that, for $k\geq k_0$, the Ricci flows $(M_k,g_k(t))$ satisfying $g_k(0)=g_k$ satisfy : $$|\operatorname{Scal}(g_k(t))|\leq\frac{1}{8t}+B\text{ for }t\in(0,T).\label{eq:3}$$ We now use Theorem \[lboundthm\] and the fact that $(M_k,g_k(0))$ has $\mathcal{C}$-curvature bounded from below by $-\operatorname{I}$ to find $T'>0$ and $K>0$ such that, for $t\in(0,T')$, $$\operatorname{R}(g_k(t))\geq_\mathcal{C} -K\operatorname{I}.\label{eq:2}$$ Since this implies that the Ricci curvature of $(M_k,g_k(t)))$ is bounded from below by $-(n-1)K$ on $[0,T')$, we can apply Lemma 6.1 in [@MS2009]. We get, for some constant $c>0$, that for $k\geq k_0$, $x,y\in M_k$ and $0<s\leq t <T'$ : $$d_{g_k(s)}(x,y)-c(\sqrt{t}-\sqrt{s})\leq d_{g_k(t)}\leq e^{c(t-s)}d_{g_k(s)} \label{eq:4}$$ where $d_{g_k(t)}$ is the distance function of $(M_k,g_k(t))$. Consider now a subsequence of the sequence $(M_k,g_k(t))_{t\in(0,T')}$ which converges in the sense of Cheeger-Gromov-Hamilton to a Ricci flow $(\tilde{M},\tilde{g}(t))_{t\in(0,T')}$. This flow also satisfies estimates , and . As in the proof of Theorem 9.2 in [@MS2009], we can prove that the distances $d_{\tilde{g}(t)}$ uniformly converge as $t$ goes to zero to some distance $\tilde{d}$, which define the usual manifold topology on $\tilde{M}$, and that $(\tilde{M},\tilde{d})$ is isometric to the GH-limit $(M,g)$ of the sequence $(M_k,g_k)$. In particular, $M$ and $\tilde{M}$ are homeomorphic. Manifolds with almost nonnegative $\mathcal{C}$-curvature. {#sec:manif-with-almost} ---------------------------------------------------------- We now proove Theorem \[almostNNCC\]. By contradiction, take a sequence of counterexamples $(M_k,g_k)$ satisfying $\operatorname{R}\geq_\mathcal{C}-{\varepsilon}_k\operatorname{I}$, where ${\varepsilon}_k$ goes to $0$, and the required bounds on the diameter and injectivity radius. We assume that none of the $M_k$ admits a metric with nonnegative $\mathcal{C}$-curvature. Without loss of generality, we assume that ${\varepsilon}_k\leq 1$. Since the injectivity radius and the Ricci curvature are bounded from below, we can use Anderson-Cheeger theorem [@AndersonCheeger2 Theorem 0.3]. It gives us, for any ${\varepsilon}>0$, some $r>0$ such that every ball $B$ of radius less than $r_0$ admit an harmonic coordinate chart ${\varphi}_B:B\to\mathbb{R}^n$ with : $$\frac{1}{1+{\varepsilon}}{\varphi}_B^*\delta\leq g\leq (1+{\varepsilon}){\varphi}_B^*\delta$$ on $B$, where $\delta$ is the euclidean metric on $\mathbb{R}^n$. If we choose ${\varepsilon}$ small enough, this control will give us an almost Euclidean isoperimetric estimate on balls of radius less than $r_0$. Consider the sequence of Ricci flows $(M_k,g_k(t))$ such that $g_k(0)=g_k$. Pseudolocality gives : - each $(M_k,g_k(t))$ exists at least on $[0,T)$ where $T$ does not depend on $k$, - for $t\in(0,T)$, $|\operatorname{Scal}(g_k(t))|\leq \frac{1}{8t}+B$, where $B$ does not depend on $k$. - the Ricci flows $(M_k,g_k(t))_{t\in(0,T)}$ form a precompact sequence in the sense of Cheeger-Gromov-Hamilton. We can then apply Theorem \[lboundthm\] to have that on some time interval $(0,T')\subset(0,T)$, $\operatorname{R}(g_k(t))\geq_\mathcal{C} -K{\varepsilon}_k\operatorname{I}$. Let $(M,g(t))_{t\in(0,T')}$ be the limit of a convergent subsequence of $(M_k,g_k(t))_{t\in(0,T')}$, it satisfies $\operatorname{R}(g(t))\geq_\mathcal{C} 0$ for $t\in(0,T')$. Now, since the Ricci curvature is bounded from below in time and along the sequence, we can find some constant $C$ such that : $$\operatorname{diam}(M_k,g_k(t))\leq e^{Ct}\operatorname{diam}(M_k,g_k(t))\leq e^{Ct}D$$ for all $k\in\mathbb{N}$ and $t\in(0,T')$. This implies that $M$ is compact. Hence, we have a subsequence of $(M_k,g_k)$ all of whose elements are diffeomorphic to $M$, in particular, these elements admit a metric with non-negative $\mathcal{C}$-curvature. This is a contradiction. Stronger results when operators in $\mathcal{C}$ have nonnegative sectional curvature {#sec:manif-with-mathc} ===================================================================================== [**We now assume that $\mathcal{C}$ contains the cone of curvature operators whose sectional curvature is nonnegative.**]{} As in the previous proofs, the crucial point is to get an $A/t+B$ bound on the scalar curvature. We first state a lemma which gives this bound when one has almost euclidean volume and $\mathcal{C}$-curvature bounded from below at the initial time. This lemma is a stronger version of Proposition 5.5 in [@2011arXiv1107.0606C]. \[lemalmosteuc\] For any dimension $n$, any $A\in(0,A_0(n))$, there exists $\kappa>0$, $\delta>0$, $\tilde{\kappa}>0$ and $T>0$ such that if $(M^n,g)$ is a compact Riemannian manifold such that : - $\operatorname{R}\geq_\mathcal{C}-\kappa\operatorname{I}$, - $\forall x\in M^n,\quad \operatorname{vol}_g(B_g(x,1))\geq (1-\delta)\omega_n$, where $\omega_n$ is the volume of the unit ball in $\mathbb{R}^n$. Then the Ricci flow $(M^n,g(t))$ with initial condition $(M^n,g)$ exists at least on $[0,T)$ and satisfies : - $\forall t\in(0,T)\quad \|\operatorname{R}(g(t))\|\leq \frac{A}{t}$, - $\forall t\in(0,T)\quad \operatorname{R}(g(t))\geq-\tilde{\kappa}\operatorname{I}$. Since $|\operatorname{Scal}|\leq 2\sqrt{n}\|\operatorname{R}\|$, we set $A_0(n)=\frac{1}{8\sqrt{n}}$. This ensures that if $\|\operatorname{R}\|\leq \frac{A}{t}$ we have the right estimate on the scalar curvature to apply Theorem \[lboundthm\]. The proof goes by contradiction. Fix $n\in\mathbb{N}$ and $A<A_0(n)$. Assume we can find a sequence of manifolds $(M_i,g_i)_{i\in\mathbb{N}}$ such that : - $\operatorname{R}(g_i)\geq_\mathcal{C}-\delta_i\operatorname{I}$, - $\forall x\in M,\quad \operatorname{vol}_{g_i}(B_{g_i}(x,1))\geq (1-\delta_i)\omega_n$, for some sequence $(\delta_i)_{i\in\mathbb{N}}$ going to zero. And assume furthermore that the sequence $(t_i)_{i\in\mathbb{N}}$ defined by $t_i=\sup\{\ t>0\ |\ \forall s\leq t,\ s\|R(g_i(s))\|\leq A\ \}$ goes to zero. Taking $i$ large enough, we can assume that $\delta_i\leq 1$ and $t_i$ is less than the time $T$ given by Theorem \[lboundthm\], this ensures that for all $t\in[0,t_i]$ : $$\operatorname{R}(g_i(t))\geq_\mathcal{C} -K\delta_i \operatorname{I}.$$ With this lower bound, we can repeat word for word the proof of Proposition 5.5 in [@2011arXiv1107.0606C] and get the $\frac{A}{t}$ bound on the norm of the curvature operator on some time interval. The lower bound on $\mathcal{C}$-curvature is now given by Theorem \[lboundthm\]. We now proove Theorem \[GHRFnnsec\]. The Ricci flow of $(X,d)$ is constructed as limit of the Ricci flows of the $(M_i,g_i)$, as in [@MS2009]. Fix $\kappa$ and $\delta$ such that Lemma \[lemalmosteuc\] apply with $A=\frac{1}{16\sqrt{n}}$. Consider a sequence $(M_i,g_i)$ satifying the assumption of the theorem. Using Lemma \[lemalmosteuc\], we have that the Ricci flows $(M_i,g_i(t))$ exist at least on $[0,T)$ and satisfy, for any $t$ in $[0,T)$ : - $\|\operatorname{R}(g_i(t))\|\leq\frac{1}{16\sqrt{n}t}$, - $\operatorname{R}(g_i(t))\geq_\mathcal{C}-K\operatorname{I}$. In addition, at time $t=0$, we have that any unit ball in any of the $M_i$’s has volume at least $(1-\delta)\omega_n$. This allow us to apply Lemma 6.1 and Corolarry 6.2 in [@MS2009] to get that, on some possibly smaller time interval $[0,T']$, we have the estimates, for some constant $C>0$ depending only on $\kappa$ and $\delta$ : - $\forall x\in M_i,\ \operatorname{vol}_{g_i}(B_{g_i}(x,1))\geq\frac{(1-\delta)\omega_n}{2}$, - for $0<s\leq t\leq T'$, $d_{g_i(s)}-C(\sqrt{t}-\sqrt{s})\leq d_{g_i(t)}\leq e^{C(t-s)}d_{g_i(s)}$, where $d_{g_i(t)}$ is the distance on $M_i$ induced by the metric $g_i(t)$. We then argue as in the proof of Theorem 9.2 in [@MS2009] and get that the sequence of Ricci flows $(M_i,g_i(t))_{t\in(0,T')}$ has a convergent subsequence whose limit $(M,g(t))_{t\in(0,T')}$ is a Ricci flow of the Gromov-Hausdorff limit $(X,d)$ of the sequence $(M_i,g_i)$ in the sense that it satisfies the conclusions of Theorem \[GHRFnnsec\]. We now go on with Theorem \[almostnnccnnsc\]. Let $\delta>0$ and $\kappa>0$ be the constants given by Lemma \[lemalmosteuc\] with $A=\frac{1}{16\sqrt{n}}$. Fix $D>0$. As in the proof of Theorem \[almostNNCC\], consider a sequence of manifolds $(M_i,g_i)$ with : - $\forall x\in M_i,\ \operatorname{vol}_{g_i}(B_{g_i}(x,1))\geq(1-\delta)\omega_n$, - $\operatorname{R}\geq_\mathcal{C}\ -{\varepsilon}_i\operatorname{I}$, - $\operatorname{diam}(M_i,g_i)\leq D$, where ${\varepsilon}_i$ goes to $0$ as $i$ goes to infinity. Assume furthermore that none of the $M_i$ admits a metric with nonnegative $\mathcal{C}$-curvature. Without loss of generality, we can assume that ${\varepsilon}_i\leq\min(\kappa,1)$. Arguing as in the proofs of Theorem \[GHRFnnsec\] and Theorem \[almostNNCC\], we get that the Ricci flows $(M_i,g_i(t))$ starting at $(M_i,g_i)$ exist at least on $[0,T)$, form a precompact family in the sense of Cheeger-Gromov-Hamilton, and satisfy : - $\operatorname{R}\geq_\mathcal{C} -K{\varepsilon}_i\operatorname{I}$, - $\operatorname{diam}(M_i,g_i(t))\leq e^{Ct}D$. In particular any limit of a convergent subsequence will be compact and have nonnegative $\mathcal{C}$-curvature. Thus the sequence contains manifolds which admit metrics with $\mathcal{C}$-nonnegative curvature. This is a contradiction. [10]{} Michael T. Anderson and Jeff Cheeger. -compactness for manifolds with [R]{}icci curvature and injectivity radius bounded below. , 35(2):265–281, 1992. Vincent Bayle. . PhD thesis, Université Joseph Fourier, 2003. <http://tel.archives-ouvertes.fr/tel-00004317/en/>. 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Non-negative [R]{}icci curvature on closed manifolds under [R]{}icci flow. , 139(2):675–685, 2011. Mario J. Micallef and McKenzie Y. Wang. Metrics with nonnegative isotropic curvature. , 72(3):649–672, 1993. Huy T. Nguyen. Isotropic curvature and the [R]{}icci flow. , (3):536–558, 2010. G. [Perelman]{}. . , November 2002. M. [Simon]{}. . , March 2009. Miles Simon. Ricci flow of almost non-negatively curved three manifolds. , 630:177–217, 2009. B. [Wilking]{}. . , November 2010. [Institut Fourier, 100 rue des maths, 38402 St Martin d’Hères.]{} [*Email adress:*]{} [thomas.richard@ujf-grenoble.fr]{} [^1]: We will drop the $\operatorname{I}$’s in the next inequalities, here a real number $\alpha$ should be viewed as the operator $\alpha \operatorname{I}$.

--- abstract: 'The positive partial transpose test is one of the main criteria for detecting entanglement, and the set of states with positive partial transpose is considered as an approximation of the set of separable states. However, we do not know to what extent this criterion, as well as the approximation, are efficient. In this paper, we show that the positive partial transpose test gives no bound on the distance of a density matrix from separable states. More precisely, we prove that, as the dimension of the space tends to infinity, the maximum trace distance of a positive partial transpose state from separable states tends to $1$. Using similar techniques, we show that the same result holds for other well-known separability criteria such as reduction criterion, majorization criterion and symmetric extension criterion. We also bring evidence that the set of positive partial transpose states and separable states have totally different shapes.' author: - | Salman Beigi[^1] Peter W. Shor[^2]\ *[Department of Mathematics, Massachusetts Institute of Technology]{}* title: Approximating the Set of Separable States Using the Positive Partial Transpose Test --- Introduction ============ The problem of detecting entanglement has been focused in quantum information theory for many years. The problem is: given a bipartite mixed state $\rho_{AB}$, decide whether this state is entangled or separable. The first attack toward solving this problem is the following observation due to Peres and the Horodeckis, [@peres; @23]. If $\rho_{AB}=\sum_i p_i\,\rho_{A_i}\otimes \rho_{B_i} $ is separable, then $[\rho_{AB}]^{T_B}=\sum_i p_i\rho_{A_i}\otimes [\rho_{B_i}]^T$, where $M^T$ denotes the transpose of matrix $M$, is also a quantum state, and thus is a positive semi-definite matrix. Therefore, if $\rho_{AB}$ is separable, its partial transpose, $[\rho_{AB}]^{T_B}$, should be positive semi-definite. The Horodeckis have proved that this criterion characterizes all separable states in dimensions $2\times 2$ and $2\times 3$, [@23]. However, there are entangled states in dimension $3\times 3$ with a positive partial transpose, [@bound]. Although the set of positive partial transpose states (PPT states) does not coincide with the set of separable states, it is usually considered as an approximation of this set. For example in [@distance] the distance of an arbitrary state from PPT states has been computed to estimate the distance from separable states. Also in [@geometry] the geometry of the set of PPT states has been studied to understand the properties of the set of separable states. However, we do not know how efficient these approximations are. For instance, given an upper bound on the distance of a state from PPT states, does it give an upper bound on the distance of the state from separable states? We can think of this problem in the point of view of complexity theory. Gurvits [@gurvits] has proved that given a bipartite density matrix $\rho_{AB}$, it is $\NP$-hard to decide whether this state is separable or entangled. An approximate formulation of this problem is the following. Given a bipartite density matrix $\rho_{AB}$ and $\epsilon>0$, decide whether there exists a separable state in the $\epsilon$-neighborhood (in trace distance) of $\rho_{AB}$. Gurvits has established a reduction from Knapsack to this problem, and has proved the $\NP$-hardness of the separability problem only for exponentially small $\epsilon$. However, as mentioned in [@salman], we can replace Knapsack with 2-out-of-4-SAT, and get to the $\NP$-hardness for an inverse polynomial $\epsilon$. Also, Gharibian [@sevag] has shown the same result using a reduction from the Clique problem. Now, the question is that how large $\epsilon$ can be while getting to the same result. For example, is there an efficient algorithm to decide whether the distance of a given state from separable states is less than $1/3$, or it is an $\NP$-hard problem? Equivalently, is there a separability test such that if a state passes the test then it is $1/3$-close to the set of separable states? The converse of this question is what we are looking for. That is, given a separability criterion, if a state passes this test can we claim a non-trivial upper bound on the distance of this state from separable states? In this paper, we prove that the answer for the PPT criterion, as well as other well-known separability tests such as reduction criterion [@reduction], majorization criterion [@majorization] and symmetric extension criterion [@symmetric1; @symmetric2], is no. More precisely, we prove the following theorem. \[thm:main\] Let $\mathcal{H}$ be a bipartite Hilbert space. If the dimension of each subsystem of $\mathcal{H}$ is large enough, there exists a PPT state acting on $\mathcal{H}$ whose trace distance from separable states is greater than $1-\epsilon$, for an arbitrary $\epsilon >0$. Main Ideas ---------- Let $\mathcal{H}=\mathcal{H}^A \otimes \mathcal{H}^B$ be a bipartite Hilbert space. We want to find PPT states $\rho^{(n)} \in \mathcal{H}^{\otimes n}$ such that the trace distance of $\rho^{(n)}$ from separable states is close to 1, for enough large numbers $n$. Suppose $\rho$ is an entangled PPT state. Then $\rho^{\otimes n}$ is entangled and also PPT. We claim that the sequence of states $\rho^{(n)}=\rho^{\otimes n}$ works for us. The intuition is that for two different quantum states $\rho$ and $\sigma$, the trace distance of $\rho^{\otimes n}$ and $\sigma^{\otimes n}$ tends to 1 as $n$ tends to infinity. However, in this problem $\sigma$ is not a fixed state and ranges over all separable states. Also, it is not obvious (and may not hold)[^3] that the closest separable state to $\rho^{\otimes n}$ is of the form $\sigma^{\otimes n}$. Another idea is to use entanglement distillation. Suppose the state $\rho$ is distillable. It means that, having arbitrary many copies of $\rho$, using an LOCC map, we can obtain arbitrary many EPR pairs. Notice that LOCC maps send separable states to separable states, and the trace distance decreases under trace preserving quantum operations. Therefore, the distance of $\rho^{\otimes n}$ from separable states is bounded from below by the distance of EPR$^{\otimes m}$ from separable states, which we know is close to 1 for large numbers $m$. Therefore, if $\rho$ is distillable then the trace distance of $\rho^{\otimes n}$ from separable states tends to 1. It is well-known that PPT states are not distillable under LOCC maps. So we cannot use this idea directly. On the other hand, in this argument, the only property of LOCC maps that we use, is that they send separable states to separable states. So, we may replace LOCC maps with [*non-entangling maps*]{}, the maps that send every separable state to a separable state. Due to the seminal work of Brandao and Plenio [@brandao1; @brandao2] every entangled state is distillable under [*asymptotically*]{} non-entangling maps[^4]. Hence, by replacing LOCC maps with asymptotically non-entangling maps and repeating the previous argument, we conclude that the trace distance of $\rho^{\otimes n}$ from separable states tends to 1. Although this idea gives a full proof of Theorem \[thm:main\], we do not present it in this paper. Instead, we use more fundamental techniques, namely, [*quantum state tomography*]{} and [*quantum de Finetti theorem*]{} [@renner; @povm]. In fact, these two techniques are the basic ideas of the results of [@brandao1; @brandao2] that we mentioned above. Since $\rho^{\otimes (n+k)}$ is a symmetric state, we may assume that the closest separable state to $\rho^{\otimes (n+k)}$ is also symmetric. Then by tracing out $k$ registers[^5] and using the finite quantum de Finetti theorem we conclude that the trace distance of $\rho^{\otimes (n+k)}$ from separable states is lower bounded by the trace distance of $\rho^{\otimes n}$ from separable states of the form $$\label{eq:1} \sum_i p_i\, \sigma_i^{\otimes n}.$$ Since such a state is separable and $\rho$ is not separable, the sum of $p_i$’s for which $\sigma_i$ is close to $\rho$ cannot be large. On the other, if $n$ is large, using quantum state tomography one can distinguish $\rho^{\otimes n}$ from $\sigma_i^{\otimes n}$, where $\sigma_i$ is far from $\rho$. Therefore, the trace distance of $\rho^{\otimes n}$ and a separable state of the form of Eq. (\[eq:1\]) is close to $1$ for enough large $n$. Notice that, in both of these arguments the only property of PPT states that we use, is that if $\rho$ and $\sigma$ are PPT, then $\rho \otimes \sigma$ is also PPT. So, we can conclude the same result for any separability test the satisfies this property. Preliminaries {#sec:prel} ============= A pure state $\vert \psi\rangle \in \mathcal{H}^A \otimes \mathcal{H}^B$ is called [*separable*]{} if it can be written of the form $\vert \psi\rangle = \vert \psi_A\rangle \otimes \vert \psi_B \rangle$, where $\vert \psi_A\rangle \in \mathcal{H}^A$ and $\vert \psi_B\rangle \in \mathcal{H}^B$. A density matrix acting on $\mathcal{H}^A \otimes \mathcal{H}^B$ is called separable if it can be written as a convex combination of separable pure states $\vert \psi\rangle\langle \psi\vert$. We denote the set of separable states by $\SEP$. For two quantum states $\rho$ and $\sigma$ we denote their trace distance by $$\label{eq:tr-dist} \|\rho - \sigma\|_{\Tr}=\frac{1}{2}\, \Tr | \rho - \sigma |,$$ where $|X|= \sqrt{X^{\dagger}X}$. Assume that $\dim \mathcal{H}_A=\dim \mathcal{H}_B=d$, and fix an orthonormal basis $\vert 1\rangle, \dots ,\vert d\rangle$ for both of Hilbert spaces. Then the partial transpose of matrices acting on $\mathcal{H}^A \otimes \mathcal{H}^B$ is a linear map defined by $(M_A\otimes N_B )^{T_B}=M_A\otimes N_B^T$. It is clear that if $\rho_{AB}$ is a separable state then $\rho_{AB}^{T_B}$ is also a density matrix and then positive semi-definite. However, it does not hold for an arbitrary state. For example, the partial transpose of the maximally entangled state is not positive semi-definite. To see that, let $\Phi(d)$ to be the maximally entangled state on $\mathcal{H}$ $$\label{eq:epr} \Phi(d)=\frac{1}{d}\sum_{i,j=1}^d \vert i,i\rangle \langle j, j\vert.$$ We have $$\begin{aligned} \Phi(d)^{T_B} & = & \frac{1}{d} \sum_{i,j} \vert i\rangle \langle j\vert \otimes \vert j\rangle\langle i\vert \\ & = & \frac{1}{d}I -\frac{1}{d} \sum_{i\neq j} \vert i\rangle \langle i\vert \otimes \vert j\rangle \langle j\vert +\frac{1}{d}\sum_{i\neq j} \vert i\rangle \langle j\vert \otimes \vert j\rangle \langle i\vert\\ & = & \frac{1}{d}I - \frac{2}{d} \sum_{i<j} \vert \phi_{ij}\rangle \langle \phi_{ij}\vert,\end{aligned}$$ where $$\label{eq:phi} \vert \phi_{ij}\rangle=\frac{1}{\sqrt{2}}(\vert i\rangle\vert j\rangle -\vert j\rangle \vert i\rangle).$$ Therefore, positive partial transpose is a test to detect entanglement [@peres; @23]. More formally, if we denote the set of density matrices with a positive semi-definite partial transpose by PPT, then $\SEP\subseteq\PPT$. Here is a list of some other separability criteria, see [@survey]. - Reduction criterion, [@reduction]: $I\otimes \rho_B \geq \rho_{AB}$, where $\rho_{B}=\Tr_{A} (\rho_{AB})$. Here, by $M\geq N$ we mean $M-N$ is a positive semi-definite matrix. - Entropic criterion, [@entropic]: $S_\alpha(\rho_{AB}) \geq S_\alpha(\rho_A)$ for $\alpha=2$ and in the limit $\alpha \rightarrow 1$, where $S_\alpha(\rho)=\frac{1}{1-\alpha}\log \Tr(\rho^\alpha)$. - Majorization criterion, [@majorization]: $\lambda_{\rho_{A}}^\downarrow \succ \lambda_{\rho_{AB}}^\downarrow $, where $\lambda_\rho^\downarrow$ is the list of eigenvalues of $\rho$ in non-increasing order, and $y \succ x$ means that, for any $k$, the sum of the first $k$ entries of list $x$ is less than or equal to that of list $y$. - Cross norm criterion, [@cross1; @cross2]: $\Tr | \mathcal{U}(\rho_{AB})|\, \leq 1$, where $\mathcal{U}$ is a linear map defined by $\mathcal{U}(M\otimes N)=v(M)v(N)^T$, relative to a fixed basis, and $v(X)=(col_1(X)^T, \dots , col_{d}(X)^T )^T$, where $col_i(X)$ is the $i$-th column of $X$. All of these criteria for separability are necessary conditions but not sufficient. Doherty et al. [@symmetric1; @symmetric2] have introduced a hierarchy of separability criteria which are both necessary and sufficient. Let $\rho_{AB}=\sum_i p_i\, \sigma_i\otimes \tau_i$ be a separable state. Then $$\rho_{AB_1B_2\cdots B_k} = \sum_i p_i\, \sigma_i\otimes \tau_i^{\otimes k}$$ is an extension of $\rho^{AB}$, meaning that $\rho_{AB}=\Tr_{B_2\cdots B_k} (\rho_{AB_1\cdots B_k})$. Also it is symmetric, meaning that it is unchanged under any permutation of subsystems $B_i$. More precisely, for any permutation $\pi$ of $k$ objects, if we define the linear map $P_\pi$ by $P_\pi \vert \psi_1\rangle \otimes \cdots \otimes \vert \psi_k\rangle = \vert \psi_{\pi(1)}\rangle \otimes \cdots \otimes \vert \psi_{\pi(k)}\rangle$, we have $$\label{eq:symmetric} P_{\pi}^{B_1\dots B_k}\, \rho_{AB_1B_2\cdots B_k}\, P_{\pi}^{B_1\dots B_k}= \rho_{AB_1B_2\cdots B_k}.$$ If such an extension exists, we say that $\rho_{AB}$ has a symmetric extension to $k$ copies. Doherty el al. have proved that a quantum state is separable iff it has a symmetric extension to $k$ copies for any number $k$, [@symmetric1; @symmetric2]. Also, they have shown that the problem of checking whether a given state has a symmetric extension to $k$ copies, for a fixed $k$, can be expressed as a semi-definite programming, and can be solved efficiently[^6]. So we get to another separability test. - Symmetric extension criterion: If $\rho_{AB}$ is separable, then it has a symmetric extension to $k$ copies. Quantum State Tomography {#sec:tomography} ------------------------ An informationally complete POVM on $\mathcal{H}$ is a set of positive semi-definite operators $\{M_n\}$ forming a basis for the space of hermitian matrices on $\mathcal{H}$, and such that $\sum_n M_n=I$. In [@povm] there is an explicit construction of an informationally complete POVM in any dimension. Such a POVM is useful for quantum state tomography. Suppose $\{M_n^*\}$ is the dual of basis $\{M_n\}$. That is $\Tr(M_nM_m^*)=\delta_{mn}$, where $\delta_{mn}$ is the Kronecker delta function. For any hermitian operator $X$ we have $$X=\sum_n \Tr(XM_n)\,M_n^*.$$ Therefore, having some copies of the state $\rho$, by measuring $\rho$ using the POVM $\{M_n\}$, we can approximate $\Tr(\rho M_n)$ and then find the matrix representation of $\rho$. Assume that $\mathcal{H}=\mathcal{H}^A\otimes \mathcal{H}^B$ is a bipartite Hilbert space. If $\{P_n\}$ and $\{Q_m\}$ are informationally complete POVM’s on $\mathcal{H}^A$ and $\mathcal{H}^B$, respectively, then it is easy to see that $\{P_n\otimes Q_m\}$ is an informationally complete POVM on $\mathcal{H}$. Therefore, if the state $\rho_{AB}$ is shared between two far apart parties $A$ and $B$, they still can perform quantum state tomography. Also, if the state $\rho_{AB}$ is separable, then all the states during the process are separable as well. Quantum de Finetti Theorem -------------------------- As in Eq. (\[eq:symmetric\]), a quantum state $\rho^{(n)}$ acting on $\mathcal{H}^{\otimes n}$ is called symmetric if $P_\pi \rho^{(n)} P_\pi=\rho^{(n)}$ for any permutation $\pi$ of $n$ objects. A symmetric state is called [*$k$-exchangeable*]{} if it has a symmetric extension to $n+k$ registers. That is a symmetric state $\rho^{(n+k)}$ such that $\Tr_{1,\dots,k}\rho^{(n+k)}=\rho^{(n)}$. Clearly, any state of the form $\rho^{\otimes n}$ is $k$-exchangeable, for any $k$. Also any convex combination of these states is $k$-exchangeable. [*Quantum de Finetti theorem*]{} says that the converse of this observation holds. That is, if a state is $k$-exchangeable, for any $k$, it is in the convex hall of symmetric product states. Quantum de Finetti theorem gives a characterization of infinitely-exchangeable states. The following theorem, known as the finite quantum de Finetti theorem, says that if a state is $k$-exchangeable (but not necessarily $(k+1)$-exchangeable), then an approximation of the above result holds. \[thm:fqdt\] [@renner] Assume that $\rho^{(n+k)}$ is a symmetric state acting on $\mathcal{H}^{\otimes {n+k}}$. Let $\rho^{(n)}=\Tr_{1\dots k}\,\rho^{(n+k)}$ be the state obtained by tracing out the first $k$ registers. Then there exists a probability measure $\mu$ on the set of density matrices on $\mathcal{H}$ such that $$\| \rho^{(n)} - \int \mu(d\sigma)\sigma^{\otimes n}\|_{\Tr} \,\leq 2\dim\mathcal{H}\,\frac{n}{n+k}.$$ Proof of Theorem \[thm:main\] {#sec:proof} ============================= Let $\mathcal{H}=\mathcal{H}^A\otimes \mathcal{H}^B$ and assume that $d = \dim \mathcal{H} > 6$. Then there exists a PPT state $\rho_{AB}=\rho$ acting on $\mathcal{H}$ which is not separable ( For example see [@bound]). Let $$\label{eq:dist} \epsilon=\min_{\sigma\in \SEP} \|\rho - \sigma\|_{\Tr}.$$ Since $\rho$ is not separable, $\epsilon > 0$. For any number $n$, $\rho^{\otimes n}$ can be considered as a bipartite state acting on $(\mathcal{H}^A)^{\otimes n }\otimes (\mathcal{H}^B)^{\otimes n }$, and it is a PPT state. Therefore, if we prove that the trace distance of $\rho^{\otimes n }$ from separable states tends to $1$, as $n$ goes to infinity, we are done. Let $\sigma^{(n)}$ be the closest separable state to $\rho^{\otimes n}$. Since $\rho^{\otimes n}$ is a symmetric state, for any permutation $\pi$ we have $$\|\rho^{\otimes n} - P_{\pi}\sigma^{(n)} P_{\pi}\|_{\Tr} = \|\rho^{\otimes n} - \sigma^{(n)}\|_{\Tr}.$$ Hence, by triangle inequality $$\|\rho^{\otimes n} - \frac{1}{n!}\sum_{\pi}P_{\pi}\sigma^{(n)} P_{\pi}\|_{\Tr} \leq \frac{1}{n!} \sum_{\pi} \|\rho^{\otimes n} - P_{\pi}\sigma^{(n)} P_{\pi}\|_{\Tr}= \|\rho^{\otimes n} - \sigma^{(n)}\|_{\Tr},$$ and then $\|\rho^{\otimes n} - \frac{1}{n!}\sum_{\pi}P_{\pi}\sigma^{(n)} P_{\pi}\|_{\Tr} = \|\rho^{\otimes n} - \sigma^{(n)}\|_{\Tr}$. This means that, we may assume that the closest separable state to $\rho^{\otimes n}$ is symmetric. Let $\sigma^{(n+n^2)}$ be the closest symmetric separable state to $\rho^{\otimes (n+n^2)}$, and let $\Tr_{1\dots n^2}\, \sigma^{(n+n^2)}$ be the state obtained by tracing out $n^2$ registers. We have $$\label{eq:10}\| \rho^{\otimes (n+n^2)} - \sigma^{(n+n^2)} \|_{\Tr}\, \geq \, \| \rho^{\otimes n} - \Tr_{1\dots n^2}\, \sigma^{(n+n^2)}\|_{\Tr}.$$ Using the finite quantum de Finetti theorem (Theorem \[thm:fqdt\]), there exists a measure $\mu$ such that $$\label{eq:xn} \Tr_{1\dots n^2}\, \sigma^{(n+n^2)} = \int \mu(d\tau)\tau^{\otimes n} + X_n,$$ where $\|X_n\|_{\Tr}\leq 2d\,\frac{n}{n+n^2}$. Hence, using Eq. (\[eq:10\]), if we prove that $\| \rho^{\otimes n} - \big(\int \mu(d\tau)\tau^{\otimes n} + X_n\big) \|_{\Tr}$ tends to $1$, as $n$ goes to infinity, we are done. Consider an informationally complete POVM on $\mathcal{H}^A$ and $\mathcal{H}^B$, and by taking their pairwise tensor product extend them to an informationally complete POVM on $\mathcal{H}$. Now apply quantum state tomography on $(n-1)$ copies of $\rho$. The outcomes of the measurements give an approximation of $\rho$. To be more precise, let $\{ M_i \}$ be the informationally complete POVM on $\mathcal{H}$. For any sequence of outcomes $(M_{l_1}, \dots ,M_{l_{(n-1)}})$ we get to the approximation $$\label{eq:tomog-approximation} \sum_i \frac{r_i}{n-1}M_i^{\ast},$$ where $r_i$ is the number of repetition of $M_i$ in $(M_{l_1}, \dots ,M_{l_{(n-1)}})$. Let $A_n$ be the sum of $(n-1)$-tuple tensor products $M_{l_1}\otimes \cdots M_{l_{(n-1)}}$ for sequences $(M_{l_1}, \dots ,M_{l_{(n-1)}})$ whose approximations, according to Eq. (\[eq:tomog-approximation\]), are in $B_{\epsilon/2}(\rho)$, the ball of radios $\epsilon/2$ in trace distance around $\rho$. Therefore, by the law of large numbers [@dudley], $\Tr(A_n \rho^{\otimes (n-1)}) \rightarrow 1 $ as $n$ goes to infinity. Also for any $\tau$ far from $\rho$, $\Tr(A_n \tau^{\otimes (n-1)})$ tends to zero. Notice that $A_n\leq I$. Hence, $$\| \rho^{\otimes n} - \big( \int \mu(d\tau)\tau^{\otimes n} + X_n \big) \|_{\Tr} \geq \Tr\, ( I\otimes A_n \cdot \rho^{\otimes n} ) - \Tr \big[ (I\otimes A_n) \cdot \big(\int \mu(d\tau)\tau^{\otimes n} + X_n\big) \big],$$ and since $\Tr\, ( I\otimes A_n \cdot \rho^{\otimes n} ) \rightarrow 1$, if we prove that $$\Tr \big[ (I\otimes A_n) \cdot (\int \mu(d\tau)\tau^{\otimes n} + X_n) \big] \rightarrow 0,$$ as $n$ goes to infinity, we are done. By Eq. (\[eq:xn\]), $\int \mu(d\tau)\tau^{\otimes n} + X_n$ is a separable state. Also, since we can apply quantum state tomography locally (see Section \[sec:tomography\] ), at the end the outcome is a separable state. We can write the outcome, before normalization, in the form $$\int \mu(d\tau)\Tr[A_n\tau^{\otimes (n-1)}]\,\tau + \widetilde{X}_n,$$ where $\|\widetilde{X}_n\|_{\Tr} \leq 2d\,\frac{n}{n+n^2}$. Let $$Y_n = \int_{\tau\notin B_{\epsilon/2}(\rho)} \mu(d\tau)\Tr[A_n\tau^{\otimes (n-1)}]\,\tau +\widetilde{X}_n ,$$ and $$c_n= \int_{\tau\in B_{\epsilon/2}(\rho)} \mu(d\tau)\Tr[A_n\tau^{\otimes (n-1)}].$$ By the law of large numbers [@dudley] there exists $\delta_n$ such that for any $\tau\notin B_{\epsilon/2}(\rho)$ we have $$\Tr [A_n\cdot \tau^{\otimes (n-1)}]\leq \delta_n,$$ and $\delta_n \rightarrow 0$ as $n$ goes to infinity. Then $\|Y_n\|_{\Tr}\leq \delta_n + 2d\frac{n}{n+n^2}$. Now, the state $$\widetilde{\tau}= \frac{1}{c_n+\Tr (Y_n)} \big[ \int_{\tau\in B_{\epsilon/2}(\rho)} \mu(d\tau)\Tr[A_n\tau^{\otimes (n-1)}]\,\tau + Y_n \big]$$ is separable. On the other hand, by definition $$\widetilde{\rho}= \frac{1}{c_n} \int_{\tau\in B_{\epsilon/2}(\rho)} \mu(d\tau)\Tr[A_n\tau^{\otimes (n-1)}]\,\tau$$ is in the $\epsilon/2$-neighborhood of $\rho$. Using Eq. (\[eq:dist\]) we have $$\begin{aligned} \epsilon & \leq & \| \rho - \widetilde{\tau} \|_{\Tr}\\ & \leq & \frac{c_n}{c_n+\Tr(Y_n)} \cdot\| \rho -\widetilde{\rho}\|_{\Tr} + \frac{|\Tr(Y_n)|}{c_n+\Tr(Y_n)}\cdot\|\rho\|_{\Tr}+ \frac{1}{c_n+\Tr(Y_n)}\cdot\|Y_n\|_{\Tr}\\ & \leq & \frac{c_n}{c_n+\Tr(Y_n)} \cdot \frac{\epsilon}{2} + \frac{2}{c_n+\Tr(Y_n)}\cdot\|Y_n\|_{\Tr}.\end{aligned}$$ Hence, $$\epsilon c_n + \epsilon \Tr(Y_n) \,\leq \, \frac{\epsilon}{2}c_n + 2\|Y_n\|_{\Tr},$$ and then $$c_n\,\leq \, \frac{2(2+\epsilon)\|Y_n\|_{\Tr}}{\epsilon} \,\leq \, 6 \epsilon^{-1} [\delta_n+ 2d\frac{n}{n+n^2}].$$ Putting everything together we find that $$\begin{aligned} \Tr \big[ (I\otimes A_n) \cdot (\int \mu(d\tau)\tau^{\otimes n} + X_n)] & = &\Tr \big[ \int_{\tau\in B_{\epsilon/2}(\rho)} \mu(d\tau)\Tr[A_n\tau^{\otimes (n-1)}]\,\tau + Y_n \big] \\ & \leq & c_n + \| Y_n \|_{\Tr} \\ & \leq & (6\epsilon^{-1}+1) \cdot ( \delta_n+ 2d\frac{n}{n+n^2}).\end{aligned}$$ Therefore $$\Tr \big[ (I\otimes A_n) \cdot (\int \mu(d\tau)\tau^{\otimes n} + X_n)] \rightarrow 0,$$ as $n$ goes to infinity. We are done. Geometry of the Set of Separable States ======================================= Theorem \[thm:main\] tells us that estimating the distance of a bipartite state from separable state by the distance from PPT states is not a good approximation. However, one may say the set of PPT states may be a reasonable approximation for the set of separable states in a geometrical point of view. For instance, two spheres centered at origin with radiuses $1$ and $2$ are far from each other, while they have the same geometric properties up to a scaler factor. In the following theorem we show that the set of separable states relative to the set of PPT states is not of this form. By Theorem \[thm:main\] the maximum distance of a PPT state from the boundary of the set of separable states is close to $1$. We can think of this problem in another direction. What is the maximum distance of a state on the boundary of separable states from the boundary of PPT states? To get an intuition on this problem, we can think of the unit sphere centered at origin in $\mathbb{R}^n$, and the cube with vertices $(\pm 1, \dots , \pm 1)$. It is easy to see that the distance of any point on the sphere from points of the cube is less than $2$. However, the distance of $(1, \dots , 1)$ from sphere is $\sqrt{n}-1$. It is because sphere and cube have totally different shapes. \[thm:shape\] Assume that $\mathcal{H}=\mathcal{H}^A\otimes \mathcal{H}^B$, and $\dim \mathcal{H}^A=\dim \mathcal{H}^B=d$. Then for any separable state $\rho$ acting on $\mathcal{H}$ there exists a state $\sigma$ on the boundary of the set of PPT states such that $\|\rho - \sigma \|_{\Tr} \leq \frac{1}{\sqrt{d}}.$ Let $\sigma$ be an arbitrary PPT state, and $\Phi(d)$ be the maximally entangled state defined in Eq. (\[eq:epr\]). Then the fidelity of $\sigma$ and $\Phi(d)$ is $$F(\sigma, \Phi(d))=[\Tr\, \sigma\,\Phi(d) ]^{1/2} = [\Tr\, \sigma^{T_B}\,\Phi(d)^{T_B} ]^{1/2} = [\Tr\, \sigma^{T_B}\, ( \frac{1}{d}I -\frac{2}{d}\sum_{i<j} \vert \phi_{ij}\rangle \langle \phi_{ij}\vert ) ]^{1/2},$$ where $\vert \phi_{ij}\rangle $ is defined in Eq. (\[eq:phi\]). Now, using the fact that $\rho^{T_B}$ is positive semi-definite we have $$F(\sigma, \Phi(d)) \leq \frac{1}{\sqrt{d}}.$$ Therefore, by the well-known inequality between fidelity and trace distance, [@chuang] page 416, we have $$\label{eq:sqrt} \| \sigma -\Phi(d)\|_{\Tr} \geq 1- F(\sigma, \Phi(d)) \geq 1- \frac{1}{\sqrt{d}}.$$ Let $\rho$ be an arbitrary separable state. Define $\rho_t= (1-t) \rho + t\Phi(d)$. Then $\rho_0=\rho$ is separable and then PPT, and $\rho_1=\Phi(d)$. Hence, there exists $0 \leq c \leq 1$ such that $\rho_c$ is on the boundary of PPT states. Then we have $$\|\rho - \rho_c\|_{\Tr} = \|\rho - \Phi(d)\|_{\Tr} - \|\rho_c - \Phi(d)\|_{\Tr}\leq 1-(1- \frac{1}{\sqrt{d}})= \frac{1}{\sqrt{d}} ,$$ where in the last inequality we use Eq. (\[eq:sqrt\]). Generalization to Other Separability Criteria ============================================= By the result of Section \[sec:proof\], if the dimension of the space is enough large, there exists a PPT state arbitrary far from separable states. In the proof, our candidate for such a state is $\rho^{\otimes n}$, where $\rho$ is an entangled PPT state. Indeed, the only property of the set of PPT states that we use, is that this set is closed under tensor product. Therefore, the same argument as in the proof of Theorem \[thm:main\], gives us the following general theorem. \[thm:general\] Assume that $C$ is a necessary but not sufficient separability criterion such that if $\rho$ and $\sigma$ satisfy $C$, then $\rho\otimes \sigma$ satisfies $C$ as well. Then for any $\epsilon>0$ there exists a state $\rho$ that satisfies $C$, and whose trace distance from separable states is at least $1-\epsilon$. Let $\rho$ be an entangled state that satisfies $C$. Then $\rho^{\otimes n}$ satisfies $C$, and by the proof of Theorem \[thm:main\], the trace distance of $\rho^{\otimes n}$ from separable states, tends to $1$ as $n$ goes to infinity. In the following theorem we prove that all separability criteria mentioned in Section \[sec:prel\] satisfy the assumption of Theorem \[thm:general\]. \[thm:example\] For any of the separability criteria mentioned in Section \[sec:prel\] there exists an entangled state that passes that test while it is arbitrary far, in trace distance, from separable states. By Theorem \[thm:general\] it is sufficent to prove that those separability criteria are closed under tensor product. - Reduction criterion: Let $X, Y, Z$ and $W$ be positive semi-definite matrices such that $X\geq Y$ and $Z\geq W$. Then $(X-Y)\otimes (Z+W)$ and $(X+Y)\otimes (Z-W)$ are positive semi-definite. Therefore $X\otimes Z -Y\otimes W = \frac{1}{2}[ (X-Y)\otimes (Z+W) + (X+Y)\otimes (Z-W) ]$ is positive semi-definite. It means that if $X\geq Y$ and $Z\geq W$, then $X\otimes Z\geq Y\otimes U$. Now assume that $\rho_{AB}$ and $\sigma_{AB}$ pass reduction criterion. Therefore $\rho_A\otimes I \geq \rho_{AB}$ and $\sigma_A\otimes I \geq \sigma_{AB}$, and then $ \rho_{A}\otimes \sigma_A \otimes I \geq \rho_{AB}\otimes \sigma_{AB}$. Hence, $\rho_{AB}\otimes \sigma_{AB}$ passes reduction criterion. - Entropic criterion: It follows easily from $S_\alpha(\rho\otimes \sigma)=S_\alpha(\rho)+ S_{\alpha}(\sigma)$. - Majorization criterion: $x \prec y$ if and only if there exists a doubly-stochastic matrix[^7] $D$ such that $x=Dy$, see [@chuang] page 575. Therefore, if $x \prec y$ and $x' \prec y'$, there exist $D$ and $D'$ such that $x=Dy$ and $x'=D'y'$. Hence $x\otimes x'=(D\otimes D')(y\otimes y')$ and then $x\otimes x'\prec y\otimes y'$. The proof follows easily using this property. - Cross norm criterion: Using $v(X\otimes X')=v(X)\otimes v(X')$ we have $\mathcal{U}((X\otimes X')\otimes (Y\otimes Y')) = \mathcal{U}(X\otimes Y)\otimes \mathcal{U}(X'\otimes Y')$. The proof follows from this equation. - Symmetric extension criterion: If $\rho^{(k)}$ and $\sigma^{(k)}$ are symmetric extensions of $\rho$ and $\sigma$ to $k$ copies, respectively, then $\rho^{(k)}\otimes \sigma^{(k)}$ is a symmetric extension of $\rho\otimes \sigma$ to $k$ copies. Conclusion ========== In this paper we have proved that for any separability criterion that is closed under tensor product, meaning that $\rho\otimes \sigma$ passes the test if $\rho$ and $\sigma$ pass the test, the set of states that pass the test is not a good approximation of the set of separable states. In other words, all well-known algorithms for detecting entanglement, give no bound on the distance of a state from separable states. For the special case of positive partial transpose test, using Theorem \[thm:shape\], we have shown that the set of PPT states and separable states have totally different shapes. An interesting question to answer is to find a separability criterion that is stronger than the known ones and also is not closed under tensor product. This problem may clarify the complexity of separability problem: Is it $\NP$-hard to decide whether there exists a separable state whose trace distance from a given state is less than a constant $c$? [**Acknowledgement.**]{} SB is thankful of Barbara Terhal for useful discussions. [10]{} Salman Beigi, [*$\NP$ vs $QMA_{\log}(2)$*]{}, arXiv:0810.5109 C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin and B. M. Terhal, [*Unextendible Product Bases and Bound Entanglement*]{}, Phys. Rev. Lett. 82 (1999) 5385 Fernando G.S.L. Brandao, [*Entanglement Theory and the Quantum Simulation of Many-Body Physics*]{}, PhD thesis, arXiv:0810.0026 Fernando G.S.L. Brandao, Martin B. Plenio, [*Entanglement Theory and the Second Law of Thermodynamics*]{}, Nature Physics 4, 873 (2008) K. Chen and L.-A. Wu, [*A matrix realignment method for recognizing entanglement*]{} Quant. Inf. Comp., 3:193, 2003. M. Christandl,. R. König, G. Mitchison and R. Renner, [*One-and-a-Half Quantum de Finetti Theorems*]{}, Communications in Mathematical Physics, Volume 273, Issue 2, pp.473-498 J. Dehaene, B. De Moor and F. Verstraete, [*On the geometry of entangled states*]{}, Journal of Modern Optics, Volume 49, Number 8, July 10, 2002 , pp. 1277-1287(11) A. C. Doherty, P. A. Parrilo and F.M. Spedalieri, [*Distinguishing separable and entangled states*]{}, Phys. Rev. Lett., 88:187904, 2002. A. C. Doherty, P. A. Parrilo and F. M. Spedalieri, [*Complete family of separability criteria*]{}, Phys. Rev. A, 69:022308, 2004. R. M. Dudley, [*Real Analysis and Probability*]{}, Cambridge University Press (2002). Sevag Gharibian, [*Strong NP-Hardness of the Quantum Separability Problem*]{}, arXiv:0810.4507 Leonid Gurvits, [*Classical deterministic complexity of Edmonds’ problem and Quantum Entanglement*]{}, quant-ph/0303055 M. Horodecki and P. Horodecki, [*Reduction criterion of separability and limits for a class of distillation protocols*]{}, Phys. Rev. A, 59:4206, 1999. R. Horodecki, P. Horodecki and M. Horodecki, [*Quantum $\alpha$-entropy inequalities: independent condition for local realism?*]{} Phys. Lett. A, 210:377–381, 1996. M. Horodecki, P. Horodecki and R. Horodecki, [*Separability of mixed states: necessary and sufficient conditions*]{}, Physics Letters A, v. 223, p. 1-8, 1996. Lawrence M. Ioannou, [*Computational complexity of the quantum separability problem*]{}, Quantum Information and Computation, Vol. 7, No. 4 (2007) 335-370 R. König and R. Renner, [*A de Finetti representation for finite symmetric quantum states*]{} J. Math. Phys. 46, 122108 (2005). M. A. Nielsen and I. L. Chuang, [*Quantum Computation and Quantum Information*]{}, Cambridge University Press, Cambridge, 2000 M. Nielsen and J. Kempe, [*Separable states are more disordered globally than locally*]{}, Phys. Rev. Lett., 86:5184–7, 2001. O. Rudolph, [*Further results on the cross norm criterion for separability*]{}, quant-ph/0202121. A. Peres, [*Separability criterion for density matrices*]{}, Phys. Rev. Lett., 77:1413–1415, 1996. S. Szarek, I. Bengtsson and K. Zyczkowski, [*On the structure of the body of states with positive partial transpose*]{}, J. Phys. A 39 L119-L126 (2006) K.G.H. Vollbrecht and R.F. Werner, [*Entanglement Measures under Symmetry*]{}, Phys. Rev. A 64, 062307 (2001). [^1]: salman@mit.edu [^2]: shor@math.mit.edu [^3]: If we replace the trace distance with $E_R(\rho)$, the relative entropy of entanglement, this property does not hold [@relent]. [^4]: This is because the entanglement of distillation under asymptotically non-entangling maps is equal to the regularized relative entropy of entanglement, and this measure of entanglement is faithful, meaning that it is non-zero for every entangled state. [^5]: Notice that partial trace decreases the trace distance. [^6]: Notice that knowing that a state has a symmetric extension to $k$ copies, for a fixed $k$, gives us no upper bound on the distance of the state from separable states. Indeed, to get a non-trivial upper bound $k$ has to be of the order of the dimension of the state. It is because the upper bound on the distance from separable states comes from the finite quantum de Finetti theorem, and this theorem gives a trivial bound for a constant $k$. See [@renner] and [@povm] for finite de Finetti theorem. Also see Theorem \[thm:general\] of the present paper. [^7]: A matrix is called doubly-stochastic if all of whose entries are positive, and the sum of entries on any row and column is equal to $1$.

--- abstract: 'We consider Monte Carlo methods for simulating solutions to the analogue of the Dirichlet boundary-value problem in which the Laplacian is replaced by the fractional Laplacian and boundary conditions are replaced by conditions on the exterior of the domain. Specifically, we consider the analogue of the so-called ‘walk-on-spheres’ algorithm. In the diffusive setting, this entails sampling the path of Brownian motion as it uniformly exits a sequence of spheres maximally inscribed in the domain. As this algorithm would otherwise never end, it is truncated when the ‘walk-on-spheres’ comes within $\varepsilon>0$ of the boundary. In the setting of the fractional Laplacian, the role of Brownian motion is replaced by an isotropic $\alpha$-stable process with $\alpha\in(0,2)$. A significant difference to the Brownian setting is that the stable processes will exit spheres by a jump rather than hitting their boundary. This difference ensures that disconnected domains may be considered and that, unlike the diffusive setting, the algorithm ends after an almost surely finite number of steps.' author: - '[Andreas E. Kyprianou]{}[^1] $^,$[^2]' - 'Ana Osojnik[^3]' - '[Tony Shardlow$^*$]{}' bibliography: - 'sphere\_stepping.bib' title: | Unbiased ‘walk-on-spheres’ Monte Carlo methods\ for the fractional Laplacian --- =1 Introduction ============ We start by recalling the classical Dirichlet problem in $d$-dimensions and re-examining a, now, classical Monte Carlo algorithm that is used to numerically simulate its solution. Suppose that $D$ is a domain in $\mathbb{R}^d$, $d\geq 2$, with sufficiently smooth boundary. We are interested in finding $u\colon D\to \mathbb{R}$ such that $$\begin{gathered} \begin{aligned} \Delta u(x) & = 0, & \qquad x & \in D, \\ u(x) & = {g}(x), & x & \in \partial D, \end{aligned}\label{Dirichlet}\end{gathered}$$ where ${g}$ is a given continuous function on the boundary. Feynman–Kac representation tells us that, for example, if $u\in C^2(\overline{D} )$ is a solution to , then $$\label{FK} u(x) = \mathbb{E}_x[{g}(W_{\tau_D}) ], \qquad x\in D,$$ where $\tau_D \coloneqq \inf\{t>0 : W_t \not\in D\}$ and $W\coloneqq (W_t, t\geq 0)$ is standard $d$-dimensional Brownian motion with probabilities $(\mathbb{P}_x, x\in\mathbb{R}^d)$. The representation suggests that solutions to can be generated numerically via straightforward Monte Carlo simulations of the path of $W$ until first exit from $D$. That is to say, if $(W^{i}_t, t\leq \tau^{i}_D)$, $i \in \mathbb{N}$ are [*iid*]{}copies of $(W_t, t\leq \tau_D)$ issued from $x\in D$, then, by the [strong law of large numbers]{}, $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n {g}(W^{i}_{\tau^{i}_D}) = u(x),\qquad \text{almost surely.} \label{FKMC}$$ For practical purposes, since it is impossible to take the limit, one truncates the series of estimates for large $n$ and the [central limit theorem]{}gives $\mathcal{O}(1/n)$ upper bounds on the variance of the $n$-term sum, which serves as a numerical error estimate. Although forming the fundamental basis of most Monte Carlo methods for diffusive Dirichlet-type problems, is an inefficient numerical approach. Least of all, this is because the Monte Carlo simulation of $u(x)$ is independent for each $x\in D$. Moreover, it is unclear how exactly to simulate the path of a Brownian motion on its first exit from $D$, that is to say, the quantity $W_{\tau_D}$. This is because of the fractal properties of Brownian motion, making its path difficult to simulate. This introduces additional numerical errors over and above that of Monte Carlo simulation. A method proposed by [@Mu], for the case that $D$ is convex, sub-samples special points along the path of Brownian motion to the boundary of the domain $D$. The method does not require a complete simulation of its path and takes advantage of the distributional symmetry of Brownian motion. In order to describe the so-called ‘walk-on-spheres’, we need to first introduce some notation. We may thus set $\rho_0 = x$ for $x\in D$ and define $r_1$ to be the radius of the largest sphere inscribed in $D$ that is centred at $x$. This sphere we will call $S_1 = \{y\in\mathbb{R}^d\colon |y-\rho_0| = r_1\}$. To avoid special cases, we henceforth assume that the surface area of $S_1\cap \partial D$ is zero (this excludes, for example, the case that $x = 0$ and $D$ is a sphere centred at the origin). Now set $\rho_1\in D$ to be a point uniformly distributed on $S_1$ and note that, given the assumption in the previous sentence, $\mathbb{P}_x(\rho_1\in \partial D) = 0$. Construct the remainder of the sequence $(\rho_n,\,n \geq 1)$ inductively. Given $\rho_{n-1}$, we define the radius, $r_n$, of the largest sphere inscribed in $D$ that is centred at $\rho_{n-1}$. Calling this sphere $S_n$, we have that $S_n = \{y\in \mathbb{R}^d \colon |y-\rho_{n-1}| =r_n\}$. We now select $\rho_n$ to be a point that is uniformly positioned on $S_n$. Once again, we note that if $\rho_{n-1}\in D$ almost surely, then the uniform distribution of both $\rho_{n-1}$ and $\rho_{n}$ ensures that $\mathbb{P}(\rho_{n}\in \partial D) = 0$. Consequently, the sequence $\rho_n$ continues for all $n\geq1$. In the case that $\rho_n$ approaches the boundary, the sequence of spheres $S_n$ become arbitrarily small in size. Thanks to the strong Markov property and the stationary and independent increments of Brownian motion, it is straightforward to prove the following result. Fix $x\in D$ and define $\rho'_1 = W_{\tau_{S'_1}}$, where $\tau_{S'_1}=\inf\{t>0 \colon W_t \in S'_1 \}$ and $S'_1$ is the largest sphere, centred at $x$, inscribed in $D$. For $n\geq 2$, given $\rho'_{n-1}\in D$, let $\rho'_n = W_{\tau_{S'_n}}$, where $\tau_{S'_n}=\inf\{t>0 \colon W_t \in S'_n \}$ and $S'_n$ is the largest sphere, centred at $\rho'_{n-1}$. Then the sequences $(\rho_n, n\geq 0)$ and $(\rho'_n, n\geq 0)$ have the same law. As an immediate consequence, $\lim_{n\to\infty}\rho_n$ almost surely exists and, moreover, it it equal in distribution to $W_{\tau_D}$. The sequence $\rho\coloneqq (\rho_n, n\geq 0)$ may now replace the role of $(W_t, t\leq \tau_D)$ in , and hence in (\[FKMC\]), albeit that one must stop the sequence $\rho$ at some finite $N$. By picking a threshold $\varepsilon>0$, we can choose $N(\varepsilon)$ as a cutoff for the sequence $\rho$ such that $N(\varepsilon) = \min\{n\geq 0: \inf_{z\in\partial D}|\rho_n -z|\leq \varepsilon\}$. Intuitively, one is inwardly ‘thickening’ the boundary $\partial D$ with an ‘$\varepsilon$-skin’ and stopping once the walk-on-spheres hits the $\varepsilon$-skin. As the sequence $\rho$ is random, $N(\varepsilon)$ is also random. Starting with Theorem 6.6 of [@Mu] and the classical computations in [@Mo], it is known that $ \mathbb{E}_x[N(\varepsilon)] = \mathcal{O}(|\!\log\varepsilon|). $ To be more precise, we have the following result. \[muller\] Suppose that $D$ is a convex domain. There exist constants $c_1,c_2>0$ such that $\mathbb{E}_x[N(\varepsilon)] \leq c_1\,\abs{\log \varepsilon} + c_2$, $\varepsilon\in(0,1)$. The Monte Carlo simulation (\[FKMC\]) can now be replaced by one based on simulating the quantity $ {g}(\rho_{N(\varepsilon)})$, $\rho_0 = x\in D$, which, in turn, is justified by the [strong law of large numbers]{}: $$\label{WoSMC1} \lim_{n \to\infty} \frac{1}{n}\sum_{i = 1}^n {g}(\rho^{i}_{N^{i}(\varepsilon)}) = \mathbb{E}_x \bp{{g}(\rho_{N(\varepsilon)})} \approx \mathbb{E}_x[{g}(W_{\tau_D})]= u(x),\qquad{\text{a.s.,}}$$ where $\varepsilon>0$ is some threshold and $(\rho^{i}_n, n\leq N^{i}(\varepsilon))$, $i\geq 0$ are [*iid*]{}copies of the walk-on-spheres process stopped at a distance $\varepsilon$ or smaller from $\partial D$. Formally speaking, a convention is required to evaluate ${g}$ just inside the boundary $\partial D$ in (\[WoSMC1\]). In many cases, ${g}$ can be evaluated without introducing any additional bias [@Given1997-qr; @Hwang2001-wc]. The Laplacian serves as the infinitesimal generator of Brownian motion, in the sense that, for appropriately smooth functions $\phi\colon \mathbb{R}^d\to \mathbb{R}$,$$\lim_{t\to0} \frac{\mathbb{E}_x[\phi(W_t)] - \phi(x)}{t} =\frac{1}{2}\Delta\phi(x), \qquad x\in\mathbb{R}^d. \label{IG}$$ Intuitively speaking, this explains an underlying connection between the Dirichlet problem (\[Dirichlet\]) and the Feynman–Kac representation of the solution (\[FK\]). In this paper, we consider the analogue of (\[Dirichlet\]) when the operator $ \Delta/2$ is replaced by the fractional Laplacian $-(-\Delta)^{\alpha/2}$ for $\alpha\in (0,2)$. In this case, the fractional Laplacian corresponds, in the same sense as (\[IG\]), to an isotropic stable Lévy process with index $\alpha$. This is a strong Markov process with stationary and independent increments, say $X = (X_t, t\geq 0)$ with probabilities $(\mathbb{P}_x, x\in\mathbb{R}^d)$, whose semi-group is represented by the Fourier transform $$\mathbb{E}_0\bp{{\rm e}^{{\rm i}\langle\theta ,X_t\rangle}} = {\rm e}^{-|\theta|^\alpha t}, \qquad \theta\in \mathbb{R}^d, \;t\geq 0,$$ where $\langle \cdot,\cdot \rangle$ represents the usual Euclidian inner product. Stable processes enjoy an isotropy in the following sense: if $U$ is any orthogonal matrix in $\mathbb{R}^{d\times d}$, then $(UX_t, t\geq 0)$ under $\mathbb{P}_0$ has the same law as $(X, \mathbb{P}_0)$. Moreover, we have the following important scaling property: for all $c>0$, $$\label{levyscaling} ((cX_{c^{-\alpha} t}, t\geq 0), \mathbb{P}_0) \text{ is equal in law to }((X_t, t\geq 0), \mathbb{P}_0).$$ In dimension two or greater, the operator $-(-\Delta)^{\alpha/2}$ can be expressed in the form $$-(-\Delta)^{\alpha/2} u(x) =-\frac{2^\alpha \,\Gamma((d+\alpha)/2)}{\pi^{d/2}\,\Gamma(-\alpha/2)} \lim_{\varepsilon\downarrow0}\int_{\mathbb{R}^d\backslash B(0,\varepsilon)}\frac{[u(y)- u(x)]}{|y-x|^{d+\alpha}}\,{\rm d}y,\qquad x\in \mathbb{R}^d,$$ where $B(0,\varepsilon) = \{x\in\mathbb{R}^d: |x|<\varepsilon\}$ and $u$ is smooth enough for the limit to make sense. Noting that $-(-\Delta)^{\alpha/2}$ is no longer a local operator, the analogous formulation of (\[Dirichlet\]) needs a little more care. In particular, the boundary condition on the domain $D$ is no longer stated on $\partial D$, but must now be stated on the complement of $D$, written $D^{\rm c}$. To avoid pathological cases, we must assume throughout that $D^{\rm c}$ has positive $d$-dimensional Lebesgue measure. The Dirichlet problem for $-(-\Delta)^{\alpha/2}$ requires one to find $u\colon D \to\mathbb{R}$ such that $$\begin{gathered} \begin{aligned} -(-\Delta )^{\alpha/2}u(x) & = 0, & \qquad x & \in D, & \\ u(x) & = {g}(x), & x & \in D^{\rm c}, \end{aligned} \label{aDirichlet}\end{gathered}$$ where ${g}$ is a suitably regular function. The fractional Dirichlet problem and variants thereof appear in many applications, in particular in physical settings where anomalous dynamics occur and where the spread of mass grows faster than linearly in time. Examples include turbulent fluids, contaminant transport in fractured rocks, chaotic dynamics and disordered quantum ensembles; see [@FracDy; @AnTr; @flights]. The numerical analysis of is no less deserving than in the diffusive setting. Just as with the classical Dirichlet setting, the solution to has a Feynman–Kac representation, expressed as an expectation at first exit from $D$ of the associated stable process. The theorem below is proved in this paper in a probabilistic way. Similar statements and proofs we found in the existing literature take a more analytical perspective. See for example the review in [@bucur] as well as the monographs [@BH], [@BV] and [@BBKRSV], the articles [@B99], [@R-O1] and [@R-O2] and references therein. We say a real-valued function $\phi$ on a Borel set $S\subset \mathbb{R}^d$ belongs to $L^1_\alpha(S)$ if it is a measurable function that satisfies $$\int_{S}\frac{|\phi(x)|}{1+|x|^{\alpha + d}}\,{\rm d}x <\infty. \label{i-test}$$ \[corr\] For dimension $d\geq 2$, suppose that $D$ is a bounded domain in $\mathbb{R}^d$ and that ${g}$ is a continuous function in $L^1_\alpha(D^\mathrm{c})$. Then there exists a unique continuous solution to in $L^1_\alpha(\mathbb{R}^d)$, which is given by $$u(x) = \mathbb{E}_x[{g}(X_{\sigma_D})],\qquad x\in D,$$ where $X = (X_t, t\geq 0)$ is an isotropic stable Lévy process with index $\alpha$ and $\sigma_D = \inf\{t>0: X_t\not\in D\}$. The case that $D$ is a ball can be found, for example, in Theorem 2.10 of [@bucur]. We exclude the case $d=1$ because convex domains are intervals for which exact solutions are known; see again [@bucur] or the forthcoming Theorem \[BGR\] lifted from [@BGR]. Theorem \[corr\] follows in fact as a corollary of a more general result stated later in Theorem \[hasacorr\], which is proved in the Appendix. In this article, our objective is to demonstrate that the walk-on-spheres method may also be extended to the setting of the Dirichlet problem with fractional Laplacian. In particular, we will show that, thanks to various distributional and path properties of stable processes, notably spatial homogeneity, isotropy, self-similarity and that it exits $D$ by a jump, simulations can be made unbiased, without the need to truncate the algorithm at an $\varepsilon$ tolerance. Whilst there exist many methods for numerically examining the fractional Dirichlet problem , which mostly appeal to classical methodology for diffusive operators, see for example [@NOS; @HO; @DEG; @ZRK; @BS; @SB; @1608.08443] to name some but not all of the existing literature, we believe that no other work appeals to the walk-on-spheres algorithm in this context. The remainder of this paper is structured as follows. In the next section, we give a brief historical review of Theorem \[muller\] and its proofs as well as providing a new, short proof. In Section \[stable\_paths\], we show how an old result of [@BGR] can be used to give an exact simulation of the paths of stable processes. In Section \[WoSfL\], we introduce the walk-on-spheres algorithm for the fractional-Laplacian Dirichlet problem. We start with domains $D$ that are convex but not necessarily bounded. Our main result shows that the walk-on-spheres algorithm ends in an almost-surely finite number of steps (without the need of approximation), which can be stochastically bounded by a geometric distribution. Moreover, the parameter of this distribution does not depend on the starting point of the walk-on-spheres algorithm. Section \[non\_convex\] looks at extensions to non-convex domains. In Section \[inhomogenous\], we consider a fractional Poisson equation, where an inhomogeneous term is introduced on the right-hand side of the fractional-Laplacian Dirichlet problem (\[aDirichlet\]). Appealing to related results concerning the resolvent of stable processes until first exit from the unit ball, we are able to develop the walk-on-spheres algorithm further. Finally in Section \[numerics\], we discuss some numerical experiments to illustrate the methods developed as well as their implementation. The classical setting ===================== As promised above, we give a brief historical review of the classical walk-on-spheres algorithm and, below, for completeness, we provide a proof of Theorem \[muller\], which, to the authors’ knowledge, is new. The walk-on-spheres algorithm was first derived by [@Mu]. In Theorem 6.1 of his article, Muller claims that one can compare $\mathbb{E}_x[N(\varepsilon)]$ with the mean number of steps of a walk-on-spheres process that is stopped when it reaches an $\varepsilon$-skin of the tangent hyperplane that passes through a point on $\partial D$ that is closest to $x$. Although the claim is correct (indeed the proof that we give for our main result Theorem \[main\] below provides the basis for an alternative justification of this fact), it is not entirely clear from Muller’s reasoning. [@Mo] uses Muller’s comparison of the mean number of steps to prove Theorem \[muller\]. He considers the total expected occupation of an appropriately time-changed version of Brownian motion when crossing each sphere of the walk until touching the aforementioned $\varepsilon$-skin of the tangent hyperplane. Using the self-similarity of Brownian motion, Motoo argues that the time-change during passage to the boundary of each sphere is such that the expected occupation across each step is uniformly bounded below. It follows that the sum of these weighted expected occupations can be bounded below by $\mathbb{E}_x[N(\varepsilon)]$. On the other hand, the aforesaid sum can also be bounded above by the total expected time-changed occupation until exiting the half-space (as defined by the tangent plane), which can be computed explicitly, thereby providing the $|\!\log\varepsilon|$ comparison. Following the foundational work of Muller and Motoo, there have been many reproofs and generalisations of the original algorithm to different processes and domain types. Notable in this respect is the work of [@Mi] and [@BB] who consider non-convex domains and [@Sa], who appeals to renewal theory to analyse the growth in $\varepsilon$ of the mean number of steps to completion of the walk-on-spheres algorithm. His method also allows for variants of the algorithm in which the sphere sizes do not need to be optimally inscribed in $D$. Later, [@ST] gives an elementary proof of the $|\!\log \varepsilon|$ bound. Mascagni and co-authors have extensively developed the walk-on-spheres algorithm in applications; see for example [@HMG; @Given2001-lu; @Given2002-vs; @Mackoy2013-es; @Hwang2001-wp]. We break the proof into two parts. In the first part, we analyse the walk-on-spheres process over one step, by considering the distance of the next point in the algorithm from the orthogonal tangent hyperplane of the first point. (Note the existence of a tangent hyperplane requires convexity of the domain.) In the second part of the proof, we use this analysis to build a supermartingale, from which the desired result follows via optional stopping. For the first part of the proof, we start by introducing notation. For any $x = (x_1, \dots, x_d)\in\mathbb{R}^d$ such that $x_1>0$, let us write $V(x) = \{(z_1, \dots, z_d)\in\mathbb{R}^d\colon z_1>0\}$ for the open half-space containing $x$ and denote its boundary $\partial V(x) = \{(z_1, \dots, z_d)\in\mathbb{R}^d\colon z_1=0\}$. Suppose that we choose our coordinate system so that $x\in D$ is such that $\rho_0 = x = (x_1, 0,\dots, 0)$ and $\partial V(\rho_0)$ is a tangent hyperplane to both $D$ and $S_1$. This assumption comes at no cost as, thanks to isotropy and spatial homogeneity of Brownian motion. Let us define $\zeta_0$, the orthogonal distance of $\rho_0$ from $\partial V(\rho_0)$. With the assumed choice of coordinate system, write $\zeta_0 \coloneqq r_1 = x_1 = |x| = |\rho_0|$ and define $$\zeta_1 = \min\Bigl\{\varepsilon, \inf_{z\in\partial V(\rho_0)}|\rho_1-z|\Bigr\};$$ that is, the minimum of $\varepsilon$ and the orthogonal distance of $\rho_1$ from $\partial V(\rho_0)$. Next, define $\theta_1$, the angle that subtends at $\rho_0$ between $\rho_1$ and the origin $(0,\dots,0)$ and recall that symmetry implies that $\theta_1$ is uniformly distributed on $[0,2\pi]$. Simple geometric considerations tell us that $$\zeta_1=x_1 - r_1 \sin\left(\frac{\pi}{2} - \theta_1\right) = \zeta_0 - \zeta_0 \sin\left(\frac{\pi}{2} - \theta_1\right) = \zeta_0(1-\cos(\theta_1)). \label{zeta1}$$ This provides an implicit expression for $\theta_1$ in terms of the orthogonal distance $\rho_0$ from the nearest tangent hyperplane. See Figure \[fig:class\_proofa\]. ![Geometric setting of the proof[]{data-label="fig:class_proofa"}](proof1-fig1_new){width="0.8\linewidth"} Assuming that $\zeta_0>\varepsilon$, thanks to isotropic symmetry, the walk-on-sphere algorithm will end at the first step if $\theta_1$ lies in a certain critical interval dictated by the choice of skin thickness $\varepsilon$. We can compute this critical (and obviously) symmetric interval as a function of $\zeta_0$, say $(-\theta^*(\zeta_0), \theta^*(\zeta_0))$, where $$\theta^*(\zeta_0) = \arccos\pp{\frac{\zeta_0 - \varepsilon}{\zeta_0}}. \label{theta*}$$ A quantity that will be of interest to us in order to complete the proof is the expectation $ \mathbb{E}_{x}[\sqrt{\zeta_1}] = \mathbb{E}_{\rho_0}[\sqrt{\zeta_1}]. $ To this end, we compute $$\begin{aligned} \mathbb{E}_{\rho_0}\left[\sqrt{\zeta_1}\right] & \leq \sqrt{\varepsilon}\,\mathbb{P}_{\rho_0}\pp{\theta_1\in (-\theta^*(\zeta_0), \theta^*(\zeta_0))}+ \mathbb{E}_{\rho_0}\left[ \mathbf{1}_{(\theta_1\not\in (-\theta^*(\zeta_0), \theta^*(\zeta_0)))}\sqrt{\zeta_1}\right]\notag\\ & = \sqrt{\varepsilon}\, \frac{\theta^*(\zeta_0)}{\pi} + \frac{1}{\pi}\int_{\theta^*(\zeta_0)}^\pi \sqrt{\zeta_0(1-\cos(u))}\,{\rm d}u\notag\\ & \eqqcolon\Lambda({\varepsilon}/{\zeta_0})\,\sqrt{\zeta_0}, \label{ineq} \end{aligned}$$ where $\mathbf{1}_S$ denotes the indicator function on the set $S$. Using the primitive ${\displaystyle\int} \sqrt{1- \cos(u)}\,{\rm d}u$ $ =-2\, \sqrt{1-\cos(u)}\,\cot(u/2)$, we have $$\Lambda(u) =\sqrt{u}\,\frac{\arccos(1-u)}{\pi} +\frac{2}{\pi}\sqrt{u} \,\cot\pp{\frac{\arccos(1-u)}{2}}.$$ One easily verifies that there is a constant $\lambda\in(0,1)$ such that $\sup_{u\in [0,1]}\Lambda(u)<\lambda$. Next we move to the second part of the proof. At each step of the walk-on-spheres, we can construct the quantities $\zeta_{n+1}$, the orthogonal distance of $\rho_{n+1}$ to the tangential hyperplane that passes through the closest point on $\partial D$ to $\rho_n$; and $\theta_n$, the angle that is subtended at $\rho_n$ between the aforesaid point and $\rho_{n+1}$. Note that $\varepsilon$ is an absorbing state for the sequence $(\zeta_n, n\geq 0)$ in the sense that, if $\zeta_n = \varepsilon$, then $\zeta_{n+k} = \varepsilon$ for all $k\geq 0$. We may thus write $N(\varepsilon) \leq N'(\varepsilon):=\min\{n\geq 0: \zeta_n = \varepsilon\}$. By the strong Markov property and the spatial homogeneity of Brownian motion given the analysis leading to (\[ineq\]), we have, on $\{n<N(\varepsilon)\}$, $$\mathbb{E}\left[\left.\sqrt{\zeta_{(n+1)\wedge N(\varepsilon)}}\,\right|\zeta_0, \dots, \zeta_n\right] = \mathbb{E}\left[\left.\sqrt{\zeta_{(n+1)\wedge N(\varepsilon)}}\,\right| \zeta_n\right] \leq \Lambda({\varepsilon}/{\zeta_n})\sqrt{\zeta_n}< \lambda \sqrt{\zeta_n}.$$ As a consequence the process $\left(\lambda^{-(n\wedge N(\varepsilon))}\sqrt{\zeta_{n\wedge N(\varepsilon)}}, n\geq 0\right)$ is a supermartingale. The optional-sampling theorem and Jensen’s inequality give us $$\varepsilon \lambda^{-\mathbb{E}_x[N'(\varepsilon)]}\geq \mathbb{E}_{x}[\lambda^{-N'(\varepsilon)}\varepsilon]\leq \sqrt{r_1}, \qquad x\in D.$$ The result now follows by taking logarithms. Exact simulation of stable paths {#stable_paths} ================================ The key ingredient to the walk-on-spheres in the Brownian setting is the knowledge that spheres are exited continuously and uniformly on the boundary of spheres. In the stable setting, the inclusion of path discontinuities means that the process will exit a sphere by a jump. The analogous key observation that makes our analysis possible is the following result, which gives the distribution of a stable process issued from the origin, when it first exits a unit sphere. \[BGR\] Suppose that $B(0,1)$ is a unit ball centred at the origin and write $\sigma_{B(0,1)} = \inf\{t>0 : X_t \not\in B(0,1)\}$. Then, $$\mathbb{P}_0(X_{\sigma_{B(0,1)}}\in \mathrm{d}y) = \pi^{-(d/2+1)}\,\Gamma(d/2)\,\sin(\pi\alpha/2)\,\left|1-|y|^2\right|^{-\alpha/2}|y|^{-d}\,{\rm d}y, \qquad |y|>1.$$ This result provides a method of constructing precise sample paths of stable processes in phase space (i.e. exploring sample paths as ordered subsets of $\mathbb{R}^d$ rather than as functions $[0,\infty]\to \mathbb{R}^d$). Choose a tolerance $\epsilon$ and initial point $X_0 =x$. Denote by $E_1$ a sampling from the distribution given in Theorem \[BGR\]. This gives the exit from a ball of radius one when $X$ is issued from the origin. By the scaling property and the stationary and independent increments, $x +\epsilon\, E_1$ is distributed as the exit position from a ball of radius $\epsilon$ centred at $x$ when the process is issued from $x$. Hence, we define $X_1=x + \epsilon\, E_1$ and then, inductively for $n\geq 1$, generate $X_{n+1}$ as the exit point of the ball centred on $X_n$ with radius $\epsilon$ by noting this is equal in distribution to $X_n + \epsilon \, E_{n+1}$, where $E_{n+1}$ is an [*iid*]{}copy of $E_1$. It is important to remark for later that the value of $\epsilon$ in this algorithm does not need to be fixed and may vary with each step. Note, however, the method does not generate the corresponding time to exit from each ball. Therefore, the sample paths that are produced, whilst being exact in the distribution of points that the stable process will pass through, cannot be represented graphically in time as there is only an equal mean duration to exiting each sphere. If the tolerance $\epsilon$ is altered on each step, then even this mean duration feature is lost. The method is used to generate Figure \[fig:sample\_path\]. ![Example sample paths for the $\alpha$-stable Levy process generated by using the exit distribution in Theorem \[BGR\] for spheres of radius $10^{-6}$. Rows shows sample paths in two- and three-dimensions for $\alpha=0.9$ (left) and $\alpha=1.8$ (right). The yellow lines indicate jumps of the process and blue dots show where the process has been.[]{data-label="fig:sample_path"}](test5_alpha0_9 "fig:") ![Example sample paths for the $\alpha$-stable Levy process generated by using the exit distribution in Theorem \[BGR\] for spheres of radius $10^{-6}$. Rows shows sample paths in two- and three-dimensions for $\alpha=0.9$ (left) and $\alpha=1.8$ (right). The yellow lines indicate jumps of the process and blue dots show where the process has been.[]{data-label="fig:sample_path"}](test5_alpha1_8 "fig:")\ ![Example sample paths for the $\alpha$-stable Levy process generated by using the exit distribution in Theorem \[BGR\] for spheres of radius $10^{-6}$. Rows shows sample paths in two- and three-dimensions for $\alpha=0.9$ (left) and $\alpha=1.8$ (right). The yellow lines indicate jumps of the process and blue dots show where the process has been.[]{data-label="fig:sample_path"}](test5_3d_alpha0_9 "fig:") ![Example sample paths for the $\alpha$-stable Levy process generated by using the exit distribution in Theorem \[BGR\] for spheres of radius $10^{-6}$. Rows shows sample paths in two- and three-dimensions for $\alpha=0.9$ (left) and $\alpha=1.8$ (right). The yellow lines indicate jumps of the process and blue dots show where the process has been.[]{data-label="fig:sample_path"}](test5_3d_alpha1_8 "fig:") On account of classical Feynman–Kac representation, simulation of solutions to parabolic and elliptic equations involving the fractional Laplacian, and more generally the infinitesimal generator of a Lévy process are synonymous with the simulation of the paths of the associated stochastic process. On account of the fact that such equations occur naturally in mathematical finance in connection with (exotic) option pricing, there are already many numerical and stochastic methods in existence for the general Lévy setting. The reader is referred, for example, to the books [@Cont; @Leven] and the references therein. Other sources offering simulation techniques can be found, e.g. [@Polish; @Rosinski1; @Rosinski2; @Rosinski3]. Similarly to works in mathematical finance, they are mostly focused on the approximation of the stable process (and indeed the general Lévy process) by a compound Poisson process or a power-series representation of the path, with a diffusive component to mimic the effect of small jumps. To our knowledge, however, the walk-on-spheres approach to path simulation has not been used in the context of simulating stable processes to date, nor, as alluded to above, to the end of numerically solving Dirichlet-type problems for the fractional Laplacian. Walk-on-spheres for the fractional Laplacian {#WoSfL} ============================================ We start by describing the walk-on-spheres for the fractional-Laplacian Dirichlet problem (\[aDirichlet\]) on a convex domain $D$. The domain $D$ may be unbounded, as long as $D^{\textrm{c}}$ has non-zero measure (even though Theorem \[corr\] requires boundedness). Fix $x\in D$. The walk-on-spheres $(\rho_n$, $n\geq 0)$, with $\rho_0 = x$ is defined in a similar way to the Brownian setting in the sense that, given $\rho_{n-1}$, the distribution of $\rho_{n}$ is selected according to an independent copy of $X_{\sigma_{B_n}}$ under $\mathbb{P}_{\rho_{n-1}}$, where $B_n = \{x\in\mathbb{R}^d\colon |x - \rho_{n-1}|< r_n\}$ and $\sigma_{B_n} = \inf\{t>0: X_t\not\in B_n\}$. The algorithm comes to an end at the random index $N = \min\{n\geq 0\colon \rho_n\not\in D\}$, again using the standard understanding that $\min\emptyset \coloneqq \infty$. See for example the depiction in Figure \[4stepsonly\]. ![Steps of the walks-on-sphere algorithm until exiting the convex domain $D$ in the stable setting. In this realisation, $N = 3$.[]{data-label="4stepsonly"}](4stepsonly_new){width="0.7\linewidth"} Even though the domain $D$ may be unbounded, our main result predicts that, irrespective of the point of issue of the algorithm, there will always be at most a geometrically distributed number of steps (whose parameter also does not depend on the point of issue) before the algorithm ends. \[main\] Suppose that $D$ is a convex domain. For all $x\in D$, there exists a constant $p = p(\alpha,d)>0$ (independent of $x$ and $D$) and a real-valued random variable $\Gamma$ such that $N\leq \Gamma$ almost surely, where $$\mathbb{P}(\Gamma = k ) = (1-p)^{k-1}p, \qquad k\in\mathbb{N}.$$ There are a number of remarks that we can make from the conclusion above. - Although $\Gamma$ has the same distribution for each $x\in D$, it is not the same random variable for each $x\in D$. As we shall see in the proof of the above theorem, the inequality $N\leq \Gamma$ is derived by comparing each step of the walk-on-spheres algorithm with a sequence of Bernoulli random variables. This sequence of Bernoulli random variables are defined up to null sets which may be different under each $\mathbb{P}_x$. Therefore, whilst the distribution of $\Gamma$ does not depend on $x$, its null sets do. - The stochastic domination in Theorem \[main\] is much stronger than the usual comparison of the mean number of steps. Indeed, whilst it immediately implies that $\mathbb{E}_x[N] = 1/p$, we can also deduce that there is an exponentially decaying tail in the distribution of the number of steps. Specifically, for any $x\in D$, $$\mathbb{P}(N >n ) \leq \mathbb{P}(\Gamma >n ) = (1-p)^n, \qquad n\in\mathbb{N}.$$ - The randomness in the geometric random variables $\Gamma$ is heavily correlated to $N$. The fact that each of the $\Gamma$ are geometrically distributed has the advantage that $$\sup_{x\in D}\mathbb{E}_x[N] \leq \sup_{x\in D}\mathbb{E}_x[\Gamma] = \frac{1}{p}.$$ However, it is less clear what kind of distributional properties can be said of the random variable $ \sup_{x\in D}\Gamma, $ which almost surely upper bounds $\sup_{x\in D}N$. Finally, it is worth stating formally that the walk-on-spheres algorithm is unbiased and therefore, providing $\mathbb{E}_x[{g}(X_{\tau_D})]<\infty$, the [strong law of large numbers]{}applies and a straightforward Monte Carlo simulation of the solution to is possible. Moreover, providing $\mathbb{E}_x[{g}(X_{\tau_D})^2]<\infty$, the [central limit theorem]{}offers the rate of convergence. \[rate1\] When $D$ is bounded and convex and ${g}$ is continuous and in $ L^1_\alpha(D^{\mathrm{c}})$, $$\label{WoSMC2} \lim_{n \to\infty} \frac{1}{n}\sum_{i = 1}^n {g}(\rho^{i}_{N^{i}}) = \mathbb{E}_x[{g}(\rho_{N})]=\mathbb{E}_x[{g}(X_{\tau_D})] = u(x),$$ almost surely where $(\rho^{i}_n, n\leq N^{i})$, $i\geq 1$ are [*iid*]{}copies of the walk-on-spheres with $\rho_0^i = x\in D$, $i\geq 1$ and $u(x)$ is the solution to (\[aDirichlet\]). Moreover, when $$\int_{D^\mathrm{c}}\frac{{g}(x)^2}{1+|x|^{\alpha+d}}\,{\rm d}x<\infty, \label{f2}$$ then $\operatorname{Var}({g}(\rho_N))<\infty$ and, in the sense of weak convergence, $$\lim_{n\to\infty}n^{1/2}\left(\frac 1n \sum_{i=1}^n {g}(\rho^{i}_{N^{i}})- u(x)\right)= \operatorname{Normal}(0, \operatorname{Var}({g}(\rho_N))).$$ The first part is a straightforward consequence of the earlier mentioned [strong law of large numbers]{}and the fact that Theorem \[corr\] ensures that $\mathbb{E}_x[{g}(\rho_N)]=\mathbb{E}_x[{g}(X_{\tau_D})]<\infty$. For the second part, we need to show that implies $\mathbb{E}_x[{g}(\rho_N)^2]=\mathbb{E}_x[{g}(X_{\tau_D})^2]<\infty$. However, if we consider the computation in (\[worksforsquared\]) of the Appendix, which shows that $\mathbb{E}_x[{g}(X_{\tau_D})]<\infty$ when ${g}$ is continuous and in $L^1_{\alpha}(D^\mathrm{c})$, then it is easy to see that the same statement holds replacing ${g}$ by ${g}^2$. Under finiteness of the second moment, the [central limit theorem]{}completes the proof. We now return to the proof of Theorem \[main\]. Our approach is to break it into several parts. For convenience, we shall henceforth write $X^{(x)}=(X^{(x)}(t)\colon t\geq 0)$ to indicate the dependency of $X$ on its initial position $X_0 = x$ (equivalent to writing $(X, \mathbb{P}_x)$). For any $x = (x_1, \dots, x_d)\in\mathbb{R}^d$ such that $x_1>0$, we have $V(x) = \{(z_1, \dots, z_d)\in\mathbb{R}^d \colon z_1>0\}$ for the open half-space containing $x$ and denote its boundary $\partial V(x) = \{(z_1, \dots, z_d)\in\mathbb{R}^d \colon z_1=0\}$. For any Borel set $A\subset\mathbb{R}^d$, we write $ \sigma_A = \inf\{t>0 \colon X_t\not\in A\}. $ We will typically use in place of $A$ the set $V(x)$ as well as $B(x,1) = \{z\in\mathbb{R}^d\colon |z- x|<1\}$, the unit ball centred at $x\in\mathbb{R}^d$. Finally write ${\rm\bf i} = (1,0, \dots, 0)\in\mathbb{R}^d$. \[scaled\] Without loss of generality (by appealing to the spatial homogeneity of $X$ which allows us to appropriately choose our coordinate system) suppose that $x= |x|\,{ \rm\bf i}\in D$ is such that $\partial V(x)$ is a tangent hyperplane to both $D$ and $B_1$. Then $X^{(x)}_{\sigma_{B_1}}$ is equal in distribution to $|x|\, X^{(\rm\bf i)}_{\sigma_{B({\rm\bf i},1)}}$ and $X^{(x)}_{\sigma_{V(x)}}$ is equal in distribution to $|x|\,X^{(\rm\bf i)}_{\sigma_{V(\mathbf{i}) }}$. The scaling property of $X$ ensures that we can write $$X^{(x)}_{s} = |x|\hat{X}^{(\mathbf{i})}_{|x|^{-\alpha}s}, \qquad s\geq 0, \label{scaling}$$ where $\hat{X}^{(x)}$ is equal in law to $X^{(x)}$. Note that $$\begin{aligned} \sigma_{B_1} & = \inf\Bp{t> 0\colon {X}^{(x)}(t)\not\in B(x,\abs{x}) }\notag \\ & = |x|^{\alpha}\,\inf\Bp{|x|^{-\alpha}t> 0\colon |x| \hat{X}^{(\mathbf{i})}(|x|^{-\alpha}t)\not\in B( x,\abs{x}) }\notag \\ & = |x|^{\alpha}\,\inf\Bp{u> 0\colon \hat{X}^{(\mathbf{i})}(u)\not\in B({\rm\bf i},1) }\notag \\ & \eqqcolon |x|^{\alpha}\,\hat{\sigma}_{B({\rm\bf i},1)}. \label{tscale} \end{aligned}$$ It follows that $$X^{(x)}_{\sigma_{B_1}} = |x| \hat{X}^{(\mathbf{i})}_{|x|^{-\alpha} |x|^{\alpha} \hat{\sigma}_{B({\rm\bf i},1) }}\,{\buildrel d \over =}\, |x| {X}^{(\mathbf{i})}_{\sigma_{B({\rm\bf i},1)}}, \label{scaleB1}$$ as required. The proof of the second claim follows the same steps and is omitted for the sake of brevity. An important consequence of the previous result is the comparison between the first exit from the largest sphere in $D$ centred at $x$ and the first exit from the tangent hyperplane to the latter sphere. Recall that $B_n = \{z\in\mathbb{R}^d\colon |z - \rho_{n-1}|< r_n\}$ denotes the $n$th sphere. \[indicators\]Suppose that $x\in D$ is such that $\partial V(x)$ is a tangent hyperplane to both $D$ and $B_1$. Define under $\mathbb{P}_x$ the indicator random variables $${I}_D = \mathbf{1}_{\{X_{\sigma_{B_1}} \not\in D \}}\quad\text{ and } \quad {I}_V=\mathbf{1}_{\{X_{\sigma_{B_1}} \not\in V(x) \}}.$$ Then $\mathbb{P}_x(I_D\geq I_V)= 1$ and, independently of $x\in D$, $\mathbb{P}_x(I_V = 1) =p(\alpha,d)$, where $$\begin{aligned} p(\alpha,d) & \coloneqq \mathbb{P}_{\mathbf i}(X_{\sigma_{B({\rm\bf i},1)}}\not\in V({\rm\bf i})) \\ & =\frac{\Gamma(d/2)}{\pi^{(d+2)/2}}\,\sin(\pi\alpha/2)\,\int_{x_1<-1}\left|1- |x|^2\right|^{-\alpha/2}|x|^{-d}\,{\rm d}x, \end{aligned}$$ which is a number in $(0,1)$. The inequality follows from the inclusion $D\subset V(x)$. The formula for $p(\alpha,d)$ uses the coordinate system and scaling property of stable processes in Lemma \[scaled\] as well as the identity for the first exit from a sphere given by Theorem \[BGR\]. We are now ready to prove our main result. Suppose we condition on the previous positions of the walk-on-spheres, $\rho_0,\dots, \rho_{k-1}$ as well as on the event $\{N>k-1\}$. Thanks to stationary and independent increments as well as isotropy in the law of a stable process, we can always choose a coordinate system, or equivalently reorient $D$ in such a way that $\rho_k = |\rho_k|{\rm\bf i}$. This has the implication that, with the aforesaid conditioning, the random variable $\mathbf{1}_{\{N = k\}}$ is independent of $\rho_0,\dots, \rho_{k-1}$ and equal in law to $I_D(\rho_{k-1})$, where we have abused our original notation to indicate the initial position of $X$ in the definition of $I_D$. Similarly, with the same abuse of notation, the event $I_V(\rho_{k-1})$ is independent of $\rho_0,\dots, \rho_{k-1}$ and equal in law to a Bernoulli random variable with probability of success $p = p(\alpha, d)$. In particular, the sequence $I_V(\rho_k)$, $k\geq 0$ is a sequence of Bernoulli trials. That is to say, if we define $$\Gamma = \min\{k\geq 1\colon I_V(\rho_k) = 1 \},$$ then it is geometrically distributed with parameter $p$. Thanks to Corollary \[indicators\], we also have that $\mathbb{P}_x(I_D\geq I_V)|_{x = \rho_k}=1$, $k< N$, that is to say, $\{I_V(\rho_k)=1\}$ almost surely implies $\{I_D(\rho_k)=1\}$, for $k<N$, and hence $$\min\{n\geq 1\colon I_D(\rho_k) = 1 \}\leq \min\{n\geq 1\colon I_V(\rho_k) = 1 \}$$ almost surely. In other words, we have $N\leq \Gamma$, almost surely, as required. Non-convex domains {#non_convex} ================== The key element in the proof of Theorem \[main\] is the comparison of the event that the next step of the walk-on-spheres exits the domain $D$ with the event that the next step of the walk-on-spheres exits a larger, more regular domain. More precisely, the aforesaid regular domain is taken to be the half-space that contains $D$ with boundary hyperplane that is tangent to both the current maximal sphere and $D$. It is the use of a half-space that allows us to work with unbounded domains but which forces the assumption that $D$ is convex. With a little more care, we can remove the need for convexity without disturbing the main idea of the proof. However, this will come at the cost of insisting that $D$ is bounded. It does however, open the possibility that $D$ is not a connected domain. We give two results in this respect. For the first one, we introduce the following definition, which has previously been used in the potential analysis of stable processes; see for example [@Chen-Song]. A domain $D$ in $\mathbb{R}^d$ is said to satisfy the [*uniform exterior-cone condition*]{}, henceforth written UECC, if there exist constants $\eta > 0$, $r > 0$ and a cone $${\rm Cone}(\eta) = \{x = (x_1,\dots,x_d) \in \mathbb{R}^d\colon |x|<\eta x_1\}$$ such that, for every $z\in \partial D$, there is a cone $C_z$ with vertex $z$, isometric to ${\rm Cone}(\eta)$ satisfying $C_z \cap B(z,r) \subset D^{\mathrm{c}}$. It is well known that, for example, bounded $C^{1,1}$ domains satisfy (UECC). We need a slightly more restrictive class of domains than those respecting UECC. We say that $D$ satisfies the [*regularised uniform exterior-cone condition*]{}, written RUECC, if it is UECC and the following additional condition holds: for each $x\in D$, suppose that $\partial(x)$ is a closest point on the boundary of $D$ to $x$. Then the isometric cone that qualifies $D$ as UECC can be placed with its vertex at $\partial(x)$ and symmetrically oriented around the line that passes through $x$ and $\partial(x)$. ![A domain that satisfies the regularised uniform exterior-cone condition[]{data-label="fig:class_proof"}](cone_new){width="0.8\linewidth"} Suppose that $D$ is open and bounded (but not necessarily connected) and satisfies RUECC. Then, for each $x\in D$, there exists a random variable $\hat{\Gamma}$ such that $N\leq \hat{\Gamma}$ almost surely and $$\mathbb{P}(\hat{\Gamma} = k) = (1-\hat{q})^{k-1}\hat{q}, \qquad k\in \mathbb{N},$$ for some $\hat{q}=\hat{q}(\alpha, D)$. Reviewing the proof of Theorem \[main\], we note that it suffices to prove that, in the context of Corollary \[indicators\], for each $x\in D$, there exists a Bernoulli random variable $\hat{J}_x$ with parameter $\hat{q}$ (independent of $x$) such that $\mathbb{P}_x(I_D\geq \hat{J}_x)=1$. To this end, we recall that, without loss of generality, we may choose our coordinate system such that $x = |x|{\rm\bf i}\in D$ is such that $\partial(x) = 0$. The assumption that $D$ is bounded implies that there exists a $\eta$ such that $|x|\leq \eta$. From the definition of RUECC, we know that there exists an $r>0$ and a cone, $C_{0}$, with vertex at $0$, a closest point on $\partial D$ to $x$, which is symmetrically oriented around the line passing through $x$ and $0$, such that $C_{0, r}\coloneqq C_{0} \cap B(0,r)\subset D^{\texttt{c}}$. We have $$\begin{aligned} \mathbb{P}_{x}(X_{\sigma_{B_1}} \in C_{0, r}) & =\mathbb{P}_{\mathbf i}(X_{\sigma_{B(\mathbf{i},1)}}\in C_{0, r/|x|}) \\ & \geq \mathbb{P}_{\mathbf i}(X_{\sigma_{B(\mathbf{i},1)}}\in C_{0, r/\eta}) \\ & = \frac{\Gamma(d/2)}{\pi^{(d+2)/2}}\,\sin(\pi\alpha/2)\int_{C_{-{\rm\bf i}, (r/\eta)}}\left|1- |y|^2\right|^{-\alpha/2}|y|^{-d}\,{\rm d}y \\ & \eqqcolon\hat{q}, \end{aligned}$$ where $C_{z, u} \coloneqq [C_{0} \cap B(0,u)]-\{z\}$, for $z\in\mathbb{R}^d$ and $u>0$. Note that $\hat{q}$ is necessarily strictly positive. Taking account of scaling, we have $\mathbb{P}_x$-almost surely that $$I_D\geq \mathbf{1}_{\{|x|^{-1}X^{(x)}_{\sigma_{B_1}} \in C_{0, r/|x|} \}}\geq\mathbf{1}_{\{|x|^{-1}X^{(x)}_{\sigma_{B_1}} \in C_{0, r/\eta} \}}\eqqcolon \hat{J},$$ where $\hat{J}$ is a Bernoulli random variable with parameter $\hat{q}$. Stochastic dominance, $N\leq \hat{\Gamma}$ almost surely, follows by the same line of reasoning as in the proof of Theorem \[main\]. For the second result, we completely relax the geometrical requirements on $D$ at the expense of efficiency. With an abuse of our earlier notation, we introduce $$N(\varepsilon) = \min\Bigl\{n\geq 0\colon \rho_n\not\in D \text{ or }\inf_{z\in \partial D}|\rho_n-z|<\varepsilon \Bigr\}.$$ Intuitively, $N(\varepsilon)$ is the step that exits the inner $\varepsilon$-thickened boundary of $D$. Suppose that $D$ is open and bounded (but not necessarily connected). Then for all $x\in D$, there exists a constant $q_\varepsilon = q_\varepsilon(\alpha,D)>0$ (independent of $x$) and a random variable $\Gamma^\varepsilon$ such that $N\leq \Gamma^\varepsilon$ almost surely, where $$\mathbb{P}_x(\Gamma^\varepsilon = k ) = (1-q_\varepsilon)^{k-1}q_\varepsilon, \qquad k\in\mathbb{N}.$$ Moreover, $q_\varepsilon = \mathcal{O}(\varepsilon^\alpha)$ as $\varepsilon\downarrow 0$. In particular $$\mathbb{E}_x[N(\varepsilon)] = \mathcal{O}(\varepsilon^{-\alpha}),\qquad \text{as $\varepsilon \downarrow 0$.} \label{ea}$$ Define $$\delta\coloneqq \inf\Bp{r>0\colon D\subset B(x,r) \text{ for all } x\in D },$$ so that any sphere of radius $\delta$ centred at $x\in D$ contains $D$. Once again, we recall that, without loss of generality, we may choose our coordinate system such that $x = |x|\,{\rm\bf i}\in D$ is such that $\partial V(x)$ is a tangent hyperplane to $B_1$ and such that $0\in \partial B_1 \cap \partial V(x)\cap\partial D$. Then, taking account of scaling, and that, for all $x\in D$ such that $\inf_{z\in\partial D}|x-z|\geq \varepsilon$, with the particular choice of coordinates described above, $\delta/|x|\leq \delta/\varepsilon$, we have $$\mathbf{1}_{\{N(\varepsilon) = 1\}}\geq \mathbf{1}_{\{X^{(x)}_{\sigma_{B_1}} \not\in B(x,\delta) \}} = \mathbf{1}_{\{|x|^{-1}X^{(x)}_{\sigma_{B_1}} \not\in B({\rm\bf i},\delta/|x|) \}} \geq \mathbf{1}_{\{|x|^{-1}X^{(x)}_{\sigma_{B_1}} \not\in B({\rm\bf i},\delta/\varepsilon) \}}. $$ Recall, however, from (\[scaleB1\]) that $|x|^{-1}X^{(x)}_{\sigma_{B_1}} {\buildrel d \over =} {X}^{(\mathbf{i})}_{\sigma_{B({\rm\bf i},1)}}$. It therefore follows that, $\mathbb{P}_x$-almost surely, $$\mathbf{1}_{\{N(\varepsilon) = 1\}} \geq \mathbf{1}_{\{X^{(\mathbf{i})}_{\sigma_{B({\rm\bf i}, 1)}} \not\in B(\mathbf{i},\delta/\varepsilon) \}} \eqqcolon J^\varepsilon,$$ where $J^\varepsilon$ is a Bernoulli random variable with parameter $$q_\varepsilon (\alpha, D) = \mathbb{P}_{\mathbf i}(X_{\sigma_{B({\rm\bf i},1)}}\notin B({\rm\bf i},\delta/\varepsilon)) =\frac{\Gamma(d/2)}{\pi^{(d+2)/2}}\, \sin(\pi\alpha/2)\, \int_{|y|\geq \delta/\varepsilon}\left|1- |y|^2\right|^{-\alpha/2}|y|^{-d}\,{\rm d}y.$$ Reverting to generalised spherical polar coordinates, in particular recalling that the Jacobian with respect to Cartesian coordinates is no larger than $|x|^{d-1}$ (see [@Blum]), we can estimate $$q_\varepsilon (\alpha, D) \leq \frac{\Gamma(d/2)}{\pi^{(d+2)/2}} \, \sin(\pi\alpha/2)\,\int_{\delta/\varepsilon}^\infty r^{-(\alpha +1)}dr =\mathcal{O} (\varepsilon^\alpha).$$ Reviewing the line of reasoning in the proof of Theorem \[main\], we see that this comparison of events on the first step can be repeated at each surviving step of the algorithm to deduce the claimed result. The $\mathcal{O}(\varepsilon^{-\alpha})$ bound in can be compared with the bounds achieved by [@BB] for the classical walk-on-spheres with Brownian motion for domains with more general geometries than convex. The worst case in [@BB] is $\mathcal{O}(\varepsilon^{2-4/a})$ for a parameter $a>0$ (describing the domain’s thickness or fractal boundary). Notably in the limit $\alpha\to 2$ ($X$ converges to Brownian motion) and $a \to \infty$ (the domain loses regularity), the two agree with an $\mathcal{O}(\varepsilon^{-2})$ bound. Fractional Poisson problem {#inhomogenous} ========================== We are now interested in using the walk-on-spheres process to find the solution to the inhomogeneous version of (\[aDirichlet\]), namely $$\begin{gathered} \begin{aligned} -(-\Delta )^{\alpha/2}u(x) & =-{f}(x), & \qquad x & \in D, \\ u(x) & = {g}(x), & x & \in D^{\rm c}, \end{aligned} \label{aDirichlet_g} \end{gathered}$$ for suitably regular functions ${f}\colon D\to \mathbb{R}$ and ${g}\colon D^{\rm c} \to \mathbb R$. We want to identify a Feynman–Kac representation for solutions to for suitable assumptions on ${g}, {f}$ and $D$. Throughout this section, we adopt the setting of the following theorem. \[hasacorr\] Let $d\geq 2$ and assume that $D$ is a bounded domain in $\mathbb{R}^d$. Suppose that ${g}$ is a continuous function which belongs to $L^1_\alpha(D^\mathrm{c})$. Moreover, suppose that ${f}$ is a function in $ C^{\alpha +\varepsilon}(\overline{D})$ for some $\varepsilon>0$. Then there exists a unique continuous solution to in $L^1_\alpha(\mathbb{R}^d)$ which is given by $$u(x) = \mathbb{E}_x[{g}(X_{\sigma_D})] + \mathbb{E}_x\left[\int_0^{\sigma_D} {f}(X_s)\,{\rm d}s\right], \qquad x\in D, \label{non_homg_FK}$$ where $\sigma_D=\inf\{t>0\colon X_t\not\in D\}$. The combinations of Theorem 2.10 and 3.2 in [@bucur] treat the case that $D$ is a ball. In the more general setting, amongst others, [@B99], [@R-O1] and [@R-O2] (see also citations therein) offer results in this direction, albeit from a more analytical perspective. We give a new probabilistic proof of Theorem \[hasacorr\] in the Appendix using a method that combines the idea of walks-on-spheres with the version of Theorem \[hasacorr\] when $D$ is a ball. It is for this reason that the (otherwise unclear) need for the assumption that ${f}\in C^{\alpha +\varepsilon}(\overline{D})$ enters. Note in particular that Theorem \[corr\] follows as a corollary. We can develop the expression in in terms of the walk-on-spheres $(\rho_n, n\leq N)$, providing the basis for a Monte Carlo simulation. What will work to our advantage here is another explicit identity that appears in [@BGR]. Define $$\begin{aligned} V_r(x,{\rm d}y) & \coloneqq \int_0^\infty \mathbb{P}_x(X_t\in {\rm d}y, \, t<{\sigma_{B(x,r)}} )\,{\rm d}t, \qquad x\in\mathbb{R}^d, \;|y|<1,\; r>0. \end{aligned}$$ The expected occupation measure of the stable process prior to exiting a unit ball centred at the origin is given, for $|y|<1$, by $$\begin{aligned} V_1(0,{\rm d}y)= 2^{-\alpha}\,\pi^{-d/2}\, \frac{\Gamma(d/2)}{\Gamma(\alpha/2)^{2}}\, |y|^{\alpha -d}\, \pp{\int_0^{|y|^{-2}-1}(u+1)^{-d/2}u^{\alpha/2-1}{\rm d}u}\,{\rm d}y.\label{V} \end{aligned}$$ Whilst the above identity is presented in a probabilistic context, it has a much older history in the analysis literature. Known as Boggio’s formula, the original derivation in the setting of potential theory dates back to [@Bog]. See the discussion in [@DR; @bucur]. In the next result, we will write as a slight abuse of notation $V_r(x,{f}(\cdot)) = \int_{|y-x|<r}{f}(y)\,V_r(x,{\rm d}y)$ for bounded measurable ${f}$. \[integral\]For $x\in D$, ${g}\in L^1_\alpha(D^\mathrm{c})$ and ${f}\in C^{\alpha +\varepsilon}(\overline{D})$, we have the representation $$u(x) =\mathbb{E}_x[{g}(\rho_{N})] + \mathbb{E}_x\left[\sum_{n=0}^{N-1} r_n^{\alpha} V_{1}(0, {f}(\rho_n + r_n\cdot))\right].$$ Given the walk-on-spheres $(\rho_n, n\leq N)$ with $\rho_0 = x\in D$, define $\sigma_n$ jointly with $\rho_n$ so that, given $\rho_{n-1}$, $(\rho_n, \sigma_n)$ is equal in law to $(X_{\sigma_{B_n}}, \sigma_{B_n})$ under $\mathbb{P}_{\rho_{n-1}}$. We can now represent the second expectation on the right-hand side of (\[non\_homg\_FK\]) in the form $$\mathbb{E}_x\left[\sum_{n\geq 0} \mathbf{1}_{\{\rho_n\in D\}} \int_{0}^{\sigma_{n+1}} {f}\left(\rho_n + X^{(n+1)}_s\right)\,\mathrm{d}s\right], \qquad x\in D, \label{withrho}$$ where $X^{(n)}$ are independent copies of $(X, \mathbb{P}_0)$. Applying Fubini’s theorem, then conditioning each expectation on $\mathcal{F}_{n}\coloneqq \sigma(\rho_k\colon k\leq n)$ followed by Fubini’s theorem again, we have $$\begin{aligned} \mathbb{E}_x\left[\int_0^{\sigma_D} {f}(X_s)\,\mathrm {d}s\right] & =\sum_{n\geq 0}\mathbb{E}_x\left[ \mathbf{1}_{\{\rho_n\in D\}} \left. \mathbb{E}_{y}\left[\int_{0}^{\sigma_{B(y,r)}} {f}( X_s)\,\mathrm{d}s\right] \right|_{y = \rho_n, r = r_n}\right]\\ & =\sum_{n\geq 0}\mathbb{E}_x\left[ \mathbf{1}_{\{\rho_n\in D\}} V_{r_n}(\rho_n, {f}(\cdot)) \right]\\ & =\mathbb{E}_x\left[\sum_{n= 0}^{N-1} V_{r_n}(\rho_n, {f}(\cdot)) \right]. \end{aligned}$$ The proof is completed once we show that $V_r(x,g) = r^{\alpha}V_1(0, {f}(x + r\cdot)),$ for $r>0$, $x\in\mathbb{R}^d$ and bounded measurable ${f}$. To this end, we appeal to spatial homogeneity and the, now, familiar computations using the scaling property of stable processes: $$\begin{aligned} V_r(x,{f}(\cdot)) & = \mathbb{E}_x\left[\int_0^{\sigma_{B(x,r)}} {f}(X_t) \,{\rm d}t\right]\nonumber \\ & = \mathbb{E}_0\left[\int_0^{\sigma_{B(0,r)}} {f}(x+X_t) \,{\rm d}t\right]\nonumber \\ & = \mathbb{E}_0\left[\int_0^{\sigma_{B(0,1)}} r^{\alpha}\,{f}(x+ r\,X_s) \,{\rm d}s\right]\nonumber \\ & =\int_{|y|<1} r^{\alpha} {f}(x + r\,y)\, V_1({0,\rm d}y)\nonumber \\ & =r^\alpha \,V_1(0, {f}(x + r\,\cdot)). \label{timescale} \end{aligned}$$ The proof is now complete. Lemma \[integral\] now informs a Monte Carlo procedure based on simulating the quantity $$\chi \coloneqq {g}(\rho_{N}) + \sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot)) , \qquad x\in D,$$ which is again justified by an obvious [strong law of large numbers]{}and the [central limit theorem]{}in the spirit of Corollary \[rate1\]. \[rate2\] When $D$ is bounded and convex, ${g}$ is continuous and in $ L^1_\alpha(D^\mathrm{c})$ and ${f}$ is a function in $ C^{\alpha +\varepsilon}(\overline{D})$ for some $\varepsilon>0$, then $$\label{WoSMC3} \lim_{n \to\infty} \frac{1}{n}\sum_{i = 1}^n \chi^{i} = \mathbb{E}_x\left[ {g}(\rho_{N}) + \sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\right] = u(x),$$ almost surely where $\chi^{i}$, $i\geq 1$ are [*iid*]{}copies of $\chi$ and $u(x)$ is the solution to (\[aDirichlet\_g\]). Moreover, when $$\int_{D^\mathrm{c}}\frac{{g}(x)^2}{1+|x|^{\alpha + d}}\,{\rm d}x<\infty. \label{f22}$$ then $\operatorname{Var}(\chi)<\infty$ and, in the sense of weak convergence, $$\lim_{n\to\infty}n^{1/2}\left(\frac 1n \sum_{i=1}^n \chi^{i}- u(x)\right)= \operatorname{Normal}(0, \operatorname{Var}(\chi)).$$ Theorem \[hasacorr\] and Lemma \[integral\] ensure that the [strong law of large numbers]{}may be invoked. For the [central limit theorem]{}, we need $\mathbb{E}_x[\chi^2]<\infty$. Taking account of the fact that $\chi$ is the sum of two terms, the Cauchy–Schwarz inequality ensures that $\mathbb{E}_x[\chi^2] $ is finite if $\mathbb{E}_x\left[{g}(\rho_{N})^2\right] $ and $ \mathbb{E}_x\left[\left( \sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\right)^2\right]$ are finite. Recall that $\mathbb{E}_x\left[{g}(\rho_{N})^2\right] = \mathbb{E}_x[{g}(X_{\sigma_D})^2]$ and, from Corollary \[rate1\], that is sufficient to ensure that this expectation is bounded. Now note that, on account of the fact that ${f}$ is bounded, there exists a constant $\kappa\in(0,1)$, such that, for each $n\leq N$, appealing to (\[timescale\]), we have $r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\leq \kappa \sigma_n$, where $\sigma_n$ is the time it takes for the walk-on-spheres to exit the $n$th sphere. Thus $\sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\leq\kappa \sum_{n=0}^{N-1} \sigma_n = \kappa\,\sigma_D$. We thus have that $$\mathbb{E}_x\left[\left( \sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\right)^2\right]\leq \kappa^2 \mathbb{E}_x[\sigma_D^2].$$ However, the latter expectation can be bounded by $\mathbb{E}_x[\sigma_{B^*}^2]$, where $B^* = B(x, R)$ for some suitably large $R$ such that $D$ is compactly embedded in $B^*$. Moreover, appealing to [@GET], we know that $\mathbb{E}_x[\sigma_{B^*}^2]$ is bounded. Numerical experiments {#numerics} ===================== In the following section, all of the routines associated with the simulations are publicly available at the following repository: <https://bitbucket.org/wos_paper/wos_repo> For the Monte Carlo procedure, independent copies of the walk-on-spheres $(\rho_n, n\leq N)$ need to be simulated whereby, by the Markov property, every new point in the sequence can be expressed as $\rho_{n+1}=\rho_n+X'_{\sigma'_{B(0,{r_n})} }$, where $X'$ is an independent version of $X$ and $$\sigma'_{B(0,{r_n})} = \inf\{t>0\colon X'_t \not\in B(0,{r_n})\}.$$ In other words, $\rho_{n+1}$ is an exit point from a ball $B(0,{r_n})$ under $\mathbb{P}_0$ translated by $\rho_n$. A consequence of Lemma \[BGR\] is that the exit distribution of $X'_t$ from $B(0,r_n)$, $r_n>0$, can be, via a change of variable $y = \tilde{y}/r_n$, written as $$\mathbb{P}_0(X_{\sigma_{B(0,r_n)}}\in {\rm d}\tilde{y}) = \pi^{-(d/2+1)}\Gamma(d/2)\sin(\pi\alpha/2)\left|r_n^2-|\tilde{y}|^2\right|^{-\alpha/2}|\tilde{y}|^{-d} r_n^{\alpha}\,{\rm d}\tilde{y}, \qquad |\tilde{y}|>r_n. \label{rdy}$$ For $d=2$, it is more convenient to work with polar coordinates $(r,\theta)$ in order to separate variables in . Indeed, recalling that ${\rm d}\tilde{y}=r\,{\rm d}r\,{\rm d}\theta$, we have $$\begin{aligned} \mathbb{P}_0(X_{\sigma_{B(0,r_n)}}\in {\rm d}\tilde{y}) & = \frac{2}{\pi}\sin(\pi\alpha/2)\left(r^2-r_n^2\right)^{-\alpha/2} r_n^{\alpha}\,\frac{{\rm d}r}{r} \times \frac{{\rm d}\theta}{2\pi}\,,\qquad r>r_n.\label{DISPOL} \end{aligned}$$ From , we see that the angle $\theta$ is sampled uniformly on $[0,2\pi]$ whereas we can sample the radius $r$ via the inverse-transform sampling method. To this end, noting that $\sin(\pi\alpha/2)B(\alpha/2, 1-\alpha/2)=\pi$, the first factor on the right-hand side of is the density of a distribution with cumulative distribution function $F$. The inverse of $F$ can be identified as follows: For $x\in[0,1]$, $$\begin{aligned} F^{-1}(x)=r_n\left(I^{-1}( 1-x;\alpha/2,1-\alpha/2))\right)^{-1/2}, \end{aligned}$$ where $I^{-1}(x;z,w)$ is the inverse of the incomplete beta function $$I(x;z,w)\coloneqq\frac{1}{B(z,w)}\int_{0}^{x}u^{z-1}(1-u)^{w-1}\,{\rm d}u, \qquad x\in [0,1],$$ and $B(z,w)\coloneqq \int_{0}^{1}u^{z-1}(1-u)^{w-1}\,{\rm d}u$ is the beta function. The homogeneous part of the solution to is somewhat easier to compute than the inhomogeneous part, which additionally involves numerical computation of the integral $r_n^{\alpha}V_1(0,{f}(\rho_n+r_n\cdot))$ in . To develop this expression, we use the substitution $u=(1-t)/t$ for the integral in and hence, when $d = 2$, for $|y|<1$, $$V_1(0,{\rm d}y)=c_{2,\alpha}B(1-\alpha/2,\alpha/2)|y|^{\alpha-2}(1-I(|y|^2;1-\alpha/2,\alpha/2))$$ with $c_{2,\alpha}=2^{-\alpha}\pi^{-1}\Gamma(\alpha/2)^{-2}$. Moreover, by converting to polar coordinates $(r,\theta)$, the simulated quantity at step $n$ becomes $$\begin{aligned} & r_n^{\alpha}V_1(0,{f}(\rho_n+r_n\cdot)) \\ & =r_n^{\alpha}c_{2,\alpha}B(1-\alpha/2,\alpha/2) \int_{|y|<1} {f}(\rho_n+r_ny)|y|^{\alpha-2}\left(1-I(|y|^2;1-\alpha/2,\alpha/2)\right)\,{\rm d}y\\ & =r_n^{\alpha} c_{2,\alpha}B(1-\alpha/2,\alpha/2)2\pi \alpha^{-1}\int_{0}^{1}\int_{-\pi}^{\pi} {f}(\rho_n+r_n r(\cos\theta,\sin\theta)) \\ & \hspace{7cm}\times\left(1-I(r^2;1-\alpha/2,\alpha/2)\right)\frac{{\rm d}\theta}{2\pi}\times \alpha r^{\alpha-1}\,{\rm d}r. \end{aligned}$$ We used the Monte Carlo approach for evaluating this integral. Consider independent random variables $\Theta\sim U(-\pi,\pi)$ and $R=X^{1/\alpha}$ such that $X\sim U(0,1)$. Then $R$ has the probability density function $f_R(r)=\alpha r^{\alpha-1}$ and we want to evaluate $$r_n^{\alpha}V_1(0,{f}(\rho_n+r_n\cdot))=a_{2,\alpha} r_n^\alpha \mathbb{E}\left[\left(1-I(R^2;1-\alpha/2,\alpha/2)\right){f}(\rho_n+r_n R(\cos\Theta,\sin\Theta))\right] \label{MCEXP}$$ with $a_{2,\alpha}=\alpha^{-1}2^{-\alpha+1}\Gamma(\alpha/2)^{-2}B(1-\alpha/2,\alpha/2)$. We simulate $n_{R,\Theta}$ samples of pairs $(R,\Theta)$ and compute the sample mean of the quantity in . The quantity is evaluated more efficiently by writing $${f}(\rho_n+r_n R(\cos\Theta,\sin\Theta))= {f}(\rho_n)+[{f}(\rho_n+r_n R(\cos\Theta,\sin\Theta))-{f}(\rho_n)] \label{[]}$$ This gives two terms: one can be evaluated directly (by storing $\mathbb{E} [1-I(R^2; 1-\alpha/2, \alpha/2)]$) and the second can be evaluated using a Monte Carlo method, but with smaller variance (as the quantity in square brackets in is $\order{r_n}$). It is worth noting that a similar mixed approach using the trapezoidal rule over $\theta$ and randomising $r$ as earlier for evaluating the left-hand side of was also tested. However, results showed that the pure Monte Carlo approach is, in comparison to the mixed one, superior with regards to accuracy and computational cost. With this view, we decided to focus on the first one. Accuracy of this algorithm and its feasibility of implementation was checked with model solutions to problems of the type and they are presented below in order. Free-space Green’s function --------------------------- The free-space Green’s function for the fractional Laplacian $(-\Delta)^{\alpha/2}$ is $$G(x, y)=c_{d,\alpha} \frac{1}{|x-y|^{\alpha-d}}$$ for a constant $c_{d,\alpha}$ for $d>1$ and $\alpha\in(0,2$); see [@bucur]. If the point $y$ is chosen outside a domain $D$, then we can construct $G$ as an exact solution to the homogeneous version of the fractional Dirichlet problem in ; that is, $u(x)=G(x,y)$ for $x\in D$ and ${g}(x)=G(x,y)$ for $x\not\in D$. Figure \[fig:test4\] shows the results of applying the walk-on-spheres algorithm to evaluate $u(0.6, 0.6)$ with $10^6$ samples, where $D$ is a unit ball in $\mathbb{R}^2$ centred at the origin and $y=(2,0)$. We observe the samples ${g}(\rho_{N})$ have larger variance when $\alpha$ is small and a larger error results from the same number of samples. ![Example simulation for with exterior data ${g}(x)=G(x,y)$ with $y=(2,0)$ on the domain given by the unit ball centred at the origin, based on $10^6$ samples. The left-hand plot shows the relative error and the right-hand plot shows the sample variance. The sample variance is larger for small $\alpha$ as the process stops further away from the boundary and can see the singularity at $(2,0)$ in the exterior data. Accordingly, the relative error is higher as we are using a fixed number of samples.[]{data-label="fig:test4"}](test4_fig5 "fig:") ![Example simulation for with exterior data ${g}(x)=G(x,y)$ with $y=(2,0)$ on the domain given by the unit ball centred at the origin, based on $10^6$ samples. The left-hand plot shows the relative error and the right-hand plot shows the sample variance. The sample variance is larger for small $\alpha$ as the process stops further away from the boundary and can see the singularity at $(2,0)$ in the exterior data. Accordingly, the relative error is higher as we are using a fixed number of samples.[]{data-label="fig:test4"}](test4_fig4 "fig:")\ Gaussian data ------------- For the Poisson problem , we take $D$ to be the unit ball in $\mathbb{R}^2$, exterior data $${g}(x)=\exp(-| x-y|^2), \qquad x\in D^{\rm c},$$ for a given $y\in \mathbb{R}^2$, and zero source term ${f}=0$. We can represent the solution to in $D$ by $$u(x) =\pi^{-2} \sin(\pi \alpha/2) \int_{D^{\texttt{c}}} \left( \frac{1-|x|^2} {|y|^2-1} \right)^{\alpha/2} \frac{1} {|y-x|^2} \exp(-|x-y|^2) \,{\rm d}y, \qquad x\in D. \label{eq:ana_T4.4}$$ This integral can be computed numerically via a quadrature approximation. Here, instead of a fixed number of samples, the number of samples is taken adaptively based on a tolerance $\varepsilon$ for the computed sample standard deviation. Figure \[fig:test5\] shows the results with $y=(2,0)$ and tolerance $\varepsilon=10^{-4}$ for evaluation of $u(0.6,0.6)$ as previously. The estimator standard deviation and absolute error exhibit no obvious trend, whereas the sample variance peaks at about $\alpha=0.6$. Also at this value, the largest number of samples is needed to satisfy the tolerance. Despite the sample variance decreasing after $\alpha=0.6$, there is an increasing trend in the amount of work required. This implies that the increase in the number of steps with $\alpha$ (see Figure \[fig:steps\]) dominates and therefore a solution point of accuracy $10^{-4}$ is computationally more costly for larger values of $\alpha$. ![Example simulation with the walk-on-spheres algorithm for based on desired tolerance of $10^{-4}$. From top left to bottom right, we see the standard deviation of the estimator, the sample variance, the absolute error (using a quadrature approximation for for the reference value), and the amount work (number of samples $\times$ mean number of steps).[]{data-label="fig:test5"}](test5_fig3 "fig:") ![Example simulation with the walk-on-spheres algorithm for based on desired tolerance of $10^{-4}$. From top left to bottom right, we see the standard deviation of the estimator, the sample variance, the absolute error (using a quadrature approximation for for the reference value), and the amount work (number of samples $\times$ mean number of steps).[]{data-label="fig:test5"}](test5_fig4 "fig:")\ ![Example simulation with the walk-on-spheres algorithm for based on desired tolerance of $10^{-4}$. From top left to bottom right, we see the standard deviation of the estimator, the sample variance, the absolute error (using a quadrature approximation for for the reference value), and the amount work (number of samples $\times$ mean number of steps).[]{data-label="fig:test5"}](test5_fig2 "fig:") ![Example simulation with the walk-on-spheres algorithm for based on desired tolerance of $10^{-4}$. From top left to bottom right, we see the standard deviation of the estimator, the sample variance, the absolute error (using a quadrature approximation for for the reference value), and the amount work (number of samples $\times$ mean number of steps).[]{data-label="fig:test5"}](test5_fig7 "fig:") Non-constant source term ------------------------ Suppose that, again in the context of , we again take $D$ to be equal to the unit ball and the source term equal to $$ {f}(x)=2^{\alpha} \Gamma(2+\alpha/2) \Gamma(1+\alpha/2) (1-(1+\alpha/2) \|{x}\|^2), \qquad x\in D,$$ and zero exterior data ${g}=0$. This has the exact solution $u(x)=\max\{0, 1-\|x\|^2\}^{1+\alpha/2}$; cf. [@Dyda2012-kl]. The behaviour of the algorithm is shown in Figure \[fig:dyda\]. As expected, we again observe no obvious trend in estimator standard deviation and absolute error. The sample variance of sums of Monte Carlo-generated integrals increases with $\alpha$ as does the number of samples accordingly. Work required grows with $\alpha$ as in Figure \[fig:test5\], but with a slightly steeper trend. Notice that accuracy of $10^{-4}$ for the inhomogeneous part of the solution would demand a lot more work than the homogeneous part in Figure \[fig:test5\]. ![Example simulation with the walk-on-spheres algorithm for based on desired tolerance of $10^{-3}$ and $n_{R,\Theta}=1000$. From top left to bottom right, we see the standard deviation of the estimator, the sample variance, the absolute error, and the amount work (number of samples $\times$ mean number of steps).[]{data-label="fig:dyda"}](test3_fig3 "fig:") ![Example simulation with the walk-on-spheres algorithm for based on desired tolerance of $10^{-3}$ and $n_{R,\Theta}=1000$. From top left to bottom right, we see the standard deviation of the estimator, the sample variance, the absolute error, and the amount work (number of samples $\times$ mean number of steps).[]{data-label="fig:dyda"}](test3_fig4 "fig:")\ ![Example simulation with the walk-on-spheres algorithm for based on desired tolerance of $10^{-3}$ and $n_{R,\Theta}=1000$. From top left to bottom right, we see the standard deviation of the estimator, the sample variance, the absolute error, and the amount work (number of samples $\times$ mean number of steps).[]{data-label="fig:dyda"}](test3_fig2 "fig:") ![Example simulation with the walk-on-spheres algorithm for based on desired tolerance of $10^{-3}$ and $n_{R,\Theta}=1000$. From top left to bottom right, we see the standard deviation of the estimator, the sample variance, the absolute error, and the amount work (number of samples $\times$ mean number of steps).[]{data-label="fig:dyda"}](test3_fig7 "fig:") ![‘Swiss cheese’ domain (interior of balls).[]{data-label="fig:cheese"}](swiss_cheese_domain) Distribution of the number of steps in convex and non-convex domains -------------------------------------------------------------------- In previous sections, a large focus was put on deriving upper bounds and limiting distributions for $N$. Here we provide numerical support for these theoretical results. The walk-on-spheres algorithm was simulated inside a unit-ball domain centred at the origin as well as inside a domain of a hundred touching unit balls centred at points $(i,j)$, $i,j=-10,\dots,10$, the so-called ‘Swiss cheese’ domain as shown in Figure \[fig:cheese\]. The first represents a convex domain whereas the latter a non-convex one. The algorithm was started at a point $x=(\sqrt{0.29},-\sqrt{0.7})$ which lies very close to the boundary in both domains. This point was chosen as numerical simulations in a unit-ball domain revealed that the mean number of steps decreases with increasing distance from the boundary of the starting point. Theorem \[main\] states that $N$ is stochastically dominated by a geometric distribution with parameter $p(\alpha,d)$. In two dimensions, we are able to numerically compute $p(\alpha,2)$ since it is the solution to with $D = B(0, 1)$, ${g}(x)=\mathbf{1}_{\{x_1<-1\}}(x)$ and zero source term ${f}=0$ as deduced from Corollary \[indicators\]. We computed values of $p(\alpha,2)$ for different $\alpha$ to accuracy $10^{-4}$. The left-hand histogram in Figure \[fig:hist\] confirms stochastic dominance of $\Gamma$ and an exponentially decaying tail as stated in Remark 2 of Theorem \[main\]. However, the right-hand histogram shows that this statement fails in the particular example of the Swiss cheese domain. Moreover, the plot of the mean number of steps against $\alpha$ in Figure \[fig:steps\] shows the observed value of $\mathbb{E}_x[N]$ is bounded above by $1/p(\alpha,2)$ for the unit-ball domain. On the other hand, this is not the case for the Swiss cheese domain, where the observed value of $\mathbb{E}_x[N]$ exceeds $1/p(\alpha,2)$ for $\alpha$ in the range $(0.3,1.6)$. An explanation for why this is happening might be as follows. At larger values of $\alpha$, the path of $X$ starts resembling that of a Brownian motion (albeit with a countable infinity of arbitrarily small discontinuities). The process $X$ is started inside a ball in the Swiss cheese. When it exits this ball, its exit position is relatively close to the boundary with high probability. Therefore the exit point of the ball containing the point of issue is more likely to be in the ‘cheese’ (which would cause an end to the algorithm) and less likely to be inside another vacuous ball. Accordingly, $\mathbb{E}_x[N]$ does not deviate largely from the example of a single ball. However, for small values of $\alpha$, exit points from the sphere containing the point of issue have a higher probability to be far from the boundary, landing inside another vacuous ball, thereby requiring the algorithm to continue. In that case, the comparison with the case of exiting a single sphere breaks down. ![Histogram of the proportion of runs of the walk-on-spheres algorithm with $\alpha=1$ for which $N>n$ for the unit-ball domain (left) and the Swiss cheese domain (right). The red curve shows the tail of Geom($p(1,2)$), this is $(1-p(1,2))^n$, as in Remark 2 of Theorem \[main\].[]{data-label="fig:hist"}](test1_hist2 "fig:") ![Histogram of the proportion of runs of the walk-on-spheres algorithm with $\alpha=1$ for which $N>n$ for the unit-ball domain (left) and the Swiss cheese domain (right). The red curve shows the tail of Geom($p(1,2)$), this is $(1-p(1,2))^n$, as in Remark 2 of Theorem \[main\].[]{data-label="fig:hist"}](test6_hist2 "fig:") ![Mean number of steps for the walk-on-spheres algorithm started at $x=(\sqrt{0.29},-\sqrt{0.7})$ inside the circle domain (left) and inside the Swiss cheese domain (right). The dashed curve on both plots is $1/p(\alpha,2)$ as in Corollary \[indicators\].[]{data-label="fig:steps"}](steps_compare_ball "fig:") ![Mean number of steps for the walk-on-spheres algorithm started at $x=(\sqrt{0.29},-\sqrt{0.7})$ inside the circle domain (left) and inside the Swiss cheese domain (right). The dashed curve on both plots is $1/p(\alpha,2)$ as in Corollary \[indicators\].[]{data-label="fig:steps"}](steps_compare_cheese "fig:") Appendix: Proof of Theorem \[hasacorr\] {#appendix-proof-of-theorem-hasacorr .unnumbered} ======================================= Our proof of Theorem \[hasacorr\] uses heavily the joint conclusion of Theorems 2.10 and 3.2 in [@bucur], namely that the Theorem \[hasacorr\] is true in the case that $D$ is a ball. Our proof is otherwise constructive proving existence and uniqueness separately. [*Existence:*]{} On account of the fact that $D$ is bounded, we can define a ball of sufficiently large radius $R>0$, say $B^* = B(X_0, R)$, centred at $X_0$, such that $D$ is a subset of $B^*$ and hence $\sigma_D \leq \sigma_{B^*}$ almost surely, irrespective of the initial position of $X$, where $\sigma_{B^*} = \inf\{t>0\colon X_t\not\in B^*\}$. In particular, thanks to stationary and independent increments, this upper bound for $\sigma_D$ does not depend on $X_0$ in law and $\sup_{x\in D}\mathbb{E}_x[\sigma_D]\leq \mathbb{E}_0[\sigma_{B^*}]<\infty$. Define for convenience $\upsilon(x)={g}(x)$ for $x\in D^\mathrm{c}$ and $$\upsilon(x) = \mathbb{E}_x[{g}(X_{\sigma_D})] + \mathbb{E}_x\left[\int_0^{\sigma_D} {f}(X_s)\,{\rm d}s\right], \qquad x\in D, \label{v}$$ where ${g}$ and ${f}$ satisfy the assumptions of the theorem. We want to prove that $\upsilon$ is bounded and continuous on $\overline{D}$. For the boundedness of $\upsilon$, we prove the boundedness of the two expectations in its definition. First note that, for all $x\in D$, $$\begin{aligned} \mathbb{E}_x\bp{|{g}(X_{\sigma_D})|} & = \mathbb{E}_x\bp{\abs{{g}(X_{\sigma_D})}\mathbf{1}_{(\sigma_D = \sigma_{B^*})}} + \mathbb{E}_x\bp{\abs{{g}(X_{\sigma_D})}\mathbf{1}_{(\sigma_D <\sigma_{B^*})}} \notag \\ & \leq \mathbb{E}_x\bp{\abs{{g}(X_{\sigma_{B^*}})}} + \sup_{x\in B^*\backslash D}|{g}(x)|\notag \\ & =\mathbb{E}_0\bp{|{g}(x + B^*X_{\sigma_{B(0,1)}})|} + \sup_{x\in B^*\backslash D}|{g}(x)|\notag \\ & =\pi^{-(d/2+1)}\,\Gamma(d/2)\,\sin(\pi\alpha/2)\, \int_{|y|>1} \frac{|{g}(x + B^* y)|}{\left|1-|y|^2\right|^{\alpha/2}|y|^{d}} \,{\rm d}y+ \sup_{x\in B^*\backslash D}|{g}(x)|\notag \\ & =C \int_{\mathbb{R}^d} \frac{|{g}(z)|}{1+|z|^{d+\alpha}} \,{\rm d}y+ \sup_{x\in B^*\backslash D}|{g}(x)|<\infty, \label{worksforsquared} \end{aligned}$$ for some constant $C\in(0,\infty)$ that does not depend on $x$ (this is ensured thanks to the boundedness of $D$). In the inequality, we have used the fact that, on $\{\sigma_D <\sigma_{B^*}\}$, we have $X_{\sigma_D}\in B^*\backslash D$, moreover, that, as a continuous function on $\mathbb{R}^d$, ${g}$ is bounded in $B^*\backslash D$. In the second equality, we have used spatial homogeneity and the scaling property of stable processes. In the third equality, we have used Theorem \[BGR\]. The fourth equality follows by changing variables to $z = x + B^*y$ in the integral, appropriately estimating the denominator and the assumption that ${g}$ is continuous and in $L^1_\alpha(D^\mathrm{c})$. The boundedness of ${f}$ on $D$ and the uniform finite mean of $\sigma_D$ ensures that the second expectation in the definition of $\upsilon$ is bounded on $\overline{D}$. We claim that $\upsilon$ is continuous in $\mathbb{R}^d$ and belongs to $L^1_\alpha(\mathbb{R}^d)$. Continuity of $\upsilon$ follows thanks to path regularity of $X$, the continuity of ${g}$, the openness of $D$ and the fact that $\omega\mapsto X_{\sigma_D}(\omega)$ and $\omega\mapsto\int_0^{\sigma_D(\omega)} {f}(X_s(\omega))\,{\rm d}s$ are continuous in the Skorohod topology (for which it is important that $\omega\mapsto\sigma_D(\omega)$ is finite). Continuity is also a consequence of the classical potential analytic point of view, seeing the identity for $\upsilon$ in in terms of Riesz potentials; see for example the classical texts of [@BH] or [@L] To check that $\upsilon\in L^1_\alpha(\mathbb{R}^d)$, we need some estimates. For $x\in D^{\rm c}$, $\upsilon(x) = {g}(x)$ and hence, as ${g}\in L^1_\alpha(D^\mathrm{c})$, it suffices to check that $ \int_{D}|\upsilon(x)|/(1+|x|^{\alpha + d})\,{\rm d}x<\infty.$ However, this is trivial on account of the boundedness and continuity of $\upsilon$ on $\overline D$. Now fix $x'\in D$ and let $B(x')$ be the largest ball centred at $x'$ that is contained in $D$. A simple application of the strong Markov property tells us that $$\begin{aligned} \upsilon(x) & = \mathbb{E}_x\left[\mathbb{E}\left[\left.{g}(X_{\sigma_D}) + \int_0^{\sigma_{B(x')}} {f}(X_s)\,{\rm d}s + \int_{\sigma_{B(x')}}^{\sigma_D} {f}(X_s)\,{\rm d}s\right|\mathcal{F}_{\sigma_{B(x')}}\right]\right] \notag\\ & = \mathbb{E}_x\left[\upsilon (X_{\sigma_{B(x')}}) + \int_0^{\sigma_{B(x')}} {f}(X_s)\,{\rm d}s\right], \qquad x\in D, \label{localisedv} \end{aligned}$$ where $(\mathcal{F}_t, t\geq 0)$ is the natural filtration generated by $X$. Thanks to the fact that Theorem \[hasacorr\] is valid on balls, we see immediately that the right-hand side of is the unique solution to $$\begin{gathered} \begin{aligned} -(-\Delta u)^{\alpha/2}u(x) & =-{f}(x), & \qquad x & \in B(x'), \\ u(x) & = \upsilon(x), & x & \in B(x')^{\rm c}. \end{aligned} \label{upsilondirichlet} \end{gathered}$$ That is to say, $\upsilon$ solves . Note that it is at this point in the argument that we are using the condition ${f}\in C^{\alpha +\varepsilon}(\overline{D}) $. Since the solution to is defined on $B(x')$ and $x'$ is chosen arbitrarily in $D$, we conclude that $\upsilon$ solves $$\begin{gathered} \begin{aligned} -(-\Delta u)^{\alpha/2}u(x) & =-{f}(x), & \qquad x & \in D, \\ u(x) & = \upsilon(x), & x & \in D^{\rm c}. \end{aligned} \label{upsilondirichletonD} \end{gathered}$$ On account of the fact that $\mathbb{P}_x(\sigma_D =0) = 1$ for all $x\in D^{\rm c}$, it follows that $\upsilon = {g}$ on $D^{\rm c}$ and hence is identical to . [*Uniqueness:*]{} Suppose that $\hat{u}$ solves , then, in particular, for any $x'\in D$, it must solve $$\begin{gathered} \begin{aligned} -(-\Delta u)^{\alpha/2}u(x) & =-{f}(x), & \qquad x & \in B(x'), \\ u(x) & = \hat{u}(x), & x & \in B(x')^{\rm c}. \end{aligned} \end{gathered}$$ As we know the Feynman–Kac representation of the solution to the above fractional Poisson problem, thanks to Theorem 3.2 in [@bucur] for domains which are balls, we are forced to conclude that $$\hat{u}(x) = \mathbb{E}_x\left[\hat{u} (X_{\sigma_{B(x')}}) + \int_0^{\sigma_{B(x')}} {f}(X_s)\,{\rm d}s\right], \qquad x\in B(x'),\quad x'\in D. \label{fixedpointonspheres}$$ Here again, we are implicitly using that ${f}\in C^{\alpha +\varepsilon}(\overline{D})$ in the application of Theorem 3.2 of [@bucur]. Let us now appeal to the same notation we have used for the walk-on-spheres. Specifically, recall the sequential exit times from maximally sized balls $\sigma_{B_k}$ for the walk-on-spheres which were defined in Section \[WoSfL\]. We claim that $$M_k\eqqcolon \hat{u} (X_{\sigma_{B_k}\wedge \sigma_D}) + \int_0^{\sigma_{B_k}\wedge \sigma_D }{f}(X_s)\,{\rm d}s, \qquad k \geq 0,$$ is a martingale. To see why, note that, by the strong Markov property and then by , $$\begin{aligned} \mathbb{E}\left[M_{k+1}|\mathcal{G}_{k}\right] = & \, \mathbf{1}_{\{k<N\}}\left\{\left.\mathbb{E}_{x}\left[ \hat{u} (X_{\sigma_{B(x)}}) + \int_0^{\sigma_{B(x)}}{f}(X_s)\,{\rm d}s \right]\right|_{x = \smash{X_{\sigma_{B_k}}}} + \int_0^{\sigma_{B_k}}{f}(X_s)\,{\rm d}s\right\}\\ & + \mathbf{1}_{\{k\geq N\}}\left\{ \hat{u} (X_{\sigma_D}) + \int_0^{ \sigma_D }{f}(X_s)\,{\rm d}s \right\}\\ = & \, \mathbf{1}_{\{k<N\}}\left\{\hat{u}(X_{\sigma_{B_k}}) + \int_0^{\sigma_{B_k}}{f}(X_s)\,{\rm d}s\right\}+ \mathbf{1}_{\{k\geq N\}}\left\{ \hat{u} (X_{\sigma_D}) + \int_0^{ \sigma_D }{f}(X_s)\,{\rm d}s \right\}\\ = & \, \hat{u} (X_{\sigma_{B_k}\wedge \sigma_D}) + \int_0^{\sigma_{B_k}\wedge \sigma_D }{f}(X_s)\,{\rm d}s \\ = & \, M_k, \qquad k\geq 1, \end{aligned}$$ where $\mathcal{G}_k = \mathcal{F}_{\sigma_{B_k} \wedge\sigma_D}$, $k\geq 1$. For consistency, we may define $M_0 = \mathbb{E}_x[M_k] = \hat{u}(x)$ thanks to . Next, we appeal to the definition of $B^*$ and, in particular, that $\sigma_D\leq \sigma_{B^*}$, as well as the continuity of $\hat{u}$ to deduce that, for all $k\geq 0$, $$\begin{aligned} & \left|\hat{u} (X_{\sigma_{B_k}\wedge \sigma_D}) + \int_0^{\sigma_{B_k}\wedge \sigma_D }{f}(X_s)\,{\rm d}s\right| \\ & \leq \left|\hat{u} (X_{\sigma_{B^*}})\right|\,\mathbf{1}_{\{\sigma_{B_k}\wedge \sigma_D = \sigma_{B^*}\}} +\sup_{y\in B^*}\left|\hat{u} (y)\right|\,\mathbf{1}_{\{\sigma_{B_k}\wedge \sigma_D < \sigma_{B^*}\}} +\sup_{y\in D} \left|{f}(y)\right|\,\sigma_{D}\\ & \leq \left|{g}(X_{\sigma_{B^*}})\right| + c_1 + c_2\sigma_{B^*}, \end{aligned}$$ where $c_1,c_2$ are constants. We know that for each fixed $x\in D$, $\mathbb{E}_x[\sigma_{B^*}]<\infty$ and, moreover, from Theorem \[BGR\], after scaling (see for example ), $\mathbb{E}_x[ |{g}(X_{\sigma_{B^*}})| ]<\infty$ as ${g}\in L^1_\alpha(D^\mathrm{c})$. Dominated convergence allows us to deduce that $(M_k, k\geq 0)$ is a uniformly integrable martingale such that, for each fixed $x\in D$, $$\begin{aligned} \hat{u}(x) & = \lim_{k\to\infty}\mathbb{E}_x[M_k] \\ & = \mathbb{E}_x[\lim_{k\to\infty}M_k] \\ & = \mathbb{E}_x\left[\hat{u} (X_{\sigma_D}) + \int_0^{ \sigma_D }{f}(X_s)\,{\rm d}s\right] \\ & = \mathbb{E}_x\left[{g}(X_{\sigma_D}) + \int_0^{ \sigma_D }{f}(X_s)\,{\rm d}s\right], \end{aligned}$$ where in the final equality we have used that $\hat{u} = {g}$ on $D^{\rm c}$. Uniqueness now follows. $\square$ Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Mateusz Kwaśniki for pointing out a number of references to us and Alexander Freudenberg for a close reading of an earlier version of this manuscript. [^1]: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK. [^2]: Supported by EPSRC grant EP/L002442/1. [^3]: Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK. Email: `a.kyprianou@bath.ac.uk`, `anaosojnik@gmail.com`, `t.shardlow@bath.ac.uk`.

--- abstract: 'We study a problem of interacting fractional charges with $J_1$-$J_2$-$J_3$ Ising model on a checkerboard lattice under magnetic field. As a result of the interplay between repulsive interactions and particle density tuning by magnetic field, the fractional charges form a classical spin liquid (CSL) phase. The CSL phase is composed of degenerate spin configurations, which can be mapped to the trimer covering of dual square lattice. The CSL state shows macroscopic ground-state entropy, implying the emergence of novel quantum spin liquid phase when quantum fluctuations are turned on. In addition to the CSL phase, the system exhibits multiple magnetization plateaus, reflecting the fertile screening processes of dimer-monomer mixtures.' author: - Kunio Tokushuku - Tomonari Mizoguchi - Masafumi Udagawa bibliography: - 'checkerboard.bib' title: Trimer classical spin liquid from interacting fractional charges --- Introduction ============ Fractionalization is one of the central topics in condensed matter physics. In many-body interacting systems, low-energy excitations are usually described as quasiparticles, whose nonuniversal parameters, such as mass, have renormalized values, while their quantum numbers are preserved to be the same as those of the original particles. Fractionalization changes this canonical description considerably, by allowing the particles to split into subunits with smaller quantum numbers. Fractional excitations are hosted in a number of systems, such as one-dimensional quantum liquid[@Bethe1931; @mourigal2013fractional; @PhysRevB.41.2326; @kitaev2001unpaired], fractional quantum Hall systems[@laughlin1983anomalous; @tsui1982two], and quantum spin liquids (QSLs) [@kitaev2006anyons; @knolle2014dynamics; @doi:10.1146/annurev-conmatphys-020911-125058; @balents2010balents]. Among them, in the candidate compounds of QSLs, the role of fractional excitations has been highlighted in their thermodynamic and transport behaviors[@tokiwa2016tokiwa; @pan2016measure; @PhysRevX.1.021002; @kimura2013quantum; @Sibille2018; @PhysRevLett.122.117201; @wan2016spinon; @kourtis2016free; @PhysRevLett.120.167202; @PhysRevB.96.195127; @banerjee2016proximate; @banerjee2017neutron; @2017NatPh..13.1079D; @kasahara2018majorana; @knolle2014dynamics]. In particular, the relevant role of fractional excitations is established in classical spin liquids (CSLs). CSL corresponds to a high-temperature precursor of QSL, composed of a macroscopic number of degenerate classical states. As lowering temperatures, coherency develop among the classical states, turning CSL into QSL. One can find several essential properties of QSL already show up in CSL. Among other things, fractional excitations can be defined in CSL, usually as static objects. While the appearance of fractionalization itself requires interaction between original particles, the assembly of fractional particles also hosts nontrivial many-body problems. The problem of interacting fractional particles is usually quite difficult to treat theoretically, however, at the level of CSL, rigorous theoretical treatments are sometimes possible. In a class of CSLs defined on frustrated magnets, we can clearly divide the role of interactions: the nearest-neighbor interaction leads to the formation of CSL with emergent fractional excitations, while the farther-neighbor interactions give rise to the interaction between the fractional particles. A typical example can be found in spin ice, a material realization of three-dimensional Coulomb phase, where a fractionalized particle, called monopole, dominates the low-energy properties of the system[@castelnovo2008magnetic]. Indeed, in dipolar spin ice, monopoles exhibit interesting many-body effects, due to their Coulombic attractive interactions proportional to $-1/r$[@castelnovo2008magnetic]. This Coulomb attraction gives rise to a liquid-gas-type phase transition elusive in a magnetic system[@PhysRevLett.90.207205]. This phase transition is controlled by magnetic field as a tuning parameter, which is translated into a chemical potential of monopoles. This analogy naturally leads to the plasma description of long-distance physics, where the screening of charge plays an important role in the thermodynamic behavior of the system. It is also interesting to turn a look at fractional particles under short-range interactions. In this case, the system is more susceptible to the hardcore constraint arising from the local charge conservation. In a variant of spin-ice-type frustrated magnet, described by $J_1$-$J_2$-$J_3$ Ising model, again the nearest-neighbor interaction ($J_1$) leads to the formation of CSL, and the second- to third-neighbor interactions ($J_2, J_3$) give rise to the short-range interactions between them[@PhysRevLett.119.077207]. In this system, the fractional excitations have magnetic charges as well-defined quantum numbers, and they satisfy a conservation law. In conventional electromagnetism, same-sign charges repel each other. Meanwhile in this emergent Coulomb phase, same-sign charges sometimes attract each other. However, the conservation law strictly forbids the same-sign charges to pair-annihilate. This keen competition between the conservation law and the interactions leads to the formation of novel classical spin liquids[@PhysRevLett.119.077207; @PhysRevB.94.104416; @rau2016spin], excitations [@mizoguchi2018magnetic; @PhysRevB.98.140402], and dynamics [@PhysRevB.94.104416]. The stream of results motivates us to look into the magnetization process of this CSL. Rich behaviors of the magnetization process have been explored so far in magnetic systems, especially with geometrical frustration[@PhysRevB.67.104431; @PhysRevB.96.180401; @PhysRevLett.102.137201; @PhysRevB.94.075136; @PhysRevB.62.15067; @PhysRevLett.112.127203; @PhysRevB.83.100405; @PhysRevLett.108.057205; @PhysRevLett.110.267201; @doi:10.1143/JPSJ.69.1016; @PhysRevLett.111.137204; @PhysRevB.87.214424; @doi:10.7566/JPSJ.83.113703]. In general, the magnetic field affects the monopoles as a staggered potential and facilitates the pair-creation of monopoles. The interaction between the induced monopoles may lead to a formation of novel CSL with macroscopic degeneracy. In fact, a topologically ordered state is proposed at a low-field plateau of the kagome Heisenberg magnet[@nishimoto2013controlling]. In this paper, we investigate a magnetic phase diagram and magnetization processes of the $J_1$-$J_2$-$J_3$ Ising model on a checkerboard lattice, which is a two-dimensional analog of the pyrochlore lattice. We obtain a ground state phase diagram analytically up to small positive farther-neighbor interactions, and find that there is a series of exotic phases which are characterized by magnetic plateaus. Remarkably, at the $1/3$-magnetization plateau, we find a CSL phase, where the whole system is tiled with magnetic trimers. This state is a new-type of CSL with novel value of residual entropy. It implies that a new class of QSL’s emerges upon adding perturbations to this CSL state. The rest of this paper is organized as follows. In Sec. \[sec:model\], we present our model and methods, with a special focus on Gauss’ law, which is central to the rigorous argument we base the existence of the CSL on. In Sec. \[sec:Phase diagram\], we show the overall ground-state phase diagram and present the results at simple limiting cases. In Sec. \[sec:tCSL\], we show the existence of the trimer classical spin liquid from both the intuitive and rigorous arguments, and detail its properties. In Sec. \[sec:magnetization\], we consider a full magnetization process and show a variety of commensurate phases, which are characterized by magnetic plateaus at $M=0, 1/5, 1/3, 1/2, 2/3, 3/4$ and $1$. Finally, in Sec. \[sec:summary\], we summarize our results and provide some future perspectives. Model and Method \[sec:model\] ============================== Model ----- ![(a) A schematic picture of checkerboard lattice. The definition of $J_1$, $J_2$ and $J_3$ couplings are shown here: the site $O$ connects with red, green and blue sites with interactions, $J_1$, $J_2$, and $J_3$, respectively. The plaquettes are divided into two sublattices. The shaded (white) plaquettes belong to the sublattice A (B). (b) A dual square lattice obtained by replacing a plaquette with a site, and by connecting neighboring plaquettes with a bond. Shaded (blank) sites correspond to the plaquettes on the sublattice A (B). Magnetic charge is defined on each site of this dual lattice, and neighboring charges interact through the coupling $J$, as defined in Eq. (\[eq:defJ\]). []{data-label="fig:interaction"}](Fig1.eps){width="\hsize"} We consider the $J_1$-$J_2$-$J_3$ Ising model on a checkerboard lattice (Fig. \[fig:interaction\]). The checkerboard lattice is a corner-sharing network of square units with diagonal bonds, which we simply call “plaquettes". We consider the network composed of $N_p$ plaquettes, i.e., $2N_p$ spins, in a periodic boundary condition in both directions. The Hamiltonian of our model is defined as $$\begin{aligned} \mathcal{H}=& J_1 \sum_{\langle i,j \rangle_{\rm{n.n.}}} \sigma^{z}_i\sigma^{z}_j +J_2 \sum_{ \langle i,j \rangle_{\rm{2nd}}} \sigma^{z}_i\sigma^{z}_j \nonumber \\ &+J_3\sum_{\langle i,j \rangle_{\rm{3rd}}} \sigma^{z}_i\sigma^{z}_j - h\sum_{i}\sigma^{z}_i, \label{eq:hamiltonian}\end{aligned}$$ where $\sigma^{z}_i=\pm1$ are Ising spins and $h$ represents an external magnetic field. Throughout this paper, we consider the case of $h\geq0$ without loss of generality. We assume the nearest-neighbor (n.n.) coupling as antiferromagnetic and set its value as a unit energy ($J_1$=1). The n.n. coupling accounts for the interaction within the plaquettes. In addition to $J_1$, we introduce $J_2$ as the diagonal interaction, and $J_3$ as the other next-nearest-neighbor interactions, as shown in Fig. \[fig:interaction\]. Throughout the paper, we fix the ratio of $J_2$ and $J_3$ as $$\begin{aligned} J_2= 2J,\ \ J_3=J. \label{eq:defJ}\end{aligned}$$ This point is of special importance, since the model can be mapped to the Hamiltonian of interacting magnetic charges with a staggered potential. To describe the mapping, we first introduce a magnetic charge: $$\begin{aligned} Q_p=\eta_p S_p,\label{eq:defcharge}\end{aligned}$$ where $p$ is a label of plaquettes and we define the total spin of the plaquette $p$, as $$\begin{aligned} S_p=\sum_{i\in p}\sigma^{z}_i,\label{eq:deftotalspin}\end{aligned}$$ and the sign factor of the plaquette as $$\begin{aligned} \eta_p= \{ \begin{array}{ll} 1 & (\mathrm{for}\; A\; \mathrm{sub.}) \\ -1 & (\mathrm{for}\; B\; \mathrm{sub.}) \end{array}. \end{aligned}$$ Here, we took the dual picture and regarded the checkerboard lattice as a square lattice of plaquettes, and made the bipartite decomposition of the plaquettes into sublattice $A$ and $B$ as shown in Fig. \[fig:interaction\]. Note that there are five possible values of $Q_p$: $Q_p=0, \pm2, \pm4$ (Fig. \[fig:TopoCharge\]). Among them, we especially call the plaquette with $Q_p=0$, a vacuum plaquette, and that with $|Q_p|=4$ as a double charge. Since the neighboring plaquettes share one spin, the charges are not independent of each other. Indeed, the magnetic charge has a conserved nature, and its distribution is a subject to the lattice analog of Gauss’ law, as we will show later. In terms of $Q_p$, we can rewrite the Hamiltonian (\[eq:hamiltonian\]) as $$\begin{aligned} \mathcal{H}=\left( \frac{1}{2} -J \right) \sum_{p}Q_p^2-J\sum_{\langle p,q \rangle}Q_pQ_q - \frac{h}{2}\sum_p\eta_pQ_p + C_1, \label{eq:DumbbelHamiltonianMag}\end{aligned}$$ where $$\begin{aligned} C_1=2(J-1)N_p.\end{aligned}$$ We show the detail of this transformation in Appendix A. In Eq. (\[eq:DumbbelHamiltonianMag\]), the first term is a “self-energy" of the charges which is proportional to $Q_p^2$. The second term is the interaction between magnetic charges on nearest-neighbor plaquettes. The coupling constant of this interaction is $J$. Accordingly, same-sign charges repel (attract) each other for $J<0$ ($J>0$). The third term is a staggered potential for charges, which arises from the magnetic field. Gauss’ law ---------- The magnetic charge, $Q_p$, satisfies the conservation law. To see this, suppose that $D$ is an arbitrary set of plaquettes and $\partial D$ denotes the boundary sites of $D$ (see Fig. \[fig:gausslaw\] ). Then, the total magnetic charges inside $D$ and the spins on $\partial D$ satisfy the following relation: $$\begin{aligned} \sum_{p\in D}Q_p=\sum_{i \in \partial D}\eta_{p_{D(i)}}\sigma^z_i. \label{eq:gausslaw}\end{aligned}$$ Here, the boundary site belongs to the two plaquettes, one inside, and one outside $D$, and $p_{D(i)}$ stands for the former. Equation (\[eq:gausslaw\]) means that the total charge in a certain region equals the sum of “flux", $\eta_{p_{D(i)}}\sigma^z_i$, on its boundary. This is nothing but a lattice analog of Gauss’ law. This Gauss’ law constrains the structure of charge configuration in the system. In particular, as developed in Ref. , this law leads to the triangle inequality: $$\begin{aligned} |\sum_{p\in D}Q_p|\leq\sum_{i \in \partial D}|\eta_{p_{D(i)}}\sigma^z_i|=N_{\partial D}. \label{eq:gaussinequality}\end{aligned}$$ Namely, the amount of total charge an arbitrary region $D$ can store is bounded by $N_{\partial D}$, the number of its boundary sites. Phase diagram \[sec:Phase diagram\] =================================== Before going into the details of analyses, we first show the overall phase diagram of the present model in Fig. \[fig:PhaseDiagram\]. Each phase corresponds to a magnetization plateau, which we characterize by the magnetization per spin, $M\equiv\frac{1}{2N_p}\sum_{i}\sigma_i^z$. Here, we focus on the region up to small positive $J$. In this section, we consider the simple limiting cases, and first discuss the stability of the Coulomb phase at $h=0$, then consider the magnetization process for $J\leq0$. We also give a brief introduction to the ground states for $J>1/4$ and $J<-1/2$. ![(a) The relation between the value of magnetic charge and spin configurations on each sublattice. The red and blue circles mean up and down spins, respectively. Colors of plaquettes represent values of magnetic charges. (b)-(d) The schematic picture of ground state spin configurations in the absence of magnetic field: (b) $-1/2<J<1/4$, (c) $J<-1/2$ and (d) $J>1/4$.[]{data-label="fig:TopoCharge"}](Fig2.eps){width="\hsize"} ![ Schematic picture of the notation used to describe Gauss’ law Eq. (\[eq:gausslaw\]). The black line represents the boundary, $\partial D$, and the region inside corresponds to the domain, $D$. On the boundary, the sites with $\eta_{p_{D(i)}}=+1 (-1)$ are marked with circles (squares). We have seven up and two down spins marked with circles, while two up and three down spins marked with squares. Accordingly, we have $\sum_{i \in \partial D}\eta_{p_{D(i)}}\sigma^z_i=7\times(+1)+(-2)\times(+1)+2\times(-1)+(-3)\times(-1)=6$. The domain $D$ involves one plaquette with $Q_p=+4$, four plaquettes with $Q_p=+2$, three plaquettes with $Q_p=0$, and three plaquettes with $Q_p=-2$. Accordingly, we obtain $\sum_{p\in D}Q_p=1\times(+4)+4\times(+2)+3\times(0)+3\times(-2)=6$, which is equal to $\sum_{i \in \partial D}\eta_{p_{D(i)}}\sigma^z_i$. []{data-label="fig:gausslaw"}](Fig10.eps){width="0.7\hsize"} ![The ground state phase diagram of the Hamiltonian, Eq. (\[eq:DumbbelHamiltonianMag\]), up to small positive $J$, obtained by the analytical arguments in Secs. \[sec:tCSL\] and \[sec:magnetization\]. []{data-label="fig:PhaseDiagram"}](Fig3.eps){width="\hsize"} Zero-field states \[sec:zerofield\] ----------------------------------- Let us start with the ground states in the absence of magnetic field. Without magnetic field, Hamiltonian (\[eq:DumbbelHamiltonianMag\]) is simplified to $$\begin{aligned} \mathcal{H}=\left( \frac{1}{2}-J \right) \sum_{p}Q_p^2-J\sum_{\langle p,q \rangle}Q_pQ_q \label{eq:DumbbelHamiltonian}.\end{aligned}$$ ### $J=0$: square ice For $J=0$, where there is no interaction between charges, the model is reduced to the square-ice model only with n.n. exchange interactions whose ground state is given by the charge vacuum ($Q_p = 0$ for every $p$), or the two-dimensional Coulomb phase \[Fig. \[fig:TopoCharge\](b)\], which is a typical CSL state with macroscopic degeneracy. Lieb’s rigorous argument gives the exact value of residual entropy as $\mathcal{S}_0^{\rm SI}=\frac{1}{2}\ln W=0.215566427$ per spin, with $W=\frac{4}{3}^{\frac{3}{2}}$ [@PhysRevLett.18.692]. ### $J<0$: staggered charge ordering {#sec:negativeJ_nofield} For $J<0$, where an attractive interaction acts between opposite charges, the first term in Eq. (\[eq:DumbbelHamiltonian\]) favors a vacuum, whereas the second term favors a staggered charge ordering. To see this competition clearly, we transform Eq. (\[eq:DumbbelHamiltonian\]) into $$\begin{aligned} \mathcal{H}=\left(\frac{1}{2}+J\right)\sum_{p}Q_p^2+\frac{|J|}{2}\sum_{\langle p,q \rangle}(Q_p+Q_q)^2. \label{eq:DumbbelHamiltonian2}\end{aligned}$$ With this form, one can minimize the first and the second terms simultaneously, by setting $Q_p = -Q_q$ for $p\in$ A sub. and $q\in$ B sub., and $|Q_p|= 0\ (4)$ for $J > -1/2$ ($J < -1/2$). This solution means the Coulomb phase extends to $J = -1/2$, while the staggered magnetic charge ordering with $|Q_p| = 4$ takes over for $J < -1/2$. In the spin language, the staggered charge ordering with $|Q_p| = 4$ corresponds to the fully-polarized ferromagnetic state \[Fig. \[fig:TopoCharge\](c)\]. ### $J > 0$: long-period phase In contrast to the case of $J<0$, same-sign charges attract with each other for $J>0$. As a result, the Coulomb phase survives only up to $J=1/4$, and then the system turns to a complicated long-period spin ordering for $J>1/4$, as in the case of pyrochlore model[@PhysRevB.94.104416]. To obtain an insight into the ground state for positive $J$, let us rewrite the Hamiltonian Eq. (\[eq:DumbbelHamiltonian\]) as $$\begin{aligned} \mathcal{H}=\left(\frac{1}{2}-3J\right)\sum_{p}Q_p^2+\frac{J}{2}\sum_{\langle p,q \rangle}(Q_p-Q_q)^2. \end{aligned}$$ This expression shows the Coulomb phase is stable at least up to $J=1/6$ from a similar argument of $J<0$. It also implies proliferation of double charges at larger $J$. At first sight, it is preferable to cover the system with same signs of double charges for large $J$, however, according to the Gauss’ law, the same-sign-charge cluster accommodates at most one double charge [@PhysRevLett.119.077207]. As a result, the ordered phase contains mixed values of charges as shown in Fig. \[fig:TopoCharge\](d). The precise value of the critical point of $J=1/4$ can be obtained by the softening of the lowest excitations, in the same argument as in pyrochlore model [@PhysRevB.94.104416]. The formal proof based on the Gauss’ law is rather involved. We summarize it in Appendix B. Magnetization process at $J\leq 0$ {#sec:Mprocess_negativeJ} ---------------------------------- Next, let us look at the magnetization process for $J\leq 0$. In this case, simple and rigorous arguments are available for the magnetization process, including the simplest non-interacting limit, $J=0$. In this region, it is convenient to rewrite the Hamiltonian (\[eq:DumbbelHamiltonianMag\]) as $$\begin{aligned} \mathcal{H}=\left(\frac{1}{2} + J\right) \sum_p(Q_p-\eta_p S_h)^2-\frac{J}{2}\sum_{\langle p,q \rangle}\left( Q_p+Q_q \right)^2 + C_2,\label{eq:hhamiltonian2}\end{aligned}$$ with $C_2=\frac{N_p}{8}\bigl[16J-16-\frac{h^2}{1+2J}\bigr].$ Here, we define $$\begin{aligned} S_h\equiv\frac{h/2}{1+2J},\end{aligned}$$ Then, in analogy with the argument in Sec. \[sec:negativeJ\_nofield\], the ground state can be obtained by minimizing the two terms in Eq. (\[eq:hhamiltonian2\]), simultaneously. It is possible by minimizing $|Q_p-\eta_pS_h|$ while keeping the staggered charge alignment $Q_p+Q_q=0$ for any neighboring plaquettes, $p$ and $q$. ### Square ice: $0<h<2 + 4J$ For small $h$, we find that the zero-field square ice, or the Coulomb phase, extends in the region: $S_h<1$ ($h<2 + 4J$). ### Dimer phase: $2 +4J <h<6 +12J$ For $S_h\geq 1$, the system goes out of the Coulomb phase, and the staggered charge ordering appears. This region is divided into two cases. First, for $1<S_h<3$, (i.e., $2 +4J <h<6 +12J$), $Q_p$ is given by $$\begin{aligned} Q_p= \{ \begin{array}{ll} +2 & \mathrm{for}\; A\ \mathrm{sub.} \\ -2 & \mathrm{for}\; B\ \mathrm{sub.} \end{array}.\end{aligned}$$ This phase corresponds to the half-magnetization plateau with $M=1/2$. While this phase has the staggered charge ordering, it still keeps macroscopic degeneracy in spin degrees of freedom, forming a CSL state. Its ground-state manifold can be mapped to the dimer covering problem, in a similar way to kagome ice[@matsuhira2002new; @doi:10.1143/JPSJ.71.2365; @isakov2004magnetization]. To see this, let us consider a dual square lattice again, and place a dimer on each down spin. Then, each spin configuration on this half-magnetization plateau can be mapped to a dimer configuration on the dual square lattice (Fig. \[fig:Kagomeice\_mapping\]). The residual entropy can be exactly obtained to be $\mathcal{S}_0^{\rm hmp}=0.14578045$ per spin by counting the dimer configuration on a square lattice [@PhysRev.124.1664]. ![(a) A schematic figure of spin configuration on the half-magnetization plateau and its dimer mapping. The yellow and green plaquettes support the magnetic charge, $Q_p=+2 $ and $-2$, as introduced in Fig. \[fig:TopoCharge\]. (b) The dimer configuration on a dual square lattice, corresponding to the spin configuration in (a). []{data-label="fig:Kagomeice_mapping"}](Fig4.eps){width="\hsize"} ### The fully-polarized state: $6+12J < h$ Finally, for $3 < S_h$, ($6+12J < h$), $Q_p$ is given by $$\begin{aligned} Q_p= \{ \begin{array}{ll} +4 & \mathrm{for}\; A\ \mathrm{sub.} \\ -4 & \mathrm{for}\; B\ \mathrm{sub.} \end{array},\end{aligned}$$ which corresponds to a fully-polarized ferromagnet. Trimer CSL {#sec:tCSL} ========== At $J=0$, the system makes a direct transition from the zero-field square ice phase to the dimer phase at $h=2$. At $J>0$, between these two CSL’s, there appears another type of CSL, which we call “trimer CSL" after its structure. In this section, we will address how the dimer phase makes instability to this trimer CSL, by the two kinds of strategies: one is based on the instability analysis due to the creation of vacuum plaquette, which is more intuitive and gives a clearer picture of this trimer CSL, and the other is a rigorous argument based on Gauss’ law. Instability of the dimer phase ------------------------------ ### Nucleation of vacuum plaquette We start with the instability analysis of the dimer phase, as the magnetic field decreases. Here, we adopt the magnetic charge representation of the Hamiltonian, Eq. (\[eq:DumbbelHamiltonianMag\]), and estimate the critical magnetic field. At half-magnetization plateau, we have $|Q_p|=2$, for all the plaquettes. As the magnetic field decreases, we expect the vacuum plaquettes to nucleate. To examine this process, we take one upward spin and flip it downward, and create a pair of vacuum plaquettes. The nucleated plaquettes are dissociated, and then they are individually screened by the charged plaquettes \[Fig. \[fig:trimercovering\] (a)\], to maximize the energy gain from the interaction term. From Eq. (\[eq:hhamiltonian2\]), we can estimate the energy increase associated with this process: $$\begin{aligned} \Delta E= 4\bigl[(1+2J)S_h - (1+6J)\bigr],\end{aligned}$$ which leads to the instability at $S_h=\frac{1+6J}{1+2J}$, or $$\begin{aligned} h=2+12J,\end{aligned}$$ below which the dimer phase becomes unstable against the creation of vacuum plaquettes. ![(a) A dissociated vacuum plaquette surrounded by charged plaquettes. The yellow and green plaquettes support the magnetic charges, $Q_p=+2$ and $-2$, as introduced in Fig. \[fig:TopoCharge\]. (b) Dimer representation of the spin configuration shown in (a). The vacuum plaquette can be interpreted as an overlapping part of the two dimers. []{data-label="fig:trimercovering"}](Fig5.eps){width="\hsize"} ### Formation of the trimer CSL Next, let us look into the state just below the critical field. In this state, all the vacuum plaquettes are surrounded by four charged plaquettes with $|Q_p|=2$. See the structure inside the magenta square in Fig. \[fig:trimercovering\](a). Since this structure locally optimizes the energy, it is desirable to tile the whole lattice with as many of this local structure as possible. However, it is not straightforward to obtain the optimal tiling pattern. To gain an insight, let us adopt a dimer representation we have introduced in Sec. \[sec:Mprocess\_negativeJ\], and place a dimer on each downward spin \[Fig. \[fig:trimercovering\](b)\]. From this viewpoint, a vacuum plaquette can be regarded as an overlapping part of two dimers. Then the tiling problem can be interpreted as fully packing the dual square lattice with overlapping dimers, or “trimers", under the condition that the overlapping part (i.e., a vacuum plaquette) does not neighbor with each other (for an example of prohibited structure, see Fig. \[fig:prohibited\_trimer\]). The packing with trimers is in sharp contrast to the dimer covering problem that appears on the half-magnetization plateau. As we numerically estimate in the next subsection, the number of possible trimer configurations increases macroscopically as system size, suggesting that the state forms a CSL. Accordingly, we name this CSL as “trimer classical spin liquid" (tCSL) after its structure. ![(a) An example of prohibited trimer arrangement in the tCSL state. The magenta circle represents a touching between two neighboring vacuum plaquettes (“vacuum touching"). Bold lines show the boundary between clusters $D_+$ and $D_-$. The yellow, gray, and green plaquettes support the magnetic charges, $Q_p=+2, 0$ and $-2$, as introduced in Fig. \[fig:TopoCharge\]. (b) Dimer representation of the spin configuration shown in (a). []{data-label="fig:prohibited_trimer"}](Fig11.eps){width="\hsize"} ### Characterization of trimer CSL In this section, we will look into the detailed character of tCSL. Firstly, the formation of tCSL is associated with the magnetization plateau. The picture of trimer covering immediately leads to the magnetization value $M=1/3$ per spin, since one out of the three plaquettes forming a trimer has the total spin $S_p=0$, while the other two have $S_p=2$. Comparing with the fully polarized ferromagnet, where $S_p=4$ for all the plaquettes, we can easily obtain $M=(0+2+2)/(4\cdot3)=1/3$. Secondly, what is the ground state energy of tCSL? To see this, again from the structure of a trimer, the number of vacuum plaquettes is $N_0=N_p/3$, while that of charged plaquette is $N_2=2N_p/3$. Since each vacuum plaquette contacts with four charged plaquettes, the number of bonds connecting the vacuum and charged plaquettes is $4N_0=\frac{4}{3}N_p$. Since the total number of bonds is $2N_p$, and no bonds connect two vacuum plaquettes, the number of bonds connecting the two charged plaquettes is $2N_p-\frac{4}{3}N_p=\frac{2}{3}N_p$. Inputting all these into Eq. (\[eq:DumbbelHamiltonianMag\]), we obtain the energy of tCSL: $$\begin{aligned} E_{\rm tCSL}=\frac{2}{3}(3J-h-1)N_p. \label{eq:energy_tCSL}\end{aligned}$$ Thirdly, how large is the ground state degeneracy? To gain an insight into the origin of degeneracy, let us look at one typical trimer configuration in Fig. \[fig:trimer\_cluster\]. Looking at this configuration, one can find a system is divided into large-scale charge clusters. In a positive (negative) charge cluster, the charged plaquettes are placed only on A (B) sublattice, and the vacuum plaquettes are placed on B (A) sublattices. Inside one cluster, one finds a staggering pattern of charged and vacuum plaquettes. Only at the cluster boundaries, the charged plaquettes neighbor with each other. Meanwhile the vacuum plaquettes are never adjacent to each other. ![ (a) An example of spin configuration in the tCSL state. Bold lines show the boundary between clusters $D_+$ and $D_-$. The yellow, gray, and green plaquettes support the magnetic charge, $Q_p=+2, 0$ and $-2$, as introduced in Fig. \[fig:TopoCharge\]. (b) The trimer covering representation corresponding to the configuration shown in (a). []{data-label="fig:trimer_cluster"}](Fig9.eps){width="\hsize"} This cluster structure implies there are two origins for the ground state degeneracy: the contribution from cluster placements, and the internal spin configuration within a cluster. The existence of these two types of configurational entropy is in common with the hexamer CSL found in the cousin system of $J_1$-$J_2$-$J_3$ Ising model on the Kagome lattice, where a novel value of residual entropy was found, as well as a characteristic spin correlation with iconic half-moon pattern in magnetic structure factor[@PhysRevLett.119.077207]. Here, in order to obtain the precise value of residual entropy, we resort to a transfer matrix method, by adopting a finite strip of checkerboard lattice with variable widths $L$, which is the number of plaquettes in the column of the strip (see Fig. \[fig:interaction\]), up to 7. The detail of transfer matrix method is summarized in Appendix C. We listed the values of residual entropy $\mathcal{S}_0$ for each width on Table \[table:residualentropy\]. While the obtained value still fluctuates around $L\sim7$, they are well within the window of $\mathcal{S}_0=0.136\pm0.04$, giving the evidence for the existence of finite residual entropy. The value $\mathcal{S}_0\simeq0.136$ is smaller compared with the zero-field value, $\mathcal{S}_0^{\rm SI}=0.21556643$, but comparable to that of the dimer phase, $\mathcal{S}_0^{\rm hmp}=0.14578045$. Previously, the configurational entropy was obtained for a trimer covering problem in several contexts[@frobose1996orientational; @ghosh2007random; @lee2017resonating; @dong20183]. In Ref. , only the angular trimers are considered. The obtained value is, if translated into the current context, $\mathcal{S}_0^{\rm ang}\sim0.13846575$ per spin, comparable to our value of $\mathcal{S}_0$. At first sight, it may seem strange that the limitation to angular trimers do not make difference in larger residual entropy. However, their model allows the touching of vacuum plaquette in our context, which seems to compensate for the entropy reduction due to the limitation of the type of trimers. In Ref. , on the other hand, the authors only discuss the covering by the line trimers, and they obtained $\mathcal{S}_0^{\rm line}\sim0.07926$. The covering problems by all species of trimers are considered in Ref. . They obtained $\mathcal{S}_0^{\rm tot}\sim0.20597$. The difference between $\mathcal{S}_0^{\rm tot}$ and our $\mathcal{S}_0$ is attributed to the prohibition of vacuum touching. Width: $L$ Residual entropy ------------ ------------------ 2 0.14931329 3 0.15296086 4 0.13473800 5 0.13851784 6 0.13915249 7 0.13313547 : The width $L$ dependence of residual entropy per site, $\mathcal{S}_0$. []{data-label="table:residualentropy"} The finite residual entropy qualifies this phase as CSL. This trimer CSL is similar to the classical spin liquids found in kagome [@PhysRevLett.119.077207] and pyrochlore [@PhysRevB.94.104416] lattices previously in a sense that the same-charge attraction leads to these CSL’s. In these systems, one spin is fractionalized into a pair of magnetic charges, and the fractionalized charges make recombination into a new phase that cannot be easily inferred from the original spin degrees of freedom. While the trimer CSL has some similarities to the previous CSL’s as mentioned above, it shows a unique feature in its structural rule: the trimer covering. To the knowledge of the authors, the trimer model does not belong to known integrable models, implying a possibility that this trimer CSL may serve as a precursor of a new type of QSL. In this light, it is interesting to introduce quantum fluctuation to this system, in the form of, e.g., quantum transverse or exchange interactions. If QSL is actually realized on the basis of this trimer covering state, we expect it to show an anomalous magnetic charge correlation. In a trimer CSL, magnetic charges tend to make clusters, as mentioned above. If this tendency survives after the superposition of trimer configurations, it will present strong short-range charge correlation, quite distinct from the previously known class of QSL’s. Rigorous argument based on Gauss’ law {#sec:tCSL_fromGausslaw} ------------------------------------- In the previous subsection, we derived the structure of tCSL in an intuitive way, from the instability analysis of half-magnetization plateau. Here, we rigorously show that the tCSL state gives the ground state, on the basis of Gauss’ law we introduced as Eq. (\[eq:gausslaw\]). To this aim, we rewrite the Hamiltonian Eq. (\[eq:DumbbelHamiltonianMag\]) in a form, $$\begin{aligned} \mathcal{H} &= \left(\frac{1}{2}-J\right)\sum_p(Q_p-\eta_pS_h)^2\nonumber\\ &- J\sum_{\langle p,q\rangle}(Q_p-\eta_pS_h)(Q_q-\eta_qS_h) + C_3, \label{eq:DumbbelHamiltonianMag_modify}\end{aligned}$$ where $$\begin{aligned} C_3 =\frac{N_p}{8}\bigl[16J-16-\frac{h^2}{1+2J}\bigr]. \label{eq:DumbbelHamiltonianMag_const}\end{aligned}$$ Here, we limit ourselves to small positive $J$, and focus on the magnetic field just below the half-magnetization plateau, i.e., $h\sim2$. In the range of magnetic field under consideration, we can safely assume that all the plaquettes are occupied by 2-up 2-down or 3-up 1-down spin configuration, namely, $|Q_p|=0$ or $2$, and $Q_p = +2$ $(-2)$ exists only at the $A$ ($B$) sublattice. Under this assumption, let us define the number of charges with $|Q_p|=0\ (2)$ as $N_0$ ($N_2$), and the number of contacts between the plaquettes with $|Q_p|=0$ and $|Q_q|=2$ as $n_{20}$. Similarly, we define $n_{00}$ ($n_{22}$) as the number of contacts between two plaquettes with charge $0$ ($2$). With these quantities, we can express the Hamiltonian, Eq. (\[eq:DumbbelHamiltonianMag\_modify\]) as $$\begin{aligned} \mathcal{H}&=\left(\frac{1}{2}-J\right)S_h^2N_0+\left(\frac{1}{2}-J\right)(S_h-2)^2N_2 \nonumber \\ &+JS_h(S_h-2)n_{20}+JS_h^2n_{00}+J(S_h-2)^2n_{22}. \label{eq:Ham_charge_contact}\end{aligned}$$ The variables in this Hamiltonian are subject to several geometrical constraints. Firstly, since each plaquette has four contacts with neighboring plaquettes on a dual square lattice, we have $$\begin{aligned} 4N_0 = 2n_{00} + n_{20}, \label{eq:Geometrical_equality_N0}\end{aligned}$$ and $$\begin{aligned} 4N_2 = 2n_{22} + n_{20}. \label{eq:Geometrical_equality_N2}\end{aligned}$$ Here, the factor 2 before $n_{00}$ and $n_{22}$ correct the double counting. Secondly, in the absence of double charges ($|Q_p|=4$), the sum of $N_0$ and $N_2$ is equal to the total number of plaquettes: $$\begin{aligned} N_0 + N_2 = N_p. \label{eq:Geometrical_equality_Np}\end{aligned}$$ Combining Eqs. (\[eq:Geometrical\_equality\_N0\])-(\[eq:Geometrical\_equality\_Np\]), the contact numbers satisfy $$\begin{aligned} n_{00} + n_{22} + n_{20} = 2N_p. \label{eq:contact_sumrule}\end{aligned}$$ With Eqs. (\[eq:Geometrical\_equality\_N0\]) and (\[eq:Geometrical\_equality\_N2\]), we can eliminate $N_0$ and $N_2$ from the Hamiltonian (\[eq:Ham\_charge\_contact\]), and obtain $$\begin{aligned} \mathcal{H} &= \bigl[\frac{1}{2}(\frac{1}{2}+J)S_h^2\bigr]n_{00}+\bigl[\frac{1}{2}(\frac{1}{2}+J)(S_h-2)^2\bigr]n_{22} \nonumber \\ &+\bigl[\frac{1}{2}(\frac{1}{2}+J)(S_h-1)^2+\frac{1}{4}-\frac{3}{2}J\bigr]n_{20}\nonumber\\ &=a_{00}n_{00} + a_{22}n_{22}+ a_{20}n_{20}, \label{eq:Ham_contact}\end{aligned}$$ where $a_{00}$, $a_{22}$ and $a_{20}$ are coefficients of $n_{00}$, $n_{22}$ and $n_{20}$. Accordingly, the search for the ground state is now reduced to finding the combination of $(n_{00}, n_{22}, n_{20})$ to minimize $\mathcal{H}$, under the constraint of sum rule, Eq. (\[eq:contact\_sumrule\]). The coefficients $a_{00}$, $a_{22}$ and $a_{20}$ are plotted in Fig. \[fig:schema\_coef\]. For $h>2+12J$, $a_{22}$ is the smallest. This region corresponds to the half-magnetization plateau, where all the plaquettes are occupied with charges $|Q_p|=2$, and all the contacts are of $2$-$2$ type, accordingly. Meanwhile, this plot shows that larger $n_{20}$ is preferable for $2-4J<h<2+12J$. To satisfy this condition, at first sight, the best strategy seems to put $Q_p=2$ on all the plaquettes of A sublattice, while $Q_p=0$ on B sublattice, to make all the contacts to be of 2-0 type. However, this charge configuration obviously violates the Gauss’ law, Eq. (\[eq:gausslaw\]). ![The coefficients of the Hamiltonian, Eq. (\[eq:Ham\_contact\]). For $2-4J<h<2+12J$, the coefficient of $n_{20}$ is lower than those of $n_{00}$ and $n_{22}$.[]{data-label="fig:schema_coef"}](Fig6.eps){width="\hsize"} To find the optimal charge configuration under the Gauss’ constraint, let us define a positive (negative) charge cluster $D_+$ ($D_-$), as a maximal set of plaquettes with $Q_p=2$ placed on the A(B)-sublattice and those with $Q_p=0$ placed on the B(A)-sublattice \[Fig. \[fig:trimer\_cluster\] (a)\]. Namely, $$\begin{aligned} \mathrm{cluster\:}D_+:Q_p= \{ \begin{array}{ll} +2 & (\mathrm{for}\; A\; \mathrm{sub.}) \\ 0 & (\mathrm{for}\; B\; \mathrm{sub.}) \end{array}, \\ \mathrm{cluster\:}D_-:Q_p= \{ \begin{array}{ll} 0 & (\mathrm{for}\; A\; \mathrm{sub.}) \\ -2 & (\mathrm{for}\; B\; \mathrm{sub.}) \end{array}.\end{aligned}$$ With these definitions, clusters $D_+$ and $D_-$ always touch with each other through the 0-0 or 2-2 contacts. To see this, suppose a cluster $D_+$ has a boundary plaquette that belongs to A (B) sublattice, then it must have charge $|Q_p|=2$ (0). This plaquette neighbors with a plaquette of $D_-$ on the B (A) sublattice, which has charge $|Q_p|=2$ (0). Now, let us apply the Gauss’ law to a cluster, $D_{\alpha}$ of either type. We assume that the number of charged plaquettes inside $D_{\alpha}$ to be $N_2^{\alpha}$, and define the number of 0-0 and 2-2 contacts with neighboring clusters to be $n_{00}^{\alpha}$ and $n_{22}^{\alpha}$, respectively. The Gauss’ inequality, Eq. (\[eq:gaussinequality\]) leads to $$\begin{aligned} 2N_2^{\alpha}=|\sum_{p\in D_{\alpha}}Q_p|\leq\sum_{i\in\partial D_{\alpha}}|\eta_{p}\sigma^z_i|=n_{22}^{\alpha}+n_{00}^{\alpha}. \label{eq:application_Gaussinequality}\end{aligned}$$ By summing over all the clusters in the system, we obtain $$\begin{aligned} 2N_2\leq 2(n_{00}+n_{22}). \label{eq:Gauss_constraint}\end{aligned}$$ Note that the factor 2 of the right-hand side comes from the double counting of bonds in the summation over clusters. Combining Eqs. (\[eq:Geometrical\_equality\_N2\]) and (\[eq:Gauss\_constraint\]), we obtain $$\begin{aligned} n_{20}\leq 4n_{00} + 2n_{22}, \label{eq:Gauss_contact_inequality}\end{aligned}$$ Now, considering the relative magnitudes of coefficients depicted in Fig. \[fig:schema\_coef\], at $h\lesssim2+12J$, Eq. (\[eq:Gauss\_contact\_inequality\]) results in the optimal solution to be $n_{00}=0$ and $n_{20}=2n_{22}$. If a certain configuration satisfies this condition, it gives one of the ground states. In fact, this condition is equivalent to the trimer covering we discussed in the previous subsection. By inserting this condition into Hamiltonian (\[eq:Ham\_charge\_contact\]) with the constant term $C_3$ given by Eq. (\[eq:DumbbelHamiltonianMag\_const\]), we obtain $$\begin{aligned} E_{\rm tCSL}=\frac{2}{3}(3J-h-1)N_p, \label{eq:GSenergy}\end{aligned}$$ which exactly corresponds to what we obtained from the trimer covering picture: Eq. (\[eq:energy\_tCSL\]). This means that the tCSL states give the ground state. Conversely, it is possible to show that any member of the ground state manifold can be expressed by the trimer covering, i.e., the ground state is composed only of the tCSL states. To prove this, it is enough to show that if $n_{00}=0$ and $n_{20}=2n_{22}$ are satisfied, the corresponding charge configuration can be expressed in terms of the trimer covering. To see this, suppose a cluster $D_{\alpha}\in D_+$. To satisfy the former condition, one needs $n_{00}^{\alpha}=0$, and in addition to that, the equality must hold in the Gauss’ inequality, Eq. (\[eq:application\_Gaussinequality\]): $$\begin{aligned} 2N_2^{\alpha}=n_{22}^{\alpha}=\sum_{i\in\partial D_{\alpha}}\sigma_i^z,\end{aligned}$$ This equality requires the boundary spins to satisfy $\sigma_i^z=+1$, and the cluster $D_{\alpha}$ to neighbor with other clusters with charged plaquette. Accordingly, within the cluster $D_{\alpha}$, each charged plaquette shares its only down spin with its neighboring vacuum plaquette. Moreover, each vacuum plaquette shares its two down spins with two of its neighboring charged plaquettes. Consequently, if one places a dimer on each down spin, all the charged plaquettes are covered with one dimer, and all the vacuum plaquettes are covered with two dimers, resulting in a trimer covering. The same argument holds for a cluster $D_{\alpha}\in D_-$. Full magnetization process \[sec:magnetization\] ================================================ In this section, we will address the rest of magnetic phase diagram shown in Fig. \[fig:PhaseDiagram\] for $J>0$. We limit ourselves to the region of small $J$, again. We show the schematic picture of spin configurations at each phase in Fig. \[fig:M\_high\], and the magnetization process in Fig. \[fig:magnetization\_process\]. ![ The schematic pictures of spin configurations at each phase, corresponding to the magnetization plateaus at (a) $M=1$, (b) $M=3/4$, (c) $M=2/3$, (d) $M=1/2$, (e) $M=1/3$, (f) $M=1/5$, and (g) $M=0$. []{data-label="fig:M_high"}](Fig8.eps){width="\hsize"} ![The magnetization process for small $J>0$. There are seven phases that have $M=0, 1/5, 1/3, 1/2, 2/3, 3/4$ and $1$. The corresponding transition points are $h=2-4J, 2+6J, 2+12J, 6+4J, 6+10J$ and $6+18J$. Shaded areas are guides for the eyes.[]{data-label="fig:magnetization_process"}](Fig7.eps){width="\hsize"} Instability of high-field phase ------------------------------- We start with the instability of the high-field fully-polarized phase \[Fig. \[fig:M\_high\](a)\], with decreasing $h$. In the fully-polarized phase, all the plaquettes have total spins $S_p=4$. A single spin flip makes an adjacent pair of plaquettes with $S_p=2$. The energy increase accompanying this process is $$\begin{aligned} \Delta E= 4\bigl[(1+2J)(S_h-3)-3J\bigr].\end{aligned}$$ This instability occurs for $\Delta E<0$, i.e. $$\begin{aligned} h<6+18J.\end{aligned}$$ Below this boundary line, the system tries to maximize the number of neighboring pairs of plaquettes with $S_p=4$ and $S_p=2$. Consequently, we obtain the 4-plaquette ordering as shown in Fig. \[fig:M\_high\](b). This ordered phase corresponds to the $M=3/4$ plateau. High-field instability of dimer phase ------------------------------------- In the previous section, we addressed the low-field instability of half-magnetization plateau into tCSL. Here, we address the instability with an increasing magnetic field. In the half-magnetization plateau, all the plaquettes have $S_p=2$ uniformly. As the magnetic field increases, we expect a nucleation of double charge, $S_p=4$. The nucleation process takes qualitatively different form from the nucleation of $S_p=2$ plaquette from the fully-polarized phase. In the fully-polarized phase, the nucleated $S_p=2$ plaquettes are always paired. In contrast, in the half-magnetization plateau, the nucleation of $S_p=4$ plaquettes occurs in pair, but they can be dissociated from each other. This is a sort of fractionalization, which reflects the deconfining nature of the dimer phase at the half-magnetization plateau. As a result, the most economical excitation is a single plaquette of $S_p=4$ surrounded by $S_p=2$ plaquettes. This object costs energy $$\begin{aligned} \Delta E=4\bigl[(1+2J)(3-S_h)-4J\bigr],\end{aligned}$$ which becomes negative, if $$\begin{aligned} h>6+4J.\end{aligned}$$ Above this field, the spin configuration, as shown in Fig. \[fig:M\_high\](c), is stabilized. If we stand on the dimer picture by placing a dimer on a down spin, the resultant state consists of an assembly of monomers screened by dimers. It follows naturally that the corresponding state makes a $M=2/3$ plateau. Most simply, three-plaquette ordering realizes this state \[Fig. \[fig:M\_high\](c)\], however, introduction of “stacking fault" does not increase the energy. Accordingly, this state shows a semi-macroscopic degeneracy of the order of $2^{L}$. Obviously, this $M=2/3$ plateau cannot be continuously connected to the 4-plaquette ordering of $M=3/4$ plateau, just below the saturated phase. The transition between the two states occurs at $h=6+10J$, from a simple comparison of the energies. Instability of square ice ------------------------- Finally, we address the instability of square ice state, as increasing $h$ from 0. To this aim, rather than the dimer representation, we resort to Gauss’ inequality, following the argument in Sec. \[sec:tCSL\_fromGausslaw\]. We start with Hamiltonian (\[eq:Ham\_contact\]). From their coefficients as shown in Fig. \[fig:schema\_coef\], it follows that $n_{22}$ should be suppressed for smaller magnetic field. Meanwhile larger $n_{20}$ is preferable for $h>2-4J$. Given that $n_{22}=0$, a set of geometrical equations, Eqs. (\[eq:Geometrical\_equality\_N0\]), (\[eq:Geometrical\_equality\_N2\]), and (\[eq:contact\_sumrule\]) lead to $$\begin{aligned} 4N_0=2n_{00}+n_{20}, \\ 4N_2=n_{20}, \\ n_{00}+n_{20}=2N_p. \label{eq:Geometrical_constraint_lowfield}\end{aligned}$$ The optimal value of $n_{20}$ is constrained by the Gauss’ inequality, Eq. (\[eq:Gauss\_constraint\]) as $$\begin{aligned} 2N_2\leq 2n_{00},\end{aligned}$$ which leads to the inequality, $$\begin{aligned} n_{20}\leq 4n_{00}, \label{eq:Gaussinequality_lowfield}\end{aligned}$$ combined with the geometrical constraint, Eq. (\[eq:Geometrical\_constraint\_lowfield\]). The optimal value of $n_{20}$ corresponds to the equality of Eq. (\[eq:Gaussinequality\_lowfield\]), which leads to $$\begin{aligned} n_{00}=\frac{2}{5}N_p,\ \ n_{20}=\frac{8}{5}N_p,\end{aligned}$$ and $$\begin{aligned} N_0=\frac{3}{5}N_p,\ \ N_2=\frac{2}{5}N_p.\end{aligned}$$ This state corresponds to the $M=1/5$ plateau, and we show the configuration in Fig. \[fig:M\_high\](f). Summary \[sec:summary\] ======================= We have studied a problem of interacting fractional charges, taking the $J_1$-$J_2$-$J_3$ Ising model on a checkerboard lattice under the magnetic field. We focused on the case, where the Hamiltonian can be written in terms of the charge degrees of freedom. In particular, at small positive $J$, we found that the half-magnetization plateau destabilizes into a classical spin liquid state, as decreasing the magnetic field. The resultant CSL is expressed as an assembly of trimer covering of the dual square lattice, and we called this state the tCSL. The tCSL state corresponds to the $1/3$ magnetization plateau, and has macroscopic ground state degeneracy, which is characterized by a novel value of residual entropy. In contrast to dimer covering, which is ubiquitous in a broad area of physics including statistical mechanics and condensed matter physics, the notion of trimer covering rarely appears. It is surprising that such elusive states can be obtained, by starting from a simple local Hamiltonian considered here. As turning on a quantum fluctuation, a novel quantum spin liquid state may further be stabilized based on a well-defined microscopic model. Moreover, we showed the interactions among magnetic charges lead to a variety of magnetization plateaus in the applied magnetic field, reflecting the rich screening processes of dimer-monomer mixtures. Nontrivial magnetization processes are observed for a number of frustrated magnetic systems. In this regard, this work shows that the picture of interacting fractional charges gives a new viewpoint to the formation of magnetization plateaus. This work was supported by the JSPS KAKENHI (Nos. JP15H05852, JP16H04026, and JP17H06138), MEXT, Japan. Part of numerical calculations were carried out on the Supercomputer Center at Institute for Solid State Physics, University of Tokyo. K. T. was supported by the Japan Society for the Promotion of Science through the Program for Leading Graduate Schools (MERIT). Mapping to charge Hamiltonian ============================= In this appendix, we derive the charge representation of Hamiltonian Eq. (\[eq:DumbbelHamiltonianMag\]) from the original Hamiltonian Eq. (\[eq:hamiltonian\]). First, we rewrite Eq. (\[eq:hamiltonian\]) with $(J_1,J_2,J_3) = (1,2J,J)$ as ![The definition of charges $p, q$ and their components spins $p_i$ and $q_i$. We define sites $[p,q]$ as be shared by two charges $p$ and $q$, therefore $[p,q]=p_4=q_1$.[]{data-label="fig:apeA"}](FigApeA.eps){width="0.7\hsize"} $$\begin{aligned} \mathcal{H}=&\frac{1}{2}\sum_p(\sigma_{p_1}^z+\sigma_{p_2}^z+\sigma_{p_3}^z+\sigma_{p_4}^z)^2-2N_p \nonumber \\ &+J\sum_{\langle p,q \rangle}(\sigma_{p_1}^z+\sigma_{p_2}^z+\sigma_{p_3}^z+\sigma_{p_4}^z-\sigma_{[p,q]}^z)\nonumber \\ &\times(\sigma_{q_1}^z+\sigma_{q_2}^z+\sigma_{q_3}^z+\sigma_{q_4}^z-\sigma_{[p,q]}^z)\nonumber \\ &-\frac{h}{2}\sum_p(\sigma_{p_1}^z+\sigma_{p_2}^z+\sigma_{p_3}^z+\sigma_{p_4}^z) \label{eq:hamiltonianApeA1},\end{aligned}$$ where we name the spins on the neighboring plaquettes, $p$ and $q$, as $p_1$-$p_4$ and $q_1$-$q_4$ respectively, as shown in Fig. \[fig:apeA\]. The symbol $[p,q]$ stands for the spin shared by $p$ and $q$. For instance, $[p,q]=p_4=q_1$ in Fig. \[fig:apeA\]. The first and the second term come from the nearest-neighbor interactions while the third term comes from the second-neighbor and the third-neighbor interactions which connect spins on neighboring plaquettes. The fourth term comes from magnetic field. We transform Eq. (\[eq:hamiltonianApeA1\]) by introducing the total spin of the plaquettes $S_p=\sum_{i\in p}\sigma^{z}_i$: $$\begin{aligned} \frac{1}{2}\sum_p(\sigma_{p_1}^z+\sigma_{p_2}^z+\sigma_{p_3}^z+\sigma_{p_4}^z)^2 =& \frac{1}{2}\sum_p S_p^2, \\ \frac{h}{2}\sum_p(\sigma_{p_1}^z+\sigma_{p_2}^z+\sigma_{p_3}^z+\sigma_{p_4}^z) =&\frac{h}{2}\sum_pS_p,\end{aligned}$$ and $$\begin{aligned} &J\sum_{\langle p,q \rangle}(\sigma_{p_1}^z+\sigma_{p_2}^z+\sigma_{p_3}^z+\sigma_{p_4}^z-\sigma_{[p,q]}^z)\nonumber \\ &\times(\sigma_{q_1}^z+\sigma_{q_2}^z+\sigma_{q_3}^z+\sigma_{q_4}^z-\sigma_{[p,q]}^z)\nonumber \\ =&J\sum_{\langle p,q \rangle}(S_p-\sigma_{[p,q]}^z)(S_q-\sigma_{[p,q]}^z)\nonumber \\ =&J\sum_{\langle p,q \rangle}(S_pS_q-S_p\sigma_{[p,q]}^z-S_q\sigma_{[p,q]}^z)+J\sum_{\langle p,q \rangle}1\nonumber \\ =&J\sum_{\langle p,q \rangle}S_pS_q-J\sum_{p}S_p\sum_{q\in (\mathrm{n.n.\: of\:} p)}\sigma_{[p,q]}^z+2JN_p \nonumber \\ =&J\sum_{\langle p,q \rangle}S_pS_q-J\sum_{p}S_p^2 +2JN_p.\end{aligned}$$ By introducing charge representation $Q_p=\eta_p S_p$, finally we obtain Eq. (\[eq:DumbbelHamiltonianMag\]): $$\begin{aligned} \mathcal{H}=\left( \frac{1}{2} -J \right) \sum_{p}Q_p^2-J\sum_{\langle p,q \rangle}Q_pQ_q - \frac{h}{2}\sum_p\eta_pQ_p + C_1, \end{aligned}$$ where $C_1 = 2(J-1)N_p$. Ordered phase at $h=0$ for $J>1/4$. =================================== In this appendix, we derive the phase boundary for $h=0$ and $J>0$, by evaluating the ground-state energy using the constraint of the Gauss’ law and some geometrical identities. We show the Coulomb phase survives up to $J=1/4$, and the ordered phase in Fig. \[fig:apeC\] realizes for $J>1/4$. \]. Bold line represents one of the same-sign-charge cluster $\mathcal{D}$. The site enclosed by orange (purple) circle is an example of a site counted as $n_{i:(2.4)}^{(\mathcal{D})}$ ($n_{b:(2.4)}^{(\mathcal{D})}$). []{data-label="fig:apeC"}](FigApeC.eps){width="\hsize"} The outline of the energy evaluation is as follows. First, we decompose an arbitrary charge configuration into assembly of maximally connected same-sign-charge clusters and vacuum plaquettes. Here the maximally connected same-sign-charge cluster means that it must be connected to opposite-sign-charges or vacuum plaquettes at the boundary. By doing so, the total energy is written by the sum of the intra-cluster energy and the inter-cluster energy. Then, we optimize the energy under the geometrical constraint and the Gauss’ law, to obtain the ground-state charge configuration. To begin with, let us address an essential property of the same-sign-charge cluster (labeled as $\mathcal{D}$), which originates from the Gauss’ law. Namely, applying the triangle inequality Eq. (\[eq:gaussinequality\]) to $\mathcal{D}$, we have $$\begin{aligned} 2 N_2^{(\mathcal{D})} + 4N_4^{(\mathcal{D})} \leq n_b^{(\mathcal{D})}, \label{eq:gauss_cluster}\end{aligned}$$ where $N_2^{(\mathcal{D})}$ ($N_4^{(\mathcal{D})}$) is a number of $|Q_p|=2$ ($|Q_p|=4$) plaquettes in $\mathcal{D}$, and $n_b^{(\mathcal{D})}$ denotes the number of boundary sites of $\mathcal{D}$. Having this at hand, we now evaluate the ground state energy. To do this, we introduce the following variables. (1) $n_{i:(\ell.\ell^\prime)}^{(\mathcal{D})}$: a number of inner sites of $\mathcal{D}$-clusters which is contact points of $|Q_p| = \ell$ and $|Q_p| = \ell^\prime$. (2) $n_{b:(\ell.\ell^\prime)}^{(\mathcal{D})}$: a number of boundary sites of $\mathcal{D}$-clusters at which a plaquette with $|Q_p| = \ell$ [*inside*]{} the cluster confronts another plaquette with $|Q_p| = \ell^\prime$ [*outside*]{} the cluster. See Fig. \[fig:apeC\] for examples of sites counted as $n_{i:(2.4)}^{(\mathcal{D})}$ and $n_{b:(2.4)}^{(\mathcal{D})}$. We note that the total number of inner sites $n_i^{(\mathcal{D})}$ and that of boundary sites $n_b^{(\mathcal{D})}$ satisfy $$\begin{aligned} n_i^{(\mathcal{D})}=&n_{i:(2.2)}^{(\mathcal{D})}+n_{i:(2.4)}^{(\mathcal{D})}, \nonumber \\ n_b^{(\mathcal{D})}=&n_{b:(2.0)}^{(\mathcal{D})}+n_{b:(2.2)}^{(\mathcal{D})}+n_{b:(2.4)}^{(\mathcal{D})} \nonumber \\ &+n_{b:(4.0)}^{(\mathcal{D})}+n_{b:(4.2)}^{(\mathcal{D})}+n_{b:(4.4)}^{(\mathcal{D})}. \label{eq:identity0}\end{aligned}$$ Note that $n_{i:(4.4)}^{(\mathcal{D})} = 0$, since the same-sign charges with $|Q_p|=4$ cannot touch with each other. Using these valuables, the total energy can be written as $$\begin{aligned} E=& \left(\frac{1}{2}-J \right)\left(\sum_{\mathcal{D}} 16N_4^{(\mathcal{D})}+4N_2^{(\mathcal{D})}\right)\nonumber \\ &-J\sum_{\mathcal{D}} \left(4n_{i:(2.2)}^{(\mathcal{D})}+8n_{i:(2.4)}^{(\mathcal{D})} \right) \nonumber \\ &+\frac{J}{2}\sum_{\mathcal{D}} \left(4n_{b:(2.2)}^{(\mathcal{D})}+8n_{b:(2.4)}^{(\mathcal{D})}+8n_{b:(4.2)}^{(\mathcal{D})}+16n_{b:(4.4)}^{(\mathcal{D})} \right). \label{eq:Energy1}\end{aligned}$$ Here, the terms proportional to $n_i^{(\mathcal{D})}$ ($n_b^{(\mathcal{D})}$) correspond to intra-cluster (inter-cluster) energy. To simplify Eq. (\[eq:Energy1\]) by erasing $n_{i:(2.2)}^{(\mathcal{D})}$, we substitute the following identities, $$\begin{aligned} 4N^{(\mathcal{D})}&=2n_i^{(\mathcal{D})}+n_b^{(\mathcal{D})}, \label{eq:ND=} \\ 4N_4^{(\mathcal{D})}&=n_{i:(2.4)}^{(\mathcal{D})}+n_{b:(4.4)}^{(\mathcal{D})}+n_{b:(4.2)}^{(\mathcal{D})}+n_{b:(4.0)}^{(\mathcal{D})}, \label{eq:identity1}\end{aligned}$$ into Eq. (\[eq:Energy1\]), and then we obtain $$\begin{aligned} E=&\left(2-12J \right) \left(N_p-N^{\mathrm{vac}}\right) +2J\sum_{\mathcal{D}} n^{(\mathcal{D})}_b\nonumber \\ &+\left(\frac{3}{2}-7J \right)\sum_{\mathcal{D}} n_{i:(2.4)}^{(\mathcal{D})} \nonumber \\ &+\sum_{\mathcal{D}}\biggl[2Jn_{b:(2.2)}^{(\mathcal{D})}+4J n_{b:(2.4)}^{(\mathcal{D})} \nonumber \\ &+\left(\frac{3}{2}-3J \right) n_{b:(4.0)}^{(\mathcal{D})}+ \left(\frac{3}{2}+J \right) n_{b:(4.2)}^{(\mathcal{D})} \nonumber \\ &+\left(\frac{3}{2}+5J \right) n_{b:(4.4)}^{(\mathcal{D})}\biggr],\label{eq:Energy2} \end{aligned}$$ where $N_p=\sum_{\mathcal{D}}N_4^{(\mathcal{D})}+\sum_{\mathcal{D}}N_2^{(\mathcal{D})}+N^{\mathrm{vac}}$ is a total number of plaquettes and $N^{\mathrm{vac}}$ is a number of vacuum plaquettes. Applying the inequality of Eq. (\[eq:gauss\_cluster\]) to the right side of Eq. (\[eq:Energy2\]) and using Eq. (\[eq:identity1\]), we obtain $$\begin{aligned} E\geq&8\left(\frac{1}{4}-J \right) \left(N_p-N^{\mathrm{vac}} \right)+6\left(\frac{1}{4}-J \right) \sum_{\mathcal{D}} n_{i:(2.4)}^{(\mathcal{D})} \nonumber \\ &+\sum_{\mathcal{D}}\biggl[2Jn_{b:(2.2)}^{(\mathcal{D})}+4J n_{b:(2.4)}^{(\mathcal{D})} \nonumber \\ &+2\left(\frac{3}{4}-J \right) n_{b:(4.0)}^{(\mathcal{D})} +\left(\frac{3}{2}+2J \right) n_{b:(4.2)}^{(\mathcal{D})} \nonumber \\ &+ \left(\frac{3}{2} + 6J \right)n_{b:(4.4)}^{(\mathcal{D})}\biggr]. \label{eq:finalform2}\end{aligned}$$ For $0 < J < 1/4$, all the coefficients of $n^{(\mathcal{D})}_{i:(\ell,\ell^\prime)}$, $n^{(\mathcal{D})}_{b:(\ell,\ell^\prime)}$ and $\left(N_p-N^{\mathrm{vac}} \right)$ in the right side of Eq. (\[eq:finalform2\]) are positive, thus the energy minimization is achieved when $n^{(\mathcal{D})}_{i:(\ell,\ell^\prime)} = n^{(\mathcal{D})}_{b:(\ell,\ell^\prime)}= \left(N_p-N^{\mathrm{vac}} \right) = 0$, which is nothing but the Coulomb phase. For $J>1/4$, on the other hand, the coefficients of $n^{(\mathcal{D})}_{i:(2.4)}$ and $\left(N_p-N^{\mathrm{vac}} \right)$ become negative, indicating the instability of the Coulomb phase. How can we derive the ground state for $J>1/4$? To do this, let us first derive the upper bound of $n_{i:(2.4)}^{(\mathcal{D})}$. To this end, we point out that the following identity holds: $$\begin{aligned} 4N_2^{(\mathcal{D})}&=n_{i:(2.4)}^{(\mathcal{D})}+ 2n_{i:(2.2)}^{(\mathcal{D})}+n_{b:(2.0)}^{(\mathcal{D})}+n_{b:(2.2)}^{(\mathcal{D})}+n_{b:(2.4)}^{(\mathcal{D})}.\label{eq:identity2}\end{aligned}$$ Then, combining Eqs. (\[eq:gauss\_cluster\]), (\[eq:identity0\]), (\[eq:identity1\]), and (\[eq:identity2\]), after some of algebra, we obtain $$\begin{aligned} 3n_{i:(2.4)}^{(\mathcal{D})}\leq n_{b:(2.0)}^{(\mathcal{D})}+n_{b:(2.2)}^{(\mathcal{D})}+n_{b:(2.4)}^{(\mathcal{D})}. \label{eq:constraint2}\end{aligned}$$ Combining Eqs. (\[eq:constraint2\]) and (\[eq:finalform2\]), we have $$\begin{aligned} E\geq&8 \left(\frac{1}{4}-J \right)\biggl[N_p-N^{\mathrm{vac}}\biggr] \nonumber \\ &+\sum_{\mathcal{D}}\biggl[(\frac{1}{2}-2J) n_{b:(2.0)}^{(\mathcal{D})} + \frac{1}{2}n_{b:(2.2)}^{(\mathcal{D})}+ \left(\frac{1}{2}+2J \right) n_{b:(2.4)}^{(\mathcal{D})}\nonumber \\ &+2 \left(\frac{3}{4}-J \right)n_{b:(4.0)}^{(\mathcal{D})}+ \left(\frac{3}{2}+2J \right)n_{b:(4.2)}^{(\mathcal{D})}\nonumber \\ &+ \left(\frac{3}{2}+6J \right)n_{b:(4.4)}^{(\mathcal{D})}\biggr] . \label{eq:finalform3}\end{aligned}$$ Further, we can derive the lower bound of $N^{\mathrm{vac}}$ as $$\begin{aligned} 4N^{\mathrm{vac}} =& 2 N_{i:(0.0)} + \sum_{\mathcal{D}}n_{b:(2.0)}^{(\mathcal{D})} + \sum_{\mathcal{D}}n_{b:(4.0)}^{(\mathcal{D})} \notag \\ \geq & \sum_{\mathcal{D}}n_{b:(2.0)}^{(\mathcal{D})}+\sum_{\mathcal{D}}n_{b:(4.0)}^{(\mathcal{D})}, \label{eq:constraint3}\end{aligned}$$ where $N_{i:(0.0)}$ is the number of contacts between vacuum plaquettes. Combining Eqs. (\[eq:constraint3\]) and (\[eq:finalform3\]), we obtain $$\begin{aligned} E \geq & 8 \left(\frac{1}{4}-J \right) N_p + \sum_{\mathcal{D} }\biggl[\frac{1}{2}n_{b:(2.2)}^{(\mathcal{D})} + \left(\frac{1}{2}+2J \right) n_{b:(2.4)}^{(\mathcal{D})}\nonumber \\ &+n_{b:(4.0)}^{(\mathcal{D})}+ \left(\frac{3}{2}+2J \right) n_{b:(4.2)}^{(\mathcal{D})} +\left(\frac{3}{2}+6J \right)n_{b:(4.4)}^{(\mathcal{D})}\biggr] . \label{eq:finalform4}\end{aligned}$$ Now, we see from Eq. (\[eq:finalform4\]) that the energy minimization is achieved when $n_{b:(2.2)}^{(\mathcal{D})} = n_{b:(2.4)}^{(\mathcal{D})} = n_{b:(4.0)}^{(\mathcal{D})} = n_{b:(4.2)}^{(\mathcal{D})} = n_{b:(4.4)}^{(\mathcal{D})} =0$, and the corresponding minimum energy is $$\begin{aligned} E=(2-8J)N_p. \label{eq:idea}\end{aligned}$$ In this case, the equalities in Eqs. (\[eq:gauss\_cluster\]), (\[eq:constraint2\]) and (\[eq:constraint3\]) hold, i.e., $$\begin{aligned} 2N_2^{(\mathcal{D})} &+ 4N_4^{(\mathcal{D})} = n_b^{(\mathcal{D})},\label{eq:final_const1}\\ 3n_{i:(2.4)}^{(\mathcal{D})}&=n_i^{(\mathcal{D})(2.0)}+n_{i:(2.2)}^{(\mathcal{D})}+n_{i:(2.4)}^{(\mathcal{D})}, \label{eq:final_const2}\\ 4N^{\mathrm{vac}}&=\sum_{\mathcal{D}}n_{b:(2.0)}^{(\mathcal{D})}. \label{eq:final_const3}\end{aligned}$$ In fact, the ordered phase in Fig. \[fig:apeC\] satisfied Eqs. (\[eq:idea\])-(\[eq:final\_const3\]). Therefore it is the ground state spin configurations for $J>1/4$. Transfer matrix method ====================== In this appendix, we summarize the transfer matrix method we used for the estimation of residual entropy of tCSL. We consider the lattice geometry shown in Fig. \[fig:transfer\_matrix\](b). The lattice is composed of $LN$ plaquettes with $L$ rows and $N$ columns, and we impose periodic boundary conditions on both directions. In this method, we obtain the total number of trimer patterns by identifying the possible trimer configurations column by column. ![(a) All possible ten types of plaquettes in tCSL. (b) Schematic picture of the lattice considered in this Appendix. (c)An example configuration and definition of $p, p'$ and $q$. The configuration corresponds to $p=5$ $p'=1$ and $q=6$.[]{data-label="fig:transfer_matrix"}](FigApeB.eps){width="\hsize"} Specifically, one plaquette can be covered by a trimer in ten different ways, as shown in Fig. \[fig:transfer\_matrix\](a). Let us make a numbering of all possible trimer configurations on the $i-$th column, and denote it as $\phi_i$. Since there are ten possible configurations for each plaquette, and each column consists of $L$ plaquettes, $\phi_i$ takes $0, 1, \cdots 10^{L}-1$. However, the trimer configuration on $i$-th column restricts the possible configuration on the $(i+1)$-th column. As shown in Fig. \[fig:transfer\_matrix\](c), type $q$ trimer on a plaquette on the $i$-th column restricts the possible trimer types on adjacent plaquettes on the $(i+1)$-th column. Possible combinations of trimer types on a triplet of plaquettes, ($p,p',q$) is summarized in a table \[table:possible\]. p p’ q --------- --------- --------- (0,1,3) (1,3,5) 9 (6,8,9) 9 (2,4,5) (0,2,4) 8 (1,3,5) (6,7) (6,8,9) (6,7) 6 7 5 (6,8,9) (3,4) (7,8,9) (6,8,9) (0,6,7) 7 (1,2) (0,2,4) 8 (1,3,5) (6,7) : Each number represents type of plaquettes in Fig. \[fig:transfer\_matrix\](a). Combinations of a row are possible configuration for $p, p'$ and $q$. []{data-label="table:possible"} We account for this restriction by introducing the transfer matrix $A^{(L)}$, whose element, $A_{\phi_i,\phi_{i+1} }^{(L)}$, takes 1, if the configurations $\phi_i$ and $\phi_{i+1}$ are compatible, and 0, otherwise. In terms of this transfer matrix, the total number of trimer configurations, $W^{(L,N)}$, can be obtained as $$\begin{aligned} W^{(L,N)}=&\sum_{\phi_1}\cdots\sum_{\phi_N}A_{\phi_1,{\phi_2} }^{(L)}\cdots A_{\phi_N,{\phi_1} }^{(L)}\nonumber\\ &=\rm{tr}[A_{\phi_i,\phi_{i+1} }^{(L)}]^N=\left(\lambda_L\right)^N,\end{aligned}$$ for sufficiently large $N$. Here, $\lambda_L$ is the largest eigenvalue of $A^{(L)}$. By taking $N\to\infty$, the residual entropy per site is evaluated as $$\begin{aligned} S_0= \lim_{N\rightarrow \infty}\frac{1}{LN}\ln(W^{(L,N)}) = \frac{1}{L}\ln\lambda_L.\end{aligned}$$ We numerically evaluate $\lambda_L$ up to $L=7$ and obtain the residual entropy as listed in table \[table:residualentropy\].

--- abstract: 'Colloidal suspensions and polyelectrolyte solutions containing multivalent counterions can exhibit some very counter-intuitive behavior usually associated with the low temperature physics. There are two particularly striking phenomena resulting from strong electrostatic correlations. One is the like-charge attraction and the second is the polyion overcharging. In this contribution we will concentrate on the problem of overcharging. In particular we will explore the kinetic limitation to colloidal charge inversion in suspensions containing multivalent counterions.' address: | *Instituto de Física, Universidade Federal do Rio Grande do Sul\ Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil\ [levin@if.ufrgs.br]{}* author: - '**Yan Levin, Jeferson J. Arenzon**' title: Kinetics of charge inversion --- Introduction ============ Colloidal suspensions and polyelectrolyte solutions containing multivalent counterions can exhibit some very curious electrostatic behavior [@Le02]. It is found that under some circumstances two like-charged polyions inside suspension can actually attract one another [@Pa80; @GuJoWe84; @KjMa86; @StRo90; @CrGr94; @RoBl96; @HaLi97; @LeArSt99; @GrMaBr97; @KoLe99; @AlAmLo98; @ArStLe99; @GoHa99; @LaPiLeFe00; @DiCaLe01]. The counterion mediated attraction is responsible for the $DNA$ compaction inside the bacteriophages, viruses that infect bacteria [@Bl91; @Bl97], and for the organization of eukaryotic cytoskeleton [@TaJa96]. Another “strange” electrostatic behavior which can occur in suspensions containing multivalent counterions is the reversal of the electrophoretic mobility [@Le02; @LoSaHe83; @GoLoHe85; @Sh99a; @NgGrSh00b; @MeHoKr01; @GrNgSh02]. The first thing that is learned in a course on electrostatics is that the force produced by the electric field on a charged particle is $$\label{1} \mathbf{F}=Q \mathbf{E} \;.$$ Thus, a positively charged particle, $Q>0$, is expected to move in the direction of the applied field while a negatively charged particle, $Q<0$, will move in the direction opposite to the field. This simple picture, however, breaks down inside colloidal suspensions of low dielectric solvent or even in aqueous suspensions containing multivalent counterions. The reason for the violation of the “simple” physics learned in high-school are the strong electrostatic many-body interactions between the colloidal particles and the counterions. The reversal of electrophoretic mobility can be understood as a combination of two electrostatically driven mechanisms. Strong electrostatic interaction between colloids and counterions leads to formation of polyion-counterion complexes [@AlChGr84; @LeBaTa98; @DiBaLe01]. The existence of counterion condensation has been known for over thirty years [@Ma69; @Ma78; @Oo71], the general phenomenon is, however, much older than this and can be traced to the pioneering work of Bjerrum on ionic association inside electrolyte solutions almost $80$ years ago [@Bj26]. In aqueous suspensions with only monovalent counterions, the net charge of complexes is of the same sign as the bare charge of polyions. If the solvent is water and the counterions are monovalent, the electrostatic interactions between the condensed counterions can be neglected [@Le02], and the simplest Poisson-Boltzmann theory is sufficient to describe the polyion-counterion complexation [@AlChGr84; @TrBoAu02]. In aqueous suspensions containing multivalent counterions or in suspension of low dielectric solvents, the electrostatic energy between the condensed counterions is significantly larger than the thermal energy and the electrostatic correlations between the condensed counterions can no longer be neglected. These electrostatic correlations can lead to colloidal overcharging i.e. the net charge of the complex is of opposite sign to the charge of the bare polyion. The overcharged colloid will then move in the “wrong” direction with respect to the applied electric field [@Le02; @GrNgSh02]. Overcharging ============ To understand the phenomenon of overcharging we shall start by studying a very simple model. Consider a sphere of radius $a$ and fixed charge $-Zq$ distributed uniformly over its surface. We would like to know how many point-like $\alpha$-valent counterions, each of charge $\alpha q$, should be placed on top of this sphere in order to minimize the total electrostatic free energy [@Th04; @Le02; @Sh99a; @MeHoKr01]. When we say “counterions” we have in mind both simple multivalent ions such as $Ca^{++}$, as well as more complicated micelle-like aggregates with $\alpha$ significantly higher than one. The free energy of a complex can be written as $$\label{2} E_n=\frac{Z^2 q^2}{2 \epsilon a}- \frac{Z \alpha n q^2}{\epsilon a}+F^{\alpha \alpha}_n \;.$$ The first term is the self energy of the charged sphere, the second term is the electrostatic energy of interaction between the sphere and $n$ condensed $\alpha$-ions, and the last term is the electrostatic energy of repulsion between the condensed counterions. To calculate the free energy of repulsion, it is convenient to express $F^{\alpha \alpha}_n$ in terms of the free energy of a one component plasma ($OCP$), $n$ $\alpha$-ions on the surface of a sphere with a uniform $neutralizing$ background, $F^{OCP}_n$. The free energy of a spherical $OCP$ can be written as $$\label{3} F^{OCP}_n=F^{\alpha \alpha}_n -\frac{\alpha^2 n^2 q^2}{\epsilon a} + \frac{\alpha^2 n^2 q^2}{2\epsilon a}\;.$$ Substituting Eq. (\[3\]) into Eq. (\[2\]) the electrostatic free energy of a polyion-counterion complex becomes, $$\label{4} E_n=\frac{(Z-\alpha n)^2 q^2}{2 \epsilon a}+F^{OCP}_n \;.$$ In the strong coupling limit, corresponding to multivalent counterions or solvents of low dielectric permittivity, the free energy of the $OCP$ is well approximated by the free energy of the low temperature phase corresponding to a triangular Wigner crystal, $$\begin{aligned} \label{5} F_{n}^{OCP}=-M\frac{\alpha^2 q^2 n^{3/2} }{ 2 \epsilon a }\;. \end{aligned}$$ where $M$ is the Madelung constant. For weaker couplings, the expression for the $F_{n}^{OCP}$ can be obtained from the fits to the Monte Carlo data [@GaChCh79]. For concreteness we shall use $M=1.106$, the value appropriate for a planar Wigner crystal [@Le02]. The effective charge of a polyion-counterion complex, in units of $-q$ is $$\begin{aligned} \label{6} Z_{eff}=Z-\alpha n \;, \end{aligned}$$ The optimum number of condensed counterions is determined from the minimization of the total electrostatic free energy. We find [@Sh99a; @MeHoKr01; @Le02] $$\begin{aligned} \label{7} Z_{eff}^*=-\frac{1+\sqrt{1+4 \gamma^2 Z}}{2 \gamma^2} \approx -\frac{\sqrt Z}{\gamma}\;, \end{aligned}$$ where $$\begin{aligned} \label{8} \gamma=\frac{4}{3 M \sqrt \alpha}\;. \end{aligned}$$ We see that the optimal charge of a polyion-counterion complex is of opposite sign to the bare colloidal charge, i.e. the complex is overcharged. Inside the colloidal suspension containing multivalent counterions or solvents of low dielectric permittivity the electrophoretic mobility can, therefore, be reversed. Some care, however, must be taken in extrapolating the results of this simple model to real systems. While we have treated the counterions as condensed on top of the sphere, this is clearly not the case for real colloidal suspension. Instead the associated counterions form a layer around a colloidal particle which can be some nanometers wide. The presence of simple electrolyte also strongly affects the net charge of the polyion-$\alpha$-ion complex. Furthermore, the complex formation is a kinetic phenomenon requiring a counterion to overcome an energy barrier in order to join the already overcharged complex. The overcharging potential ========================== In the previous section we found that the minimum of the total electrostatic free energy of a polyion-$\alpha$-ion complex corresponds to an overcharged state. However, for a counterion to join an already overcharged complex it must overcome an energy barrier. The waiting time for a thermal fluctuation of sufficient strength necessary to drive a counterion over an activation barrier scales exponentially with the height of the barrier. There is, therefore, a kinetic limitation to the degree of overcharging which can prevent a thermodynamically optimum state from being reached on experimental time scale. To explore this further we have to construct an effective interaction potential between a complex and a counterion separated by distance $r$. The work necessary to bring a counterion from infinity to join a complex containing $n$ $\alpha$-ions is $$\begin{aligned} \label{9} W=\frac{dE_n}{d n}\;. \end{aligned}$$ We define the reduced electrostatic potential of a counterion on the surface of the complex as $\varphi(a)=\beta W$, where $\beta=1/k_B T$. Differentiating Eq. (\[4\]) we find $$\begin{aligned} \label{10} \varphi(a) =-\frac{(Z-\alpha n) \lambda_B \alpha }{a} - \frac{3 M \alpha^2 \sqrt n }{4 a}\;, \end{aligned}$$ where $\lambda_B=q^2/\epsilon k_B T $. The first term of Eq. (\[10\]) is the electrostatic energy of interaction between a uniform spherical charge and an $\alpha$-ion, while the second term is due to electrostatic correlations between the $\alpha$-ions. In the strong coupling limit correlational contribution to the interaction potential decay exponentially fast with the separation from the polyion surface [@RoBl96; @LaPiLeFe00; @StLeAr02]. The characteristic length is set by the average separation between the condensed counterions. More specifically we can approximate the reduced interaction potential by $$\begin{aligned} \label{11} \varphi(r) =-\frac{(Z-\alpha n) \lambda_B \alpha }{r} - \frac{3 M \alpha^2 \sqrt n }{4 a} e^{-(r-a)/\xi}\;. \end{aligned}$$ The decay of the correlational contribution is governed by the characteristic length $\xi$ which in the strong coupling limit is well approximated by [@RoBl96; @LaPiLeFe00; @StLeAr02], $$\begin{aligned} \label{12} \xi=\frac{1}{|\mathbf G|}\;, \end{aligned}$$ where $\mathbf G$ is the reciprocal lattice vector of a triangular Wigner crystal of condensed counterions. Due to strong coupling between the condensed counterions, Eq. (\[12\]) should remain a good approximation even significantly above the crystallization temperature. For a triangular Wigner crystal, $$\begin{aligned} \label{13} {|\mathbf G|}=\frac{4 \pi}{\sqrt 3 b}\;, \end{aligned}$$ where $b$ is the lattice spacing $$\begin{aligned} \label{14} b=\frac{1}{3^{1/4} \sqrt \sigma }\; \end{aligned}$$ and $\sigma=n/4\pi a^2$ is the surface density of condensed counterions. Substituting Eqs. (\[13\]) and (\[14\]) into Eq. (\[12\]), the decay length is found to be $$\begin{aligned} \label{15} \xi=\frac{3^{1/4}}{2 \sqrt{ \pi}} \frac{a}{\sqrt n}\;. \end{aligned}$$ We are now in possession of the electrostatic potential which will allow us to study the kinetics of overcharging. Kinetics of overcharging ======================== For $n<Z/\alpha$, the electrostatic potential between a counterion and a complex is purely attractive favoring further counterion condensation. Inside an electrolyte solution this tendency towards polyion-counterion association is opposed by the loss of entropy resulting from the confinement of condensed counterions near the colloidal surface. Here, however, we shall not be concerned with the role of entropy [@Le02]. For $n>Z/\alpha$ the interaction potential has two minima, one located at $r=a$ and the second at $n=\infty$. For $Z/\alpha<n<n^*$ the $r=a$ minimum is the dominant one, while for $Z>n^*$ the global minimum changes to $r=\infty$. The value of $n^*$ corresponds to the number of condensed counterions which minimize the electrostatic free energy of the complex Eq. (\[4\]), $$\begin{aligned} \label{16} n^*=\frac{Z-Z_{eff}^*}{\alpha}\;. \end{aligned}$$ ![The reduced interaction potential between a complex of $Z=4000$, $a=1000$ Å, $n=n^*$ condensed trivalent counterions, and a trivalent counterion located at distance $r$ from the center of colloid.[]{data-label="Fig1"}](fig1.ps){width="8cm"} In the case of trivalent counterions the energy barrier that a counterion needs to overcome in order to join a complex which already contains $n^*$ condensed $\alpha$-ions is less than $2k_BT$, Fig 1. Thus, for trivalent counterions there is no kinetic hindrance to reaching the optimum overcharged state. We next look at the height of the activation barrier as a function of the counterion valence, Fig. 2. ![The height of the activation barrier that an $\alpha$-ion must overcome to join an optimally overcharged complex composed of a colloid with $Z=4000$, $a=1000$ Å$\,$ and $n=n^*$ condensed $\alpha$-ions.[]{data-label="Fig2"}](fig2.ps){width="8cm"} It is clear that the height of the activation barrier grows rapidly with the increased valence of the $\alpha$-ions. In particular we see that for $\alpha=10$ the activation barrier is already some $10 k_B T$ which is probably the maximum height that a counterion can overcome on a reasonable experimental time scale. Thus, the process of overcharging by the $\alpha$-ions with $\alpha>10$ will be kinetically controlled. For example, from Eq. (\[7\]) we see that the optimal state of overcharging of a colloidal particle of $Z=4000$ and radius $a=1000$ Å$\,$ by micelles with $\alpha=25$ corresponds to $Z^*_{eff}=-271$. In practice, though, the process of overcharging will come to a stop when the barrier height reaches about $10k_B T$, implying that the complex will stop growing when the net charge is only $Z_{eff}=-70$. Conclusion ========== In this contribution we have explored the kinetic limitation to overcharging. We find that kinetics does not play an important role for overcharging by simple multivalent counterions, so that the state of optimal overcharging, Eq. (\[7\]), is accessible within an experimental time scales. On the other hand, we find that the activation barrier grows rapidly with the valence of counterions, suggesting that the extent of overcharging by micelle-like aggregates is largely kinetically controlled. The kinetic limitation to overcharging might also be important for the formation of the DNA-cationic lipid complexes. The problem of a reliable and safe mechanism for gene delivery is particularly pressing in view of the current medical applications. Strong electrostatic repulsion between a DNA and a cellular membrane inhibits transfection of a naked DNA into the cell. A way to overcome this difficulty is through the formation of overcharged complexes between the DNA and the cationic liposomes [@FeRi89; @Fe97; @Fr97; @HoMuAn98; @KuLeBa99]. These lipoplexes having a net positive charge are attracted to the cellular membrane, facilitating the genetic transfection. Finally, the presence of a simple electrolyte will have a strong influence on the overcharging. It has been demonstrated that for sufficient concentration of $\alpha$-ions, monovalent salt favors overcharging [@NgGrSh00c; @Le02]. In fact in the presence of simple electrolyte the thermodynamic state of optimum overcharging corresponds to the charge inversion of as much as $100\%$. This should be contrasted with the result of Eq. (\[7\]), which shows that in the absence of salt, the effective charge of a complex scales as a square root of the bare charge. The presence of salt will also lower the height of the activation barrier reducing the kinetic hindrance to overcharging.

--- abstract: 'We present a measurement of $CP$-violation parameters in the $\bz\to\ks\ks\ks$ decay based on a sample of $275\times 10^6$ $B\bbar$ pairs collected at the $\ufs$ resonance with the Belle detector at the KEKB energy-asymmetric $e^+e^-$ collider. One neutral $B$ meson is fully reconstructed in the decay $\bz\to\ks\ks\ks$, and the flavor of the accompanying $B$ meson is identified from its decay products. $CP$-violation parameters are obtained from the asymmetry in the distributions of the proper-time interval between the two $B$ decays: ${{\cal S}}= \SksksksResultSS$ and ${{\cal A}}= \AksksksResultSS$.' author: - 'K. Sumisawa' - 'Y. Ushiroda' - 'M. Hazumi' - 'K. Abe' - 'K. Abe' - 'I. Adachi' - 'H. Aihara' - 'Y. Asano' - 'V. Aulchenko' - 'T. Aushev' - 'A. M. Bakich' - 'U. Bitenc' - 'I. Bizjak' - 'S. Blyth' - 'A. Bondar' - 'A. Bozek' - 'M. Bračko' - 'J. Brodzicka' - 'T. E. Browder' - 'Y. Chao' - 'A. Chen' - 'K.-F. Chen' - 'W. T. Chen' - 'B. G. Cheon' - 'R. Chistov' - 'Y. Choi' - 'A. Chuvikov' - 'S. Cole' - 'J. Dalseno' - 'M. Danilov' - 'M. Dash' - 'A. Drutskoy' - 'S. Eidelman' - 'Y. Enari' - 'F. Fang' - 'S. Fratina' - 'N. Gabyshev' - 'A. Garmash' - 'T. Gershon' - 'G. Gokhroo' - 'B. Golob' - 'A. Gorišek' - 'J. Haba' - 'K. Hara' - 'T. Hara' - 'H. Hayashii' - 'T. Higuchi' - 'T. Hokuue' - 'Y. Hoshi' - 'S. Hou' - 'W.-S. Hou' - 'Y. B. Hsiung' - 'T. Iijima' - 'A. Imoto' - 'K. Inami' - 'A. Ishikawa' - 'H. Ishino' - 'R. Itoh' - 'M. Iwasaki' - 'Y. Iwasaki' - 'J. H. Kang' - 'J. S. Kang' - 'S. U. Kataoka' - 'N. Katayama' - 'H. Kawai' - 'T. Kawasaki' - 'H. R. Khan' - 'H. Kichimi' - 'H. J. Kim' - 'H. O. Kim' - 'S. K. Kim' - 'S. M. Kim' - 'K. Kinoshita' - 'S. Korpar' - 'P. Križan' - 'P. Krokovny' - 'R. Kulasiri' - 'S. Kumar' - 'C. C. Kuo' - 'A. Kusaka' - 'A. Kuzmin' - 'Y.-J. Kwon' - 'J. S. Lange' - 'G. Leder' - 'T. Lesiak' - 'S.-W. Lin' - 'F. Mandl' - 'D. Marlow' - 'T. Matsumoto' - 'A. Matyja' - 'W. Mitaroff' - 'K. Miyabayashi' - 'H. Miyake' - 'H. Miyata' - 'R. Mizuk' - 'T. Nagamine' - 'Y. Nagasaka' - 'E. Nakano' - 'M. Nakao' - 'Z. Natkaniec' - 'S. Nishida' - 'O. Nitoh' - 'T. Nozaki' - 'S. Ogawa' - 'T. Ohshima' - 'T. Okabe' - 'S. Okuno' - 'S. L. Olsen' - 'Y. Onuki' - 'W. Ostrowicz' - 'H. Ozaki' - 'C. W. Park' - 'H. Park' - 'N. Parslow' - 'L. S. Peak' - 'R. Pestotnik' - 'L. E. Piilonen' - 'M. Rozanska' - 'H. Sagawa' - 'Y. Sakai' - 'T. R. Sarangi' - 'N. Sato' - 'T. Schietinger' - 'O. Schneider' - 'R. Seuster' - 'M. E. Sevior' - 'H. Shibuya' - 'V. Sidorov' - 'J. B. Singh' - 'A. Somov' - 'R. Stamen' - 'S. Stanič' - 'M. Starič' - 'T. Sumiyoshi' - 'S. Suzuki' - 'O. Tajima' - 'F. Takasaki' - 'K. Tamai' - 'N. Tamura' - 'M. Tanaka' - 'Y. Teramoto' - 'X. C. Tian' - 'K. Trabelsi' - 'T. Tsuboyama' - 'T. Tsukamoto' - 'S. Uehara' - 'T. Uglov' - 'K. Ueno' - 'S. Uno' - 'P. Urquijo' - 'G. Varner' - 'K. E. Varvell' - 'S. Villa' - 'C. C. Wang' - 'C. H. Wang' - 'M.-Z. Wang' - 'Q. L. Xie' - 'B. D. Yabsley' - 'A. Yamaguchi' - 'H. Yamamoto' - 'Y. Yamashita' - 'M. Yamauchi' - Heyoung Yang - 'J. Zhang' - 'L. M. Zhang' - 'Z. P. Zhang' - 'V. Zhilich' - 'D. Žontar' title: | \ Measurement of Time-Dependent [$CP$]{}-Violating\ Asymmetries in $\bz\to\ks\ks\ks$ Decay --- In the Standard Model (SM), $CP$ violation arises from the Kobayashi-Maskawa phase [@Kobayashi:1973fv] in the weak-interaction quark-mixing matrix. In particular, the SM predicts to a good approximation that ${{\cal S}}= -\xi_f\sin 2\phi_1$ and ${{\cal A}}=0$ for both ${b \to c\overline{c}s}$ and ${b \to s\overline{q}q}$ transitions, where ${{\cal S}}$ (${{\cal A}}$) is a parameter for mixing-induced (direct) $CP$ violation [@bib:sanda], $\xi_f = +1 (-1)$ corresponds to $CP$-even (-odd) final states, and $\phi_1$ is one of angles of the Unitarity Triangle. Measurements of time-dependent $CP$ asymmetries in $\bz \to J/\psi \ks$ [@bib:CC] and related decay modes, which are governed by the $b \to c\overline{c}s$ transition, by the Belle [@bib:CP1_Belle; @bib:BELLE-CONF-0436] and BaBar [@bib:CP1_BaBar] collaborations already determine ${{\sin2\phi_1}}$ rather precisely; the present world average value is ${{\sin2\phi_1}}= \sinbbWAResult$ [@bib:HFAG]. $CP$-violation parameters in the flavor-changing $b \to s$ transition are sensitive to phenomena at a very high-energy scale [@bib:lucy; @Akeroyd:2004mj]. Belle measurements [@Abe:2004xp] in the decay modes $\bz\to$ $\phi\ks$, $\phi\kl$, $\kp\km\ks$, $\fzero\ks$, $\eta'\ks$, $\omega\ks$, and $\ks\piz$, which are dominated by the ${b \to s\overline{q}q}$ transition, yield ${{\sin2\phi_1}}= \SbsqqResult$ when all the modes are combined. Measurements by the BaBar collaboration also yield a similar deviation [@bib:HFAG; @bib:BaBar_sss]. To elucidate the difference in $CP$ asymmetries between ${b \to s\overline{q}q}$ and ${b \to c\overline{c}s}$ transitions, it is essential to examine additional modes that may be sensitive to the same $b \to s$ amplitude. The $\bz$ decay to $\ks\ks\ks$, which is a $\xi_f = +1$ state, is one of the most promising modes for this purpose [@Gershon:2004tk]. Since there is no $u$ quark in the final state, the decay is dominated by the ${b \to s\overline{s}s}$ transition. Its branching fraction ${{\cal B}}(\bz\to\ks\ks\ks) = \BRksksks$ was reported by Belle [@Garmash:2003er]. In this Letter, we describe a measurement of $CP$ asymmetries in the $\bz\to\ks\ks\ks$ decay. In the decay chain $\Upsilon(4S)\to \bz\bzb \to f_{CP}f_{\rm tag}$, where one of the $B$ mesons decays at time $t_{CP}$ to a $CP$ eigenstate $f_{CP}$ and the other decays at time $t_{\rm tag}$ to a final state $f_{\rm tag}$ that distinguishes between $B^0$ and $\bzb$, the decay rate has a time dependence given by $$\begin{aligned} \label{eq:psig} {\cal P}(\Delta{t}) = \frac{e^{-|\Delta{t}|/{\taubz}}}{4{\taubz}} \biggl\{1 + {\ensuremath{q}}\cdot \Bigl[ {{\cal S}}\sin({{\Delta m_d}}\Delta{t}) \nonumber \\ + {{\cal A}}\cos({{\Delta m_d}}\Delta{t}) \Bigr] \biggr\}. $$ Here $\taubz$ is the $B^0$ lifetime, ${{\Delta m_d}}$ is the mass difference between the two $B^0$ mass eigenstates, $\Delta{t}$ is the time difference $t_{CP}$ $-$ $t_{\rm tag}$, and the $b$-flavor charge is ${\ensuremath{q}}$ = +1 ($-1$) when the tagging $B$ meson is a $B^0$ ($\bzb$). At the KEKB energy-asymmetric $e^+e^-$ (3.5 on 8.0 GeV) collider [@bib:KEKB], the $\Upsilon(4S)$ resonance is produced with a Lorentz boost of $\beta\gamma=0.425$ along the $z$ direction which is antiparallel to the positron beamline. Since the $B^0$ and $\bzb$ mesons are approximately at rest in the $\Upsilon(4S)$ center-of-mass system (cms), $\Delta t$ can be determined from the displacement in $z$ between the $f_{CP}$ and $f_{\rm tag}$ decay vertices: $\Delta t \simeq (z_{CP} - z_{\rm tag})/(\beta\gamma c) \equiv \Delta z/(\beta\gamma c)$. The Belle detector [@Belle] is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber, an array of aerogel threshold Cherenkov counters, a barrel-like arrangement of time-of-flight scintillation counters, an electromagnetic calorimeter (ECL) and an iron flux-return instrumented to detect $K_L^0$ mesons and to identify muons. A 2.0cm radius beampipe and a 3-layer SVD (SVD-I) were used for a 140 fb$^{-1}$ data sample, while a 1.5 cm radius beampipe, a 4-layer silicon detector (SVD-II) [@Ushiroda] and a small-cell inner drift chamber were used for an additional 113 fb$^{-1}$ data sample. In total $275\times 10^6$ $B\bbar$ pairs were accumulated. We reconstruct the $\bz\to\ks\ks\ks$ decay in the $\kspm\kspm\kspm$ or $\kspm\kspm\kszz$ final state, where the $\pip\pim$ ($\piz\piz$) state from a $\ks$ decay is denoted as $\kspm$ ($\kszz$). Pairs of oppositely charged tracks with the $\pip\pim$ invariant mass within 0.012 GeV/$c^2$ ($\simeq 3 \sigma$) of the nominal $\ks$ mass are used to reconstruct $\kspm$ candidates. The $\pip\pim$ vertex is required to be displaced from the interaction point (IP) by a minimum transverse distance of 0.22 cm for candidates with p $>1.5$ GeV/$c$ and 0.08 cm for those with p $<1.5$ GeV/$c$, where p is the momentum of $\ks$. The angle in the transverse plane between the $\ks$ momentum vector and the direction defined by the $\ks$ vertex and the IP should be less than 0.03 rad (0.1 rad) for the high (low) momentum candidates. The mismatch in the $z$ direction at the $\ks$ vertex point for the two charged pion tracks should be less than 2.4 cm (1.8 cm) for the high (low) momentum candidates. After two good $\kspm$ candidates have been found which satisfy the criteria given above, looser requirements are applied for the third $\kspm$ candidate. The requirement on the transverse direction matching is relaxed to 0.2 rad (0.4 rad for low momentum candidates), and the mismatch of the two charged pions in the $z$ direction is required to be less than 5 cm (1 cm if both pions have hits in the SVD). To select $\kszz$ candidates, we reconstruct $\piz$ candidates from pairs of photons with $E_\gamma > 0.05$ GeV, where $E_\gamma$ is the photon energy measured with the ECL. The reconstructed $\piz$ candidate is required to have an invariant mass between 0.08 and 0.15 GeV/$c^2$ and momentum above 0.1 GeV/$c$. $\kszz $ candidates are required to have an invariant mass between 0.47 and 0.52 GeV/$c^2$, and a fit is performed with constraints on the $\ks$ vertex and $\piz$ masses to improve the $\piz\piz$ invariant mass resolution. The $\kszz$ candidate is combined with two good $\kspm$ candidates to reconstruct a $\bz$ meson. To identify $\bz\to\ks\ks\ks$ decays, we use the energy difference $\dE\equiv E_B^{\rm cms}-E_{\rm beam}^{\rm cms}$ and the beam-energy constrained mass $\mb\equiv\sqrt{(E_{\rm beam}^{\rm cms})^2- (p_B^{\rm cms})^2}$, where $E_{\rm beam}^{\rm cms}$ is the beam energy in the cms, and $E_B^{\rm cms}$ and $p_B^{\rm cms}$ are the cms energy and momentum of the reconstructed $B$ candidate, respectively. The $\bz$ meson signal region is defined as $|\dE|<0.10$ GeV for $\bz \to \kspm\kspm\kspm$, $-0.15 < \dE < 0.10$ GeV for $\bz \to \kspm\kspm\kszz$, and $5.27 < \mb < 5.29~{\rm GeV}/c^2$ for both decays. To suppress the $e^+e^- \rightarrow q\overline{q}$ continuum background ($q = u,~d,~s,~c$), we form a signal over background likelihood ratio $\rsigbkg$ by combining likelihoods for two quantities; a Fisher discriminant of modified Fox-Wolfram moments [@SFW], and the cosine of the cms $\bz$ flight direction.The requirement for $\rsigbkg$ depends both on the decay mode and on the flavor-tagging quality; after applying all other cuts, this rejects 94% of the $q\bar{q}$ background while retaining 75% of the signal. If both $\bz\to\kspm\kspm\kspm$ and $\kspm\kspm\kszz$ candidates are found in the same event, we choose the $\bz\to\kspm\kspm\kspm$ candidate. When multiple $\bz\to\kspm\kspm\kspm$ candidates are found, we prioritize those with three good $\kspm$ candidates. If more than one candidate still remain, we select the one with the smallest value for $\sum(\Delta M_{\kspm})^2$, where $\Delta M_{\kspm}$ is the difference between the reconstructed and nominal mass of $\kspm$. For multiple $\bz\to\kspm\kspm\kszz$ candidates, we select the $\kspm\kspm$ pair that has the smallest $\sum(\Delta M_{\kspm})^2$ value and the $\kszz$ candidate with the minimum $\chi^2$ of the constrained fit. We reject $\ks\ks\ks$ candidates if they are consistent with $\bz\to\chi_{c0}\ks\to(\ks\ks)\ks$ or $\bz\to D^0\ks\to(\ks\ks)\ks$ decays, i.e. if one of the $\ks$ pairs has an invariant mass within $\pm 2 \sigma$ of the $\chi_{c0}$ mass or $D^0$ mass, where $\sigma$ is the $\ks\ks$ mass resolution. Figure \[fig:mb\] shows the $\mb$ and $\dE$ distributions for the reconstructed $\bz\to\ks\ks\ks$ candidates after flavor tagging. The signal yield is determined from an unbinned two-dimensional maximum-likelihood fit to the $\dE$-$\mb$ distribution. The $\kspm\kspm\kspm$ signal distribution is modeled with a Gaussian function (a sum of two Gaussian functions) for $\mb$ ($\dE$). For $\bz\to\kspm\kspm\kszz$ decay, the signal is modeled with a two-dimensional smoothed histogram obtained from Monte Carlo (MC) events. For the continuum background, we use the ARGUS parameterization [@bib:ARGUS] for $\mb$ and a linear function for $\dE$. The fits after flavor tagging yield $\NBsigkspmkspmkspm$ $\bz\to\kspm\kspm\kspm$ events and $\NBsigkspmkspmkszz$ $\bz\to\kspm\kspm\kszz$ events for a total of $\NBsigksksks$ $\bz\to\ks\ks\ks$ events in the signal region, where the errors are statistical only. The obtained purity is $\PBkspmkspmkspm$ for the $\kspm\kspm\kspm$ and $\PBkspmkspmkszz$ for the $\kspm\kspm\kszz$ channels. We use events outside the signal region as well as a large MC sample to study the background components. The dominant background is from continuum. The contamination of $\bz\to\chi_{c0}\ks$ events in the $\bz\to\ks\ks\ks$ sample is small (less than $2.6$% at 90% C.L.). The contributions from other $B\overline{B}$ events are negligibly small. The influence of these backgrounds is treated as a source of systematic uncertainty in the $CP$ asymmetry measurement. Backgrounds from the decay $\bz\to D^0\ks$ are found to be negligible. The $b$-flavor of the accompanying $B$ meson is identified from inclusive properties of particles that are not associated with the reconstructed $\bz\to\ks\ks\ks$ candidates. The algorithm for flavor tagging is described in detail elsewhere [@bib:fbtg_nim]. We use two parameters, $q$ defined in Eq. (\[eq:psig\]) and $r$, to represent the tagging information. The parameter $r$ is an event-by-event, MC-determined flavor-tagging dilution factor that ranges from $r=0$ for no flavor discrimination to $r=1$ for unambiguous flavor assignment. It is used only to sort data into six $r$ intervals. The wrong tag fractions $w$ for each of the $r$ intervals and their differences $\Delta w$ for $\bz$ and $\bzb$ decays are determined from data [@Abe:2004xp]. The vertex position for $\bz\to\ks\ks\ks$ decays is obtained using $\kspm$ trajectories and a constraint on the IP; the IP profile ($\sigma_x\simeq100~\mu$m, $\sigma_y\simeq5~\mu$m, $\sigma_z\simeq3$ mm) is convolved with finite $\bz$ flight length in the plane perpendicular to the $z$ axis. To reconstruct the $\kspm$ trajectory with sufficient resolution, both charged pions from the $\ks$ decay are required to have enough SVD hits; at least one layer with hits on both sides and at least one additional $z$ hit in other layers for SVD-I, and at least two layers with hits on both sides for SVD-II. The reconstruction efficiency depends both on the $\kspm$ momentum and on the SVD geometry. The vertex efficiencies with SVD-II (86% for $\kspm\kspm\kspm$ and 74% for $\kspm\kspm\kszz$) are higher than those with SVD-I (79% for $\kspm\kspm\kspm$ and 62% for $\kspm\kspm\kszz$) because of the larger outer radius and the additional detector layer. The typical vertex resolution is about 97 $\mu$m (113 $\mu$m) for SVD-I (SVD-II) when two or three $\kspm$ candidates can be used. The resolution is worse when only one $\kspm$ can be used; the typical value is 152 $\mu$m (168 $\mu$m) for SVD-I (SVD-II), which is comparable to the ${f_{\rm tag}}$ vertex resolution. The determination of the vertex of the ${f_{\rm tag}}$ final state is the same as the $\bz\to\phi\ks$ analysis, and is described in detail elsewhere [@Abe:2004xp; @bib:resol_nim]. We determine ${{\cal S}}$ and ${{\cal A}}$ by performing an unbinned maximum-likelihood fit to the observed $\Dt$ distribution. The probability density function (PDF) expected for the signal distribution, ${\cal P}_{\rm sig}(\Dt;{{\cal S}},{{\cal A}},{\ensuremath{q}},w,\Delta w)$, is given by Eq. (\[eq:psig\]) after incorporating the effect of incorrect flavor assignment. The distribution is convolved with the proper-time interval resolution function, $\rsig$, which is a function of event-by-event vertex errors. We find from MC simulation that universal $\rsig$ parameters used for measurements of $CP$ asymmetries in the $\bz\to\jpsi\ks$ and related decays [@bib:BELLE-CONF-0436; @bib:resol_nim] approximately describe the resolution for the $\bz\to\ks\ks\ks$ decay. To account for differences between $\kspm$ trajectories and charged tracks, additional parameters that rescale vertex errors are introduced. When only one $\kspm$ is used in the vertex fit, these parameters are determined from a fit to the $\Dt$ distribution of $\bz\to\jpsi\ks$ candidates, where only the $\ks$ and the IP constraint are used for the vertex reconstruction. The procedure is the same as that for the $\bz\to\ks\piz$ decay and is described elsewhere [@Abe:2004xp]. For events with two or three $\kspm$ used in the vertexing, we also find from MC simulation that the resolution is well described by the same $\rsig$ parameterization with an additional correction function that depends on the number of $\kspm$ decays used for the vertex reconstruction. The form of this correction function is determined from a study using MC simulation. We determine the following likelihood value for each event $i$: $$\begin{aligned} P_i &=& (1-\fol)\int \biggl[ \fsig{\cal P}_{\rm sig}(\Dt')R_{\rm sig}(\Dt_i-\Dt') \nonumber \\ &+&(1-\fsig){\cal P}_{\rm bkg}(\Dt')R_{\rm bkg}(\Dt_i-\Dt')\biggr] d(\Dt') \nonumber \\ &+& \fol P_{\rm ol}(\Dt_i), \label{eq:likelihood}\end{aligned}$$ where $P_{\rm ol}$ is a broad Gaussian function that represents an outlier component with a small fraction $\fol$ [@bib:resol_nim]. The signal probability $\fsig$ is calculated on an event-by-event basis from the function obtained by the $\dE$-$\mb$ two-dimensional fit used to extract the signal yield. A PDF for background events, ${\cal P}_{\rm bkg}$, is modeled as a sum of exponential and prompt components, and is convolved with a sum of two Gaussians $R_{\rm bkg}$. All parameters in ${\cal P}_{\rm bkg}$ and $R_{\rm bkg}$ are determined by the fit to the $\Dt$ distribution of a background-enhanced control sample, i.e. events outside of the $\dE$-$\mb$ signal region. We fix $\tau_\bz$ and ${{\Delta m_d}}$ at their world-average values [@bib:PDG2004]. In order to reduce the statistical error on ${{\cal A}}$, we include events without vertex information. The likelihood value in this case is obtained from the function of Eq. (\[eq:likelihood\]) integrated over $\Dt_i$. The only free parameters in the final fit are ${{\cal S}}$ and ${{\cal A}}$, which are determined by maximizing the likelihood function $L = \prod_iP_i(\Dt_i;{{\cal S}},{{\cal A}})$ where the product is over all events. An unbinned maximum likelihood fit to the 167 $\bz\to\ks\ks\ks$ candidates, containing $\NBsigksksks$ $\ks\ks\ks$ signal events, yields $$\begin{aligned} {{\cal S}}&=& \SksksksResultSS,\nonumber \\ {{\cal A}}&=& \AksksksResultSS.\nonumber \end{aligned}$$ We define the raw asymmetry in each $\Dt$ interval by $(N_{+}-N_{-})/(N_{+}+N_{-})$, where $N_{+(-)}$ is the number of observed candidates with $q=+1(-1)$. The raw asymmetries in two regions of the flavor-tagging parameter $r$ are shown in Fig. \[fig:asym\]. Note that these are simple projections onto the $\Delta t$ axis and do not reflect other event-by-event information (such as the signal fraction, the wrong tag fraction and the vertex resolution), which is in fact used in the unbinned maximum-likelihood fit for ${{\cal S}}$ and ${{\cal A}}$. The systematic error is primarily due to the resolution function ($\pm 0.12$ for ${{\cal S}}$ and $\pm 0.04$ for ${{\cal A}}$), the background fractions ($\pm 0.10$ for ${{\cal S}}$ and $\pm 0.03$ for ${{\cal A}}$), fit bias ($\pm 0.08$ for ${{\cal S}}$ and $\pm 0.05$ for ${{\cal A}}$), and background modeling ($\pm 0.08$ for ${{\cal S}}$ and $\pm 0.01$ for ${{\cal A}}$). Other sources of systematic error are uncertainties in the wrong tag fraction ($\pm 0.04$ for ${{\cal S}}$ and $\pm 0.01$ for ${{\cal A}}$), physics parameters $\Delta m_d$ and $\tau_{B^0}$ ($\pm 0.01$ for ${{\cal S}}$ and $\pm 0.01$ for ${{\cal A}}$), the vertexing ($\pm 0.02$ for ${{\cal S}}$ and $\pm 0.05$ for ${{\cal A}}$), and the effect of tag side interference [@Long:2003wq] ($\pm 0.02$ for ${{\cal S}}$ and $\pm 0.02$ for ${{\cal A}}$). We add each contribution in quadrature to obtain the total systematic errors. Various cross-checks of the measurement are performed. We reconstruct $\bpm\to\ks\ks\kpm$ decays without using the charged kaon for the vertex reconstruction and apply the same fit procedure. We obtain ${{\cal S}}_{\ks\ks\kpm} = \SkskskpmResultS$ and ${{\cal A}}_{\ks\ks\kpm} = \AkskskpmResultS$, which are consistent with no $CP$ asymmetry. MC pseudo-experiments are generated to perform ensemble tests. We find that the statistical errors obtained in our measurement are all consistent with the expectations from the ensemble tests. We apply the same procedure to the $\bz\to\jpsi\ks$ sample without $\jpsi$ daughter tracks for vertex reconstruction. We obtain ${{\cal S}}_{\jpsi\ks} = +0.68\pm 0.10$(stat) and ${{\cal A}}_{\jpsi\ks} = +0.02\pm 0.04$(stat), which are in good agreement with the world average values [@bib:HFAG]. We conclude that the vertex resolution for the $\bz\to\ks\ks\ks$ decay is well understood. We use a frequentist approach [@FeldmanCousins] to determine the statistical significance of the deviation from the SM. From 1-dimensional confidence intervals for ${{\cal S}}$, the case with ${{\cal S}}= -0.73$ for $\bz\to\ks\ks\ks$ is ruled out at a 99.7% confidence level, equivalent to $2.9\sigma$ significance for Gaussian errors. In summary, we have performed the measurement of $CP$-violation parameters in the $\bz \to \ks\ks\ks$ decay based on a sample of $275\times 10^6$ $B\bbar$ pairs. The decay is dominated by the $b\to s$ flavor-changing neutral current and the $\ks\ks\ks$ final state is a $CP$ eigenstate. It is therefore sensitive to a possible new $CP$-violating phase beyond the SM. The result differs from the SM expectation by 2.9 standard deviations. We thank the KEKB group for the excellent operation of the accelerator, the KEK cryogenics group for the efficient operation of the solenoid, and the KEK computer group and the NII for valuable computing and Super-SINET network support. We acknowledge support from MEXT and JSPS (Japan); ARC and DEST (Australia); NSFC (contract No. 10175071, China); DST (India); the BK21 program of MOEHRD and the CHEP SRC program of KOSEF (Korea); KBN (contract No. 2P03B 01324, Poland); MIST (Russia); MHEST (Slovenia); SNSF (Switzerland); NSC and MOE (Taiwan); and DOE (USA). [*Note added.*]{}—As we were preparing to submit this paper, we became aware of a paper from the BaBar collaboration [@Aubert:2005dr] which reports on the branching fraction and $CP$ asymmetries in the $\bz\to\ks\ks\ks$ decay. [999]{} M. Kobayashi and T. Maskawa, Prog. Theor. Phys. [**49**]{}, 652 (1973). A. B. Carter and A. I. Sanda, Phys. Rev. D **23**, 1567 (1981); I. I. Bigi and A. I. Sanda, Nucl. Phys. **B193**, 85 (1981). Throughout this paper, the inclusion of the charge conjugate decay mode is implied unless otherwise stated. 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--- abstract: 'We have theoretically studied the photon statistical properties in a nonlinear whispering-gallery-mode microresonator coupled with two nanoparticles. By tuning the relative position of two nanoparticles, the photon statistical features of the system can be modified remarkably. Interestingly, a controllable switching between unconventional and conventional photon blockade can be realized by manipulating the angular positions of two nanoparticles. We also investigate the influence of the Kerr effect on the second order correlation function and find that there is an optimal choice for the relative position of two nanoparticles and the strength of Kerr effect that can generate strong antibunching. Furthermore, under the strong driving, two photon blockade can be achieved when the system is close to an exceptional point. Our work may provide an effective way to control photon statistical characteristics and have potential applications in quantum information science.' author: - 'Wen-An Li' title: 'Tunable photon blockade in a whispering-gallery-mode microresonator coupled with two nanoparticles' --- introduction ============ A single photon source, an indispensable device for generating photons at the single-photon level, plays a central role in diverse areas such as quantum cryptography [@1], quantum information processing [@2], single-photon transistor [@3], quantum computation [@4] and so on. The photon blockade is one of most attractive mechanisms for constructing a single photon source. In close analog to Coulomb blockade for electrons [@5; @6], the photon blockade is a striking quantum phenomenon, where the excitation of a photon blocks the transport of subsequent photons for the nonlinear cavity so that they are emitted one by one. As a consequence, the cavity can only host one photon at a time, acting as a “photon turnstile” [@7; @8]. In 2005, the photon blockade was first demonstrated experimentally with a single atom trapped in an optical cavity [@9]. Subsequently, the strong antibunching behaviors were predicted in various experimental setups including a quantum dot in a photonic crystal [@10; @11] and circuit cavity quantum electrodynamics systems [@12; @13; @14]. In these works, the observation of antibunching requires large nonlinearities with respect to the decay rate of the system, so it is known as “conventional photon blockade”. Apart from the single photon blockade, the multi-photon blockade has also attracted much interest due to its potential applications in multiphoton quantum-nonlinear optics like an $n$-photon source ($n>1$) [@15]. To date, the multi-photon blockade has been studied in various configurations [@16; @17; @18; @19; @20]. For instance, the two- and three-photon blockade can be observed in a system consisting of a cavity with Kerr nonlinearity driven by a weak classical field [@16]. The prerequisite for realizing multi-photon blockade in this system is the presence of strong nonlinearities. Another method to realize three-photon blockade is based on the collective decay of two atoms trapped in a single-mode cavity with different coupling strengths [@17]. In this scheme, the two-photon and three-photon blockades strongly depend on the location of two atoms in the strong-coupling regime. Recently, the two-photon blockade was first observed in an atom-driven cavity quantum electrodynamics system [@21]. Although many progress on the study of multi-photon blockade has been made, the accomplishment of the multi-photon blockade is still challenging in experiments. In 2010, Liew and Savona found a new mechanism for the photon blockade, where strong photon antibunching can be obtained even with nonlinearities much smaller than the decay rates of the cavity modes [@22]. This mechanism is referred to as the “unconventional photon blockade”. Its feature can be understood as destructive quantum interference between different excitation pathways from the ground state to the two-photon states. Since then, a sequence of theoretical proposals based on this mechanism were suggested in many different systems including, for example, a bimodal optical cavity with a quantum dot [@23; @24; @25; @26; @27; @28; @29], symmetric and antisymmetric modes in weakly nonlinear photonic molecules [@30], coupled optomechanical systems [@31; @32]. More recently, the unconventional photon blockade was experimentally observed in two coupled superconducting circuit resonators [@33] and in a quantum dot embedded in a bimodal micropillar cavity [@34]. In parallel, the physical systems described by non-Hermitian Hamiltonians have also attracted much interest [@rev1; @rev2; @rev3; @rev4; @rev5], because such Hamiltonians exhibit special degeneracies known as exceptional points, at which two or more eigenvalues and the corresponding eigenvectors coalesce. In 2001, the physical existence of the exceptional point was experimentally demonstrated in microwave cavities [@35]. Subsequently, a variety of unconventional effects have been observed in experiments, such as loss-induced coherence [@36; @37], unidirectional lasing [@38], wireless power transfer [@39], and exotic topological states [@40; @41]. In recent experiments [@42; @43; @44], by coupling two nanoscale scatters (i.e. nanoparticles) to a whispering-gallery-mode (WGM) micro-toroid cavity, the system can be steered in a precise and controlled manner to the exceptional point. The presence of two nanoparticles within the mode volume of the cavity leads to the asymmetric backscattering of counter-propagating optical waves, which can be adjusted by manipulating the relative position of two nanoparticles. In the vicinity of the exceptional points, some counterintuitive effects have been shown including loss-induced revival of lasing [@37], ultra-sensitive sensor [@42], chiral lasing [@43] and optomechanically induced transparency [@46]. Motivated by above works [@42; @43; @44; @46], one question that arises naturally is whether the asymmetric coupling of two counter-propagating optical waves affects the photon statistical properties of cavity modes. In the previous works [@25; @26; @27; @30], studies on the unconventional photon blockade are based on the *symmetric* coupling of the optical modes. According to the optimal conditions, the required Kerr nonlinearity decreases with increasing coupling strength of the optical modes. It means that strong photon antibunching with weak Kerr nonlinearity requires large optical coupling between optical modes, which is not easy to realize in the experiments. Here, we consider the *asymmetric* coupling of two optical modes in one resonator and study the new possibility of controlling the photon blockade by tuning the relative angular position of two nanoparticles along the circumference of the nonlinear microresonator. In fact, adjusting the relative position of two nanoparticles corresponds to the change in the relative phase of the coupling coefficients without increasing the amplitudes of coupling constants. We find that the relative phase of the coupling coefficients plays a crucial role in modifying the photon statistical properties of the system. By tuning the relative position of two nanoparticles, the photon statistical properties can be well controlled and the switching between unconventional and conventional photon blockade can be realized. We also investigate the influence of the Kerr nonlinearity strength on the photon statistics properties. Furthermore, in the vicinity of an exceptional point, two-photon blockade effect can be achieved under the strong driving. Our work, with weak nonlinearity but without requiring strong coupling between optical modes, can be realized within current experimental techniques. The remainder of the paper is organized as follows. In Sec. II, the theoretical model and Hamiltonian are described for the nonlinear WGM microresonator coupled with two nanoparticles. In Sec. III, the output power spectra of the WGM microresonator system are presented. Subsequently, in Sec. IV, the photon statistical properties of present system is analytically and numerically discussed. Finally, a summary of the main results is given in Sec. V. ![Schematic diagram of the WGM microresonator with Kerr medium coupled with two nanoparticles which is coherently driven by a pump field at frequency $\omega_L$ through an optical tapered fiber waveguide. The WGM microresonator supports two counterpropagating modes (clockwise mode $\hat{a}_{\mathrm{C}}$ and anti-clockwise mode $\hat{a}_{\mathrm{A}}$), which can be asymmetrically coupled through backscattering by two nanoparticles. $\beta$ is the relative angle between two particles.[]{data-label="fig1"}](fig1){width="40.00000%"} theoretical model ================= As shown in Fig.\[fig1\], we consider a WGM microresonator with Kerr medium coupled to an optical fiber waveguide for in- and out-coupling of light. With its circular geometry, the WGM cavity supports clockwise and anti-clockwise travelling modes with degenerate eigenfrequencies $\omega_c$ and the same decay rate $\gamma=\gamma_{\mathrm{ex}}+\gamma_{\mathrm{in}}$. $\gamma_{\mathrm{ex}}$ is the external decay rate (the outgoing coupling coefficient) from the WGM microresonator into the tapered fiber and $\gamma_{\mathrm{in}}$ is the intrinsic decay rate. Two nanoparticles are placed in the evanescent field of the resonator, which can tune the coherent backscattering of clockwise and anti-clockwise travelling modes inside the resonator. In the presence of the optical loss, the system considered here is an open system and the Hamiltonian is non-Hermitian. \[sec:level2\]Review of the two-mode approximation model -------------------------------------------------------- In order to give a full description of this open system, we first briefly review the two-mode approximation model and the eigenmode evolution in a WGM microresonator with nanoscatter-induced broken spatial symmetry. The two-mode approximation model was first phenomenologically introduced for deformed microdisk cavities [@tma1; @tma2] and was later rigorously derived for the microdisk with two scatterers [@47]. The key idea is to model the dynamics in the slowly-varying envelope approximation in the time domain with a Schrödinger-like equation $id\Psi/dt=H\Psi$. Here, $\Psi$ is the two-component column vector $(\Psi_A, \Psi_C)^T$, where the superscript $T$ indicates the matrix transpose. The complex-valued entry $\Psi_A$ ($\Psi_C$) stands for all the field amplitudes of the anti-clockwise (clockwise) propagating waves. Since the microcavity is an open system, the corresponding effective Hamiltonian, $$H= \left( \begin{matrix} \Omega & J_1 \\ J_2 & \Omega \end{matrix} \right)$$ is a $2\times2$ matrix, which is in general non-Hermitian. The real parts of the diagonal elements $\Omega$ are the frequencies and the imaginary parts are the decay rates of the resonant traveling waves. The complex-valued off-diagonal elements $J_1$ and $J_2$ are the backscattering coefficients, which describe the scattering from the clockwise (anti-clockwise) to the anti-clockwise (clockwise) travelling wave. In general, in the open system the backscattering is asymmetric, $|J_1|\neq |J_2|$, which is allowed because of the non-Hermiticity of the Hamiltonian. A short calculation shows that the complex eigenvalues of $H$ are $\Omega_{\pm}=\Omega\pm\sqrt{J_1J_2}$ and the complex (not normalized) right eigenvectors are $$\Psi_{\pm}= \left( \begin{matrix} \sqrt{J_1} \\ \pm\sqrt{J_2} \end{matrix} \right).$$ Clearly, in the case of asymmetric backscattering one component of a given eigenvector is larger than the other component. Physically, it means that the eigenvectors show an imbalance of clockwise and anti-clockwise components if the backscattering is asymmetric. For the particular case of the WGM microresonator perturbed by two scatterers the matrix elements of $H$ are determined as follows [@42; @43; @47], $$\begin{aligned} &\Omega=\omega_c-i\frac{\gamma}{2}+\sum_{j=1}^2\epsilon_j, \\ &J_1=\sum_{j=1}^2\epsilon_j e^{-i2m\beta_j}, \\ &J_2=\sum_{j=1}^2\epsilon_j e^{i2m\beta_j},\end{aligned}$$ where $m$ is the azimuthal mode number, $\beta_j$ is the angular position of scatterer $j$ and $2\epsilon_j$ is the complex frequency splitting that is introduced by scatterer $j$ alone. $\epsilon_j$ can be calculated for the single-particle-microdisk system either fully numerically (using, e.g., the finite element method (FEM) [@fem], the boundary element method (BEM) [@bem]), or analytically using the Green’s function approach [@green]. In recent experiments, $\epsilon_j$ can be adjusted by tuning the distance between the resonator and the particles. Here, we take the position of one of the nanoparticles as the reference position. For example, take the orange particle (in Fig.1) as the first particle and set its angular position to be $\beta_1=0$, then the angular position of the second particle is $\beta_2=\beta$, where $\beta$ represents the relative angular position of the two scatters. Therefore, the asymmetric backscattering coefficients of counterpropagating waves, induced by the nanoparticles, can be reduced to $$J_{1,2}=\epsilon_1+\epsilon_2 e^{\mp i 2m\beta}.$$ It is noted that the relative angular $\beta$ is of great importance, since it can modify the photon statistical properties of the system (see discussions below). Although the two-mode approximation model was given for the isolated microdisk cavity perturbed by two particles, it is still valid in the waveguide-cavity systems by assuming that there is no backscattering of light between the microcavity and the waveguides. It can be justified when the distances between cavity and waveguides are sufficiently large. Note that, the extension of the two-mode model to waveguide-cavity systems has been introduced and tested in recent experiments [@42; @43]. \[sec:level2\]The Hamiltonian of our model ------------------------------------------ Based on above discussions, we will give a theoretical description of our model. To make our scheme work, a driving laser of frequency $\omega_L$ is applied to the system via the evanescent coupling of the optical fiber and the resonator, the field amplitudes are given by $F=\sqrt{\gamma_{\mathrm{ex}} P_L/\hbar \omega_L}$, where $P_L$ is the pump power. In the frame rotating with the input field frequency $\omega_L$, the Hamiltonian of this system is described by $$\begin{aligned} \label{eq1} \nonumber \hat{H}_{\mathrm{sys}}=&\Delta(\hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}}+ \hat{a}_{\mathrm{A}}^\dag \hat{a}_{\mathrm{A}})+U (\hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}}\hat{a}_{\mathrm{C}}\\ \nonumber &+\hat{a}_{\mathrm{A}}^\dag \hat{a}_{\mathrm{A}}^\dag \hat{a}_{\mathrm{A}}\hat{a}_{\mathrm{A}}) +J_1\hat{a}_{\mathrm{C}} \hat{a}_{\mathrm{A}}^\dag+J_2\hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{A}} \\ &+iF(\hat{a}_{\mathrm{C}}^\dag-\hat{a}_{\mathrm{C}})\end{aligned}$$ where $\Delta=\Delta_c+\mathrm{Re}(\epsilon_1+\epsilon_2)$, and $\Delta_c=\omega_c-\omega_L$. The nonlinear Kerr coefficient is given by $U=\hbar \omega_c^2 c n_2/n_0^2V_{\mathrm{eff}}$, where $c$ is the speed of light in vacuum, $n_0$ and $n_2$ are the linear and nonlinear refractive index of the material and $V_\mathrm{eff}$ is the effective mode volume. $\hat{a}_{\mathrm{C}}$ ($\hat{a}_{\mathrm{A}}$) and $\hat{a}_{\mathrm{C}}^\dag$ ($\hat{a}_{\mathrm{A}}^\dag$) are the photon annihilation and creation operators of the clockwise modes (anti-clockwise modes), satisfying the bosonic commutation relations $[\hat{a}_{\mathrm{C}},\hat{a}_{\mathrm{C}}^\dag]=1$ and $[\hat{a}_{\mathrm{A}},\hat{a}_{\mathrm{A}}^\dag]=1$. In above Hamiltonian (\[eq1\]), the first term denotes the energy of the WGM microresonator in the rotating frame. The second term represents the Kerr nonlinear interaction. The third and fourth terms are the coherent coupling of the clockwise mode with anti-clockwise mode. In general, $J_1\neq J_2$, which can be tuned by the relative angular position of two nanoparticles and the distance between nanoparticles and the WGM microresonator. Due to this asymmetric coupling between two counterpropagating modes, some interesting, controllable photon statistical properties will be shown in our system. The last term describes the interaction between the cavity field and the input field. output power spectra of the WGM microresonator ============================================== Before discussing the photon statistical properties of our system, we first study the system output power spectra. As mentioned above, when two nanoparticles are placed along the circumference of the resonator, the system exhibits fully asymmetric internal backscattering. The position of each particle can be controlled by a nanopositioner, which tunes the relative position and effective size of the nanotip in the WGM fields [@42; @43]. By carefully tuning the relative positions of two particles, the system can be steered to an exceptional point. For the present system, the nanoparticles induced frequency splitting of the optical modes can be derived as $\Delta\omega=\pm\sqrt{J_1J_2}$, thus the corresponding critical value of $\beta$ can be obtained as [@46] $$\beta_c=\frac{l\pi}{2m}\mp\frac{\mathrm{arg}(\epsilon_1)-\mathrm{arg}(\epsilon_2)}{2m} \quad (l=\pm1,\pm3,...),$$ where $\mp$ corresponds to $J_1=0$ or $J_2=0$. Here, $\epsilon_j$ ($j=1,2$) are complex numbers, and we assume $|\epsilon_1|=|\epsilon_2|$. $\mathrm{arg}(\epsilon_j)$ denotes the argument of complex number $\epsilon_j$. In the vicinity of the exceptional points, some unconventional effects may occur. Thus, it is of great interest to investigate the output power spectra of such coupled system when the relative position of two particles varies. According to the Hamiltonian (\[eq1\]) above, the dynamics of the coupled system can be described by the quantum Langevin equations \[eq3\] $$\begin{aligned} \frac{d}{dt}\hat{a}_{\mathrm{C}}=&\left(-\frac{\gamma_{\mathrm{opt}}}{2}-i\Delta-2iU\hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}}\right)\hat{a}_{\mathrm{C}}-iJ_2\hat{a}_{\mathrm{A}}+F+\hat{a}_{\mathrm{in}}^{\mathrm{C}},\\ \frac{d}{dt}\hat{a}_{\mathrm{A}}=&\left(-\frac{\gamma_{\mathrm{opt}}}{2}-i\Delta-2iU\hat{a}_{\mathrm{A}}^\dag \hat{a}_{\mathrm{A}}\right)\hat{a}_{\mathrm{A}}-iJ_1\hat{a}_{\mathrm{C}}+\hat{a}_{\mathrm{in}}^{\mathrm{A}},\end{aligned}$$ where $\hat{a}_{\mathrm{in}}^{\mathrm{C}}$ and $\hat{a}_{\mathrm{in}}^{\mathrm{A}}$ are the input vacuum noises of the cavity modes, respectively. $\gamma_{\mathrm{opt}}=\gamma_{\mathrm{in}}-\mathrm{Im}(\epsilon_1+\epsilon_2)$ is the total optical loss. Under the mean-field approximation, we assume that the mean values of these noise operators are zero, i.e., $\langle \hat{a}_{\mathrm{in}}^{\mathrm{C}}\rangle=\langle \hat{a}_{\mathrm{in}}^{\mathrm{A}}\rangle=0$. Here, we are interested in the influence of relative position of two nanoparticles on the system output power spectra. Under weak Kerr nonlinearity $U\ll \gamma_{\mathrm{in}}$, we can easily omit the Kerr interaction terms in above Eq.(\[eq3\]). Furthermore, we assume that all of the time derivatives in the quantum Langevin equations are set to be zero. Thus, it is easy to obtain steady-state values of the dynamical variables as $$\begin{aligned} \langle \hat{a}_{\mathrm{C}}\rangle=\frac{F(\gamma_{\mathrm{opt}}/2+i\Delta)}{(\gamma_{\mathrm{opt}}/2+i\Delta)^2+J_1J_2},\\ \langle \hat{a}_{\mathrm{A}}\rangle=\frac{-iFJ_1}{(\gamma_{\mathrm{opt}}/2+i\Delta)^2+J_1J_2},\end{aligned}$$ Note that $J_1J_2=\epsilon_1^2+\epsilon_2^2+2\epsilon_1\epsilon_2\cos(2m\beta)$. We can see that the $\beta$-dependent optical coupling rate indeed affects the intracavity optical intensity. By using the standard input-output relations, i.e., $\langle \hat{a}_{\mathrm{out}}\rangle=\langle \hat{a}_{\mathrm{in}}\rangle-\sqrt{\gamma_{\mathrm{ex}}}\langle \hat{a}_{\mathrm{C}}\rangle$, we obtain the the normalized power forward transmission spectra $$T=\left|\frac{\langle \hat{a}_{\mathrm{out}}\rangle}{\langle \hat{a}_{\mathrm{in}}\rangle}\right|^2=\left|1-\frac{\gamma_{\mathrm{ex}}}{F}\langle \hat{a}_{\mathrm{C}}\rangle\right|^2.$$ When the Kerr terms are included, the exact expression of $\langle \hat{a}_{\mathrm{C}}\rangle$ ($\langle \hat{a}_{\mathrm{A}}\rangle$) can not be obtained generally. Therefore, we numerically calculate the solutions to equations (\[eq3\]) under the mean-field approximation and plot the transmission rate versus detuning under different relative angular position of two particles in Fig. \[fig2\](a). Here, we have selected the experimentally accessible values $\gamma_{\mathrm{ex}}/\gamma_{\mathrm{in}}=1$, $\epsilon_1/\gamma_{\mathrm{in}}=1.5-0.1i$, $\epsilon_2/\gamma_{\mathrm{in}}=1.4999-0.101489i$, $U/\gamma_{\mathrm{in}}=0.059$ and $m=4$ [@42; @43; @46]. With these parameters, the exceptional point corresponds to the angular position at $\beta_c\approx0.4$. Fig. \[fig2\](a) shows the $\beta$-dependent transmission rate with two nanoparticles, featuring a asymmetric spectrum around the resonance due to the asymmetric backscattering between the clockwise- and anticlockwise-travelling waves. More interestingly, when $\beta$ is set to be $\pi/8$ (in the vicinity of the exceptional points), the transmission spectra demonstrates only one local minimum at the resonance. For $\beta=\pi/16$, strong absorption is shown around $\Delta/\gamma_{\mathrm{in}}=2$. However, by tuning the system close to the exceptional point (namely, $\beta=\pi/8$), a transparency window emerges. Hence, an optical switching (at $\Delta/\gamma_{\mathrm{in}}=2$) can be achieved by adjusting the relative angular position of two particles. For completeness, we plot the transmission spectra versus driving field detuning and relative angle between two nanoparticles in Fig. \[fig2\](b). photon statistical properties of the WGM microresonator with two nanoparticles ============================================================================== \[sec:level2\]General formalism ------------------------------- To correctly account for the driven-dissipative character of the system, we introduce the quantum master equation for the system density matrix, $$\begin{aligned} \label{eq7} \frac{d\hat{\rho}}{dt}=-i[\hat{H}_{\mathrm{sys}}, \hat{\rho}]+\gamma_1\mathcal{L}[\hat{a}_{\mathrm{C}}]\hat{\rho}+\gamma_2\mathcal{L}[\hat{a}_{\mathrm{A}}]\hat{\rho}\end{aligned}$$ where $\mathcal{L}[\hat{x}]\hat{\rho}=\hat{x}\hat{\rho} \hat{x}^\dag-\frac{1}{2}\hat{x}^\dag \hat{x}\hat{\rho}-\frac{1}{2}\hat{\rho} \hat{x}^\dag \hat{x}$ is the Lindblad superoperator term for the collapse operator $\hat{x}$ acting on the density matrix $\hat{\rho}$ to account for losses to the environment. $\gamma_1$ and $\gamma_2$ denote the damping constant of clockwise mode and anti-clockwise mode, respectively. Here, the decay rates of the resonator modes are assumed to be equal, i.e., $\gamma_1=\gamma_2=\gamma_{\mathrm{opt}}$. The steady-state solution $\rho_{ss}$ of the density matrix $\hat{\rho}$ can be obtained by setting $d\hat{\rho}/dt=0$ in Eq. (\[eq7\]). To observe the photon blockade, we focus on the statistic properties of clockwise mode photons, which are described by the zero-delay-time second order correlation function of the steady state, defined by $$\label{eq9} g_{\mathrm{C}}^{(2)}(0)=\frac{\langle \hat{a}_{\mathrm{C}}^\dag\hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}} \hat{a}_{\mathrm{C}}\rangle}{\langle \hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}}\rangle^2}=\frac{\mathrm{Tr}\left(\rho_{ss}\hat{a}^\dag_{\mathrm{C}}\hat{a}^\dag_{\mathrm{C}}\hat{a}_{\mathrm{C}}\hat{a}_{\mathrm{C}}\right)}{[\mathrm{Tr}(\rho_{ss}\hat{a}^\dag_{\mathrm{C}}\hat{a}_{\mathrm{C}})]^2}.$$ This physical quantity emphasizes the joint probability of detecting two photons at the same time. The value of $g_{\mathrm{C}}^{(2)}(0)<1$ ($g_{\mathrm{C}}^{(2)}(0)>1$) corresponds to sub-Poisson (super-Poisson) statistics of the cavity field, which is a nonclassical (classical) effect. This effect of the sub-Poisson photon statistics is often referred to as photon antibunching. \[sec:level2\]Weak driving limit -------------------------------- If the driving field coupling $F$ is very weak, due to photon blockade, only lower energy levels of the cavity field are occupied (the total excitation number of the system doesn’t exceed 2). In this case, the truncated state of the system can be expanded as $$\begin{aligned} \label{eq10} \nonumber |\psi\rangle=&C_{00}|00\rangle+C_{10}|10\rangle+C_{01}|01\rangle \\ &+C_{11}|11\rangle+C_{20}|20\rangle+C_{02}|02\rangle.\end{aligned}$$ Here $|mn\rangle$ represents the fock state basis of the system with the number $m$ denoting the photon number in clockwise cavity mode, $n$ denoting the photon number in anti-clockwise cavity mode. $C_{mn}$ stands for the probability amplitude and $|C_{mn}|^2$ denotes occupying probability in the state $|mn\rangle$. Using Eq.(\[eq9\]) and Eq.(\[eq10\]), the second order correlation function $g_{\mathrm{C}}^{(2)}(0)$ can be approximately given as $$\label{eq11} g_{\mathrm{C}}^{(2)}(0) \simeq \frac{2|C_{20}|^2}{|C_{10}|^4}.$$ The result of Eq.(\[eq11\]) can be used to approximately describe the photon statistical properties in the weak driving limit. To obtain the coefficients $C_{mn}$ in Eq.(\[eq10\]), we can substitute the state $|\psi\rangle$ into the Schroödinger’s equation $i\frac{\partial}{\partial t}|\psi\rangle=\widetilde{H}|\psi\rangle$, where $\widetilde{H}=\hat{H}_{\mathrm{sys}}-i\frac{\gamma_{\mathrm{opt}}}{2}(\hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}}+\hat{a}_{\mathrm{A}}^\dag \hat{a}_{\mathrm{A}})$. Then, we get a set of equations for the coefficients $$\begin{aligned} &i\frac{\partial}{\partial t}C_{00}\simeq 0,\\ &i\frac{\partial}{\partial t}C_{10}=iF C_{00}+\bar{\Delta}C_{10}+J_2C_{01}-i\sqrt{2}F C_{20},\\ &i\frac{\partial}{\partial t}C_{01}=J_1C_{10}+\bar{\Delta}C_{01}-iF C_{11},\\ &i\frac{\partial}{\partial t}C_{11}=iFC_{01}+2\bar{\Delta}C_{11}+\sqrt{2}J_1C_{20}+\sqrt{2}J_2C_{02},\\ &i\frac{\partial}{\partial t}C_{20}=i\sqrt{2}FC_{10}+\sqrt{2}J_2C_{11}+2(\bar{\Delta}+U_1)C_{20},\\ &i\frac{\partial}{\partial t}C_{02}=\sqrt{2}J_1C_{11}+2(\bar{\Delta}+U_2)C_{02},\end{aligned}$$ where $\bar{\Delta}=\Delta-i\frac{\gamma_{\mathrm{opt}}}{2}$. Under the weak driving condition $F\ll\gamma_{\mathrm{in}}$, we have $|C_{00}|\gg|C_{10}|, |C_{01}|\gg|C_{20}|, |C_{11}|, |C_{02}|$, thus $C_{20}$ and $C_{11}$ can be removed in the Eq.(14b) and Eq.(14c). The vacuum state $C_{00}$ approximately has unity occupancy. Then, the steady-state solution can be found by solving the coupled equations for the coefficients. For simplicity of presentation, only $C_{10}$ and $C_{20}$ are given as below: $$\begin{aligned} \label{eq13} \nonumber &C_{10}=\frac{-iF\bar{\Delta}}{\bar{\Delta}^2-J_1J_2}, \\ &C_{20}=\frac{1}{2\sqrt{2}}\frac{F^2[J_1J_2U+2\bar{\Delta}^2(\bar{\Delta}+U)]}{(\bar{\Delta}^2-J_1J_2)[J_1J_2(\bar{\Delta}+U)-\bar{\Delta}(\bar{\Delta}+U)^2]}.\end{aligned}$$ With Eq.(\[eq11\]) and Eq.(\[eq13\]), we can approximately obtain the analytical expression of the second order correlation function and the optimal condition for the photon blockade. However, the exact expressions for the condition $g_{\mathrm{C}}^{(2)}(0)\approx 0$ are too cumbersome to be presented here. Interestingly, from Eq.(\[eq13\]), it is obvious that the second order correlation function is closely related to the relative angular position of two particles. In particular, in the vicinity of exceptional points, i.e., $\beta=\beta_c\approx0.4$, $g_{\mathrm{C}}^{(2)}(0)\simeq |\bar{\Delta}|^2/|\bar{\Delta}+U|^2$. It means that the system shows stronger photon antibunching effect as the Kerr nonlinearity increases. When $\beta\neq\beta_c$, the case becomes different. The in-depth discussions and results of numerical calculation by the master equation approach for different parameter conditions are presented in the following subsections. ![The second order correlation function $\mathrm{log}_{10}[g_{\mathrm{C}}^{(2)}(0)]$ as a function of the detuning $\Delta/\gamma_{\mathrm{in}}$ under various relative angular positions $\beta$ of two nanoparticles. The parameters have been selected the same as in Fig.\[fig2\].[]{data-label="fig3"}](fig3){width="45.00000%"} \[sec:level2\]Single photon blockade ------------------------------------ In this subsection, we study the photon statistical properties of the nonlinear WGM microresonator system with two nanoparticles by numerically solving the master equation (\[eq7\]). Figure 3 displays the second order correlation function $g_{\mathrm{C}}^{(2)}(0)$ of the cavity mode $\hat{a}_{\mathrm{C}}$ in a logarithmic scale as a function of the detuning $\Delta/\gamma_{\mathrm{in}}$ under various relative angular positions $\beta$ of two nanoparticles. Here, we consider the weak Kerr nonlinearity $U/\gamma_{\mathrm{in}}=0.059$. We can see that the profile of the second-order correlation function in a logarithmic scale $\mathrm{log}_{10}[g_{\mathrm{C}}^{(2)}(0)]$ varying with the detuning $\Delta/\gamma_{\mathrm{in}}$ exhibits a peak-dip structure. With the increasing of the detuning $\Delta$, the value of $\mathrm{log}_{10}[g_{\mathrm{C}}^{(2)}(0)]$ first arrives at the maximum and then at the minimum. For $\beta=\pi/4$, the maximum value at the peak is about $0.5$ while the minimum value at the dip is about $-3.0$. Interestingly, the photon statistics can be changed dramatically by tuning the value of relative angular position $\beta$. We find that the value of the second order correlation function $g_{\mathrm{C}}^{(2)}(0)$ is about $0.92$ when $\Delta/\gamma_{\mathrm{in}}=0.3$ and $\beta=\pi/8$. However, by tuning the parameter $\beta$ to $\pi/4$ and keeping $\Delta/\gamma_{\mathrm{in}}=0.3$, the value of $g_{\mathrm{C}}^{(2)}(0)$ rapidly decreases to $0.002$, which indicates the strong antibunching effect. ![Energy-level and transition path diagram of the WGM system with two nanoparticles. The quantum interference between different paths leads to strong antibunching effect.[]{data-label="fig4"}](fig4){width="46.00000%"} The physical grounds behind the photon antibunching under the weak Kerr effect can be explained by the effect of quantum interference between different pathways, as shown in Fig.\[fig4\]. There are two paths for the system to reach the two photon state of clockwise cavity mode :(i) the direct path, i.e., $|00\rangle\stackrel{F}{\longrightarrow}|10\rangle\stackrel{F}{\longrightarrow}|20\rangle$, and (ii) tunnel-coupling-mediated transition $|00\rangle\stackrel{F}{\longrightarrow}|10\rangle\stackrel{J_1}{\longrightarrow}|01\rangle\stackrel{F}{\longrightarrow}|11\rangle\stackrel{J_2}{\longrightarrow}|20\rangle$. With proper choice of parameters, the photons coming from the two pathways would destructively interfere. In other words, the destructive quantum interference between the direct path and the indirect path can reduce the probability in the two-photon excited state, this is known as unconventional photon blockade. In present model, strong antibunching effect can be achieved through adjusting the relative phase of coupling coefficients $J_1$ and $J_2$, instead of increasing the amplitudes of them. Thus, the relative phase $\beta$ plays a crucial role in the photon statistical properties of the system. In particular, when the system is steered close to an exceptional point (i.e., $\beta\approx0.4$), the indirect path $|00\rangle\stackrel{F}{\longrightarrow}|10\rangle\stackrel{J_1}{\longrightarrow}|01\rangle\stackrel{F}{\longrightarrow}|11\rangle\stackrel{J_2}{\longrightarrow}|20\rangle$ is blocked due to the fact that $J_1=0$ or $J_2=0$ in the vicinity of the exceptional point. Only the direct path to the two photon state is allowed and then strong antibunching requires large nonlinearities, which is just the feature of the conventional photon blockade. Accordingly, a controllable switching between the unconventional and conventional photon blockade can be realized by tuning the relative angle $\beta$. In our model, another factor affecting the photon statistical properties is the Kerr nonlinearity. Figure \[fig5\] plots the second order correlation function $g_{\mathrm{C}}^{(2)}(0)$ in a logarithmic scale versus Kerr nonlinearity $U/\gamma_{\mathrm{in}}$ under various values of $\beta$ by fixing the value of detuning at $\Delta/\gamma_{\mathrm{in}}=0.4$. In contrast to the conventional photon blockade, the value of $g_{\mathrm{C}}^{(2)}(0)$ does not always monotonically decrease with the increase of the strength of Kerr nonlinearity. It’s worth noting that there exists a local minimum value of $g_{\mathrm{C}}^{(2)}(0)$, which can be adjusted by tuning the value of $\beta$. It suggests that the photon antibunching can be further enhanced with an optimal choice for the relative position $\beta$ and Kerr coefficient $U$. In previous works [@25; @26; @27; @30], achieving strong photon antibunching with weak Kerr effect requires a large coupling strength between cavity modes. Here, we only need to tune the relative angular position $\beta$ of two nanoparticles without requiring the large coupling strength $J_{1,2}$. Note that, in the vicinity of the exceptional points, i.e. $\beta=\pi/8$, the local minimum in the curve disappears. With increasing the strength of Kerr effect, the value of $g_{\mathrm{C}}^{(2)}(0)$ monotonically decreases. The physical reason is that, at the exceptional point, quantum interference between different pathways is suppressed and the second order correlation function $g_{\mathrm{C}}^{(2)}(0)$ shows the features of conventional photon blockade. ![The second order correlation function in a logarithmic scale $\mathrm{log}_{10}[g_{\mathrm{C}}^{(2)}(0)]$ versus Kerr nonlinearity $U/\gamma_{\mathrm{in}}$ under various values of $\beta$ by fixing $\Delta/\gamma_{\mathrm{in}}=0.4$. All other parameters are given the same as in Fig.\[fig2\]. The black dotted line denotes the position where $\mathrm{log}_{10}[g_{\mathrm{C}}^{(2)}(0)]=0$.[]{data-label="fig5"}](fig5){width="42.00000%"} ![The second order (blue solid curve) and third order (red dot-dashed curve) correlation function as a function of the detuning $\Delta/\gamma_{\mathrm{in}}$ under various relative positions $\beta$ in (a)-(c); (d) The second and third order correlation function as a function of relative positions $\beta$ by fixing the detuning $\Delta/\gamma_{\mathrm{in}}=-2.8$. Here we have selected $F/\gamma_{\mathrm{in}}=2$. All other parameters are given the same as in Fig.\[fig2\]. The black dotted line denotes the position where $\mathrm{log}_{10}[g_{\mathrm{C}}^{(n)}(0)]=0$ ($n=2,3$). The two-photon bunching and three-photon antibunching can be achieved in the grey area.[]{data-label="fig6"}](fig6){width="49.00000%"} \[sec:level2\]Two photon blockade --------------------------------- Next, we consider the strong-pumping case (e.g., $F=2\gamma_{\mathrm{in}}$), where the photon excitation becomes much stronger. This allows for the implementation of two-photon blockade where the presence of two photons suppresses the addition of further photons. To demonstrate the two photon blockade effect, we plot the equal-time second order field correlation function ($g_{\mathrm{C}}^{(2)}(0)=\langle \hat{a}_{\mathrm{C}}^{\dag 2}\hat{a}_{\mathrm{C}}^2\rangle/\langle \hat{a}_{\mathrm{C}}^\dag\hat{a}_{\mathrm{C}}\rangle^2$) and third order field correlation function ($g_{\mathrm{C}}^{(3)}(0)=\langle \hat{a}_{\mathrm{C}}^{\dag 3}\hat{a}_{\mathrm{C}}^3\rangle/\langle \hat{a}_{\mathrm{C}}^\dag\hat{a}_{\mathrm{C}}\rangle^3$) in logarithmic units as a function of the normalized detuning $\Delta/\gamma_{\mathrm{in}}$ under various values of $\beta$ in Fig.\[fig6\](a)-(c). Here, the system parameters are chosen as the same as those used in Fig.\[fig2\]. It is noteworthy that, in the vicinity of the exceptional points (i.e. $\beta=\pi/8\approx \beta_c$), clear signatures of two photon blockade phenomena ($g_{\mathrm{C}}^{(2)}(0)>1$, and $g_{\mathrm{C}}^{(3)}(0)<1$) are shown in the grey area of Figure.\[fig6\](c). When $\beta\neq\beta_c$, the two photon blockade phenomena disappear and at the mean time single photon blockade appears. The physical reason is that when the system is not near the exceptional points destructive quantum interference between different pathways leads to strong photon antibunching, so the two photon bunching is greatly suppressed. On the contrary, at the exceptional points, the two-photon bunching and three-photon antibunching can be realized under the strong driving because of the uneven energy levels of the system. This feature leads to an optical switching from the single-photon blockade to the two-photon blockade by just tuning the relative angular position of two nanoparticles. To show this switching operation, we plot the second order (blue solid curve) and third order (red dot-dashed curve) field correlation functions as a function of the relative angular position $\beta/\pi$ in Fig.\[fig6\](d) by fixing the detuning $\Delta/\gamma_{\mathrm{in}}=-2.8$. Moreover, Kerr effect are also crucial for the degree of three photon antibunching. Figure \[fig7\] plots the second order and third order field correlation functions versus Kerr nonlinearity strength by fixing the relative angular position $\beta=\pi/8$ and detuning $\Delta/\gamma_{\mathrm{in}}=-2.8$. From fig.\[fig7\], we find that two photon blockade effect occurs in the grey region. Therefore, the choice of relative position $\beta$ and Kerr nonlinearity $U$ is very important for achieving the two photon blockade in the system discussed here. ![The second order (blue solid curve) and third order (red dot-dashed curve) correlation function as a function of the Kerr nonlinearity strength $U/\gamma_{\mathrm{in}}$ by setting $\Delta/\gamma_{\mathrm{in}}=-2.8$, $\beta=\pi/8$, $F/\gamma_{\mathrm{in}}=2$. All other system parameters used here are the same as in Fig.\[fig2\]. The black dotted line denotes the position where $\mathrm{log}_{10}[g_{\mathrm{C}}^{(n)}(0)]=0$ ($n=2,3$). The two-photon blockade can be achieved in the grey area.[]{data-label="fig7"}](fig7){width="40.00000%"} conclusions =========== In conclusion, we have studied the photon statistical properties in the nonlinear WGM microresonator coupled with two nanoparticles. By tuning the relative angular position $\beta$ of two nanoparticles, the photon statistical properties of the system can be well controlled and the switching between unconventional and conventional photon blockade can be achieved. We also investigate the influence of the Kerr effect on the second order correlation function and find that there is an optimal choice for relative position $\beta$ and Kerr coefficient $U$ to generate strong antibunching. Moreover, under the strong driving, two photon bunching and three photon antibunching can be achieved when the system is steered to the exceptional points. 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--- abstract: 'The neutral member of a Majorana fermion triplet $(\Sigma^+,\Sigma^0,\Sigma^-)$ is proposed as a candidate for the dark matter of the Universe. It may also serve as the seesaw anchor for obtaining a radiative neutrino mass.' --- UCRHEP-T454\ KANAZAWA-08-07\ September 2008 [**Fermion Triplet Dark Matter\ and Radiative Neutrino Mass\ **]{}  : The cosmological and astrophysical evidence [@bhs05] for dark matter (DM) is a powerful incentive for considering new particles and interactions beyond those of the standard model (SM) of quarks and leptons. Whereas most studies have concentrated on supersymmetric extensions of the SM, other excellent DM candidates abound. For example, if the SM is extended to include just one new scalar or fermion multiplet, then there are many possible DM candidates [@cfs06]. In particular, a scalar doublet $(\eta^+,\eta^0)$ odd under an exactly conserved $Z_2$ symmetry [@dm78] is a very good choice [@m06-1; @bhr06; @lnot07; @glbe07]. Such a “dark” scalar doublet is amenable to discovery at the Large Hadron Collider (LHC) [@cmr07]. It is also very useful for generating small radiative Majorana neutrino masses [@m06-1] if there exist neutral singlet fermions $N_i$ which are odd under $Z_2$. For a brief review of the further developments of this idea of “scotogenic” neutrino mass, see Ref. [@m07-1]. More recently, it has been extended to include $A_4$ tribimaximal mixing [@m08-1] as well. Now the lightest $N_i$ may also be considered a DM candidate [@kst03; @kms06; @akrsz08]. However, processes such as $\mu \to e \gamma$ impose severe constraints on the Yukawa couplings of $N_i$, making it difficult to satisfy the cosmological relic abundance required. One way to avoid this problem is to introduce additional interactions for $N_i$ [@ks06; @bm08; @s08]. Other SM singlets have also been considered [@sz85; @hllt07; @blmrs08; @m08-3; @hln08; @l08; @ckk08; @ks08; @cms08]. Whereas the canonical seesaw mechanism uses the fermion singlet $N$ so that the neutrino mass is given by $m_\nu \simeq -m_D^2/m_N$ where $m_D$ is the Dirac mass linking $\nu$ to $N$, it is not the only way to realize the generic dimension-five effective operator [@w79] $${\cal L}_5 = - {f_{ij} \over 2 \Lambda} (\nu_i \phi^0 - l_i \phi^+) (\nu_j \phi^0 - l_j \phi^+) + H.c.$$ for obtaining small Majorana neutrino masses in the SM. In fact, there are three tree-level (and three generic one-loop) realizations [@m98]. The second most often considered mechanism for neutrino mass is that of a scalar triplet $(\xi^{++},\xi^+,\xi^0)$, whereas the third tree-level realization, i.e. that of a fermion triplet $(\Sigma^+,\Sigma^0,\Sigma^-)$ [@flhj89], has not received as much attention. However, it has some rather intriguing properties. It supports a new U(1) gauge symmetry [@m02; @mr02; @bd05] and may be important for gauge-coupling unification [@m05; @bs07; @df07] in the SM. It may be probed [@mr02; @bns07; @fhs08; @aa08] at the LHC, and is being discussed in a variety of other contexts [@f07; @abbgh08; @ff08; @m08-4; @moy08]. Now suppose $\Sigma^0$ is also odd under $Z_2$, then it may become a DM candidate [@m05; @ff08] and replace $N$ in the radiative generation of neutrino mass as shown in Fig. 1. (360,120)(0,0) (90,10)(130,10) (180,10)(130,10) (180,10)(230,10) (270,10)(230,10) (155,85)(180,60)3 (205,85)(180,60)3 (180,10)(50,90,180)3 (180,10)(50,90,0)3 (110,0)\[\][$\nu_i$]{} (250,0)\[\][$\nu_j$]{} (180,0)\[\][$\Sigma^0_k$]{} (135,50)\[\][$\eta^0$]{} (230,50)\[\][$\eta^0$]{} (150,90)\[\][$\phi^{0}$]{} (217,90)\[\][$\phi^{0}$]{} The difference between $N$ and $\Sigma^0$ is that whereas the former has only Yukawa interactions in the minimal scenario, the latter has electroweak gauge interactions, i.e. $\Sigma^0 \Sigma^\pm W^\mp$, which will allow $\Sigma^0$ and $\Sigma^\pm$ to annihilate and coannihilate in the early Universe to account for the correct DM relic abundance without relying on their Yukawa couplings [@kms06]. Note that $\Sigma^\pm$ is slightly heavier than $\Sigma^0$ from electroweak radiative corrections [@cfs06]. It is also possible [@m08-2] that $\Sigma^0$ exists as DM without having anything to do with neutrino mass.  : It is well-known that gauge-coupling unification occurs for the minimal supersymmetric standard model (MSSM) but not the SM. On the other hand, the addition of $\Sigma$ improves the situation and gauge-coupling unification in the SM is possible [@m05; @bs07; @df07] with the inclusion of some other fields. Consider the one-loop renormalization-group equations governing the evolution of the three gauge couplings of the standard $SU(3)_C \times SU(2)_L \times U(1)_Y$ gauge group as functions of mass scale: $${1 \over \alpha_i(M_1)} - {1 \over \alpha_i(M_2)} = {b_i \over 2 \pi} \ln {M_2 \over M_1},$$ where $\alpha_i = g_i^2/4 \pi$ and the numbers $b_i$ are determined by the particle content of the model between $M_1$ and $M_2$. In the SM with one Higgs scalar doublet, these are given by $$\begin{aligned} SU(3)_C &:& b_C = -11 + (4/3) N_f = -7, \\ SU(2)_L &:& b_L = -22/3 + (4/3) N_F + 1/6 = -19/6, \\ U(1)_Y &:& b_Y = (4/3)N_f + 1/10 = 41/10,\end{aligned}$$ where $N_f = 3$ is the number of quark and lepton families and $b_Y$ has been normalized by the well-known factor of 3/5. Using the input [@pdg06] $$\begin{aligned} \alpha_L(M_Z) &=& (\sqrt{2}/\pi)G_F M^2_W = 0.0340, \\ \alpha_Y(M_Z) &=& \alpha_L(M_Z) \tan^2 \theta_W = 0.0102, \\ \alpha_C(M_Z) &=& 0.122,\end{aligned}$$ it is easy to check that the SM particle content is incompatible with the unification condition $$\alpha_C(M_U) = \alpha_L(M_U) = (5/3) \alpha_Y (M_U) = \alpha_U.$$ Suppose $(\Sigma^+,\Sigma^0,\Sigma^-) \sim (1,3,0)$ and $(\eta^+,\eta^0) \sim (1,2,1/2)$ are added at the scale $M_X$, together with two real scalar color octets $\zeta_{1,2} \sim (8,1,0)$, then $\Delta b_L = 2(2/3) + 1/6 = 3/2$, $\Delta b_Y = 1/10$, and $\Delta b_C = 3(2)(1/6) = 1$ between $M_X$ and $M_U$, so that Eq. (9) implies $$\ln {M_U \over M_Z} = \left( {\pi \over 45} \right) \left( {3 \over \alpha_Y(M_Z)} + {9 \over \alpha_L(M_Z)} - {14 \over \alpha_C(M_Z)} \right) = 31.0.$$ Hence $M_U \simeq 2.65 \times 10^{15}~{\rm GeV}$, which is an acceptable value [@bmm82] for suppressing the proton decay lifetime above the experimental lower bound of about $10^{32}$ years. The scale $M_X$ is determined to be about 730 GeV. Thus the new particles have a chance of being observed at the LHC. In particular, the $\zeta$ scalars would be produced in abundance at the LHC because they are color octets [@bm84; @gw07] and would decay in one loop to two gluons [@m05], i.e. $\zeta \to \zeta \zeta \to g g$.  : Consider the case where the SM is extended to include only one fermion triplet $\Sigma = (\Sigma^+,\Sigma^0,\Sigma^-) \sim (1,3,0)$ which is odd under $Z_2$ with all other fields even. In that case, $m_{\Sigma^\pm} = m_{\Sigma^0}$ at tree level, but the former is heavier than the latter from one-loop electroweak radiative corrections, namely [@cfs06] $$\Delta = m_{\Sigma^\pm} - m_{\Sigma^0} = {\alpha_L m_\Sigma \over 4 \pi} \left\{ f\left( {M_W \over m_\Sigma} \right) - \cos^2 \theta_W f\left( {M_Z \over m_\Sigma} \right) \right\},$$ where $$\begin{aligned} f(r) &=& -r^2 + r^4 \ln r + r(r^2-4)^{1/2} (1+r^2/2) \ln [-1-(r^2-4)^{1/2}r/2 + r^2/2] \nonumber \\ &\simeq& 2 \pi r - 3 r^2, ~~{\rm for}~r \ll 1.\end{aligned}$$ This splitting is positive and approaches $(\alpha_L/2)\cos \theta_W(1- \cos \theta_W)M_Z \simeq 167$ MeV for large $m_\Sigma$. This means that $\Sigma^\pm$ is allowed to decay into $\Sigma^0$ plus a virtual $W^\pm$ which then converts into $\pi^\pm$ or leptons. The relic abundance of $\Sigma^0$ is determined by the annihilation and coannihilation of itself and $\Sigma^\pm$. These cross sections are dominated by their $s$-wave contributions. For $\Sigma^0 \Sigma^0 \to W^+ W^-$ through $\Sigma^\pm$ exchange, $$\sigma (\Sigma^0\Sigma^0)|v| \simeq {2 \pi \alpha_L^2 \over m_\Sigma^2},$$ where $v$ is the relative velocity of the incident particles in their center of mass and $m_\Sigma \gg \Delta$ is assumed. As for coannihilation, several processes have to be included: $\Sigma^0 \Sigma^\pm \to W^0 W^\pm$ through $\Sigma^\pm$ exchange and $\Sigma^0 \Sigma^\pm \to W^\pm \to \bar{f} f', ~W^\pm W^0,~W^\pm H$, as well as $\Sigma^+ \Sigma^-\to W^0W^0$ through $\Sigma^\pm$ exchange, $\Sigma^+ \Sigma^- \to W^+W^-$ through $\Sigma^0$ exchange, $\Sigma^+ \Sigma^- \to W^0 \to \bar{f} f, ~W^+ W^-,~W^0 H$, and $\Sigma^\pm \Sigma^\pm \to W^\pm W^\pm$ through $\Sigma^0$ exchange. They are also easily calculated to be $$\sigma (\Sigma^0\Sigma^\pm)|v| \simeq {29 \pi \alpha_L^2 \over 8 m_\Sigma^2}, \qquad \sigma (\Sigma^+ \Sigma^-)|v| \simeq {37 \pi \alpha_L^2 \over 8 m_\Sigma^2}, \qquad \sigma (\Sigma^\pm \Sigma^\pm)|v| \simeq { \pi \alpha_L^2 \over m_\Sigma^2}.$$ In the above, we have kept only the $a_{ij}$ coefficients in the relative-velocity expansion of the cross section: $\sigma_{ij}|v| = a_{ij} + b_{ij}v^2$. Note that $\sigma (\Sigma^0\Sigma^0)|v|$ is smaller than $\sigma (\Sigma^0\Sigma^\pm)|v|$ and $\sigma (\Sigma^+\Sigma^-)|v|$. This means that $\Sigma^\pm$ contributes importantly to the relic abundance of $\Sigma^0$. Using the method developed in Ref. [@gs91] to take coannihilation into account, we calculate below the relic abundance of $\Sigma^0$ as a function of $m_\Sigma$ and $\Delta$. The decoupling temperature $T_f$ of $\Sigma^0$ is estimated by using the effective cross section $\sigma_{\rm eff}$ and the effective degrees of freedom $g_{\rm eff}$ from the condition $$x = \ln {0.038~g_{\rm eff}~M_{\rm Pl}~m_\Sigma~\langle\sigma_{\rm eff}|v|\rangle \over \sqrt{g_\ast x}},$$ where $x=m_\Sigma/T$, $g_\ast = 106.75$ is the SM number of relativistic degrees of freedom at $T_f$, $M_{\rm Pl}=1.22\times 10^{19}$ GeV is the Planck mass, and $$\begin{aligned} \langle \sigma_{\rm eff}|v| \rangle &=& {g_0^2 \over g_{\rm eff}^2} \sigma (\Sigma^0\Sigma^0) + 4{g_0 g_\pm \over g_{\rm eff}^2} \sigma (\Sigma^0\Sigma^\pm) (1+\epsilon)^{3/2} \exp ({-\epsilon x}) \nonumber \\ &+& {g_{\pm}^2 \over g_{\rm eff}^2} [2 \sigma (\Sigma^+ \Sigma^-) + 2 \sigma (\Sigma^\pm \Sigma^\pm)](1+\epsilon)^2 \exp ({-2\epsilon x}), \nonumber \\ g_{\rm eff} &=& g_0+2 g_\pm(1+\epsilon)^{3/2} \exp ({-\epsilon x}),\end{aligned}$$ with $g_0=g_\pm=2$ and $\epsilon = \Delta/m_\Sigma$. The relic abundance is then given by $$\Omega h^2={1.04\times 10^9 x_f \over g_\ast^{1/2}M_{\rm Pl}({\rm GeV})I_a},$$ where $I_a=x_f\int^\infty_{x_f} a_{\rm eff}x^{-2}dx$, $x_f = m_\Sigma/T_f$, and $a_{\rm eff}$ is extracted from $\sigma_{\rm eff}|v|=a_{\rm eff} +b_{\rm eff}v^2$. Using the observational data $\Omega h^2=0.11\pm 0.006$ [@dm03], we find $m_{\Sigma^0}$ to be in the range 2.28 to 2.42 TeV. Here the electroweak radiative contribution to $\Delta$ is already at its asymptotic value of about 167 MeV and its effect on $m_{\Sigma^0}$ is negligible. There is no tree-level contribution to $\Delta$ unless a Higgs triplet $(s^+,s^0,s^-)$ is added [@m05] with $\langle s^0 \rangle \neq 0$. However, this value should be less than about 1 GeV to conform to precision electroweak measurements; hence $\Delta$ would still be negligible and our result is unchanged.  : To have scotogenic neutrino masses, consider now the addition of the dark scalar doublet $\eta$ and the specific choice of one $\Sigma$ and two $N$’s, then under the assumption $ m^2_\Sigma, m^2_N \ll m^2_\eta$, the resulting radiative masses are given by [@m06-1] $$({\cal M}_\nu)_{\alpha\beta}={\lambda_5 v^2\over 8\pi^2} \sum_{j=0,1,2}{h_{\alpha j}h_{\beta j}M_j \over m_\eta^2}, \label{mass}$$ where $M_0=m_\Sigma$, $M_{1,2} = m_{N_{1,2}}$, $h_{\alpha j}$ are their Yukawa couplings, $v = \langle \phi^0 \rangle$, and $\lambda_5$ is the scalar coupling in the quartic term $(\lambda_5/2)(\Phi^\dagger \eta)^2 + H.c.$ which splits Re($\eta^0$) and Im($\eta^0$). Since $\lambda_5$ and $m_\eta$ are adjustable, it is clear that realistic neutrino masses may be obtained for $h \sim 10^{-2}$, in which case processes such as $\mu \to e \gamma$ are well below their experimental upper bounds. The problem with $N$ as dark matter is the requirement of $h > 1$ for it to have a large enough annihilation cross section [@kms06]. Looking at the form of Eq. (18), it is clear that it is possible to have a one-to-one correspondence betweeen the neutrino mass eigenvalues $m_{1,2,3}$ and the seesaw anchor masses $M_{0,1,2}$. As an [*anstaz*]{}, let the $3 \times 3$ Yukawa coupling matrix linking $e,\mu, \tau$ to $M_{0,1,2}$ be given by $$h_{\alpha j} = \pmatrix{\sqrt{2/3} & 1/\sqrt{3} & 0 \cr -1/\sqrt{6} & 1/\sqrt{3} & -1/\sqrt{2} \cr -1/\sqrt{6} & 1/\sqrt{3} & 1/\sqrt{2}} \pmatrix{h_0 & 0 & 0 \cr 0 & h_1 & 0 \cr 0 & 0 & h_2},$$ then the tribimaximal mixing of neutrinos is obtained, and their mass eigenvalues are $$m_{i+1} = {\lambda_5 v^2 h_i^2 M_i \over 8 \pi^2 m_\eta^2},~~~i = 0,1,2.$$  : Since $\Sigma^0$ has gauge interactions, its relic abundance is adequately accounted for. There is no need for it to have large Yukawa couplings, as is in the case [@kms06] of choosing the singlet fermion $N$ as dark matter, where $m_\eta$ must also be close to $m_N$. This means radiative flavor-changing decays are easily suppressed. In the above example, using the experimental upper bound of $1.2 \times 10^{-11}$ on the branching fraction of $\mu \to e \gamma$, this corresponds to the condition $$| |h_0|^2 - |h_1|^2 | < 0.77 (m_\eta/2.35~{\rm TeV})^2.$$ Since $h$ is not required to be large and $\eta$ should be heavier than $\Sigma$, the tension between the constraints of dark-matter relic abundance and flavor-changing radiative decays is removed.  : In this paper we have proposed the addition of a fermion triplet $(\Sigma^+,\Sigma^0,\Sigma^-)$ to the standard model of quarks and leptons. We consider $\Sigma^0$ as a dark-matter candidate, being odd under an exactly conserved $Z_2$ symmetry. We show that with $\Sigma^\pm$ slightly heavier than $\Sigma^0$ from electroweak radiative corrections, $m_\Sigma^0 \sim 2.35$ TeV yields the correct dark-matter relic abundance from the annihilation and coannihilation of $\Sigma$ through gauge interactions. 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--- abstract: 'We study the set $M(X)$ of full non-atomic Borel (finite or infinite) measures on a non-compact locally compact Cantor set $X$. For an infinite measure $\mu \in M(X)$, the set $\mathfrak{M}_\mu = \{x \in X : \mbox{for any compact open set } U \ni x \mbox{ we have } \mu(U) = \infty \}$ is called defective. We call $\mu$ *non-defective* if $\mu(\mathfrak{M}_\mu) = 0$. The class $M^0(X) \subset M(X)$ consists of probability measures and infinite non-defective measures. We classify measures $\mu$ from $M^0(X)$ with respect to a homeomorphism. The notions of goodness and compact open values set $S(\mu)$ are defined. A criterion when two good measures from $M^0(X)$ are homeomorphic is given. For any group-like $D \subset [0,1)$ we find a good probability measure $\mu$ on $X$ such that $S(\mu) = D$. For any group-like $D \subset [0,\infty)$ and any locally compact, zero-dimensional, metric space $A$ we find a good non-defective measure $\mu$ on $X$ such that $S(\mu) = D$ and $\mathfrak{M}_\mu$ is homeomorphic to $A$. We consider compactifications $cX$ of $X$ and give a criterion when a good measure $\mu \in M^0(X)$ can be extended to a good measure on $cX$.' author: - | O. Karpel\ Institute for Low Temperature Physics,\ 47 Lenin Avenue, 61103 Kharkov, Ukraine\ (e-mail: helen.karpel@gmail.com) title: Good measures on locally compact Cantor sets --- Introduction ============ The problem of classification of Borel finite or infinite measures on topological spaces has a long history. Two measures $\mu$ and $\nu$ defined on Borel subsets of a topological space $X$ are called *homeomorphic* if there exists a self-homeomorphism $h$ of $X$ such that $\mu = \nu\circ h$, i.e. $\mu(E) = \nu(h(E))$ for every Borel subset $E$ of $X$. The topological properties of the space $X$ are important for the classification of measures up to a homeomorphism. For instance, Oxtoby and Ulam [@Oxt-Ul] gave a criterion for a Borel probability measure on the finite-dimensional cube to be homeomorphic to the Lebesgue measure. Similar results were obtained for various manifolds (see [@Alp-Pr; @Oxt-Pr]). A Cantor set (or Cantor space) is a non-empty zero-dimensional compact perfect metric space. For Cantor sets the situation is much more difficult than for connected spaces. During the last decade, in the papers [@Akin3; @Austin; @S.B.O.K.; @D-M-Y; @Yingst] the Borel probability measures on Cantor sets were studied. In [@K], infinite Borel measures on Cantor sets were considered. For many applications in dynamical systems the state space is only locally compact. In this paper, we study Borel both finite and infinite measures on non-compact locally compact Cantor sets. It is possible to construct uncountably many full (the measure of every non-empty open set is positive) non-atomic measures on the Cantor set $X$ which are pairwise non-homeomorphic (see [@Akin1]). This fact is due to the existence of a countable base of clopen subsets of a Cantor set. The *clopen values set* $S(\mu)$ is the set of finite values of a measure $\mu$ on all clopen subsets of $X$. This set provides an invariant for homeomorphic measures, although it is not a complete invariant. For the class of the so called *good* probability measures, $S(\mu)$ *is* a complete invariant. By definition, a full non-atomic probability or non-defective measure $\mu$ is good if whenever $U$, $V$ are clopen sets with $\mu(U) < \mu(V)$, there exists a clopen subset $W$ of $V$ such that $\mu(W) = \mu(U)$ (see [@Akin2; @K]). Good probability measures are exactly invariant measures of uniquely ergodic minimal homeomorphisms of Cantor sets (see [@Akin2], [@GW]). For an infinite Borel measure $\mu$ on a Cantor set $X$, denote by $\mathfrak{M}_\mu$ the set of all points in $X$ whose clopen neighbourhoods have only infinite measures. The full non-atomic infinite measures $\mu$ such that $\mu(\mathfrak{M}_\mu) = 0$ are called *non-defective*. These measures arise as ergodic invariant measures for homeomorphisms of a Cantor set and the theory of good probability measures can be extended to the case of non-defective measures (see [@K]). In Section 2, we define a good probability measure and a good non-defective measure on a non-compact locally compact Cantor set $X$ and extend the results concerning good measures on Cantor sets to non-compact locally compact Cantor sets. For a Borel measure $\mu$ on $X$, the set $S(\mu)$ is defined as a set of all finite values of $\mu$ on the compact open sets. The defective set $\mathfrak{M}_\mu$ is the set of all points $x$ in $X$ such that every compact open neighbourhood of $x$ has infinite measure. We prove the criterion when two good measures on non-compact locally compact Cantor sets are homeomorphic. For every group-like subset $D \subset [0,1)$ we find a good probability measure $\mu$ on a non-compact locally compact Cantor set such that $S(\mu) = D$. For every group-like subset $D \subset [0,\infty)$ and any locally compact, zero-dimensional, metric space $A$ (including $A = \emptyset$) we find a good non-defective measure $\mu$ on a non-compact locally compact Cantor set such that $S(\mu) = D$ and $\mathfrak{M}_\mu$ is homeomorphic to $A$. In Section 3, compactifications of non-compact locally compact Cantor sets are studied. We investigate whether compactification can be used to classify measures on non-compact locally compact Cantor sets. We consider only the compactifications which are Cantor sets and extend measure $\mu$ by giving the remainder of compactification a zero measure. It turns out that in some cases good measure can be extended to a good measure on a Cantor set, while in other cases the extension always produces a measure which is not good. The extensions of a non-good measure are always non-good. After compactification of a non-compact locally compact Cantor set, new compact open sets are obtained. We study how the compact open values set changes. Based on this study, we give a criterion when a good measure on a non-compact locally compact Cantor set stays good after the compactification. Section 4 illustrates the results of Sections 2 and 3 with the examples. For instance, the Haar measure on the set of $p$-adic numbers and the invariant measure for $(C,F)$-construction are good. We give examples of good ergodic invariant measures on the generating open dense subset of a path space of stationary Bratteli diagrams such that any compactification gives a non-good measure. Measures on locally compact Cantor sets ======================================= Let $X$ be a non-compact locally compact metrizable space with no isolated points and with a (countable) basis of compact and open sets. Hence $X$ is totally disconnected. The set $X$ is called a *non-compact locally compact Cantor set*. Every two non-compact locally compact Cantor sets are homeomorphic (see [@D2]). Take a countable family of compact open subsets $O_n \subset X$ such that $X = \bigcup_{n=1}^\infty O_n$. Denote $X_1 = O_1$, $X_2 = O_2 \setminus O_1$, $X_3 = O_3 \setminus (O_1 \cup O_2)$,... The subsets $X_n$ are compact, open, pairwise disjoint and $X = \bigcup_{n=1}^\infty X_n$. Since $X$ is non-compact, we may assume without loss in generality that all $X_n$ are nonempty. Since $X$ has no isolated points, every $X_n$ has the same property. Thus, we represent $X$ as a disjoint union of a countable family of compact Cantor sets $X_n$. Recall that a Borel measure on a locally compact Cantor space is called *full* if every non-empty open set has a positive measure. It is easy to see that for a non-atomic measure $\mu$ the support of $\mu$ in the induced topology is a locally compact Cantor set. We can consider measures on their supports to obtain full measures. Denote by $M(X)$ the set of full non-atomic Borel measures on $X$. Then $M(X) = M_f(X) \sqcup M_\infty(X)$, where $M_f(X) = \{\mu \in M(X) : \mu(X) < \infty\}$ and $M_\infty(X) = \{\mu \in M(X) : \mu(X) = \infty\}$. For a measure $\mu \in M_\infty(X)$, denote $\mathfrak{M}_\mu = \{x \in X : \mbox{for any compact and open set } U \ni x \mbox{ we have } \mu(U) = \infty \}$. It will be shown that $\mathfrak{M}_\mu$ is a Borel set. Denote by $M_\infty^{0}(X) = \{\mu \in M_\infty(X) : \mu(\mathfrak{M}_\mu) = 0\}$. Let $M^0(X) = M_f(X) \sqcup M_\infty^{0}(X)$. Throughout the paper we will consider only measures from $M^0(X)$. We normalize the measures from $M_f(X)$ so that $\mu(X) = 1$ for any $\mu \in M_f(X)$. Recall that $\mu \in M^0(X)$ is *locally finite* if every point of X has a neighbourhood of finite measure. The properties of measures from the class $M^0(X)$ are collected in the following proposition. \[basic\_prop\] Let $\mu \in M^0(X)$. Then \(1) The measure $\mu$ is locally finite if and only if $\mathfrak{M}_\mu = \emptyset$, \(2) The set $X \setminus \mathfrak{M}_\mu$ is open. The set $\mathfrak{M}_\mu$ is $F_\sigma$. \(3) For any compact open set $U$ with $\mu(U) = \infty$ and any $a > 0$ there exists a compact open subset $V \subset U$ such that $a \leq \mu(V) < \infty$. \(4) The set $\mathfrak{M}_\mu$ is nowhere dense. \(5) $X = \bigsqcup_{i = 1}^\infty V_i \bigsqcup \mathfrak{M}_\mu$, where each $V_i$ is a compact open set of finite measure and $\mathfrak{M}_\mu$ is a nowhere dense $F_\sigma$ and has zero measure. The measure $\mu$ is $\sigma$-finite. \(6) $\mu$ is uniquely determined by its values on the algebra of compact open sets. **Proof.** (1) The condition $\mathfrak{M}_\mu = \emptyset$ means that every point $x \in X$ has a compact open neighbourhood of finite measure. Hence $\mu$ is locally finite and vise versa. \(2) We have $X \setminus \mathfrak{M}_\mu = \{x \in X : \mbox{ there exists a compact open set } U_x \ni x \mbox{ such that } \mu(U_x) < \infty \}$. Then for every point $x \in X \setminus \mathfrak{M}_\mu$ we have $U_x \subset X \setminus \mathfrak{M}_\mu$. Hence $X \setminus \mathfrak{M}_\mu$ is open. Therefore, for every $n \in \mathbb{N}$ the set $X_n \setminus \mathfrak{M}_\mu$ is open and $X_n \cap \mathfrak{M}_\mu$ is closed. Then $\mathfrak{M}_\mu = \bigsqcup_{n \in \mathbb{N}} \; (X_n \cap \mathfrak{M}_\mu)$ is $F_\sigma$ set. \(3) Let $U$ be a non-empty compact open subset of $X$ such that $\mu(U) = \infty$. Since $\mu \in M^{0}(X)$, we have $\mu(U) = \mu(U \setminus \mathfrak{M}_{\mu})$. Since $U$ is open, the set $U \setminus \mathfrak{M}_{\mu} = U \cap (X \setminus \mathfrak{M}_{\mu})$ is open. There are only countably many compact open subsets in $X$, hence the open set $U \setminus \mathfrak{M}_{\mu}$ can be represented as a disjoint union of compact open subsets $\{U_i\}_{i \in \mathbb{N}}$ of finite measure. We have $\mu(U) = \sum_{i=0}^{\infty} \mu(U_i) = \infty$, hence for every $a \in \mathbb{R}$ there is a compact open subset $V = \bigsqcup_{i=0}^{N}U_i$ such that $a \leq \mu(V) < \infty$. \(4) Let $U$ be a compact open subset of $X$. It suffices to show that there exists a non-empty compact open subset $V \subset U$ such that $V \cap \mathfrak{M}_{\mu} = \emptyset$. If $\mu(U) < \infty$ then $U \cap \mathfrak{M}_{\mu} = \emptyset$. Otherwise, by (3), there exists a compact open subset $V \subset U$ such that $0 < \mu(V) < \infty$. Obviously, $V \cap \mathfrak{M}_{\mu} = \emptyset$. \(5) follows from the proof of (3). \(6) follows from (5). $\blacksquare$ For a measure $\mu \in M^0(X)$ define the *compact open values set* as the set of all finite values of the measure $\mu$ on the compact open sets: $$S(\mu) = \{\mu(U):\,U\mbox{ is compact open in } X \mbox{ and } \mu(U) < \infty\}.$$ For each measure $\mu \in M^0(X)$, the set $S(\mu)$ is a countable dense subset of the interval $[0, \mu(X))$. Indeed, the set $S(\mu)$ is dense in $[0, \mu(V)]$ for every compact open set $V$ of finite measure (see [@Akin1]). By Proposition \[basic\_prop\], $S(\mu)$ is dense in $[0, \mu(X))$. Let $X_{1}$, $X_{2}$ be two non-compact locally compact Cantor sets. It is said that measures $\mu_{1} \in M(X_{1})$ and $\mu_{2} \in M(X_{2})$ are *homeomorphic* if there exists a homeomorphism $h \colon X_{1} \rightarrow X_{2}$ such that $\mu_{1}(E) = \mu_{2}(h(E))$ for every Borel subset $E \subset X_1$. Clearly, $S(\mu_{1}) = S(\mu_{2})$ for any homeomorphic measures $\mu_1$ and $\mu_2$. We call two Borel infinite measures $\mu_1 \in M^0_\infty(X_{1})$ and $\mu_2 \in M^0_\infty(X_{2})$ *weakly homeomorphic* if there exists a homeomorphism $h \colon X_{1} \rightarrow X_{2}$ and a constant $C>0$ such that $\mu_{1}(E) = C \mu_{2}(h(E))$ for every Borel subset $E \subset X_1$. Then $S(\mu_{1}) = C S(\mu_{2})$. Let $D$ be a dense countable subset of the interval $[0,a)$ where $a \in (0, \infty]$. Then $D$ is called *group-like* if there exists an additive subgroup $G$ of $\mathbb{R}$ such that $D = G \cap [0, a)$. It is easy to see that $D$ is group-like if and only if for any $\alpha, \beta \in D$ such that $\alpha \leq \beta$ we have $\beta - \alpha \in D$ (see [@Akin2; @K]). Let $X$ be a locally compact Cantor space (either compact or non-compact) and $\mu\in M^0(X)$. A compact open subset $V$ of $X$ is called *good* for $\mu$ (or just good when the measure is understood) if for every compact open subset $U$ of $X$ with $\mu(U) < \mu(V)$, there exists a compact open set $W$ such that $W \subset V$ and $\mu(W) = \mu(U)$. A measure $\mu$ is called *good* if every compact open subset of $X$ is good for $\mu$. If $\mu \in M^0(X)$ is a good measure and $\nu \in M^0(X)$ is (weakly) homeomorphic to $\mu$ then, obviously, $\nu$ is good. It is easy to see that in the case of compact Cantor set the definition of a good measure coincides with the one given in [@Akin2]. For a compact open subset $U \subset X$ let $\mu|_U$ be the restriction of the measure $\mu$ to the Cantor space $U$. Then the set $U$ is good if and only if $S(\mu|_U) = S(\mu|_X) \cap [0, \mu(U)]$. Denote by $H_{\mu}(X)$ the group of all homeomorphisms of a space $X$ preserving the measure $\mu$. The action of $H_{\mu}(X)$ on $X$ is called *transitive* if for every $x_{1}, x_{2} \in X$ there exists $h \in H_{\mu}(X)$ such that $h(x_{1}) = x_{2}$. The action is called *topologically transitive* if there exists a dense orbit, i.e. there is $x \in X$ such that the set $O(x) = \{h(x) : h \in H_\mu(X)\}$ is dense in $X$. We extend naturally the notion of partition basis introduced in [@Akin3]. A *partition basis* $\mathcal{B}$ for a non-compact locally compact Cantor set $X$ is a collection of compact open subsets of $X$ such that every non-empty compact open subset of $X$ can be partitioned by elements of $\mathcal{B}$. The properties of good measures on non-compact locally compact Cantor sets are gathered in the following proposition. The proofs for the measures on compact Cantor spaces can be found in [@Akin2; @Akin3; @K]. \[many\] Let $X$ be a locally compact Cantor space (either compact or non-compact). Let $\mu \in M^0(X)$. Then \(a) If $\mu$ is good and $C > 0$ then $C \mu$ is good and $S(C \mu) = C S(\mu)$. \(b) If $\mu$ is good and $U$ is a non-empty compact open subset of $X$ then the measure $\mu|_U$ is good and $S(\mu|_U) = S(\mu) \cap [0, \mu(U)]$. \(c) $\mu$ is good if and only if every compact open subset of finite measure is good. \(d) $\mu$ is good if and only if for every non-empty compact open subset $U$ of finite measure, the measure $\mu|_U$ is good. \(e) If $\mu$ is good then $S(\mu)$ is group-like. \(f) If a compact open set $U$ admits a partition by good compact open subsets then $U$ is good. \(g) The measure $\mu$ is good if and only if there exists a partition basis $\mathcal B$ consisting of compact open sets which are good for $\mu$. \(h) If $\mu$ is good, then the group $H_\mu(X)$ acts transitively on $X \setminus \mathfrak{M_\mu}$. In particular, the group $H_\mu(X)$ acts topologically transitively on $X$. \(i) If $\mu$ is a good measure on $X$ and $\nu$ is the counting measure on $\{1,2,...,n\}$ then $\mu \times \nu$ is a good measure on $X \times \{1,2,...,n\}$. **Proof**. (a), (b) are clear. \(c) Suppose that every compact open subset of finite measure is good. Let $V$ be any compact open set with $\mu(V) = \infty$ and $U$ be a compact open set with $\mu(U) < \infty$. By Proposition \[basic\_prop\], there exists a compact open subset $W \subset V$ such that $\mu(U) \leq \mu(W) < \infty$. By assumption, $W$ is good. Hence there exists a compact open set $W_1\subset W$ with $\mu(W_1) = \mu(U)$ and $V$ is good. \(d) Suppose that for every non-empty compact open subset $U$ of finite measure, the measure $\mu|_U$ is good. We prove that every compact open subset of finite measure is good, then use (c). Let $U$, $V$ be compact open sets with $0 < \mu(U) < \mu(V) < \infty$. Set $W = U \cup V$. Then $W$ is a compact open set of finite measure. Since $\mu|_W$ is good, there exists $W_1 \subset V$ such that $\mu(W_1) = \mu(U)$. \(e) If $\mu$ is good then for any $\alpha, \beta \in S(\mu)$ such that $\beta - \alpha \geq 0$, we have $\beta - \alpha \in S(\mu)$. Hence $S(\mu)$ is group-like. (f) See [@Akin3] for the case of finite measure and [@K] for infinite measure. \(g) If there exists a partition basis $\mathcal B$ consisting of compact open sets which are good for $\mu$, then, by (f), every compact open set is good. \(h) For any $x, y \in X \setminus \mathfrak{M_\mu}$ there exists a compact open set $U$ of finite measure such that $x, y \in U$. By (d), the measure $\mu|_U$ is a good finite measure on a Cantor space $U$. By Theorem 2.13 in [@Akin2], there exists a homeomorphism $h \colon U \rightarrow U$ which preserves $\mu$ and $h(x) = y$. Define $h_1 \in H_\mu(X)$ to be $h$ on $U$ and the identity on $X \setminus U$. For every $x \in X\setminus\mathfrak{M}_\mu$ we have $O(x) = X \setminus\mathfrak{M}_\mu$. By Proposition \[basic\_prop\], the set $X \setminus\mathfrak{M}_\mu$ is dense in $X$. Hence $H_\mu(X)$ acts topologically transitively on $X$. \(i) The rectangular compact open sets $U \times \{z\}$, where $U$ is compact open in $X$ and $z \in \{1,2,...,n\}$, form a partition basis for $X \times \{1,2,...,n\}$. Since $\mu \times \nu (U \times \{z\}) = \mu(U)$, these sets are good. The measure $\mu$ is good by (g). $\blacksquare$ For $G$ an additive subgroup of $\mathbb{R}$ we call a positive real number $\delta$ a *divisor* of $G$ if $\delta G = G$. The set of all divisors of $G$ is called $Div(G)$. By a full measure on a discrete countable topological space $Y$ we mean a measure $\nu$ such that $0 < \nu(\{y\}) < \infty$ for every $y \in Y$. We will use the following theorem for $Y = \mathbb{Z}$, but the proof stays correct for any discrete countable topological space $Y$. \[good\_product\] Let $\mu$ be a good measure on a non-compact locally compact Cantor space $X$. Let $\nu$ be a full measure on $\mathbb{Z}$, where $\mathbb{Z}$ is endowed with discrete topology. Let $G$ be an additive subgroup of $\mathbb{R}$ generated by $S(\mu)$. Then $\mu \times \nu$ is good on $X \times \mathbb{Z}$ if and only if there exists $C > 0$ such that $\nu(\{i\}) \in C \cdot Div(G)$ for every $i \in \mathbb{Z}$. **Proof.** Lets prove the “if” part. Suppose $\mu$ is good on $X$ and $\nu(\{i\}) \in C \cdot Div(G)$ for some $C > 0$ and every $i \in \mathbb{Z}$. By Proposition \[many\] (g), it suffices to prove that a compact open set of the form $U \times \{i\}$ is good for any compact open $U \subset X$ and any $i \in \mathbb{Z}$. Thus, it suffices to show that $S(\mu \times \nu|_{U \times \{i\}}) = S(\mu \times \nu|_{X \times \mathbb{Z}}) \cap [0, \mu \times \nu (U \times \{i\})]$. The inclusion $S(\mu \times \nu|_{U \times \{i\}}) \subset S(\mu \times \nu|_{X \times \mathbb{Z}}) \cap [0, \mu \times \nu (U \times \{i\})]$ is always true, hence we need to prove the inverse inclusion. We have $S(\mu \times \nu|_{U \times\{i\}}) = \nu(\{i\}) S(\mu|_U) = C \delta S(\mu|_U)$ for some $\delta \in Div(G)$. Since $\mu$ is good on $X$, we obtain $S(\mu|_U) = G \cap [0, \mu(U)]$. Hence $S(\mu \times \nu|_{U \times\{i\}}) = C G \cap [0, C \delta \mu(U)] = C G \cap [0, \mu \times \nu (U \times \{i\})]$. Note that $C \delta \mu(U) \in CG$ because $\delta \in Div(G)$. Therefore, it suffices to prove that $S(\mu \times \nu|_{X \times \mathbb{Z}}) \subset C G$. The set $S(\mu \times \nu|_{X \times \mathbb{Z}})$ consists of all finite sums $\sum_{i,j} \mu(U_i) \nu(\{j\})$, where each $U_i$ is a compact open set in $X$ and $j \in \mathbb{Z}$. We have $\sum_{i,j} \mu(U_i) \nu(\{j\}) = \sum_{i,j} \mu(U_i) C \delta_j \subset CG$, here $\delta_i \in Div(G)$. Hence $S(\mu \times \nu|_{U \times\{i\}}) \supset S(\mu \times \nu|_{X \times \mathbb{Z}}) \cap [0, \mu \times \nu (U \times \{i\})]$ and $U \times \{i\}$ is good. Now we prove the “only if part”. Suppose that $\mu \times \nu$ is good on $X \times \mathbb{Z}$. Then for any $i \in \mathbb{Z}$ we have $S(\mu \times \nu|_{X \times\{i\}}) = S(\mu \times \nu|_{X \times \mathbb{Z}}) \cap [0, \mu \times \nu(X \times\{i\})]$. Note that $S(\mu \times \nu|_{X \times\{i\}} = \nu(\{i\}) S(\mu|_X)$. Denote by $\widetilde{G}$ the additive subgroup of $\mathbb{R}$ generated by $S(\mu \times \nu|_{X \times \mathbb{Z}})$. Let $\alpha = \nu(\{i\})$. Then $\alpha G = \widetilde{G}$. Let $j \in \mathbb{Z}$ and $\beta = \nu(\{j\})$. By the same arguments, we have $\beta G = \widetilde{G}$. Then $\frac{\alpha}{\beta} \in Div(G)$. Indeed, $\frac{\alpha}{\beta} G = \frac{1}{\beta} \widetilde{G} = G$. Hence $\alpha = \beta \delta$, where $\delta \in Div(G)$. Set $C = \nu(\{j\})$. Then for every $i \in \mathbb{Z}$ we have $\nu(\{i\}) = C \delta_i$ where $\delta_i = \frac{\nu(\{i\})}{\nu(\{j\})} \in Div (G)$. $\blacksquare$ Let $X$, $Y$ be non-compact locally compact Cantor sets. If $\mu \in M^0(X)$, $\nu \in M^0(Y)$ are good measures, then the product $\mu \times \nu$ is a good measure on $X \times Y$ and $$S(\mu \times \nu) = \left\{\sum_{i=0}^N \alpha_i \cdot \beta_i : \alpha_i \in S(\mu), \beta_i \in S(\nu), N \in \mathbb{N}\right\} \cap [0, \mu(X)\times \nu(Y)).$$ **Proof.** Let $X = \bigsqcup_{m = 1}^{\infty} X_n$ and $Y = \bigsqcup_{n = 1}^{\infty} Y_n$, where each $X_n$, $Y_n$ is a Cantor set. Then $X \times Y = \bigsqcup_{m, n = 1}^{\infty} X_m \times Y_n$ and $\mu \times \nu |_{X_m \times Y_n} = \mu|_{X_n} \times \nu|_{Y_n}$. Since $\mu|_{X_n}$ and $\nu|_{Y_n}$ are good finite or non-defective measures on a Cantor set, the measure $\mu \times \nu |_{X_m \times Y_n}$ is good by Theorem 2.8 ([@Akin3]), Theorem 2.10 ([@K]). By Proposition \[many\], $\mu \times \nu$ is good on $X \times Y$. $\blacksquare$ \[krit\_homeo\_good\] Let $X$, $Y$ be non-compact locally compact Cantor spaces. Let $\mu \in M^0(X)$ and $\nu \in M^0(Y)$ be good measures. Let $S(\mu) = S(\nu)$. Let $\mathfrak{M}$ be the defective set for $\mu$ and $\mathfrak{N}$ be the defective set for $\nu$. Assume that there is a homeomorphism $h \colon \mathfrak{M} \rightarrow \mathfrak{N}$ where the sets $\mathfrak{M}$ and $\mathfrak{N}$ are endowed with the induced topologies. Then there exists a homeomorphism $\widetilde{h} \colon X \rightarrow Y$ which extends $h$ such that $\mu = \nu \circ \widetilde{h}$. Conversely, if $\mu \in M^0(X)$ and $\nu \in M^0(Y)$ are good homeomorphic measures then $S(\mu) = S(\nu)$ and there is a homeomorphism $h \colon \mathfrak{M} \rightarrow \mathfrak{N}$. **Proof.** The second part of the Theorem is clear. We prove the first part. Let $X = \bigsqcup_{i=1}^\infty X_i$ and $Y = \bigsqcup_{j=1}^\infty Y_j$ where $X_i$, $Y_j$ are compact Cantor spaces. First, consider the case when $\mathfrak{M} = \mathfrak{N} = \emptyset$, i.e. the measures $\mu$, $\nu$ are either finite of infinite locally finite measures. Since $S(\mu) = S(\nu)$, we have $\mu(X_1) \in S(\nu)$. There exists $n \in \mathbb{N}$ such that $\nu(\bigsqcup_{j=1}^{n-1} Y_j) \leq \mu(X_1) < \nu(\bigsqcup_{j=1}^{n} Y_j)$. Since $S(\nu)$ is group-like, we see that $\mu(X_1) - \nu(\bigsqcup_{j=1}^{n-1} Y_j) \in S(\nu)$. Since $\nu$ is good, there exists a compact open subset $W \subset Y_n$ such that $\nu(W) = \mu(X_1) - \nu(\bigsqcup_{j=1}^{n-1} Y_j)$. Hence $Z = \bigsqcup_{j=1}^{n-1} Y_j \sqcup W$ is a compact Cantor set and $\mu(X_1) = \nu(Z)$. By Theorem 2.9 ([@Akin2]), there exists a homeomorphism $h_1 \colon X_1\rightarrow Z$ such that $\mu |_{X_1} = \nu |_{Z} \circ h_1$. Set $\widetilde{h}|_{X_1} = h_1$. Consider $(Y_n \setminus W) \bigsqcup_{j=n+1}^\infty Y_j$ instead of $Y$ and $\bigsqcup_{i=2}^\infty X_i$ instead of $X$. Reverse the roles of $X$ and $Y$. Proceed in the same way using $Y_n \setminus W$ instead of $X_1$. Thus, we obtain countably many homeomorphisms $\{h_i\}_{i=1}^\infty$. Given $x \in X$, set $\widetilde{h}(x) = h_i(x)$ for the corresponding $h_i$. Then $\widetilde{h} \colon X \rightarrow Y$ is a homeomorphism which maps $\mu$ into $\nu$. Now, let $\mathfrak{M} \neq \emptyset$. If $\mu(X_1) < \infty$, we proceed as in the previous case. If $\mu(X_1) = \infty$ then $X_1 \cap \mathfrak{M} \neq \emptyset$. Then $h(X_1 \cap \mathfrak{M})$ is a compact open subset of $\mathfrak{N}$ in the induced topology. Hence there exists a compact open set $W \subset Y$ such that $W \cap \mathfrak{N} = h(X_1 \cap \mathfrak{M})$. Then, by Theorem 2.11 ([@K]), the sets $X_1$ and $W$ are homeomorphic via measure preserving homeomorphism $h_1$ and $h_1|_{X_1 \cap \mathfrak{M}} = h$. Since $W$ is compact, there exists $N$ such that $W \subset \bigsqcup_{n=1}^N Y_n$. Reverse the roles of $X$ and $Y$ and consider $\bigsqcup_{n=1}^N Y_n \setminus W$ instead of $X_1$. $\blacksquare$ The corollary for weakly homeomorphic measures follows: \[krit\_weak\_homeo\_good\] Let $\mu \in M_{\infty}^0(X)$ and $\nu \in M_{\infty}^0(Y)$ be good infinite measures on non-compact locally compact Cantor sets $X$ and $Y$. Let $\mathfrak{M}$ be the defective set for $\mu$ and $\mathfrak{N}$ be the defective set for $\nu$. Then $\mu$ is weakly homeomorphic to $\nu$ if and only if the following conditions hold: \(1) There exists $c > 0$ such that $S(\mu) = c S(\nu)$, \(2) There exists a homeomorphism $h \colon \mathfrak{M} \rightarrow \mathfrak{N}$ where the sets $\mathfrak{M}$ and $\mathfrak{N}$ are endowed with the induced topologies. Let $\mu \in M^0_{\infty}(X)$ be a good measure on a non-compact locally compact Cantor set $X$ and $V$ be any compact open subset of $X$ with $\mu(V) < \infty$. Then $\mu$ on $X$ is homeomorphic to $\mu$ on $X \setminus V$. Let $S(\mu) = G \cap [0, \infty)$. Then $\mu$ is homeomorphic to $c \mu$ if and only if $c \in Div(G)$. Let $\mu$ be a good finite or non-defective measure on a non-compact locally compact Cantor set $X$. Let $U$, $V$ be two compact open subsets of $X$ such that $\mu(U) = \nu(V) < \infty$. Then there is $h \in H_\mu(X)$ such that $h(U) = V$. **Proof.** Set $Y = U \cup V$. Then $Y$ is a Cantor set with $\mu(Y) < \infty$. By Proposition 2.11 in [@Akin2], there exists a self-homeomorphism $h$ of $Y$ such that $h(U) = V$ and $h$ preserves $\mu$. Set $h$ to be identity on $X \setminus Y$. $\blacksquare$ Let $\mu$ and $\nu$ be good non-defective measures on non-compact locally compact Cantor sets $X$ and $Y$. Let $\mathfrak{M}$ be the defective set for $\mu$ and $\mathfrak{N}$ be the defective set for $\nu$. If there exist compact open sets $U \subset X$ and $V \subset Y$ such that $\mu(U) = \nu(V) < \infty$ and $\mu|U$ is homeomorphic to $\nu|V$, then $\mu$ is homeomorphic to $\nu$ if and only if $\mathfrak{M}$ and $\mathfrak{N}$ (with the induced topologies) are homeomorphic. **Proof.** Let $\gamma = \mu(U) = \nu (V)$. Since $\mu|U$ is homeomorphic to $\nu|V$, we have $S(\mu|U) = S(\nu|V)$. Since $\mu$ and $\nu$ are good, we have $S(\mu) \cap [0, \gamma] = S(\nu) \cap [0, \gamma]$ by Proposition \[many\]. Since $S(\mu)$ and $S(\nu)$ are group-like, we obtain $S(\mu) = S(\nu)$. $\blacksquare$ \[goodSmu\] Let $\mu \in M^{0}(X)$ be a good measure on a non-compact locally compact Cantor set $X$. Then the compact open values set $S(\mu)$ is group-like and the defective set $\mathfrak{M}_\mu$ is a locally compact, zero-dimensional, metric space (including $\emptyset$). Conversely, for every countable dense group-like subset $D$ of $[0, 1)$, there is a good probability measure $\mu$ on a non-compact locally compact Cantor set such that $S(\mu) = D$. For every countable dense group-like subset $D$ of $[0, \infty)$ and any locally compact, zero-dimensional, metric space $A$ (including $A = \emptyset$) there is a good non-defective measure $\mu$ on a non-compact locally compact Cantor set such that $S(\mu) = D$ and $\mathfrak{M}_\mu$ is homeomorphic to $A$. **Proof**. The first part of the theorem follows from Propositions \[basic\_prop\], \[many\]. We prove the second part. First, consider the case of finite measure. Let $D \subset [0,1)$ be a countable dense group-like subset. Then there exist a strictly increasing sequence $\{\gamma_n\}_{n=1}^{\infty} \subset D$ such that $\lim_{n \rightarrow \infty} \gamma_n = 1$. For $n = 1,2,...$ set $\delta_n = \gamma_{n} - \gamma_{n-1}$. Denote by $S_n = D \cap [0, \delta_n]$. Then $D_n = \frac{1}{\delta_n} (D \cap [0,\delta_n])$ is a group-like subset of $[0,1]$ with $1 \in D_n$. In [@Akin2], it was proved that there exists a good probability measure $\mu_n$ on a Cantor set $X_n$ such that $S(\mu_n|_{X_n}) = D_n$. The measure $\nu_n = \delta_n \mu_n$ is a good finite measure on $X_n$ with $S(\nu_n|_{X_n}) = D \cap [0,\delta_n]$. Set $X = \bigsqcup_{n=1}^{\infty} X_n$ and let $\mu|_{X_n} = \nu_n$. Then $\mu$ is a good probability measure on a non-compact locally compact Cantor space $X$ and $S(\mu|_{X}) = D$. Now consider the case of infinite measure. Let $\gamma \in D$. Since $D \subset [0, \infty)$ is group-like, we see that $\frac{1}{\gamma}D \cap [0,1]$ is a group-like subset of $[0,1]$. In [@Akin2] it was proved that there exists a good probability measure $\mu_1$ on a Cantor space $Y$ with $S(\mu_1) = \frac{1}{\gamma}D \cap [0,1]$. Set $\mu = \gamma \mu_1$. Then $\mu$ is a good finite measure on $Y$ and $S(\mu) = D \cap [0,\gamma]$. Endow the set $\mathbb{Z}$ with discrete topology. Let $\nu$ be a counting measure on $\mathbb{Z}$. Set $X = Y \times \mathbb{Z}$ and $\widetilde{\mu} = \mu \times \nu$. Then, by Theorem \[good\_product\], $\widetilde{\mu}$ is good with $S(\widetilde{\mu}) = D$ and $\mathfrak{M}_{\widetilde{\mu}} = \emptyset$. Suppose $A$ is a non-empty compact zero-dimensional, metric space. Then, by Theorem 2.15 ([@K]), there exists a good non-defective measure $\mu$ on a Cantor space $Y$ such that $S(\mu) = D$ and $\mathfrak{M}_\mu$ is homeomorphic to $A$. By the above, there exists a good locally finite measure $\nu$ on a non-compact locally compact set $X$ with $S(\nu) = D$ and $\mathfrak{M}_\nu = \emptyset$. Set $Z = Y \sqcup X$ and $\widetilde{\mu}|_{Y} = \mu$, $\widetilde{\mu}|_{X} = \nu$. Then $\widetilde{\mu}$ is good on a non-compact locally compact Cantor set $Z$ with $S(\widetilde{\mu}) = D$ and $\mathfrak{M}_{\widetilde{\mu}}$ is homeomorphic to $A$. Suppose that $A$ is a non-empty, non-compact, locally compact, zero-dimensional metric space. Then $A = \bigsqcup_{n=1}^{\infty} A_n$ where each $A_n$ is a non-empty, compact, zero-dimensional metric space. By Theorem 2.15 ([@K]), for every $n = 1,2,...$ there exists a good non-defective measure $\mu_n$ on a Cantor set $Y_n$ such that $S(\mu_n) = D$ and $\mathfrak{M}_{\mu_n}$ is homeomorphic to $A_n$. Set $X = \bigsqcup_{n=1}^{\infty} Y_n$ and $\mu|_{Y_n} = \mu_n$. Then $\mu$ is good on a non-compact locally compact Cantor set $X$ with $S(\mu) = D$ and $\mathfrak{M}_{\widetilde{\mu}}$ is homeomorphic to $A$. $\blacksquare$ \[invarhomeo\] Let $D$ be a countable dense group-like subset of $[0, \infty)$. Then there exists an aperiodic homeomorphism of a non-compact locally compact Cantor set with good non-defective invariant measure $\widetilde{\mu}$ such that $S(\widetilde{\mu}) = D$. **Proof**. We use the construction similar to the one in the proof of Theorem \[goodSmu\]. Let $\mu$ be a good measure on a Cantor set $Y$ with $S(\mu) = D \cap [0,\gamma]$ for some $\gamma \in D$. Let $\nu$ be a counting measure on $\mathbb{Z}$. Set $\widetilde{\mu} = \mu \times \nu$ on $X = Y \times \mathbb{Z}$. Then $\widetilde{\mu}$ is a good non-defective measure on a non-compact locally compact Cantor set $X$ with $S(\widetilde{\mu}) = D$. Since the measure $\mu$ is a good finite measure on $Y$, there exists a minimal homeomorphism $T \colon Y \rightarrow Y$ such that $\mu$ is invariant for $T$ (see [@Akin2]). Let $T_1(x,n) = (Tx, n+1)$. Then $T_1$ is aperiodic homeomorphism of $X$. The measure $\widetilde{\mu}$ is invariant for $T_1$. $\blacksquare$ The measure $\widetilde{\mu}$ built in Corollary \[invarhomeo\] is invariant for any skew-product with the base $(Y,T)$ and cocycle acting on $\mathbb{Z}$. Let $X$ be a non-compact locally compact Cantor set. Then there exist continuum distinct classes of homeomorphic good measures in $M_f(X)$. There also exist continuum distinct classes of weakly homeomorphic good measures in $M_\infty^0(X)$. **Proof.** There exist uncountably many distinct group-like subsets $\{D_\alpha\}_{\alpha \in \Lambda}$ of $[0,1]$. By Theorem \[goodSmu\], for each $D_\alpha$ there exists a good probability measure $\mu_\alpha$ on $X$ such that $S(\mu_\alpha) = D_\alpha$. By Theorem \[krit\_homeo\_good\], the measures $\{\mu_\alpha\}_{\alpha \in \Lambda}$ are pairwise non-homeomorphic. Let $Y$ be a compact Cantor set. Let $\mu$ be a non-defective measure on $Y$. Denote by $[\mu]$ the class of weak equivalence of $\mu$ in the set of all non-defective measures on $Y$. There exist continuum distinct classes $[\mu_\alpha]$ of weakly homeomorphic good non-defective measures on a Cantor set $Y$ (see Theorem 2.18 in [@K]). Moreover, if there exists $C>0$ such that $G(S(\mu_\alpha)) = C G(S(\mu_\beta))$ then $\mu_\beta \in [\mu_\alpha]$. Let $\nu$ be a counting measure on $\mathbb{Z}$. Then, by Theorem \[good\_product\], $\mu_\alpha \times \nu$ is a good measure on a non-compact locally compact Cantor set $Y \times \mathbb{Z}$ and $G(S(\mu_\alpha \times \nu)) = G(S(\mu_\alpha))$. Hence, by Corollary \[krit\_weak\_homeo\_good\], the measures $\mu_\alpha \times \nu$ and $\mu_\beta \times \nu$ are weakly homeomorphic if and only if $\mu_\beta \in [\mu_\alpha]$. $\blacksquare$ If $\mu$ is Haar measure for some topological group structure on a non-compact locally compact Cantor space $X$ then $\mu$ is a good measure on $X$. **Proof.** The ball $B$ centered at the identity in the invariant ultrametric is a compact open subgroup of $X$. Since $\mu$ is translation-invariant, by Proposition \[many\], it suffices to show that $\mu|_B$ is good for every such ball $B$. Since the restriction of $\mu$ on $B$ is a Haar measure on a compact Cantor space, $\mu|_B$ is good by Proposition 2.4 in [@Akin3]. $\blacksquare$ From measures on non-compact spaces to measures on compact spaces and back again ================================================================================ Let $X$ be a non-compact locally compact Cantor space. A *compactification* of $X$ is a pair $(Y,c)$ where $Y$ is a compact space and $c \colon X \rightarrow Y$ is a homeomorphic embedding of $X$ into $Y$ (i.e. $c \colon X \rightarrow c(X)$ is a homeomorphism) such that $\overline{c(X)} = Y$, where $\overline{c(X)}$ is the closure of $c(X)$. In the paper, by compactification we will mean not only a pair $(Y,c)$ but also the compact space $Y$ in which $X$ can be embedded as a dense subset. We will denote the compactifications of a space $X$ by symbols $cX$, $\omega X$, etc., where $c$, $\omega$ are the corresponding homeomorphic embeddings. Let $\mu \in M^0(X)$. We will consider only such compactifications $cX$ that $cX$ is a Cantor set. Since $c$ is a homeomorphism, the measure $\mu$ on $X$ passes to a homeomorphic measure on $c(X)$. Since we are interested in the classification of measures up to homeomorphisms, we can identify the set $c(X)$ with $X$. Hence $X$ can be considered as an open dense subset of $cX$. The set $cX \setminus X$ is called the *remainder* of compactification. As far as $X$ is locally compact, the remainder $cX \setminus X$ is closed in $cX$ for every compactification $cX$ (see [@E]). Since $\overline{X} = cX$, the set $cX \setminus X$ is a closed nowhere dense subset of $cX$. Compactifications $c_1X$ and $c_2X$ of a space $X$ are *equivalent* if there exists a homeomorphism $f \colon c_1X \rightarrow c_2X$ such that $fc_1(x) = c_2(x)$ for every $x \in X$. We shall identify equivalent compactifications. For any space $X$ one can consider the family $\mathcal{C}(X)$ of all compactifications of $X$. The order relation on $\mathcal{C}(X)$ is defined as follows: $c_2X \leq c_1X$ if there exists a continuous map $f \colon c_1X \rightarrow c_2X$ such that $fc_1 = c_2$. Then we have $f(c_1(X)) = c_2(X)$ and $f(c_1X \setminus c_1(X)) = c_2X \setminus c_2(X)$. \[Alex\] Every non-compact locally compact space $X$ has a compactification $\omega X$ with one-point remainder. This compactification is the smallest element in the set of all compactifications $\mathcal{C}(X)$ with respect to the order $\leq$. The topology on $\omega X$ is defined as follows. Denote by $\{\infty\}$ the point $\omega X \setminus X$. Open sets in $\omega X$ are the sets of the form $\{\infty\} \cup (X \setminus F)$, where $F$ is a compact subspace of $X$, together with all sets that are open in $X$. For any Borel measure $\nu$ on the set $cX \setminus X$ with the induced topology, $\widetilde{\mu} = \mu + \nu$ is a Borel measure on $cX$ such that $\widetilde{\mu}|_X = \mu$. Since the aim of compactification is the study of a measure $\mu$ on a locally compact set $X$, we will consider only such extensions $\widetilde{\mu}$ on $cX$ that $\mu(cX \setminus X) = 0$. \[Smu\_diff\_comp\] Let $X$ be a non-compact locally compact Cantor set and $\mu \in M^0(X)$. Let $c_1X$, $c_2X$ be the compactifications of $X$ such that $c_1X \leq c_2X$. Denote by $\mu_1$ the extension of $\mu$ on $c_1 X$ and by $\mu_2$ the extension of $\mu$ on $c_2 X$. Then $S(\mu) \subseteq S(\mu_1) \subseteq S(\mu_{2})$. **Proof.** Since $c_1X \leq c_2X$, there exists a continuous map $f \colon c_2X \rightarrow c_1 X$ such that $f(c_2 X \setminus X) = c_1 X \setminus X$ and $fc_2(x) = c_1(x)$ for any $x \in X$. Since $f$ is continuous, it suffices to prove that $f$ preserves measure, that is $\mu_1(V) = \mu_2(f^{-1}(V))$ for any compact open $V \subset X$. Recall that we can identify $c_i(X)$ with $X$. Hence we can consider $f$ as an identity on $X \subset c_iX$ and $f$ preserves measure. That is, for every compact open subset $U$ of $X$ we have $\mu(U) = \mu_1(U) = \mu_2(U)$. Hence $S(\mu) \subseteq S(\mu_1)$. Since $\mu(c_iX \setminus X) = 0$, the measure of any clopen subset of $c_iX$ is the sum of measures of compact open subsets of $X$. Hence the measures of all clopen sets are preserved. Thus, $S(\mu_1) \subseteq S(\mu_{2})$. $\blacksquare$ We can consider the homeomorphic embedding of a set $X$ into a non-compact locally compact Cantor set $Y$ such that $\mu(Y \setminus X) = 0$. Then, by the same arguments as above, the inclusion $S(\mu|_X) \subseteq S(\mu|_Y)$ holds. \[krit\_good\] Let $X$ be a non-compact locally compact Cantor set and $\mu \in M^0(X)$ be a good measure. Let $cX$ be any compactification of $X$. Then $\mu$ is good on $cX$ if and only if $S(\mu|_{cX}) \cap [0, \mu(X)) = S(\mu|_X)$. **Proof**. First, we prove the “if” part. Let $V$ be a clopen set in $cX$. Consider two cases. First, let $V \cap (cX \setminus X) = \emptyset$. Then $V$ is a compact open subset of $X$. Since $\mu$ is good on $X$ and $S(\mu|_{cX}) \cap [0, \mu(X)) = S(\mu|_X)$, we see that $V$ stays good in $cX$. Now, suppose that $V \cap (cX \setminus X) \neq \emptyset$. Then $V \cap X$ is an open set and $\mu(V) = \mu(V \cap X) = \mu(\bigsqcup_{n=1}^{\infty} V_n)$ where each $V_n$ is a compact open set in $X$. Let $U$ be any compact open subset of $X$ with $\mu(U) < \mu(V)$. Then there exists $N \in \mathbb{N}$ such that $\mu(U) < \mu(\bigsqcup_{n = 1}^{N} V_n)$. The set $Z = \bigsqcup_{n = 1}^{N} V_n$ is a compact open subset of $X$. Since $S(\mu|_{cX}) \cap [0, \mu(X)) = S(\mu|_X)$, we have $\mu(U) \in S(\mu|_X)$. Since $\mu$ is good on $X$, there exists a compact open subset $W \subset Z$ such that $\mu(W) = \mu(U)$. Now we prove the “only if” part. Assume the converse. Suppose that $\mu$ is good and the equality does not hold. Then there exists $\gamma \in (0,\mu(X))$ such that $\gamma \in S(\mu|_{cX}) \setminus S(\mu|_X)$. Since $S(\mu|_X)$ is dense in $(0,\mu(X))$, there exists a compact open subset $U \subset X$ such that $\mu(U) > \gamma$. Hence $\gamma \in S(\mu|_{cX}) \cap [0, \mu(U)]$ and $\gamma \not \in S(\mu|_U)$. Thus $U$ is not good and we get a contradiction. $\blacksquare$ By Proposition \[basic\_prop\], the set $X \setminus \mathfrak{M}_\mu$ is a non-compact locally compact Cantor set and $\overline{X \setminus \mathfrak{M}_\mu} = X$. Thus, the set $X \setminus \mathfrak{M}_\mu$ can be homeomorphically embedded into $X$ and then into some compactification $cX$. After embedding $X \setminus \mathfrak{M}_\mu$ into $X$, we add only compact open sets of infinite measure. Hence if $\mu$ was good on $X \setminus \mathfrak{M}_\mu$, it remains good on $X$ and $S(\mu|_{X \setminus \mathfrak{M}_\mu}) = S(\mu|_X)$. We can consider $X$ as a step towards compactification of $X\setminus \mathfrak{M}_\mu$ and include $\mathfrak{M}_\mu$ into $cX \setminus X$. The measure $\mu \in M^0(X)$ is locally finite on $X \setminus \mathfrak{M}_\mu$, so we can consider only locally finite measures among infinite ones. If $\mu$ is not good on a locally compact Cantor set $X$ then clearly $\mu$ is not good on any compactification $cX$. Let $\mu$ be a good infinite locally finite measure on a non-compact locally compact Cantor set $X$. Then $\mu$ is good on $\omega X$. **Proof.** By definition of topology on $\omega X$, the “new” open sets have compact complement. Since $\mu$ is locally finite on $X$, the measure of compact subsets of $X$ is finite. Hence the measure of each new clopen set is infinite. By Theorem \[krit\_good\], $\mu$ is good on $\omega X$. $\blacksquare$ \[gamma\] Let $\mu$ be a good measure on a non-compact locally compact Cantor set $X$. Then for any $\gamma \in [0,\mu(X))$ there exists a compactification $cX$ such that $\gamma \in S(\mu|_{cX})$. **Proof.** The set $S(\mu|_{X})$ is dense in $[0,\mu(X))$. Hence for every $\gamma \in [0,\mu(X)$ there exist $\{\gamma_n\}_{n=1}^{\infty} \subset S(\mu|_{X})$ such that $\lim_{n\rightarrow \infty} \gamma_n = \gamma$. Since $\mu$ is good, there exist disjoint compact open subsets $\{U_n\}_{n=1}^{\infty}$ such that $\mu(U_n) = \gamma_n$. Then $U = \bigsqcup_{n=1}^{\infty} U_n$ is a non-compact locally compact Cantor set. Consider the compactification $cX = \omega U \sqcup c(X \setminus U)$, where $c(X \setminus U)$ is any compactification of $X \setminus U$. Then $\omega U$ is a clopen set in $cX$ and $\mu(\omega U) = \gamma \in S(\mu|_{cX})$. $\blacksquare$ From Theorems \[krit\_good\], \[gamma\] the corollary follows: For any measure $\mu$ on a non-compact locally compact Cantor space $X$ there exists a compactification $cX$ such that $\mu$ is not good on $cX$. If a measure $\mu \in M^0(X)$ is a good probability measure then, by Theorem \[krit\_good\], the measure $\mu$ is good on $cX$ if and only if $S(\mu|_{cX}) = S(\mu|_X) \cup \{1\}$. \[1\] Let $X$ be a non-compact locally compact Cantor set and $\mu \in M_f(X)$. If there exists a compactification $cX$ such that $S(\mu|_{cX}) = S(\mu|_X) \cup \{1\}$ then $1 \in G(S(\mu|_X))$. **Proof**. Let $\gamma \in S(\mu|_{cX}) \cap (0,1)$. Since the complement of a clopen set is a clopen set, we have $1 - \gamma \in S(\mu|_{cX})$. Since $S(\mu|_{cX}) = S(\mu|_X) \cup \{1\}$, we have $\gamma, 1 - \gamma \in S(\mu|_{X})$. Hence $1 \in G(S(\mu|_X))$. $\blacksquare$ Thus, if $1 \not \in G(S(\mu|_X))$ then for any compactification $cX$ the set $S(\mu|_X)$ cannot be preserved after the extension. The examples are given in the last section. The corollary follows from Proposition \[1\] and Theorem \[krit\_good\]. Let $\mu$ be a probability measure on a non-compact locally compact Cantor set $X$ and $1 \not \in G(S(\mu|_X))$. Then for any compactification $cX$ of $X$, $\mu$ is not good on $cX$. \[goodAlex\] Let $\mu$ be a good probability measure on a non-compact locally compact Cantor set $X$. Then $\mu$ is good on Alexandroff compactification $\omega X$ if and only if $1 \in G(S(\mu|_X))$. **Proof**. By Proposition \[1\] and Theorem \[krit\_good\], if $\mu$ is good on $\omega X$ then $1 \in G(S(\mu|_X))$. Suppose $\mu$ is good on $X$ and $1 \in G(S(\mu|_X))$. Since $\mu$ is good, any compact open subset of $\mu$ is good, hence for every compact open $U \subset X$ we have $\mu(U) = G(S(\mu|_X)) \cap [0, \mu(U)] = S(\mu)\cap [0, \mu(U)]$. Every clopen subset of $\omega X$ has a compact open subset of $X$ as a complement. Hence for every clopen $V \subset \omega X$ we see that $\mu(V) = 1 - \mu(X \setminus V) \in G(S(\mu|_X)) \cap (0,1) = S(\mu|_X)$. So, $S(\mu|_{\omega X}) = S(\mu|_X) \cup \{1\}$. Hence $\mu$ is good on $\omega X$ by Theorem \[krit\_good\]. $\blacksquare$ For a Cantor set $Y$ denote by $M^0(Y)$ the set of all either finite or non-defective measures on $Y$ (see [@K]). Since an open dense subset of a Cantor set is a locally compact Cantor set, the corollary follows: \[good\_subs\] Let $Y$ be a (compact) Cantor set and measure $\mu \in M^0(Y)$. Let $X \subset Y$ be an open dense subset of $Y$ of full measure. If $\mu$ is good on $Y$ then $\mu$ is good on $X$. **Proof**. The set $X$ is a locally compact Cantor set and $Y$ is a compactification of $X$. Any compact open subset $U$ of $X$ is a clopen subset of $Y$ and all clopen subsets of $U$ are compact open sets. Thus, a $\mu|_Y$-good compact open set in $X$ is, a fortiory, $\mu|_X$-good. $\blacksquare$ Thus, the extensions of a non-good measure are always non-good. The corollary follows from Lemma \[Smu\_diff\_comp\], Theorem \[krit\_good\] and Corollary \[good\_subs\]. \[2compactifications\] Let $X$ be a non-compact locally compact Cantor set and $\mu \in M^0(X)$. Let $c_1X$, $c_2X$ be compactifications of $X$ such that $c_1X \geq c_2X$. Let $\mu$ be good on $c_1X$. Then $\mu$ is good on $c_2X$. Moreover, if $\mu \in M_f(X)$ then $\mu|_{c_1X}$ is homeomorphic to $\mu|_{c_2X}$. Recall that Alexandroff compactification $\omega X$ is the smallest element in the set of all compactifications of $X$. Hence, if $\mu$ is not good on $\omega X$ then $\mu$ is not good on any compactification $cX$ of $X$. The following theorem can be proved using the results of Akin [@Akin2] for measures on compact sets. Let $X$, $Y$ be non-compact locally compact Cantor spaces, and $\mu \in M^0_f(X)$, $\nu \in M^0_f(Y)$ be good measures such that their extensions to $\omega X$, $\omega Y$ are good. Then $\mu|_X$ and $\nu|_Y$ are homeomorphic if and only if $S(\mu|_X) = S(\nu|_Y)$. **Proof.** The “only if” part is trivial, we prove the “if” part. Since $\mu|_{\omega X}$ and $\nu|_{\omega Y}$ are good by Theorem \[krit\_good\], we have $S(\mu|_{\omega X}) = S(\nu|_{\omega Y})$. Denote by $x_0 = \omega X \setminus X$ and $y_0 = \omega Y \setminus Y$. By Theorem 2.9 [@Akin2], there exists a homeomorphism $f \colon \omega X \rightarrow \omega Y$ such that $f_*\mu = \nu$ and $f(x_0) = y_0$. Hence $f(X) = Y$ and the theorem is proved. $\blacksquare$ In Example 1, we present a class of good measures on non-compact locally compact Cantor sets such that these measures are not good on the Alexandroff compactifications. Thus, these measures are not good on any compactification of the corresponding non-compact locally compact Cantor sets. Examples ======== Let $B$ be a non-simple stationary Bratteli diagram with the matrix $A$ transpose to the incidence matrix. Let $\mu$ be an ergodic $\mathcal{R}$-invariant measure on $B$ (see [@S.B.O.K.; @S.B.; @K]). Let $\alpha$ be the class of vertices that defines $\mu$. Then $\mu$ is good as a measure on a non-compact locally compact set $X_\alpha$. The measure $\mu$ on $X_\alpha$ can be either finite or infinite, but it is always locally finite. The set $X_B$ is a compactification of $X_\alpha$. Since $\mu$ is ergodic, we have $\mu(X_B \setminus X_\alpha) = 0$. In [@S.B.O.K.; @K] the criteria of goodness for probability or non-defective measure $\mu$ on $X_B$ were proved in terms of the Perron-Frobenius eigenvalue and eigenvector of $A$ corresponding to $\mu$ (see Theorem 3.5 [@S.B.O.K.] for probability measures and Corollary 3.4 [@K] for infinite measures). It is easy to see that these criteria are particular cases of Theorem \[krit\_good\]. We consider now a class of stationary Bratteli diagrams and give a criterion when a measure $\mu$ from this class is good on the Alexandroff compactification $\omega X_\alpha$. Fix an integer $N \geq 3$ and let $$F_N = \begin{pmatrix} 2 & 0 & 0\\ 1 & N & 1\\ 1 & 1 & N \\ \end{pmatrix}$$ be the incidence matrix of the Bratteli diagram $B_N$. For $A_N = F_N^T$ we easily find the Perron-Frobenius eigenvalue $\lambda = N+1$ and the corresponding probability eigenvector $$x = \left(\frac{1}{N},\ \frac{N-1}{2N},\ \frac{N-1}{2N}\right)^T.$$ Let $\mu_N$ be the measure on $B_N$ determined by $\lambda$ and the eigenvector $x$. The measure $\mu_N$ is good on $\omega X_\alpha$ if and only if for there exists $R \in \mathbb{N}$ such that $\frac{2(N+1)^R}{N-1}$ is an integer. This is possible if and only if $N = 2^k+1$, $k \in \mathbb{N}$. For instance, the measure $\mu_N$ is good on $\omega X_\alpha$ for $N = 3, 5$ but is not good for $N = 4$. Note that the criterion for goodness on $\omega X_\alpha$ here is the same as for goodness on $X_B$. This example is a particular case of more general result (the notation from [@S.B.O.K.] is used below): Let $B$ be a stationary Bratteli diagram defined by a distinguished eigenvalue $\lambda$ of the matrix $A = F^T$. Denote by $x = (x_1,...,x_n)^T$ the corresponding reduced vector. Let the vertices $2, \ldots, n$ belong to the distinguished class $\alpha$ corresponding to $\mu$. Then $\mu$ is good on $X_B$ if and only if $\mu$ is good on $\omega X_\alpha$. **Proof.** By Theorem \[Alex\] and Corollary \[2compactifications\], if $\mu$ is good on $X_B$ then $\mu$ is good on $\omega X_\alpha$. We prove the converse. By Theorem 3.5 in [@S.B.O.K.] and Theorem \[goodAlex\], it suffices to prove that $1 \in G(S(\mu|_{X_\alpha}))$ only if there exists $R \in \mathbb{N}$ such that $\lambda^R x_1 \in H(x_2,...,x_n)$. Note that $G(S(\mu|_{X_\alpha})) = \left(\bigcup_{N=0}^\infty \frac{1}{\lambda^N} H(x_2,...,x_n) \right)$, where $H(x_2,...,x_n)$ is an additive group generated by $x_2,...,x_n$. Suppose that $1 \in G(S(\mu|_{X_\alpha}))$. Since $\sum_{i=1}^n x_i = 1$, we see that $x_1 \in G(S(\mu|_{X_\alpha}))$, hence there exists $R \in \mathbb{N}$ such that $\lambda^R x_1 \in H(x_2,...,x_n)$. $\blacksquare$ Return to a general case of ergodic invariant measures on stationary Bratteli diagrams. If $\mu$ is a probability measure on $X_\alpha$ and $S(\mu|_{X_\alpha}) \cup \{1\} = S(\mu|_{X_B})$ then, by Lemma \[Smu\_diff\_comp\], we have $S(\mu|_{\omega X_\alpha}) = S(\mu|_{X_B})$. By Theorem \[krit\_good\], the measure $\mu$ is good on $\omega X_\alpha$. Hence $\mu|_{\omega X_\alpha}$ is homeomorphic to $\mu|_{X_B}$ (see [@Akin2]). If $\mu$ is infinite, then the measures $\mu|_{\omega X_\alpha}$ and $\mu|_{X_B}$ are not homeomorphic since $\mathfrak{M}_{\mu|_{\omega X_\alpha}}$ is one point and $\mathfrak{M}_{\mu|_{X_B}}$ is a Cantor set (see [@K]). Let $X$ be a Cantor space and $\mu$ be a good probability measure on $X$ with $S(\mu) = \{\frac{m}{2^n} : m \in \mathbb{N} \cap [0,2^n], n \in \mathbb{N}\}$ (for example a Bernoulli measure $\beta (\frac{1}{2},\frac{1}{2})$). Clearly, $\mu_n = \frac{1}{2^n}\mu$ is a good measure for $n \in \mathbb{N}$ with $S(\mu_n) = \frac{1}{2^n} S(\mu) \subset S(\mu)$. Let $\{X_n, \mu_n\}_{n=1}^{\infty}$ be a sequence of Cantor spaces with measures $\mu_n$. Let $A = \bigsqcup_{n=1}^{\infty} X_n$ be the disjoint union of $X_n$. Denote by $\nu$ a measure on $A$ such that $\nu|_{X_n} = \mu_n$. Then $\nu$ is a good measure on a locally compact Cantor space $A$ with $S(\nu) = S(\mu) \cap [0,1)$. Consider the one-point compactification $\omega A$ and the extension $\nu_1$ of $\nu$ to $\omega A$. We add to $S(\nu)$ the measures of sets which contain $\{\infty\}$ and have a compact open complement. Hence we add the set $\Gamma = \{1 - \gamma : \gamma \in S(\nu)\}$. Since $\Gamma \subset S(\nu) \cup \{1\}$, we have $S(\nu_1) = S(\nu) \cup \{1\}$. By Theorem \[krit\_good\], the measure $\nu_1$ is good on $\omega A$. Consider the two-point compactification of $A$. Let $A = A_1 \sqcup A_2$ where $A_1 = \bigsqcup_{k = 1}^{\infty} X_{2k - 1}$ and $A_2 = \bigsqcup_{j = 1}^{\infty} X_{2j}$. Then $cA = \omega A_1 \sqcup \omega A_2$ is a two-point compactification of $A$. Let $\nu_2$ be the extension of $\nu$ to $cA$. Then $\nu_2(A_1) = \frac{2}{3} \not \in S(\nu)$. Hence, by Theorem \[krit\_good\], the measure $\nu_2$ is not good on $cA$. In the same example, we can make a two-point compactification which preserves $S(\nu|_A)$. Since $\mu_n$ is good for $n \in \mathbb{N}$, there is a compact open partition $X_n^{(1)} \sqcup X_n^{(2)} = X_n$ such that $\mu_n (X_n^{(i)}) = \frac{1}{2^{n+1}}$ for $i = 1,2$. Let $B_i = \bigsqcup_{n=1}^{\infty} X_n^{(i)}$ for $i = 1,2$. Consider $\widetilde{c}A = \omega B_1 \sqcup \omega B_2$. Then it can be proved the same way as above that $S(\nu|_{\widetilde{c}A}) = S(\nu|_A) \cup \{1\}$. Let $\mu = \beta(\frac{1}{3}, \frac{2}{3})$ be a Bernoulli (product) measure on Cantor space $Y = \{0,1\}^\mathbb{N}$ generated by the initial distribution $p(0) = \frac{1}{3}$, $p(1) = \frac{2}{3}$. Then $\mu$ is not good but $S(\mu) = \{\frac{a}{3^n} : a \in \mathbb{N} \cap [0, 3^n], n \in \mathbb{N}\}$ is group-like (see [@Akin1]). Let $X$ be any open dense subset of $Y$ such that $\mu(Y \setminus X) = 0$. Thus, $Y$ is a compactification of a non-compact locally compact Cantor space $X$ and $\mu$ extends from $X$ to $Y$. Then $\mu$ is not good on $X$. The compact open subsets of $X$ are exactly the clopen subsets of $Y$ that lie in $X$. The compact open subset of $X$ is a union of the finite number of compact open cylinders. Consider any compact open cylinder $U = \{a_0,...,a_n,*\}$ which consists of all points in $z \in Y$ such that $z_i = a_i$ for $0 \leq i \leq n$. Then $U$ is a disjoint union of two subcylinders $V_1 = \{a_0,...,a_n,0,*\}$ and $V_2 = \{a_0,...,a_n,1,*\}$ with $\mu(V_2) = 2 \mu(V_1)$. Let the numerator of the fraction $\mu(V_1)$ be $2^k$. Then the numerator of the fraction $\mu(V_2)$ is $2^{k+1}$. Moreover, for any compact open $W \subset V_2$ the numerator of the fraction $\mu(W)$ will be divisible by $2^{k+1}$. Since $S(\mu)$ contains only finite sums of measures of cylinder compact open sets and the denominators of elements of $S(\mu)$ are the powers of $3$, there is no compact open subset $W \subset V_2$ such that $\mu(W) = \mu(V_1)$. Hence $\mu$ is not good on $X$. Moreover, let $x = \{00...\}$ be a point in $Y$ which consists only of zeroes. Then $S(\mu |_{Y \setminus\{x\}})\varsubsetneq S(\mu |_Y)$ while $S(\mu |_{Y \setminus\{y\}}) = S(\mu |_Y)$ for any $y \neq x$. Consider the case $y \neq x$. Let, for instance, $y = \{111....\}$, all other cases are proved in the same way. Let $U_n = \{z \in Y : z_0 = ... = z_{n-1} = 1, z_{n} = 0\}$. Then $Y \setminus\{y\} = \bigsqcup_{n=1}^\infty U_n \sqcup \{0*\}$. Denote by $S_N = \mu(\bigsqcup_{n=1}^N U_n)$ and $S_0 = 0$. Then $\lim_{N \rightarrow \infty} S_N = \frac{2}{3}$. Let $G = \{\frac{a}{3^n} : a \in \mathbb{Z}, n \in \mathbb{N}\}$. Then $G$ is an additive subgroup of reals and $S(\mu|_Y) = G \cap [0,1]$. We prove that $S(\mu |_{Y \setminus\{y\}}) = G \cap [0,1)$, i.e. for every $n \in \mathbb{N}$ and $a = 0,..., 3^n - 1$ there exists a compact open set $W$ in $Y \setminus \{y\}$ such that $\mu(W) = \frac{a}{3^n}$. Indeed, we have $S(\mu|_{\{0*\}}) = G \cap [0, \frac{1}{3}]$ and $[0,1) = \cup_{n=0}^\infty [S_N, S_N + \frac{1}{3}]$. Hence $G \cap [0,1) = \cup_{n=0}^\infty (G \cap [S_N, S_N + \frac{1}{3}])$. For every $\gamma \in G$ there exists $N\in \mathbb{N}$ such that $\gamma \in [S_N, S_N + \frac{1}{3}]$. There exists a compact open subset $W_0$ of $\{0*\}$ such that $\mu(W_0) = \gamma - S_N$. Set $W = U_N \sqcup W_0$. Then $W$ is a compact open subset of $Y \setminus \{y\}$ and $\mu(W) = \gamma$. Now consider the set $Y \setminus\{x\}$. Every cylinder that lies in $Y \setminus\{x\}$ has even numerator, hence $S(\mu |_{Y \setminus\{x\}})\varsubsetneq S(\mu |_Y)$. It can be proved in the same way as above that $S(\mu |_{Y \setminus\{x\}}) = \{\frac{2k}{3^n} : k \in \mathbb{N}\} \cap [0,1)$. Denote by $|A|$ the cardinality of a set $A$. Given two subsets $E, F \subset \mathbb{Z}$, by $E+F$ we mean $\{e+f | e \in E, f \in F\}$ (for more details see [@D1; @D2]). Let $\{F_n\}_{n=1}^\infty, \{C_n\}_{n=1}^\infty \subset \mathbb{Z}$ such that for each $n$ \(1) $|F_n|<\infty$, $|C_n|<\infty$, \(2) $|C_n| > 1$, \(3) $F_n + C_n + \{-1,0,1\} \subset F_{n+1}$, \(4) $(F_n + c) \cap (F_n + c') = \emptyset$ for all $c \neq c' \in C_{n+1}$. Set $X_n = F_n \times \prod_{k > n} C_k$ and endow $X_n$ with a product topology. By (1),(2), each $X_n$ is a Cantor space. For each $n$, define a map $i_{n,n+1} \colon X_{n} \rightarrow X_{n + 1}$ such that $$i_{n,n+1}(f_n, c_{n+1}, c_{n+2},...) = (f_n + c_{n+1}, c_{n+2},...).$$ By (1), (2) each $i_{n,n+1}$ is a well defined injective continuous map. Since $X_n$ is compact, we see that $i_{n,n+1}$ is a homeomorphism between $X_n$ and $i_{n,n+1}(X_n)$. So the embedding $i_{n,n+1}$ preserves topology. The set $i_{n, n+1}(X_n)$ is a clopen subset of $X_{n+1}$. Let $i_{m,n} \colon X_m \rightarrow X_n$ such that $i_{m,n} = i_{n, n-1} \circ i_{n-1, n-2} \circ ... \circ i_{m+1,m}$ for $m < n$ and $i_{n,n} = id$. Denote by $X$ the topological inductive limit of the sequence $(X_n, i_{n,n+1})$. Then $X = \bigcup_{n=1}^\infty X_n$. Since $i_{m,n} = i_{n, n-1} \circ i_{n-1, n-2} \circ ... \circ i_{m+1,m}$ for $m < n$, we can write $X_1 \subset X_2 \subset ...$ The set $X$ is a non-compact locally compact Cantor set. The Borel $\sigma$-algebra on $X$ is generated by cylinder sets $[A]_n = \{x \in X : x = (f_n, c_{n+1},c_{n+2},...) \in X_n \mbox{ and } f_n \in A\}$. There exists a canonical measure on $X$. Let $\kappa_n$ stand for the equidistribution on $C_n$ and let $\nu_n = \frac{|F_n|}{|C_1|...|C_n|}$ on $F_n$. The product measure on $X_n$ is defined as $\mu_n = \nu_n \times \kappa_{n+1} \times \kappa_{n+2}\times ...$ and a $\sigma$-finite measure $\mu$ on $X$ is defined by restrictions $\mu|_{X_n} = \mu_n$. The measure $\mu$ is a unique up to scaling ergodic locally finite invariant measure for a minimal self-homeomorphism of $X$ (for more details see [@D1; @D2]). For every two compact open subsets $U,V \subset X$ there exists $n \in \mathbb{N}$ such that $U, V \subset X_n$. The measure $\mu$ is obviously good, since the restriction of $\mu$ onto $X_n$ is just infinite product of equidistributed measures on $F_n$ and $C_m$, $m > n$. We have $S(\mu) = \{\frac{a}{|C_1|...|C_n|} : a, n \in \mathbb{N}\} \cap [0, \mu(X))$. Let $p$ be a prime number and $\mathbb{Q}_p$ be the set of $p$-adic numbers. Endowed with the $p$-adic norm, the set $\mathbb{Q}_p$ is a non-compact locally compact Cantor space. Then the Haar measure $\mu$ on $\mathbb{Q}_p$ is good and $S(\mu) = \{n p^{\gamma} | n \in \mathbb{N}, \gamma \in \mathbb{Z}\}$. **Acknowledgement** I am grateful to my advisor Sergey Bezuglyi for giving me the idea of this work, for many helpful discussions and for reading the preliminary versions of this paper. [99]{} *E.Akin*, Measures on Cantor space, “Topology Proc.”, **24** (1999), 1 - 34. *E.Akin*, Good measures on Cantor space, “Trans. Amer. Math. Soc.”, **357** (2005), 2681 - 2722. *E. Akin, R. Dougherty, R.D. Mauldin, A. Yingst*, Which Bernoulli measures are good measures? “Colloq. 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Amer. Math. Soc.” **359** (2007), 6155 - 6166 *R. Engelking,* General Topology, Berlin: Heldermann, 1989. *E. Glasner, B. Weiss,* Weak orbital equivalence of minimal Cantor systems, “Internat. J. Math.” **6** (1995), 559 - 579. *O. Karpel,* Infinite measures on Cantor spaces, J Difference Equ. Appl., DOI:10.1080/10236198.2011.620955. *J.C.Oxtoby, V.S.Prasad,* Homeomorphic measures in the Hilbert Cube, “Pacific J. Math.”, **77** (1978), 483 - 497. *J.C.Oxtoby, S.M.Ulam*, Measure preserving homeomorphisms and metrical transitivity, “Ann. Math.”, **42** (1941), 874 - 920. *Andrew Q. Yingst*, A characterization of homeomorphic Bernoulli trial measures, “Trans. Amer. Math. Soc.”, **360** (2008), 1103 - 1131.

--- author: - | CARLOS GERSHENSON\ Vrije Universiteit Brussel title: 'A General Methodology for Designing Self-Organizing Systems' --- Author’s address: Krijgskundestraat 33 B-1160 Brussel, Belgium cgershen@vub.ac.be http://homepages.vub.ac.be/cgershen Introduction ============ Over the last half a century, much research in different areas has employed self-organizing systems to solve complex problems, e.g. [Ashby1956,Beer1966,BonabeauEtAl1999,EngineeringSOS2004,ZambonelliRana2005]{}. Recently, particular methodologies using the concepts of self-organization have been proposed in different areas, such as software engineering [WooldridgeEtAl2000,ZambonelliEtAl2003]{}, electrical engineering [RamamoorthyEtAl1993]{}, and collaborative support [@JonesEtAl1994]. However, there is as yet no general framework for constructing self-organizing systems. Different vocabularies are used in different areas, and with different goals. In this paper, I present an attempt to develop a general methodology that will be useful for designing and controlling *complex* systems [@Bar-Yam1997]. The proposed methodology, as with any methodology, does not provide ready-made solutions to problems. Rather, it provides a *conceptual framework*, a *language,* to assist the solution of problems. Also, many current problem solutions can be *described* as proposed. I am not suggesting new solutions, but an alternative way of thinking about them. As an example, many standardization efforts have been advanced in recent years, such as ontologies required for the Semantic Web [@Berners-LeeEtAl2001], or FIPA standards. I am not insinuating that standards are not necessary. Without them engineering would be chaos. But as they are now, they cannot predict future requirements. They are developed with a static frame of mind. They are not adaptive. What this work suggests is a way of introducing the expectation of change into the development process to be able to cope with the unexpected beforehand, in problem domains where this is desired. The paper is organized as follows: in the next section, notions of complexity and self-organization are discussed. In Section [secConceptualFw]{}, original concepts are presented. These will be used in the Methodology, exposed in Section \[secMethodology\]. In Section [secSOTL]{}, a case study concerning self-organizing traffic lights is used to illustrate the steps of the Methodology. Discussion and conclusions follow in Sections \[secDiscussion\] and \[secConclusions\]. Complexity and Self-organization ================================ There is no general definition of *complexity*, since the concept achieves different meanings in different contexts [@Edmonds1999]. Still, we can say that a system is complex if it consists of several *interacting* elements [@Simon1996], so that the behavior of the system will be difficult to deduce from the behavior of the parts. This occurs when there are many parts, and/or when there are many interactions between the parts. Typical examples of complex systems are a living cell, a society, an economy, an ecosystem, the Internet, the weather, a brain, and a city. These all consist of numerous elements whose interactions produce a global behavior that cannot be reduced to the behavior of their separate components [@GershensonHeylighen2005]. For example, a cell is considered a living system, but the elements that conform it are not alive. The properties of life arise from the complex dynamical *interactions* of the components. The properties of a system that are not present at the lower level (such as life), but are a product of the interactions of elements, are sometimes called *emergent* [@Anderson1972]. Another example can be seen with gold: it has properties, such as temperature, malleability, conductivity, and color, that emerge from the interactions of the gold atoms, since atoms do not have these properties. Even when there is no general definition or measure of complexity, a relative *notion* of complexity can be useful: \[theorem\][Notion]{} The complexity of a system scales with the number of its elements, the number of interactions between them, the complexities of the elements, and the complexities of the interactions [@Gershenson2002a]:[^1] $$C_{sys} \sim \#\overline{E} \#\overline{I} \sum_{j=0}^{\#\overline{E}}C_{e_{j}} \sum_{k=0}^{\#\overline{I}}C_{i_{k}} \label{eqComplexity}$$ The complexity on an interaction $C_{i}$ can be measured as the number of different possible interactions two elements can have.[^2] The problem of a strict definition of complexity lies in the fact that there is no way of drawing a line between simple and complex systems independently of a context. For example, the *dynamics* of a system can be simple (ordered), complex, or chaotic, having a complex structure. Cellular automata and random Boolean networks are a clear example of this, where moreover, the interactions of their components are quite simple. On the other hand, a *structurally* simple system can have complex and chaotic dynamics. For this case, the damped pendulum is a common example. Nevertheless, for practical purposes, the above notion will suffice, since it allows the comparison of the complexity of one system with another under a common frame of reference. Notice that the notion is recursive, so a basic level needs to be set contextually for comparing two systems. The term *self-organization* has been used in different areas with different meanings, as is cybernetics [@vonFoerster1960; @Ashby1962], thermodynamics [@NicolisPrigogine1977], biology [@CamazineEtAl2003], mathematics [@Lendaris1964], computing [@HeylighenGershenson2003], information theory [@Shalizi2001], synergetics [@Haken1981], and others [@SkarCoveney2003] (for a general overview, see [Heylighen2003sos]{}). However, the use of the term is subtle, since any dynamical system can be said to be self-organizing or not, depending partly on the observer [@GershensonHeylighen2003a; @Ashby1962]: If we decide to call a “preferred" state or set of states (i.e. attractor) of a system “organized", then the dynamics will lead to a self-organization of the system. It is not necessary to enter into a philosophical debate on the theoretical aspects of self-organization to work with it, so a practical notion will suffice: \[theorem\][Notion]{} A system *described* as self-organizing is one in which elements *interact* in order to achieve *dynamically* a global function or behavior. This function or behavior is not imposed by one single or a few elements, nor determined hierarchically. It is achieved *autonomously* as the elements interact with one another. These interactions produce feedbacks that regulate the system. All the previously mentioned examples of complex systems fulfill the definition of self-organization. More precisely, the question can be formulated as follows: *when is it useful to describe a system as self-organizing?* This will be when the system or environment is very dynamic and/or unpredictable. If we want the system to solve a problem, it is useful to describe a complex system as self-organizing when the “solution" is not known beforehand and/or is changing constantly. Then, the solution is dynamically strived for by the elements of the system. In this way, systems can adapt quickly to unforeseen changes as elements interact locally. In theory, a centralized approach could also solve the problem, but in practice such an approach would require too much time to compute the solution and would not be able to keep the pace with the changes in the system and its environment. In engineering, a self-organizing system would be one in which elements are designed in order to solve *dynamically* a problem or perform a function at the system level. Thus, the elements need to divide, but also integrate, the problem. For example, the parts of a car are designed to perform a function at the system level: to drive. However, the parts of a (normal) car do not change their behavior in time, so it might be redundant to call a car self-organizing. On the other hand, a swarm of robots [@DorigoEtAl2004] will be conveniently described as self-organizing, since each element of the swarm can change its behavior depending on the current situation. It should be noted that all engineered self-organizing systems are to a certain degree *autonomous*, since part of their actual behavior will not be determined by a designer. In order to understand self-organizing systems, two or more *levels of abstraction* [@Gershenson2002a] should be considered: elements (lower level) organize in a system (higher level), which can in turn organize with other systems to form a larger system (even higher level). The understanding of the system’s behavior will come from the relations observed between the descriptions at different levels. Note that the levels, and therefore also the terminology, can change according to the interests of the observer. For example, in some circumstances, it might be useful to refer to cells as elements (e.g. bacterial colonies); in others, as systems (e.g. genetic regulation); and in others still, as systems coordinating with other systems (e.g. morphogenesis). A system can cope with an unpredictable environment *autonomously* using different but closely related approaches: - **Adaptation** (learning, evolution) [@Holland1995]. The system changes its behavior to cope with the change. - **Anticipation** (cognition) [@Rosen1985]. The system predicts a change to cope with, and adjusts its behavior accordingly. This is a special case of adaptation, where the system does not require to experience a situation before responding to it. - **Robustness** [@vonNeumann1956; @Jen2005]. A system is robust if it continues to function in the face of perturbations [@Wagner2005]. This can be achieved with modularity [@Simon1996; @Watson2002], degeneracy [@FernandezSole2003], distributed robustness [@Wagner2004], or redundancy [@GershensonEtAl2006]. Successful self-organizing systems will use combinations of the these approaches to maintain their integrity in a changing and unexpected environment. Adaptation will enable the system to modify itself to “fit" better within the environment. Robustness will allow the system to withstand changes without losing its function or purpose, and thus allowing it to adapt. Anticipation will prepare the system for changes before these occur, adapting the system without it being perturbed. We can see that all of them should be taken into account while engineering self-organizing systems. In the following section, further concepts will be introduced that will be necessary to apply the methodology . The Conceptual Framework {#secConceptualFw} ======================== Elements of a complex system interact with each other. The actions of one element therefore affect other elements, directly or indirectly. For example, an animal can kill another animal directly, or indirectly cause its starvation by consuming its resources. These interactions can have negative, neutral, or positive effects on the system [@HeylighenCampbell1995]. Now, intuitively thinking, it may be that the “smoothening" of local interactions, i.e. the minimization of “interferences" or “friction" will lead to global improvement. But is this always the case? To answer this question, the terminology of multi-agent systems [Maes1994,WooldridgeJennings1995,Wooldridge2002,Schweitzer2003]{} can be used. We can say that: \[theorem\][Notion]{} An agent is a description of an entity that *acts* on its environment. Examples of this can be a trader acting on a market, a school of fish acting on a coral reef, or a computer acting on a network. Thus, every element, and every system, can be seen as agents with *goals* and behaviors thriving to reach those goals. The behavior of agents can affect (positively, negatively, or neutrally) the fulfillment of the goals of other agents, thereby establishing a relation. The *satisfaction* or fulfillment of the goals of an agent can be represented using a variable $% \sigma \in \lbrack 0,1]$.[^3] Relating this to the higher level, the satisfaction of a system $\sigma _{sys}$ can be recursively represented as a function $f: \mathbb{R} \rightarrow \lbrack 0..1]$ of the satisfaction of the $n$ elements conforming it: $$\sigma _{sys}=f\left( \sigma _{1},\sigma _{2},...,\sigma _{n},w_{0},w_{1},w_{2},...,w_{n}\right) \label{eqSigmaSys}$$ where $w_{0}$ is a bias and the other weights determine the importance given to each $\sigma _{i}$. If the system is homogeneous, then $f$ will be the weighted sum of $\sigma _{i}$, $w_{i}=\frac{1}{n}\forall i\neq 0$, $w_{0}=0$. Note that this would be very similar to the activation function used in many artificial neural networks [@Rojas1996]. For heterogenous systems, $f$ may be a nonlinear function. Nevertheless, the weights $w_{i}$’s are determined *tautologically* by the importance of the $\sigma $ of each element to the satisfaction of the system. Thus, it is a useful tautology to say that maximizing individual $\sigma $’s, adjusting individual behaviors (and thus relations), will maximize $\sigma _{sys}$. If several elements decrease $% \sigma _{sys}$ as they increase their $\sigma $, we would not consider them as part of the system. It is important to note that this is independent of the potential nonlinearity of $f$. An example can be seen with the immune system. It categorizes molecules and micro-organisms as akin or alien [VazVarela1978]{}. If they are considered as alien, they are attacked. Auto-immune diseases arise when this categorization is erroneous, and the immune system attacks vital elements of the organism. On the other hand, if pathogens are considered as part of the body, they are not attacked. Another example is provided by cancer. Carcinogenic cells can be seen as “rebel", and no longer part of the body, since their goals differ from the goal of the organism. Healthy cells are described easily as part of an organism. But when they turn carcinogenic, they can better be described as parasitic. The tautology is also useful because it gives a general mathematical representation for system satisfaction, which is independent of a particular system. A reductionist approach would assume that maximizing the satisfaction of the elements of a system would also maximize the satisfaction of the system. However, this is not always the case, since some elements can “take advantage" of other elements. Thus, we need to concentrate *also* on the interactions of the elements. If the model of a system considers more than two levels, then the $\sigma $ of higher levels will be recursively determined by the $\sigma $’s of lower levels. However, the $f$’s most probably will be very different on each level. Certainly, an important question remains: how do we determine the function $% f $ and the weights $w_{i}$’s? To this question there is no complete answer. One option would be to approximate $f$ numerically [@DeWolfEtAl2005]. An explicit $f$ may be difficult to find, but an approximation can be very useful. Another method consists of *lesioning* the system[^4]: removing or altering elements of the system, and observing the effect on $\sigma _{sys}$. Through analyzing the effects of different lesions, the function $f$ can be reconstructed and the weights $w_{i}$’s obtained. If a small change $\Delta \sigma _{i}$ in any $\sigma _{i}$ produces a change $\Delta \sigma _{sys}\geq \Delta \sigma _{i}$, the system can be said to be *fragile*. What could then be done to maximize $\sigma _{sys}$? How can we relate the $% \sigma _{i}$’s and avoid conflicts between elements? This is not an obvious task, for it implies bounding the agents’ behaviors that reduce other $% \sigma _{i}$’s, while preserving their functionality. Not only should the interference or friction between elements be minimized, but the synergy [Haken1981]{} or “positive interference" should also be promoted. Dealing with complex systems, it is not feasible to tell each element what to do or how to do it, but their behaviors need to be constrained or modified so that their goals will be reached, blocking the goals of other elements as little as possible. These constraints can be called *mediators* [Michod2003,Heylighen2003]{}. They can be imposed from the top down, developed from the bottom up, be part of the environment, or be embedded as an *aspect* [@tenHaafEtAl2002 Ch. 3] of the system. An example can be found in city traffic: traffic lights, signals and rules mediate among drivers, trying to minimize their conflicts, which result from the competition for limited resources, i.e. space to drive through. The role of a mediator is to arbitrate among the elements of a system, to minimize interferences and frictions and maximize synergy. Therefore, the efficiency of the mediator can be measured directly using $\sigma _{sys}$. Individually, we can measure the “friction" $\phi _{i}\in \lbrack -1,1]$ that agent $i$ causes in the rest of the system, relating the change in satisfaction $\Delta \sigma _{i}$ of element $i$ and the change in satisfaction of the system $\Delta \sigma _{sys}$: $$\phi _{i}=\frac{-\Delta \sigma _{i}-\Delta \sigma _{sys}\left( n-1\right) }{n% }. \label{eqFriction}$$ Friction occurs when the increase of satisfaction of one element causes a decrease in the satisfaction of some other elements that is greater than the increase. Note that $\phi _{i}=0$ does imply that there is no conflict, since one agent can “get" the satisfaction proportionally to the “loss" of satisfaction of (an)other agent(s). Negative friction would imply synergy, e.g. when $\Delta \sigma _{i}\geq 0$ while other elements also increase their $\sigma $. The role of a mediator would be to maximize $\sigma _{sys}$ by minimizing $\phi _{i}$’s. With this approach, friction can be seen as a type of *interaction* between elements. Thus, the problem can be put in a different way: how can we find/develop/evolve efficient mediators for a given system? One answer to this question is the methodology proposed in this paper. The answer will not be complete, since we cannot have precise knowledge of $f$ for large evolving complex systems. This is because the evolution of the system will change its own $f$ [@Kauffman2000], and the relationships among different $\sigma _{i}$’s. Therefore, predictions cannot be complete. However, the methodology proposes to follow steps to increase the understanding (and consequently the control) of the system and the relations between its elements. The goal is to identify conflicts and diminish them without creating new ones. This will increase the $\sigma _{i}$’s and thus $% \sigma _{sys}$. The precision of $f$ is not so relevant if this is achieved. It should be noted that the timescale chosen for measuring $\Delta \sigma _{i}$ is very important, since at short timescales the satisfaction can decrease, while on the long run it will increase. In other words, there can be a short term “sacrifice" to harvest a long term “reward". If the timescale is too small, a system might get stuck in a “local optimum", since all possible actions would decrease its satisfaction on the short term. But in some cases the long term benefit should be considered for maximization. A way of measuring the slow change of $\sigma _{i}$ would be with its integral over time for a certain interval $\Delta t$: $$\int_{t}^{t+\Delta t}\sigma _{i}dt. \label{eqIntegralSigma}$$ Another way of dealing with the local optima is to use neutral changes to explore alternative solutions [@Kimura1983]. Before going into further detail, it is worth noting that this is not a reductionist approach. Smoothing out local interactions will not provide straightforward clues as to what will occur at the higher level. Therefore, the system should be observed at both levels: making local and global changes, observing local and global behaviors, and analyzing how one affects the other. Concurrently, the *dependence* $\epsilon $ $\in \lbrack -1,1]$ of an element to the system can be measured by calculating the difference of the satisfaction $\sigma _{i}$ when the element interacts within the system and its satisfaction $\widetilde{\sigma _{i}}$ when the element is isolated. $$\epsilon =\sigma _{i}-\widetilde{\sigma _{i}}. \label{eqDependence}$$ In this way, full dependence is given when the satisfaction of the element within the system $\sigma _{i}$ is maximal and its satisfaction $\widetilde{\sigma _{i}}$ is minimal when the element is isolated. A negative $\epsilon $ would imply that the element would be more satisfied on its own and is actually “enslaved" by the system. Now we can use the dependences of the elements to a system to measure the *integration* $\tau $ $\in \lbrack -1,1]$ of a system, which can be seen also as a gradual measure of a meta-system transition (MST) [@Turchin1977]. $$\tau =\frac{1}{n}\sum\limits_{i=1}^{n}\epsilon _{i}. \label{eqIntegration-MST}$$ A MST is a gradual process, but it will be complete when elements are not able to reach their goals on their own, i.e. $\overline{\sigma _{i}}% \rightarrow 0$. Examples include cells in multi-cellular organisms and mitochondria in eukaryotes. In an evolutionary process, natural (multilevel [@Michod1997; @Lenaerts2003]) selection will tend to increase $\tau $ because this implies higher satisfaction both for the system and its elements (systems with a negative $\tau $ are not viable). Relations and mediators that contribute to this process will be selected, since higher $% \sigma $’s imply more chances of survival and reproduction. Human designers and engineers also select relations and mediators that increase the $\sigma $’s of elements and systems. Therefore, we can see that evolution will tend, in the long run, towards synergetic relationships [@Corning2003], even if resources are scarce. In the next section, the steps suggested for developing a self-organizing system are presented, using the concepts described in this section. The Methodology {#secMethodology} =============== The proposed methodology meets the requirements of a system, i.e. what the system should do, and enables the designer to produce a system that fulfills the requirements. The methodology includes the following steps: Representation, Modeling, Simulation, Application, and Evaluation, which will be exposed in the following subsections. Figure \[diagram\] presents these steps. These steps should not necessarily be followed one by one, since the stages merge with each other. There is also backtracking, when the designer needs to return to an earlier stage for reconsideration before finishing a cycle. This methodology should not be seen as a recipe that provides ready-made solutions, but rather as a guideline to direct the search for them. The stages proposed are not new, and similar to those proposed by iterative and incremental development methodologies. Still, it should be noted that the active feedback between stages within each iteration can help in the design of systems ready to face uncertainties in complex problem domains. The novelty of the methodology lies in the *vocabulary* used to describe self-organizing systems. Representation -------------- The goal of this step is to develop a *specification* (which might be tentative) of the components of the system. The designer should always remember the distinction between model and modeled. A model is an abstraction/description of a “real" system. Still, there can be several descriptions of the same system [Gershenson2002a,GershensonHeylighen2005]{}, and we cannot say that one is better than another independently of a context. There are many possible representations of a system. According to the *constraints* and *requirements*, which may be incomplete, the designer should choose an appropriate vocabulary (metaphors to speak about the system), abstraction levels, granularity, variables, and interactions that need to be taken into account. Certainly, these will also depend on the experience of the designer. The choice between different approaches can depend more on the expertise of the designer than on the benefits of the approaches. Even when there is a wide diversity of possible systems, a general approach for developing a Representation can be abstracted. The designer should try to divide a system into elements by identifying semi-independent modules, with internal goals and dynamics, and with few interactions with their environment. Since interactions in a model will increase the complexity of the model, we should group “clusters" of interacting variables into elements, and then study a minimal number of interactions between elements. The first constraints that help us are space and time. It is useful to group variables that are close to each other (i.e. interacting constantly) and consider them as elements that relate to other elements in occasional interactions. Multiscale analysis [@Bar-Yam2005] is a promising method for identifying levels and variables useful in a Representation. Since the proposed methodology considers elements as agents, another useful criterion for delimiting them is the identification of goals. These will be useful in the Modeling to measure the satisfaction $\sigma $ of the elements. We can look at genes as an example: groups of nucleotides co-occur and interact with other groups and with proteins. Genes are identified by observing nucleotides that keep close together and act together to perform a function. The fulfillment of this function can be seen as a goal of the gene. Dividing the system into modules also divides the problem it needs to solve, so a complex task will be able to be processed in parallel by different modules. Certainly, the integration of the “solutions" given by each module arises as a new problem. Nevertheless, modularity in a system also increases its robustness and adaptability [@Simon1996; @Watson2002; @FernandezSole2003]. The representation should consider at least two levels of abstraction, but if there are many variables and interactions in the system, more levels can be contemplated. Since elements and systems can be seen as agents, we can refer to all of them as $x$-agents, where $x$ denotes the level of abstraction relative to the simplest elements. For example, a three-layered abstraction would contemplate elements (0-agents) forming systems that are elements (subsystems, 1-agents) of a greater system (meta-system, 2-agents). If we are interested in modeling a research institute, 0-agents would be researchers, 1-agents would be research groups, and the research institute would be a 2-agent. Each of these have goals and satisfactions ($\sigma ^{x}$) that can be described and interrelated. For engineering purposes, the satisfaction of the highest level, i.e. the satisfaction of the system that is being designed, will be determined by the tasks expected from it. If these are fulfilled, then it can be said that the system is “satisfied". Thus, the designer should concentrate on engineering elements that will strive to reach this satisfaction. If there are few elements or interactions in the Representation, there will be low complexity, and therefore stable dynamics. The system might be better described using traditional approaches, since the current approach might prove redundant. A large variety of elements and/or interactions might imply a high complexity. Then, the Representation should be revised before entering the Modeling stage. Modeling -------- In science, models should ideally be as simple as possible, and predict as much as possible [@Shalizi2001]. These models will provide a better understanding of a phenomenon than complicated models. Therefore, a good model requires a good Representation. The “elegance" of the model will depend very much on the metaphors we use to speak about the system. If the model turns out to be cumbersome, the Representation should be revised. The Modeling should specify a Control* *mechanism that will ensure that the system does what it is required to do. Since we are interested in self-organizing systems, the Control will be *internal* and *distributed*. If the problem is too complex, it can be divided into different subproblems. The Modeling should also consider different trade-offs for the system. ### Control mechanism The Control mechanism can be seen as a *mediator* [@Heylighen2003] ensuring the proper interaction of the elements of the system, and one that should produce the desired performance. However, one cannot have a strict control over a self-organizing system. Rather, the system should be *steered* [@Wiener1948]. In a sense, self-organizing systems are like teenagers: they cannot be tightly controlled since they have their own goals. We can only attempt to steer their actions, trying to keep their internal variables under certain boundaries, so that the systems/teenagers do not “break" (in Ashby’s sense [@Ashby1947]). To develop a Control, the designer should find aspect systems, subsystems, or constraints that will prevent the negative interferences between elements (friction) and promote positive interferences (synergy). In other words, the designer should search for ways of minimizing frictions $\phi _{i}$’s that will result in maximization of the global satisfaction $\sigma _{sys}$. The performance of different mediators can be measured using equation ([eqSigmaSys]{}). The Control mechanism should be *adaptive*. Since the system is dynamic and there are several interactions within the system and with its environment, the Control mechanism should be able to cope with the changes within and outside the system, in other words, *robust*. An adaptive Control will be efficient in more contexts than a static one. In other words, the Control should be *active* in the search of solutions. A static Control will not be able to cope with the complexity of the system. There are several methods for developing an adaptive Control, e.g. [@SastryBodson1994]. But these should be applied in a distributed way, in an attempt to reduce friction and promote synergy. Different methods for reducing friction in a system can be identified. In the following cases, an agent A negatively affected by the behavior of an agent B will be considered[^5]: - **Tolerance**. This can be seen as the acceptance of others and their goals. A can tolerate B by modifying itself to reduce the friction caused by B, and therefore increase $\sigma _{A}$. This can be done by moving to another location, finding more resources, or making internal changes. - **Courtesy**. This would be the opposite case to Tolerance. B should modify its behavior not to reduce $\sigma _{A}$. - **Compromise**. A combination of Courtesy and Tolerance: both agents A and B should modify their behaviors to reduce the friction. This is a good alternative when both elements cause friction to each other. This will be common when A and B are similar, as in homogeneous systems. - **Imposition**. This could be seen as forced Courtesy. The behavior of B could be changed by force. The Control could achieve this by constraining B or imposing internal changes. - **Eradication**. As a special case of Imposition, B can be eradicated. This certainly would decrease $\sigma _{B}$, but can be an alternative when either $\sigma _{B}$ does not contribute much to $\sigma _{sys}$, or when the friction caused by B in the rest of the system is very high. - **Apoptosis**. B can “commit suicide". This would be a special case of Courtesy, where B would destroy itself for the sake of the system. The difference between Compromise/Apoptosis and Imposition/Eradication is that in the former cases, change is triggered by the agent itself, whereas in the latter the change is imposed from the “outside" by a mediator. Tolerance and Compromise could be generated by an agent or by a mediator. Different methods for reducing friction can be used for different problems. A good Control will select those in which other $\sigma $’s are not reduced more than the gain in $\sigma $’s. The choice of the method will also depend on the importance of different elements for the system. Since more important elements contribute more to $\sigma _{sys}$, these elements can be given preference by the Control in some cases. Different methods for increasing synergy can also be identified. These will consist of taking measures to increase $\sigma _{sys}$, even if some $\sigma $’s are reduced: - **Cooperation**. Two or more agents adapt their behavior for the benefit of the whole. This might or might not reduce some $\sigma $’s. - **Individualism**. An agent can choose to increase its $\sigma $ if it increases $\sigma _{sys}$. A mediator should prevent increases in $% \sigma $’s if these reduce $\sigma _{sys}$, i.e. friction. - **Altruism**. An agent can choose to sacrifice an increase of its $\sigma $ or to reduce its $\sigma $ in order to increase $\sigma _{sys}$. This would make sense only if the *relative* increase of $\sigma _{sys}$ is greater than the decrease of the $\sigma $ of the altruistic agent. A mediator should prevent wasted altruism. - **Exploitation**. This would be forced Altruism: an agent is driven to reduce its $\sigma $ to increase $\sigma _{sys}$. A common way of reducing friction is to enable agents to learn via reinforcement [@KaelblingEtAl1996]. With this method, agents tend to repeat actions that bring them satisfaction and avoid the ones that reduce it. Evolutionary approaches, such as genetic algorithms [@Mitchell1996], can also reduce friction and promote synergy. However, these tend to be “blind", in the sense that variations are made randomly, and only their effects are evaluated. With the criteria presented above, the search for solutions can be guided with a certain aim. However, if the relationship between the satisfaction of the elements and the satisfaction of the system is too obscure, “blind" methods remain a good alternative. In general, the Control should explore different alternatives, trying to constantly increase $\sigma _{sys}$ by minimizing friction and maximizing synergy. This is a constant process, since a self-organizing system is in a dynamic environment, producing “solutions" for the current situation. Note that a mediator will not necessarily maximize the satisfaction of the agents. However, it should try to do so for the system. ### Dividing the problem If the system is to deal with many parameters, then it can be seen as a *cognitive* system [@Gershenson2004]. It must “know" or “anticipate" what to do according to the current situation and previous history. Thus, the main problem, i.e. what the elements should do, could be divided into the problems of communication, cooperation, and coordination [GershensonHeylighen2004]{}. For a system to self-organize, its elements need to *communicate*: they need to “understand" what other elements, or mediators, “want" to tell them. This is easy if the interactions are simple: sensors can give *meaning* to the behaviors of other elements. But as interactions turn more complex, the *cognition* [@Gershenson2004] required by the elements should also be increased. New meanings can be learned [@Steels1998; @DeJong2000]to adapt to the changing conditions. These can be represented as “concepts" [@Gardenfors2000], or encoded, e.g., in the weights of a learning neural network [@Rojas1996]. The precise implementation and philosophical interpretations are not relevant if the outcome is the desired one. The problem of *cooperation* has been widely studied using game theory [@Axelrod1984]. There are several ways of promoting cooperation, especially if the system is designed. To mention mention only two of them: the use of tags [@RioloEtAl2001; @HalesEdmonds2003] and multiple levels of selection [@Michod1997; @Lenaerts2003] have proven to yield cooperative behavior. This will certainly reduce friction and therefore increase $\sigma _{sys}$. Elements of a system should *coordinate* while reducing friction, not to obstruct each other. An important aspect of coordination is the *division of labour*. This can promote synergy, since different elements can specialize in what they are good at and *trust[^6]* others to do what they are good at [@Gaines1994; @DiMarzoSerugendo2004]. This process will yield a higher $\sigma _{sys}$ compared to the case when every element is meant to perform all functions independently of how well each element performs each function. A good Control will promote division of labour by mediating a balance between *specialization* and *integration*: elements should devote more time doing what they are best at, but should still take into account the rest of the system. Another aspect of coordination is the *workflow*: if some tasks are prerequisites of other tasks, a mediator should synchronize the agents to minimize waiting times. ### Trade-offs A system needs to be able to cope with the complexity of its domain to achieve its goals. There are several trade-offs that can be identified to reach a balance and cope better with this complexity: - **Complexity of Elements/Interactions**. The complexity of the system required to cope with the complexity of its domain can be tackled at one end of the spectrum by complex elements with few/simple interactions, and at the other by simple elements with several/complex interactions. - **Quality/Quantity**. A system can consist at one extreme of a few complex elements, and at the other of several simple elements. - **Economy/Redundancy**. Solving a problem with as few elements as possible is economical. But a minimal system is very fragile. Redundancy is one way of favoring the *robustness* of the system [vonNeumann1966,FernandezSole2003,Wagner2004,GershensonEtAl2006]{}. Still, too much redundancy can also reduce the speed of adaptation and increase costs for maintaining the system. - **Homogeneity/Heterogeneity**. A homogeneous system will be easier to understand and control. A heterogenous system will be able to cope with more complexity with less elements, and will be able to adapt more quickly to sudden changes. If there is a system of ten agents each able to solve ten tasks, a homogeneous system will be able to solve more than ten tasks robustly. A fully heterogeneous system would be able to solve more than a hundred tasks, but it would be fragile if one agent failed. Heterogeneity also brings diversity, that can accelerate the speed of exploration, adaptation, and evolution, since different solutions can be sought in parallel. The diversity is also related to the amount of variety of perturbations that the system can cope with [@Ashby1956], i.e. robustness. - **System/Context**. The processing and storage of information can be carried out internally by the system, or the system can exploit its environment “throwing" complexity into it, i.e. allow it to store or process information [@GershensonEtAl2003a]. - **Ability/Clarity**. A powerful system will solve a number of complex problems, but it will not be very useful if the functioning of the system cannot be understood. Designers should be able to understand the system in order to be able to control it [@Schweitzer2003]. - **Generality/Particularity**. An abstract Modeling will enable the designer to apply the Modeling in different contexts. However, particular details should be considered to make the implementation feasible. There are only very relative ways of measuring the above mentioned trade-offs. However, they should be kept in mind during the development of the system. In a particular system, the trade-offs will become clearer once the Simulation is underway. They can then be reconsidered and the Modeling updated. Simulation ---------- The aim here is to build computer simulation(s) that implement the model(s) developed in the Modeling stage, and test different scenarios and mediator strategies. The Simulation development should proceed in stages: from abstract to particular. First, an abstract scenario should be used to test the main concepts developed during the Modeling. Only when these are tested and refined, should details be included in the Simulation. This is because particular details take time to develop, and there is no sense in investing before knowing wether the Modeling is on the right track. Details can also influence the result of the Simulation, so they should be put off until a time when the main mechanisms are understood. The Simulation should compare the proposed solutions with traditional approaches. This is to be sure that applying self-organization in the system brings any benefit. Ideally, the designer should develop more than one Control to test in the simulation. A rock-scissors-paper situation could arise, where no Control is best in all situations (also considering classic controls). The designer can then adjust or combine the Controls, and then compare again in the Simulation. Experiments conducted with the aid of the Simulation should go from simple to extensive. Simple experiments will show proof of concepts, and their results can be used to improve the Modeling. Once this is robust, extensive studies should be made to be certain of the performance of the system under different conditions. Based on the Simulation results and insights, the Modeling and Representation can be improved. A Simulation should be mature before taking the implementation into the real world. Application ----------- The role of this stage is basically to use the developed and tested model(s) in a real system. If this is a software system, the transition will not be so difficult. On the other hand, the transition to a real system can expose artifacts of a naive Simulation. A useful way to develop robust Simulations consists in adding some noise into the system [@Jakobi1997]. Good theoretical solutions can be very difficult/expensive/impossible to implement (e.g. if they involve instantaneous access to information, mind reading, teleportation, etc.). The feasibility of the Application should be taken into account during the whole design process. In other words, the designer should have an *implementation bias* in all the Methodology stages. If the proposed system turned out to be too expensive or complicated, all the designer’s efforts would be fruitless. If the system is developed for a client, there should be feedback between developers and clients during the whole process [@Cotton1996] to avoid client dissatisfaction once the system is implemented. The *legacy* of previous systems should also be considered for the design [ValckenaersEtAl2003]{}: if the current implementation is to be modified but not completely replaced, the designer is limited by the capabilities of the old system. Constraints permitting, a pilot study should be made before engaging in a full Application, to detect incongruences and unexpected issues between the Simulation or Modeling stages and the Application. With the results of this pilot study, the Simulation, Modeling, and Representation can be fine-tuned. Evaluation ---------- Once the Application is underway, the performance of new system should be measured and compared with the performances of the previous system(s). Constraints permitting, efforts should be continued to try to improve the system, since the requirements it has to meet will certainly change with time (e.g. changes of demand, capacity, etc.). The system will be more adaptive if it does not consider its solution as the best once and for all, and is able to change itself according to its performance and the changing requirements. Notes on the methodology ------------------------ - All returning arrows in the Figure \[diagram\] are given because it is not possible to predict the outcome of strategies before they have been tried out. All information and eventualities cannot be abstracted, nor emergent properties predicted before they have been observed. Emergent properties are *a posteriori*. - The proposed Methodology will be useful for unpredictable problem domains, where all the possible system’s situations cannot be considered beforehand. It could also be useful for creative tasks, where a self-organizing system can explore the design space in an alternative way. - Most methodologies in the literature apply to software systems, e.g. [@JacobsonEtAl1999; @Jennings2000]. This one is more general, since it is domain independent. The principles presented are such that can be applied to any domain for developing a functioning self-organizing system: Any system can be modeled as a group of agents, with satisfactions depending on their goals. The question is *when* is it useful to use this Methodology. Only application of the Methodology will provide an answer to this question. It should be noted that several approaches have been proposed in parallel, e.g. [@CaperaEtAl2003; @DeWolfHolvoet2005], that, as the present work, and are part of the ongoing effort by the research community to understand self-organizing systems. - The proposed Methodology is not quite a spiral model [@Boehm1988], because the last stage does not need to be reached to return to the first. This is, there is no need to deploy a working version (finish a cycle/iteration) before revisiting a previous stage, as for example in extreme programming. Rather, the Methodology is a *backtracking* model, where the designer can always return to previous stages. - It is not necessary to understand a solution before testing it. In many cases understanding can come only after testing, i.e., the global behavior of the system is irreducible. Certainly, understanding the causes of a phenomenon will allow the modelers to have a greater control over it. A detailed diagram of the different substeps of the Methodology can be appreciated in Figure \[diagram-detailed\]. Case Study: Self-organizing Traffic Lights {#secSOTL} ========================================== Recent work on self-organizing traffic lights [@Gershenson2005] will be used to illustrate the flow through the different steps of the Methodology. These traffic lights are called self-organizing because the global performance is given by the local rules followed by each traffic light: they are unaware of the state of other intersections and still manage to achieve global coordination. Traffic modeling has increased the understanding of this complex phenomenon [PrigogineHerman1971,Traffic95,Traffic97,Traffic99,Helbing1997,HelbingHuberman1998]{}. Even when vehicles can follow simple rules, their local interactions generate global patterns that cannot be reduced to individual behaviors. Controlling traffic lights in a city is not an easy task: it requires the coordination of a multitude of components; the components affect one another; furthermore, these components do not operate at the same pace over time. Traffic flows and densities change constantly. Therefore, this problem is suitable to be tackled by self-organization. A centralized system could also perform the task, but in practice the amount of computation required to process all the data from a city is too great to be able to respond in real time. Thus, a self-organizing system seems to be a promising alternative. **Requirements**. The goal is to develop a feasible and efficient traffic light control system. **Representation**. The traffic light system can be modelled on two levels: the vehicle level and the city level. These are easy to identify because vehicles are objects that move through the city, establishing clear spatiotemporal distinctions. The goal of the vehicles is to flow as fast as possible, so their “satisfaction" $\sigma $ can be measured in terms of their average speed and average waiting time at a red light. Cars will have a maximum $\sigma $ if they go as fast as they are allowed, and do not stop at intersections. $\sigma $ would be zero if a car stops indefinitely. The goal of the traffic light system on the city level is to enable vehicles to flow as fast as possible, while mediating their conflicts for space and time at intersections. This would minimize fuel consumption, noise, pollution, and stress in the population. The satisfaction of the city can be measured in terms of the average speeds and average waiting times of all vehicles (i.e. average of $\sigma _{i},\ \forall i$), and with the average percentage of stationary cars. $\sigma _{sys}$ will be maximum if all cars go as fast as possible, and are able to flow through the city without stopping. If a traffic jam occurs and all the vehicles stop, then $\sigma _{sys}$ would be minimal. **Modeling**. Now the problem for the Control can be formulated: find a mechanism that will coordinate traffic lights so that these will mediate between vehicles to reduce their friction (i.e. try to prevent them from arriving at the same time at crossings). This will maximize the satisfactions of the vehicles and of the city ($\sigma _{i}$’s and $\sigma _{sys}$). Since all vehicles contribute equally to $\sigma _{sys}$, ideally the Control should minimize frictions via Compromise. **Simulation**. A simple simulation was developed in NetLogo [Wilensky1999]{}, a multi-agent modeling environment. The “Gridlock" model [@WilenskyStroup2002] was extended to implement different traffic control strategies. It consists of an abstract traffic grid with intersections between cyclic single-lane arteries of two types: vertical or horizontal (similar to the scenarios of [@BML1992; @BrockfeldEtAl2001]). Cars only flow in a straight line, either eastbound or southbound. Each crossroad has traffic lights that allow traffic flow in only one of the intersecting arteries with a green light. Yellow or red lights stop the traffic. The light sequence for a given artery is green-yellow-red-green. Cars simply try to go at a maximum speed of 1 “patch" per timestep, but stop when a car or a red or yellow light is in front of them. Time is discrete, but not space. A “patch" is a square of the environment the size of a car. The simulation can be tested at the URL http://homepages.vub.ac.be/cgershen/sos/SOTL/SOTL.html . At first, a tentative model was implemented. The idea was unsuccessful. However, after refining the model, an efficient method was discovered, named *sotl-request*. **Modeling**. In the *sotl-request* method, each traffic light keeps a count $\kappa _{i}$ of the number of cars times time steps ($c\ast ts $) approaching *only* the red light, independently of the status or speed of the cars (i.e. moving or stopped). $\kappa _{i}$ can be seen as the integral of waiting/approaching cars over time. When $\kappa _{i}$ reaches a threshold $\theta $, the opposing green light turns yellow, and the following time step it turns red with $\kappa _{i}=0$ , while the red light which counted turns green. In this way, if there are more cars approaching or waiting before a red light, the light will turn green faster than if there are only few cars. This simple mechanism achieves self-organization as follows: if there is a single or just a few cars, these will be made to stop for a longer period before a red light. This gives time for other cars to join them. As more cars join the group, cars will be made to wait shorter periods before a red light. Once there are enough cars, the red light will turn green even before the oncoming cars reach the intersection, thereby generating “green waves". Having “platoons" or “convoys" of cars moving together improves traffic flow, compared to a homogeneous distribution of cars, since there are large empty areas between platoons, which can be used by crossing platoons with few interferences. The *sotl-request* method has no phase or internal clock. Traffic lights change only when the above conditions are met. If no cars are approaching a red light, the complementary light can remain green. **Representation**. It becomes clear now that it would be useful to consider traffic lights as agents as well. Their goal is to “get rid" of cars as quickly as possible. To do so, they should avoid having green lights on empty streets and red lights on streets with high traffic density. Since the satisfactions of the traffic lights and vehicles are complementary, they should interact via Cooperation to achieve synergy. Also, $\sigma _{sys}$ could be formulated in terms of the satisfactions of traffic lights, vehicles, or both. **Modeling**. Two classic methods were implemented to compare their performance with *sotl-request*: *marching* and *optim*. *Marching* is a very simple method. All traffic lights “march in step": all green lights are either southbound or eastbound, synchronized in time. Intersections have a phase $\varphi _{i}$, which counts time steps. $\varphi _{i}$ is reset to zero when the phase reaches a period value $p$. When $% \varphi _{i}==0$, red lights turn green, and yellow lights turn red. Green lights turn yellow one time step earlier, i.e. when $\varphi ==p-1$. A full cycle of an intersection consists of $2p$ time steps. “Marching" intersections are such that $\varphi _{i}==\varphi _{j},\forall i,j$. The *optim* method is implemented trying to set phases $\varphi _{i}$ of traffic lights so that, as soon as a red light turns green, a car that was made to stop would find the following traffic lights green. In other words, a fixed solution is obtained so that *green waves* flow to the southeast. The simulation environment has a radius of $r$ square patches, so that these can be identified with coordinates $(x_{i},y_{i}),$ $% x_{i},y_{i}\in \lbrack -r,r]$. Therefore, each artery consists of $2r+1$ patches. In order to synchronize all the intersections, red lights should turn green and yellow lights should turn red when $$\varphi _{i}==round(\frac{2r+x_{i}-y_{i}}{4})$$ and green lights should turn to yellow the previous time step. The period should be $p=r+3$. The three is added as an extra margin for the reaction and acceleration times of cars (found to be best, for low densities, by trial and error). These two methods are *non-adaptive*, in the sense that their behavior is dictated beforehand, and they do not consider the actual state of the traffic. Therefore, there cannot be Cooperation between vehicles and traffic lights, since the latter ones have fixed behaviors. On the other hand, traffic lights under the *sotl-request* method are sensitive to the current traffic condition, and can therefore respond to the needs of the incoming vehicles. **Simulation**. Preliminary experiments have shown that *sotl-request*, compared with the two traditional methods, achieves very good results for low traffic densities, but very poor results for high traffic densities. This is because depending on the value of $\theta $, high traffic densities can cause the traffic lights to change too fast. This obstructs traffic flow. A new model was developed, taking this factor into account. **Modeling**. The *sotl-phase* method takes *sotl-request* and only adds the following constraint: a traffic light will not be changed if the time passed since the last light change is less than a minimum phase, i.e. $\varphi _{i}<\varphi _{\min }$. Once $\varphi _{i}\geq \varphi _{\min } $, the lights will change if/when $\kappa _{i}$ $\geq $ $\theta $. This prevents the fast changing of lights[^7]. **Simulation**. *Sotl-phase* performed a bit less effectively than *sotl-request* for very low traffic densities, but still much better than the classic methods. However, *sotl-phase* outperformed them also for high densities. An unexpected phenomenon was observed: for certain traffic densities, *sotl-phase* achieved *full synchronization*, in the sense that no car stopped. Therefore, speeds were maximal and there were no waiting times nor sopped cars. Thus, satisfaction was maximal for vehicles, traffic lights, and the city. Still, this is not a realistic situation, because full synchronization is achieved due to the toroidal topology of the simulation environment. The full synchronization is achieved because platoons are promoted by the traffic lights, and platoons can move faster through the city modulating traffic lights. If two platoons are approaching an intersection, *sotl-phase* will stop one of them, and allow the other to pass without stopping. The latter platoon keeps its phase as it goes around the torus, and the former adjusts its speed until it finds a phase that does not cause a conflict with another platoon. **Modeling**. Understanding the behavior of the platoons, it can be seen that there is a favorable system/context trade-off. There is no need of direct communication between traffic lights, since information can actually be sent via platoons of vehicles. The traffic lights communicate *stigmergically* [@TheraulazBonabeau1999], i.e. via their environment, where the vehicles are conceptualized as the environment of traffic lights. **Simulation**. With encouraging results, changes were made to the Simulation to make it more realistic. Thus, a scenario similar to the one of [@FaietaHuberman1993] was developed. Traffic flow in four directions was introduced, alternating streets. This is, arteries still consist of one lane, but the directions alternate: southbound-northbound in vertical roads, and eastbound-westbound in horizontal roads. Also, the possibility of having more cars flowing in particular directions was introduced. Peak hour traffic can be simulated like this, regulating the percentages of cars that will flow in different roads. An option to switch off the torus in the simulation was added. Finally, a probability of turning at an intersection $ P_{turn}$ was included. Therefore, when a car reaches an intersection, it will have a probability $P_{turn}$ of reducing its speed and turning in the direction of the crossing street. This can cause cars to leave platoons, which are more stable when $P_{turn}=0$. The results of experiments in the more realistic Simulation confirmed the previous ones: self-organizing methods outperform classic ones. There can still be full synchronization with alternating streets, but not without a torus or with $P_{turn}>0$. **Modeling**. Another method was developed, *sotl-platoon*, adding two restrictions to *sotl-phase* for regulating the size of platoons. Before changing a red light to green, *sotl-platoon* checks if a platoon is not crossing through, not to break it. More precisely, a red light is not changed to green if on the crossing street there is at least one car approaching at $\omega $ patches from the intersection. This keeps platoons together. For high densities, this restriction alone would cause havoc, since large platoons would block the traffic flow of intersecting streets. To avoid this, a second restriction is introduced. The first restriction is not taken into account if there are more than $\mu $ cars approaching the intersection. Like this, long platoons can be broken, and the restriction only comes into place if a platoon will soon be through an intersection. **Simulation**. *Sotl-platoon* manages to keep platoons together, achieving full synchronization commonly for a wide density range, more effectively than *sotl-phase* (when the torus is active). This is because the restrictions of *sotl-platoon* prevent the breaking of platoons when these would leave few cars behind, with a small time cost for waiting vehicles. Still, this cost is much lower than breaking a platoon and waiting for separated vehicles to join back again so that they can switch red lights to green before reaching an intersection. However, for high traffic densities platoons aggregate too much, making traffic jams more probable. The *sotl-platoon* method fails when a platoon waiting to cross a street is long enough to reach the previous intersection, but not long enough to cut its tail. This will prevent waiting cars from advancing, until more cars join the long platoon. This failure could probably be avoided introducing further restrictions. In more realistic experiments (four directions, no torus, $P_{turn}=0.1$), *sotl-platoon* gives on average 30% (up to 40%) more average speed, half the stopped cars, and seven times less average waiting times than non-responsive methods. Complete results and graphics of the experiments discussed here can be found in [@Gershenson2005]. **Representation**. If priority is to be given to certain vehicles (e.g. public transport, emergency), weights can be added to give more importance to some $\sigma _{i}$’s. A meso-level might be considered, where properties of platoons can be observed: their behaviors, performance, and satisfaction and the relationships of these with the vehicle and city levels could enhance the understanding of the self-organizing traffic lights and even improve them. **Simulation**. Streets of varying distances between crossings were tested, and all the self-organizing methods maintained their good performance. Still more realistic simulations should be made before moving to the Implementation, because of the cost of such a system. At least, multiple-street intersections, multiple-lane streets, lane changing, different driving behaviors, and non homogeneous streets should be considered. **Application**. The proposed system has not been implemented yet. Still, it is feasible to do so, since there is the sensor technology to implement the discussed methods in an affordable way. Currently, a more realistic simulation is being developed in cooperation with the Brussels Ministry of Mobility and Public Works to study its potential application in the city of Brussels. A pilot study should be made before applying it widely, to fine tune different parameters and methods. External factors, e.g. pedestrians and cyclists, could also affect the performance of the system. Pedestrians could be taken into account considering them as cars approaching a red light. For example, a button could be used to inform the intersection of their presence, and this would contribute to the count $\kappa _{i}$. A mixed strategy between different methods could be considered, e.g. *sotl-platoon* for low and medium densities, and *sotl-phase* or *marching* for high densities. **Evaluation**. If a city deploys a self-organizing traffic light system, it should be monitored and compared with previous systems. This will help to improve the system. If the system would be an affordable success, its implementation in other cities would be promoted. Discussion {#secDiscussion} ========== As could be seen in the case study, the backtracking between different steps in the Methodology is necessary because the behavior of the system cannot be predicted from the Modeling, i.e. it is not reducible. It might be possible to reason about all possible outcomes of simple systems, and then to implement the solution. But when complexity needs to be dealt with, a mutual feedback between experience and reasoning needs to be established, since reasoning alone cannot process all the information required to predict the behavior of a complex system [@Edmonds2005]. For this same reason, it would be preferable for the Control to be distributed. Even when a central supercomputer could possibly solve a problem in real time, the information delay caused by data transmission and integration can reduce the efficiency of the system. Also, a distributed Control will be more robust, in as much as if a module malfunctions, the rest of the system can still provide reliable solutions. If a central Control fails, the whole system will stop working. The Simulation and Experiments are strictly necessary in the design of self-organizing systems [@Edmonds2005]. This is because their performance cannot be evaluated by purely formal methods [@EdmondsBryson2004]. Still, formal methods are necessary in the first stages of the Methodology. I am not suggesting a trial-and-error search. But since the behavior of a complex system in a complex environment cannot be predicted completely, the models need to be contrasted with experimentation before they can be validated. This Methodology suggests one possible path for finding solutions. Now the reader might wonder whether the proposed Methodology is a *top-down* or a *bottom-up* approach. And the answer is: it is both and neither, since (at least) higher and lower levels of abstraction need to be considered simultaneously. The approach tests different local behaviors, and observes local and global (and meso) performances, for local and global (and meso) requirements. Thus, the Methodology can be seen as a *multi-level* approach. Since “conflicts" between agents need to be solved at more than one level, the Control strategies should be carefully chosen and tested. A situation as in the prisoner’s dilemma [@Axelrod1984] might easily arise, when the “best" solution on one level/timescale is not the best solution on another level/timescale. Many frictions between agents are due to faulty communication, especially in social and political relations. If agents do not know the goals of others, it will be much more difficult to coordinate and increase $\sigma _{sys}$. For example, in a social system, knowing what people or corporations need to fulfill their goals is not so obvious. Still, with emerging technologies, social systems perform better in this respect. Already in the early 1970s, the project Cybersin in Chile followed this path [@MillerMedina2005]: it kept a daily log of productions and requirements from all over the country (e.g. mines, factories, etc.), in order to distribute products where they were needed most; and as quickly as possible. Another step towards providing faster response to the needs of both individuals and social systems can be found in e-government [@LayneLee2001]. A company should also follow these principles to be able to adapt as quickly as possible. It needs to develop “sensors” to perceive the satisfactions and conflicts of agents at different levels of abstraction, and needs to develop fast ways of adapting to emerging conflicts, as well as to changing economic environment. A tempting solution might be to develop a homogeneous system since, e.g., homogeneous societies have fewer conflicts [@Durkheim1893]. This is because all the elements of a homogeneous system pursue the same goals. Thus, less diversity is easier to control. However, less diversity will be less able to adapt to sudden changes. Nevertheless, societies cannot be *made* homogeneous without generating conflicts since people are already diverse, and therefore already have a diversity of goals. The legacy [@ValckenaersEtAl2003] of social systems gives less freedom to a designer, since some goals are already within the system. A social Control/mediator needs to satisfy these while trying to satisfy those of the social system. Conclusions {#secConclusions} =========== This paper suggests a conceptual framework and a general methodology for designing and controlling self-organizing systems. The Methodology proposes the exploration for proper Control mechanisms/mediators/constraints that will reduce frictions and promote synergy so that elements will dynamically reach a robust and efficient solution. The proposed Methodology is general, but certainly it is not the only way to *describe* self-organizing systems. Even if this paper is aimed mainly at engineers, it is rather philosophical. It presents no concrete results, but *ideas* that can be exploited to produce them. Certainly, these ideas have their roots in current practices, and many of them are not novel. Still, the aim of this work is not for novelty but for synthesis. The Methodology strives to build artificial systems. Still, these could be used to understand natural systems using the synthetic method [@Steels1993]. Therefore, the ideas presented here are potentially useful not only for engineering, but also for science. The backtracking ideology is also applicable to this Methodology: it will be improved once applied, through learning from experience. This Methodology is not final, but evolving. The more this Methodology is used, and in a wider variety of areas, the more potentially useful its abstractions will be. For example, would it be a good strategy to minimize the standard deviation of $\sigma $’s? This might possibly increase stability and reduce the probability of conflict, but this strategy, as any other, needs to be tested before it can be properly understood. It is worth noting that apart from self-organizing traffic lights [@Gershenson2005], the Methodology is currently being used to develop Ambient Intelligence protocols [@GershensonHeylighen2004] and to study self-organizing bureaucracies. Any system is liable to make mistakes (and *will* make them in an unpredictable environment). But a good system will *learn* from its mistakes. This is the basis for adaptation. It is pointless to attempt to build a “perfect" system, since it is not possible to predict future interactions with its environment. What should be done is to build systems that can adapt to their unexpected future and are robust enough not to be destroyed in the attempt. Self-organization provides one way to achieve this, but there is still much to be done to harness its full potential. I should like to thank Hugues Bersini, Marco Dorigo, Erden Göktepe, Dirk Helbing, Francis Heylighen, Diana Mangalagiu, Peter McBurney, Juan Julián Merelo, Marko Rodriguez, Frank Schweitzer, Sorin Solomon, and Franco Zambonelli for interesting discussions and comments. I also wish to thank Michael Whitburn for proof-reading an earlier version of the manuscript. This research was partly supported by the Consejo Nacional de Ciencia y Teconolgía (CONACyT) of México. [^1]: This can be confirmed mathematically in certain systems. As a general example, random Boolean networks [Kauffman1969,Kauffman1993,Gershenson2004c]{} show clearly that the complexity of the network increases with the number of elements and the number of interactions. [^2]: Certainly, the number of possible interactions for certain elements is impossible to enumerate or measure. [^3]: In some cases, $\sigma $ could be seen as a “fitness" [HeylighenCampbell1995]{}. However, in most genetic algorithms [Mitchell1996]{} a fitness function is imposed from the outside, whereas $% \sigma $ is a property of the agents, that can change with time. [^4]: This method has been used widely to detect functions in complex systems such as genetic regulatory networks and nervous systems. [^5]: Even when equation \[eqFriction\] relates the satisfaction of an element to the satisfaction of the system, this can also be used for the relation between satisfactions of two elements, when $\Delta \sigma _{i}=0$ for all other elements. [^6]: Trust is also important for communication and cooperation. [^7]: A similar method has been used successfully in the United Kingdom for some time, but for isolated intersections [@VincentYoung1986].

--- abstract: 'In this preceedings paper we report on a calculation of graphene’s Landau levels in a magnetic field. Our calculations are based on a self-consistent Hartree-Fock approximation for graphene’s massless-Dirac continuum model. We find that because of graphene’s chiral band structure interactions not only shift Landau-level energies, as in a non-relativistic electron gas, but also alter Landau level wavefunctions. We comment on the subtle continuum model regularization procedure necessary to correctly maintain the lattice-model’s particle hole symmetry properties.' address: - | Physics Department, University of Texas at Austin, 1 University Station\ Austin, Texas, 78712, USA\ National High Magnetic Field Laboratory, Florida State University, 1800 E Paul Dirac Dr\ Tallahassee, Florida, 32310, USA barlas@magnet.fsu.edu - | Physics Department, University of California at San Diego, 9500 Gilman Drive\ La Jolla, California, 92093, USA leewc@physics.ucsd.edu - | Department of Physics, Tohoku University\ Sendai, 980-8578, Japan nomura@cmpt.phys.tohoku.ac.jp - | Physics Department, University of Texas at Austin, 1 University Station\ Austin, Texas, 78712, USA\ macd@physics.utexas.edu author: - YAFIS BARLAS - 'WEI-CHENG LEE' - KENTATO NOMURA - ALLAN H MACDONALD title: 'RENORMALIZED LANDAU LEVELS AND PARTICLE-HOLE SYMMETRY IN GRAPHENE' --- Introduction ============ Graphene, a one-atom-thick two-dimensional crystal of carbon atoms arranged in a honeycomb lattice, is a gapless semiconductor with an unusual massless Dirac-fermion band structure that has long attracted theoretical interest [@semenoff; @haldane]. The low-energy properties of graphene are characterized by quasiparticle dispersion [@earlyguy] linear in momentum and by vanishing density-of-states at the neutral system Fermi energy. The band eigenstates can be considiered as sublattice-pseudospin spinors and have a chiral property which qualitatively alters the way in which electron-electron interactions influence electronic properties. In particular electron-electron interactions lead to a logarithimic enhancement of the Fermi velocity in doped and undoped graphene related to a lack of screening at the Dirac point [@gin; @polinissc; @vafek].\ In the presence of a magnetic field, graphene’s electronic structure also changes in a nontrivial way when compared to the non-relativistic two dimensional electron gas (2DEG) case, leading to the so-called half-quantized Hall effect [@qhtheory; @qhexpt] in which the plateau values of the Hall conductivity are given by $\sigma_{xy} = 4(n+1/2)(e^{2}/h)$. Plateau conductivity values are separated by $4e^{2}/h$ because of the fourfold degeneracy due to valley and spin. In this paper we analyze the effect of electron-electron interactions on graphene’s LL spectrum. We show that because of the chiral nature of graphene’s band structure, interactions not-only shift Landau level energies but also alter Landau-level wavefunctions. Self Consistent Hartree-Fock Approximation ========================================== In this section we use the self-consistent Hartree-Fock approximation (SCHFA) to study the effect of electron-electron interactions on the LL spectium of graphene within the massless Dirac-fermion(MDF) model. The low-energy properties of graphene can be adequately described by a MDF model: $$\label{MDF} \mathcal{H}_{\vec{p}} = v \vec{\sigma} \cdot p$$ where $\sigma^{i}$’s are Pauli matrices acting on graphene’s psuedospin degrees of freedom, $\vec{p}$ is a two dimensional vector measured from the Dirac points. (As we will discuss later this model requires especially subtle ultraviolet regularizations in order to yield physically correct predictions.) In the presence of a unifrom magnetic field $\vec{B} = B \hat{z}$ applied in a direction perpendicular to the plane of the graphene sheet \[MDF\] is modified by $\vec{p} \to \vec{\pi} = \vec{p} -(e/c)\vec{A}$ where $\vec{A}$ is the vector potential with $\vec{B} =\vec{\nabla} \times \vec{A}$. Defining the usual raising and lowering LL operators $\vec{a^{\dagger}}$ and $a$, with $a^{\dagger} = (l_{B}/\sqrt{2}\hbar) \pi$, where $l_{B} = (\hbar c/eB)^{1/2} = 25.66/\sqrt{(B}[\rm{T}])\rm{nm}$ is the magnetic length, we can identify a zero-energy eigenstate given by $a \phi_{0} = 0$ and finite-energy chiral eigenatates $n$ and $\bar{n} = -n$ with eigenenergies $\varepsilon_{n} = sgn(n) \sqrt{\frac{2 \hbar v^{2}e B |n|}{c}}$. In the Landau gauge $ \vec{A} = (0,Bx,0 )$, the corresponding eigenvectors are $$|\psi_{n \neq 0 , X} \rangle = \frac{1}{\sqrt{2}}\left( \begin{array}{c} -i sgn(n) \phi_{|n|-1,X}\\ \phi_{|n|,X} \end{array} \right), \qquad |\psi_{n = 0 , X} \rangle = \left( \begin{array}{c} 0 \\ \phi_{0,X}\end{array} \right),$$ where $\phi_{n,X}$ is a Landau-gauge eigenstate of a non-relatvistic electron gas and $X$ denotes the guiding center degree of freedom within a LL. Projecting the interacting Hamiltonian onto the LL basis gives: $$\mathcal{H}_{e-e} = \frac{1}{2} \sum_{\vec{q}} v_{q} : \hat{\rho}(-\vec{q}) \hat{\rho}(\vec{q}):,$$ where $v_{q} = 2 \pi e^{2}/ \epsilon |q|$ is the two-dimensional Fourier transform of the Coulomb interaction and $$\hat{\rho}(-\vec{q}) = \sum_{n X n' X'} c^{\dagger}_{n X \tau s} c_{n' X' \tau' s'} \delta (q_{y} + X - X') e^{i q_{x}(X'-\frac{q_{y}}{2})}. F^{R}_{n,n'}(\vec{q}), \label{density}$$ In Eq.( \[density\]) $c^{\dagger}_{n X \tau s} /c_{n X \tau s}$ are creation/annihilation operator for particles in LL $n$ at guiding center $X$ for valley $\tau $ and spin $s$. For notational simplicity we have assumed $l_{B} =1$; we will however restore these length units in the final results. $F^{R}_{n,n'}(\vec{q})$ is referred to as graphene’s relativistic form factor and captures the orbital and sublattice pseudospin character of the LL orbitals [@kentaroQHF] ($n \neq 0 $ and $n' \neq 0$): $$F^{R}_{n,n'}(\vec{q}) = \frac{1}{2} \big[ F_{|n|,|n'|}(\vec{q}) + sgn(nn') F_{|n|-1,|n'|-1}(\vec{q}) \big],$$ $ F_{|n|,|n'|}(\vec{q})$ is the well known form factor for an ordinary 2DEG [@allanearlypapers] in the presence of a perpendicular magnetic field: $$F_{n,n'}(\vec{q}) = \bigg\{ \begin{array}{cc} \sqrt{\frac{n'!}{n!}} \big[i\frac{q_{x} -i q_{y}}{\sqrt{2}} \big]^{n-n'} L_{n'}^{n-n'}(\frac{q^{2}}{2}) e^{-q^{2}/4} & n \geq n'\\ \sqrt{\frac{n!}{n'!}} \big[i\frac{q_{x} + i q_{y}}{\sqrt{2}} \big]^{n'-n} L_{n}^{n'-n}(\frac{q^{2}}{2}) e^{-q^{2}/4} & n' > n, \end{array}$$ and the form factor for the lowest LL is just $F_{00}^{R}(q) = F(q)= e^{-q^{2}/4}$. Finally $$F^{R}_{0,n}(\vec{q}) = \frac{1}{\sqrt{2}}F_{0,n}(\vec{q}) = \sqrt{\frac{1}{2n!}} \big[i\frac{q_{x} -i q_{y}}{\sqrt{2}} \big]^{n} L^{n}(\frac{q^{2}}{2}) e^{-q^{2}/4}$$ Here $ L_{n'}^{n-n'} $ are the associated Laguerre polynomials. In the Hartree-Fock approximation the effective single-particle Hamiltonian depends on the density matrix. In the case of Landau-level systems, the density-matrix is usefully parameterized as follows: $$\Delta^{n,n'}_{\tau, \tau',s,s'} (\vec{q})= \frac{1}{N_{\phi}} \sum_{X,X'} \langle c^{\dagger}_{n',X', \tau', s'}c_{n,X, \tau, s} \rangle \delta(q_{y}+X'-X). \exp^{- iq_{x} X}$$ The Hartree-Fock theory Hamiltonian is expressed in terms of $\Delta$ as follows: [^1] $$\begin{aligned} \label{finalHFham} \langle n,X,\tau,s | \mathcal{H}_{e-e} |n',X',\tau',s' \rangle &=& \sum_{n_{1},n_{2}} \sum_{q} \bigg[ \frac{1}{2 \pi l_{B}^{2}} V_{n,n',n_{2},n_{1}} (\vec{q}) \Delta^{n_{2},n_{1}}_{\tau", \tau",s",s"}(\vec{q}) \delta^{ss'}_{\tau \tau'} \\ \nonumber &-& \frac{1}{L^{2}} \sum_{p} V_{n_{2},n',n,n_{1}} (\vec{p}) \Delta^{n_{2},n_{1}}_{\tau, \tau',s,s'}(\vec{q}) \exp^{i (\vec{p} \times \vec{q})l_{B}^{2} \cdot \hat{z}} \bigg] \exp^{i q_{x} X'} \delta(q_{y}l_{B}^{2} + X - X')\end{aligned}$$ here we restore $l_{B} $ and define $$V_{n_{1},n_{2},n_{3},n_{4}}(\vec{q}) = v_{q} F^{R}_{n_{1},n_{4}}(\vec{q})F^{R}_{n_{2},n_{3}}(-\vec{q})$$ For the purposes of this paper we assume that the translational invariance is not broken. In this case $\Delta$ is non-zero only for $\vec{q}=0$. It follows that $$\Delta^{n_{1},n_{2}}_{\tau \tau' s s'}(\vec{q}) =\Delta^{n_{1},n_{2}}_{\tau \tau' s s'}(0) \delta_{\vec{q}=0} \delta(|n_{1}| - |n_{2}|).$$ The $\delta$ function which sets mixing between states with different values of $|n|$ can be seen to follow from spatial isotropy in the continuum model. Because both positive and negative values of $n$ occur in graphene this restriction does not forbid mixing of states with positive and negative $n$ by interactions. The fact that Landau-level wavefunctions are altered by interactions in graphene is the main difference between relativistic and non-relativistic cases. Assuming no broken translational symmetry, the first term in (\[finalHFham\]) is just the constant Hartree (electrostatic) potential which can be absorbed in the zero of energy. Dropping this term yields $$\langle n,X,\tau,s | \mathcal{H} |n',X',\tau',s' \rangle = - \sum_{|n_{1}| = |n_{2}|} \frac{1}{L^{2}} \sum_{\vec{q}} v_{q} F^{R}_{n_{2},n'}(\vec{q})F^{R}_{n,n_{1}}(-\vec{q}) \Delta^{n_{1},n_{2}}_{\tau \tau' s s'}(0) \delta(X-X')$$ The mixing between $n$ and $\bar{n}$ LL is due to the fact that the self-energy is not diagonal in the pseudospin as can be seen by examining spatial isotropy consequences more closely. Isotropy is encoded in the form factors in the above equation: $$F^{R}_{m,n'}(\vec{q})F^{R}_{n,m}(-\vec{q}) = F^{R}_{m,n'}(\vec{q}) \bar{F}^{R}_{m,n}(\vec{q}) \sim \exp^{-i \varphi (|n'|-|n|)}$$ yielding an angular integral that is proportinal to $\delta(|n'| - |n|) $ not $\delta (n'-n)$. The other delta function gives $X=X'$. Assuming no spin or valley broken symmetries we can also assume that different valleys can be considered independently. This yields (suppressing the spin and valley indices): $$\label{finaleeham} \langle n |\mathcal{H}_{e-e}| n' \rangle = -\sum_{|n_{1}| = |n_{2}|} \frac{1}{L^{2}} \sum_{\vec{q}} v_{q} F^{R}_{n_{2},n'}(\vec{q})F^{R}_{n,n_{1}}(-\vec{q}) \Delta^{n_{1},n_{2}}.$$ Notice that this matrix element is non-zero only if $|n|=|n'|$. To compute the matrix elements we have to employ a LL index cutoff reminisent of the high-energy cutoff used in for MDF description of graphene at zero magnetic field. This ultraviolet cutoff plays a role because of the unbounded negative energy sea of the massless Dirac model. Following the procedure used at zero magnetic field, we choose a LL cutoff, a maximum value for $|n|$, M based on the physically natural cutoff of momentum at an inverse lattice constant scale, $k_c \sim 1/a$, and on the semi-classical relationship between momentum and Landau-level index. This yields $$\hbar v_{F} k_{c} = \sqrt{2} \frac{\hbar v_{F}}{l_{B}} \sqrt{M}.$$ Using $k_{c} \sim 1/a$ where $a= 0.246nm$ is graphene honeycomb lattice constant we get a magnetic field dependent cutoff $$M \sim \frac{5000}{B[T]}$$ where $B$ is the magnitude of the magnetic field. We can write the mean field Hartree-Fock hamiltoninan in the $(n,\bar{n})$-sector as a two level system: $$\mathcal{H}_{MF} = \mathcal{E} + \vec{\mathcal{B}} \cdot \vec{\sigma} \label{hmf}$$ where the pseudospin electric $\mathcal{E}$ and magnetic $\mathcal{B}$ field in the $(n,\bar{n})$-space $\mathcal{E} = - \alpha_{gr}(\sqrt{\pi}/2) (V_{n} + \bar{V}_{n})/2 $, $\mathcal{B}_{x} = - \alpha_{gr}(\sqrt{\pi}/2) \tilde{V}_{n}$ and $\mathcal{B}_{z} = \sqrt{n} - \alpha_{gr}(\sqrt{\pi}/2) ((V_{n} - \bar{V}_{n})/2)$ ,depend on the interaction matrix elements $$V_{n} = \frac{l_{\rm B}}{e^{2}} \sqrt{\frac{2}{\pi}} \langle n| H_{e-e} | n \rangle, \qquad \bar{V}_{n} = \frac{l_{\rm B}}{e^{2}} \sqrt{\frac{2}{\pi}} \langle \bar{n}| H_{e-e} | \bar{n} \rangle, \qquad \tilde{V}_{n} = \frac{l_{\rm B}}{e^{2}} \sqrt{\frac{2}{\pi}} \langle n| H_{e-e} | \bar{n} \rangle, \label{twobytwo}$$ $\alpha_{gr} = e^{2}/(\hbar v_{F})$ is graphene’s coupling constant. Results and Discussions ======================= \[figure1\] We must now address the regularization procedures that are necessary to extract useful predictive results from these Hartree-Fock calculations. The exchange energies $V_{n}$ and $\bar{V}_{n}$ in Eq.(\[twobytwo\]) both diverge with cutoff $M$ like $\sqrt{M}$ while $\mathcal{B}_{z}$ diverges like $\ln{M}$ and $\mathcal{B}_{x}$ goes to zero like $1/\sqrt{M}$. These large exchange energies are indeed physical because graphene’s $\pi$-electron system has a high density of electrons, close to one per honeycomb lattice unit cell. This large energy is however neither easily measureable or of any great interest. Instead we want to use our Hartree-Fock theory to calculate the spacing of energy levels near the Dirac point and their dependences on the total Landau level filling factor. Progress can be made by simply choosing a convenient zero of energy, as we do in the zero-field case. We propose using the energy of the $n=0$ Landau level, $\langle 0 | H_{e-e} | 0\rangle$ [*evaluated for a neutral graphene sheet*]{} as the zero of energy. In zero field the analogous choice solves all problems, but that is not true in the massless Dirac model case as we now discuss. In order to explain the problem which remains and our resolution of this problem we go to a more microscopic level by considering properties not of the massless Dirac model but of the one-band nearest neighbour tight-binding model for graphene’s $\pi$-orbitals. This model has the particle-hole symmetry property that when the Hamiltonian acts on a wavefunction which is restricted to one sublattice of the bipartite honeycomb lattice, it produces a wavefunction confined to the other sublattice. From this property it readily follows that eigenstates of the band Hamiltonian occur in positive and negative energy pairs which have opposite intersublattice phases and, importantly for the Hatree-Fock calculations, that the density-matrix of a neutral graphene sheet is just half of the trivial density matrix of a state in which all $\pi$-orbital states are full, $$\Delta_{i',i}^{neutral} = \frac{1}{2} \; \delta_{i',i'} .$$ This property is preserved in a magnetic field and implies that the role of generalized Hubbard model interactions at the neutrality point in Hartree-Fock theory is simply to shift all energy levels by an irrelevant constant. This property is independent of the dependence of the interaction on site-separation. When translated to the continuum model, this property implies that for the case of a neutral graphene sheet both $\mathcal{E}$ in Eq.( \[hmf\]) should be independent of $n$ and equal to $\langle 0 | H_{e-e} | 0\rangle$ and that $\mathcal{B}_{x}$ should vanish. Although the regularization procedure discussed above recovers this result with errors that vanish with cutoff like $1/\sqrt{M}$, the particle-hole symmetry property is so essential to the observed properties of graphene sheets that these errors are uncomfortably large at practical values of $M$. We therefore propose the following regularization procedure for Hartree-Fock Landau level calculations for the massless Dirac model of graphene: i) Solve the Hartree-Fock equations for neutral graphene by setting $\Delta_{0,0}=1/2$, and $\Delta_{n,n}=(1-sgn(n))/2$ for $n \ne 0$. ii) For the neutral case choose $\langle 0 | H_{e-e} | 0\rangle$ as the zero of energy and set $\mathcal{E}$ and $\mathcal{B}_{x}$ to zero to compensate for the violation of particle-hole symmetry caused by a finite $M$ cut-off iii) In the case of charged graphene sheets $\Delta_{n',n}$ must be determined by a self consistent calculation in which $\mathcal{E}$ and $\mathcal{B}_{x}$ are replaced by the the difference between their neutral and charged system values. The Hartree-Fock energy levels of a neutral graphene sheet obtained by following i) are illustrated in Fig.( \[figure1\]). The logarithmic dependence of all levels on cut-off $M$ in this figure is expected to appear experimentally as a weak logarithmic correction to the $\sqrt{B}$ dependence of Landau level energies on field expected for non-interacting electrons. This work was supported by the Welch Foundation and by the National Science Foundation under grant DMR-0606489. [0]{} G. W. Semenoff [*Phys. Rev. Lett.*]{} [**53**]{}, 2449, (1984). F. D. M. Haldane [*Phys. Rev. Lett.*]{} [**61**]{}, 2015, (1988). P. R. Wallace [*Phys. Rev. B*]{} [**71**]{}, 622, (1947). J. Gonzales, F. Guinea and M. A. H. Vozmediano [*Phys. Rev. B*]{} [**59**]{}, 2474, (1999). M. Polini [*et. al.*]{} [*Solid State Comm.*]{} [**143**]{}, 58, (2007). O. Vafek [*Phys. Rev. Lett.*]{} [**99**]{}, 047002, (2007). K. S. Novoselov [*et. al.*]{} [*Nature*]{} [**438**]{}, 197, (2005); Y. B. Zhang [ *et. al.*]{} [*Nature*]{} [**438**]{}, 201, (2005) V. P. Gusynin and S. G. Sharapov [*Phys. Rev. Lett.*]{} [**95**]{}, 146801, (2005) K. Nomura and A. H. MacDonald [*Phys. Rev. Lett.*]{} [**96**]{}, 256602, (2006). A. H MacDonald [*J. Phys. C: Solid State Phys.*]{} [**18**]{}, 1003, (1985). [^1]: The details of this calculation are similar to the one in Ref. 10.

--- author: - '[**A. Soleiman$^{1}$ and Nabil L. Youssef$^{\,2}$**]{}' title: '[**Some Types of Recurrence in Finsler geometry**]{}' --- [$^{1}$Department of Mathematics, Faculty of Science,\ Benha University, Benha, Egypt\ amr.hassan@fsci.bu.edu.eg, amrsoleiman@yahoo.com]{} [$^{2}$Department of Mathematics, Faculty of Science,\ Cairo University, Giza, Egypt\ nlyoussef@sci.cu.edu.eg, nlyoussef2003@yahoo.fr]{} *Dedicated to the memory of Waleed A. Elsayed* [**Abstract.**]{} The pullback approach to global Finsler geometry is adopted. Three classes of recurrence in Finsler geometry are introduced and investigated: simple recurrence, Ricci recurrence and concircular recurrence. Each of these classes consists of four types of recurrence. The interrelationships between the different types of recurrence are studied. The generalized concircular recurrence, as a new concept, is singled out. [**Keywords:**]{} recurrent; generalized recurrent; Ricci recurrent; generalized Ricci recurrent; concircularly recurrent; generalized concircularly recurrent. [**MSC 2010**]{}: 53C60, 53B40, 58B20. . Many types of recurrent Riemannian manifolds have been studied by many authors (e.g., [@R3; @R4; @R2; @R5; @R1]). On the other hand, some types of recurrent Finsler spaces have been also studied (e.g., [@F3; @F2; @F1]). In this paper, we gather all known types of Finsler recurrence (related to Cartan connection), besides some new ones, in a single general setting. We study *intrinsically* three classes of recurrence: simple recurrence, Ricci recurrence and concircular recurrence. Each of these classes consists of four types of recurrence. The interrelationships between the different types of recurrence are investigated. A special emphasis is focused on the new concept of generalized concircular recurrence. At the end of the paper we provide a concise diagram presenting the relationships among the different types of Finsler recurrences treated. In this section, we give a brief account of the basic concepts of the pullback approach to intrinsic Finsler geometry necessary for this work. For more details, we refer to [@r58; @r86; @r94; @r96]. We shall use the notations of [@r86]. In what follows, we denote by $\pi: {{\cal T}}M\longrightarrow M$ the subbundle of nonzero vectors tangent to $M$, $\mathfrak{F}(TM)$ the algebra of $C^\infty$ functions on $TM$, ${\mathfrak{X}(\pi (M))}$ the $\mathfrak{F}(TM)$-module of differentiable sections of the pullback bundle $\pi^{-1}(T M)$. The elements of $\mathfrak{X}(\pi (M))$ will be called $\pi$-vector fields and will be denoted by barred letters $\overline{X} $. The tensor fields on $\pi^{-1}(TM)$ will be called $\pi$-tensor fields. The fundamental $\pi$-vector field is the $\pi$-vector field $\overline{\eta}$ defined by $\overline{\eta}(u)=(u,u)$ for all $u\in TM$. We have the following short exact sequence of vector bundles $$0\longrightarrow \pi^{-1}(TM)\stackrel{\gamma}\longrightarrow T({{\cal T}}M)\stackrel{\rho}\longrightarrow \pi^{-1}(TM)\longrightarrow 0 ,\vspace{-0.1cm}$$ with the well known definitions of the bundle morphisms $\rho$ and $\gamma$. The vector space $V_u ({{\cal T}}M)= \{ X \in T_u ({{\cal T}}M) : d\pi(X)=0 \}$ is the vertical space to $M$ at $u$. Let $D$ be a linear connection on the pullback bundle $\pi^{-1}(TM)$. We associate with $D$ the map $K:T {{\cal T}}M\longrightarrow \pi^{-1}(TM):X\longmapsto D_X \overline{\eta} ,$ called the connection map of $D$. The vector space $H_u ({{\cal T}}M)= \{ X \in T_u ({{\cal T}}M) : K(X)=0 \}$ is called the horizontal space to $M$ at $u$ . The connection $D$ is said to be regular if $$T_u ({{\cal T}}M)=V_u ({{\cal T}}M)\oplus H_u ({{\cal T}}M) \,\,\, \forall \, u\in {{\cal T}}M.$$ If $M$ is endowed with a regular connection, then the vector bundle maps $ \gamma,\, \rho |_{H({{\cal T}}M)}$ and $K |_{V({{\cal T}}M)}$ are vector bundle isomorphisms. The map $\beta:=(\rho |_{H({{\cal T}}M)})^{-1}$ will be called the horizontal map of the connection $D$. The horizontal ((h)h-) and mixed ((h)hv-) torsion tensors of $D$, denoted by $Q $ and $ T $ respectively, are defined by $$Q (\overline{X},\overline{Y})=\textbf{T}(\beta \overline{X}\beta \overline{Y}), \, \,\,\, T(\overline{X},\overline{Y})=\textbf{T}(\gamma \overline{X},\beta \overline{Y}) \quad \forall \, \overline{X},\overline{Y}\in\mathfrak{X} (\pi (M)),\vspace{-0.2cm}$$ where $\textbf{T}$ is the (classical) torsion tensor field associated with $D$. The horizontal (h-), mixed (hv-) and vertical (v-) curvature tensors of $D$, denoted by $R$, $P$ and $S$ respectively, are defined by $$R(\overline{X},\overline{Y})\overline{Z}=\textbf{K}(\beta \overline{X}\beta \overline{Y})\overline{Z},\quad {P}(\overline{X},\overline{Y})\overline{Z}=\textbf{K}(\beta \overline{X},\gamma \overline{Y})\overline{Z},\quad {S}(\overline{X},\overline{Y})\overline{Z}=\textbf{K}(\gamma \overline{X},\gamma \overline{Y})\overline{Z},$$ where $\textbf{K}$ is the (classical) curvature tensor field associated with $D$. The contracted curvature tensors of $D$, denoted by $\widehat{{R}}$, $\widehat{ {P}}$ and $\widehat{ {S}}$ (known also as the (v)h-, (v)hv- and (v)v-torsion tensors respectively), are defined by $$\widehat{ {R}}(\overline{X},\overline{Y})={ {R}}(\overline{X},\overline{Y})\overline{\eta},\quad \widehat{ {P}}(\overline{X},\overline{Y})={ {P}}(\overline{X},\overline{Y})\overline{\eta},\quad \widehat{ {S}}(\overline{X},\overline{Y})={ {S}}(\overline{X},\overline{Y})\overline{\eta}.$$ [*[@r94]*]{} \[th.1\] Let $(M,L)$ be a Finsler manifold and $g$ the Finsler metric defined by $L$. There exists a unique regular connection $\nabla$ on $\pi^{-1}(TM)$ such that (a) : $\nabla$ is metric[*:*]{} $\nabla g=0$, (b) : The (h)h-torsion of $\nabla$ vanishes[*:*]{} $Q=0 $, (c) : The (h)hv-torsion $T$ of $\nabla$ satisfies*:* $g(T(\overline{X},\overline{Y}), \overline{Z})=g(T(\overline{X},\overline{Z}),\overline{Y})$. Such a connection is called the Cartan connection associated with the Finsler manifold $(M,L)$. The only linear connection we deal with in this paper is the Cartan connection. In this section, we introduce three classes of recurrent Finsler spaces which will be the object of our investigation in the next sections. These notions are defined in Riemannian geometry [@R3; @R4; @R2; @R5; @R1]. We extend them to the Finslerian case. For a Finsler manifold $(M,L)$, we set the following notations: $$\begin{aligned} \stackrel{h}\nabla&:&\text{the $h$-covariant derivatives associated with Cartan connection},\\ \text{Ric} &:& \text{the horizontal Ricci tensor of Cartan connection},\\ r &:& \text{the horizontal scalar curvature of Cartan connection},\\ G(\overline{X},\overline{Y})\overline{Z} &:=& g(\overline{X},\overline{Z}) \overline{Y}-g(\overline{Y},\overline{Z})\overline{X},\\ C &:=& {R}-\frac{r}{n(n-1)}\, {G}: \text{ the concircular curvature tensor},\\ \textbf{R}(\overline{X},\overline{Y},\overline{Z},\overline{W}) &:=& g(R(X,Y)Z,W),\\ \textbf{G}(\overline{X},\overline{Y},\overline{Z},\overline{W}) &:=& g(G(X,Y)Z,W),\\ \textbf{C}(\overline{X},\overline{Y},\overline{Z},\overline{W}) &:=& g(C(X,Y)Z,W).\end{aligned}$$ A Finsler manifold is said to be horizontally integrable if its horizonal distribution is completely integrable (or, equivalently, $\widehat{R}=0)$. \[def.1a\] Let $(M,L)$ be a Finsler manifold of dimension $n\geq3$ with non-zero $h$-curvature tensor ${R}$. Then, $(M,L)$ is said to be: (a) : recurrent Finsler manifold $(F_{n})$ if $\stackrel{h}{\nabla} {R}= A\otimes {R}$, (b) : 2-recurrent Finsler manifold $(2F_{n})$ if $\stackrel{h}{\nabla}\stackrel{h}{\nabla} {R}= \alpha\otimes {R}$, (c) : generalized recurrent Finsler manifold $(GF_{n})$ if $\stackrel{h}{\nabla} {R}= A\otimes {R}+B\otimes {G}$, (d) : generalized 2-recurrent Finsler manifold $(G(2F_{n}))$ if $\stackrel{h}{\nabla}\stackrel{h}{\nabla} {R}= \alpha\otimes {R}+\mu \otimes {G}$, where $A$ and $B$ (resp. $\alpha$ and $\mu$ ) are non-zero scalar 1-forms (resp. 2-forms) on $TM$, and positively homogenous of degree zero in $y$, called the recurrence forms. In particular, if $\stackrel{h}{\nabla} {R}=0$, then $(M,L)$ is called symmetric. \[def.2a\] Let $(M,L)$ be a Finsler manifold of dimension $n\geq3$ with non-zero horizontal Ricci tensor $\emph{\text{Ric}}$. Then, $(M,L)$ is said to be: (a) : Ricci recurrent Finsler manifold $(RF_{n})$ if $\stackrel{h}{\nabla} \emph{\text{Ric}}= A\otimes \emph{\text{Ric}}$, (b) : 2-Ricci recurrent Finsler manifold $(2RF_{n})$ if $\stackrel{h}{\nabla}\stackrel{h}{\nabla} \emph{\text{Ric}}= \alpha\otimes \emph{\text{Ric}}$, (c) : generalized Ricci recurrent Finsler manifold $(GRF_{n})$ if $\stackrel{h}{\nabla} \emph{\text{Ric}}= A\otimes \emph{\text{Ric}}+B\otimes {g}$, (d) : generalized 2-Ricci recurrent Finsler manifold $(G(2RF_{n}))$ if $$\stackrel{h}{\nabla}\stackrel{h}{\nabla} \emph{\text{Ric}}= \alpha\otimes \emph{\text{Ric}}+\mu \otimes {g},$$ where $A$ and $B$ (resp. $\alpha$ and $\mu$ ) are as given in Definition \[def.1a\]. In particular, if $\stackrel{h}{\nabla} \emph{\text{Ric}}=0$, then $(M,L)$ is called Ricci symmetric. \[def.3a\] Let $(M,L)$ be a Finsler manifold of dimension $n\geq3$ with non-zero concircular curvature tensor $C$. Then, $(M,L)$ is said to be: (a) : concircularly recurrent Finsler manifold $(CF_{n})$ if $\stackrel{h}{\nabla} {C}= A\otimes {C},$ (b) : 2-concircularly recurrent Finsler manifold $(2CF_{n})$ if $\stackrel{h}{\nabla}\stackrel{h}{\nabla} {C}= \alpha\otimes {C},$ (c) : generalized concircularly recurrent Finsler manifold $(GCF_{n})$ if $$\stackrel{h}{\nabla} {C}= A\otimes {C}+B\otimes {G},$$ (d) : generalized 2-concircularly recurrent Finsler manifold $(G(2CF_{n}))$ if $$\stackrel{h}{\nabla}\stackrel{h}{\nabla} {C}= \alpha\otimes {C}+\mu \otimes {G},$$ where $A$ and $B$ (resp. $\alpha$ and $\mu$ ) are as given in Definition \[def.1a\]. In particular, if $\stackrel{h}{\nabla} {C}=0$, then $(M,L)$ is called concircularly symmetric. We quote the following two Lemmas from [@F1]; they are very useful in the sequel. \[lem.1\] For a horizontally integrable Finsler manifold, we have: (a) : $\mathfrak{S}_{\overline{X},\overline{Y},\overline{Z}}\, \{{R}(\overline{X},\overline{Y})\overline{Z}\}=0.$ [^1] (b) : $\textbf{R}(\overline{X},\overline{Y},\overline{Z},\overline{W})=\textbf{R}(\overline{Z},\overline{W},\overline{X},\overline{Y}).$ (c) : $\mathfrak{S}_{\overline{X},\overline{Y},\overline{Z}} \,\{{(\stackrel{h}{\nabla}R)}(\overline{X},\overline{Y},\overline{Z},\overline{W})\}=0.$ (d) : The horizontal Ricci tensors $Ric$ is symmetric. (e) : $\mathfrak{S}_{\overline{U},\overline{V};\,\,\overline{W},\overline{X};\,\,\overline{Y},\overline{Z}}{\left\{({R} (\overline{U},\overline{V})\textbf{R})(\overline{W},\overline{X},\overline{Y},\overline{Z})\right\}}=0.$ [^2] (f) : $\mathfrak{S}_{\overline{U},\overline{V};\,\,\overline{W},\overline{X};\,\,\overline{Y},\overline{Z}}{\left\{({R} (\overline{U},\overline{V})\textbf{C})(\overline{W},\overline{X},\overline{Y},\overline{Z})\right\}}=0$. (g) : $ (\stackrel{h}{\nabla}\stackrel{h}{\nabla}\omega)(\overline{Y},\overline{X},\overline{Z})- (\stackrel{h}{\nabla}\stackrel{h}{\nabla}\omega)(\overline{X},\overline{Y},\overline{Z})= ( {R}(\overline{X},\overline{Y})\omega)(\overline{Z})$; $\omega$ is a $\pi$(1)-form. \[lem.2\] Let $(M,L)$ be a horizontally integrable Finsler manifold and let $\omega$ be a $\pi$(2)-form. If any one of the following relations holds $$\begin{aligned} \mathfrak{S}_{\overline{U},\overline{V};\,\overline{W},\overline{X};\,\overline{Y},\overline{Z}}{\left\{\omega(\overline{U},\overline{V})\textbf{R}(\overline{W},\overline{X},\overline{Y},\overline{Z}) \right\}}&=&0,\\ \mathfrak{S}_{\overline{U},\overline{V};\,\overline{W},\overline{X};\,\overline{Y},\overline{Z}}{\left\{\omega(\overline{U},\overline{V})\textbf{C}(\overline{W},\overline{X},\overline{Y},\overline{Z}) \right\}}&=&0,\\ \mathfrak{S}_{\overline{U},\overline{V};\,\overline{W},\overline{X};\,\overline{Y},\overline{Z}}{\left\{\omega(\overline{U},\overline{V})\textbf{G}(\overline{W},\overline{X},\overline{Y},\overline{Z}) \right\}}&=&0,\end{aligned}$$ then $\omega$ vanishes identically. \[prop.1\] Let $(M,L)$ be a horizontally integrable Finsler manifold of dimension $n\geq3$. If $(M,L)$ is recurrent (resp. 2-recurrent) with recurrence form $A$ ( resp. $\alpha$), then we have: (a) : $\mathfrak{S}_{\overline{X},\overline{Y},\overline{Z}} \,\{{(A\otimes R)}(\overline{X},\overline{Y},\overline{Z},\overline{W})\}=0$ (b) : $\stackrel{h}{\nabla} A$ *(*resp. $\alpha$*)* is symmetric, (c) : $R(\overline{X},\overline{Y}){\bf R}=0$. The proof follows from Definition \[def.1a\] together with Lemmas \[lem.1\] and \[lem.2\]. \[thm.1\] If $(M,L)$ is a recurrent Finsler manifold of dimension $n\geq3$ with recurrence form $A$, then (a) : $(M,L)$ is $RF_{n} $. (b) : $(M,L)$ is $ CF_{n} $ provided that $r\neq0$. (c) : $(M,L)$ is $ 2F_{n} $ *(* resp. $2RF_{n}$*)* provided that $\stackrel{h}{\nabla}A+A\otimes A\neq0$.   **(a)** is clear from the definitions of recurrence and Ricci recurrence. **(b)** As $(M,L)$ is a recurrent Finsler manifold, $ \stackrel{h}{\nabla} {R}= A\otimes {R}$, with $A\neq 0 $. Hence, $\stackrel{h}{\nabla} {r}= rA $, with ${r\neq0}$ by assumption. Consequently, $$\begin{aligned} \stackrel{h}{\nabla} {C}&=& \stackrel{h}{\nabla} {\left\{{R}-\frac{r}{n(n-1)}\, {G}\right\}} =\,\, \stackrel{h}{\nabla}{R}-\frac{\stackrel{h}{\nabla}r}{n(n-1)}\otimes G, {\quad\text{since} \stackrel{h}{\nabla}{G}=0} \nonumber \\ &{=}& A\otimes {R}-\frac{rA}{n(n-1)}\otimes G = A\otimes C \nonumber . \end{aligned}$$ **(c)** Using $\stackrel{h}{\nabla} {R}= A\otimes {R}$, we have $$\label{e1} \stackrel{h}{\nabla} \stackrel{h}{\nabla} {R}= \stackrel{h}{\nabla}A\otimes {R}+A\otimes \stackrel{h}{\nabla}{R} =(\stackrel{h}{\nabla}A +A\otimes A)\otimes {R} = \alpha \otimes {R},$$ where $\alpha:=\stackrel{h}{\nabla}A +A\otimes A$. Hence, $(M,L)$ is $2F_{n}$ provided that $\alpha\neq0$. Similarly, one can show that $(M,L)$ is $2RF_{n}$. One can easily show that, the sufficient condition for a Ricci recurrent Finsler manifold of dimension $n\geq3$ with recurrence form $A$ to be a 2-Ricci recurrent Finsler manifold is that $\stackrel{h}{\nabla}A +A\otimes A \neq0 $. \[prop.2\] Let $(M,L)$ be a horizontally integrable concircularly recurrent (resp. 2-concircularly recurrent) Finsler manifold of dimension $n\geq3$ with recurrence form $A$ (resp. $\alpha$), then we have: (a) : $\stackrel{h}{\nabla} A$ *(*resp. $\alpha$*)* is symmetric, (b) : $R(\overline{X},\overline{Y}){\bf C}=0$, (c) : $R(\overline{X},\overline{Y}){\bf R}=0$. The proof follows from Definition \[def.3a\] together with Lemmas \[lem.1\] and \[lem.2\], after some calculations. \[thm.2\] If $(M,L)$ is a concircularly recurrent Finsler manifold of dimension $n\geq3$ with recurrence form $A$, then (a) : $(M,L)$ is $ 2CF_{n} $ provided that $\stackrel{h}{\nabla}A+A\otimes A\neq0$. (b) : $(M,L)$ is $ GF_{n} $ *(*resp. $ GRF_{n}$*)* provided that $\stackrel{h}{\nabla}r-rA\neq0$. (c) : $(M,L)$ is $ F_{n} $ provided that $\widehat{R}=0$.   **(a)** Let $(M,L)$ be concircularly recurrent, then $\stackrel{h}{\nabla} {C}= A\otimes {C}$, [with]{} $ A\neq 0 $. Consequently, $$\stackrel{h}{\nabla} \stackrel{h}{\nabla} {C}= \stackrel{h}{\nabla}A\otimes {C}+A\otimes \stackrel{h}{\nabla}{C} = (\stackrel{h}{\nabla}A +A\otimes A)\otimes {C} = \alpha \otimes {C},$$ where $\alpha:=\stackrel{h}{\nabla}A+A\otimes A$. Hence, $(M,L)$ is a $2CF_{n}$ if $\alpha\neq0$. **(b)** As $\stackrel{h}{\nabla} {C}= A\otimes {C}$, ${C}= {R}-\frac{r}{n(n-1)}\, {G}$ and $\stackrel{h}{\nabla} {G}=0$, we get $$\begin{aligned} \stackrel{h}{\nabla}{R}&{=}&A\otimes {R}+\frac{1}{n(n-1)}\{\stackrel{h}{\nabla}r-rA\}\otimes {G} \label{eq.3}\\ \stackrel{h}{\nabla}{R}&{=}& A\otimes {R}+B\otimes {G} \nonumber , \end{aligned}$$ where $B=\frac{1}{n(n-1)}(\stackrel{h}{\nabla}{r}-rA)$. Since $A\neq0$, then, $(M,L)$ is $GF_{n}$ if $B\neq0$. Now, taking the trace of both sides of (\[eq.3\]), one gets $$\stackrel{h}{\nabla}{Ric}= A\otimes {Ric}+B_{1}\otimes g,$$ where $B_{1}=\frac{1}{n}\{\stackrel{h}{\nabla}r-rA\}$. Hence, $(M,L)$ is $GRF_{n}$ if $B_{1}\neq0$. **(c)** Follows from Theorem C of [@F1]. \[thm.3\] If $(M,L)$ is a generalized recurrent Finsler manifold of dimension $n\geq3$ with recurrence forms $A $ and $B$, then (a) : $(M,L)$ is $G(2F_{n}) $ provided that $\stackrel{h}{\nabla}A +A\otimes A\neq0$ and $\stackrel{h}{\nabla}B +A\otimes B\neq0$. (b) : $(M,L)$ is $CF_{n}$ provided that $r\neq0$. (c) : $(M,L)$ is $2CF_{n} $ provided that $\stackrel{h}{\nabla}A +A\otimes A\neq0$ and $r\neq0$. (d) : $(M,L)$ is $GRF_{n} $ (e) : $(M,L)$ is $ F_{n} $ provided that $\widehat{R}=0 $ and $r\neq0$.   **(a)** Let $(M,L)$ be a generalized recurrent Finsler manifold, then $ \stackrel{h}{\nabla} {R}= A\otimes {R}+B\otimes {G}$, with $A\neq 0 \neq B $. Consequently, $$\begin{aligned} \stackrel{h}{\nabla} \stackrel{h}{\nabla} {R}&{=}& (\stackrel{h}{\nabla}A\otimes {R}+A\otimes \stackrel{h}{\nabla}{R}) +\stackrel{h}{\nabla}B\otimes {G}, \,\, \text{since} \stackrel{h}{\nabla}G=0 \nonumber \\ &{=}& (\stackrel{h}{\nabla}A\otimes {R} +A\otimes A\otimes {R}+A\otimes B\otimes {G}) +\stackrel{h}{\nabla}B\otimes {G} \nonumber \\ &{=}& (\stackrel{h}{\nabla}A +A\otimes A)\otimes {R}+(A\otimes B +\stackrel{h}{\nabla}B)\otimes {G} \label{eq.aa} \\ &{=}& \alpha\otimes {R}+\mu\otimes {G} \nonumber , \end{aligned}$$ where $\alpha:=\stackrel{h}{\nabla}A +A\otimes A$ and $\mu:=\stackrel{h}{\nabla}B+A\otimes B$ are non-zero scalar 2-forms. **(b)** By double contraction of $ \stackrel{h}{\nabla} {R}= A\otimes {R}+B\otimes {G}$, we get $$\begin{aligned} \label{eq.con} \stackrel{h}{\nabla} {r}&=& rA + n(n-1)B . \end{aligned}$$ Consequently, $$\begin{aligned} \stackrel{h}{\nabla} {C}&=& \stackrel{h}{\nabla} {\left\{{R}-\frac{r}{n(n-1)}\, {G}\right\}}=\,\,\stackrel{h}{\nabla}{R}-\frac{\stackrel{h}{\nabla}r}{n(n-1)}\otimes G \nonumber \\ &\stackrel{(\ref{eq.con})}{=}& A\otimes {R}-\frac{rA}{n(n-1)}\otimes G = A\otimes C \label{eq.con1} . \end{aligned}$$ **(c)** follows from **(b)**, **(d)** is trivial, **(e)** follows from **(b)** and Theorem \[thm.2\]. One can easily show that, the sufficient condition for a generalized Ricci recurrent Finsler manifold of dimension $n\geq3$, with recurrence forms $A$ and $B$, to be a generalized 2-Ricci recurrent Finsler manifold is that $\stackrel{h}{\nabla}A +A\otimes A\neq0$ and $\stackrel{h}{\nabla}B +A\otimes B\neq0$. \[prop.3\] Let $(M,L)$ be a horizontally integrable generalized recurrent Finsler manifold of dimension $n\geq3$ with recurrence forms $A, B$ and scalar curvature $r$, then we have: (a) : $\mathfrak{S}_{\overline{X},\overline{Y},\overline{Z}} \,\{{(A\otimes R+B\otimes G)}(\overline{X},\overline{Y},\overline{Z},\overline{W})\}=0$. (b) : $\overline{d}A$ and $\overline{d}B+A\wedge B$ vanish identically. (c) : $R(\overline{X},\overline{Y}){\bf R}=0$,\ where $\overline{d}A(\overline{X},\overline{Y}):=(\stackrel{h}{\nabla}{A})(\overline{X},\overline{Y}) -(\stackrel{h}{\nabla}{A})(\overline{Y},\overline{X})$. The proof of [(**a)**]{} is easy.\ Now, we prove [(**b)**]{}. Let $(M,L)$ be horizontally integrable and generalized recurrent with recurrence forms $A$ and $B$. Then, by Theorem \[thm.3\], $(M,L)$ is concirculary recurrent, i.e., $ \stackrel{h}{\nabla}C=A\otimes C$, by (\[eq.con1\]). Consequently, $\stackrel{h}{\nabla}\textbf{C}=A\otimes\textbf{ C}$. Hence, $$\stackrel{h}{\nabla}\stackrel{h}{\nabla}\textbf{C}= (\stackrel{h}{\nabla}A)\otimes \textbf{C} +A\otimes \stackrel{h}{\nabla}\textbf{C} =(\stackrel{h}{\nabla}A+A\otimes A)\otimes \textbf{C}.$$ From which, taking into account Lemma \[lem.1\][**(g)**]{}, we obtain $$\begin{aligned} {R}(\overline{U},\overline{V})\textbf{C}&=&-({\overline{d}}A)(\overline{U},\overline{V})\textbf{C}.\label{eq.16a}\end{aligned}$$ Hence, using Lemma \[lem.1\](**f**), it follows that $$\mathfrak{S}_{\overline{U},\overline{V};\,\overline{W},\overline{X};\,\overline{Y},\overline{Z}} {\left\{{\overline{d}}A(\overline{U},\overline{V})\textbf{C}(\overline{W},\overline{X},\overline{Y},\overline{Z}) \right\}}=0.$$ From which, together with Lemma \[lem.2\], we conclude that $$\label{da} {\overline{d}}A=0$$ On the other hand, from (\[eq.aa\]), we obtain $$\begin{aligned} \stackrel{h}{\nabla}\stackrel{h}{\nabla}\textbf{R}&=& (\stackrel{h}{\nabla}A+A\otimes A)\otimes \textbf{R} +(A\otimes B+ \stackrel{h}{\nabla}B)\otimes\textbf{G}.\end{aligned}$$ From which, taking into account (\[da\]) and Lemma \[lem.1\], we get $$\begin{aligned} {R}(\overline{U},\overline{V})\textbf{R}&=& -(\overline{d}B+A\wedge B)(\overline{U},\overline{V})\textbf{G}.\label{eq.16}\end{aligned}$$ Hence, from Lemma \[lem.1\](**f**), we obtain $$\begin{aligned} \mathfrak{S}_{\overline{U},\overline{V};\,\overline{W},\overline{X};\,\overline{Y},\overline{Z}} {\left\{(\overline{d}B+A\wedge B)(\overline{U},\overline{V})\textbf{G}(\overline{W},\overline{X},\overline{Y},\overline{Z}) \right\}}&=&0.\end{aligned}$$ Therefore, $\overline{d}B+A\wedge B$ vanishes identically. Finally, the proof of ([**c**]{}) follows from ([**b**]{}) and (\[eq.16\]). \[prop.3a\] Let $(M,L)$ be a horizontally integrable generalized 2-recurrent Finsler manifold of dimension $n\geq3$ with recurrence forms $\alpha, \mu$ and non-zero constant scalar curvature $r$, then we have: (a) : $\alpha$ and $\mu$ are symmetric scalar 2-forms. (b) : $R(\overline{X},\overline{Y}){\bf R}=0$. In this section, we study a new type of Finsler recurrence, namely the generalized concircular recurrence, which generalizes the concircular recurrence investigated in [@F1] by the present authors. \[before last\] Let $(M,L)$ be a generalized concircularly recurrent Finsler manifold of dimension $n\geq3$ with recurrence forms $A, B$ and scalar curvature $r$, then (a) : $(M,L)$ is $G(2CF_{n}) $ provided that $\stackrel{h}{\nabla}A +A\otimes A\neq0$ and $\stackrel{h}{\nabla}B +A\otimes B\neq0$, (b) : $(M,L)$ is $GF_{n} $ *(*resp. $GRF_{n}$*)* provided that $B-\frac{rA}{n(n-1)}+\frac{\stackrel{h}{\nabla}{r}}{n(n-1)}\neq0$.   **(a)** Let $(M,L)$ be generalized concircularly recurrent. Then, $\stackrel{h}{\nabla} {C}= A\otimes {C}+B\otimes {G},$ with $ A\neq 0 \neq B$. Consequently, $$\begin{aligned} \stackrel{h}{\nabla} \stackrel{h}{\nabla} {C}&{=}& (\stackrel{h}{\nabla}A\otimes {C}+A\otimes \stackrel{h}{\nabla}{C}) +\stackrel{h}{\nabla}B\otimes {G}, \nonumber \\ &=& (\stackrel{h}{\nabla}A\otimes {C} +A\otimes A\otimes {C}+A\otimes B\otimes {G}) +\stackrel{h}{\nabla}B\otimes {G} \nonumber \\ &{=}& (\stackrel{h}{\nabla}A +A\otimes A)\otimes {C}+(\stackrel{h}{\nabla}B+A\otimes B)\otimes {G} \nonumber \\ &{=}& \alpha\otimes {C}+\mu\otimes {G} \nonumber . \end{aligned}$$ where $\alpha:=\stackrel{h}{\nabla}A +A\otimes A$ and $\mu:=\stackrel{h}{\nabla}B+A\otimes B$. If $\alpha$ and $\mu$ are none-zero, then $(M,L)$ is $G(2CF_{n})$. **(b)** As $\stackrel{h}{\nabla} {C}= A\otimes {C}+B\otimes {G},$ with $ A\neq 0 \neq B$, then $$\begin{aligned} \stackrel{h}{\nabla} ({R}-\frac{r}{n(n-1)}\, {G})&{=}& A\otimes ({R}-\frac{r}{n(n-1)}\, G)+B\otimes {G} \nonumber \\ \stackrel{h}{\nabla}{R}-\frac{\stackrel{h}{\nabla}{r}}{n(n-1)}\otimes {G}&{=}& A\otimes ({R}-\frac{r}{n(n-1)}\, G)+B\otimes {G}, \,\, \text{since} \stackrel{h}{\nabla}G=0 \nonumber \\ \stackrel{h}{\nabla}{R}&{=}& A\otimes {R}+(B-\frac{rA}{n(n-1)}+\frac{\stackrel{h}{\nabla}{r}}{n(n-1)})\otimes {G} \label{eq.2}\\ \stackrel{h}{\nabla}{R}&{=}& A\otimes {R}+B_{1}\otimes {G}\nonumber , \end{aligned}$$ where $B_{1}:=B-\frac{rA}{n(n-1)}+\frac{\stackrel{h}{\nabla}{r}}{n(n-1)}$. Since $B_{1}\neq0$, then $(M,L)$ is $GF_{n}$. On the other hand, from (\[eq.2\]), we obtain $$\begin{aligned} \stackrel{h}{\nabla}\text{Ric}&{=}&A\otimes \text{Ric}+\frac{1}{n}(n(n-1)B-rA+\stackrel{h}{\nabla}r)\otimes {g} \label{new}\\ &{=}& A\otimes \text{Ric}+B_{2}\otimes {g} \nonumber , \end{aligned}$$ where $B_2=(n-1)B_1$. This completes the proof. \[thm.1c\] Let $(M,L)$ be a horizontally integrable generalized concircularly recurrent Finsler manifold with recurrence forms $A$ and $B$. The scalar curvature $r$ of $(M,L)$ is horizontally parallel if and only if $\, 2rA=2n\,A\circ \text{\em{Ric}}_{o}-n(n-1)(n-2)B$, where ${\text{\em{Ric}}}_o$ is defined by $g({\text{\em{Ric}}}_{o}\overline{X},\overline{Y}) := \text{\em{Ric}}(\overline{X},\overline{Y})$. By (\[eq.2\]) and Lemma \[lem.1\][**(c)**]{}, we obtain $$\begin{aligned} && A(\overline{W})R(\overline{X},\overline{Y})\overline{Z}+A(\overline{X})R(\overline{Y},\overline{W})\overline{Z} +A(\overline{Y})R(\overline{W},\overline{X})\overline{Z}\\ &&+\frac{1}{n(n-1)}\{n(n-1)B(\overline{W})- rA(\overline{W})+\stackrel{h}{\nabla}{r}(\overline{W})\} \{g(\overline{X},\overline{Z})\overline{Y}-g(\overline{Y},\overline{Z})\overline{X}\}\\ &&+\frac{1}{n(n-1)}\{n(n-1)B(\overline{X})- rA(\overline{X})+\stackrel{h}{\nabla}{r}(\overline{X})\} \{g(\overline{Y},\overline{Z})\overline{W}-g(\overline{W},\overline{Z})\overline{Y}\}\\ &&+\frac{1}{n(n-1)}\{n(n-1)B(\overline{Y})- rA(\overline{Y})+\stackrel{h}{\nabla}{r}(\overline{Y})\} \{g(\overline{W},\overline{Z})\overline{X}-g(\overline{X},\overline{Z})\overline{W}\}=0.\end{aligned}$$ Contracting the above relation with respect to ${\overline{Y}}$, given that $g(\overline{X},\overline{\sigma}):=A(\overline{X})$, we get $$\begin{aligned} && A(\overline{W}){\text{{Ric}}}(\overline{X},\overline{Z})-A(\overline{X}){\text{{Ric}}}(\overline{W},\overline{Z}) +\textbf{R}(\overline{W},\overline{X},\overline{Z}, \overline{\sigma})\\ &&+\frac{1}{n}\{n(n-1)B(\overline{W})- rA(\overline{W})+\stackrel{h}{\nabla}{r}(\overline{W})\} g(\overline{X},\overline{Z})\\ &&-\frac{1}{n}\{n(n-1)B(\overline{X})- rA(\overline{X})+\stackrel{h}{\nabla}{r}(\overline{X})\} g(\overline{Z},\overline{W})\\ &&+\frac{1}{n(n-1)}\{n(n-1)B(\overline{X})- rA(\overline{X})+\stackrel{h}{\nabla}{r}(\overline{X})\} g(\overline{W},\overline{Z})\\ &&-\frac{1}{n(n-1)}\{n(n-1)B(\overline{W})- rA(\overline{W})+\stackrel{h}{\nabla}{r}(\overline{W})\} g(\overline{X},\overline{Z})=0.\end{aligned}$$ This Relation reduces to $$\begin{aligned} && A(\overline{W})\text{Ric}(\overline{X},\overline{Z})-A(\overline{X})\text{Ric}(\overline{W},\overline{Z}) +\textbf{R}(\overline{W},\overline{X},\overline{Z}, \overline{\sigma})\\ &&+\frac{n-2}{n(n-1)}\{n(n-1)B(\overline{W})- rA(\overline{W})+\stackrel{h}{\nabla}{r}(\overline{W})\} g(\overline{X},\overline{Z})\\ &&-\frac{n-2}{n(n-1)}\{n(n-1)B(\overline{X})- rA(\overline{X})+\stackrel{h}{\nabla}{r}(\overline{X})\} g(\overline{Z},\overline{W})=0.\end{aligned}$$ Contracting the above relation with respect to $\overline{X}$ and $\overline{Z}$, we obtain $$2rA(\overline{W})-2nA(\text{Ric}_{o}\overline{W})+n(n-1)(n-2)B(\overline{W})+(n-2)\stackrel{h}{\nabla}{r}(\overline{W})=0.$$ Hence, the result follows. \[last\] Let $(M,L)$ be a horizontally integrable generalized concircularly recurrent Finsler manifold with recurrence forms $A$ and $B$. If the scalar curvature $r$ of $(M,L)$ is constant, then $(M,L)$ is a $GRF_{n}$. If the scalar curvature $r$ of $(M,L)$ is constant, then $\stackrel{h}{\nabla}r=0$. Hence, in view of Lemma \[thm.1c\], we get $$\label{new2} rA=nA\circ {\text{{Ric}}}_{o}-\frac{n(n-1)(n-2)}{2}B.$$ From which, together (\[new\]), we obtain $$\begin{aligned} \stackrel{h}{\nabla}{\text{Ric}}&{=}& A\otimes {\text{Ric}}+\left(\frac{n(n-1)}{2}B-A\circ \text{Ric}_{o}\right)\otimes {g} \nonumber \\ &=&A\otimes {\text{Ric}}+D \otimes g \label{feq} , \end{aligned}$$ where $D:=\frac{n(n-1)}{2}B-A\circ \text{Ric}_{o}$. Now, we show that $D\neq0$. Assume the contrary, then $$A\circ \text{Ric}_{o}=\frac{n(n-1)}{2}B.$$ Substituting into (\[new2\]), we obtain $$\label{tt} rA=n(n-1)B.$$ From which together with (\[eq.2\]), noting that the scaler curvature $r$ is constant, we get $rA=0$. Hence, again by (\[tt\]), $B=0$. This is a contradiction.\ Therefore, by (\[feq\]), $(M,L)$ is $GRF_{n}$ (as $D\neq0$). Both Theorem \[before last\](b) and Theorem \[last\] state roughly that, under certain conditions, a $GCF_n$ manifold is a $GRF_n$ manifold. The difference between the two results is that in Theorem \[before last\](b) the condition is posed on the recurrence forms $A$ and $B$, whereas in Theorem \[last\] the condition is posed on the geometric structure of the underlying manifold ($r$ is constant, $\widehat{R}=0$). **Concluding Remarks** Three classes of recurrence in Finsler geometry are introduced and investigated. The interrelationships among these classes of recurrence are studied. The following diagram presents concisely the most important results of the paper, where an arrow means “if ... then”. Here are some comments on this table: - Continuous arrows represent results (theorems) of the paper. Dashed arrows represent examples of results that can be deduced from continuous arrows. - Conditions posed on the recurrence forms are not written in the diagram. The written conditions are those posed on the geometric structure of the underlying manifold. - One can deduce the following result from the table: $$F_n\stackrel{r\neq0, \widehat{R}=0}\Longleftrightarrow CF_n$$ This is one of the main result of [@F1]. - Among other new important results that can be deduced from the table, we set: $$GF_n\Longleftrightarrow CF_n$$ $$F_n\stackrel{r\neq0, \widehat{R}=0}\Longleftrightarrow GF_n$$ $$GCF_n\stackrel{r\neq0}\Longrightarrow CF_n$$ [21]{} H. Akbar-Zadeh, *Initiation to global Finsler geometry*, Elsevier, 2006. U. C. De, N. Guha and D. Kamilya, *On generalized Ricci-recurrent manifolds*, Tensor, N. S., **56** (1995), 312-317. Y. B. Maralabhavi and M. Rathnamma, *On generalized recurrent manifold*, Indian J. Pure Appl. Math., **30** (1999), 1167-1171. M. Matsumoto, *On $h$-isotropic and $\textsc{C}^{h}$-recurrent Finsler spaces*, J. Math. Kyoto Univ., **11** (1971), 1-9. R. S. Mishra and H. D. Pande, *Recurrent Finsler spaces*, J. Ind. Math. Soc. **32** (1968) 17-22. E. M. Patterson, *Some theorems on Ricci recurrent spaces*, J. London Math. Soc., **27** (1952), 287-295. H. Singh and Q. Khan, *On generalized recurrent Riemannian manifolds*, Publ. Math. Debrecen, **56** (2000), 87-95. A. G. Walker, *On Ruses’s spaces of recurrent curvature*, Proc. London Math. Soc., **52** (1950), 36-64. Nabil L. Youssef and A. Soleiman, *On concircularly recurrent Finsler manifolds*, Balkan J. Geom. Appl., **18**, **1** (2013), 101-113. arXiv: 0704.0053 \[math. DG\]. Nabil L. Youssef, S. H. Abed and A. Soleiman, *A global approach to the theory of special Finsler manifolds*, J. Math. Kyoto Univ., **48**, **4** (2008), 857-893. arXiv: 0704.0053 \[math. DG\]. [to3em]{}, *A global approach to the theory of connections in Finsler geometry*, Tensor, N. S., **71**, **3** (2009), 187-208. arXiv: 0801.3220 \[math.DG\]. [to3em]{}, *Geometric objects associated with the fundumental connections in Finsler geometry*, J. Egypt. Math. Soc., **18**, **1** (2010), 67-90. arXiv: 0805.2489 \[math.DG\]. [^1]: $\mathfrak{S}_{\overline{X},\overline{Y},\overline{Z}}$ denotes the cyclic sum over ${\overline{X},\overline{Y},\overline{Z}}$. [^2]: $\mathfrak{S}_{\overline{U},\overline{V};\,\,\overline{W},\overline{X};\,\, \overline{Y},\overline{Z}}$ denotes the cyclic sum over the three pairs of arguments $\overline{U},\overline{V};\,\, \overline{W},\overline{X};\,\,\overline{Y},\overline{Z}$.

--- abstract: 'We introduce a 3-manifold invariant $\hat{Z}_b^G(q)$ valued in integer-coefficient power series and an invariant of knot complements $F_K^G({\bf x},q)$ valued in multi-variable series that depend on the choice of gauge group $G$. This generalizes the earlier works of Gukov-Pei-Putrov-Vafa [@GPPV] and Gukov-Manolescu [@GM] which correspond to $G=SU(2)$ case. As in the $SU(2)$ case, there is a surgery formula relating $F_K^G({\bf x},q)$ to $\hat{Z}_b^G(q)$ of a Dehn surgery of the knot. We provide explicit calculations of $\hat{Z}_b^G(q)$ for negative definite plumbings and $F_K^G({\bf x},q)$ for torus knots. Furthermore, we present a specialization of $F_K^G({\bf x},q)$ to symmetric representations which should satisfy a recurrence given by the quantum A-polynomial for symmetric representations.' address: 'Division of Physics, Mathematics and Astronomy, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125' author: - Sunghyuk Park bibliography: - 'higher\_rank.bib' date: 'September 19, 2019' title: 'Higher rank $\hat{Z}$ and $F_K$' --- Introduction ============ Categorification of the Chern-Simons theory is one of the most exciting open questions in quantum topology. While homology theories categorifying quantum link invariants are fairly well-understood by now, whether there is a homology theory categorifying the Witten-Reshetikhin-Tureav (WRT) invariants is still widely open. One approach to this problem comes from physics. According to recent works of Gukov-Putrov-Vafa [@GPV] and Gukov-Pei-Putrov-Vafa [@GPPV], we can decompose WRT invariants into categorifiable “homological blocks” often denoted by $\hat{Z}_b(q)$. These are integer coefficient $q$-series and are supposed to be the (graded) Euler characteristic of a conjectural homological invariant $\mathcal{H}^{i,j}_{b,{\rm BPS}}$ so that $$\hat{Z}_b(q) = \sum_{i,j}(-1)^i q^j \text{rank }\mathcal{H}^{i,j}_{b,{\rm BPS}}.$$ More recently Gukov-Manolescu [@GM] studied an analog of $\hat{Z}_b(q)$ for knot complements, denoted by $F_K(x,q)$, and demonstrated that $\hat{Z}_b(q)$ behaves well under cutting and gluing along a torus boundary. More than a decade ago, Dunfield-Gukov-Rasmussen [@DGR] made a fascinating conjecture, also partly motivated from physics, that $\mathfrak{sl}(N)$ link homologies should be equipped with a family of differentials $d_N$ for $N\in \mathbb{Z}$. This conjecture revealed many hidden structures of the link homologies and could be used to compute Khovanov homology very efficiently. Now, a very natural (and ambitious) question is this : Is there an analogous story for 3-manifolds? The purpose of this paper is to define categorifiable objects $\hat{Z}_b^G(q)$ and $F_K^G({\bf x},q)$ for arbitrary gauge group $G$. We believe that our work will serve as a stepping stone towards the analogous story of large $N$ on 3-manifold side. Organization of the paper {#organization-of-the-paper .unnumbered} ------------------------- In Section 2 we list a few notational conventions we will use throughout this paper. In Section 3 we define higher rank $\hat{Z}_b(q)$ for (weakly) negative definite plumbed 3-manifolds. We provide many examples, hoping it will serve as a good reference. In Section 4 we introduce higher rank $F_K({\bf x},q)$ and give an explicit formula for torus knots. Higher rank surgery formula (without proof) is also presented. In Section 5 we specialize our higher rank $F_K$ to symmetric representations. We numerically check for some examples that it satisfies the quantum volume conjecture. Appendix A is on a geometric meaning of the label $b$ of $\hat{Z}_b$, and Appendix B is a derivation of definition (\[integralZhat\]). Acknowledgments {#acknowledgments .unnumbered} --------------- We would like to thank Sergei Gukov for his invaluable guidance and Nikita Sopenko for his kind help with Mathematica, as well as Francesca Ferrari, Sarah Harrison and Ciprian Manolescu for helpful conversations. The author was supported by Kwanjeong Educational Foundation. Notations and conventions ========================= We follow the convention used in [@GM] for knots, 3-manifolds, and colored Jones polynomials. Throughout this article, $G$ is a semisimple Lie group, $Q$ is the root lattice, $P$ is the weight lattice, and $W$ is the Weyl group. $\rho$ denotes the Weyl vector (half-sum of positive roots), and $\alpha$ and $\omega$ will be reserved for roots and fundamental weights. We use $\rm B$ for the linking matrix of a plumbed 3-manifold. For a multi-index monomial, we use the following notation $$x^\beta := \prod_{1\leq i\leq r}x_i^{(\beta,\omega_i)}$$ where $r = \text{rank }G$ and $\beta\in P$. When it comes to $q$-series, we mostly don’t bother to fix the overall $q$-power, and just use the notation $\cong$ for equivalence up to sign and overall $q$-power. Higher rank $\hat{Z}_b$ ======================= Definition for plumbings ------------------------ We present here a formula for $\hat{Z}$ for (weakly) negative definite plumbed manifolds, with arbitrary gauge group $G$. We use the same good old Gauss sum reciprocity to deduce this definition. (See appendix \[Gauss\] for derivation.) : \[Zhatsgm\] For a plumbed 3-manifold $Y$ with weakly negative definite linking matrix $\mathrm{B}$, define[^1] $$\boxed{ \hat{Z}_b^G(Y;q) := \pm \frac{q^{\frac{3\sigma-\mathrm{Tr}\,\mathrm{B}}{2}(\rho,\rho)}}{|W|^{b_1(\Gamma)}} {\rm v.p.}\int_{|x_{vi}|=1}\prod_{v\in V}\prod_{1\leq i\leq r}\frac{dx_{vi}}{2\pi i x_{vi}}\left(\sum_{w\in W}(-1)^{l(w)} x_v^{w(\rho)}\right)^{2-\deg v}\, \Theta_b^{-\mathrm{B}}(x^{-1},q) \label{integralZhat} }$$ where $$\Theta_b^{-\mathrm{B}}(x^{-1},q) := \sum_{\ell\in \mathrm{B} Q^{V}+b}q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)}\prod_{v\in V}x_v^{-\ell_v}. \label{Theta}$$ In particular, in case $G = SU(N)$, this takes the following simple form : $$\label{Zhatb} \hat{Z}_b^{SU(N)}(Y;q) := \pm \frac{q^{\frac{3\sigma -\mathrm{Tr}\,\mathrm{B}}{2}\frac{N^3-N}{12}}}{|W|^{b_1(\Gamma)}} {\rm v.p.}\oint_{|x_{vi}|=1}\prod_{v\in V}\prod_{1\leq i\leq N-1}\frac{dx_{vi}}{2\pi i x_{vi}}\, F_{3d}(x)\Theta_{2d}^{b}(x,q)$$ with $$\begin{aligned} F_{3d}(x) &:= \prod_{v\in V}\left(\sum_{w\in W}(-1)^{l(w)} \prod_{1\leq i\leq N-1}x_{vi}^{(\omega_i,w(\rho))}\right)^{2-\deg v}\nonumber\\ &= \prod_{v\in V}\left( \prod_{1\leq i < j\leq N}(y_{vi}^{1/2}y_{vj}^{-1/2}-y_{vi}^{-1/2}y_{vj}^{1/2}) \right)^{2-\deg v}\\ \Theta_{2d}^{b}(x,q)&:=\sum_{\ell\in \mathrm{B} Q^{V}+b}q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)}\prod_{v\in V}\prod_{1\leq i\leq N-1}x_{vi}^{-(\omega_i,\ell_{v})}, \end{aligned}$$ where $x_{i} = \frac{y_{i}}{y_{i+1}}$. Here “v.p.” denotes the principal value integral. That is, taking average over $W$ number of deformed contours, each corresponding to a Weyl chamber. For instance, the deformed contour corresponding to a permutation $\sigma\in W \cong S_{N}$ is $$|y_{\sigma(1)}| < |y_{\sigma(2)}| < \cdots < |y_{\sigma(N)}|. \label{SU(N)contour}$$ These $\hat{Z}_b^G$ are topological invariants. $\hat{Z}_b^G$ defined above is invariant under Neumann moves (a.k.a. 3d Kirby moves). The proof is analogous to that of the $SU(2)$ version. See Proposition 4.6 in [@GM]. Examples and explicit calculations ---------------------------------- [**Comparison with Hee-Joong Chung’s paper.**]{} We find agreement with Chung’s result [@C] for many examples. (In the following, we present $\hat{Z}$’s up to $\cong$ equivalence.) - $Y = M(-1;\frac{1}{2},\frac{1}{3},\frac{1}{7})$. In this case $|H_1(Y)| = 1$ and there’s only one homological block. $G$ $\hat{Z}_0(Y)$ --------- ---------------------------------------------------------------------------------------------- $SU(2)$ $1 -q -q^5 +q^{10} -q^{11} +q^{18} +q^{30} -q^{41} +q^{43} -q^{56} -q^{76} +q^{93} -\cdots$ $SU(3)$ $1 -2q +2q^3 +q^4 -2q^5 -2q^8 + 4q^9 + 2q^{10} - 4q^{11} +2q^{13} -6q^{14} +2q^{15} -\cdots$ $SU(4)$ $1 -3q +q^2 +3q^3 -3q^5 -q^6 -q^7 -5q^8 +15q^9 +5q^{10} -11q^{11} -q^{12} +\cdots$ - $Y = M(-1;\frac{1}{2},\frac{1}{5},\frac{2}{7})$. Again, $|H_1(Y)| = 1$ and there’s only one homological block. $G$ $\hat{Z}_0(Y)$ --------- ------------------------------------------------------------------------------------------------------ $SU(2)$ $1 -q^{3} -q^{5} +q^{12} -q^{23} +q^{36} +q^{42} -q^{59} +q^{81} -q^{104} -q^{114} +q^{141} -\cdots$ $SU(3)$ $1 -2q^3 -2q^5 +2q^6 +2q^9 +q^{12} -2q^{14} +2q^{15} -2q^{18} -3q^{20} +6q^{21} -4q^{23} -\cdots$ $SU(4)$ $1 -3q^3 -3q^5 +5q^6 -q^7 +2q^8 +3q^9 -q^{10} -q^{12} +2q^{13} -6q^{14} +2q^{15} -\cdots$ - $Y = M(-1;\frac{1}{3},\frac{1}{5},\frac{3}{7})$. In this case $|H_1(Y)| = 4$. $G$ $\hat{Z}_b(Y)$ --------- ---------------------------------------------------------------------------------------------------------------------------- $SU(2)$ $1 +q^4 +q^{16} -q^{68} +q^{144} -q^{260} -q^{320} -q^{356} +q^{484} +q^{528} +q^{612} -q^{832} +\cdots$ $-\frac{1}{2}q^{15/4}(1 +q^6 +q^{10} +q^{12} -q^{44} -q^{48} -q^{58} -q^{88} +q^{122} +q^{164} +q^{182} +q^{190} -\cdots)$ $q^{13/2}(1 -q^{32} -q^{56} -q^{72} +q^{136} +q^{160} +q^{208} -q^{344} +q^{496} -q^{696} -q^{792} -q^{848} +\cdots)$ $SU(3)$ $1 +3q^4 +2q^{12} +3q^{16} +2q^{28} +2q^{48} +2q^{52} +q^{64} +4q^{68} +4q^{80} +4q^{92} - \cdots$ $-\frac{1}{6}q^{-7/4}(2 +2q +2q^3 -4q^5 +2q^6 -2q^7 +4q^9 +2q^{10} +4q^{12} +2q^{13} -2q^{14} +\cdots)$ $-\frac{1}{3}q^{-7}(1 +2q^6 -2q^8 +2q^{12} -2q^{14} +2q^{18} -2q^{20} +q^{24} -2q^{26} +2q^{28} +4q^{30} +\cdots)$ $\frac{1}{3}q^{-21/4}(1 +q -q^2 +q^4 -q^5 +q^7 +q^{10} +q^{13} -2q^{14} -q^{15} +2q^{16} +q^{19} +\cdots) \times 2$ $SU(4)$ $1 +q^{12} +8q^{16} +3q^{20} +16q^{24} +11q^{28} +15q^{32} +4q^{36} +26q^{40} +5q^{44} +\cdots$ $-\frac{1}{12}q^{-15/4}(1 +2q -2q^2 +2q^4 -2q^5 +q^6 +4q^7 -2q^8 +3q^{10} +6q^{13} -\cdots)$ $\frac{1}{6}q^{-1}(2 -2q^2 -2q^4 -3q^6 -2q^8 -5q^{10} -14q^{12} -5q^{14} -4q^{16} -12q^{18} +\cdots)$ $\frac{1}{12}q^{-1/4}(2 +2q -2q^2 +2q^3 +q^4 -2q^5 +6q^6 +2q^7 +8q^8 +6q^9 -\cdots) \times 2$ $-q^{3/2}(1 +3q^4 +2q^8 +6q^{12} +6q^{16} +4q^{20} +9q^{24} +9q^{28} +11q^{32} +9q^{36} +\cdots) \times 2$ $q^{-2}(1 +2q^4 +3q^8 +6q^{16} +6q^{20} +11q^{24} +17q^{32} +10q^{36} +9q^{40} +14q^{48} +\cdots)$ $-\frac{1}{12}q^{-15/4}(2 +q -2q^2 +q^3 +2q^4 -q^5 +4q^6 +4q^7 +2q^8 -2q^9 +\cdots)$ $\frac{1}{6}q^{-11/2}(1 +3q^6 -4q^8 +7q^{12} -2q^{14} -q^{16} +6q^{18} +3q^{20} +7q^{22} +4q^{24} +\cdots)$ Observe that [@C] agrees with our example computations except in the $SU(4)$ case of the last example. Our homological blocks are more refined in a sense that Chung’s $\hat{Z}_{1}$ is the sum of our 2nd and 7th blocks and Chung’s $\hat{Z}_{3}$ is the sum of our 1st and 6th blocks. This example illustrates that in general we can’t simply decompose $Z_a$ into $\hat{Z}_b$’s by just collecting terms whose $q$-powers differ by an integer.[^2] [**Other examples and higher rank false theta functions.** ]{} - $Y=S_0^3(K_n)$. The 0-surgery on twist knots are probably the simplest examples. For instance, $G$ $\hat{Z}_0(S_0^3(\mathbf{5}_2))$ --------- ---------------------------------------------------------------------------------------------------------------- $SU(2)$ $\frac{1}{2!}(1 -q +q^3 -q^6 +q^{10} -q^{15} +q^{21} -q^{28} +q^{36} -q^{45} +q^{55} -q^{66} +q^{78} -\cdots)$ $SU(3)$ $\frac{1}{3!}(1 -2q +2q^3 +q^4 -4q^6 +2q^9 +2q^{10} +q^{12} -2q^{13} -4q^{15} +2q^{18} +2q^{19} +\cdots)$ $SU(4)$ $\frac{1}{4!}(1 -3q +q^2 +4q^3 -2q^4 +q^5 -5q^6 -2q^7 +3q^8 +2q^9 +9q^{10} -2q^{11} -\cdots)$ $SU(5)$ $\frac{1}{5!}(1 -4q +3q^2 +6q^3 -7q^4 -2q^5 +2q^7 -2q^8 +6q^9 +15q^{10} -12q^{11} -23q^{12} +\cdots)$ Indeed, for every positive twist knot $K_p$ the following is easy to deduce from our definition (\[integralZhat\]). \[twist knot prop\] $$\label{twist knot Zhat} \hat{Z}_0^G(S_0^3(K_p)) \cong \frac{1}{|W|}\sum_{\ell\in P_+\cap (Q+\rho)} N_\ell \sum_{w\in W}(-1)^{l(w)}q^{\frac{1}{2}||\sqrt{p}\ell - \frac{1}{\sqrt{p}}w(\rho)||^2} =: \frac{1}{|W|}\chi_{p,\rho}$$ where $$N_\ell := \sum_{w\in W}(-1)^{l(w)}K(w(\ell))$$ and $K(\beta)$ denotes the Kostant partition function.[^3] Note that $\chi_{p,\rho}$ is exactly the higher rank false theta function (a character of the log-VOA $W^0(p)_Q$) given in equation (1.2) of [@BM]! Similarly for double twist knots $K_{m,n}$ with $m,n>0$,[^4] $$\label{double twist knot Zhat} \hat{Z}_0^G(S_0^3(K_{m,n})) \cong \frac{1}{|W|}\chi_{m,\rho}\chi_{n,\rho}.$$ The 0-surgery on $K_p$ has a simple plumbing description as shown in Figure \[double twist knot figure\]. (-2,0) node\[above\][$m$]{} circle(0.5ex)– (0,0) node\[above\][$0$]{} circle(0.5ex) (2,0) node\[above\][$0$]{} circle(0.5ex)– (4,0) node\[above\][$n$]{} circle(0.5ex); (0,0) edge\[bend left\] node\[midway,above\][$+$]{} (2,0); (0,0) edge\[bend right\] node\[midway,below\][$-$]{} (2,0); (7,0) node\[above\][$-1$]{} circle(0.5ex) (9,0) node\[above\][$0$]{} circle(0.5ex)– (11,0) node\[above\][$p$]{} circle(0.5ex); (7,0) edge\[bend left\] node\[midway,above\][$+$]{} (9,0); (7,0) edge\[bend right\] node\[midway,below\][$-$]{} (9,0); The linking matrix and its inverse are $$\mathrm{B} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & p\end{pmatrix} \quad,\quad \mathrm{B}^{-1} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -p & 1 \\ 0 & 1 & 0\end{pmatrix}$$ There is a single trivalent vertex with $0$ framing. This contributes the following factor in $F_{3d}(x)$ : $$\qty(\sum_{w\in W}(-1)^{l(w)}x_0^{w(\rho)})^{-1} = \frac{1}{|W|}\sum_{\ell_0\in P_+ \cap (Q+\rho)}N_{\ell_0} \sum_{w\in W}(-1)^{l(w)}x_0^{w(\ell_0)}.$$ For $\ell = (0,\ell_0,\ell_p)^t$, $$q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)} = q^{\frac{1}{2}||\sqrt{p}\ell_0 - \frac{1}{\sqrt{p}}\ell_p||^2 -\frac{1}{2p}||\ell_p||^2}.$$ Applying (\[integralZhat\]), it is straightforward to get (\[twist knot Zhat\]). Using a plumbing description of the 0-surgery on $K_{m,n}$ (Figure \[double twist knot figure\]), it is easy to derive (\[double twist knot Zhat\]) as well. - $Y=\Sigma(p,q,r)$. For convenience let’s define the following notation for higher rank false theta functions : $$\chi_{p,\beta}^{G} := \sum_{\ell\in P_+\cap (Q+\rho)} N_\ell \sum_{w\in W}(-1)^{l(w)}q^{\frac{1}{2}||\sqrt{p}\ell - \frac{1}{\sqrt{p}}w(\beta)||^2}$$ Note that for $SU(2)$, this notation is related to the usual notation of false theta functions as follows : $$\chi_{p,n\rho}^{SU(2)} = \Psi_{p,p-n},\text{ for }n = 1, \cdots, p-1.$$ For every Brieskorn sphere $Y = \Sigma(p,q,r)$ with $0 < p < q < r$ pairwise relatively prime, we have \[Brieskorn prop\] $$\hat{Z}_0^G(\Sigma(p,q,r)) \cong \sum_{(w_1,w_2)\in W^2}(-1)^{l(w_1w_2)}\chi_{pqr,qr\rho + prw_1(\rho) + pqw_2(\rho)}$$ That is, it is a sum of $|W|^2$ number of higher rank false theta functions.[^5] The proof is analogous to that of Proposition 4.8 in [@GM]. Note that we didn’t have to treat $\Sigma(2,3,5)$ separately. In this sense, using $\chi_{p,\beta}$ as false theta functions is more natural than using $\Psi_{p,n}$. - $Y = M(a_0;\frac{a_1}{b_1},\frac{a_2}{b_2},\frac{a_3}{b_3})$. Let $b_1,b_2,b_3>0$ and assume that $Y$ has negative orbifold number; i.e. $$e = a_0 + \sum_{i=1}^{3}\frac{a_i}{b_i} < 0.$$ Assume further that the central meridian is trivial in homology; i.e. $$e\, \mathrm{lcm}(b_1,b_2,b_3) = -1.$$ Then their $\hat{Z}_b$’s can be expressed as signed sum of higher rank false theta functions : \[some Seifert prop\] $$\label{some Seifert Zhat} \hat{Z}_b^G(M(a_0;\frac{a_1}{b_1},\frac{a_2}{b_2},\frac{a_3}{b_3})) \cong \sum_{(w_1,w_2)\in W^2}\mathbf{1}_b(w_1,w_2) (-1)^{l(w_1w_2)}\chi_{\frac{b_1b_2b_3}{|H_1|},\frac{b_2b_3}{|H_1|}\rho + \frac{b_1b_3}{|H_1|}w_1(\rho) + \frac{b_1b_2}{|H_1|}w_2(\rho)}$$ where $$\mathbf{1}_b(w_1,w_2) := \begin{cases} 1 &\text{ if } \ell(\rho,\rho,w_1(\rho),w_2(\rho)) \in \mathrm{B}Q^V+b\\ 0 &\text{ otherwise}\end{cases}$$ Observe that Proposition \[some Seifert prop\] is a slight generalization of Proposition \[Brieskorn prop\]. $Y$ can be described as a star-shaped plumbing with 3 legs. The only vertices whose degree is not 2 are the central vertex and the terminal vertices. Denote by $\ell(\ell_0,\ell_1,\ell_2,\ell_3)$ an element $\ell\in \mathrm{B}Q^V + b$ such that $$\ell_v = \begin{cases} \ell_0 & v \text{ is the central vertex}\\ \ell_1, \ell_2, \ell_3 & v \text{ is the corresponding terminal vertex} \\ 0 & \text{Otherwise}\end{cases}$$ Then for any $\ell$ with $\ell_1,\ell_2,\ell_3 \in W(\rho)$, $$q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)} = q^{\frac{1}{2|H_1|}||\sqrt{b_1b_2b_3}\ell_0 - \frac{1}{\sqrt{b_1b_2b_3}}(b_2b_3\ell_1 + b_3b_1\ell_2 + b_1b_2\ell_3)||^2 + C}$$ for some constant $C$ independent of $\ell$. Applying the definition (\[integralZhat\]), it is straightforward to obtain (\[some Seifert Zhat\]). Note that the assumption $e\,\text{lcm}(b_1,b_2,b_3)=-1$ was introduced so that $$\ell(\rho,\rho,w_1(\rho),w_2(\rho))\in \mathrm{B}Q^V+b \Leftrightarrow \ell(\rho + Q,\rho,w_1(\rho),w_2(\rho))\in \mathrm{B}Q^V+b$$ Naturally, there are some open questions regarding modularity of these $q$-series : What are the quantum modularity properties of the higher rank $\hat{Z}$? More specifically we can ask What are the “mock side” of the higher rank false theta functions? (i.e. what is the higher rank $\hat{Z}$ of the orientation reversed 3-manifold?) Higher rank $F_K$ ================= Review of $SU(2)$ case ---------------------- In [@GM], a new knot invariant $F_K(x,q)$ was defined for weakly negative definite knots and conjectured for all knots. Roughly, $F_K$ is a series $$F_K(x,q) = \frac{1}{2!}\sum_{m=1}^{\infty} (x^{m/2}-x^{-m/2})f_m(q)$$ where $m$ is odd and each $f_m(q)$ is a Laurant series with $\mathbb{Z}$-coefficient. One of the most important property of $F_K(x,q)$ is that $$F_K(x,q) \xrightarrow{q\rightarrow 1} \frac{x^{1/2}-x^{-1/2}}{\Delta_K(x)}$$ where $\Delta_K(x) = \nabla_K(x^{1/2}-x^{-1/2})$ is the Alexander polynomial of $K$, and $\frac{1}{\Delta_K(x)}$ should be understood as an average over expansions near $x=0$ and $x=\infty$. Higher rank $F_K$ ----------------- Let’s study higher rank $F_K(\mathbf{x},q)$. The higher rank analog of $F_K$ should be of the following form : $$F_{K}^G(\mathbf{x},q) = \frac{1}{|W|}\sum_{\beta\in P_+\cap (Q+\rho)}f_\beta^G(q)\sum_{w\in W}(-1)^{l(w)}x^{w(\beta)}.$$ This can be computed by ’t Hooft resumming the (higher rank) colored Jones polynomials and should be annihilated by the (higher rank) quantum A-polynomial. [**Right-handed trefoil with $G = SU(3)$.**]{} The first few $f_\beta^{SU(3)}$ are (up to overall sign and $q$-power) $$\begin{aligned} f_{(1,1)} &= -q,\quad f_{(4,1)} = -q^2,\quad f_{(5,2)} = -2q^3,\quad f_{(7,1)} = -q^4,\quad f_{(5,5)} = q^5,\quad \nonumber\\ f_{(7,4)} &= 2q^6,\quad f_{(10,1)} = -q^7,\quad f_{(8,5)} = q^8,\quad f_{(11,2)} = -2q^9,\quad f_{(7,7)} = -q^9,\quad \nonumber\\ f_{(13,1)} &= -q^{11},\quad f_{(11,5)} = q^{12},\quad \cdots\end{aligned}$$ where we have written $\beta$ in the fundamental weights basis. (Because $f_{(m,n)} = f_{(n,m)}$, we have only written those terms with $m \geq n$.) The $q$-power of this $f_\beta$ is, up to overall constant, $$\frac{(\beta,\beta)}{12}.\label{q-power}$$ In the $q\rightarrow 1$ limit we have $$F_{\mathbf{3}_1^r}^{SU(3)}(x_1,x_2,1) = \frac{x_1^{1/2}-x_1^{-1/2}}{x_1+x_1^{-1}-1}\frac{x_2^{1/2}-x_2^{-1/2}}{x_2+x_2^{-1}-1}\frac{x_1^{1/2}x_2^{1/2}-x_1^{-1/2}x_2^{-1/2}}{x_1x_2+x_1^{-1}x_2^{-1}-1}.\label{classicallimit}$$ Alternatively, we can start with $SU(3)$ colored Jones polynomials of the trefoil [@GMV] and ‘t Hooft resum : $$J_{\mathbf{3}_1^r,n_1,n_2}^{SU(3)}(q=e^\hbar) \xrightarrow{\text{`t Hooft resum}}\xrightarrow{\hbar \rightarrow 0} \frac{1}{x_1+x_1^{-1}-1}\frac{1}{x_2+x_2^{-1}-1}\frac{1}{x_1x_2+x_1^{-1}x_2^{-1}-1}.$$ Note that (\[q-power\]) and (\[classicallimit\]) completely characterize $F_{\mathbf{3}_1^r}(x_1,x_2,q)$. In general the following version of Melvin-Morton conjecture was proven by D. Bar-Natan and S. Garoufalidis in [@BG] : For arbitrary gauge group $G$ of rank $r$, the semi-classical limit of $F_K(x_1,\cdots,x_r,q)$ should be $$\boxed{ F_K^G(x_1, \cdots, x_r,1) = \prod_{\alpha\in \Delta^+}\frac{x^{\alpha/2}-x^{-\alpha/2}}{\Delta_K(x^\alpha)} }$$ For torus knots, we found the following explicit expression : For $K = T_{s,t}$, $f_\beta(q)$ is a monomial of degree $\frac{(\beta,\beta)}{2st}$, up to an overall $q$-power. More precisely, $$\label{torus knot F_K} F_{T_{s,t}}^G \cong \frac{1}{|W|}\sum_{\beta\in P_+ \cap (Q+\rho)}\sum_{w\in W}(-1)^{l(w)}x^{w(\beta)}\sum_{(w_1,w_2)\in W^2}(-1)^{l(w_1w_2)}\mathbf{1}(\beta,w_1,w_2)N_{\frac{1}{st}(\beta + tw_1(\rho)+sw_2(\rho))} q^{\frac{(\beta,\beta)}{2st}}$$ where $$\mathbf{1}(\beta,w_1,w_2):=\begin{cases} 1 &\text{ if }\frac{1}{st}(\beta + tw_1(\rho) + sw_2(\rho)) \in P_+\cap (Q+\rho)\\ 0&\text{ otherwise}\end{cases}$$ This can be derived either directly from (\[integralZhat\]) by using plumbing description or by reverse-engineering using the higher rank surgery formula that we discuss below. Here we present a direct derivation. Recall from [@GM] that the complement of $T_{s,t}$ has a plumbing description as in Figure \[torus knot figure\], where $0<t'<t$, $0<s'<s$ are chosen such that $st'\equiv -1 (\text{mod }t)$ and $ts' \equiv -1 (\text{mod }s)$. (-2,0) node\[above\][$-\frac{t}{t'}$]{} circle(0.5ex)– (0,0) node\[above\][$-1$]{} circle(0.5ex) – (2,0) node\[above\][$-\frac{s}{s'}$]{} circle(0.5ex); (0,-2) node\[below\][$-st$]{} circle(0.5ex)–(0,0); The linking matrix is $$\mathrm{B} = \begin{pmatrix} -st & 1 & 0 & 0 \\ 1 & -1 & 1 & 1 \\ 0 & 1 & (-\frac{t}{t'}) & 0 \\ 0 & 1 & 0 & (-\frac{s}{s'})\end{pmatrix}$$ where $(-\frac{t}{t'})$ and $(-\frac{s}{s'})$ should be understood as block matrices corresponding to the continued fractions. To compute the integral (\[integralZhat\]) with $x_{-st}$ left unintegrated, we just have to replace the theta function $\Theta^{-\mathrm{B}}(x^{-1},q)$ with $$\Theta^{-\mathrm{B}'}(x^{-1},q) \cong \sum_{\alpha\in Q^{V'}}q^{-\frac{1}{2}(\alpha,\mathrm{B}'\alpha)-(\alpha,\delta)}\prod_{v\in V'}x_v^{-(\mathrm{B}'\alpha + \delta)}\cdot x_{-st}^{-\alpha_{-1}-\rho}$$ where $V' = V \setminus \{v_{-st}\}$ and $\mathrm{B}'$ is the corresponding sub-linking matrix. Set $\beta = -\alpha_{-1}-\rho$. We need to multiply $\Theta^{-\mathrm{B}'}(x^{-1},q)$ with $$\prod_{v\in V'}\prod_{1\leq i\leq r}\qty(\sum_{w\in W}(-1)^{l(w)}x_v^{w(\rho)})^{2-\deg v}$$ and take the constant term with respect to variables $x_v$, $v\in V'$. As $2-\deg v$ is non-zero for only 3 vertices (the central vertex $v_{-1}$ and the 2 terminal vertices) it is pretty easy to compute. The only contributions come from those $\alpha$’s such that $\mathrm{B}'\alpha + \delta$ takes values $w_1(\rho), w_2(\rho)$ on the terminal vertices for some $w_1,w_2\in W$, a value in $Q+\rho$ in the central vertex, and $0$ on all the other vertices. Using simple linear algebra, it is easy to check that for those $\alpha$’s, $$q^{-\frac{1}{2}(\alpha,\mathrm{B}'\alpha) - (\alpha,\delta)} = q^{\frac{(\beta,\beta)}{2st}+C}$$ for some constant $C$ independent of $\alpha$, and that $\frac{1}{st}(\beta + tw_1(\rho) + sw_2(\rho))$ is the value of $\mathrm{B}'\alpha + \delta$ on the central vertex. This proves (\[torus knot F\_K\]). Just as in [@GM], we can use surgery formula for these higher rank $F_K$ to compute higher rank $\hat{Z}_b(S_{p/r}^3(K))$. For instance, surgery on $\mathbf{3}_1^r$ gives us the following $\hat{Z}$’s ($SU(3)$ analog of the table 1 in [@GM]) : $r$ $S_{-1/r}^3(\mathbf{3}_1^r)$ $\hat{Z}_0^{SU(3)}(S_{-1/r}^3(\mathbf{3}_1^r))$ ----- ------------------------------ ---------------------------------------------------------------------------------------------------- $1$ $\Sigma(2,3,7)$ $1 -2q +2q^3 +q^4 -2q^5 -2q^8 + 4q^9 + 2q^{10} - 4q^{11} +2q^{13} -6q^{14} +2q^{15} -\cdots$ $2$ $\Sigma(2,3,13)$ $1 -2q +2q^3 -q^4 +2q^{10} -2q^{11} -2q^{14} +2q^{16} +2q^{19} -2q^{20} +4q^{21} -4q^{23} -\cdots$ $3$ $\Sigma(2,3,19)$ $1 -2q +2q^3 -q^4 +2q^{16} -2q^{17} -2q^{20} +2q^{22} +2q^{25} -2q^{26} +4q^{33} -4q^{35} -\cdots$ $4$ $\Sigma(2,3,25)$ $1 -2q +2q^3 -q^4 +2q^{22} -2q^{23} -2q^{26} +2q^{28} +2q^{31} -2q^{32} +4q^{45} -4q^{47} -\cdots$ $5$ $\Sigma(2,3,31)$ $1 -2q +2q^3 -q^4 +2q^{28} -2q^{29} -2q^{32} +2q^{34} +2q^{37} -2q^{38} +4q^{57} -4q^{59} -\cdots$ $r$ $\Sigma(2,3,6r+1)$ $\sum_{(w_1,w_2)\in W^2}(-1)^{l(w_1w_2)}\chi_{36r+6, 3(6r+1)w_1(\rho)+ 2(6r+1)w_2(\rho)+ 6\rho}$ In fact it is easy to check that for $K=T_{s,t}$, $$\begin{aligned} \mathcal{L}_{-1/r}\left[\prod_{\alpha\in \Delta^+}(x^{\frac{\alpha}{2r}}-x^{-\frac{\alpha}{2r}})F_K(\mathbf{x},q) \right] &\cong \sum_{(w_1,w_2)\in W^2}(-1)^{l(w_1w_2)}\chi_{st(rst+1),t(rst+1)w_1(\rho) + s(rst+1)w_2(\rho) +st\rho}\\ &\cong \hat{Z}_0^G(\Sigma(s,t,rst+1))\end{aligned}$$ We conjecture the following surgery formula (analogous to Conjecture 1.7 of [@GM]) : Let $K\subset S^3$ be a knot. Then $$\boxed{ \hat{Z}_b^G(S_{p/r}^3(K)) \cong \mathcal{L}_{p/r}^{(b)}\left[\prod_{\alpha\in \Delta^+}(x^{\frac{\alpha}{2r}}-x^{-\frac{\alpha}{2r}})F_K^G(\mathbf{x},q) \right] }$$ whenever the RHS makes sense. Moreover we conjecture that our 0-surgery formula in [@CGPS] holds for higher rank as well : Let $K\subset S^3$ be a knot. Then $$\boxed{ \hat{Z}_0^G(S_{0}^3(K)) \cong \frac{1}{|W|}f_\rho^G(K) }$$ Symmetric representations and large $N$ ======================================= Specialization to symmetric representations ------------------------------------------- In this section we present a specialization of $F_K^G({\bf x},q)$ to symmetric representations. We restrict our attention to $G=SU(N)$. Although the unreduced version seems to be better-behaving, to get $F_K$ with symmetric colorings (symmetric powers of the defining representation), we need to use the reduced version : $$F_K^{\rm red}(\mathbf{x},q) := \frac{1}{|W|}\sum_{\beta\in P_+ \cap (Q+\rho)}f_\beta(q)\frac{\sum_{w\in W}(-1)^{l(w)}x^{w(\beta)}}{\sum_{w\in W}(-1)^{l(w)}x^{w(\rho)}}$$ In particular, the symmetrically colored $F_K$ corresponds to the following specialization : $$F_K^{\rm sym}(x,q) := F_K^{\rm red}((x,q,\cdots,q),q)$$ That is, we set $x_2 = \cdots = x_r = q$. A version of quantum volume conjecture [@FGS] states that this should be annihilated by the symmetrically colored quantum A-polynomial[^6] : $$\label{QuantVolConj} \boxed{ \hat{A}^{\rm sym}(\hat{x},\hat{y},q)F_K^{\rm sym}(x,q) = 0 }$$ [**Right-handed trefoil.**]{} For the right-handed trefoil, $F_{\mathbf{3}_1^r}^{\rm sym}(x,q)$ for $SU(N)$ with the first few values of $N$ look like the following : - $SU(2)$ $$\begin{aligned} F_{\mathbf{3}_1^r}^{\rm sym}(x,q) &\cong \frac{1}{2}\left[ (-q + q^2 + q^3 - q^6 -q^8 + q^{13} + q^{16} - \cdots) \right.\\ &\quad + (x + x^{-1})(q^2 + q^3 - q^6 -q^8 + q^{13} + q^{16} - \cdots)\\ &\quad + (x^2 + x^{-2})(q^2 + q^3 - q^6 -q^8 + q^{13} + q^{16} - \cdots)\\ &\quad + (x^3 + x^{-3})(q^3 - q^6 -q^8 + q^{13} + q^{16} - \cdots)\\ &\quad + (x^4 + x^{-4})(-q^6 - q^8 + q^{13} + q^{16} - \cdots)\\ &\quad\left. + \cdots \right] \end{aligned}$$ - $SU(3)$ $$\begin{aligned} F_{\mathbf{3}_1^r}^{\rm sym}(x,q) &\cong \frac{1}{2}\left[(-2q -2q^2 +2q^4 +4q^5 +4q^6 +4q^7 +2q^8 -2q^{10} -4q^{11} -6q^{12} +\cdots) \right.\\ &\quad + (q^{1/2}x+q^{-1/2}x^{-1})q^{1/2}(-1 -2q -q^2 +q^3 +3q^4 +4q^5 +4q^6 +3q^7 +q^8 -q^9 +\cdots)\\ &\quad + (qx^2 + q^{-1}x^{-2})(-q -q^2 +2q^4 +3q^5 +4q^6 +3q^7 + 2q^8 -2q^{10} +\cdots)\\ &\quad + (q^{3/2}x^3 + q^{-3/2}x^{-3})q^{1/2}(q^3 +2q^4 +3q^5 +3q^6 +2q^7 +q^8 +\cdots)\\ &\quad + (q^2x^4 + q^{-2}x^{-4})(q^3 +q^4 +2q^5 +2q^6 +2q^7 +q^8 +\cdots)\\ &\quad\left. +\cdots \right] \end{aligned}$$ - $SU(4)$ $$\begin{aligned} F_{\mathbf{3}_1^r}^{\rm sym}(x,q) &\cong \frac{1}{2}\left[(q^{-2} +q^{-1} -2 -4q -8q^2 -7q^3 -7q^4 +\cdots) \right.\\ &\quad + (qx + q^{-1}x^{-1})(q^{-2} -1 -5q -6q^2 -8q^3 -5q^4 -2q^5 +\cdots)\\ &\quad + (q^2x^2 + q^{-2}x^{-2})(-2 -3q -6q^2 -5q^3 -5q^4 +4q^6 +\cdots)\\ &\quad + (q^3x^3 + q^{-3}x^{-3})(-q^{-1} -1 -3q -3q^2 -4q^3 -2q^4 +5q^6 +9q^7 +\cdots)\\ &\quad + (q^4x^4 + q^{-4}x^{-4})(-1 -q -2q^2 -q^3 -q^4 +2q^5 +4q^6 +8q^7 +11q^8 +\cdots)\\ &\quad\left. + \cdots \right] \end{aligned}$$ Note that the overall factor is $\frac{1}{2}$ instead of $\frac{1}{N!}$. This is due to reduction of the Weyl symmetry to $\mathbb{Z}_2$. It is easy to verify (\[QuantVolConj\]) numerically in this case, using the super-A-polynomial for the right-handed trefoil $$\hat{A}^{\rm super}(\hat{x},\hat{y},a,q) = a_0 + a_1\hat{y} + a_2\hat{y}^2$$ where $$\begin{aligned} a_0 &= -\frac{(-1+\hat{x})(-1+aq\hat{x}^2)}{a\hat{x}^3(-1+a\hat{x})(-q + a\hat{x}^2)}\\ a_1 &= \frac{(-1+a\hat{x}^2)(-a^2\hat{x}^2 + aq^3\hat{x}^2 +aq\hat{x}(1+\hat{x}+a(-1+\hat{x})\hat{x})-q^2(1+a^2\hat{x}^4))}{a^2q\hat{x}^3(-1+a\hat{x})(-q + a\hat{x}^2)}\\ a_2 &= 1\end{aligned}$$ with $a$ specialized to $q^N$. Future direction : large $N$ ---------------------------- We end with some intriguing open questions. Is there a HOMFLYPT version of $F_K$? Conjecturally, there is a HOMFLYPT version $F_K^{\rm super}(x,q,a)$ such that we can recover $F_K^{SU(N),{\rm sym}}(x,q)$ as a certain specialization (e.g. $a\rightarrow q^N$), and $$\boxed{\hat{A}^{\rm super}(\hat{x},\hat{y},a,q)F_K^{\rm super}(x,q,a) = 0.}$$ We expect to have an expansion of the form $$F_{K}^{\rm super}(x,q,a) \cong \frac{1}{2}\sum_{n\in \mathbb{Z}}(q^{-1} a^{1/2} x)^n f_n(q,a)$$ with $f_{-n}(q,a)=f_{n}(q,a)$, and that it has the following Weyl symmetry : $$F_{K}^{\rm super}(x^{-1},q,a) = F_{K}^{\rm super}(a^{-1}q^2x,q,a).$$ We can also ask another, much conjectural question regarding homology theories : What’s $\mathcal{H}_{b,{\rm BPS}}^G$ categorifying $\hat{Z}_b^G(q)$? Is there a family of differentials analogous to that of [@DGR]? We will pursue these questions in our future work. $\mathrm{Spin}^{T^\vee}$-structures {#spint} =================================== In this section we define a generalization of $\mathrm{Spin}^c$-structures which we will call $\mathrm{Spin}^{T^\vee}$-structures. These are what the labels $b$ of $\hat{Z}_b^G$ are geometrically. Let $G$ be a Lie group and let $T < G$ be a maximal torus, and let $T^\vee \cong \mathfrak{h}^*/P$ be its dual. First, the group $\mathrm{Spin}^{T^\vee}(n)$ is defined to be $$\mathrm{Spin}^{T^\vee}(n) := \mathrm{Spin}(n) \times_{\mathbb{Z}_2}T^\vee.$$ A $\mathrm{Spin}^{T^\vee}$-structure $\mathfrak{s}$ on an $n$-manifold $M^n$ is a principal $\mathrm{Spin}^{T^\vee}(n)$-bundle with an isomorphism between the natural $SO(n)$-bundle associated to $\mathfrak{s}$ and the frame bundle of $M^n$. Note that the Weyl group naturally acts on $\mathrm{Spin}^{T^\vee}(n)$. It is important that we regard $T^\vee \cong \mathfrak{h}^*/P$ as a variety, not just a topological torus. There’s a natural first Chern class map $$\begin{aligned} \mathrm{Spin}^{T^\vee}(M) &\rightarrow H^2(M;P) \\ \mathfrak{s}&\mapsto c_1(\mathfrak{s}).\end{aligned}$$ The following propositions are analogous to those of $\mathrm{Spin}^c$-structures. $\mathrm{Spin}^{T^\vee}(M)$ is affinely isomorphic to $H^2(M;Q)$ On a 4-manifold $W$ with boundary $Y$, $\mathrm{Spin}^{T^\vee}(W)$ is canonically isomorphic to the set $\mathcal{C}_W\subseteq H^2(W;P)$ of characteristic covectors of $H_2(W,Y;Q)$; i.e. $\nu\in \mathcal{C}_W\subseteq H^2(W;P)$ iff $$(\nu,x) = \frac{1}{2}\langle x,x\rangle\,\mathrm{mod}\,2$$ for any $x\in H_2(W,Y;Q)$. Good old Gauss sum reciprocity {#Gauss} ============================== In this section we derive a formula which generalizes appendix A of [@GPPV] to any graph manifold with weakly negative definite linking matrix, with arbitrary gauge group $G$. Let $Y$ be a graph manifold obtained by a plumbing on a connected decorated graph $\Gamma$. (Each vertex $v$ of $\Gamma$ is decorated by an integer $a_v$ ‘framing’ and a nonnegative integer $g_v$ ‘genus’. We have set $g_v=0$ in Definition \[integralZhat\].) Then the WRT invariant $Z_{G_k}(Y)$ can be computed by [^7] $$Z_{G_k}(Y) \cong \sum_{\rm colorings}\prod_{v\in V}\mathcal{V}_v\prod_{e\in E}\mathcal{E}_e \label{eq2.1}$$ where the vertex and the edge factors are[^8] $$\mathcal{V} = t_{\lambda\lambda}^{a_v} s_{\rho\lambda}^{2-2g_v-\deg v}$$ $$\mathcal{E} = s_{\mu\lambda}.$$ Here the $s, t$ matrices are as usual [^9] $$s_{\lambda \mu} = \frac{i^{|\Delta_+|}}{|P/kQ^\vee|^{1/2}}\sum_{w\in W}(-1)^{l(w)}q^{(w(\lambda),\mu)},$$ $$t_{\lambda \mu} = \delta_{\lambda \mu} q^{\frac{1}{2}(\lambda,\lambda)}q^{-\frac{1}{2}(\rho,\rho)}$$ with $q=e^{\frac{2\pi i}{m k}}$ and set of (shifted) colors being $$\lambda,\mu\in C = \{\lambda\in P_+ + \rho \,\vert\, (\lambda,\theta^\vee) < k\}.$$ Then these $s,t$ matrices are invariant (up to sign) under the action of the affine Weyl group $$W^a = W \ltimes kQ^\vee.$$ In fact, $C$ is simply the fundamental domain $P/W^a$. Using this fact, we can manipulate the form of $Z_{G_k}(Y)$ to write it in a Gauss sum reciprocity-friendly way : $$\begin{aligned} (\ref{eq2.1}) &= \frac{1}{|W^V|} \sum_{\text{coloring}\in W(C)^{V}}\prod_{v\in V}\mathcal{V}_v\prod_{e\in E}\mathcal{E}_e \nonumber\\ &= \frac{1}{|W^V|} q^{-\frac{\sum a_i}{2}(\rho,\rho)}\left( \frac{i^{|\Delta_+|}}{|P/kQ^\vee|^{1/2}}\right)^{|V| + 1 - b_1(\Gamma)} \nonumber\\ &\quad\times \sum_{\lambda\in W(C)^{V}}\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v}q^{\frac{a_v}{2}(\lambda_v,\lambda_v)} \prod_{(u_1,u_2)\in E}\sum_{w\in W}(-1)^{l(w)}q^{(w(\lambda_{u_1}),\lambda_{u_2})} \nonumber\\ &= \frac{1}{|W|^{b_1(\Gamma)+1}} q^{-\frac{\sum a_i}{2}(\rho,\rho)}\left( \frac{i^{|\Delta_+|}}{|P/kQ^\vee|^{1/2}}\right)^{|V| + 1 - b_1(\Gamma)} \sum_{\lambda\in W(C)^{V}}\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v} \nonumber\\ &\quad \times\sum_{s\in W^{b_1(\Gamma)}}(-1)^{l(s)}q^{\frac{1}{2}(\lambda, \mathrm{B}_s\lambda)} \label{midstep}\end{aligned}$$ Here, the self-adjoint matrix $\mathrm{B}_s$ is the symmetrization $\frac{1}{2}(\mathrm{B}_s^0 + {\mathrm{B}_s^0}^{\dagger})$ of a bilinear form $\mathrm{B}_s^0$ on $P^{V}$ characterized by the following properties : - For each $v$, $(\mu_v,\mathrm{B}_s^0\lambda_v) = a_v(\mu_v,\lambda_v)$ - For each $u\neq v\in V$ with $(u,v)\not\in E$, $(\mu_u, \mathrm{B}_s^0 \lambda_v) = 0$ - For each $u\neq v\in V$ with $(u,v)\in E$, $(\mu_u,\mathrm{B}_s^0\lambda_v) = (\mu_u,w_{uv}(\lambda_v))$ for some $w_{uv}\in W$ - For each cycle $c=[v_1, \cdots, v_m]\in H_1(\Gamma)$, $\prod_{i=1}^{m}w_{v_i v_{i+1}} = s_c\in W$. So far we have expressed the WRT invariant in the following form : $$Z_{G_k}(Y) = \sum_{s\in W^{b_1(\Gamma)}}(-1)^{l(s)}Z_{G_k}(Y,s).$$ Each $Z_{G_k}(Y,s)$ can be considered as the contribution from almost-Abelian flat connections twisted by $s$ [@CGPS]. In the following, let’s restrict our attention to the Abelian sector ($s=\text{id}$) for simplicity. In this case $\mathrm{B}_s = \mathrm{B}$ is the usual linking matrix. To extend the range of summation $W(C)$ to $P/kQ^\vee$, i.e. to make sense of the summation even for colors upon which the action of $W^a$ is not free, we need to regularize the linear term $\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v}$. Let $\omega_1, \cdots, \omega_r\in P$ be a $\mathbb{Z}$-linear basis of $P$ (e.g. fundamental weights). We can then write $\lambda_v = \sum_{i = 1}^{r}n_{vi}\omega_i$ for some $n_{v1}, \cdots, n_{vr}\in \mathbb{Z}$. Thanks to Weyl denominator formula, we have $$\begin{aligned} \prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v} &= \prod_{v\in V} \left( \prod_{\alpha\in \Delta_+}(q^{\frac{(\lambda_v,\alpha)}{2}} - q^{-\frac{(\lambda_v,\alpha)}{2}}) \right)^{2-2g_v-\deg v} \nonumber\\ &= \prod_{v\in V} \left( \prod_{\alpha\in \Delta_+}(\prod_{1\leq i\leq r}x_{vi}^{\frac{(\omega_i,\alpha)}{2}} - \prod_{1\leq i\leq r}x_{vi}^{-\frac{(\omega_i,\alpha)}{2}}) \right)^{2-2g_v-\deg v}\bigg|_{x_{vi} = q^{n_{vi}}}\label{regularize}\end{aligned}$$ In case $\deg v > 2$, this expression can be singular only when $$\left| \prod_{1\leq i\leq r}y_i^{\frac{(\omega_i,\alpha)}{2}}\right| = 1$$ for some $\alpha\in \Delta_+$. In terms of new variables $z_i := \log |x_i|$, this is simply $$\sum_{1\leq i\leq r} (\omega_i,\alpha) z_i = 0.$$ These are precisely the walls (hyperplanes) for Weyl reflections. Deforming the origin $z_1 = \cdots = z_r = 0$ to a complement of these walls is the same as a choice of a Weyl chamber. Moreover, for each choice of such a Weyl chamber, we can expand (\[regularize\]) as a geometric series. Therefore, we can regularize the linear term by taking an average of $|W|$ number of $q$-series, each determined by a choice of a Weyl chamber. To sum up, we can re-express the linear term in the following form : $$\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v} \overset{\text{regularize}}{=\joinrel=\joinrel=} \frac{1}{|W^V|} \sum_{\ell\in \delta + Q^{V}} n_\ell \,q^{(\lambda,\ell)}$$ with $\delta_v = (2-2g_v-\deg v)\rho \mod Q$ and $n_\ell\in \mathbb{Z}$. With this regularization in hand, we extend the range of summation and apply the Gauss sum reciprocity.[^10] Let $n\in \mathbb{Z}^+$ be such that $nP\subseteq Q^\vee\subseteq P$. Then we have[^11] $$\begin{aligned} Z_{G_k}(Y,\text{id}) &\Rightarrow \frac{1}{|W|^{b_1(\Gamma)+1}} q^{-\frac{\sum a_i}{2}(\rho,\rho)}\left( \frac{i^{|\Delta_+|}}{|P/kQ^\vee|^{1/2}}\right)^{|V| + 1 - b_1(\Gamma)} \frac{1}{|(Q^\vee/nP)^V|} \nonumber\\ &\quad\times \sum_{\lambda \in P^{V}/nkP^{V}}q^{\frac{1}{2}(\lambda,\mathrm{B}\lambda)} \cdot \left(\frac{1}{|W^V|} \sum_{\ell\in \delta + Q^{V}}n_\ell q^{(\lambda,\ell)}\right) \nonumber\\ &= \frac{(-1)^{|\Delta_+||V|}}{|W|^{|V|+b_1(\Gamma)+1}}\left(\frac{i^{|\Delta_+|}}{|P/Q^\vee|^{1/2}} \right)^{-|V|-b_1(\Gamma)+1} q^{-\frac{\sum a_i}{2}(\rho,\rho)}k^{\frac{r}{2}(b_1(\Gamma) - 1)} \nonumber\\ &\quad\times \frac{e^{\frac{\pi i}{4}\sigma(\mathrm{B})}}{|\det \mathrm{B}|^{1/2}}\sum_{a\in (P^\bullet)^{V}/\mathrm{B}(P^\bullet)^{V}}e^{-\pi i k(a,\mathrm{B}^{-1}a)}\sum_{b\in (Q^{V}+\delta)/\mathrm{B}Q^{V}}e^{-2\pi i(a,\mathrm{B}^{-1}b)} \sum_{\ell\in \mathrm{B}Q^{V}+b}n_\ell q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)}\label{rawresult}\end{aligned}$$ Note that $a$ and $b$ takes values in different sets : $a\in (P^\bullet)^{V}/\mathrm{B}(P^\bullet)^{V}$ while $b\in (Q^{V}+\delta)/\mathrm{B}Q^{V}$. The $a$ labels should be understood as ‘Abelian flat connections’ and the $b$ labels are ‘$\mathrm{Spin}^{T^\vee}$-structures’ of Section \[spint\]. Generalizing the definition of $\hat{Z}_b$ in appendix A of [@GPPV], we define $\hat{Z}_b^G$ for a weakly negative definite graph manifold $Y$ with gauge group $G$ as follows : $$\hat{Z}_b^G(Y;q) :\cong |W|^{-|V|-b_1(\Gamma)}q^{-\frac{\mathrm{Tr}\,\mathrm{B}}{2}(\rho,\rho)}\sum_{\ell\in \mathrm{B}Q^{V} + b}n_\ell q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)}\in |W|^{-|V|-b_1(\Gamma)} q^{\Delta_b}\mathbb{Z}[[q]] \label{Zhatdef}$$ where $$b\in (Q^{V}+\delta)/\mathrm{B}Q^{V},$$ $$\Delta_b = -\frac{\mathrm{Tr}\,\mathrm{B}}{2}(\rho,\rho) + \min_{\ell\in \mathrm{B}Q^{V}+b}-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)\in \mathbb{Q}.$$ Here $\Delta_b$ should be thought of as a relative conformal weight among different $b$’s for the same 3-manifold. This is because we ignored the framing factor in (\[eq2.1\]), which shifts the overall $q$-power. It is easy to check that this definition is equivalent to Definition \[Zhatsgm\]. Finally, it is also clear from our derivation (\[rawresult\]) that the higher rank analog of the unfolded $S_{ab}$ matrix (appearing in Conjecture 2.6 of [@GPPV]) for the Abelian sector is simply $$S_{ab} = e^{-2\pi i (a,\mathrm{B}^{-1}b)}.$$ Of course for graph manifolds with $b_1(\Gamma)>0$ there are different sectors corresponding to each $s\in W^{b_1(\Gamma)}$, and we have to combine all of them to compare with the WRT invariant. See [@CGPS]. [^1]: By considering homology orientations, we should be able to fix the sign as well. [^2]: Still, it is possible to derive our formula (\[integralZhat\]) from Mariño’s Chern-Simons matrix model or vice versa in case of Seifert manifolds. This is because Gaussian measure is the same as Laplace transform accompanied by $S_{ab}$ matrix. [^3]: E.g. for $SU(2)$, $N_\ell^{A_1} = \text{sgn}((\ell,\alpha_1))$, and for $SU(3)$, $N^{A_2}_\ell = \mathrm{sgn}(\prod_{\alpha\in \Delta_+}(\ell,\alpha)) \min\{|(\ell,\alpha_1)|,|(\ell,\alpha_2)|\}$. [^4]: In our notation, $K_{m,n}$ denotes the double twist knot with $m$ and $n$ full twists. [^5]: That $\hat{Z}$’s for Brieskorn spheres should be expressed as sums of higher rank false theta functions was envisaged earlier in [@CCFGH]. [^6]: Of course in principle there should be quantum A-polynomials for all colors, with rank number of $x$ and $y$ variables, which annihilate $F_K^G({\bf x},q)$. However for simplicity we only consider symmetric representations here. [^7]: Up to a framing factor, which is an integer power of $\zeta^3 = e^{2\pi i \frac{(k-h^\vee)\dim G/k}{24}} = e^{\frac{\dim G}{12} \pi i} q^{-\frac{m h^\vee\dim G}{24}}$. [^8]: Here, $2-2g_v-\deg v$ should probably be understood as the Euler characteristic of the Riemann surface $\Sigma_{g_v,\deg v}$ of genus $g_v$ with $\deg v$ punctures. [^9]: We follow the convention and notations used in [@BK]. In particular, $P$ is the weight lattice, $Q$ is the root lattice, and $Q^\vee$ is the coroot lattice. [^10]: In fact, this process of extending the range of colors and regularizing it is quite subtle, and in fact this may not be a strict equality for some values of $k$. We will deal with this subtlety in more detail elsewhere. Anyway the end result will be a topological invariant. [^11]: See [@DT] for the version of Gauss sum reciprocity formula we use.

--- abstract: 'This paper explores the fundamental properties of distributed minimization of a sum of functions with each function only known to one node, and a pre-specified level of node knowledge and computational capacity. We define the optimization information each node receives from its objective function, the neighboring information each node receives from its neighbors, and the computational capacity each node can take advantage of in controlling its state. It is proven that there exist a neighboring information way and a control law that guarantee global optimal consensus if and only if the solution sets of the local objective functions admit a nonempty intersection set for fixed strongly connected graphs. Then we show that for any tolerated error, we can find a control law that guarantees global optimal consensus within this error for fixed, bidirectional, and connected graphs under mild conditions. For time-varying graphs, we show that optimal consensus can always be achieved as long as the graph is uniformly jointly strongly connected and the nonempty intersection condition holds. The results illustrate that nonempty intersection for the local optimal solution sets is a critical condition for successful distributed optimization for a large class of algorithms.' author: - 'Guodong Shi, Alexandre Proutiere and Karl Henrik Johansson[^1]' title: | **Distributed Optimization: Convergence Conditions\ from a Dynamical System Perspective[^2]** --- [**Keywords:**]{} Distributed optimization, Dynamical Systems, Multi-agent systems, Optimal consensus Introduction ============ Motivation ---------- Distributed optimization is on finding a global optimum using local information exchange and cooperative computation over a network. In such problems, there is a global objective function to be minimized, say, and each node in the network can only observe part of the objective. The update dynamics is executed through an update equation implemented in each node of the network, based on the information received from the local objective and the neighbors. The literature has not to sufficient extent studied the real meaning of “distributed" optimization, or the [*level*]{} of distribution possible for convergence. Some algorithms converge faster than others, while they depend on more information exchange and a more complex iteration rule. For a precise study of the level of distribution for optimization methods, the way nodes share information, and the computational capacity of each node should be specified. Thus, an interesting question arises: fixing the knowledge set and the computational capacity, what is the best performance of any distributed algorithm? In this paper, we investigate the fundamental performance limits of distributed algorithms when the constraints on how nodes exchange information and on their computational capacity are fixed. We address these limits from a dynamical system point of view and characterize some fundamental conditions on the global objective function for a distributed solution to exist. Related Works ------------- Distributed optimization is a classical topic in applied mathematics with several excellent textbooks, e.g., [@book1; @book2; @book3]. Assuming that some estimate of the subgradient for each component of the overall objective function can be passed over the network from one node to another via deterministic or randomized iteration, a class of subgradient-based incremental algorithms was investigated in [@solodov; @rabbat; @nedic01; @bjsiam; @ram]. A series of results were established combining consensus and subgradient computation. This idea can be traced back to 1980s to the pioneering work [@tsi]. A subgradient method for fixed undirected topology was given in [@bj08]. Then in [@nedic09], convergence bounds for time-varying graphs with various connectivity assumptions were shown. This work was then extended to a constrained optimization case in [@nedic10], where each agent is assumed to always lie in a particular convex set. Consensus and optimization were shown to be guaranteed when each node makes a projection onto its own set at each step. Following the ideas of [@nedic10], a randomized discrete-time algorithm and a deterministic continuous-time algorithm were presented for optimal consensus in [@shirandom] and [@shitac], respectively, where in both cases the goal is to form a consensus within the intersection of the optimal solution sets of the local objective functions. An augmented Lagrangian algorithm was presented for constrained optimization with directed gossip communication in [@jmf]. An alternative approach was presented in [@lu1], where the nodes keep their gradient sum equal to zero during the iteration by utilizing gossiping. Dynamical system solutions to distributed optimization problem have been considered for more than fifty years. The Arrow-Hurwicz-Uzawa flow was shown to converge to the set of saddle points for a constrained convex optimization problem [@ahu]. In [@brockett], a simple and elegant continuous-time protocol was presented to solve linear programming problems. More recently, in [@elia], a continuous-time solution having second-order node dynamics was proposed for solving distributed optimization problems for fixed bidirectional graphs. In [@eben], a smooth vector field was shown to be able to drive the system trajectory to converge to the saddle point of the Lagrangian of a convex and constrained optimization problem. In [@shitac], a network of first-order dynamical system was proposed to solve convex intersection computation problems with directed time-varying communication graphs. Besides optimization, a continuous-time interpretation to discrete-time algorithms was discussed for recursive stochastic algorithms in [@ljung]. Consensus algorithms have been proven to be useful in the design of distributed optimization methods [@nedic09; @nedic10; @shirandom; @shitac; @elia; @lu1]. Consensus methods have also been extensively studied for both discrete-time and continuous-time models in the past decade, some references related to the current paper include [@tsi; @jad03; @julien2; @lwang; @lin07; @ren05; @mar; @caoming1; @mor; @nedic08; @shi09; @shi11]. Main Contribution ----------------- This paper considers the following distributed optimization model. The network consists of $N$ nodes with directed communication. Each node $i$ has a convex objective function $f_i: \mathds{R}^m\rightarrow \mathds{R}$. The goal of the network is to reach consensus meanwhile minimizing the function $\sum_{i=1}^N f_i$. At any time $t$, each node $i$ observes the gradient of $f_i$ at its current state $g_i(t)$ and the neighboring information $n_i(t)$ from its neighbors. The map $n_i(t)$ is zero when the nodes state is equal to all its neighbors’ state. The evolution of the nodes’ states is given by a first-order integrator with right-hand side being a control law $\mathcal{J}(n_i,g_i)$ taking feedback from $g_i(t)$ and $n_i(t)$. We assume $\mathcal{J}(n_i,g_i)$ to be injective in $g_i$ when $n_i$ takes value zero. The main results we obtain are stated as follows: - We prove that there exists a neighboring information rule $n_i$ and a control law $\mathcal{J}$ guaranteeing global optimal consensus if and only if the intersection of the solution sets of $f_i,i=1,\dots,N$, is nonempty intersection set for fixed strongly connected graphs. - We show that given any $\epsilon>0$, there exists a control law $\mathcal{J}$ that guarantees global optimal consensus with error no larger than $\epsilon$ for fixed, bidirectional, and connected graphs under mild conditions. - We show that optimal consensus can always be achieved for time-varying graphs as long as the graph is uniformly jointly strongly connected and the nonempty intersection condition above holds. We conclude that the nonempty intersection of the solution sets of the local objectives seems to be a fundamental condition for distributed optimization. Paper Organization ------------------ In Section 2, some preliminary mathematical concepts and lemmas are introduced. In Section 3, we formulate the considered optimization model, node dynamics, and define the problem of interest. Section 4 focuses on fixed graphs. A necessary and sufficient condition is presented for the exact solution of optimal consensus, and then approximate solutions are investigated as $\epsilon$-optimal consensus. Section 5 is on time-varying graphs, and we show optimal consensus under uniformly jointly strongly connected graphs. Finally, in Section 6 some concluding remarks are given. Preliminaries ============= In this section, we introduce some notations and provide preliminary results that will be used in the rest of the paper. Directed Graphs --------------- A directed graph (digraph) $\mathcal {G}=(\mathcal {V}, \mathcal {E})$ consists of a finite set $\mathcal{V}$ of nodes and an arc set $\mathcal {E}$, where an arc is an ordered pair of distinct nodes of $\mathcal {V}$ [@god]. An element $(i,j)\in\mathcal {E}$ describes an arc which leaves $i$ and enters $j$. A [*walk*]{} in $\mathcal {G}$ is an alternating sequence $\mathcal {W}: i_{1}e_{1}i_{2}e_{2}\dots e_{m-1}i_{m}$ of nodes $i_{\kappa}$ and arcs $e_{\kappa}=(i_{\kappa},i_{\kappa+1})\in\mathcal {E}$ for $\kappa=1,2,\dots,m-1$. A walk is called a [*path*]{} if the nodes of the walk are distinct, and a path from $i$ to $j$ is denoted as $i\rightarrow j$. $\mathcal {G}$ is said to be [*strongly connected*]{} if it contains path $i\rightarrow j$ and $j\rightarrow i$ for every pair of nodes $i$ and $j$. A digraph $\mathcal {G}$ is called [*bidirectional*]{} when for any two nodes $i$ and $j$, $(i,j)\in\mathcal{E}$ if and only if $(j,i)\in\mathcal{E}$. Ignoring the direction of the arcs, the connectivity of a bidirectional digraph is transformed to that of the corresponding undirected graph. A time-varying graph is defined as $\mathcal {G}_{\sigma(t)}=(\mathcal {V},\mathcal {E}_{\sigma(t)})$ where $\sigma:[0,+\infty)\rightarrow \mathcal {Q}$ denotes a piecewise constant function, where $\mathcal {Q}$ is a finite set containing all possible graphs with node set $\mathcal{V}$. Moreover, the joint graph of $\mathcal {G}_{\sigma(t)}$ in time interval $[t_1,t_2)$ with $t_1<t_2\leq +\infty$ is denoted as $\mathcal {G}([t_1,t_2))= \cup_{t\in[t_1,t_2)} \mathcal {G}(t)=(\mathcal {V},\cup_{t\in[t_1,t_2)}\mathcal {E}_{\sigma(t)})$. Dini Derivatives ---------------- The upper [*Dini derivative*]{} of a continuous function $h: (a,b)\to \mathds{R}$ ($-\infty\leq a<b\leq \infty$) at $t$ is defined as $$D^+h(t)=\limsup_{s\to 0^+} \frac{h(t+s)-h(t)}{s}.$$ When $h$ is continuous on $(a,b)$, $h$ is non-increasing on $(a,b)$ if and only if $ D^+h(t)\leq 0$ for any $t\in (a,b)$. The next result is convenient for the calculation of the Dini derivative [@dan; @lin07]. \[lemdini\] Let $V_i(t,x): \mathds{R}\times \mathds{R}^d \to \mathds{R}\;(i=1,\dots,n)$ be $C^1$ and $V(t,x)=\max_{i=1,\dots,n}V_i(t,x)$. If $ \mathcal{I}(t)=\{i\in \{1,2,\dots,n\}\,:\,V(t,x(t))=V_i(t,x(t))\}$ is the set of indices where the maximum is reached at $t$, then $ D^+V(t,x(t))=\max_{i\in\mathcal{ I}(t)}\dot{V}_i(t,x(t)). $ Limit Sets ---------- Consider the following autonomous system $$\label{i1} \dot{x}=f(x),$$ where $f:\mathds{R}^d\rightarrow \mathds{R}^d$ is a continuous function. Let $x(t)$ be a solution of (\[i1\]) with initial condition $x(t_0)=x^0$. Then $\Omega_0\subset \mathds{R}^d$ is called a [*positively invariant set*]{} of (\[i1\]) if, for any $t_0\in\mathds{R}$ and any $x^0\in\Omega_0$, we have $x(t)\in\Omega_0$, $t\geq t_0$, along every solution $x(t)$ of (\[i1\]). We call $y$ a $\omega$-limit point of $x(t)$ if there exists a sequence $\{t_k\}$ with $\lim_{k\rightarrow \infty}t_k=\infty$ such that $$\lim_{k\rightarrow \infty}x(t_k)=y.$$ The set of all $\omega$-limit points of $x(t)$ is called the $\omega$-limit set of $x(t)$, and is denoted as $\Lambda^+\big(x(t)\big)$. The following lemma is well-known [@rou]. \[leminvariant\] Let $x(t)$ be a solution of (\[i1\]). Then $\Lambda^+\big(x(t)\big)$ is positively invariant. Moreover, if $x(t)$ is contained in a compact set, then $\Lambda^+\big(x(t)\big)\neq \emptyset$. Convex Analysis --------------- A set $K\subset \mathds{R}^d$ is said to be [*convex*]{} if $(1-\lambda)x+\lambda y\in K$ whenever $x\in K,y\in K$ and $0\leq\lambda \leq1$. For any set $S\subset \mathds{R}^d$, the intersection of all convex sets containing $S$ is called the [*convex hull*]{} of $S$, denoted by $co(S)$. Let $K$ be a closed convex subset in $\mathds{R}^d$ and denote $|x|_K\doteq\inf_{y\in K}| x-y |$ as the distance between $x\in \mathds{R}^d$ and $K$, where $|\cdot|$ is the Euclidean norm. There is a unique element ${P}_{K}(x)\in K$ satisfying $|x-{P}_{K}(x)|=|x|_K$ associated to any $x\in \mathds{R}^d$ [@aubin]. The map ${P}_{K}$ is called the [*projector*]{} onto $K$. The following lemma holds [@aubin]. \[lemconvex\] (i). $\langle {P}_{K}(x)-x,{P}_{K}(x)-y\rangle\leq 0,\quad \forall y\in K$. (ii). $|{P}_{K}(x)-{P}_{K}(y)|\leq|x-y|, x,y\in \mathds{R}^d$. \(iii) $|x|_K^2$ is continuously differentiable at $x$ with $\nabla |x|_K^2=2\big(x-{P}_{K}(x)\big)$. Let $f: \mathds{R}^d\rightarrow \mathds{R}$ be a real-valued function. We call $f$ a convex function if for any $x,y\in\mathds{R}^d$ and $0\leq\lambda \leq1$, it holds that $f\big((1-\lambda)x+\lambda y\big)\leq (1-\lambda)f(x)+\lambda f(y)$. The following lemma states some well-known properties for convex functions. \[lemfunction\] Let $f:\mathds{R}^d\rightarrow \mathds{R}\in C^1$ be a convex function. (i). $f(x)\geq f(y)+\big\langle x-y, \nabla f(y)\big\rangle$. (ii). Any local minimum is a global minimum, i.e., $\arg \min f=\big\{z: \nabla f(z)=0 \big\}$. Problem Definition ================== Objective --------- Consider a network with node set $\mathcal {V}=\{1,2,\dots,N\}$ modeled in general as a directed graph $\mathcal{G}=(\mathcal {V}, \mathcal {E})$. A node $j$ is said to be a [*neighbor*]{} of $i$ at time $t$ when there is an arc $(j, i)\in \mathcal {E}$, and we denote $\mathcal{N}_i$ the set of neighbors for node $i$. Node $i$ is associated with a cost function $f_i: \mathds{R}^m\rightarrow \mathds{R}, m>0$ which is observed by node $i$ only. The objective for the network is to cooperatively solve the optimization problem $$\label{1} \begin{array}{cl} \mathop{\rm minimize}\ & \sum_{i=1}^N f_i(z) \\ \textrm{subject to} & z\in \mathds{R}^m. \end{array}$$ We impose the following assumption on the functions $f_i, i=1,\dots,N$. [**A1.**]{} For all $i=1,\dots,N$, we have (i) $f_i\in C^1$; (ii) $f_i$ is a convex function; (iii) $\arg \min f_i\neq \emptyset$. Problem (\[1\]) is equivalent with the following problem: $$\label{16} \begin{array}{cl} \mathop{\rm minimize}\ & \sum_{i=1}^N f_i(z_i) \\ \textrm{subject to} & z_i\in \mathds{R}^m \\ & z_1=\dots=z_N. \end{array}$$ From (\[16\]) we see that consensus algorithms are a natural mean for solving the optimization problem (\[1\]). Information Flow ---------------- The state of node $i$ at time $t$ is denoted as $x_i(t)\in \mathds{R}^m$. We define the information flow for node $i$ as follows. - The local optimization information $g_i(t)$ node $i$ receives from its objective $f_i$ at time $t$ is the gradient of $f_i$ at its current state, i.e., $$\begin{aligned} g_i(t)\doteq \nabla f_i\big(x_i(t)\big).\end{aligned}$$ - The neighboring information $n_i(t)$ node $i$ receives from its neighbors at time $t$ is $$\begin{aligned} n_i(t)\doteq \hbar_i\big(x_i(t), x_j(t): j \in \mathcal{N}_i \big),\end{aligned}$$ where $\hbar_i: \mathds{R}^m\times \mathds{R}^{m|\mathcal{N}_i|}\rightarrow \mathds{R}^l$ is a continuous function, $|\mathcal{N}_i|$ denotes the number of elements in $\mathcal{N}_i$, and $l$ is a given integer indicating the dimension of the neighboring information. Let $\hbar= \hbar_1 \otimes \dots \otimes \hbar_N: \mathds{R}^{m(1+|\mathcal{N}_1|)} \times \dots \times \mathds{R}^{m(1+|\mathcal{N}_N|)} \rightarrow \mathds{R}^{Nl}$ denote the direct sum of $\hbar_i, i=1,\dots,N$. Then $\hbar$ represents the rule of all neighboring information flow over the whole network. We impose the following assumption. [**A2.**]{} $\hbar\in \mathscr{R} \doteq \Big\{ h_1 \otimes \dots \otimes h_N$:   $h_i$: $ \mathds{R}^{m(1+|\mathcal{N}_i|)} \mapsto\mathds{R}^{l}$ and $h_i\equiv0$ within the local consensus manifold $\big\{x_i=x_{j}: j \in \mathcal{N}_i \big\}$ for all $i\in\mathcal{V}\Big\}$. Assumption A2 is to say that the neighboring information a node receives from its neighbors becomes trivial when the node is in the same state as all its neighbors. This is a quite natural assumption in the literature on distributed averaging and optimization algorithms [@jad03; @mor; @saber04; @nedic09; @nedic10]. Computational Capacity ---------------------- We adopt a dynamical system model to define the way nodes update their respective states. The evolution of the nodes’ states is restricted to be a first-order integrator: $$\label{2} \dot{x}_i=u_i, \quad i=1,\dots,N,$$ where the right-hand side $u_i$ is interpreted as a control input and the control law is characterized as $$\begin{aligned} \label{5} u_i= \mathcal {J} \big(n_i,g_i\big),\ i=1,\dots,N\end{aligned}$$ with $\mathcal{J}: \mathds{R}^l\times \mathds{R}^m\rightarrow \mathds{R}^m$. For the control law $\mathcal{J}$, we impose the following assumption. [**A3.**]{} $\mathcal{J} \in \mathscr{C}\doteq \big\{\mathcal{F}(\cdot,\cdot)\in C^0:\ \mathds{R}^l\times \mathds{R}^m\rightarrow \mathds{R}^m,\ \mathcal{F}(0,\cdot)\ \mbox{is injective}\big\}$. Assumption A3 indicates that the control law applied in each node should have the same structure, irrespectively of individual local optimization information or neighboring information. Note that our network model is homogeneous because one cannot tell the difference from one node to another. We assume that the control law $\mathcal{J}(0,\cdot)$ is injective, so each node takes different response to different gradient information on the local consensus manifold. Again, Assumption A3 is widely applied in the literature [@jad03; @mor; @saber04; @nedic09; @nedic10]. Problem ------- Let $x(t)=(x_1^T(t),\dots,x_N^T(t))^T\in \mathds{R}^{mN}$ be the trajectory of system (\[2\]) with control law (\[5\]) for initial condition $x^0=x(t_0)$. Denote $F(z)=\sum_{i=1}^N f_i(z)$. We introduce the following definition. Global [*optimal consensus*]{} of (\[2\])–(\[5\]) is achieved if for all $x^0\in \mathds{R}^{mN}$, we have $$\label{3} \limsup_{t\rightarrow +\infty} F\big(x_i(t)\big)= \min_{z\in \mathds{R}^m} F(z)$$ and $$\label{4} \lim_{t\rightarrow +\infty} \big |x_i(t)-x_j(t)\big|=0,\quad i,j=1,\dots,N.$$ The problem considered in this paper is to characterize conditions on the control law $\mathcal{J}$ under which global optimal consensus is achieved. In Section 4 this is done for fixed graphs and in Section 5 for time-varying graphs. Fixed Graphs ============ In this section, we consider the possibility of solving optimal consensus using control law (\[5\]) under fixed communication graphs. We first discuss whether exact optimal consensus can be reached for directed graphs. Then we show the existence of an approximate solution for optimal consensus over bidirectional graphs. Exact Solution -------------- We make an assumption on the solution set of $F=\sum_{i=1}^N f_i$. [**A4.** ]{} $\arg \min F(z)\neq\emptyset$ is a bounded set. The main result on the existence of a control law solving optimal consensus is stated as follows. \[thm1\]Assume that A1 and A4 hold. Let the communication graph $\mathcal{G}$ be fixed and strongly connected. There exist a neighboring information rule $\hbar \in \mathscr{R}$ and a control law $\mathcal{J}\in \mathscr{C}$ such that global optimal consensus is achieved if and only if $$\begin{aligned} \label{intersection} \bigcap_{i=1}^N \arg \min f_i(z)\neq \emptyset.\end{aligned}$$ According to Theorem \[thm1\], the optimal solution sets of $f_i$, $i=1,\dots,N$, having nonempty intersection is a critical condition for the existence of a control law (\[5\]) that solves the optimal consensus problem. Condition (\[intersection\]) is obviously a strong constraint which in general does not hold. Therefore, basically Theorem \[thm1\] suggests that exact solution of optimal consensus is seldom possible for the given model. It follows from the proof below that the necessity statement of Theorem \[thm1\] relies only on the fact that the limit set of an autonomous system is invariant. It is straightforward to verify that for a discrete-time autonomous dynamical system defined by $$\begin{aligned} y_{k+1}=f(y_k)\end{aligned}$$ with $f$ a continuous function, its limit set is invariant. Therefore, if we consider a model with discrete-time update as $$\begin{aligned} \label{203} x_i(k+1)=x_i(k)+u_i(k)\end{aligned}$$ with $$\begin{aligned} \label{204} u_i(k)=\mathcal{J}\big(n_i(k),g_i(k)\big),\end{aligned}$$ where $n_i$, $g_i$, and $\mathcal{J}$ agree with the definitions above, the necessity statement of Theorem \[thm1\] still holds. However, the sufficiency statement of Theorem \[thm1\] may in general not hold for discrete-time updates since even for the centralized optimization problem, there is not always an algorithm with constant step size which can solve the problem exactly, cf., [@bertsekas]. In [@nedic09], a discrete-time algorithm was provided for solving (\[1\]), where the structure of the nodes’ update is the sum of a consensus term averaging the neighbors’ states, and a subgradient term of the local objective function with a fixed step size. It is easy to see that the algorithm in [@nedic09] can be rewritten as (\[203\]) and (\[204\]) as long as the graph is fixed and the step size is constant. All the properties we impose on the information flow and update dynamics are kept. Convergence bounds were established for the case with constant step size in [@nedic09]. Theorem \[thm1\] shows that proposing a convergence bound is in general the best we can do for algorithms like the one developed in [@nedic09], and the result also explains why a time-varying step size may be necessary in distributed optimization algorithms, as in [@nedic10]. In the rest of this subsection, we first give the proof of the necessity claim of Theorem \[thm1\], and then we present a simple proof for the sufficiency part with bidirectional graphs. The sufficiency part of Theorem \[thm1\] in fact follows from the upcoming conclusion, Theorem \[thm4\], which does not rely on Assumption A4. ### Necessity We now prove the necessity statement in Theorem \[thm1\] by a contradiction argument. Suppose $\bigcap_{i=1}^N \arg \min f_i(z)= \emptyset$ and there exists a distributed control in the form of (\[5\]), say $\mathcal {J}_0 \big(n_i,g_i\big)$, under which global optimal consensus is reached for certain neighboring information flow $n_i$ satisfying Assumption A2. Let $x(t)$ be a trajectory of system (\[2\]) with control $\mathcal {J}_0 \big(n_i,g_i\big)$ and $\Lambda^+(x(t))$ be its $\omega$-limit set. The definition of optimal consensus leads to that $x(t)$ converges to the bounded set $\Big( \arg \min F(z)\Big)^N\bigcap \mathcal {M}$, where $\Big( \arg \min F(z)\Big)^N$ denotes the $N$’th power set of $ \arg \min F(z)$ and $\mathcal{M}$ denotes the consensus manifold, defined by $$\begin{aligned} \mathcal{M}\doteq \big\{x=(x_1^T\dots x_N^T)^T: \ x_1=\dots=x_N;\ x_i\in \mathds{R}^m, i=1,\dots,N\big\}.\end{aligned}$$ Therefore, each trajectory $x(t)$ is contained in a compact set. Based on Lemma \[leminvariant\], we conclude that $\Lambda^+(x(t))\neq \emptyset$ and $$\begin{aligned} \label{6} \Lambda^+(x(t))\subseteq \Big( \arg \min F(z)\Big)^N\bigcap \mathcal {M},\end{aligned}$$ Moreover, $\Lambda^+(x(t))$ is positively invariant since system (\[2\]) is autonomous under control $\mathcal {J}_0 \big(n_i,g_i\big)$ when the communication graph is fixed. This is to say, any trajectory of system (\[2\]) under control $\mathcal {J}_0 \big(n_i,g_i\big)$ must stay within $\Lambda^+(x(t))$ for any initial value in $\Lambda^+(x(t))$. Now we take $y\in\Lambda^+(x(t))$. Then we have $y\in \Big( \arg \min F(z)\Big)^N\bigcap \mathcal {M}$ according to (\[6\]), and thus $y=(z_\ast^T \dots z_\ast^T)^T$ for some $z_\ast\in \arg \min F(z)$. With Assumption A1, the convexity of the $f_i$’s implies that $$\begin{aligned} \arg \min F(z)=\big\{z\in\mathds{R}^m:\ \sum_{i=1}^N \nabla f_i(z)=0 \big\}.\end{aligned}$$ On the other hand, we have $$\bigcap_{i=1}^N \arg \min f_i(z)= \bigcap_{i=1}^N \big\{z\in\mathds{R}^m:\ \nabla f_i(z)=0 \big\}= \emptyset.$$ Therefore, there exists two indices $i_1,i_2\in \{1,\dots,N\}$ with $i_1\neq i_2$ such that $$\begin{aligned} \nabla f_{i_1}(z_\ast)\neq \nabla f_{i_2}(z_\ast).\end{aligned}$$ Consider the solution of (\[2\]) under control $\mathcal {J}_0 \big(n_i,g_i\big)$ for initial time $t_0$ and initial value $y$. The fact that $y$ belongs to the consensus manifold guarantees $$\begin{aligned} n_{i_1}(t_0)=n_{i_2}(t_0)=0.\end{aligned}$$ With Assumption A4, we have $$\begin{aligned} \mathcal {J}_0\big(n_{i_1}(t_0), g_{i_1}(t_0)\big)= \mathcal {J}_0\big(0, \nabla f_{i_1}(z_\ast)\big) \neq \mathcal {J}_0\big(0, \nabla f_{i_2}(z_\ast)\big)= \mathcal {J}_0\big(n_{i_2}(t_0), g_{i_2}(t_0)\big).\end{aligned}$$ This implies $\dot{x}_{i_1}(t_0)\neq \dot{x}_{i_2}(t_0)$. As a result, there exists a constant $\varepsilon>0$ such that $x_{i_1}(t)\neq x_{i_2}(t)$ for $t\in (t_0,t_0+\varepsilon)$. In other word, the trajectory will leave the set $$\Big( \arg \min F(z)\Big)^N\bigcap \mathcal {M}$$ for $(t_0,t_0+\varepsilon)$, and therefore will also leave the set $\Lambda^+(x(t))$. This contradicts the fact that $\Lambda^+(x(t))$ is positively invariant. The necessity part of Theorem \[thm1\] has been proved. ### Sufficiency: Bidirectional Case We now provide an alternative proof of sufficiency for bidirectional graphs, which is based on some geometrical intuition of the vector field. Note that compared to the proof of Theorem \[thm4\] on directed graphs, this proof uses completely different arguments which indeed cannot be applied to directed graphs. Therefore, we believe the proof given in the following is interesting at its own right, because it reveals some fundamental difference between directed and bidirectional graphs. Let $a_{ij}>0$ be a constant marking the weight of arc $(j,i)$. We will show that the particular neighboring information flow $$n_i=\sum\limits_{j \in \mathcal{N}_i}a_{ij}\big(x_j-x_i\big)$$ and control law $$\begin{aligned} \label{7} \mathcal{J}_\star(n_i,g_i)=n_i-g_i=\sum\limits_{j \in \mathcal{N}_i}a_{ij}\big(x_j-x_i\big)-\nabla f_i\big(x_i\big)\end{aligned}$$ ensure global optimal consensus for system (\[2\]). Note that (\[7\]) is indeed a continuous-time version of the algorithm proposed in [@nedic09]. We suppose $\mathcal{G}$ is bidirectional. In this case, we have $a_{ij}=a_{ji}$ for all $i$ and $j$, and we use unordered pair $\{i,j\}$ to denote the edge between node $i$ and $j$. Noticing that $$\begin{aligned} \mathcal{J}_\star(n_i,g_i)=\sum\limits_{j \in \mathcal{N}_i}a_{ij}\big(x_j-x_i\big)-\nabla f_i\big(x_i\big)=-\nabla_{x_i} \Big(\frac{1}{2}\sum\limits_{j \in \mathcal{N}_i}a_{ij}\big|x_j-x_i\big|^2+f_i(x_i)\Big),\end{aligned}$$ we have that $(\ref{7})$ indeed solves the following convex problem $$\begin{aligned} \label{a1} \begin{array}{cl} \mathop{\rm minimize}\ & F_{\mathcal{G}}(x)\doteq\sum_{i=1}^N f_i(x_i)+ \frac{1}{2}\sum_{\{j,i\}\in\mathcal{E}}a_{ij}\big|x_j-x_i\big|^2 \\ \textrm{subject to} & x_i\in \mathds{R}^m,\ i=1,\dots,N. \end{array}\end{aligned}$$ We establish the following lemma relating the solution sets of problems (\[1\]) and (\[a1\]). \[lem2\] Suppose $\bigcap_{i=1}^N \arg \min f_i(z)\neq \emptyset$. Suppose also the communication graph $\mathcal{G}$ is fixed, bidirectional, and connected. Then we have $$\begin{aligned} \arg \min F_{\mathcal{G}}(x)=\Big( \bigcap_{i=1}^N \arg \min f_i(z)\Big)^N\bigcap \mathcal {M}=\Big( \arg \min F(z)\Big)^N\bigcap \mathcal {M}.\end{aligned}$$ [*Proof.*]{} When $\bigcap_{i=1}^N \arg \min f_i(z)\neq \emptyset$, it is straightforward to see that $$\arg \min F(z)= \bigcap_{i=1}^N \arg \min f_i(z).$$ Now take $x_\ast=(p_\ast^T \dots p_\ast^T)^T\in \Big( \bigcap_{i=1}^N \arg \min f_i(z)\Big)^N\bigcap \mathcal {M}$, where $p_\ast\in \bigcap_{i=1}^N \arg \min f_i(z)$. First we have $x_\ast\in \arg \min_x \sum_{i=1}^N f_i(x_i) $. Second we have $x_\ast\in \arg \min_x \frac{1}{2}\sum_{\{j,i\}\in\mathcal{E}}a_{ij}\big|x_j-x_i\big|^2$. Therefore, we conclude that $x_\ast\in \arg \min F_{\mathcal{G}}(x) $. This gives $$\begin{aligned} \label{11} \arg \min F_{\mathcal{G}}(x) \supseteq\Big( \bigcap_{i=1}^N \arg \min f_i(z)\Big)^N\bigcap \mathcal {M}.\end{aligned}$$ On the other hand, convexity gives $$\begin{aligned} \arg \min F_{\mathcal{G}}(x)=\bigg\{x:\ -(L\otimes I_m) x= \Big( \big(\nabla f_1(x_1)\big)^T \dots \big(\nabla f_N(x_N)\big)^T \Big)^T\bigg\},\end{aligned}$$ where $\otimes$ represents the Kronecker product, $I_m$ is the identity matrix in $\mathds{R}^m$, and $L=D-A$ is the Laplacian of the graph $\mathcal{G}$ with $A=[a_{ij}]$ and $D={\rm diag}(d_1,\dots,d_N)$, where $d_i=\sum_{j=1}^n a_{ij}$. Noticing that $$(\mathbf{1}_N^T\otimes I_m) (L\otimes I_m) =\mathbf{1}_N^TL\otimes I_m=0,$$ where $\mathbf{1}_N=(1\dots 1)^T \in \mathds{R}^N$, we have $$\begin{aligned} \label{8} \Big(\mathbf{1}_N^T \otimes I_m\Big) \Big( \big(\nabla f_1(x_1)\big)^T \dots \big(\nabla f_N(x_N)\big)^T \Big)^T=\sum_{i=1}^N \nabla f_i(x_i)=0\end{aligned}$$ for any $x\in\arg \min F_{\mathcal{G}}(x) $. Now take $x^\ast=(q_1^T \dots q_N^T)^T\in\arg \min F_{\mathcal{G}}(x) $. Suppose there exist two indices $i_\ast$ and $j_\ast$ such that $$\nabla f_{i_\ast}(q_{i_\ast})\neq \nabla f_{j_\ast}(q_{j_\ast}).$$ Then at least one of $\nabla f_{i_\ast}(q_{i_\ast})$ and $\nabla f_{j_\ast}(q_{j_\ast})$ must be nonzero. Taking $\hat{p}\in \bigcap_{i=1}^N \arg \min f_i(z)$, we have $$\sum_{i=1}^N f_i(q_i)>\sum_{i=1}^Nf_i(\hat{p})$$ because for $x=(x_1^T \dots x_N^T)^T\in\arg \min \sum_{i=1}^N f_i(x_i)$, we have $\nabla f_i(x_i)=0, i=1,\dots,N$. Consequently, for $w_\ast=(\hat{p}^T \dots \hat{p}^T)^T$, we have $$F_{\mathcal{G}}(x^\ast)>F_{\mathcal{G}}(w_\ast),$$ which is impossible according to the definition of $x^\ast$ so that such $i_\ast$ and $j_\ast$ cannot exist. In light of (\[8\]), this immediately implies $$\nabla f_i(q_i)=0,\ i=1,\dots,N,$$ or equivalently $$\begin{aligned} \label{10} q_i\in \arg \min f_i(z),\ i=1,\dots,N\end{aligned}$$ for all $x^\ast=(q_1^T\dots q_N^T)^T\in\arg \min F_{\mathcal{G}}(x)$. Therefore, we conclude from (\[10\]) that $$\sum_{i=1}^N f_i(q_i)=\sum_{i=1}^Nf_i(p_\ast),$$ and this implies $$\sum_{\{j,i\}\in\mathcal{E}}a_{ij}\big|q_j-q_i\big|^2=0$$ as long as $x^\ast=(q_1^T \dots q_N^T)^T\in\arg \min F_{\mathcal{G}}(x)$. The connectivity of the communication graph thus further guarantees that $q_1=\dots=q_N$, so we have proved that $ x^\ast\in \Big( \bigcap_{i=1}^N \arg \min f_i(z)\Big)^N\bigcap \mathcal {M}.$ Consequently, we obtain $$\begin{aligned} \label{12} \arg \min F_{\mathcal{G}}(x) \subseteq\Big( \bigcap_{i=1}^N \arg \min f_i(z)\Big)^N\bigcap \mathcal {M}.\end{aligned}$$ The desired lemma holds from (\[11\]) and (\[12\]). $\square$ Now since $F_{\mathcal{G}}(x)$ is a convex function and we have $\dot{x}=\nabla F_{\mathcal{G}}(x)$ for system (\[2\]) with control (\[7\]), we conclude that $$\lim_{t\rightarrow \infty}{\rm dist}\big(x(t),{\arg \min F_{\mathcal{G}}(x) }\big)=0.$$ Lemma \[lem2\] ensures $$\lim_{t\rightarrow \infty}{\rm dist}\bigg(x(t),\Big( \bigcap_{i=1}^N \arg \min f_i(z)\Big)^N\bigcap \mathcal {M}\bigg)=0$$ if $\mathcal{G}$ is bidirectional and connected. Equivalently, global optimal consensus is reached. We see from the proof above that the construction of $F_{\mathcal{G}}(x)$ is critical because the convergence argument is based on the fact that the gradient of $F_{\mathcal{G}}(x)$ is consistent with the communication graph. It can be easily verified that finding such a function is in general impossible for directed graphs. Approximate Solution -------------------- Theorem \[thm1\] indicates that optimal consensus is impossible no matter how the control law $\mathcal{J}$ is chosen from $\mathscr{C}$ as long as the nonempty intersection condition (\[intersection\]) is not fulfilled. In this subsection, we discuss the approximate solution of the optimal consensus problem in the absence of (\[intersection\]). We introduce the following definition. Global [*$\epsilon$-optimal consensus*]{} is achieved if for all $x^0\in \mathds{R}^{mN}$, we have $$\label{3a} \limsup_{t\rightarrow +\infty} F\big(x_i(t)\big)\leq \min_{z\in \mathds{R}^m} F(z)+\epsilon$$ and $$\label{4a} \lim_{t\rightarrow +\infty} \big |x_i(t)-x_j(t)\big|\leq \epsilon,\quad i,j=1,\dots,N.$$ Denoting $F_{\mathcal{G}}(x;K)=\sum_{i=1}^N f_i(x_i)+ \frac{K}{2}\sum_{\{j,i\}\in\mathcal{E}}a_{ij}\big|x_j-x_i\big|^2$, we impose the following assumption. [**A5.** ]{} (i) $\arg \min F(z)\neq\emptyset$; (ii) $\arg \min F_{\mathcal{G}}(x;K)\neq\emptyset$ for all $K\geq 0$; (iii) $\bigcup_{K\geq 0} \arg \min F_{\mathcal{G}}(x;K)$ is bounded. For $\epsilon$-optimal consensus, we present the following result. \[thm2\] Assume that A1 and A5 hold. Let the communication graph $\mathcal{G}$ be fixed, bidirectional, and connected. Then for any $\epsilon>0$, there exist a neighboring information rule $\hbar \in \mathscr{R}$ and a control law $\mathcal{J}\in \mathscr{C}$ such that global $\epsilon$-optimal consensus is achieved. [*Proof.* ]{} Again, let $a_{ij}>0$ be any constant marking the weight of arc $(j,i)$ and $a_{ij}=a_{ji}$ for all $(i,j)\in \mathcal{E}$. Fix $\epsilon$. We will show that under neighboring information flow $$n_i=\sum\limits_{j \in \mathcal{N}_i}a_{ij}\big(x_j-x_i\big),$$ there exists a constant $K_\epsilon>0$ such that the control law $$\begin{aligned} \label{50} u_i=\mathcal{J}_{K_\epsilon}(n_i,g_i)\doteq K_\epsilon n_i-g_i\end{aligned}$$ guarantees global $\epsilon$-optimal consensus. It is straightforward to see that $$\begin{aligned} \mathcal{J}_{K}(n_i,g_i)=K\sum\limits_{j \in \mathcal{N}_i}a_{ij}\big(x_j-x_i\big)-\nabla f_i\big(x_i\big)=-\nabla_{x_i} \Big(\frac{K}{2}\sum\limits_{j \in \mathcal{N}_i}a_{ij}\big|x_j-x_i\big|^2+f_i(x_i)\Big).\end{aligned}$$ System (\[2\]) with control law $u_i=\mathcal{J}_K(n_i,g_i)$ can be written into the following compact form $$\begin{aligned} \label{90} \dot{x}=-\nabla F_{\mathcal{G}}(x;K),\ \ x=(x_1^T \dots x_N^T)^T\in \mathds{R}^{mN}.\end{aligned}$$ Then the convexity of $F_{\mathcal{G}}(x;K)$ ensures that control law $\mathcal{J}_{K}(n_i,g_i)$ asymptotically solves the convex optimization problem $$\begin{aligned} \label{ka1} \begin{array}{cl} \mathop{\rm minimize}\ & F_{\mathcal{G}}(x;K)=\sum_{i=1}^N f_i(x_i)+ \frac{K}{2}\sum_{\{j,i\}\in\mathcal{E}}a_{ij}\big|x_j-x_i\big|^2 \\ \textrm{subject to} & x_i\in \mathds{R}^m,\ i=1,\dots,N. \end{array}\end{aligned}$$ Convexity gives $$\begin{aligned} \label{92} \arg \min F_{\mathcal{G}}(x;K)=\bigg\{x:\ -K(L\otimes I_m) x= \Big( \big(\nabla f_1(x_1)\big)^T \dots \big(\nabla f_N(x_N)\big)^T \Big)^T\bigg\}.\end{aligned}$$ Under Assumptions A1 and A5, we have that $$\begin{aligned} L_0\doteq \sup \Big\{ \big|\nabla \tilde{F}(x)\big|:\ x\in \bigcup_{K\geq0} \arg \min F_{\mathcal{G}}(x;K)\Big\}\end{aligned}$$ is a finite number, where $\tilde{F}(x)=\sum_{i=1}^N f_i(x_i)$. We also define $$\begin{aligned} D_0\doteq \sup \Big\{ \big|z_\ast-x_i\big|:\ i=1,\dots,N,\ x\in \bigcup_{K\geq 0} \arg \min F_{\mathcal{G}}(x;K)\Big\},\end{aligned}$$ where $z_\ast\in \arg \min F$ is an arbitrarily chosen point. Let $p=(p_1^T \dots p_N^T)^T \in \arg \min F_{\mathcal{G}}(x;K)$ with $p_i\in \mathds{R}^m, i=1, \dots,N$. Since the graph is bidirectional and connected, we can sort the eigenvalues of the Laplacian $L\otimes I_m$ as $$0=\lambda_1=\dots=\lambda_{m} <\lambda_{m+1}\leq \dots \leq\lambda_{mN}.$$ Let $l_1\dots,l_{mN}$ be the orthonormal basis of $\mathds{R}^{mN}$ formed by the right eigenvectors of $L\otimes I_m$, where $l_1,\dots,l_m$ are eigenvectors corresponding to the zero eigenvalue. Suppose $p=\sum_{k=1}^{mN}c_k l_{k}$ with $c_k\in\mathds{R}, k=1,\dots,mN$. According to (\[92\]), we have $$\begin{aligned} \Big|K(L\otimes I_m)p\Big|^2= K^2\Big|\sum_{k=m+1}^{mN} c_k \lambda_k l_k\Big|^2=K^2\sum_{k=m+1}^{mN} c_k^2 \lambda_k^2 \leq L_0^2,\end{aligned}$$ which yields $$\begin{aligned} \label{94} \sum_{k=m+1}^{mN} c_k^2 \leq \Big(\frac{L_0}{K\lambda_2^\ast}\Big)^2,\end{aligned}$$ where $\lambda_2^\ast>0$ denotes the second smallest eigenvalue of $L$. Now recall that $$\begin{aligned} \mathcal{M}\doteq \big\{x=(x_1^T\dots x_N^T)^T: \ x_1=\dots=x_N;\ x_i\in \mathds{R}^m, i=1,\dots,N\big\}.\end{aligned}$$ is the consensus manifold. Noticing that $\mathcal{M}={\rm span} \{l_1,\dots, l_m\}$, we conclude from (\[94\]) that $$\begin{aligned} \label{100} \sum_{k=m+1}^{mN} c_k^2 =\Big| \sum_{k=m+1}^{mN} c_k l_k \Big|^2=| p |_{\mathcal{M}}^2=\sum_{i=1}^N \Big|p_i-\frac{\sum_{i=1}^{N} p_i}{N}\Big|^2\leq \Big(\frac{L_0}{K\lambda_2^\ast}\Big)^2.\end{aligned}$$ The last equality in (\[100\]) is due to the fact that $\mathbf{1}_N \otimes\Big( \frac{\sum_{i=1}^{N} p_i}{N}\Big)$ is the projection of $p$ on to $\mathcal{M}$. Thus, for any $\varsigma >0$, there is $K_1(\varsigma)>0$ such that when $K\geq K_1(\varsigma)$, $$\begin{aligned} \label{101} \Big|p_i-p_{\rm ave}\Big|\leq \varsigma,\ i=1,\dots,N\end{aligned}$$ and $$\begin{aligned} |F(p_i)-F(p_{\rm ave})\Big|\leq \varsigma,\ i=1,\dots,N,\end{aligned}$$ where $p_{\rm ave}=\frac{\sum_{i=1}^N p_i}{N}$. On the other hand, with (\[92\]), we have $$\begin{aligned} \label{102} \sum_{i=1}^N \nabla f_i(p_i)=\sum_{i=1}^N \nabla f_i(p_{\rm ave}+\hat{p}_i)=0,\end{aligned}$$ where $\hat{p}_i=p_i-p_{\rm ave}$. Now according to (\[101\]) and (\[102\]), since each $f_i\in C^1$, for any $\varsigma >0$, there is $K_2(\varsigma)>0$ such that when $K\geq K_2(\varsigma)$, $$\begin{aligned} \Big|\sum_{i=1}^N \nabla f_i(p_{\rm ave}) \Big|\leq \frac{\varsigma}{D_0}.\end{aligned}$$ This implies $$\begin{aligned} F(p_{\rm ave})\leq F(z_\ast)+|z_\ast-p_{\rm ave}|\times \Big|\sum_{i=1}^N \nabla f_i(p_{\rm ave}) \Big|\leq F(z_\ast)+\varsigma.\end{aligned}$$ Therefore, for any $\epsilon>0$, we can take $K_0=\max\{K_1(\epsilon/2), K_2(\epsilon/2) \}$. Then when $K\geq K_0$, we have $$\begin{aligned} |p_i-p_j|\leq \epsilon;\ \ F(p_i)\leq \min_z F(z)+\epsilon\end{aligned}$$ for all $i$ and $j$. Now that $F_{\mathcal{G}}(x;K)$ is a convex function and observing (\[90\]), every limit point of system (\[2\]) with control law $\mathcal{J}_K(n_i,g_i)$ is contained in the set $\arg \min F_{\mathcal{G}}(x;K)$. Noting that $p$ is arbitrarily chosen from $\arg \min F_{\mathcal{G}}(x;K)$, $\epsilon$-optimal consensus is achieved as long as we choose $K_\epsilon\geq K_0$. This completes the proof. $\square$ Theorem \[thm2\] can be compared to the results given in [@solodov], where a discrete-time incremental algorithm with constant step size was shown to be able to reach an $\epsilon$-approximate solution of (\[1\]). Incremental algorithms relies on global iteration along each local objective function alternatively [@solodov; @ram; @bjsiam]. They are therefore fundamentally different with the model we discuss. For the discrete-time algorithm proposed in [@nedic09], a bound of the convergence error was expressed explicitly as a function of the fixed step size. However, this bound will not vanish as the fixed step size tends to zero or infinity [@nedic09]. Note that the parameter $K$ in the control law $\mathcal{J}_{K}(n_i,g_i)$ can be viewed as a step size. As shown in Theorem \[thm2\], the convergence error vanishes as $K$ tends to infinity, which is essentially different with the discrete-time case in [@nedic09]. From Theorems \[thm1\] and \[thm2\], we conclude that even though without the nonempty intersection condition (\[intersection\]), it is impossible to reach exact optimal consensus via control law of the form of (\[5\]), it is still possible to find a control law that guarantees approximate optimal consensus with arbitrary accuracy. Discussion: Global vs. Local ---------------------------- A fundamental question in distributed optimization is whether global optimization can be obtained by neighboring information flow and cooperative computation. We have the following observation. - Note that in this paper, to determine a proper $K$ in (\[50\]) for a given $\epsilon$ relies on knowledge of the structure of the network, and the information of all $f_i, i=1,\dots,N$. Finding a proper control law for $\epsilon$-optimal consensus requires thus [*global*]{} knowledge of the network. Apparently also the nonempty intersection condition in Theorem \[thm1\] is a [*global*]{} constraint. - Incremental algorithms with constant step size have been shown to be able to reach $\epsilon$-optimal solution for any error bound $\epsilon$ as long as the step size is sufficiently small, e.g., [@solodov; @nedic01; @bjsiam]. In an incremental algorithm, iteration is carried out by only one node alternatively on each local objective function, which is is equivalent to the fact that the $N$ nodes perform the iteration, but any node can access the states of all other nodes. Therefore, it means that the underlying graph is indeed complete, which is certainly a [*global*]{} constraint. - One can also use time-varying step size. In [@nedic10], it was shown that global optimization can be achieved by a algorithm combining consensus algorithm and subgradient computation with a time-varying step size. However, this time-varying step size must be applied to all nodes homogeneously, which makes it a [*global*]{} parameter. From the above observations, we can conclude that in general for distributed optimization methods, some [ global]{} information (or constraint) is somehow inevitable to guarantee a global (exact or $\epsilon$-approximate) convergence. This reveals some fundamental limit of distributed information collection and algorithm design. Assumption Feasibility ---------------------- This subsection discusses the feasibility of Assumptions A4 and A5 and shows that some mild conditions are enough to ensure A4 and A5. Let A1 hold. If $\tilde{F}(x)=\sum_{i=1}^N f_i(x_i)$ is coercive, i.e., $\tilde{F}(x)\rightarrow \infty$ as long as $|x|\rightarrow \infty$, then A4 and A5 hold. [*Proof.*]{} Assume that A1 holds. a). Since $\tilde{F}(x)=\sum_{i=1}^N f_i(x_i)$ is coercive, it follows straightforwardly that $F(z)=\sum_{i=1}^Nf_i(z)$ is also coercive. As a result, $\arg \min F(z)\neq \emptyset$ is a bounded set. Thus, A4 and A5.(i) hold. b). Observing that $\frac{K}{2}\sum_{\{j,i\}\in\mathcal{E}}a_{ij}\big|x_j-x_i\big|^2\geq 0$ for all $x=(x_1^T \dots x_N^T)^T\in \mathds{R}^{mN}$ and that $\tilde{F}(x)=\sum_{i=1}^N f_i(x_i)$ is coercive, we obtain that $\arg \min F_{\mathcal{G}}(x;K)\neq\emptyset$ for all $K\geq 0$. Thus, A5.(ii) holds. c). Based on a), we can denote $F_\ast=\min_z F(z)=F(z_\ast)$. Since $\sum_{i=1}^Nf_i(x_i)$ is coercive, there exists a constant $M(F_\ast)>0$ such that $\sum_{i=1}^N f_i(x_i)> F_\ast$ for all $|x|>M$. This implies $$\begin{aligned} F_{\mathcal{G}}(x;K)> F_{\mathcal{G}}(\mathbf{1}_N\otimes z_\ast;K)=F_\ast\end{aligned}$$ for all $|x|> M$. That is to say, the global minimum of $F_{\mathcal{G}}(x;K)$ is reached within the set $\{|x|\leq M\}$ for all $K>0$. Therefore, we have $$\begin{aligned} \bigcup_{K\geq 0} \arg \min F_{\mathcal{G}}(x;K)\subseteq \big\{|x|\leq M\big\}.\end{aligned}$$ This proves A5.(iii). $\square$ Next, we propose another case when A4 and A5 hold. Let A1 hold. Suppose each $\arg \min f_i$ is bounded and the argument space for each $f_i$ is $\mathds{R}$, i.e., $m=1$. Then A4 and A5 holds. [*Proof.*]{} Assume that A1 holds. a). Let $x_i^\ast \in \arg \min f_i$. Denote $y_\ast=\min \{x_1^\ast,\dots,x_N^\ast\}$. Then for any $i=1,\dots,N$, we have $$\begin{aligned} 0\geq f_i(x_i^\ast)-f_i(y_\ast)\geq (x_i^\ast -y_\ast)\nabla f_i(y_\ast)\end{aligned}$$ according to inequality (i) of Lemma \[lemfunction\]. This immediately yields $\nabla f_i(y_\ast)\leq 0$ for all $i=1,\dots,N$. Thus, for any $y<y_\ast$, we have $$\begin{aligned} F(y)-F(y_\ast)\geq (y -y_\ast)\nabla F(y_\ast)=\sum_{i=1}^N (y -y_\ast) \nabla f_i(y_\ast)\geq 0,\end{aligned}$$ which implies $F(y)\geq F(y_\ast)$ for all $y<y_\ast$. A symmetric analysis leads to that $F(y)\geq F(y^\ast)$ for all $y>y^\ast$ with $y^\ast=\max \{x_1^\ast,\dots,x_N^\ast\}$. Therefore, we obtain $ F(y)\geq \min\{F(y_\ast), F(y^\ast)\}$ for all $y\neq [y_\ast, y^\ast]$. This implies that a global minimum is reached within the interval $[y_\ast, y^\ast]={\rm co}\{x_1^\ast,\dots,x_N^\ast\}$ and A5.(i) thus follows. If $\arg \min f_i$ is bounded for $i=1,\dots,N$, there exist $b_i\leq d_i, i=1,\dots, N$ such that $\arg \min f_i=[b_i, d_i]$. Define $b_\ast =\min\{b_1,\dots,b_N\}$ and $d^\ast= \max\{d_1,\dots,d_N\}$. Following a similar argument we have $\arg \min F \subseteq [b_\ast, d^\ast]$. Thus A4 holds. b). Introduce the following cube in $\mathds{R}^N$: $$\mathcal{C}_\ast^\eta\doteq \Big\{x=(x_1^T \dots x_N^T)^T: \ x_i \in [y_\ast-\eta, y^\ast+\eta],i=1,\dots,N\Big\},$$ where $\eta>0$ is a given constant. [*Claim.*]{} For any $K\geq 0$, $\mathcal{C}_\ast^\eta$ is an invariant set of system (\[2\]) under control law $\mathcal{J}_K(n_i,g_i)$. Define $\Psi(x(t))=\max_{i\in\mathcal{V}} x_i(t)$. Then based on Lemma \[lemdini\], we have $$\begin{aligned} D^+\Psi(x(t))&=\max_{i\in \mathcal{I}_0(t)} \frac{d}{dt}x_i(t)\nonumber\\ &=\max_{i\in \mathcal{I}_0(t)} \sum\limits_{j \in \mathcal{N}_i}a_{ij}\big(x_j-x_i\big)-\nabla f_i\big(x_i\big) \nonumber\\ &\leq \max_{i\in \mathcal{I}_0(t)} \Big[-\nabla f_i\big(x_i\big) \Big],\end{aligned}$$ where $\mathcal{I}_0(t)$ denotes the index set which contains all the nodes reaching the maximum for $\Psi(x(t))$. Since $$\begin{aligned} 0\geq f_i(x_i^\ast)-f_i(y_\ast+\eta)\geq (x_i^\ast -y_\ast-\eta)\nabla f_i(y_\ast+\eta),\ i=1,\dots,N\end{aligned}$$ we have $\nabla f_i( y^\ast+\eta)\geq 0$ for all $i=1,\dots,N$. As a result, we obtain $$\begin{aligned} D^+\Psi(x(t))\Big|_{\Psi(x(t))=y^\ast+\eta}\leq 0,\end{aligned}$$ which implies $\Psi(x(t))\leq y^\ast+\eta$ for all $t\geq t_0$ under initial condition $\Psi(x(t_0))\leq y^\ast+\eta$. Similar analysis ensures that $\min_{i\in\mathcal{V}} x_i(t)\geq y^\ast-\eta$ for all $t\geq t_0$ as long as $\min_{i\in\mathcal{V}} x_i(t_0)\geq y^\ast-\eta$. This proves the claim. Note that every trajectory of system (\[2\]) under control law $\mathcal{J}_K(n_i,g_i)$ asymptotically solves (\[ka1\]). This immediately leads to that $ F_{\mathcal{G}}(x;K)$ reaches its minimum within $\mathcal{C}_\ast^\eta$ for any $K\geq 0$ since $\mathcal{C}_\ast^\eta$ is an invariant set. Then A5.(ii) holds straightforwardly. c). Since $\arg \min f_i$ is bounded for $i=1,\dots,N$, there exist $b_i\leq d_i, i=1,\dots, N$ such that $\arg \min f_i=[b_i, d_i]$. Define $b_\ast =\min\{b_1,\dots,b_N\}$ and $d^\ast= \max\{d_1,\dots,d_N\}$. We will prove the conclusion by showing $\arg \min F_{\mathcal{G}}(x;K) \subseteq \mathcal{C}_\ast$ for all $K\geq 0$, where $$\mathcal{C}_\ast\doteq \Big\{x=(x_1^T \dots x_N^T)^T: \ x_i \in [b_\ast, d^\ast],i=1,\dots,N\Big\}.$$ Let $z=(z_1 \dots,z_N)^T\in \arg \min F_{\mathcal{G}}(x;K)$. First we show $\max\{z_1,\dots,z_N\} \leq d^\ast$ by a contradiction argument. Suppose $\max\{z_1,\dots,z_N\} > d^\ast$. Now let $i_1,\dots, i_k$ be the nodes reaching the maximum state, i.e., $z_{i_1}=\dots=z_{i_k}=\max\{z_1,\dots,z_N\}$. There will be two cases. - Let $k=N$. We have $z_1=\dots=z_N=y$ in this case. Then for all $i$ and $x_i^\ast \in \arg \min f_i$, we have $$\begin{aligned} 0>f_i(x_i^\ast)-f_i(y)\geq (x_i^\ast -y) \nabla f_i(y)\end{aligned}$$ which yields $ \nabla f_i(y)>0, i=1,\dots,N$ since $y>d^\ast$. This immediately leads to $$\begin{aligned} F_{\mathcal{G}}(z;K)=F(y)>\min F \geq \min F_{\mathcal{G}}(z;K),\end{aligned}$$ which contradicts the fact that $z\in \arg \min F_{\mathcal{G}}(x;K)$. - Let $k<N$. Then we denote $s_\ast=\max\big \{ z_i: i\notin\{i_1,\dots,i_k\}, i=1,\dots,N \big\}$, which is actually the second largest value in $\{z_1,\dots,z_N\}$. We define a new point $\hat{z}=(\hat{z}_1 \dots,\hat{z}_N)^T$ by $\hat{z}_i=z_i, i\notin\{i_1,\dots,i_k\}$ and $$\begin{aligned} \hat{z}_i=\begin{cases} d^\ast, & \mbox{if $s_\ast<d^\ast$}\\ s_\ast, & \mbox{otherwise} \end{cases}\end{aligned}$$ for $i\in \{i_1,\dots,i_k\}$. Then it is easy to obtain that $F_{\mathcal{G}}(z;K)>F_{\mathcal{G}}(\hat{z};K)$, which again contradicts the choice of $z$. Therefore, we have proved that $\max\{z_1,\dots,z_N\} \leq d^\ast$. Based on a symmetric analysis we also have $\min\{z_1,\dots,z_N\} \geq b_\ast$. Therefore, we obtain $\arg \min F_{\mathcal{G}}(x;K) \subseteq \mathcal{C}_\ast$ for all $K\geq 0$ and A5.(iii) follows. $\square$ Time-varying Graphs =================== Now we consider time-varying graphs. The communication in the multi-agent network is modeled as $\mathcal {G}_{\sigma(t)}=(\mathcal {V},\mathcal {E}_{\sigma(t)})$ with $\sigma:[0,+\infty)\rightarrow \mathcal {Q}$ being a piecewise constant function, where $\mathcal {Q}$ is a finite set indicating all possible graphs. In this case the neighbor set for each node is time-varying, and we let $\mathcal{N}_i(\sigma(t))$ represent the set of agent $i$’s neighbors at time $t$. As usual in the literature [@jad03; @lin07; @shi09], an assumption is given to how fast $\mathcal {G}_{\sigma(t)}$ can vary. [**A6.**]{} [*(Dwell Time)*]{} There is a lower bound $\tau_D>0$ between two consecutive switching time instants of $\sigma(t)$. We have the following definition. \(i) $\mathcal {G}_{\sigma(t)}$ is said to be [*uniformly jointly strongly connected*]{} if there exists a constant $T>0$ such that $\mathcal {G}([t,t+T))$ is strongly connected for any $t\geq0$. \(ii) $\mathcal {G}_{\sigma(t)}$ is said to be [*uniformly jointly quasi-strongly connected*]{} if there exists a constant $T>0$ such that $\mathcal {G}([t,t+T))$ has a spanning tree for any $t\geq0$. With time-varying graphs, $$\begin{aligned} n_i(t)\doteq \hbar_i\big(x_i(t), x_j(t): j \in \mathcal{N}_i(\sigma(t)) \big).\end{aligned}$$ where $\hbar_i: \mathds{R}^m\times \mathds{R}^{m|\mathcal{N}_i(\sigma(t))|}\rightarrow \mathds{R}^l$ is now piecewise defined. As a result, assumption A2 is transformed to the following piecewise version. [**A7.**]{} $\hbar\in \mathscr{R}_\ast \doteq \Big\{ h_1 \otimes \dots \otimes h_N$: $h_i$ maps $ \mathds{R}^{m(1+|\mathcal{N}_i(\sigma(t))|)}$ to $\mathds{R}^{l}$ on each time interval when $\sigma(t)$ is constant, and $h_i\equiv0$ within the time-varying local consensus manifold $\big\{x_i=x_{j}: j \in \mathcal{N}_i (\sigma(t))\big\}$ for all $i\in\mathcal{V}\Big\}$. For optimal consensus with time-varying graphs, we present the following result. \[thm3\] Suppose A1 and A6 hold and $\mathcal {G}_{\sigma(t)}$ is uniformly jointly strongly connected. Suppose $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$ contains at least one interior point. Then there exist a neighboring information rule $\hbar \in \mathscr{R}_\ast$ and a control law $\mathcal{J}\in \mathscr{C}$ such that global optimal consensus is achieved and $$\begin{aligned} \label{17} \lim_{t\rightarrow\infty}x_i(t)=x_\ast. \end{aligned}$$ for some $x_\ast\in \bigcap_{i=1}^N \arg \min f_i$. Note that (\[17\]) is indeed a stronger conclusion than our definition of optimal consensus as Theorem \[thm3\] guarantees that all the node states converge to a common point in the global solution set of $F(z)$. We will see from the proof of Theorem \[thm3\] that this state convergence highly relies on the existence of an interior point of $\bigcap_{i=1}^N \arg \min f_i$. In the absence of such an interior point condition, it turns out that optimal consensus still stands. We present another theorem stating the fact. \[thm4\] Suppose A1 and A6 hold and $\mathcal {G}_{\sigma(t)}$ is uniformly jointly strongly connected. Suppose also $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$. Then there exist a neighboring information rule $\hbar \in \mathscr{R}_\ast$ and a control law $\mathcal{J}\in \mathscr{C}$ such that global optimal consensus is achieved. The proofs of Theorems \[thm3\] and \[thm4\] rely on the following neighboring information flow $$\begin{aligned} \label{201} n_i=\sum\limits_{j \in \mathcal{N}_i(\sigma(t))}a_{ij}(t)\big(x_j-x_i\big),\end{aligned}$$ where $a_{ij}(t)>0$ is any weight function associated with arc $(j,i)$. The resulting control law is $$\begin{aligned} \mathcal{J}_\star(n_i,g_i)=n_i-g_i.\end{aligned}$$ An assumption is made on each $a_{ij}(t),i,j=1,2,...,N$. [**A8.**]{} [*(Weights Rule)*]{} (i) Each $a_{ij}(t)$ is piece-wise continuous and $a_{ij}(t)\geq0$ for all $i$ and $j$. (ii). There are $a^\ast>0$ and $a_\ast>0$ such that $ a_\ast\leq a_{ij}(t)\leq a^\ast,\quad t\in \mathds{R}^+.$ Preliminary Lemmas ------------------ We establish three useful lemmas in this subsection. Suppose $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$ and take $z_\ast\in \bigcap_{i=1}^N \arg \min f_i$. We define $$\begin{aligned} V_i(t)=\big|x_i(t)-z_\ast\big|^2,\ i=1,\dots,N,\end{aligned}$$ and $$\begin{aligned} V(t)=\max_{i=1,\dots,N} V_i(t).\end{aligned}$$ The following lemma holds with the proof in Appendix A.1. \[lemmono\] Let A1 and A8 hold. Suppose $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$. Then along any trajectory of system (\[2\]) with neighboring information (\[201\]) and control law $\mathcal{J}_\star(n_i,g_i)$, we have $D^+V(t)\leq 0$ for all $t\in \mathds{R}^+$. A direct consequence of Lemma \[lemmono\] is that when $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$, we have $$\begin{aligned} \lim_{t\rightarrow \infty} V(t)=d_\ast^2\end{aligned}$$ for some $d_\ast\geq 0$ along any trajectory of system (\[2\]) with control law $\mathcal{J}_\star(n_i,g_i)$. However, it is still unclear whether $V_i(t)$ converges or not. We establish another lemma indicating that with proper connectivity condition for the communication graph, all $V_i(t)$’s have the same limit $d_\ast^2$. The proof can be found in Appendix A.2. \[lemlimit\] Let A1, A6, and A8 hold. Suppose $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$ and $\mathcal {G}_{\sigma(t)}$ is uniformly jointly strongly connected. Then along any trajectory of system (\[2\]) with neighboring information (\[201\]) and control law $\mathcal{J}_\star(n_i,g_i)$, we have $\lim_{t\rightarrow \infty} V_i(t)=d_\ast^2 $ for all $i$. The next lemma shows that each node will reach its own optimum along the trajectories of system (\[2\]) under control law $\mathcal{J}_\star(n_i,g_i)$. The proof is in Appendix A.3. \[lemnodeoptimum\] Let A1, A6, and A8 hold. Suppose $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$ and $\mathcal {G}_{\sigma(t)}$ is uniformly jointly strongly connected. Then along any trajectory of system (\[2\]) with control law $\mathcal{J}_\star(n_i,g_i)$, we have $\limsup_{t\rightarrow \infty} \big| x_i(t)\big|_{\arg\min f_i}=0$ for all $i$. Proof of Theorem \[thm3\] ------------------------- The proof of Theorem \[thm3\] relies on the following lemma. \[lemunique\] Let $z_1,\dots,z_{m+1}\in\mathds{R}^m$ and $d_1,\dots,d_{m+1}\in\mathds{R}^+$. Suppose there exist solutions to equations (with variable $y$) $$\label{23} \begin{cases} |y-z_1|^2 =d_1;\\ \ \ \ \ \ \ \vdots\\ |y-z_{m+1}|^2 =d_{m+1}. \end{cases}$$ Then the solution is unique if ${\rm rank}\big(z_2-z_1, \dots, z_{m+1}-z_1\big)=m$. [*Proof.*]{} Take $j>1$ and let $y$ be a solution to the equations. Noticing that $$\langle y-z_1,y-z_1\rangle=d_1; \quad \langle y-z_j,y-z_j\rangle=d_j$$ we obtain $$\begin{aligned} \langle y,z_j-z_1\rangle= \frac{1}{2}\Big(d_1-d_j+|z_j|^2-|z_1|^2\Big), \ j=2,\dots,m+1.\end{aligned}$$ The desired conclusion follows immediately. $\square$ We now prove Theorem \[thm3\]. Let $r_\star=(r_1^T \dots r_N^T)^T$ be a limit point of a trajectory of system (\[2\]) with control law $\mathcal{J}_\star(n_i,g_i)$. We first show consensus. Based on Lemma \[lemlimit\], we have $\lim_{t\rightarrow \infty} V_i(t)=d_\ast$ for all $z_\ast\in\bigcap_{i=1}^N \arg \min f_i$. This is to say, $|r_i-z_\ast|=d_\ast$ for all $i$ and $z_\ast\in\bigcap_{i=1}^N \arg \min f_i$. Since $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$ contains at least one interior point, it is obvious to see that we can find $z_1,\dots,z_{m+1}\in\bigcap_{i=1}^N \arg \min f_i$ with ${\rm rank}\big(z_2-z_1, \dots, z_{m+1}-z_1\big)=m$ and $d_1,\dots,d_{m+1}\in\mathds{R}^+$, such that each $r_i, i=1,\dots,N$ is a solution of equations (\[23\]). Then based on Lemma \[lemunique\], we conclude that $r_1=\dots=r_N$. Next, with Lemma \[lemnodeoptimum\], we have $| r_i|_{\arg\min f_i}=0$. This implies that $r_1=\dots=r_N\in\bigcap_{i=1}^N \arg \min f_i$, i.e., optimal consensus is achieved. We turn to state convergence. We only need to show that $r_\star$ is unique along any trajectory of system (\[2\]) with neighboring information (\[201\]) and control law $\mathcal{J}_\star(n_i,g_i)$. Now suppose $r_\star^1=\mathbf{1}_N\otimes r^1$ and $r_\star^2=\mathbf{1}_N\otimes r^2$ are two different limit points with $r^1\neq r^2 \in \bigcap_{i=1}^N \arg \min f_i$. According to the definition of a limit point, we have that for any $\varepsilon>0$, there exists a time instant $t_\varepsilon$ such that $|x_i(t_\varepsilon)-r^1|\leq \varepsilon$ for all $i$. Note that Lemma \[lemmono\] indicates that the disc $B(r^1,\varepsilon)=\{y: |y-r^1|\leq \varepsilon\}$ is an invariant set for initial time $t_\varepsilon$. While taking $\varepsilon={|r^1-r^2|}/{4}$, we see that $r^2\notin B(r^1,|r^1-r^2|/{4})$. Thus, $r^2$ cannot be a limit point. Now since the limit point is unique, we denote it as $\mathbf{1}_N\otimes x_\ast$ with $x_\ast\in \bigcap_{i=1}^N \arg \min f_i$. Then we have $\lim_{t\rightarrow\infty}x_i(t)=x_\ast$ for all $i=1,\dots,N$. This completes the proof. Proof of Theorem \[thm4\] ------------------------- In this subsection, we prove Theorem \[thm4\]. We need the following lemma on robust consensus, which can be found in [@shicdc]. \[lemrobust\] Consider a network with node set $\mathcal{V}=\{1,\dots,N\}$ with time-varying communication graph $\mathcal{G}_{\sigma(t)}$. Let the dynamics of node $i$ be $$\begin{aligned} \dot{x}_i=\sum\limits_{j \in \mathcal{N}_i(\sigma(t))}a_{ij}(t)\big(x_j-x_i\big)+w_i(t),\end{aligned}$$ where $w_i(t)$ is a piecewise continuous function. Suppose A6 and A8 hold and $\mathcal{G}_{\sigma(t)}$ is uniformly jointly quasi-strongly connected. Then we have $$\lim_{t\rightarrow +\infty} \big |x_i(t)-x_j(t)\big|=0,\quad i,j=1,\dots,N$$ if $\lim_{t\rightarrow \infty}w_i(t)=0$ for all $i$. Lemma \[lemnodeoptimum\] indicates that $\limsup_{t\rightarrow \infty} \big| x_i(t)\big|_{\arg\min f_i}=0$ for all $i$, which yields $$\begin{aligned} \lim_{t\rightarrow \infty}\nabla f_i\big(x_i(t)\big)=0\end{aligned}$$ for all $i$ according to Assumption A1. Then the consensus part in the definition of optimal consensus follows immediately from Lemma \[lemrobust\]. Again by Lemma \[lemnodeoptimum\], we further conclude that $\limsup_{t\rightarrow \infty} {\rm dist}\big( x_i(t), \bigcap_{i=1}^N \arg \min f_i\big)=0$. The desired conclusion thus follows. Conclusions =========== Various algorithms have been proposed in the literature for the distributed minimization of $\sum_{i=1}^N f_i$ with $f_i$ only known to node $i$. This paper explored some fundamental properties for distributed methods given a certain level of node knowledge, computational capacity, and information flow. It was proven that there exists a control law that ensures global optimal consensus if and only if $\arg \min f_i, i=1,\dots,N$, admit a nonempty intersection set for fixed strongly connected graphs. We also showed that for any error bound, we can find a control law which guarantees global optimal consensus within this bound for fixed, bidirectional, and connected graphs under some mild conditions such as that $f_i$ is coercive for some $i$. For time-varying graphs, it was proven that optimal consensus can always be achieved as long as the graph is uniformly jointly strongly connected and the nonempty intersection condition holds. It was then concluded that nonempty intersection for the local optimal solution sets is a critical condition for distributed optimization using consensus processing. More challenges lie in exploring the corresponding limit of performance for high-order schemes, the optimal structure of the underlying communication graph for distributed optimization, and the fundamental communication complexity required for global convergence. Acknowledgment {#acknowledgment .unnumbered} =============== The authors would like to thank Prof. Angelia Nedić for helpful discussions and for pointing out relevant literature. Appendix {#appendix .unnumbered} ======== **A.1   Proof of Lemma \[lemmono\]** {#a.1-proof-of-lemma-lemmono .unnumbered} ------------------------------------- Based on Lemma \[lemdini\], we have $$\begin{aligned} \label{20} D^+V(t)&=\max_{i\in \mathcal{I}(t)} \frac{d}{dt}V_i(t)\nonumber\\ &=\max_{i\in \mathcal{I}(t)} 2\Big\langle x_i(t)-z_\ast, \sum\limits_{j \in \mathcal{N}_i(\sigma(t))}a_{ij}(t)\big(x_j-x_i\big)-\nabla f_i\big(x_i\big)\Big\rangle,\end{aligned}$$ where $\mathcal{I}(t)$ denotes the index set which contains all the nodes reaching the maximum for $V(t)$. Let $m\in\mathcal{I}(t)$. Denote $$Z_t=\big\{z:\ |z-z_\ast|\leq \sqrt{V(t)} \big\}$$ as the disk centered at $z_\ast$ with radius $\sqrt{V(t)}$. Take $y=x_m(t)+(x_m(t)-z_\ast)$. Then from some simple Euclidean geometry it is obvious to see that $P_{Z_t}(y)=x_m(t)$, where $P_{Z_t}$ is the [*projector*]{} onto $Z_t$. Thus, for all $j\in\mathcal{N}_{m}(\sigma(t))$, we obtain $$\begin{aligned} \label{18} \big\langle x_m(t)-z_\ast,x_j(t)-x_m(t)\big\rangle&=\big\langle y-x_m(t),x_j(t)-x_m(t)\big\rangle\nonumber\\ &=\big\langle y-P_{Z_t}(y),x_j(t)-P_{Z_t}(y)\big\rangle\nonumber\\ &\leq 0\end{aligned}$$ according to inequality (i) in Lemma \[lemconvex\] since $x_j(t)\in Z_t$. On the other hand, based on inequality (i) in Lemma \[lemfunction\], we also have $$\begin{aligned} \label{19} \big\langle x_m(t)-z_\ast,-\nabla f_m\big(x_m(t)\big)\big\rangle\leq f_m(z_\ast)-f_m\big(x_m(t)\big) \leq 0\end{aligned}$$ in light of the definition of $z_\ast$. With (\[20\]), (\[18\]) and (\[19\]), we conclude that $$\begin{aligned} D^+ V(t) =\max_{i\in \mathcal{I}(t)} 2\big\langle x_i(t)-z_\ast, \sum\limits_{j \in \mathcal{N}_i(\sigma(t))}a_{ij}(t)\big(x_j-x_i\big)-\nabla f_i\big(x_i\big)\big\rangle\leq 0,\end{aligned}$$ which completes the proof. $\square$ **A.2   Proof of Lemma \[lemlimit\]** {#a.2-proof-of-lemma-lemlimit .unnumbered} -------------------------------------- In order to prove the desired conclusion, we just need to show $\liminf_{t\rightarrow \infty} V_i(t)=d_\ast^2$ for all $i$. With Lemma \[lemmono\], we conclude that $\forall\varepsilon>0, \exists M(\varepsilon)>0$, s.t., $$\begin{aligned} \label{24} \sqrt{V_i(t)}\leq d_\ast+\varepsilon\end{aligned}$$ for all $i$ and $t\geq M$. For all $t\geq M$ and all $i,j\in\mathcal{V}$, we have $$\begin{aligned} \label{25} \big\langle x_i(t)-z_\ast,x_j(t)-x_i(t) \big\rangle\leq-V_i(t)+(d_\ast+\varepsilon)\sqrt{V_i(t)}.\end{aligned}$$ If $x_i(t)=z_\ast$ (\[25\]) follows trivially from (\[24\]). Otherwise we take $y_\ast= z_\ast+ (d_\ast+\varepsilon)\frac{x_i(t)-z_\ast}{|x_i(t)-z_\ast|}$ and $B_t=\big\{z: |z-z_\ast|\leq d_\ast+\varepsilon\big \}$. Here $B_t$ is the disk centered at $z_\ast$ with radius $d_\ast+\varepsilon$, and $y_\ast$ is a point within the boundary of $B_t$ and falls the same line with $z_\ast$ and $x_{i_0}(t)$. Take also $q_\ast=y_\ast+x_i(t)-z_\ast$. Then we have $$\begin{aligned} \big\langle x_i(t)-z_\ast,x_j(t)-y_\ast \big\rangle&=\big\langle q_\ast-y_\ast,x_j(t)-y_\ast \big\rangle\nonumber\\ &=\big\langle q_\ast-P_{B_t}(q_\ast),x_j(t)-P_{B_t}(q_\ast) \big\rangle\nonumber\\ &\leq 0\end{aligned}$$ according to inequality (i) in Lemma \[lemconvex\], which leads to $$\begin{aligned} \big\langle x_i(t)-z_\ast,x_j(t)-x_i(t) \big\rangle&=\big\langle x_i(t)-z_\ast,x_j(t)-y_\ast \big\rangle+\big\langle x_i(t)-z_\ast,y_\ast-x_i(t) \big\rangle\nonumber\\ &\leq\big\langle x_i(t)-z_\ast,y_\ast-x_i(t)\big\rangle\nonumber\\ &=-V_i(t)+(d_\ast+\varepsilon)\sqrt{V_i(t)}.\end{aligned}$$ This proves the claim. Now suppose there exists $i_0\in\mathcal{V}$ with $\liminf_{t\rightarrow \infty} V_i(t)=\theta_{i_0}^2<d_\ast^2$. Then we can find a time sequence $\{t_k\}_1^\infty$ with $\lim_{k\rightarrow \infty}t_k =\infty$ such that $$\begin{aligned} \label{28} \sqrt{V_{i_0}(t_k)}\leq \frac{\theta_{i_0}+d_\ast}{2}.\end{aligned}$$ We divide the rest of the proof into three steps. [*Step 1.*]{} Take $t_{k_0}>M$. We bound $V_{i_0}(t)$ in this step. With the weights rule A8, (\[25\]) and inequality (i) in Lemma \[lemfunction\], we see that $$\begin{aligned} \label{45} \frac{d}{dt} V_{i_0}(t)&=2\Big \langle x_{i_0}(t)-z_\ast, \sum\limits_{j \in \mathcal{N}_{i_0}(\sigma(t))}a_{i_0j}(t)\big(x_j-x_{i_0}\big)-\nabla f_{i_0}\big(x_{i_0}(t)\big) \Big\rangle\nonumber\\ &\leq 2\sum\limits_{j \in \mathcal{N}_{i_0}(\sigma(t))}a_{i_0j}(t) \Big \langle x_{i_0}(t)-z_\ast,x_j(t)-x_{i_0}(t) \Big\rangle+f_{i_0}\big(z_\ast\big)-f_{i_0}\big(x_{i_0}(t)\big)\nonumber\\ &\leq 2(N-1)a^\ast\Big(-V_{i_0}(t)+(d_\ast+\varepsilon)\sqrt{V_{i_0}(t)}\Big),\end{aligned}$$ for all $t\geq t_{k_0}$, which implies $$\begin{aligned} \label{26} \frac{d}{dt}\sqrt{V_{i_0}(t)} \leq -(N-1)a^\ast\Big(\sqrt{V_{i_0}(t)}-(d_\ast+\varepsilon)\Big),\ \ t\geq t_{k_0}.\end{aligned}$$ In light of Grönwall’s inequality, (\[28\]) and (\[26\]) yield $$\begin{aligned} \label{30} \sqrt{V_{i_0}(t)} &\leq e^{-(N-1)^2a^\ast T_D}\sqrt{V_{i_0}(t_{k_0})}+\Big(1-e^{-(N-1)^2a^\ast T_D}\Big)(d_\ast+\varepsilon)\nonumber\\ &\leq \frac{e^{-(N-1)^2a^\ast T_D}}{2} \theta_{i_0}+\Big(1-\frac{e^{-(N-1)^2a^\ast T_D}}{2}\Big)(d_\ast+\varepsilon)\nonumber\\ &\doteq \Lambda_\ast.\end{aligned}$$ for all $t\in[t_{k_0}, t_{k_0}+(N-1)T_D]$ with $T_D=T+\tau_D$, where $T$ comes from the definition of uniformly jointly strongly connected graphs and $\tau_D$ represents the dwell time. [*Step 2.*]{} Since the graph is uniformly jointly strongly connected, we can find an instant $\hat{t}\in[t_{k_0},t_{k_0}+T]$ and another node $i_1\in\mathcal{V}$ such that $(i_0,i_1)\in\mathcal{G}_{\sigma(t)}$ for $t\in[\hat{t}, \hat{t}+\tau_D]$. In this step, we continue to bound $V_{i_1}(t)$. Similar to (\[25\]), for all $t\geq M$ and all $i,j\in\mathcal{V}$, we also have $$\begin{aligned} \label{29} \big\langle x_i(t)-z_\ast,x_j(t)-x_i(t) \big\rangle\leq-\sqrt{V_i(t)}\Big(\sqrt{V_i(t)}-\sqrt{V_j(t)}\Big)\end{aligned}$$ when $V_j(t)\leq V_i(t)$. Then based on (\[25\]), (\[30\]), and (\[29\]), we obtain $$\begin{aligned} \label{31} \frac{d}{dt} V_{i_1}(t) &\leq 2\sum\limits_{j \in \mathcal{N}_{i_1}(\sigma(t))}a_{i_1j}(t) \Big \langle x_{i_1}(t)-z_\ast,x_j(t)-x_{i_1}(t) \Big\rangle\nonumber\\ &= 2\sum\limits_{j \in \mathcal{N}_{i_1}(\sigma(t))\setminus \{i_0\}}a_{i_1j}(t) \Big \langle x_{i_1}(t)-z_\ast,x_j(t)-x_{i_1}(t) \Big\rangle+2a_{i_1i_0}(t) \Big \langle x_{i_1}(t)-z_\ast,x_{i_0}(t)-x_{i_1}(t) \Big\rangle\nonumber\\ &\leq 2(N-2)a^\ast\Big(-V_{i_1}(t)+(d_\ast+\varepsilon)\sqrt{V_{i_1}(t)}\Big)-2a_\ast\sqrt{V_{i_1}(t)}\Big(\sqrt{V_{i_1}(t)}-\sqrt{V_{i_0}(t)}\Big)\nonumber\\ &\leq- 2\Big((N-2)a^\ast+a_\ast\Big)V_{i_1}(t) +2\sqrt{V_{i_1}(t)} \Big((N-2)a^\ast(d_\ast+\varepsilon)+\Lambda_\ast a_\ast\Big)\end{aligned}$$ for $t\in[\hat{t},\hat{t}+\tau_D]$, where without loss of generality we assume $V_{i_1}(t)\geq V_{i_0}(t)$ during all $t\in[\hat{t},\hat{t}+\tau_D]$. Then (\[31\]) gives $$\begin{aligned} \frac{d}{dt} \sqrt{V_{i_1}(t)} &\leq- \Big((N-2)a^\ast+a_\ast\Big)\sqrt{V_{i_1}(t)} + \Big((N-2)a^\ast(d_\ast+\varepsilon)+\Lambda_\ast a_\ast\Big), t\in[\hat{t},\hat{t}+\tau_D]\end{aligned}$$ which yields $$\begin{aligned} \sqrt{V_{i_1}(\hat{t}+\tau_D)}&\leq e^{- \big((N-2)a^\ast+a_\ast\big)\tau_D}(d_\ast+\varepsilon)+\Big(1-e^{- \big((N-2)a^\ast+a_\ast\big)\tau_D}\Big)\frac{(N-2)a^\ast(d_\ast+\varepsilon)+\Lambda_\ast a_\ast}{(N-2)a^\ast+a_\ast}\nonumber\\ &=\frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\times\frac{e^{-(N-1)^2a^\ast T_D}}{2} \theta_{i_0}\nonumber\\ &\ \ \ \ \ \ \ \ \ \ \ +\Big(1-\frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\times\frac{e^{-(N-1)^2a^\ast T_D}}{2}\Big)(d_\ast+\varepsilon)\end{aligned}$$ again by Grönwall’s inequality and some simple algebra. Next, applying the estimate of node $i_0$ in step 1 on $i_1$ during time interval $[\hat{t}+\tau_D,t_{k_0}+(N-1)T_D]$, we arrive at $$\begin{aligned} \sqrt{V_{i_1}(t)}&\leq \frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\times\frac{e^{-2(N-1)^2a^\ast T_D}}{2} \theta_{i_0}\nonumber\\ &\ \ \ \ \ \ \ \ \ \ \ +\Big(1-\frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\times\frac{e^{-2(N-1)^2a^\ast T_D}}{2}\Big)(d_\ast+\varepsilon)\end{aligned}$$ for all $t\in[t_{k_0}+T_D, t_{k_0}+(N-1)T_D]$. [*Step 3.*]{} Noticing that the graph is uniformly jointly strongly connected, the analysis of steps 1 and 2 can be repeatedly applied to nodes $i_3,\dots,i_{N-1}$, and eventually we have that for all $i_0,\dots,i_{N-1}$, $$\begin{aligned} \sqrt{V_{i_m}\big( t_{k_0}+(N-1)T_D\big)}&\leq \Big(\frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\Big)^{N-2}\times\frac{e^{-(N-1)^3a^\ast T_D}}{2} \theta_{i_0}\nonumber\\ &\ \ \ \ \ \ \ \ \ \ \ +\Bigg(1-\Big(\frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\Big)^{N-2}\times\frac{e^{-(N-1)^3a^\ast T_D}}{2} \Bigg)(d_\ast+\varepsilon)\nonumber\\ &<d_\ast\end{aligned}$$ for sufficiently small $\varepsilon$ because $\theta_{i_0}<d_\ast$ and $$\Big(\frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\Big)^{N-2}\times\frac{e^{-(N-1)^3a^\ast T_D}}{2} <1$$ is a constant. This immediately leads to that $$\begin{aligned} V\big(t_{k_0}+(N-1)T_D\big)<d_\ast,\end{aligned}$$ which contradicts the definition of $d_\ast$. This completes the proof. **A.3   Proof of Lemma \[lemnodeoptimum\]** {#a.3-proof-of-lemma-lemnodeoptimum .unnumbered} -------------------------------------------- With Lemma \[lemlimit\], we have that $\lim_{t\rightarrow \infty} V_i(t)=d_\ast^2$ for all $i\in\mathcal{V}$. Thus, $\forall\varepsilon>0, \exists M(\varepsilon)>0$, s.t., $$\begin{aligned} \label{40} d_\ast\leq \sqrt{V_i(t)}\leq d_\ast+\varepsilon\end{aligned}$$ for all $i$ and $t\geq M$. If $d_\ast=0$, the desired conclusion follows straightforwardly. Now we suppose $d_\ast>0$. Assume that there exists a node $i_0$ satisfying $\limsup_{t\rightarrow \infty} \big| x_{i_0}(t)\big|_{\arg\min f_{i_0}}>0$. Then we can find a time sequence $\{t_k\}_1^\infty$ with $\lim_{k\rightarrow \infty}t_k =\infty$ and a constant $\delta$ such that $$\begin{aligned} \label{42} \big| x_{i_0}(t_k)\big|_{\arg\min f_{i_0}}\geq\delta, \ k=1,\dots.\end{aligned}$$ Denote also $B_1\doteq\big\{z: |z-z_\ast|\leq d_\ast+1\big \}$ and $G_1=\max\big\{ \nabla f_{i_0}(y):\ y\in B_1\big\}$. Assumption A1 ensures that $G_1$ is a finite number since $B_1$ is compact. By taking $\varepsilon=1$ in (\[40\]), we see that $x_i(t)\in B_1$ for all $i$ and $t\geq M(1)$. As a result, we have $$\begin{aligned} \label{41} \Big|\frac{d}{dt}{x}_{i_0}(t)\Big|=\Big|\sum_{j\in\mathcal{N}_{i_0}(\sigma(t))} a_{i_0 j}(t)(x_j-x_{i_0})+\nabla f_{i_0}(x_{i_0})\Big|\leq 2(n-1) a^\ast (d_\ast+1)+G_1.\end{aligned}$$ Combining (\[42\]) and (\[41\]), we conclude that $$\begin{aligned} \label{43} \big| x_{i_0}(t)\big|_{\arg\min f_{i_0}}\geq \frac{\delta}{2}, \ t\in[t_k,t_k+\tau],\end{aligned}$$ for all $k=1,\dots$, where by definition $\tau=\frac{\delta}{2\big( 2(n-1) a^\ast (d_\ast+1)+G_1\big)}$. Now we introduce $$D_\delta\doteq \min \Big\{f_{i_0}(y)-f_{i_0}(z_\ast):\ \big| x_{i_0}(t)\big|_{\arg\min f_{i_0}}\geq \frac{\delta}{2}\ {\rm and}\ y\in B_1\Big\}.$$ Then we know $D_\delta >0$ again by the continuity of $f_{i_0}$. According to (\[45\]), (\[40\]), and (\[43\]), we obtain $$\begin{aligned} \frac{d}{dt} V_{i_0}(t) &\leq 2(N-1)a^\ast\Big(-V_{i_0}(t)+(d_\ast+\varepsilon)\sqrt{V_{i_0}(t)}\Big)+f_{i_0}\big(z_\ast\big)-f_{i_0}\big(x_{i_0}(t)\big)\nonumber\\ &\leq 2(N-1)a^\ast (d_\ast+\varepsilon)\varepsilon -D_\delta,\end{aligned}$$ for $t\in [t_k,t_k+\tau]$, $k=1,\dots$. This leads to $$\begin{aligned} \label{46} V_{i_0}(t_k+\tau)&\leq V_{i_0}(t_k)+\Big(2(N-1)a^\ast (d_\ast+\varepsilon)\varepsilon -D_\delta\Big)\tau \nonumber\\ &\leq d_\ast+\varepsilon+\Big(2(N-1)a^\ast (d_\ast+\varepsilon)\varepsilon -D_\delta\Big)\tau\nonumber\\ &< d_\ast\end{aligned}$$ as long as $\varepsilon$ is sufficiently small so that $$\varepsilon \Big(1+2(N-1)a^\ast (d_\ast+\varepsilon)\Big) <D_\delta \tau.$$ We see that (\[46\]) contradicts (\[40\]). The desired conclusion thus follows. [99]{} S. Boyd and L. Vandenberghe. [*Convex Optimization*]{}. New York, NY: Cambridge University Press, 2004. D. P. Bertsekas. 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--- abstract: 'Deep learning based natural language processing model is proven powerful, but need large-scale dataset. Due to the significant gap between the real-world tasks and existing Chinese corpus, in this paper, we introduce a large-scale corpus of informal Chinese. This corpus contains around 37 million book reviews and 50 thousand netizen’s comments to the news. We explore the informal words frequencies of the corpus and show the difference between our corpus and the existing ones. The corpus can be further used to train deep learning based natural language processing tasks such as Chinese word segmentation, sentiment analysis.' author: - | Jianyu Zhao\ School of Data Science\ Fudan University, China\ `18210980101@fudan.edu.cn`\ `zhaojianyu@gsqtec.com`\ Zhuoran Ji\ GSQ Tec.\ Shen Zhen, China\ `jizhuoran@gsqtec.com`\ bibliography: - 'references.bib' title: 'LSICC: A Large Scale Informal Chinese Corpus' --- Introduction ============ Deep learning has been the mainstay for natural language processing, ranging from text summarization [@paulus2017deep] to sentiment analysis [@zhang2018deep] to text generation [@sutskever2014sequence] and automated question-answering system [@yu2014deep]. Unlike traditional rule-based methods, the scale and quality of the corpus significantly influence the performance of the deep learning models. In Chinese NLP field, there are many famous large-scale corpora with high quality, such as Baidu Encyclopedia, People’s Daily News and Sina Weibo News. Various powerful Chinese deep learning models are trained on these corpora [@li2018analogical], [@min2015bosonnlp], [@cui2016consensus], [@nallapati2016abstractive], [@gu2016incorporating]. However, most Chinese corpora are in written Chinese, while most real-world deep learning based NLP systems deal with informal Chinese, such as products reviews, netizens’ opinions, and microblogs. There are great gaps between informal Chinese and written Chinese, especially in words usages and sentences structures. The pre-trained deep learning model trained from written Chinese corpus, such as words embedding and Chinese words segmentation tools, may perform badly on tasks with informal Chinese. To address this issue, we introduce LSICC, a large-scale corpus of informal Chinese. Containing around 37 million book reviews and 50 thousand netizens’ opinions to news, LSICC is a typical informal Chinese corpus. Most sentences of LSICC are in spoken Chinese and even Internet slang. As far as we know, LSICC is the first large-scale, well-formatted, cleansed corpus focusing on informal Chinese. This paper makes the following contributions: 1. collect a large scale corpus of informal Chinese 2. filter out the informationless data items 3. compare the proportions of informal words in several corpus Informal Chinese ================ Informal Chinese, including spoken Chinese and Chinese Internet Slang, has a substantial difference with the formal one, in both grammar and words usage. In this section, we discuss the difference between formal Chinese and informal Chinese. Spoken Chinese -------------- For most language, there are differences between the spoken one and the written one. In Chinese, the gap is even more significant due to the long history of written Chinese. Similar to another language, spoken Chinese sometimes does not follow the rules as strictly as written Chinese, especially for the elliptical sentences. For example, in spoken Chinese, the subjects sometimes are omitted. In addition to the grammar, the usage of the words influences the neural network based Chinese natural language processing model most. There are various interchangeable words pairs between written Chinese and spoken Chinese, such as [UTF8]{}[gbsn]{}“脑袋” and [UTF8]{}[gbsn]{}“头部” , which both mean “head” in Chinese. The two words in each interchangeable words pair usually have almost the same meanings, but the one in written Chinese is more formal, while the one in spoken Chinese is informal. Internet Slang -------------- Born in the 1990s, Chinese Internet slang refers to various kinds of slang created by netizens and used in chat rooms, social networking services, and online community. Nowadays, Chinese Internet slang is not little memes within internet ingroup, but becoming popular language style of all Chinese speakers. From 2012, Xinhuanet selects “Top 10 Chinese Internet Slang” [@topten] every year, and Chinese Internet slang is used even by Chinese official institutions. The first kind of Internet slang is the phonetic substitution, whose pronunciation is same or similar to the formal phrase. For example, in Internet slang, people may use [UTF8]{}[gbsn]{}“神马” to replace [UTF8]{}[gbsn]{}“什么” . Both [UTF8]{}[gbsn]{}“神马” and [UTF8]{}[gbsn]{}“什么” are pronounced as “cien ma” and has the meaning of “what”. However, in written Chinese, [UTF8]{}[gbsn]{}“神马” means “horse-god”, while [UTF8]{}[gbsn]{}“什么” means “what”. Transliteration is also a primary way to form Internet slang. As the words are transliterated from another language, both the meaning and pronunciation of the transliterated words are similar to the source language. For example, [UTF8]{}[gbsn]{}“伐木累” is transliterated from English word “family” and only used as Chinese Internet slang [@Li:2008:MMR:1613715.1613849]. Meanwhile, Internet slang is also created by giving new meanings to the old words. For example, in written Chinese, [UTF8]{}[gbsn]{}“酱油” means “soy s sauce”. However, in the Chinese Internet slang, it refers to “passing by". Data Collection =============== LSICC collects book reviews from DouBan Dushu and netizen’s opinions from Chiphell. This section describes these two datasets and pre-processing methods briefly. DouBan DuShu ------------ DouBan DuShu[^1] is a Chinese website where users can share their reviews about various kinds of books. Most of the users on this website are unprofessional book reviewers. Therefore, the comments are usually spoken Chinese or even Internet slang. In addition to the comments, users can mark the books from one star to 5 stars according to the quality of the books. We have collected more than 37 million short comments from about 18 thousand books with 1 million users. The great number of users provide diversities of the language styles, from moderate formal to informal. An example of the data item is shown in table \[douban\]. [l|l|l]{} Key & Description & Value Example\ Book Name & The name of the book & [UTF8]{}[gbsn]{}理想国 \ User Name & Who gives the comment (anonymized) & 399\ Tag & The tag the book belongs to & [UTF8]{}[gbsn]{}思想 \ Comment & Content of the comment & [UTF8]{}[gbsn]{}我是国师的脑残粉 \ Star & Stars given to the book (from 1 star to 5 stars) & 5 stars\ Date & When the comment posted & 2018-08-21\ Like & Count of “like” on the comment & 0\ Chiphell -------- Chiphell [^2] is a web portal where netizens share their views to news and discuss within groupuscule. We have collected discussion forums from several subjects, such as computer hardware, motors and clothes. There are more than 50 thousand discussions in the corpus. Similar to the DouBan DuShu corpus, most of the sentences collected from Chiphell are informal Chinese and some of them are in particular domains. An example from each subject is shown in table \[chh\]. [l|p[6cm]{}|p[6cm]{}]{} Subject & Topic & Example\ News & [UTF8]{}[gbsn]{}美机场航空业希望修改客机降落的Emoji表情：机头朝下不吉利 & [UTF8]{}[gbsn]{}那我还说改完的意思是无限复飞呢,飞到没油不又gg了 \ Computer Hardware & [UTF8]{}[gbsn]{}请问现在大船货除开3610还有其他性价比的大船大容量吗 & [UTF8]{}[gbsn]{}我1T的PM1633。。卖1300都木有人接 \ Mobile Phones & [UTF8]{}[gbsn]{}努比亚X 综合讨论帖 & [UTF8]{}[gbsn]{}MIX3辣鸡被友商各种吊打 \ Clothes & [UTF8]{}[gbsn]{}程序媛的皮艺生活 & [UTF8]{}[gbsn]{}花点时间在复杂又感兴趣的事情上是一件快乐又有成就感的体验 \ Data Pre-processing ------------------- In addition to the raw dataset, we extracted the comments and preprocessed them to provide a clean, formal formatted and comprehensive Chinese corpus. After carefully investigate the raw text, mainly three preprocessing methods are applied: 1. convert Traditional Chinese to Simplified Chinese 2. remove over-short comments (less than 4 characters) 3. add identifier to special characters, such as special signs, English words and emoticons Experiments =========== To further explore the informal Chinese corpus, we calculate the proportion of informal words in the corpus. The experiment is conducted on Weibo News [@hu2015lcsts], Sougou News, People’s Daily [@yu2001guideline] and the LSICC. We manually collected 70 informal words as the benchmark, which covers both spoken Chinese words and Chinese network slang words. We counted the frequencies of informal words and the number of total words to calculate the proportion of the informal words in the whole corpus. As shown in table \[proportion\], the LSICC has the highest proportion of the informal words, which is more than two times the second highest one, the Weibi News. Noted that the more formal the media is, the lower the proportion of the informal words in it. Corpus Informal Words Total Words Proportion ---------------- ---------------- ------------- ------------ LSICC 621807 705231306 8.82 Weibo News 46831 125082112 3.74 Sougou News 1238 14160148 0.87 People’s Daily 25 3482887 0.07 : Proportion of the informal words in each corpus \[proportion\] The result indicated that the gap between the language that the real-world natural language models deal with the existing corpora is significant. Using the vector representations extracted from the corpus of formal Chinese as the word embedding may attribute to poor performance. Conclusions and Future Work =========================== We constructed a large-scale Informal Chinese dataset and conducted a basic words frequency statistic experiment on it. Compared to the existing Chinese corpus, LSICC is more typical dataset for real-world natural language processing tasks, especially for sentiment analysis. As a next step, we should conduct embedding extraction Chinese words segmentation and sentiment analysis on LSICC. Meanwhile, as the raw information, such as the usernames and book names is kept, LSICC can also be used to build recommendation systems and explore social network. [^1]: available on: https://github.com/JaniceZhao/Douban-Dushu-Dataset.git [^2]: available on: https://github.com/JaniceZhao/Chinese-Forum-Corpus.git

--- abstract: 'Let $E$ be an elliptic curve over the rationals. Let $L$ be an infinite Galois extension of the rationals with uniformly bounded local degrees at almost all primes. We will consider the infinite extension $L(E_{\text{tor}})$ of the rationals where we adjoin all coordinates of torsion points of $E$. In this paper we will prove an effective (and in the non-CM case even explicit) lower bound for the height of non-zero elements in $L(E_{\text{tor}})$ that are not a root of unity.' address: 'Department of Mathematics, Universitetsparken 5, 2100 Copenhagen Denmark ' author: - Linda Frey bibliography: - 'Bibliographie.bib' title: 'Small Heights in Large non-Abelian Extensions' --- Introduction {#Introduction} ============ Let $E$ be an elliptic curve defined over ${\mathds Q}$, let $\mu_\infty$ be the set of all roots of unity and let ${\mathds Q}(E_\text{tor})$ be the smallest field extension of ${\mathds Q}$ that contains all coordinates of torsion points of $E$. In 2013 Habegger [@MR3090783] showed that in ${\mathds Q}(E_\text{tor})^* \setminus \mu_\infty$ the height is bounded from below by a positive constant. In an earlier paper [@2017arXiv171204214F], the author proves an explicit lower height bound in that case. We will generalize these results and allow larger base fields as follows. \[generalization\] Let $E$ be an elliptic curve over ${\mathds Q}$. Let $L$ be a (possibly infinite) Galois extension of ${\mathds Q}$ with uniformly bounded local degrees by $d\in{\mathds N}$. Then the height in $L(E_{\text{tor}})^* \setminus \mu_\infty$ is bounded from below by a positive constant. We can even give an explicit formula for such a lower height bound. \[generalization\] Let $E$ be an elliptic curve over ${\mathds Q}$. Let $L$ be a (possibly infinite) Galois extension of ${\mathds Q}$ with uniformly bounded local degrees by $d\in{\mathds N}$. Let $p$ be a prime such that $p$ is surjective, supersingular and greater than $\max(2d+2,\exp(\operatorname{Gal}(L/{\mathds Q})))$. Then for any $\alpha \in L(E_{\text{tor}})^* \setminus \mu_\infty$ we have $h(\alpha) \geq \frac{(\log p)^4}{p^{5p^4}}$. For the definitions of surjective and supersingular see section \[NotNFC\]. Remark that, by [@MR3009657], we have that $\max(2d+2,\exp(\operatorname{Gal}(L/{\mathds Q})))$ is always finite. Furthermore, in an earlier paper [@2017arXiv171204214F], the author proves an explicit upper bound for such a prime that only depends on the $j$-invariant (or the conductor, respectively) of $E$ and $\max(2d+2,\exp(\operatorname{Gal}(L/{\mathds Q})))$.\ The proof of our Theorem \[generalization\] involves the theory of local fields, ramification theory and Galois theory. In his proof, Habegger makes heavy use of the Frobenius. In our generalized case, we can not always be sure that there exists a lift of the Frobenius. We will work around that by taking suitable powers of suitable morphisms. Another key ingredient in Habegger’s proof are non-split Cartan subgroups. In our proof we can completely work around that by considering the unramified and the tamely ramified case together.\ Acknowledgements {#acknowledgements .unnumbered} ---------------- I am very thankful for all the people who helped me write this article, in particular the following. I thank Philipp Habegger who proposed this problem to me. He gave me many helpful comments and productive input. I thank Francesco Amoroso for giving me helpful comments and helping me solve the CM case. I thank Gabriel Dill for his amazing accurateness and great patience while reading my manuscripts over and over again. I thank Waltraut Lederle for helping me with my group theory issues. This research was done during my PhD[^1] at Universität Basel in the DFG project 223746744 “Heights and unlikely intersections” and written up during my SNF grant Early.PostDoc Mobility at the University of Copenhagen.\ Infinite base fields ==================== Let $L$ be a Galois extension of ${\mathds Q}$, $S$ a set of prime numbers and $d \in{\mathds N}$. We say that $L$ has *uniformly bounded local degrees* above $S$ by $d$ if and only if for all primes $p \in S$ and $v$ extending $p$, we have $[L_v:{\mathds Q}_p] \leq d$. Here, $L_v$ is the $v$-adic completion of $L$. Our goal is proving the following theorem. Let $E$ be an elliptic curve over ${\mathds Q}$ and let $L/{\mathds Q}$ be a Galois extension with uniformly bounded local degrees above all but finitely many primes. Then $L(E_{\text{tor}})$ has the Bogomolov property. We will use the following result of Checcoli to make use of the uniform boundedness. \[Sara\] Let $L/{\mathds Q}$ be a Galois extension. Then the following conditions are equivalent: - $L$ has uniformly bounded local degrees above every prime. - $L$ has uniformly bounded local degrees above all but finitely many primes. - $\operatorname{Gal}(L/{\mathds Q})$ has finite exponent. Remark that uniformly bounded means that the degrees are bounded independently of $p$. In a paper of Checcoli and Zannier [@MR2755687] there is also the implication $(2) \Rightarrow (3)$ from the above theorem. But since we need the stronger implication $(1) \Rightarrow (3)$, we use the result of Checcoli. A field that fulfills these properties is for a fixed $d$ any subextension of ${\mathds Q}^{(d)} \subset \overline{{\mathds Q}}$, which is the compositum of all number fields of degree at most $d$ over ${\mathds Q}$. We will first prove the CM case since it follows from Theorem 1.5 of [@MR3182009] before we handle the more complicated non-CM case. Let $E$ be an elliptic curve with complex multiplication over ${\mathds Q}$ and let $L/{\mathds Q}$ be a Galois extension with uniformly bounded local degrees above all but finitely many primes by $d$. Then $L(E_{\text{tor}})$ has the Bogomolov property and there exists an effectively computable bound only depending on $d$. Let $L_0$ be the CM field of $E$ and consider the following diagram & L(E\_)\ [Q]{}(E\_) &\ & L\ L\_0 &\ [Q]{} & Consider the restriction $f: G \to \operatorname{Gal}({\mathds Q}(E_\text{tor})/L_0) \times \operatorname{Gal}(L/L_0), \sigma \mapsto (\sigma|_{{\mathds Q}(E_\text{tor})}, \sigma|_L)$. It is injective because $L(E_\text{tor})$ is the compositum of $L$ and ${\mathds Q}(E_\text{tor})$, hence the only element that maps to the identity is the identity itself.\ We want to show that $\operatorname{Gal}(L(E_\text{tor})/L) =:H$ is contained in the center of$\operatorname{Gal}(L(E_\text{tor})/L_0) =: G$ since that will allow us to use Theorem 1.5 of [@MR3182009] and immediately yield that $L(E_\text{tor})$ is Bogomolov.\ Let $\sigma \in G$ and $\tau \in H$. Then $\sigma \tau |_{{\mathds Q}(E_\text{tor})} = \tau \sigma |_{{\mathds Q}(E_\text{tor})}$ since $G$ is abelian (because of $E$ having complex multiplication). Furthermore $\sigma \tau |_L = \sigma |_L$ since $\tau$ acts as the identity on $L$ and $\sigma |_L = \tau \sigma |_L$ since the image of $\sigma$ is inside $L$ (because $L/L_0$ is Galois).\ The bound in [@MR3182009] is effectively computable and only depends on the place $v$ above which we have uniformly bounded local degrees, $d$ and the degree $[L_0:{\mathds Q}]$. Since we have uniformly bounded local degrees and $[L_0:{\mathds Q}] = 2$, the bound only depends on $d$. The same thing can be done for any CM abelian variety. Local preliminaries {#NotNFC} =================== For the rest of this paper we will fix an elliptic curve $E$ over ${\mathds Q}$ without complex multiplication and with $j$-invariant $j_E$. Furthermore, fix a field $L$ with the properties from Theorem \[Sara\] and call $d$ the uniform bound for the local degrees. We use the notation $F(N) = F(E[N])$ for a field $F$ and a natural number $N$. We need the following properties of a prime $p$. $$\begin{aligned} (P1) & \quad p \text{ is supersingular} \label{P1}\\ (P2) & \quad p \text{ is surjective} \label{P2}\\ (P3) & \quad p \geq \max(2d+2, \exp(\operatorname{Gal}(L/{\mathds Q}))) \label{P3}\\ (P4) & \quad j_E \not\equiv 0, 1728 {\text{ mod }}p \label{P4}\end{aligned}$$ We will fix a prime $p$ such that $p$ fulfills properties (P1), (P2), (P3) and (P4). For $N\in {\mathds N}$ we let $N = p^n M$ where $M$ and $p$ are coprime. We want to consider every field as a subfield of a fixed algebraic closure $\overline{{\mathds Q}_p}$ of ${\mathds Q}_p$. With ${\mathds Q}_q$ we denote the unique quadratic unramified extension. The proof goes as follows: For an element $\alpha \in L(E_{\text{tor}})$ we will fix a finite Galois extension $K/{\mathds Q}$ such that $K(E_{\text{tor}})$ contains $\alpha$ and $K\subset L \subset \overline{{\mathds Q}_p}$. Set $q=p^2$ and call ${\mathds Q}_q$ the unique quadratic unramified extension of ${\mathds Q}_p$. Then we fix a Galois extension $F$ of ${\mathds Q}_q$ such that: ${\mathds Q}_q \subset F \subset \overline{{\mathds Q}_p}$, the $v$-adic completion of $K$ is contained in $F$ (where $v$ extends $p$) and $[F:{\mathds Q}_p]$ is uniformly bounded by $2d$ (since it is possible that we have to choose $F$ larger than $K_v$ so that it contains ${\mathds Q}_q$). Since we consider all fields as subfields of $\overline{{\mathds Q}_p}$ we can restrict the $p$-adic valuation of $\overline{{\mathds Q}_p}$ to any subfield. Since all fields are Galois, the completion with respect to any place above $p$ will be the same. For a natural number $N$, we consider $F(N)$ and deal with two cases: the wildly ramified case where $p^2 \mid N$ and the tamely ramified case where $p^2 \nmid N$. We start with a few technical lemmas. \[equal\] Let $p^2 | N$. We have ${\mathds Q}_q(N)\cap F = {\mathds Q}_q (N/p) \cap F$. Recall that $p > d \geq [F:{\mathds Q}_p]$. By Lemma 3.4 (v) of [@MR3090783] we know that $$\begin{aligned} \operatorname{Gal}({\mathds Q}_q(N)/{\mathds Q}_q(N/p)) \cong ({\mathds Z}/p{\mathds Z})^2\end{aligned}$$ in the case of $p^2 \mid N$. We consider the following diagram where the numbers next to the lines describe the degrees of the extensions: \[dasRichtigeDiagramm\] By the multiplicativity of the degree in a tower of field extensions, we have that $$\begin{aligned} [{\mathds Q}_q (N/p)({\mathds Q}_q (N) \cap F) : {\mathds Q}_q (N/p)] \text{ divides } p^2\end{aligned}$$ and by the above diagram $$\begin{aligned} [{\mathds Q}_q (N/p)({\mathds Q}_q (N) \cap F) : {\mathds Q}_q (N/p)] = [{\mathds Q}_q(N)\cap F : {\mathds Q}_q (N/p) \cap F] \leq d < p.\end{aligned}$$ Hence $[{\mathds Q}_q(N)\cap F : {\mathds Q}_q (N/p) \cap F]$ must be one and the fields are equal. \[Lemma3.3wild\] Let $p^2 \mid N$. Then the extension $F(p^n)/F$ is abelian. Furthermore, $$\begin{aligned} \operatorname{Gal}(F(p^n)/F(p^{n-1})) \cong ({\mathds Z}/p{\mathds Z})^2.\end{aligned}$$ Consider the following diagram: We get that $\operatorname{Gal}(F(p^n)/F(p^{n-1}))$ is isomorphic to a subgroup of $\operatorname{Gal}({\mathds Q}_q(p^n)/{\mathds Q}_q(p^{n-1}))$ of index at most $[F:{\mathds Q}_q]$. Since by Lemma 3.4 (v) of [@MR3090783] $\operatorname{Gal}({\mathds Q}_q(p^n)/{\mathds Q}_q(p^{n-1}))$ has order $p^2$ and $[F:{\mathds Q}_q]$ is strictly less than $p$, we must have $\operatorname{Gal}(F(p^n)/F(p^{n-1})) \cong \operatorname{Gal}({\mathds Q}_q(p^n)/{\mathds Q}_q(p^{n-1}))$. By Lemma 3.3 (i) of [@MR3090783], we get $\operatorname{Gal}(F(p^n)/F(p^{n-1})) \cong ({\mathds Z}/p{\mathds Z})^2$. To prove that $F(p^n)/F$ is abelian, we look at the following diagram So by [@MR3090783], Lemma 3.4 (iv), $\operatorname{Gal}(F(p^n)/F)$ is isomorphic to a subgroup of $\operatorname{Gal}({\mathds Q}_q(p^n)/{\mathds Q}_q)$ which is isomorphic to ${\mathds Z}/(q-1){\mathds Z}\times ({\mathds Z}/p^{n-1}{\mathds Z})^{2}$, hence both Galois groups have to be abelian. \[ramindex\] Let $p^2 \mid N$. The ramification index of the extension $F(p^n)/F$ is a multiple of $q^{n-1}$ and a divisor of $q^{n-1}(q-1)$. The extension $F(p^n)/F(p^{n-1})$ is totally ramified and its Galois group is isomorphic to $\operatorname{Gal}({\mathds Q}_q(p^n)/{\mathds Q}_q(p^{n-1})) \cong ({\mathds Z}/p{\mathds Z})^2$. In particular, $F(p^n)/F(p)$ is totally ramified. We consider the following diagram: & F(p\^n) = [Q]{}\_q(p\^n)F &\ [Q]{}\_q(p\^n) && F\ & [Q]{}\_q(p\^n) F&\ & [Q]{}\_q & We want to equip this diagram with the ramification indices. From Lemma 3.3 (i) of [@MR3090783], we know that ${\mathds Q}_q(p^n)/{\mathds Q}_q$ is totally ramified of degree $(q-1)q^{n-1}$. By construction, the extension $F/{\mathds Q}_q$ has degree (hence ramification index) at most $d$ which is less than $p$. Since $\operatorname{Gal}(F(p^n)/{\mathds Q}_q(p^n))$ is isomorphic to a subgroup of $\operatorname{Gal}(F/{\mathds Q}_q)$, its degree has to be at most $d$ hence also the ramification index. So we get the following diagram. & F(p\^n) = [Q]{}\_q(p\^n)F &\ [Q]{}\_q(p\^n) && F\ & [Q]{}\_q(p\^n) F&\ & [Q]{}\_q & This shows that the ramification index of $F(p^n)/{\mathds Q}_q$ is a multiple of the ramification degree of ${\mathds Q}_q(p^n)/{\mathds Q}_q$ which is $(q-1)q^{n-1}$. But since the ramification degree of $F/{\mathds Q}_q$ is at most $d$ which is coprime to $p$, we get that the ramification degree of $F(p^n)/F$ has to be a multiple of $q^{n-1}$. With a similar diagram we can show that $F(p^n)/F(p^{n-1})$ is totally ramified. Recall that by Lemma \[equal\] we have ${\mathds Q}_q(p^{n-1})\cap F = {\mathds Q}_q (p^n)\cap F$. Hence also ${\mathds Q}_q(p^{n-1}) = {\mathds Q}_q(p^{n-1}) \cap F(p^{n-1}) = {\mathds Q}_q(p^n) \cap F(p^{n-1})$. \[diagram1\] & F(p\^n) = [Q]{}\_q(p\^n)F(p\^[n-1]{}) &\ [Q]{}\_q(p\^n)&& F(p\^[n-1]{})\ & [Q]{}\_q(p\^[n]{}) F(p\^[n-1]{})= [Q]{}\_q (p\^[n-1]{}) F(p\^[n-1]{}) = [Q]{}\_q(p\^[n-1]{})& The ramification index of ${\mathds Q}_q(p^n)/{\mathds Q}_q(p^{n-1})$ is exactly $q$ and the ramification index of $F(p^n)/{\mathds Q}_q(p^n)$ is at most its degree $[F(p^n):{\mathds Q}_q(p^n)] \leq [F:{\mathds Q}_q] < p$, hence not divisible by $p$. The same works for $F(p^{n-1})/{\mathds Q}_q(p^{n-1})$. By looking at the divisibility we see that the ramification index of $F(p^n)/F(p^{n-1})$ also has to be $q$, hence it is totally ramified. Furthermore, $\operatorname{Gal}(F(p^n)/F(p^{n-1}))$ is isomorphic to $\operatorname{Gal}({\mathds Q}_q(p^n)/{\mathds Q}_q(p^{n-1}))$. The following lemma is the analogue of Lemma 3.4 of [@MR3090783]. \[Lemma3.4\] The following statements hold. (i) The compositum $F(p^n)F(M)$ is $F(N)$. (ii) The extension $F(N)/F(p^n)$ is unramified. (iii) The Galois group $\operatorname{Gal}(F(N)/F(M))$ is abelian. (iv) If $n \geq 2$, then $\operatorname{Gal}(F(N)/F(N/p)) \cong \operatorname{Gal}(F(p^n)/F(p^{n-1})) \cong ({\mathds Z}/p{\mathds Z})^2$ and the extension $F(N)/F(N/p)$ is totally ramified. (v) The image of the representation $\operatorname{Gal}(F(p^n)/F) \to \operatorname{Aut}(E[p^n])$ contains multiplication by $M^{[F:{\mathds Q}_q]}$. (vi) The Galois group $\operatorname{Gal}(F(p)/F)$ is isomorphic to a subgroup of ${\mathds Z}/(q-1){\mathds Z}$. Every $N$-torsion point is the sum of a $p^n$-torsion point and an $M$-torsion point. Hence, the composition $F(p^n)F(M)$ has to be equal to $F(N)$ which is the statement in (i).\ For (ii) we consider the following diagram: Since the extension ${\mathds Q}_q(N) /{\mathds Q}_q(p^n)$ is unramified by Lemma 3.4 (ii) [@MR3090783], the subextension ${\mathds Q}_q(N)/{\mathds Q}_q(N) \cap F(p^n)$ also has to be unramified. Hence by [@MR1697859] Proposition 7.2, the extension $F(N)/F(p^n)$ extension also has to be unramified.\ For (iii) we consider the following diagram: & F(N) = [Q]{}\_q(p\^n)F(M) &\ [Q]{}\_q(p\^n) && F(M)\ & [Q]{}\_q(p\^n) F(M)&\ & [Q]{}\_q & So $\operatorname{Gal}(F(N)/F(M))$ is isomorphic to a subgroup of $\operatorname{Gal}({\mathds Q}_q(p^n)/{\mathds Q}_q)$ which is by [@MR3090783], Lemma 3.4 (iv), isomorphic to ${\mathds Z}/(q-1){\mathds Z}\times ({\mathds Z}/p^{n-1} {\mathds Z})^2$, so it has to be abelian.\ For (iv) we recall Lemma 3.4 (iv) of [@MR3090783]: $$\begin{aligned} \operatorname{Gal}({\mathds Q}_q(N)/{\mathds Q}_q(N/p)) \cong \begin{cases} ({\mathds Z}/p{\mathds Z})^2 &\mbox{ if } n\geq 2,\\ {\mathds Z}/(q-1){\mathds Z}&\mbox{ if } n = 1. \end{cases}\end{aligned}$$ Let now $n\geq 2$. We want to use Lemma 2.1 (i) of [@MR3090783] with the unramified extension $F(N/p)/F(p^{n-1})$ (see Lemma \[Lemma3.4\] (ii)) and the totally ramified extension $F(p^n)/F(p^{n-1})$ (see Lemma \[ramindex\]). We get $F(p^{n-1}) = F(p^n)\cap F(N/p)$ and with the following diagram we can use Lemma \[Lemma3.3wild\] to get $$\begin{aligned} \operatorname{Gal}(F(N)/F(N/p)) \cong \operatorname{Gal}(F(p^n)/F(p^{n-1})) \cong ({\mathds Z}/p{\mathds Z})^2.\end{aligned}$$ With Lemma (ii) 2.1 of [@MR3090783], we get that the extension $F(N)/F(N/p)$ is totally ramified.\ We now come to part (v). By Lemma 3.3 (iii) of [@MR3090783], the image of the representation $\operatorname{Gal}({\mathds Q}_q(p^n)/{\mathds Q}_q) \to \operatorname{Aut}(E[p^n])$ contains multiplication by $M$. Let us call $\sigma$ the preimage of multiplication by $M$. We have a representation $\operatorname{Gal}(F(p^n)/{\mathds Q}_q) \to \operatorname{Aut}(E[p^n])$ that is compatible to the above one and we can choose an element in $\operatorname{Gal}(F(p^n)/{\mathds Q}_q)$ that restricts to $\sigma$ in $\operatorname{Gal}({\mathds Q}_q(p^n)/{\mathds Q}_q)$. We will call this element also $\sigma$. Since $\operatorname{Gal}(F(p^n)/F)$ is a normal subgroup of $\operatorname{Gal}(F(p^n)/{\mathds Q}_q)$, we can look at the projection $f: \operatorname{Gal}(F(p^n)/{\mathds Q}_q) \to \operatorname{Gal}(F(p^n)/{\mathds Q}_q) / \operatorname{Gal}(F(p^n)/F)$. The index of $\operatorname{Gal}(F(p^n)/{\mathds Q}_q)$ in $\operatorname{Gal}(F(p^n)/F)$ is equal to $[F:{\mathds Q}_q]$. So $$\begin{aligned} f(\sigma^{[F:{\mathds Q}_q]}) = f(\sigma)^{[F:{\mathds Q}_q]} = id.\end{aligned}$$ Hence $\sigma^{[F:{\mathds Q}_q]} $ is an element of $\operatorname{Gal}(F(p^n)/F)$ and it will act as multiplication by $M^{[F:{\mathds Q}_q]} $ on $E[p^n]$.\ For (vi) we consider the following diagram. & F(p) = [Q]{}\_q(p)F &\ [Q]{}\_q(p) && F\ & [Q]{}\_q(p) F&\ & [Q]{}\_q & By [@MR3090783], Lemma 3.4 (iv), we know that $\operatorname{Gal}({\mathds Q}_q(p)/{\mathds Q}_q) \cong {\mathds Z}/(q-1){\mathds Z}$. So $\operatorname{Gal}(F(p)/F)$ has to be isomorphic to a subgroup of ${\mathds Z}/(q-1){\mathds Z}$. Recall the definition of the higher ramification groups: $$\begin{aligned} G_i (L/K) := \{ \sigma \in \operatorname{Gal}(L/K) | \forall a \in {\mathcal{O}}_K \text{ we have } w(\sigma (a) - a) \geq i+1 \}.\end{aligned}$$ \[HigherRamificationGroup\] Let $p^2 \mid N$. Then there is $s\geq q^{n-1}-1$ such that $$\begin{aligned} \operatorname{Gal}(F(N)/F(N/p)) \subset G_s (F(N)/F).\end{aligned}$$ First, we want to show that $\operatorname{Gal}(F(p^n)/F(p^{n-1})) \subset G_s (F(p^n)/{\mathds Q}_q)$ for some $s\geq q^{n-1}-1$.\ By Lemma 3.3 (ii) of [@MR3090783], we know that $$\begin{aligned} \operatorname{Gal}({\mathds Q}_q(p^n)/{\mathds Q}_q(p^{n-1})) = G_{q^{n-1}-1}({\mathds Q}_q(p^n)/{\mathds Q}_q).\end{aligned}$$ So we take an element $\psi$ of $\operatorname{Gal}(F(p^n)/F(p^{n-1}))$ and look at the restriction to ${\mathds Q}_q(p^n)$ which will be an element of $\operatorname{Gal}({\mathds Q}_q(p^n)/{\mathds Q}_q(p^{n-1}))$ and hence of $G_{q^{n-1}-1}({\mathds Q}_q(p^n)/{\mathds Q}_q)$. We will use Herbrand’s Theorem (Theorem 10.7 of [@MR1697859]) which says that for any $s \geq -1$ $$\begin{aligned} (G_s (F(p^n)/{\mathds Q}_q) \operatorname{Gal}(F(p^n)/{\mathds Q}_q(p^n)))/\operatorname{Gal}(F(p^n)/{\mathds Q}_q(p^n)) = G_t ({\mathds Q}_q(p^n)/{\mathds Q}_q)\end{aligned}$$ where $t$ depends on $s$. By Proposition IV.12 of [@MR554237] $t$ is given by a continuous and increasing function of $s$ that maps $0$ to $0$ and goes to infinity as $s$ goes to infinity. By the piecewiese linearity seen in the equation on p. 73 of [@MR554237], we see that for $t=q^{n-1}-1$ we can find $s$ such that the above is true and $s\geq t$.\ Now since the restriction $\psi|_{{\mathds Q}_q(p^n)}$ is an element of $G_{q^{n-1}-1} ({\mathds Q}_q(p^n)/{\mathds Q}_q)$ we find $\sigma_1 \in G_s (F(p^n)/{\mathds Q}_q)$ and $\sigma_2 \in \operatorname{Gal}(F(p^n)/{\mathds Q}_q(p^n))$ such that $\psi = \sigma_1 \sigma_2$. Since $G_s(F(p^n)/{\mathds Q}_q)$ is a normal subgroup of $\operatorname{Gal}(F(p^n)/{\mathds Q}_q)$, we can consider $\operatorname{Gal}(F(p^n)/{\mathds Q}_q)/G_s(F(p^n)/{\mathds Q}_q)$. We want to consider the homomorphism of groups $$\begin{aligned} f: \operatorname{Gal}(F(p^n)/{\mathds Q}_q) \to \operatorname{Gal}(F(p^n)/{\mathds Q}_q)/G_s (F(p^n)/{\mathds Q}_q).\end{aligned}$$ Since $\sigma_1 \in G_s (F(p^n)/{\mathds Q}_q)$ we have $f(\sigma_1) = \operatorname{id}$. Furthermore, $$\begin{aligned} f(\sigma_1 \sigma_2)^{[F(p^n):{\mathds Q}_q(p^n)]} &= (f(\sigma_1) f(\sigma_2))^{[F(p^n):{\mathds Q}_q(p^n)]}\\ &= f(\sigma_2)^{[F(p^n):{\mathds Q}_q(p^n)]}\\ &= f(\sigma_2^{[F(p^n):{\mathds Q}_q(p^n)]})\\ & = f(\operatorname{id})\\ & = \operatorname{id}.\end{aligned}$$ So with $e := [F:{\mathds Q}_q]!$ we can make sure that $(\sigma_1 \sigma_2)^e \in G_s (F(p^n)/{\mathds Q}_q)$. But since $\psi$ was in $\operatorname{Gal}(F(p^n)/F(p^{n-1}))$ which is by Lemma \[Lemma3.4\] (iv) isomorphic to $({\mathds Z}/p{\mathds Z})^2$ and $e$ is coprime to the order of $\operatorname{Gal}(F(p^n)/F(p^{n-1}))$, we can find $\tilde{\psi}\in \operatorname{Gal}(F(p^n)/F(p^{n-1}))$ such that $\tilde{\psi}^e = \psi$.\ Hence we get that $$\begin{aligned} \label{HigherRamification1} \operatorname{Gal}(F(p^n)/F(p^{n-1})) \subset G_s (F(p^n)/{\mathds Q}_q) = G\end{aligned}$$ and we showed that there exists $s\geq q^{n-1}-1$ such that $G_s (F(p^n)/{\mathds Q}_q)$ has order at least $p^2$. Now by Lemma 2.1 (iii) of [@MR3090783] and with $F(N/p)/F(p^{n-1})$ unramified and $F(p^n)/F(p^{n-1})$ totally ramified, we have $$\begin{aligned} \label{HigherRamification2} \operatorname{Gal}(F(N)/F(N/p)) \cap G_s(F(N)/F(p^{n-1})) \cong G_s (F(p^n)/F(p^{n-1}))\end{aligned}$$ by restriction. By Lemma \[Lemma3.4\] (iv), $\operatorname{Gal}(F(N)/F(N/p))$ must have order $p^2$ and since $G_s (F(p^n)/F(p^{n-1}))$ also has order $p^2$, they have to be isomorphic by restriction. By set theory, we then get $$\begin{aligned} \operatorname{Gal}(F(N)/F(N/p))& = \operatorname{Gal}(F(N)/F(N/p)) \cap G_s(F(N)/F(p^{n-1})) \nonumber \\ \label{HigherRamification3}&\subset G_s(F(N)/F(p^{n-1})).\end{aligned}$$ By the formal definition of the higher ramification group we get that $$\begin{aligned} G_s(F(N)/F(p^{n-1})) = G_s (F(N)/F) \cap \operatorname{Gal}(F(N)/F(p^{n-1})) \subset G_s(F(N)/F).\end{aligned}$$ Hence $\operatorname{Gal}(F(N)/F(N/p))$ is a subgroup of $G_s(F(N)/F)$ which is what we wanted to show. \[Lemma3.5\] Let $n\geq 2$. We have $F(N) \cap \mu_{p^\infty} = \mu_{p^n}$. Since by Lemma 3.5 of [@MR3090783] ${\mathds Q}_q(N) \cap \mu_{p^\infty} = \mu_{p^n}$, we have $F(N) \supset \mu_{p^n}$ and we only have to show “$\subset$”. We will closely follow Habegger’s proof of Lemma 3.5 of [@MR3090783] and first show that $F(p^n) \cap \mu_{p^\infty} = \mu_{p^n}$. Let $\zeta \in F(p^n)$ be a root of unity of order $p^{n^\prime}$ with $n^\prime \geq n$. By restricting we get a surjective homomorphism $$\begin{aligned} \operatorname{Gal}(F(p^n)/F) \twoheadrightarrow \operatorname{Gal}(F(\zeta)/F).\end{aligned}$$ We will later prove that the left group is isomorphic to $({\mathds Z}/p^{n-1} {\mathds Z})^2 \times A$ where $A$ is a subgroup of ${\mathds Z}/(q-1){\mathds Z}$. The right part is isomorphic to a subgroup of $\operatorname{Gal}({\mathds Q}_p(\zeta)/{\mathds Q}_p)$ which itself is isomorphic to ${\mathds Z}/p^{n^\prime-1}{\mathds Z}\times {\mathds Z}/(p-1){\mathds Z}$ by Proposition II.7.13 of [@MR1697859]. Remark that ${\mathds Z}/p^{n^\prime-1}{\mathds Z}\times {\mathds Z}/(p-1){\mathds Z}$ is cyclic since $p-1$ and $p$ are coprime, hence all subgroups are direct products of subgroups. Since the index of $\operatorname{Gal}(F(\zeta)/F)$ in $\operatorname{Gal}({\mathds Q}_p(\zeta)/{\mathds Q}_p)$ can be at most $[F:{\mathds Q}_p]$ which is less than $p$, we must have that $\operatorname{Gal}(F(\zeta)/F)$ is actually isomorphic to ${\mathds Z}/p^{n^\prime-1}{\mathds Z}\times A$ where $A$ is a subgroup of ${\mathds Z}/(p-1){\mathds Z}$. Recall that $\operatorname{Gal}F(p^n)/F)$ is isomorphic to a subgroup of $\operatorname{Gal}({\mathds Q}_q(p^n)/{\mathds Q}_q) \cong ({\mathds Z}/p^{n-1}{\mathds Z})^2 \times {\mathds Z}/(p-1){\mathds Z}$. So the homomorphism can only be surjective if it maps $({\mathds Z}/p^{n-1} {\mathds Z})^2$ surjectively to ${\mathds Z}/p^{n^\prime-1}{\mathds Z}$ which is only possible when $n \geq n^\prime$. Together with $n^\prime \geq n$ we get that $n^\prime = n$. Let now $\zeta \in F(N)$ be a root of unity of order $p^{n^\prime}$ with $n^\prime \geq n$. The extension $F(N)/F(p^n)$ is unramified, hence also $F(p^n)(\zeta)/F(p^n)$. By the properties of the Weil pairing we know that $\zeta \in {\mathds Q}_p(p^{n^\prime}) \subset F(p^{n^\prime})$. By Lemma \[ramindex\], the extension $F(p^{n^\prime})/F(p^n)$ is totally ramified and so is $F(p^n)(\zeta)/F(p^n)$. Hence this extension must be trivial and we have $\zeta\in F(p^n)$.\ So let us now prove that $\operatorname{Gal}(F(p^n)/F) \cong ({\mathds Z}/p^{n-1} {\mathds Z})^2 \times A$. Recall that $p$ and $[F:{\mathds Q}_q]$ are coprime and consider the following diagrams: & F(p\^n) & & & F(p\^n) &\ [Q]{}\_q(p\^n) &&F(p) & [Q]{}\_q(p\^n) &&F\ &[Q]{}\_q(p\^n) F(p) & & &[Q]{}\_q(p\^n) F &\ &[Q]{}\_q (p) & & & [Q]{}\_q & The diagram on the right hand side shows that $\operatorname{Gal}(F(p^n)/F)$ is isomorphic to a subgroup $({\mathds Z}/p^{n-1} {\mathds Z})^2 \times {\mathds Z}/(q-1){\mathds Z}$. The diagram on the left hand side shows that $\operatorname{Gal}(F(p^n)/F(p))$ is isomorphic to a subgoup of $({\mathds Z}/p^{n-1} {\mathds Z})^2$. By Goursat’s Lemma [@MR1508819] and since their orders are equal, the groups have to be isomorphic. Recall that $N=p^n M$. Let $\psi\in\operatorname{Gal}(F(N)/F(N/p))$ and $\xi\in F(N) \cap \mu_M$. Then $\psi(\xi)=\xi$. By Proposition II 7.12 of [@MR1697859], the extension $F(\xi)/F$ is unramified. Now we want to prove that $F(\xi) \subset F(N/p)$ (and hence $\psi(\xi)=\xi$). We know that $F(\xi)/F$ is unramified, hence $F(N/p)(\xi)/F(N/p)$ is also unramified. Furthermore, by Lemma \[Lemma3.4\] (iv), $F(N)/F(N/p)$ is totally ramified, hence as a subextension, $F(N/p)(\xi)/F(N/p)$ also has to be totally ramified. But totally ramified and unramified extensions are trivial and we get that $F(N/p)(\xi) = F(N/p)$, hence $F(\xi)\subset F(N/p)$. \[Lemma3.6\] Let $N=p^n M$ with $n\geq 2$. If $\psi \in \operatorname{Gal}(F(N)/F(N/p))$ and $\alpha\in F(N)\setminus \{0\}$ such that $\frac{\psi(\alpha)}{\alpha}\in \mu_\infty$, then $$\begin{aligned} \frac{\psi(\alpha)}{\alpha} \in \mu_q.\end{aligned}$$ We will follow the analogous proof of Lemma 3.6 in [@MR3090783] very closely and only change it where we need to use generalized results of this section. We write $x^\psi$ for $\psi(x)$ if $x\in F(N)$, hence $\frac{\psi(\alpha)}{\alpha} = \alpha^{\psi-1}$. Let $N^\prime$ denote the order of $\beta := \alpha^{\psi-1}$ and decompose it as $N^\prime = p^{n^\prime} M^\prime$ with nonnegative $n^\prime$ and $M^\prime$ and $p$ coprime. Then $\xi := \beta^{p^{n^\prime}}$ has order $M^\prime$. By the lemma above, $\xi$ is fixed by $\psi$. The order of $\beta^{M^\prime}$ is $p^{n^\prime}$. Hence $n^\prime \leq n$ by the above Lemma \[Lemma3.5\]. For the same reason we have $\beta^{pM^\prime}\in F(N/p)$, hence $\psi$ fixes $\beta^{pM^\prime}$. Let us write $1= ap^{n^\prime} + bM^\prime$ with $a$ and $b$ integers. Then $\beta=\xi^a\beta^{bM^\prime}$ and so $\psi$ fixes $\beta^p$ since it fixes $\xi$ and $\beta^{pM^\prime}$. Let $t$ denote the order of $\psi$ as an element of $\operatorname{Gal}(F(N)/F(N/p))$. Then $$\begin{aligned} 1 = \alpha^{p(\psi^t-1)} = \alpha^{p(\psi-1)(\psi^{t-1}+...+\psi+1)} = \beta^{p(\psi^{t-1}+...+\psi+1)} = \beta^{pt}.\end{aligned}$$ By Lemma \[Lemma3.4\] (iv) the order $t$ divides $p$ and the statement follows. The tamely ramified case ======================== Again, we fix $E$, $L$, $K$ and $p$ as in section \[NotNFC\].\ Remark that the tamely ramified case includes the unramified case. For the whole section let $p^2 \nmid N$ and $\varphi_q \in \operatorname{Gal}({\mathds Q}_q^{\text{unr}}/{\mathds Q}_q)$ be the lift of the Frobenius squared as in [@MR3090783]. For $p\nmid N$ we let $\tilde{F}: = F$ and for $p\mid N$ we let $\tilde{F} := F(p)$. Recall that the extension $F/{\mathds Q}_q$ is Galois. Recall that $N=p^n M$. \[LemmaFrobTame\] Let $\mathcal{E}$ be a multiple of $[F:{\mathds Q}_q](q-1)$. We have - $\varphi_q^{\mathcal{E}} |_{\tilde{F}\cap {\mathds Q}_q^{\text{unr}}} = \operatorname{id}$. - There exists $\tilde{\varphi}$ in $\operatorname{Gal}(\tilde{F}(M)/\tilde{F})$ such that the restriction $\tilde{\varphi}|_{(\tilde{F} \cap {\mathds Q}_q^{\text{unr}})(M)}$ coincides with the restriction $\varphi_q^\mathcal{E}|_{(\tilde{F} \cap {\mathds Q}_q^{\text{unr}})(M)}$. - For $\tilde{\varphi}$ from (ii) we have that $\tilde{\varphi}|_{K(N)}$ lies in the center of $\operatorname{Gal}(K(N)/{\mathds Q})$. - The extension $\tilde{F} (M)/(\tilde{F} \cap {\mathds Q}_q^{\text{unr}})(M)$ is totally ramified. - The ramification index of $\tilde{F}(M)/{\mathds Q}_q$ is at most $(q-1)[F:{\mathds Q}_q] \leq \mathcal{E}$. \(i) We have that $[F:{\mathds Q}_p]$ is a multiple of $|\operatorname{Gal}({\mathcal{O}}_F/{\mathfrak{P}}/ {\mathcal{O}}_{{\mathds Q}_q} / {\mathfrak{p}})|$ where ${\mathfrak{P}}$ and ${\mathfrak{p}}$ are the maximal ideals of ${\mathcal{O}}_F$ and ${\mathcal{O}}_{{\mathds Q}_q}$, respectively. By Lemma \[Lemma3.4\] (vi) we have that $\operatorname{Gal}(F(p)/F) \subset {\mathds Z}/(q-1){\mathds Z}$ hence in the case of $p|N$ we have that $[\tilde{F}:{\mathds Q}_q]$ is a divisor of $[F:{\mathds Q}_q](q-1)$ which divides $\mathcal{E}$ and whenever $p\nmid N$, we still have that $[\tilde{F}:{\mathds Q}_q] |$ divides $\mathcal{E}$. So $\mathcal{E}$ is always a multiple of the local degree $[\tilde{F}:{\mathds Q}_q]$. After restriction $\varphi_q|_{\tilde{F}\cap{\mathds Q}_q^{\text{unr}}}$ is an element of the Galois group $\operatorname{Gal}(\tilde{F}\cap {\mathds Q}_q^{\text{unr}}/{\mathds Q}_q)$. But the order of this group is a divisor of $\mathcal{E}$ since $\operatorname{Gal}(\tilde{F}\cap {\mathds Q}_q^{\text{unr}} / {\mathds Q}_q)$ is a quotient of $\operatorname{Gal}(\tilde{F}/{\mathds Q}_q)$ which has order dividing $(q-1)[F:{\mathds Q}_q]$. Hence, the $\mathcal{E}$-th power of $\varphi_q$ has to be the identity on $\tilde{F} \cap {\mathds Q}_q^{\text{unr}}$.\ (ii) First, we want to show that $(\tilde{F} \cap {\mathds Q}_q^\text{unr}) (M) \cap \tilde{F} = \tilde{F}\cap {\mathds Q}_q^\text{unr}$. The inclusion $(\tilde{F} \cap {\mathds Q}_q^\text{unr}) (M) \cap \tilde{F} \supset \tilde{F}\cap {\mathds Q}_q^\text{unr}$ is obvious and we have to prove “$\subset$”. By Lemma 3.1 of [@MR3090783], the extension ${\mathds Q}_q(M)/{\mathds Q}_q$ is unramified, hence ${\mathds Q}_q^{\text{unr}} (M) = {\mathds Q}_q^{\text{unr}}$. We have $(\tilde{F} \cap {\mathds Q}_q^{\text{unr}})(M) \cap \tilde{F} \subset {\mathds Q}_q^{\text{unr}} (M) \cap \tilde{F} = {\mathds Q}_q^{\text{unr}} \cap \tilde{F}$. We consider the following diagram: [Q]{}\_q\^ &&(M) &&\ &([Q]{}\_q\^)(M) & &\ [Q]{}\_q(M) &&( [Q]{}\_q\^) (M) = [Q]{}\_q\^ &\ &[Q]{}\_q(M)([Q]{}\_q\^) &&\ &[Q]{}\_q && Recall that $\mathcal{E}$ is a multiple of $(q-1)[F:{\mathds Q}_p]$ and by (i) $\varphi_q^{\mathcal{E}} |_{\tilde{F}\cap {\mathds Q}_q^\text{unr}}$ is trivial. Hence $\varphi_q^{\mathcal{E}} \in \operatorname{Gal}({\mathds Q}_q^\text{unr} / \tilde{F} \cap {\mathds Q}_q^\text{unr})$. By the diagram, the Galois group $\operatorname{Gal}( (\tilde{F}\cap{\mathds Q}_q^{\text{unr}})(M)/\tilde{F}\cap {\mathds Q}_q^\text{unr})$ is isomorphic to $\operatorname{Gal}(\tilde{F}(M)/\tilde{F})$ and we call $\tilde{\varphi}$ the image of $\varphi_q |_{\tilde{F}\cap {\mathds Q}_q^{\text{unr}}}$ under that isomorphism. Note that in the case of $p\mid N$, $\tilde{\varphi}$ acts trivially on $\tilde{F} = F(p) \supset K(p)$. In the case of $p\nmid N$, $\tilde{\varphi}$ acts trivially on $F \supset K$.\ (iii) Recall that $K\subset F$ and hence by the above paragraph $\tilde{\varphi}$ acts trivially on $K$ and $K(p)$, in the cases $p\nmid N$ and $p\mid N$ respectively. We will distinguish the two cases $p\nmid N$ and $p|N$.\ For $p|N$ we already remarked that $\tilde{\varphi}|_{K(p)}$ is the identity and we now want to show that $\tilde{\varphi} |_{K(N)}$ lies in the center of $\operatorname{Gal}(K(N)/{\mathds Q})$. Consider the following diagram & K(N) &\ K(p) && [Q]{}(M)\ & K(p) [Q]{}(M)\ &[Q]{}& which shows that $\operatorname{Gal}(K(N)/{\mathds Q})$ is by restriction isomorphic to a subgroup of $\operatorname{Gal}({\mathds Q}(M)/{\mathds Q}) \times \operatorname{Gal}(K(p)/{\mathds Q})$. Now the proof of Lemma 5.1 of [@MR3090783] shows that $\tilde{\varphi}$ lies in the center of $\operatorname{Gal}({\mathds Q}(M)/{\mathds Q})$. Together with $\tilde{\varphi}$ acting trivially on $K(p)$, we get that it lies in the center of $\operatorname{Gal}(K(N)/{\mathds Q})$.\ Now let $p\nmid N$. We do the same as above, considering $K$ instead of $K(p)$. Consider the diagram: & K(M) &\ K && [Q]{}(M)\ & K [Q]{}(M)\ &[Q]{}& And again: $\operatorname{Gal}(K(M)/{\mathds Q})$ is by restriction isomorphic to a subgroup of $\operatorname{Gal}({\mathds Q}(M)/{\mathds Q}) \times \operatorname{Gal}(K/{\mathds Q})$ and since $\tilde{\varphi}$ acts trivially on $K$ and lies in the center of $\operatorname{Gal}({\mathds Q}(M)/{\mathds Q})$, it also lies in the center of $\operatorname{Gal}(K(M)/{\mathds Q})$.\ (iv) We will use Lemma 2.1 (ii) of [@MR3090783] again. Since $\tilde{F}/\tilde{F}\cap {\mathds Q}_p^{\text{unr}}$ is totally ramified and $(\tilde{F}\cap {\mathds Q}_p^{\text{unr}})(M)/\tilde{F} \cap {\mathds Q}_p^{\text{unr}}$ is unramified, the extension $\tilde{F}(M)/(\tilde{F}\cap {\mathds Q}_p^{\text{unr}})(M)$ is also totally ramified.\ (v) We consider the following diagram Since the extension ${\mathds Q}_q(M)/{\mathds Q}_q$ is unramified (Chapter VII [@MR2514094]), the only contribution to the ramification degree of $\tilde{F} / {\mathds Q}_q$ can come from the extension $\tilde{F}(M)/{\mathds Q}_q (M)$. Since the Galois group of the said extension is a subgroup of $\operatorname{Gal}(\tilde{F}/{\mathds Q}_q)$, it has degree at most $(q-1)[F:{\mathds Q}_q]$, hence also the ramification degree cannot be larger. Recall that since we view $F$ as a subfield of $\overline{{\mathds Q}_p}$, we can consider $|\alpha|_p$ for $\alpha\in F$. \[Restklassenk\] Let $L/K$ be a totally ramified extension of fields with $K \subset L \subset \overline{{\mathds Q}_p}$ and $[L:{\mathds Q}_p]$, $[K:{\mathds Q}_p]$ finite. Then for every $\alpha\in {\mathcal{O}}_L$ there exists $\beta\in {\mathcal{O}}_K$ such that $|\alpha - \beta|_p < 1$. Since the field extension is totally ramified, the residue fields are equal. Consider $\alpha$ as an element in the residue field of $L$. Take any $\beta \in {\mathcal{O}}_K$ in the same residue class as $\alpha$. Then, as $\alpha$ and $\beta$ are in the same residue class, their difference $\alpha - \beta$ is zero in the residue field. This means $\alpha - \beta$ is an element of the maximal ideal, hence $|\alpha-\beta|_p$ has to be smaller than one. Let $\alpha \in \tilde{F} (M)^*$ with $|\alpha|_p \leq 1$. Then for $\tilde{\varphi}$ and $\mathcal{E}$ as in Lemma \[LemmaFrobTame\] we have $$\begin{aligned} | \tilde{\varphi}(\alpha) - \alpha^{q^\mathcal{E}}|_p \leq p^{-\frac1{\mathcal{E}}}.\end{aligned}$$ Let $\alpha \in \tilde{F} (M)$ with $|\alpha|_p \leq 1$. Then by Lemma \[Restklassenk\] and \[LemmaFrobTame\] (iv) we find $\beta\in (\tilde{F}\cap{\mathds Q}_q^{\text{unr}})(M)$ with $|\beta|_p \leq 1$ and $|\alpha - \beta |_p < 1$. Now $| \tilde{\varphi}(\alpha)-\tilde{\varphi}(\beta)|_p = |\alpha - \beta|_p$ since Galois automorphisms do not change the valuation. Furthermore, we have $$\begin{aligned} (\alpha^{q^\mathcal{E}} - \beta^{q^\mathcal{E}}) &= (\alpha - \beta)(\alpha^{{q^\mathcal{E}}-1} + \alpha^{{q^{\mathcal{E}}}-2}\beta + ... + \alpha\beta^{{q^{\mathcal{E}}}-2} + \alpha^{{q^{\mathcal{E}}}-1})\\\end{aligned}$$ hence $$\begin{aligned} |\alpha^{q^{\mathcal{E}}} - \beta^{q^{\mathcal{E}}}|_p &= |\alpha - \beta|_p |\alpha^{{q^\mathcal{E}}-1} + \alpha^{{q^\mathcal{E}}-2}\beta + ... + \alpha\beta^{{q^\mathcal{E}}-2} + \alpha^{{q^\mathcal{E}}-1}|_p\\ &\leq |\alpha - \beta|_p \max(|\alpha^{{q^\mathcal{E}}-1}|_p,|\alpha^{{q^\mathcal{E}}-2}\beta|_p, ..., |\alpha\beta^{{q^\mathcal{E}}-2}|_p, |\alpha^{{q^\mathcal{E}}-1}|_p)\\ &\leq |\alpha - \beta|_p \\ & < 1.\end{aligned}$$ Now consider $| \tilde{\varphi}(\beta) - \beta^{q^\mathcal{E}}|_p$. Since $\beta \in (\tilde{F}\cap{\mathds Q}_q^{\text{unr}})(M)$, we can apply Lemma \[LemmaFrobTame\] (ii) and get that $\tilde{\varphi}$ acts as $\tilde{\varphi}(\beta)$ is equal to $\beta^{q^\mathcal{E}}$ in the residue field. Again, as in the proof of the above lemma, this means that their difference is an element of the maximal ideal in $(\tilde{F}\cap{\mathds Q}_q^{\text{unr}})(M)$, which means that $| \tilde{\varphi}(\beta) - \beta^{q^\mathcal{E}}|_p < 1$. So we have $$\begin{aligned} | \tilde{\varphi}(\alpha) - \alpha^{q^\mathcal{E}}|_p & = | \tilde{\varphi}(\alpha) - \tilde{\varphi}(\beta)+ \tilde{\varphi}(\beta)- \beta^{q^\mathcal{E}}+\beta^{q^\mathcal{E}}- \alpha^{q^\mathcal{E}}|_p \\ & \leq \max(| \tilde{\varphi}(\alpha) - \tilde{\varphi}(\beta)|_p, | \tilde{\varphi}(\beta)- \beta^{q^\mathcal{E}}|_p, |\beta^{q^\mathcal{E}} - \alpha^{q^\mathcal{E}}|_p)\\ &= \max(|\alpha-\beta|_p, | \tilde{\varphi}(\beta)- \beta^{q^\mathcal{E}}|_p)\\ &< 1.\end{aligned}$$ Since the valuation is discrete and we bounded the ramification degree in Lemma \[LemmaFrobTame\] (v), it has to be at most $p^{-\frac1{\mathcal{E}}}$ which proves the statement. We recall a result of an earlier paper of the author. \[sumexpl\] Let $\delta < \frac12$ and let $\beta \in \overline{{\mathds Q}^*} \setminus \mu_\infty$ be such that $[{\mathds Q}(\beta):{\mathds Q}] \geq 16$ and $h(\beta)^\frac12 \leq \frac12$. Then we have $$\begin{aligned} \frac{1}{[{\mathds Q}(\beta):{\mathds Q}]} \sum_{\tau : {\mathds Q}(\beta) \hookrightarrow {\mathds C}} \log |\tau (\beta) -1 | \leq \frac{40}{\delta^4} h(\beta)^{\frac12-\delta}.\label{boundthesuminexpl}\end{aligned}$$ \[Lemma4.1NFC\] Let $\alpha \in \tilde{F} (M)^*$. Then for $\tilde{\varphi}$ as in Lemma \[LemmaFrobTame\] we have $$\begin{aligned} | \tilde{\varphi} (\alpha) - \alpha^{q^\mathcal{E}}|_p \leq p^{-\frac1{\mathcal{E}}} \max (1, | \tilde{\varphi}(\alpha)|_p) \max (1, |\alpha|_p)^{q^\mathcal{E}} .\end{aligned}$$ For $|\alpha|_p \leq 1$ this is the above lemma. Let now $|\alpha|_p > 1$ and consider $\alpha^{-1}$. Then we can use the ultrametric triangle inequality and with the above lemma we get $$\begin{aligned} |\alpha^{-q^\mathcal{E}}(\tilde{\varphi}(\alpha)-\alpha^{q^\mathcal{E}})|_p = |(\alpha^{-q^\mathcal{E}}-\tilde{\varphi}(\alpha^{-1})) \tilde{\varphi}(\alpha)|_p \leq p^{-\frac1{\mathcal{E}}}| \tilde{\varphi}(\alpha)|_p\end{aligned}$$ which gives the desired result. Recall that an element $\sigma \in \operatorname{Gal}(K(N)/{\mathds Q})$ acts on the places of $K(N)$ by $|\cdot|_{\sigma v}= |\sigma^{-1} (\cdot)|_v$. \[Lemma6.3\] Let $p^2 \nmid N$. Let $\alpha\in K(N) \backslash \mu_\infty$ be non-zero. Then $$\begin{aligned} \label{Equation6.10} h(\alpha) \geq \left(\frac{\log p}{\mathcal{E}(1+q^\mathcal{E})(1+ 5\cdot2^{11})}\right)^4.\end{aligned}$$ We follow the proof of Lemma 5.1 of [@MR3090783] closely. Let $x = \tilde{\varphi}|_{K(N)}(\alpha) - \alpha^{q^\mathcal{E}} \in K(N)$ where $\tilde{\varphi}|_{K(N)}$ is the lift of the Frobenius from before. This is nonzero since otherwise we would get $h(\alpha) = h(\tilde{\varphi}|_{K(N)}(\alpha)) = h(\alpha^{q^\mathcal{E}}) = {q^\mathcal{E}} h(\alpha)$ hence $h(\alpha) = 0$ which contradicts our assumption on $\alpha$. So we can use the product formula $$\begin{aligned} \sum_w d_w \log |x|_w = 0\end{aligned}$$ where the sum is over all places of $K(N)$. Let $w$ be a finite place of $K(N)$ above $p$. Then $w= \sigma^{-1} v$ for some $\sigma \in \operatorname{Gal}(K(N)/{\mathds Q})$ and $v$ a place above $p$ because this Galois group acts transitively on the places of $K(N)$ above $p$. By Lemma \[LemmaFrobTame\] (iii) $\tilde{\varphi}|_{K(N)}$ and $\sigma$ commute and we get $$\begin{aligned} |x|_w = |\sigma(\tilde{\varphi}|_{K(N)}(\alpha))-\sigma(\alpha)^{q^\mathcal{E}}|_v = | \tilde{\varphi}|_{K(N)}(\sigma(\alpha))-\sigma(\alpha)^{q^\mathcal{E}}|_v.\end{aligned}$$ Now we estimate the right-hand side from above using Lemma \[Lemma4.1NFC\] applied to $\sigma(\alpha)$ $$\begin{aligned} |x|_w & = |\tilde{\varphi}| _{K(N)}(\alpha) - \alpha|_w \\ & = |\sigma(\tilde{\varphi}|_{K(N)}(\alpha)) - \sigma (\alpha) |_v\\ & = |\tilde{\varphi}|_{K(N)}(\sigma(\alpha)) - \sigma (\alpha) |_v \\ & \leq p^{-\frac1{\mathcal{E}}} \max(1,| \tilde{\varphi}|_{K(N)}(\sigma(\alpha))|_v)\max(1,|\sigma(\alpha)|_v)^{q^\mathcal{E}}\\ & = p^{-\frac1{\mathcal{E}}} \max(1,|\sigma(\tilde{\varphi}|_{K(N)}(\alpha))|_v)\max(1,|\sigma(\alpha)|_v)^{q^\mathcal{E}}\\ & = p^{-\frac1{\mathcal{E}}} \max(1,| \tilde{\varphi}|_{K(N)}(\alpha)|_w)\max(1,|\alpha|_w)^{q^\mathcal{E}}.\end{aligned}$$ For an arbitrary finite place $w$ of $K(N)$, the ultrametric triangle inequality gives $$\begin{aligned} |x|_w \leq \max(| \tilde{\varphi}|_{K(N)}(\alpha)|_w,|\alpha^{q^\mathcal{E}}|_w) \leq \max(1,| \tilde{\varphi}|_{K(N)}(\alpha)|_w)\max(1,|\alpha|_w)^{q^\mathcal{E}}.\end{aligned}$$ For the infinite places $w$ we have to take a little detour. We define $$\begin{aligned} \beta = \frac{\tilde{\varphi}|_{K(N)}(\alpha)}{\alpha^{q^\mathcal{E}}} \in \overline{{\mathds Q}} \setminus \{1\}\end{aligned}$$ and bound $$\begin{aligned} |x|_w &= |\beta -1|_w |\alpha^{q^\mathcal{E}}|_w \leq |\beta -1|_w \max(1,|\alpha^{q^\mathcal{E}}|_w) \\ &\leq |\beta -1|_w \max(1,| \tilde{\varphi}|_{K(N)}(\alpha)|_w) \max(1,|\alpha^{q^\mathcal{E}}|_w)\end{aligned}$$ instead. We get $$\begin{aligned} 0 = &\sum_w d_w \log |x|_w \\ = &\sum_{w|p} d_w \log |x|_w + \sum_{w\nmid p, w \nmid \infty} d_w \log |x|_w + \sum_{w|\infty} d_w \log |x|_w\\ \leq& \sum_{w|p} d_w \log (p^{-\frac1{\mathcal{E}}} \max(1,| \tilde{\varphi}|_{K(N)}(\alpha)|_w)\max(1,|\alpha^{q^\mathcal{E}}|_w))\\ &+ \sum_{w\nmid p, w \nmid \infty} d_w \log (\max(1,| \tilde{\varphi}|_{K(N)}(\alpha)|_w)\max(1,|\alpha^{q^\mathcal{E}}|_w)) \\ &+ \sum_{w|\infty} d_w \log (|\beta -1|_w \max(1,| \tilde{\varphi}|_{K(N)}(\alpha)|_w) \max(1,|\alpha^{q^\mathcal{E}}|_w)).\end{aligned}$$ After dividing by $[K(N):{\mathds Q}]$ this gives $$\begin{aligned} \frac{\log p}{\mathcal{E}} - \frac1{[K(N):{\mathds Q}]} \sum_{w|\infty} d_w \log |\beta -1|_w \leq (1+q^\mathcal{E})h(\alpha).\label{boundNFC1}\end{aligned}$$ Let us now assume that $h(\beta) \leq \frac14, [ {\mathds Q}(\beta):{\mathds Q}] \geq 16$ and $h(\alpha) \leq 1$. This is without loss of generality since otherwise the conclusion of the Lemma is clear. By Lemma \[sumexpl\] with $\delta = \frac14$ we get $$\begin{aligned} \frac1{[K(N):{\mathds Q}]} \sum_{\tau: {\mathds Q}(\beta) \to {\mathds C}} \log |\tau(\beta) -1| & = \frac1{[{\mathds Q}(\beta):{\mathds Q}]} \sum_{\tau: {\mathds Q}(\beta) \to {\mathds C}} \log |\tau(\beta) -1|\\ & \leq 5\cdot2^{11} h(\beta)^{\frac14}\\ & \leq 5\cdot2^{11} ((1+q^\mathcal{E})h(\alpha))^{\frac14}.\end{aligned}$$ Together with estimate we get $$\begin{aligned} \frac{\log p}{\mathcal{E}} - 5\cdot2^{11}((1+q^\mathcal{E})h(\alpha))^{\frac14} &\leq (1+q^\mathcal{E})h(\alpha).\end{aligned}$$ Hence $$\begin{aligned} \frac{\log p}{\mathcal{E}} &\leq (1+q^\mathcal{E})(1+ 5\cdot2^{11}) h(\alpha)^{\frac14}\end{aligned}$$ which gives $$\begin{aligned} \left(\frac{\log p}{\mathcal{E}(1+q^\mathcal{E})(1+ 5\cdot2^{11})}\right)^4 &\leq h(\alpha).\end{aligned}$$ The wildly ramified case ======================== Again, we fix $E$, $F$, $K$, $p$ and $p^2 = q$ as in section \[NotNFC\]. For this whole section we will only consider the case where $p^2|N$. Let $v$ be the place of $F$ above $p$. Recall that we considered $F$ as a subfield of $\overline{{\mathds Q}_p}$, hence for an element $\alpha\in F$ we can consider $|\alpha|_p$. Let $\psi \in \operatorname{Gal}({\mathds Q}_q(N)/{\mathds Q}_q(N/p))$. Then $\psi|_{F\cap {\mathds Q}_q (N)} = \operatorname{id}$. By Lemma \[equal\] we have ${\mathds Q}_q(N)\cap F = {\mathds Q}_q (N/p) \cap F$. Since $\psi \in \operatorname{Gal}({\mathds Q}_q(N)/{\mathds Q}_q(N/p))$, $\psi$ must be the identity on ${\mathds Q}_q(N/p)$, hence also on ${\mathds Q}_q(N/p) \cap F = {\mathds Q}_q(N) \cap F$. \[Lemma4.2NFC\] Let $\alpha \in F(N)$. Then $$\begin{aligned} |\psi(\alpha)^q-\alpha^q|_p \leq p^{-1} \max (1, |\psi(\alpha)|_p)^q \max(1,|\alpha|_p)^q\end{aligned}$$ for all $\psi \in \operatorname{Gal}(F(N)/F(N/p))$. First, we suppose $|\alpha|_p \leq 1$. Let $\psi\in \operatorname{Gal}(F(N)/F(N/p)) $ and consider the restriction $\psi|_{F(p^n)} \in \operatorname{Gal}(F(p^n)/F(p^{n-1}))$. By Lemma \[HigherRamificationGroup\] this is an element of $G_i(F(N)/F)$ for $i=q^{n-1}-1$. By the definition of the ramification group, this means $$\begin{aligned} \psi(\alpha) -\alpha \in {\mathfrak{P}}^{q^{n-1}}\end{aligned}$$ where ${\mathfrak{P}}$ is the maximal ideal in the ring of integers of $F(N)$. By Lemmas \[ramindex\] and \[Lemma3.4\] (ii) the ramification index $e$ of $F(N)/F$ is at most $q^{n-1}(q-1) \leq q^n$. Therefore, $(\psi(\alpha)-\alpha)^q \in {\mathfrak{P}}^{q^n} \subset {\mathfrak{P}}^e$. Since $p \equiv 0 {\text{ mod }}{\mathfrak{P}}^e$ we conclude $$\begin{aligned} (\psi(\alpha)-\alpha)^q \equiv \psi(\alpha)^q - \alpha^q {\text{ mod }}{\mathfrak{P}}^e.\end{aligned}$$ This leads to $|\psi(\alpha^q)-\alpha^q|_p \leq |p|_p = p^{-1}$. Hence the statement follows if $|\alpha|_p \leq 1$. Now for $|\alpha|_p > 1$ consider $\alpha^{-1}$ with $|\alpha^{-1}|_p \leq 1$. We get $|\psi(\alpha)^{-q}-\alpha^{-q}|_p \leq p^{-1}$ and $$\begin{aligned} |\alpha^{-q}(\psi(\alpha)^q-\alpha^q)|_p = |(\alpha^{-q}-\psi(\alpha)^{-q})\psi(\alpha)^q|_p \leq p^{-1}|\psi(\alpha)^q|_p.\end{aligned}$$ After multiplying by $|\alpha^q|_p$ we obtain our statement. \[Lemma5.2\] Let $\psi\in \operatorname{Gal}(K(N) / K(N/p))$ and $v$ be the place of $K(N)$ above $p$. Let $$G = \{ \sigma \in \operatorname{Gal}(K(N) / {\mathds Q}) | \sigma \psi \sigma^{-1} = \psi\}$$ be the centralizer of $\psi$. Then $$|Gv| \geq\frac{[K(N):{\mathds Q}]}{p^4 d_v}.$$ Let $H := \operatorname{Gal}(K (N) /K (N/p))$, it is a normal subgroup of $\operatorname{Gal}(K(N) / {\mathds Q})$. The orbit of $\psi$ under conjugation by $\operatorname{Gal}(K(N)/{\mathds Q})$ is contained in $H$. The stabilizer of this action is $G$ so we can use the orbit-stabilizer theorem. Furthermore, by the proof of Lemma 5.2 of [@MR3090783], we have $|\operatorname{Gal}({\mathds Q}(N)/{\mathds Q}(N/p))| \leq p^4$. Since $H$ is isomorphic to a subgroup of that group, we have $|H| \leq p^4$. We get $$\begin{aligned} |G| \geq \frac{|\operatorname{Gal}(K(N)/{\mathds Q})|}{|H|} = \frac{[K(N):{\mathds Q}]}{|H|} \geq \frac{[K(N):{\mathds Q}]}{p^4}.\end{aligned}$$ Furthermore, again by the orbit-stabilizer theorem, for a place $v$ of $K(N)$ above $p$ we have $$\begin{aligned} \label{4.21} |Gv| = \frac{|G|}{|\operatorname{Stab}_G (v)|} \geq \frac{[K(N):{\mathds Q}]}{p^4 |\operatorname{Stab}_G (v)|}.\end{aligned}$$ The Galois group $\operatorname{Gal}(K(N)/{\mathds Q})$ acts transitively on all places of $K(N)$ lying above $p$ and the total number of such places is $\frac{[K(N):{\mathds Q}]}{d_v}$ since $K(N)$ is a Galois extension of ${\mathds Q}$. The number of places is by the orbit-stabilizer theorem again the same as $$\frac{|\operatorname{Gal}(K(N)/{\mathds Q})|}{|\operatorname{Stab}_{\operatorname{Gal}(K(N)/{\mathds Q})} (v)|}.$$ This gives us the following inequality: $$|\operatorname{Stab}_G (v)| \leq |\operatorname{Stab}_{\operatorname{Gal}(K(N)/{\mathds Q})} (v)| = d_v.$$ After inserting this in equation we get the desired result $$\begin{aligned} |Gv| &\geq \frac{[K(N):{\mathds Q}]}{p^4 |\operatorname{Stab}_G (v)|} \geq \frac{[K(N):{\mathds Q}]}{p^4 d_v}.\end{aligned}$$ The next height bound is the analogue of Lemma 5.3 of [@MR3090783]. \[Lemma5.3\] Let $\alpha\in K(N) \backslash \mu_\infty$ be non-zero and let $n\geq 2$ be the greatest integer with $p^n \mid N$. If $\alpha^q\notin F(N/p)$, then $$\begin{aligned} \label{Estimate5.8} h(\alpha) \geq \frac{(\log p)^4}{4\cdot 10^6 p^{32}}.\end{aligned}$$ By hypothesis we may chose $\psi \in \operatorname{Gal}(F(N)/F(N/p))$ with $\psi (\alpha^q) \neq \alpha^q$. We let $$x = \psi(\alpha^q) - \alpha^q$$ and observe $x\neq 0$ by our choice of $\psi$. So $$\begin{aligned} \sum_v d_v \log |x|_v = 0 \label{Summe5.9}\end{aligned}$$ by the product formula. Say $G= \{ \sigma \in \operatorname{Gal}(K(N) / {\mathds Q}) | \sigma \psi \sigma^{-1} = \psi\}$ as in Lemma \[Lemma5.2\] and $v$ is the place of $K(N)$ coming from the restriction of the $p$-adic valuation on $\overline{{\mathds Q}_p}$ to $K(N)$. For $\sigma\in G$ we have $$|(\sigma\psi\sigma^{-1}) (\alpha^q) - \alpha^q|_{\sigma v} = |\psi(\sigma^{-1}(\alpha^q))- \sigma^{-1}(\alpha^q)|_p.$$ We may apply Lemma \[Lemma4.2NFC\] to $\sigma^{-1}(\alpha)$. This implies $$\begin{aligned} |(\sigma\psi\sigma^{-1})(\alpha^q)-\alpha^q|_{\sigma v} & = |\psi(\sigma^{-1}(\alpha))^q- \sigma^{-1}(\alpha)^q|_v\\ & \leq p^{-1}\max \{1,|\psi(\sigma^{-1}(\alpha))|_v\}^q\max\{1,|\sigma^{-1}(\alpha)|_v\}^q\\ &\leq p^{-1}\max \{1,|(\sigma\psi\sigma^{-1})(\alpha)|_{\sigma v}\}^q\max\{1,|\alpha|_{\sigma v}\}^q.\end{aligned}$$ Now $\sigma\psi\sigma^{-1} = \psi$ since $\sigma\in G$. Therefore, $$\begin{aligned} |x|_w \leq p^{-1}\max\{1,|\psi(\alpha)|_w\}^q\max\{1,|\alpha|_w\}^q \text{ for all } w\in Gv. \label{Estimate5.10}\end{aligned}$$ If $w$ is an arbitrary finite place of $K(N)$, the ultrametric triangle inequality implies $$\begin{aligned} |x|_w \leq \max\{1,|\psi(\alpha)|_w\}^q\max\{1,|\alpha|_w\}^q. \label{Estimate5.11}\end{aligned}$$ Now let $w$ be an infinite place. We define $$\beta = \frac{\psi(\alpha^q)}{\alpha^q} \in\overline{{\mathds Q}} \backslash \{1\}$$ and bound the following expression instead: $$\begin{aligned} |x|_w = |\beta - 1|_w |\alpha^q|_w \leq |\beta - 1|_w \max\{1,|\alpha|_w\}^q. \label{Estimate5.12}\end{aligned}$$ We split the sum up into the finite places in $Gv$, the remaining finite places, and the infinite places and the continue like in the proof of Lemma\[Lemma6.3\]. The estimates , and together with the product formula imply $$\begin{aligned} 0 \leq &\sum_{w\in Gv} d_w (\log p^{-1} +q\log(\max\{1,|\psi(\alpha)|_w\}\max\{1,|\alpha|_w\})) \nonumber\\ &+ \sum_{w\nmid\infty, w\notin Gv} d_w q \log(\max\{1,|\psi(\alpha)|_w\}\max\{1,|\alpha|_w\}) \nonumber\\ &+ \nonumber \sum_{w\mid\infty} d_w (\log|\beta-1|_w + q\log\max\{1,|\alpha|_w\})\\ = &\sum_{w\in Gv} d_w \log p^{-1} \nonumber\\ &+ \sum_{w\nmid\infty} d_w q \log(\max\{1,|\psi(\alpha)|_w\}\max\{1,|\alpha|_w\}) \nonumber\\ \nonumber&+ \sum_{w\mid\infty} d_w (\log|\beta-1|_w + q\log\max\{1,|\alpha|_w\})\\ \leq &\sum_{w\in Gv} d_w \log p^{-1} \label{Estimate5.13}\\ &+ \sum_{w} d_w q \log(\max\{1,|\psi(\alpha)|_w\}\max\{1,|\alpha|_w\}) \nonumber\\ \nonumber&+ \sum_{w\mid\infty} d_w \log|\beta-1|_w.\end{aligned}$$ Moreover, since the action of the Galois group is transitive and all fields here are Galois over ${\mathds Q}$, all local degrees $d_w$ with $w \in Gv$ equal $d_v$. So $$\begin{aligned} \sum_{w\in Gv} d_w \log p^{-1} = d_v \log p^{-1} |Gv| \leq -d_v \log p \frac{ [K(N):{\mathds Q}]}{p^4 d_v} = - \frac{\log p}{p^4} [K(N):{\mathds Q}]\end{aligned}$$ by Lemma \[Lemma5.2\]. We use this estimate together with and after dividing by $[K(N):{\mathds Q}]$ we obtain $$\begin{aligned} 0 \leq -\frac{\log p}{ p^4} + \frac1{[K(N):{\mathds Q}]}\left( \sum_{w\mid\infty} d_w \log |\beta - 1|_w \right) + qh(\psi(\alpha)) + qh(\alpha).\end{aligned}$$ Also, $h(\psi(\alpha)) = h(\alpha)$ and $q = p^2$, hence $$\begin{aligned} \frac{\log p}{p^4} \leq \frac1{[{\mathds Q}(\beta):{\mathds Q}]}\left( \sum_{\tau: {\mathds Q}(\beta) \hookrightarrow {\mathds C}} \log |\tau(\beta) - 1| \right) + 2p^2h(\alpha).\end{aligned}$$ By construction we certainly have $\beta \neq 0,1$ and in order to apply Lemma \[sumexpl\] it remains to show that $\beta$ is not a root of unity. If we assume the contrary, then $\frac{\psi(\alpha)}{\alpha}$ will be a root of unity too. Lemma \[Lemma3.6\] implies $\left(\frac{\psi(\alpha)}{\alpha}\right)^q = 1$ which contradicts our assumption on $\alpha$. We have $h(\beta) \leq h(\psi(\alpha^q)) + h(\alpha^q) \leq 2p^2 h(\alpha)$. Assuming $h(\beta) \leq \frac14$ and $h(\alpha) \ leq 1$ (which we can do since otherwise we would have a lower bound for $h(\alpha)$ that is better than the claim), we apply Lemma \[sumexpl\] with $\delta = \frac14$ and get: $$\begin{aligned} \frac{1}{[{\mathds Q}(\beta):{\mathds Q}]} \sum_{\tau : {\mathds Q}(\beta) \hookrightarrow {\mathds C}} \log |\tau (\beta) -1 | &\leq 5\cdot2^{11} h(\beta)^{\frac14}\\ &\leq 5\cdot2^{11} (2p^2 h(\alpha))^{\frac14}\end{aligned}$$ and hence $$\begin{aligned} \frac{\log p}{p^4} &\leq 5\cdot2^{11} (2p^2 h(\alpha))^\frac14 + 2p^2 h(\alpha)\\ & \leq 5\cdot2^{11} (2p^2 h(\alpha))^\frac14 + 2p^2 h(\alpha)^\frac14\\ & \leq (5\cdot2^{11} (2p^2)^\frac14+2p^2) h(\alpha)^\frac14.\end{aligned}$$ We solve the above inequality for $h(\alpha)$: $$\begin{aligned} h(\alpha) & > \left(\frac{\log p}{(5\cdot2^{11} (2p^2)^{\frac14}+2p^2)p^4}\right)^4\\ & \geq \left(\frac{\log p}{(5\cdot2^{11} 2^\frac14 p^\frac12+2p^4)p^4}\right)^4\\ & \geq \left(\frac{\log p}{(6090 p^{-\frac72}+1)2p^8}\right)^4\\ & \geq \left(\frac{\log p}{46 p^8}\right)^4\\ &= \frac{(\log p)^4}{4\cdot 10^6 p^{32}}.\end{aligned}$$ Descent and the final bound =========================== Again, we fix $E$, $L$ and $p$ as in section \[NotNFC\]. Let also $\mathcal{E}$ be a multiple of $[F:{\mathds Q}_p](q-1)$. Now we want to turn the conditional bound in the ramified case in an unconditional bound using some descent method. First, we construct a useful automorphism of $K(N)/{\mathds Q}$. \[Lemma6.4\] Let $n\geq 0$ be the greatest integer with $p^n\mid N$. There exists $\sigma_F\in\operatorname{Gal}(F (N)/F)$, lying in the center of $\operatorname{Gal}(K(N)/K)$ such that $\sigma_F(\zeta)=\zeta^{4^{[F:{\mathds Q}_q]}}$ for all $\zeta\in\mu_{p^n}$. Moreover, $\sigma_F$ acts on $E[p^n]$ as multiplication by $2^{[F:{\mathds Q}_q]}$. Before we prove this Lemma, let us recall that by Lemma \[Lemma3.5\] $F(N)$ contains $\mu_{p^n}$.\ Since $p$ is odd, Lemma \[Lemma3.4\] (v) implies that there is $\sigma_F^\prime \in \operatorname{Gal}(F (p^n)/F)$ that acts on $E[p^n]$ as multiplication by $2^{[F:{\mathds Q}_q]} $. Since the Weil pairing $\langle\cdot,\cdot\rangle$ is surjective, we can find for every root of unity $\zeta\in \mu_{p^n}$ points $P,Q \in E[p^n]$ such that $\langle P,Q\rangle = \zeta$. Now $\sigma^\prime_F (\langle P,Q\rangle ) = \langle \sigma^\prime_F (P), \sigma^\prime_F (Q) \rangle = \langle 2^{[F:{\mathds Q}_q]} P, 2^{[F:{\mathds Q}_q]} Q \rangle = \langle P,Q \rangle^{4^{[F:{\mathds Q}_q]}}$. Hence $\sigma^\prime_F$ acts on $\mu_{p^n}$ as raising to the $4^{[F:{\mathds Q}_q]}$-th power.\ We will now lift the automorphism $\sigma_F^{\prime q-1}$ to $\sigma_F^{q-1}\in\operatorname{Gal}(F (N)/F (M))$. For that we consider the following diagram & F(N) &\ F(p\^n) && F(M)\ & F(p\^n)F(M) &\ &F& and we will prove that $\sigma_F^{\prime q-1}|_{F(p^n)\cap F(M)}$ is the identity. We know that $F(M)/F$ is unramified by Lemma \[Lemma3.4\] (ii), hence its subextension $F(p^n) \cap F(M) /F$ is also unramified. But on the other hand, $F(p^n)/F(p)$ is totally ramified by Lemma \[ramindex\], hence $F(p^n) \cap F(M)$ has to be a subfield of $F(p)$ which has degree $q-1$ over $F$. Hence we get that $[F(p^n) \cap F(M) : F]$ divides $q-1$. So $\sigma_F^{\prime q-1}$ is already in $\operatorname{Gal}(F(p^n)/F(p^n)\cap F(M))$ which is by the above diagram isomorphic to $\operatorname{Gal}(F(N)/F(M))$. We will call the image of $\sigma_F^{\prime q-1}$ under this isomorphism $\sigma_F^{q-1}$.\ Taking the sum of points gives an isomorphism between $E[p^n] \times E[M]$ and $E[N]$ which is compatible with the action of $\operatorname{Gal}(K(N)/K)$. Since $\sigma_F $ acts as multiplication by $2^{[F:{\mathds Q}_q]}$ on $E[p^n]$ and trivially on $E[M]$ and $F$ (hence also on $K$), it must lie in the center of $\operatorname{Gal}(K(N)/K)$. Some group theory ================= For $p\neq 2$ the vector space $V := \{ A\in \operatorname{Mat}_2 ({\mathds F}_p) | \operatorname{Tr}A = 0 \}$ has only trivial linear subspaces that are invariant under conjugation with $\operatorname{SL}_2 ({\mathds F}_p)$. Let $U$ be a non-trivial linear subspace of $V$ that is invariant under conjugation by $\operatorname{SL}_2 ({\mathds F}_p)$. By considering the non-degenerate scalar product $\langle A,B \rangle := \operatorname{Tr}(A^T B)$ on $V$ we can show that $U^\perp$ is also invariant: Let $A \in U^\perp$ and $B\in U$. Then for any $S \in \operatorname{SL}_2 ({\mathds F}_p)$ we have $\operatorname{Tr}((SAS^{-1})^T B) = \operatorname{Tr}(S^{-T} (A^T S^T B)) = \operatorname{Tr}((A^T S^T B) S^{-T} ) = \operatorname{Tr}(A^T (S^T B S^{-T})) = \operatorname{Tr}(A^T B^\prime)$ for some $B^\prime \in U$ since $U$ is invariant under conjugation. But then $\operatorname{Tr}(A^T B^\prime) = 0$, hence $SAS^{-1} \in U^\perp$.\ We know that $V$ has dimension three. Now if $U$ is an invariant linear subspace of dimension $2$, its orthogonal complement has to be of dimension one and we get that $V$ has only trivial invariant subvector spaces if and only if it does not have a one-dimensional invariant subvector space which is what we will prove now.\ Let $U \subset V$ be invariant under conjugation by $\operatorname{SL}_2 ({\mathds F}_p)$ and of dimension one. Then there must be a matrix $A = \begin{pmatrix} a & b \\ c & -a \end{pmatrix}$ non-zero, such that $$U = \{ 0, A, 2A, \ldots, (p-1)A \}.$$ Consider $S = \begin{pmatrix} 1 & 1 \\0 & 1 \end{pmatrix}$ and $$S \begin{pmatrix} a & b \\c & -a \end{pmatrix} S^{-1} = \begin{pmatrix} a+c & -2a-c+b \\c & -a-c \end{pmatrix} \stackrel{!}{=} \lambda \begin{pmatrix} a & b \\c & -a \end{pmatrix}.$$ So in order for $U$ to be invariant, we have to have $c=0$ and $\lambda = 1$, which also gives $a=0$. Hence $U$ must be the matrices of the form $\begin{pmatrix} 0 & b \\0 & 0 \end{pmatrix}$. Let us assume that space of matrices is invariant. Then the orthogonal complement is $$U^\perp = \left\{ \begin{pmatrix} x & 0 \\y & z \end{pmatrix} \in \operatorname{Mat}_2 ({\mathds F}_p)\right\}.$$ But here we can again find that conjugation by $S$ does not stay within $U^\perp$. Let $A = \begin{pmatrix} 0 & 0 \\1 & 0 \end{pmatrix} \in U^\perp$. Then $$SAS^{-1} = \begin{pmatrix} 1 & -1 \\1 & -1 \end{pmatrix} \notin U^\perp.$$ We excluded all possibilities of one-dimensional invariant linear subspaces and proved the lemma. \[containsSL2\] Let $p$ be as in section \[NotNFC\]. Then $\rho(\operatorname{Gal}(K(p)/K))$ contains $ \operatorname{SL}_2 ({\mathds F}_p)$. By property (P2), we have $\rho(\operatorname{Gal}(K(p)/{\mathds Q})) = \operatorname{GL}_2({\mathds F}_p)$. Consider the normal subgroup $N:= \operatorname{Gal}(K(p)/K)$ of $\operatorname{Gal}(K(p)/{\mathds Q})$. Then $\operatorname{Gal}(K(p)/{\mathds Q})/N$, which is isomorphic to $\operatorname{Gal}(K/{\mathds Q})$, has exponent $\exp(\operatorname{Gal}(K/{\mathds Q}))$. So also $\rho(\operatorname{Gal}(K(p)/{\mathds Q}))/\rho(N)$ has exponent dividing $\exp(\operatorname{Gal}(K/{\mathds Q}))$. Consider $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \in \operatorname{SL}_2({\mathds F}_p)$. Hence $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \in \rho(\operatorname{Gal}(K(p)/{\mathds Q}))$. We take the $\exp(\operatorname{Gal}(K/{\mathds Q})) $-th power of this matrix and get an element of $\rho(N)$ (recall that $\exp(\operatorname{Gal}(K/{\mathds Q}))$ is coprime to $p$): $$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^{\exp(\operatorname{Gal}(K/{\mathds Q}))} = \begin{pmatrix} 1 & \exp(\operatorname{Gal}(K/{\mathds Q})) \\ 0 & 1 \end{pmatrix} \in \rho(N) \cap \operatorname{SL}_2({\mathds F}_p).$$ Since $\rho(N)$ is normal in $\rho(\operatorname{Gal}(K(p)/{\mathds Q}))$, then also $\rho(N) \cap \operatorname{SL}_2({\mathds F}_p)$ is normal in $\operatorname{SL}_2({\mathds F}_p)$. By Theorem 8.4 of [@MR1878556], the only normal subgroups of $\operatorname{SL}_2({\mathds F}_p)$ are $\{1\}, \{\pm 1 \}, \operatorname{SL}_2({\mathds F}_p)$. But since we found one element in $\rho (N) \cap \operatorname{SL}_2 ({\mathds F}_p)$ that is not in $\{\pm 1 \}$, we get $\rho(N) \cap \operatorname{SL}_2 ({\mathds F}_p) = \operatorname{SL}_2({\mathds F}_p)$. \[isMat2\] Let $p\geq 3$ and $G$ be a subgroup of $\operatorname{Mat}_2({\mathds F}_p)$ of order $p^2$. Let $V$ be the subgroup of $\operatorname{Mat}_2({\mathds F}_p)$ generated by $A B A^{-1}$ where $B$ varies over $G$ and $A$ varies over $\operatorname{SL}_2({\mathds F}_p)$. Then $V=\operatorname{Mat}_2({\mathds F}_p)$. Since $G$ has more than $p$ elements, there must be a non-scalar matrix in $G$. So let $B\in G$ be a non-scalar matrix. Then since scalar matrices are the only matrices that commute with all elements on $\operatorname{SL}_2({\mathds F}_p)$, there must be $A\in \operatorname{SL}_2({\mathds F}_p)$ such that $A B A^{-1} \neq B$. Then $\operatorname{Tr}(A B A^{-1} - B) = \operatorname{Tr}(A B A^{-1}) - \operatorname{Tr}(B) = \operatorname{Tr}(B) - \operatorname{Tr}(B) = 0$ and $V^0 := \{ B \in V | \operatorname{Tr}(B) = 0 \} \neq \{ 0 \}$. Since the action by conjugation of $\operatorname{SL}_2 ({\mathds F}_p)$ on $\{ B \in \operatorname{Mat}_2 ({\mathds F}_p) | \operatorname{Tr}(B) = 0 \} $ leaves only the trivial subvector spaces invariant and $V^0$ is not just the zero vector, we find that $V^0 = \{ B \in \operatorname{Mat}_2({\mathds F}_p) | \operatorname{Tr}(B) = 0 \}$ which has dimension $3$. Now since for $p > 2$ the identity matrix is an element of $V$, but not of $V^0$, we have $V \supsetneq V^0$. But since $V^0$ is an ${\mathds F}_p$-vector space of dimension $3$ (hence has order $p^3$) and $V$ is strictly larger than $V^0$ (hence has to have order strictly larger than $p^3$), we get $V=\operatorname{Mat}_2({\mathds F}_p)$. The actual descent ================== \[Lemma6.2\] Let $G := \operatorname{Gal}(F(N)/F(N/p))$. Suppose $E$ and $p$ satisfy (P1) and (P2). We assume $p^2\mid N$. Then: - The subgroup of $H := \operatorname{Gal}(K(N)/K)$ generated by the conjugates of $G$ equals $\operatorname{Gal}(K(N)/K(N/p))$. - If $\alpha\in K(N)$ with $\sigma(\alpha)\in F(N/p)$ for all $\sigma\in\operatorname{Gal}(K(N)/K)$, then $\alpha\in K(N/p).$ \(i) We will follow closely the proof of Habegger’s Lemma 6.2 but we will not use the concept of non-split Cartan subgroups as in Habegger’s proof. Instead we will use the lemmas above to show that the group $H$ is large enough. We have $$\begin{aligned} \label{6.3} H \subset \operatorname{Gal}(K(N)/K(N/p))\end{aligned}$$ and we will show the equality.\ We now want to look at the Galois representations and choose a basis for each $E[N]$ that is compatible with the diagram below. Let $$\tilde{\rho} : \operatorname{Gal}(K(p)/K) \to \operatorname{GL}_2 ({\mathds F}_p) \text{ and } \rho: \operatorname{Gal}(K(N)/K) \to \operatorname{GL}_2 ({\mathds Z}/p^n{\mathds Z})$$ and put them into the following commutative diagram: $$\begin{aligned} \label{Bild6.4} \begin{xy}\xymatrix{ &&\operatorname{Gal}(K(N)/K) \ar[d] \ar[r]^\rho & \operatorname{GL}_2 ({\mathds Z}/p^n{\mathds Z}) \ar[d] \\ &&\operatorname{Gal}(K(N/p)/K) \ar[d] \ar[r]^{\rho|_{K(N/p)}} & \operatorname{GL}_2 ({\mathds Z}/p^{n-1}{\mathds Z})\ar[d]\\ &&\operatorname{Gal}(K(p)/K) \ar[r]^{\tilde{\rho}} & \operatorname{GL}_2 ({\mathds F}_p) } \end{xy}\end{aligned}$$ The right vertical arrows are the natural surjections and the left vertical arrows are induced by the restrictions. By Lemma \[containsSL2\] we know that $\tilde{\rho}(\operatorname{Gal}(K(p)/K))$ contains $\operatorname{SL}_2({\mathds F}_p)$. We will now construct a homomorphism ${\mathcal L}$ from $\operatorname{Gal}(K(N)/K(N/p))$ to $\operatorname{Mat}_2 ({\mathds F}_p)$ which will firstly show by its injectivity that $|H| \leq p^4$ and secondly, through Lemma \[isMat2\], show equality.\ If $\sigma\in\operatorname{Gal}(K(N)/K(N/p))$ then $\rho (\sigma)$ is represented by $1+p^{n-1} {\mathcal L}^\prime (\sigma)$ with ${\mathcal L}^\prime (\sigma)\in\operatorname{Mat}_2 ({\mathds Z})$. Moreover, ${\mathcal L}^\prime (\sigma)$ is well-defined modulo $p\operatorname{Mat}_2 ({\mathds Z})$. We obtain by reduction mod $p$ a “logarithm” ${\mathcal L}: \operatorname{Gal}(K(N)/K(N/p)) \to \operatorname{Mat}_2 ({\mathds F}_p)$. The name comes from the following property: Let $\sigma_1, \sigma_2 \in \operatorname{Gal}(K(N)/K(N/p))$, then $\rho (\sigma_1 \sigma_2) = (1+p^{n-1} {\mathcal L}(\sigma_1))(1+p^{n-1} {\mathcal L}(\sigma_2)) \equiv 1+p^{n-1}({\mathcal L}(\sigma_1)+{\mathcal L}(\sigma_2)) {\text{ mod }}p^n \operatorname{Mat}_2({\mathds Z}) $ where we let ${\mathcal L}$ be the reduction of ${\mathcal L}^\prime$ modulo $p\operatorname{Mat}_2 ({\mathds Z})$. So ${\mathcal L}(\sigma_1\sigma_2)={\mathcal L}(\sigma_1) + {\mathcal L}(\sigma_2)$, hence ${\mathcal L}$ is a group homomorphism. We want to show that ${\mathcal L}$ is injective. Let $\sigma \in \operatorname{Gal}(K(N)/K(N/p))$ be such that ${\mathcal L}(\sigma) = \operatorname{id}$ in $\operatorname{Mat}_2 {\mathds F}_p$. This means that $\rho(\sigma) = 1$ in $\operatorname{GL}_2 ({\mathds Z}/p^n {\mathds Z})$. We look at the diagram $\eqref{Bild6.4}$ and see that this means that $\sigma$ fixes $K(p^n)$. Since it is an element of $\operatorname{Gal}(K(N)/K(N/p)$, it also fixes $K(N/p)$, hence fixes $K(N)$. So $\sigma \in \operatorname{Gal}(K(N)/K(N/p))$ is the identity and ${\mathcal L}$ is injective. Hence we get $$\begin{aligned} [K(N):K(N/p)] \leq |\operatorname{Mat}_2({\mathds F}_p)| = p^4. \label{6.7}\end{aligned}$$ Remark that since the Galois extension $K(N/p)/K$ is normal, the Galois group $\operatorname{Gal}(K(N)/K(N/p))$ is a normal subgroup of $\operatorname{Gal}(K(N)/K)$, hence if $\sigma\in \operatorname{Gal}(K(N)/K)$ and $\psi \in G$ we have $\sigma\psi\sigma^{-1}\in\operatorname{Gal}(K(N)/K(N/p))$. Then $$\begin{aligned} \rho (\sigma\psi\sigma^{-1}) \equiv 1+p^{n-1}\tilde{\rho}(\sigma) {\mathcal L}^\prime(\psi)\tilde{\rho}(\sigma)^{-1} {\text{ mod }}p^n\operatorname{Mat}_2({\mathds Z}).\end{aligned}$$ Hence $$\begin{aligned} {\mathcal L}(\sigma\psi\sigma^{-1}) = \tilde{\rho}(\sigma) {\mathcal L}(\psi)\tilde{\rho}(\sigma)^{-1}. \label{equation6.8}\end{aligned}$$ By Lemma \[Lemma3.4\] (iv), $G$ has order $p^2$, so $|{\mathcal L}(G)| = p^2$ by injectivity of ${\mathcal L}$.\ We recall that by Lemma \[containsSL2\] the image of $\tilde{\rho}$ contains $\operatorname{SL}_2 ({\mathds F}_p)$. So the definition of $H$ and equation imply that conjugating a matrix in ${\mathcal L}(G)$ by an element of $\operatorname{SL}_2({\mathds F}_p)$ stays within ${\mathcal L}(H)$. We want to apply Lemma \[isMat2\] to ${\mathcal L}(G)$ to deduce ${\mathcal L}(H) = \operatorname{Mat}_2({\mathds F}_p)$. Then $|{\mathcal L}(H)| = |H| = p^4$ and by , $H$ has to be equal to $\operatorname{Gal}(K(N)/K(N/p))$. For applying Lemma \[isMat2\] we have to prove that ${\mathcal L}(G)$ contains a non-zero scalar matrix:\ By Lemma \[Lemma3.4\] (v) we know that the image of the Galois representation $\operatorname{Gal}(F(p^n)/F) \to \operatorname{Aut}E[p^n]$ contains multiplication by $M^{[F:{\mathds Q}_q]}$ for any $M$ coprime to $p$. We now want to construct an element in $\operatorname{Gal}(F(p^n)/F(p^{n-1}))$ whose image is scalar multiplication. Let $M$ be a generator of $({\mathds Z}/p^n{\mathds Z})^*$, then the multiplication by $M$ will have order $(p-1)p^{n-1}$ in $\operatorname{Aut}E[p^n]$.\ Now by Lemma \[Lemma6.4\] we know that there exists $\sigma_F^\prime\in\operatorname{Gal}(F(p^n)/F)$ such that its image is the multiplication by $M^{[F:{\mathds Q}_q]}$. We want to show that $\sigma_F^{\prime (q-1)p^{n-2}} $ is an element of $\operatorname{Gal}(F(p^n)/F(p^{n-1}))$ and that it is not trivial. We start with the non-triviality. Remark that $\operatorname{Gal}(F(p^n)/F)/\operatorname{Gal}(F(p^n)/F(p^{n-1})) \cong \operatorname{Gal}(F(p^{n-1})/F)$. Since $\operatorname{Gal}(F(p^n)/F)$ is isomorphic to $A\times ({\mathds Z}/p^{n-1}{\mathds Z})^2$ where $A$ is a subgroup of ${\mathds Z}/(q-1){\mathds Z}$ (see the proof of Lemma \[Lemma3.5\]), its exponent will be $ap^{n-1}$ where $a\mid (q-1)$. But $[F:{\mathds Q}_q]$ is not a multiple of $ap^{n-1}$ since $[F:{\mathds Q}_q]$ is coprime to $p$. Hence $\sigma_F^{\prime (q-1)p^{n-2}}$ is not trivial.\ Now consider $\operatorname{Gal}(F(p^{n-1})/F)$. This group is isomorphic to $A\times ({\mathds Z}/p^{n-2}{\mathds Z})^2$ where $A$ is a subgroup of ${\mathds Z}/(q-1){\mathds Z}$ and its exponent will be $ap^{n-2}$ where $a\mid (q-1)$. On the other hand, $[F:{\mathds Q}_q](q-1)p^{n-2}$ is now a multiple of $(q-1)p^{n-2}$, hence the restriction of $\sigma_F^{\prime (q-1)p^{n-2}}$ to $F(p^{n-1}))$ is trivial which means that it has to be in $\operatorname{Gal}(F(p^n)/F(p^{n-1}))$.\ By Lemma \[Lemma3.4\] (iv), we have $\operatorname{Gal}(F(p^n)/F(p^{n-1}))\cong \operatorname{Gal}(F(N)/F(N/p))$, hence we can find $\sigma_F \in \operatorname{Gal}(F(N)/F(N/p))$ that gets mapped to $\sigma_F^\prime$ under that isomorphism. We can apply ${\mathcal L}$ to find that ${\mathcal L}(\operatorname{Gal}(F(N)/F(N/p)))$ contains an element that acts as scalar multiplication on the $p^n$-torsion points, hence has to be a scalar matrix.\ (ii) Now we proceed with the second part. Let $\alpha\in K(N)$ with $\sigma(\alpha)\in F(N/p)$ for all $\sigma\in\operatorname{Gal}(K(N)/K)$. Since we can invert elements of Galois groups, it makes sense to consider $\sigma^{-1}$ whenever $\sigma\in\operatorname{Gal}(K(N)/K)$ and with the first part of the Lemma we get that the group generated by $\sigma\psi\sigma^{-1}$ equals $\operatorname{Gal}(K(N)/K(N/p))$. Since $\alpha$ is fixed by such a $\sigma\psi\sigma^{-1}$, it has to be in $K(N/p)$ which is what we wanted to show. The technique of the descent used in the following theorem has been developed by Amoroso and Zannier in Section 4 of [@MR2651944]. \[Proposition6.1\] Let $E$ be an elliptic curve over ${\mathds Q}$ without complex multiplication. Let $L$ be a Galois extension of ${\mathds Q}$. Suppose there exists $d\in{\mathds N}$ such that $L$ has uniformly bounded local degrees above all but finitely many primes where $d$ is the said uniform bound. Then there is a prime number $p$ satisfying , , and . If $\alpha\in L(E_{\text{tor}})^* \backslash \mu_\infty$, then $$\begin{aligned} \label{Equation6.11} h(\alpha) \geq \frac{(\log p)^4}{p^{5p^4}}.\end{aligned}$$ Again, we follow here the analogous proof of Proposition 6.1 of [@MR3090783] closely. Since $E$ does not have complex multiplication, its $j$-invariant is neither $0$, nor $1728$. So the reduction of $E$ at $p$ is an elliptic curve with $j$-invariant neither $0$, nor $1728$ for all but finitely many primes $p$. By a Theorem of Serre [@MR0387283], all but finitely many of these $p$ satisfy (P2). Furthermore, by [@MR903384], there are infinitely many supersingular primes for an elliptic curve over ${\mathds Q}$. We may thus fix a prime $p$ satisfying (P1), (P2), (P3) and (P4) and set $q=p^2$.\ Recall the following facts thet we fixed in the beginning of the chapter: Let $\alpha \in L(E_{\text{tor}})^* \backslash \mu_\infty $. Then $\alpha \in K(N)$ for some $N=p^n M$ with $M\in{\mathds N}$ coprime to $p$, $n$ a nonnegative integer and $K \subset L$ a number field that is Galois over ${\mathds Q}$. Let ${\mathds Q}_q$ be the unique quadratic unramified extension of ${\mathds Q}_p$. Then we fix a finite Galois extension $F/{\mathds Q}_q$ with ${\mathds Q}_q \subset F \subset \overline{{\mathds Q}_p}$ such that the $v$-adic completion of $K$ is contained in $F$ (where $v$ extends $p$) and $[F:{\mathds Q}_p]$ is uniformly bounded by $d$. Let furthermore $\mathcal{E} = (q-1)[F:{\mathds Q}_q] \exp(\operatorname{Gal}(L/{\mathds Q}))$.\ We take $\sigma_F\in \operatorname{Gal}(F(N)/F)$ as in Lemma \[Lemma6.4\]. If we are in the case of $p^2 \nmid N$, we can artificially choose an element in $\operatorname{Gal}(F(p^2N)/F)$ and restrict it to $F(N)$. Then we define $$\begin{aligned} \label{Equation6.12} \gamma = \frac{\sigma_F (\alpha)}{\alpha^{4^{[F:{\mathds Q}_q]}}} \in K(N).\end{aligned}$$ By the properties of the height we get $$\begin{aligned} \label{Equation6.13} h(\gamma) \leq h(\sigma_F (\alpha)) + h(\alpha^{4^{[F:{\mathds Q}_q]}}) = (4^{[F:{\mathds Q}_q]} +1) h(\alpha).\end{aligned}$$ Let us start with the case of $n\geq 2$, hence $p^2 \mid N$. Since $\gamma\in K(N) \subset F(N)$, hence $\sigma (\gamma) \in K(N) \subset F(N)$ for all $\sigma\in\operatorname{Gal}(K(N)/K)$, there is a least integer $n^\prime \leq n$ such that $\sigma(\gamma)\in F (p^{n^\prime}M)$ for all $\sigma\in\operatorname{Gal}(K(N)/K)$. Lemma \[Lemma6.2\] implies that then also $\gamma\in K(p^{n^\prime}M)$.\ By minimality of $n^\prime$ there is a $\sigma\in\operatorname{Gal}(K(N)/K)$ such that $\sigma(\gamma)\notin F (p^{n^\prime -1}M)$. We will split this up into two cases: First $n^\prime \geq 2$ and second $n^\prime \leq 1$. We start with $n^\prime \geq 2$. We apply $\sigma$ to and obtain $$\begin{aligned} \label{Equation6.14} \sigma(\gamma) = \frac{\sigma(\sigma_F (\alpha))}{\sigma(\alpha^{4^{[F:{\mathds Q}_q]}})} = \frac{\sigma_F(\sigma (\alpha))}{\sigma(\alpha)^{4^{[F:{\mathds Q}_q]}}}\end{aligned}$$ since $\sigma_F$ lies in the center of $\operatorname{Gal}(K(N)/K)$ by Lemma \[Lemma6.4\]. Next we want to apply Lemma \[Lemma5.3\] to $\sigma(\gamma)$, so we must verify that $\sigma(\gamma)^q\notin F(p^{n^\prime -1}M)$. We will show this by contradiction, so assume $\sigma(\gamma)^q\in F(p^{n^\prime -1}M)$. Since $\sigma(\gamma)\notin F (p^{n^\prime -1}M)$, there is $\psi \in \operatorname{Gal}(F (N)/F(p^{n^\prime-1}M))$ such that $\psi(\sigma(\gamma)) \neq \sigma(\gamma)$. Furthermore, $\psi(\sigma(\gamma)^q) = \sigma(\gamma)^q$ by our assumption and so $$\begin{aligned} \label{Equation6.15} \psi(\sigma(\gamma)) = \xi \sigma(\gamma) \text{ for some } \xi^q = 1 \text{ while } \xi \neq 1.\end{aligned}$$ We apply $\psi$ to equation and obtain $$\begin{aligned} \psi(\sigma(\gamma)) & = \frac{\psi(\sigma_F(\sigma (\alpha)))}{\psi(\sigma(\alpha)^{4^{[F:{\mathds Q}_q]}})}\\ & = \frac{\sigma_F(\psi (\sigma(\alpha)))}{\psi(\sigma(\alpha))^{4^{[F:{\mathds Q}_q]}}} \end{aligned}$$ since $\psi$ commutes with $\sigma_F$ (by Lemma \[Lemma6.4\]). We define $\eta = \frac{\psi(\sigma(\alpha))}{\sigma(\alpha)} \neq 0$ and get, by and , $$\begin{aligned} \xi & = \frac{\psi(\sigma(\gamma))}{\sigma(\gamma)} \nonumber\\ & = \frac{\psi\left( \frac{\sigma_F(\sigma(\alpha))}{\sigma(\alpha)^{4^{[F:{\mathds Q}_q]}}} \right)}{\frac{\sigma_F(\sigma(\alpha))}{\sigma(\alpha)^{4^{[F:{\mathds Q}_q]}}}} \nonumber\\ & = \frac{\psi(\sigma_F(\sigma(\alpha)))\sigma(\alpha)^{4^{[F:{\mathds Q}_q]}}}{\psi(\sigma(\alpha))^{4^{[F:{\mathds Q}_q]}} \sigma_F (\sigma(\alpha))} \nonumber\\ & = \frac{\sigma(\alpha)^{4^{[F:{\mathds Q}_q]}}} {\psi(\sigma(\alpha))^{4^{[F:{\mathds Q}_q]}}} \frac{\psi(\sigma_F(\sigma(\alpha)))}{\sigma_F (\sigma(\alpha))} \nonumber\\ & = \frac{\sigma(\alpha)^{4^{[F:{\mathds Q}_q]}}}{\psi(\sigma(\alpha))^{4^{[F:{\mathds Q}_q]}}} \frac{\sigma_F(\psi(\sigma(\alpha)))}{\sigma_F (\sigma(\alpha))} \nonumber\\ & = \eta^{-4^{[F:{\mathds Q}_q]}} \sigma_F(\eta)\nonumber\\ & = \frac{\sigma_F(\eta)}{\eta^{4^{[F:{\mathds Q}_q]}}} \label{XiEqualsXiPrime}.\end{aligned}$$ Since $\xi$ is a root of unity, we have $$4^{[F:{\mathds Q}_q]} h(\eta) = h(\eta^{4^{[F:{\mathds Q}_q]}}) = h(\xi \eta^{4^{[F:{\mathds Q}_q]}}) = h(\sigma_F (\eta)) = h(\eta),$$ so $h(\eta) = 0$ and by Kronecker’s Theorem, $\eta$ is a root of unity. We now fix $\widetilde{M}\in{\mathds N}$ coprime to $p$ such that $\eta^{\widetilde{M}}\in\mu_{p^\infty}$. Lemma \[Lemma3.5\] now implies that $\eta^{\widetilde{M}}$ is already in $\mu_{p^n}$ and by Lemma \[Lemma6.4\] we have $\sigma_F (\eta^{\widetilde{M}}) =(\eta^{\widetilde{M}})^{4^{[F:{\mathds Q}_q]}}$. And we get $$\begin{aligned} \sigma_F (\eta) = \xi^\prime \eta^{4^{[F:{\mathds Q}_q]}}\end{aligned}$$ for some $\xi^\prime$ such that $\xi^{\prime{\widetilde{M}}} = 1$. Using equation and we get that $\xi = \xi^\prime$ and hence $\xi^q = \xi^{\widetilde{M}} = 1$. But since $\widetilde{M}$ and $q$ are coprime we must have $\xi = 1$ which is a contradiction. So our assumption on $\sigma(\gamma)^q$ is false and we get $\sigma(\gamma)^q \notin F (p^{n^\prime-1}M)$. Hence we can apply Lemma \[Lemma5.3\] and get the following lower bound for the height of $\sigma(\gamma)$ $$\begin{aligned} \frac{(\log p)^4}{4\cdot 10^6 p^{32}} \leq h(\sigma (\gamma)) = h(\gamma) \leq (4^{[F:{\mathds Q}_q]}+1) h(\alpha) \label{ramifiedHeight}.\end{aligned}$$ This was the case of $n^\prime \geq 2$. Let us now assume that $n^\prime \leq 1$, which allows us to descent to the tamely ramified case. Recall that $\gamma = \frac{\sigma_F (\alpha)}{\alpha^{4^{[F:{\mathds Q}_q]}}}$. We do a descent as in the totally ramified case: There is a least integer $n^\prime \leq 1$ such that $\sigma(\gamma)\in F (p^{n^\prime}M)$ for all $\sigma\in\operatorname{Gal}(K(N)/K)$. Lemma \[Lemma6.2\] implies that then also $\gamma\in K(p^{n^\prime}M)$. We will treat this case together with the case $n \leq 1$ where we do not need a descent at all.\ Since $h(\sigma(\gamma)) = h(\gamma)$ we can in both cases compute the height of $\gamma$ where $\gamma$ will be an element of $K(p^{n^\prime}M)$ where $n^\prime \leq 1$. We want to apply Lemma \[Lemma6.3\], so we will prove that $\gamma \neq 0$ and not a root of unity. If so, we would have $4^{[F:{\mathds Q}_q]} h(\alpha) = h(\alpha^{4^{[F:{\mathds Q}_q]}}) = h(\gamma \alpha^{4^{[F:{\mathds Q}_q]}}) = h(\sigma_F (\alpha)) = h(\alpha)$ by the properties of the height and hence $h(\alpha) = 0$. By Kronecker’s Theorem this either means $\alpha =0$ or $\alpha\in\mu_\infty$. But this is a contradiction to our assumption on $\alpha$. Hence Lemma \[Lemma6.3\] gives $$\begin{aligned} h(\gamma) \geq \left(\frac{\log p}{\mathcal{E}(1+q^\mathcal{E})(1+ \frac1{2^{10}})}\right)^4.\end{aligned}$$ Moreover, we can use inequality and $\mathcal{E} \leq \frac{p^4}{2}$ to get $$\begin{aligned} h(\alpha) & \geq \frac1{4^{[F:{\mathds Q}_q]}+1} \left(\frac{\log p}{\mathcal{E}(1+q^\mathcal{E})(1+ \frac1{2^{10}})}\right)^4\\ & \geq \frac{(\log p)^4}{(4^{[F:{\mathds Q}_q]}\mathcal{E}q^\mathcal{E}2)^4}\\ & \geq \frac{(\log p)^4}{(4^{\frac{p^4}{2}} \frac{p^4}{2}p^{p^4}2)^4}\\ & \geq \frac{(\log p)^4}{(4^{\frac{p^4}{2}} p^{p^4+4})^4} \\ & \geq \frac{(\log p)^4}{4^{2p^4} p^{4p^4+16}} \\ & \geq \frac{(\log p)^4}{p^{\frac{\log 4}{\log p}2p^4 + 4p^4+16}} \\ & \geq \frac{(\log p)^4}{p^{5p^4}}.\end{aligned}$$ Now we have to put the tamely (inequality above) and totally ramified (inequality ) case into one bound.\ We get $h(\alpha) \geq \max\left(\frac{(\log p)^4}{(4^{[F:{\mathds Q}_q]}+1)\cdot 2\cdot 10^3 p^{32}}, \frac{(\log p)^4}{p^{5p^4}}\right) = \frac{(\log p)^4}{p^{5p^4}}$. [^1]: This paper is the second part of the PhD thesis of the author. The first part of the thesis is the paper [@2017arXiv171204214F].

--- abstract: 'The magnetic and transport properties have been investigated for the composite polycrystalline manganites, (1-x)$La_{2/3}Ca_{1/3}MnO_3$/(x)yttria-stabilized zirconia ( (1-x)LCMO/(x)YSZ ), at various YSZ fractions, x, ranging from 0 % to 15 %. The ac magnetic susceptibility, $\chi(T)$, DC magnetization, $M(T)$, temperature dependent resistivity, $\rho(T)$ and thermoelectric power (TEP), $S(T)$, have been measured. It was found, surprisingly, that a TEP peak showed up in the magnetic transition region for the sample with the x even as little as 0.75 %. The magnetic transition temperature reaches the minimum value as x increases from 0 % to 4.5 % and goes up as x increases further. Several possible factors such as the effect of strain, the finite size effect, and the effect of magnetic tunnelling coupling, [*etc.*]{}, in affecting the above physical properties of the composite manganites have been studied carefully. The strain induced by the YSZ/LCMO boundary layer (BL) was identified as the leading factor to account for the x dependence of these properties. It demonstrates that the effect of strain could be important in the bulk manganites as in the film sample.' address: | (1) Department of Physics and State Key Lab for Mesoscopic Physics, Peking University, Beijing 100871, P.R.China\ (2) Electron microscopy laboratory, Peking University, Beijing 100871, P.R.China author: - 'Wei Liu$^{1}$, Chinping Chen$^{1}$[@chen], Xinfeng Wang$^{1}$, Jun Zhao$^{1}$, Xiangyu Xu$^{1}$, Jun Xu$^{1,2}$, and Shousheng Yan$^{1}$' title: ' Effect of strain on the magnetic and transport properties of the composite manganites, La$_{2/3}$Ca$_{1/3}$MnO$_3$/yttria-stabilized zirconia' --- Introduction ============ The perovskite manganites, $A_{1-y}B_yMnO_3$, in which A is for the rare earth trivalent cation and B for the alkaline divalent one, exhibit complicated phases at various temperatures and hole doping concentrations, y. Due to the important magnetic application potentials and fundamental research interests, tremendous activities in the physics community have focused in this area during the past decade. With an appropriate doping concentration y, an FM transition takes place at the Curie point, $T_C$, with the decreasing temperature. It is accompanied with the metal-insulator (MI) transition at $T_P$. A colossal magneto-resistance (CMR) slao occurs around this temperature. These properties can be explained with the double exchange (DE) mechanism, [@Zener51; @Anderson55; @Goodenough55; @deGennes60]. Recently, growing attentions have been focused on the effect of strain arising from the interface or surface states of the CMR thin film [@Dulli00; @Bibes01; @Bibes03]. The variation of the magnetic transition temperature, $T_C$, was interpreted as attributed to the strain. However, it is usually difficult to separate the strain-induced effect from the finite size confinement one with the thin films [@Zhang01; @Ziese02; @Andres03]. On the other hand, the effect of strain is usually overlooked in the bulk samples. We would like to demonstrate that the effect of the BL strain with the polycrystalline composite manganites, (1-x)LCMO/(x)YSZ, is a very important factor for the variation of the physical properties such as $T_C$, the temperature dependent resistivity, $\rho(T)$, and the $S(T)$ behavior, [*etc.*]{}. Sample preparation ================== A double-staged process was applied in preparing the LCMO/YSZ samples [@Yuan02]. In the first stage, the LCMO nano-sized powder was produced by the sol-gel method and then sintered at 1300$\ ^{o}C$ for 10 hours to form crystals of about 20 $\mu$m. In the second stage, the thus-obtained LCMO powders were mixed with the YSZ powders of about 2 $\mu$m for the heat treatment at 1350$\ ^{o}C$ for another 10 hours. The X-ray diffraction (XRD) was performed by a Philip x’ pert diffractometer using the Cu K$_{\alpha}$ line (1.54056 $\AA$). The XRD spectra are presented in Fig. \[XRD\]. The YSZ phase was identified for the samples with x exceeding 4.5 %. The lattice constants of the LCMO phase remain unchanged within 0.001 $\AA$ for all of the YSZ mixing concentration, x. This indicates that, within the detection sensitivity of XRD, the bulk LCMO composition was not modified during the heat treatment due to any possible diffusion of ions from the YSZ composition. We have prepared the samples of LCMO annealed at 1400$\ ^{o}C$ without the mixing of YSZ as well. This would demonstrate the widely investigated disorder effect resulting from different annealing conditions. The morphology of the sample was investigated by a scanning electron microscope (SEM) performed on the system of FEI STRATA DB235 focus ion beam (FIB) electron microscope. It revealed that at low mixing concentration, x $<$ 4.5 %, the LCMO phase formed large cluster surrounded by a thin layer of YSZ component at a thickness of the order of 10 nm. The BL area increases while the cluster size of the LCMO phase decreases with the increasing YSZ fraction. The interconnecting paths between the adjacent LCMO clusters separated by the YSZ layers would reduce accordingly. On the other hand, at the mixing fraction, x, exceeding about 4.5 %, the YSZ phase forms cluster-like structure by itself. Thus, the BL area decreases correspondingly. This observation is consistent with the previous report[@Yuan02]. Within the detection sensitivities of the XRD and SEM, no indication of the inter diffusion between the LCMO and the YSZ phases exists. The two phases, hence, form solid mixture with the existence of the BL strain in between. Since the YSZ phase is insulating and non-magnetic. the LCMO/YSZ composites are, therefore, appropriate for the investigation of the magnetic and transport properties under the BL strain. Experiment ========== The dc magnetization, $M(T)$, and ac magnetic susceptibility, $\chi(T) = \chi'+ i\chi''$, were performed using Quantum Design PPMS and MPMS, respectively. The ac susceptibility measurement was carried out with the excitation field of 10 Oe at 113 Hz under a few Oe of dc background field. The applied field for the $M(T)$ measurement, including the field-cooled (FC) and zero-field-cooled (ZFC), is 50 Oe, while the field applied during the cooling stage before the FC measurement is 3000 Oe. The magnetic transition temperatures, $T_C(dc)$ and $T_C(ac)$, are obtained using the definition, $dM/dT$ and $d\chi'/dT$. These two transition temperatures agree with each other well within a few Kelvin, as plotted in Fig. \[mag1\]. Also plotted in the same figure are the metal-insulator (MI) transition temperature, $T_P$, determined by the $\rho(T)$ measurement, and the magnetic transition temperature width, $\Delta T$, defined by the difference of temperatures at which $d\chi'/dT$ = 0. The correlation of the $T_C$ with the BL effect is apparent in the figure. The lowest $T_C$ occurs at x = 4.5 %, corresponding to the minimum BL inferred from the SEM observation. In order to investigate the effect of thermal treatment on $T_C$, the sample of x = 0 annealed at 1400$\ ^{o}C$ was measured as well for the magnetic transition temperature, $T_C$. It is 260 K using the same maximum slope criteria described above. The depression of $T_C$ down to about 245 K with the sample annealed at 1300$\ ^{o}C$ accounts for about 6 % effect. This is a well studied effect and is attributed to the disorder associated with the polycrystalline grain size distribution, [@Sanchez96; @Gupta96]. The out-of-phase component, $\chi''(T)$, is shown in Fig. \[chi2\]. Two characteristic dissipation features, the peak at high temperature, $T_{DH}$, and the bump at low temperature, $T_{DL}$, appear for each sample and are depicted in the inset of Fig. \[chi2\] as a function of the YSZ fraction, x. $T_{DH}$ is lower than $T_C$ by a few Kelvins. It is resulting from the energy dissipation process associated with the critical spin fluctuation near the FM phase transition. The x-dependence of $T_{DH}$ is therefore similar to that of $T_C$. On the other hand, $T_{DL}$ appears around 70 K for all of the samples, including the one with x = 0. This indicates that the LT bump is independent of the BL. In fact, similar bumps in $\chi''(T)$ at $T < T_C$ have been observed in many previous experiments, attributed to the magnetic inhomogeneity [@Moreira98]. Hence, the x independence of $T_{DL}$ is an indication that the characteristic crystalline grain size associated with the LCMO phase stays unaltered with the mixing of the YSZ component. The field-cooled (FC) and zero-field-cooled (ZFC) dc magnetization measurements on the chosen samples, x = 0 %, 1.25 %, 4.5 % and 15 %, were performed to investigate the magnetic inhomogeneity. Fig. \[FC-ZFC\] shows the normalized magnetization, M(T)/M$_m$, versus the reduced temperature, T/$T_C$, where M$_m$ is the maximum magnetization occurring near the freezing temperature at which the FC and ZFC branches of the $M(T)$ curves diverge. There is no appreciable difference for the x = 0 % sample from the other ones, indicating that the magnetic disorder revealed by this measurement is ascribed to the polycrystalline grains, independent of the BL. This is of the same origin causing the LT bump in the $\chi''(T)$ measurements. Note that, there is no structure in the $M(T)$ curve or the in-phase component, $\chi'(T)$ (not shown here), at the temperature near the LT bump. It indicates that it is not resulting from a magnetic phase transition. The $\rho(T)$ measurement was carried out from 80 K to 300 K in a home-made insert-probe by a standard 4-probe dc techniques using cold-pressed indium as the electrical contacts. The typical contact resistances is on the order of a few $\Omega$ with the applied current on the order of a few mA. The $\rho(T)$ curves are published in Fig. 1 by Liu [*et al*]{} [@Liu03]. In the region below $T_C$, the $\rho(T)$ behaviors are analyzed with the various scattering processes of the electrons by the function $\rho(T) = \rho_0 + AT^a + BT^b$, where a = 2 or 3/2, and b = 5 or 9/2. The maximum fitting range in temperature for a stable result is from the lowest temperature of the measurement to $T \sim$ 0.8 $T_C$. The $AT^a$ term with a = 2 or 3/2 gives equally good fitting of the experimental data. The coefficient of the fitting, A, corresponding to a = 3/2 or 2 also exhibits identical x dependence. It is difficult to distinguish which of the following two processes is the more important one, the $T^2$ term for the electron-electron scattering or the $T^{3/2}$ term for the scattering of electrons by the disordered spin glass component [@SpinGlass93]. The x dependence of this term is represented by the coefficient $A$, calculated using a = 2, versus x and plotted in Fig. \[RT\]. The variation with different samples is within a factor of 4. Similarly, an equally good fitting is obtained with the $BT^b$ term using b = 5 for the electron-phonon scattering or b = 9/2 for the electron-magnon scattering within the framework of DE mechanism [@Kubo72]. Since the $B$-variation versus x is the same using b = 9/2 or 5, the ratio of $B/A$ is plotted in the inset of Fig. \[RT\] using b = 5. Also plotted in the same inset is the $\rho_0/A$ ratio, by the open squares. The x dependence of $\rho_0$ associated with the residual or disorder scattering process is only slightly higher than that of the $AT^a$ term. From the above analysis for the various scattering processes, the $BT^b$ term is affected most profoundly by the presence of the LCMO/YSZ BL layer, indicating that the BL-induced strain has a strong effect on the electron-phonon coupling strength. The temperature dependent TEP, $S(T)$, was measured with the series of LCMO/YSZ samples by the home-made insert-probe using the dc differential technique. The electrical contacts were made by the cold-pressed indium. The sample was installed across two temperature platforms. One was regulated at temperature $T$, while the other controlled to vary within $T$ + $\Delta T$. A continuous voltage output, $\Delta V$, taken by Keithley 2010 multimeter was recorded with the corresponding $\Delta T$, typically a few tenths of a kelvin, changing slowly. The slope of the linear relation between $\Delta V$ and $\Delta T$, with the correction of the contribution from the Cu leads, would give the measured TEP of the sample. Abrupt TEP jumps occur during the magnetic phase transition for the x $>$ 0 % samples, but not for the sample with x = 0 %, see Fig. \[TEP\]. This demonstrates clearly a strong correlation of the jumps with the existence of the BL, and was interpreted as due to the magnetic inhomogeneity induced by the BL strain [@Liu03]. Similar TEP jump with the magneto-TEP effect has been observed in the thin film CMR manganites also [@Jaime96; @Jaime99]. The substrate strain unavoidably caused the magnetic inhomogeneity in the sample. Under the applied magnetic field, the inhomogeneous magnetic component has been reduced. The TEP jump was therefore suppressed to show the magneto-TEP effect. Discussion ========== The non-magnetic, insulating YSZ component intermixing in the LCMO causes variations in the magnetic and transport properties of the manganites. Most of the interesting features in the physical properties under current investigations are strongly correlated with the LCMO/YSZ BL. Several effects would be introduced on the samples by the existence of the BL. However, only a leading one is accounted for the observed x dependence. The strain induced by the BL would be the major factor identified in the present work. Usually, the effect of strain on the physical properties of the manganites, especially on the $T_C$ variations, is studied with the films. However, for thickness under a few hundred nanometers, the finite size confinement effect would become important to superimpose on the effect of strain. With the substrate strain, the interface magnetic inhomogeneity has been directly observed by the techniques of NMR [@Bibes01] or X-ray photoemission spectroscopy [@Dulli00]. Nonetheless, for these films, the finite size effect seems to be the leading factors in the depression of $T_C$, dominating the effect of strain under discussion. Fig. \[FiniteSize\] displays the shift of $T_C$ versus film thickness, $d$, summarized from many of the previous experiments for various thin films grown on different substrates[@Bibes01; @Andres03; @Gupta96; @Bibes02; @Campillo01; @Snyder96; @Huhtinen02; @Xiong96; @Rao98]. The results follows very well the law of finite size confinement[@Fisher72; @FiniteSize00], $|T_C(\infty)-T_C(d)|/T_C(\infty) = (\xi_0/d)^\lambda$ with $\lambda$ = 1 in consistent with the result from the mean field theory[@Zhang01], where $\xi_0 =$ 6 nm is the correlation length at T = 0 K, $T_C(d)$ is the transition temperature for a film of thickness, $d$, and $T_C(\infty)$ is that for the corresponding bulk samples. Note that the dispersion of the data points in Fig. \[FiniteSize\] indicates that the effect of the substrate strain and the crystallinity of the films superimposed on top of the confinement effect are non-negligible. This is reasonable since these points are summarized from various experiments performed by different laboratories. The above result indicates that the finite size effect is the leading factor for the suppression of $T_C$ with the thin film samples at a thickness less than a few hundred nanometers, even with the obvious coexistence of the substrate strain often suggested as the solely factor. For an LCMO cluster enclosed by the YSZ component at the small YSZ fraction, x $\leq$ 4.5 %, $T_C$ drops dramatically with the increasing x. In this region, the cluster size is on the order of several tens of micrometers. This is simply too large for the finite size confinement effect to occur according to the analysis for the thin films. For the YSZ serving as a non-magnetic separation between the LCMO phase, the reduction in the effective magnetic coupling is unlikely the major factor for the x dependence of the $T_C$ depression either. This effect would more or less level off at a layer thickness of a few nm according to the previous experiment[@Sirena03]. In the present work, the non-magnetic YSZ layer is at least 10 nm in thickness, reaching the region of saturation for such an effect. The effect of intergranular magnetic tunnelling coupling, which is beyond the DE mechanism, is unlikely the major factor either responsible for the observed properties. In this case, the depression of $T_C$ is less than 5 % with $T_P$ lower than $T_C$ by a temperature depending on the extent of the intergranular coupling strength. [@Sanchez96; @Gupta96; @Mahesh96; @Mahendrian96; @Hwang96; @Yuan03]. The main features of the present results, see Fig. \[mag1\], do not fit the description above since $T_C$ is depressed by more than 20 % with the varying x and $T_P$ follows it closely, see Fig. \[mag1\]. Furthermore, the insulating YSZ layer with a thickness of more than 10 nm is too thick for the electrical current to tunnel through at LT to show metallic property. In the polycrystalline LCMO/YSZ composite system, the LCMO cluster is larger than 10 $\mu$m. The ratio of the boundary strained layer over the volume depends on the thickness of the strained layer. It is possible for the spatial relaxation of an interfacial strain to extend over a distance of 1 $\mu$m [@Soh02], and results in a non-negligible volume fraction of the boundary strained layer in the bulk LCMO component. According to the previous reports, $T_C$ would be seriously depressed by the biaxial strain resulting from the substrate lattice mismatch. Merely 1 % of the biaxial strain would cause an order of 10 % variation in $T_C$[@Millis98b], as demonstrated by the experiment of ultrasound spectroscopy[@Darling98]. Yet, such a low level of strain would go undetected by the usual experimental techniques such as the XRD analysis. The magnetic anisotropy or inhomogeneity caused by the strain would explain the depression of $T_C$, and the corresponding broadening of the magnetic transition, $\Delta T$. In the analysis of $\rho(T)$ at $T < T_C$, the residual term, $\rho_0$, and the $AT^a$ term exhibit much less structure dependence than the $BT^b$ term. This is a strong evidence supporting that the electron-phonon coupling strength is modified by the existence of the BL. Since the Jahn-Teller (J-T) phonon mode depends strongly on the biaxial strain of the lattice [@Millis98b], it is reasonable to infer that the x dependence of the $BT^b$ term is caused by the biaxial strain, which affects the magnetic transition, $T_C$ as well. The strong correlation of the TEP jump during the magnetic transition with the presence of the BL is interpreted as of magnetic origin[@Liu03], and can be reasonably explained by the magnetic inhomogeneity induced by the strain. At LT, $S(T)$ shows a typical metallic behavior with a small absolute value. As the temperature increases toward the HT region, the fraction of the PM component having the semiconducting property increases. For the x = 0 sample, the change in the PM fraction relative to the FM one is smooth, showing a smooth transition in $S(T)$. On the other hand, the introduction of the BL with the x $>$ 0 samples would cause an extra contribution from the inhomogeneous magnetism, resulting in an abrupt deviation of $S(T)$ from the metallic region. Interestingly, in the previou work on the series of samples with constant valence, $Pr_{0.7}Ca_{0.3-x}Sr_xMnO_3$ [@Hejtmanek96], the temperature-dependent TEP behavior has been demonstrated to correlate strongly with the magnetic transition. Especially, a TEP jump begins at the temperature near the ferromagnetic-antiferromagnetic (AF) transition as shown in Fig.4 by Hejtmanek [*et al*]{} [@Hejtmanek96]. According to the present picture in explaining the TEP behavior, the occurrence of the AF component within the FM matrix is responsible for the abrupt jump of the TEP. A noteworthy point, however, is that the cause of the inhomogeneous distribution of the magnetism for the $Pr_{0.7}Ca_{0.3-x}Sr_xMnO_3$ samples is attributed to the FM-AF transition, quite different from the existence of the BL-induced strain in the presence work. Conclusion ========== In conclusion, the YSZ component in the LCMO/YSZ composite materials induces a large effect on the various physical properties such as the variations of $T_C$ and $T_P$, the broadening of magnetic transition, the pronounced TEP jump during the magnetic transition, and the resistivity variation in the LT FM phase, [*etc.*]{}. The BL-induced strain plays a crucial role in the explanation of the observed properties. In this respect, the effect of strain is not only important in the manganite film already reported by some of the recent works, but also has a profound effect in the bulk sample, as demonstrated by the present work. Acknowledgement =============== We are grateful to Prof. Songliu Yuan of the Department of Physics, Huazhong University of Science and Technology, Wuhan, China, for providing us with the samples and for the helpful discussions. One of the authors, C.P. Chen, would also like to appreciate Prof. Tong-han Lin of the Department of Physics, Peking University, Beijing, China, for some of the points raised and the fruitful discussions. e-mail address : cpchen@pku.edu.cn, TEL : +86-10-62751751 C. Zener, Phys. Rev. [**82**]{}, 403 (1951) P. Anderson and H. Hasegawa, Phys. Rev. [**100**]{}, 675 (1955) J. B. Goodenough, Phys. Rev. [**100**]{}, 564 (1955) P. G. de Gennes, Phys. Rev. [**118**]{}, 141 (1960) Hani Dulli, E.W. Plummer, P.A. Dowben, Jaewu Choi, and S.-H. Liou, Appl. Phys. Lett. [**77**]{}, 570 (2000) M. Bibes, Ll. Balcells, S. Valencia, J. Fontcuberta, M. Wojcik, E. Jedryka, and S. Nadolski, Phys. Rev. Lett. [**87**]{}, 67210 (2001) M. Bibes, Ll. Balcells, J. Foncuberta, M. Wojcik, S. 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--- abstract: 'In this paper, we study the class of Jordan dialgebras (also called quasi-Jordan algebras). We develop an approach for reducing problems on dialgebras to the case of ordinary algebras. It is shown that straightforward generalizations of the classical Cohn’s, Shirshov’s, and Macdonald’s Theorems do not hold for dialgebras. However, we prove dialgebraic analogues of these statements. Also, we study multilinear special identities which hold in all special Jordan algebras and do not hold in all Jordan algebras. We find a natural correspondence between special identities for ordinary algebras and dialgebras.' author: - 'Vasily Voronin[^1]' title: Special and exceptional Jordan dialgebras --- INTRODUCTION {#introduction .unnumbered} ============ One of the most important classes of nonassociative algebras is the class of Lie algebras defined by the anti-commutativity and Jacobi identities $x^2=0$, $(xy)z+(zx)y+(yz)x=0$. It is well-known that every associative algebra $A$ turns into a Lie algebra with respect to the new product $[a,b]=ab-ba$, $a,b\in A$. The Lie algebra obtained is denoted by $A^{(-)}$. The classical Poincaré—Birkhoff—Witt Theorem implies that every Lie algebra can be embedded into $A^{(-)}$ for an appropriate associative algebra $A$. Leibniz algebras introduced in [@Loday:93; @Cuvier:94] are the most popular non-commutative analogues of Lie algebras. An algebra $(L, [\cdot,\cdot])$ is said to be a (right) Leibniz algebra if the product $[\cdot,\cdot]\colon L\times L\to L$ satisfies the following (right) Leibniz identity: $$\label{eq:IdOfLeibnizAlgebras} [[x,y],z]=[[x,z],y]+[x,[y,z]].$$ To get an analogue of the Poincaré—Birkhoff—Witt Theorem for Leibniz algebras, the notion of an associative dialgebra was introduced in [@LodayPirashvili:93]. Namely, an associative dialgebra is a linear space $D$ with two bilinear operations $\vdash,\dashv\colon D\times D\to D$ satisfying certain axioms. The new product $[x,y]=x\dashv y-y\vdash x$, $x,y\in D$, satisfies (\[eq:IdOfLeibnizAlgebras\]), so $D$ is a Leibniz algebra with respect to this new product. The Leibniz algebra obtained is denoted by $D^{(-)}$. As it was shown in [@Loday:01; @AymonGrivel:03], every Leibniz algebra can be embeddable into $D^{(-)}$ for an appropriate associative dialgebra $D$. Another important class of nonassociative algebras is the class of Jordan algebras defined by the commutativity and Jordan identity $(x^2y)x=x^2(yx)$. It is well-known that if $A$ is an associative algebra over a field on characteristic $\not=2$ then $A$ with respect to the new product $a\circ b=\frac{1}{2}(ab+ba)$ is a Jordan algebra denoted by $A^{(+)}$. For Jordan algebras, the analogue of the Poincaré—Birkhoff—Witt theorem is not true: There exist Jordan algebras that can not be embedded into $A^{(+)}$ for any associative algebra $A$. Therefore, the following notion makes sense: If a Jordan algebra $J$ is a subalgebra of $A^{(+)}$ for some associative algebra $A$ then it is said to be a special Jordan algebra. The notion of a Jordan dialgebra was introduced in [@Kol:08] as a particular example of a general algebraic definition of what is a variety of dialgebras. This general operadic approach leads to three identities defining the variety of Jordan dialgebras. Independently, the notion of quasi-Jordan algebra emerged in [@VelasquezFelipe:08] as the variety of some non-commutative analogues of Jordan algebras. Namely, if one considers an associative dialgebra $D$ with respect to a new product $x\circ y=\frac{1}{2}(x\dashv y+y\vdash x)$, $x,y\in D$, then the algebra obtained is a quasi-Jordan algebra. In [@VelasquezFelipe:08], two identities were stated to define the variety of quasi-Jordan algebras. Later in [@Br:08], the third (missing) one was noticed, so the notions of quasi-Jordan algebras and Jordan dialgebras went to coincidence. In [@Br:09], the natural notions of a special Jordan dialgebra and of a special identity (s-identity, for short) were introduced. An s-identity of Jordan dialgebras is an identity which holds in all special Jordan dialgebras but does not hold in some Jordan dialgebra. In this note, we show the correspondence between multilinear s-identities of Jordan algebras and Jordan dialgebras (Theorem \[thm:CorrespSId\]). In particular, one of the main results of [@Br:09] follows from this theorem. Also, several natural problems were posed in [@Br:09]: How to generalize the classical statements known for Jordan algebras to the case of dialgebras. This paper is devoted to the solution of all these problems. We prove the analogues of the following theorems: - Cohn’s Theorem [@Cohn:54] on the characterization of elements of free special Jordan algebra as symmetric elements of free associative algebra. - Cohn’s example [@Cohn:54] of an exceptional Jordan algebra which is a homomorphic image of two-generated special Jordan algebra. In particular, the class of special Jordan algebras is not a variety. - Shirshov’s Theorem [@Zhevl:78] on the speciality of two-generated Jordan algebra. - Macdonald’s Theorem [@Zhevl:78] on special identities in three variables. The main method of study is the following. Given a Jordan dialgebra $J$, we build two Jordan algebras $\bar J$ and $\widehat J$ as described in [@Pozh:09]. The classical theorems hold for these Jordan algebras, and their properties allow to make conclusions about the dialgebra $J$. Moreover, the theory of conformal algebras [@Kac:96] is deeply involved into considerations. PRELIMINARIES ============= Dialgebras {#subsec:DefDialg} ---------- A linear space $D$ with two bilinear operations $\vdash,\dashv\colon D\times D\to D$ is called a *dialgebra*. The base field is denoted by $\Bbbk$. A dialgebra is *associative* if it satisfies the identities $$\label{eq:0-DialgebraDef} (x{\mathbin\dashv}y){\mathbin\vdash}z=(x{\mathbin\vdash}y){\mathbin\vdash}z,\quad x{\mathbin\dashv}(y{\mathbin\vdash}z)=x{\mathbin\dashv}(y{\mathbin\dashv}z)$$ and $$\begin{gathered}(x,y,z)_\vdash:=(x{\mathbin\vdash}y){\mathbin\vdash}z-x{\mathbin\vdash}(y{\mathbin\vdash}z)=0,\\ (x,y,z)_\dashv:=(x{\mathbin\dashv}y){\mathbin\dashv}z-x{\mathbin\dashv}(y{\mathbin\dashv}z)=0,\\ (x,y,z)_\times:=(x{\mathbin\vdash}y){\mathbin\dashv}z-x{\mathbin\vdash}(y{\mathbin\dashv}z)=0. \end{gathered}$$ This class of dialgebras is well investigated in [@Loday:01]. Recently, some interesting structural results on associative dialgebras were presented in [@Gonzalez:10]. A dialgebra that satisfies the identities (\[eq:0-DialgebraDef\]), is called a *0-dialgebra.* If $D$ is a 0-dialgebra then the subspace $D_0={\mathop{\mathrm{Span}}\nolimits}\{a{\mathbin\vdash}b-a{\mathbin\dashv}b\mid a,b\in D\}$ is an ideal of $D$ and the quotient dialgebra $\bar D=D/D_0$ can be identified with an ordinary algebra. The space $D$ may be endowed with left and right actions of $\bar D$: $$\bar a\cdot x=a{\mathbin\vdash}x,\quad x\cdot\bar a=x{\mathbin\dashv}a,\quad x,\,a\in D,$$ where $\bar a$ denotes the image of $a$ in $\bar D$. Let $A$ be an algebra that acts on a linear space $M$ via some operations $\circ\colon A\times M\to M$ and $\circ\colon M\times A\to M$. In this case, we can define the algebra $(A\oplus M,\circ)$, where the product $\circ$ is given by the formula $(a+m)\circ(b+n)=ab+(a\circ n+m\circ b)$, that is, $M\circ M=0$. The algebra obtained is called the split null extension of $A$ by means of $M$. We have seen before that we can define actions of the algebra $\bar D$ on the dialgebra $D$, so the split null extension $\bar D\oplus D$ is defined. We will denote it by $\widehat D$. In any dialgebra $D$ a *dimonomial* is an expression of the form $w=(a_1\ldots a_n)$, where $a_1,\ldots,a_n\in D$ and parentheses indicate some placement of parentheses with some choice of operations. By induction we can define the *central letter* $c(w)$ of a dimonomial: if $w\in D$, then $c(w)=w$, otherwise $c(w_1{\mathbin\vdash}w_2)=c(w_2)$ and $c(w_1{\mathbin\dashv}w_2)=c(w_1)$. Let $c(w)=a_k$. If $D$ is 0-dialgebra, then $w=(a_1{\mathbin\vdash}\ldots{\mathbin\vdash}a_{k-1}{\mathbin\vdash}a_k{\mathbin\dashv}a_{k+1}{\mathbin\dashv}\ldots{\mathbin\dashv}a_n)$ for the same parenthesizing as in $(a_1\ldots a_n)$. We will denote this $w$ by $(a_1\ldots a_{k-1}\dot a_k a_{k+1}\ldots a_n)$. In an associative dialgebra parenthesizing does not matter, so it is reasonable to use the notation $w=a_1\ldots a_{k-1}\dot a_k a_{k+1}\ldots a_n$, where the dot indicates the central letter. Let $X$ be a set of generators. Obviously, the basis of the free dialgebra ${\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle$ generated by $X$ consists of dimonomials with a free placement of parentheses and a free choice of operations. It is clear that the basis of the free 0-dialgebra ${\mathrm{Di}}{\mathrm{Alg}}0\, \langle X\rangle$ is the set of dimonomials $(a_1\ldots a_{k-1}\dot a_k a_{k+1}\ldots a_n)$ where $k=1,\ldots,n$ and $a_1,\ldots,a_n\in X$. Finally, the basis of the free associative dialgebra ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ consists of dimonomials $a_1\ldots a_{k-1}\dot a_k a_{k+1}\ldots a_n$ (see [@Loday:01]). If $X=\{x_1,\ldots,x_n\}$ then every dipolynomial $f\in {\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ can be presented as a sum $f=f_1+\ldots+f_n$, where each $f_i$ collects all those dimonomials with central letter $x_i$, $i=1,\dots, n$. Jordan dialgebras ----------------- Let us consider the class of Jordan dialgebras over a field $\Bbbk$ such that ${\mathop{\mathrm{char}}\nolimits}\Bbbk\not=2,3$. In this case, the variety of Jordan algebras ${\mathrm{Jord}}$ over the field $\Bbbk$ is defined by the following multilinear identities $$x_1x_2=x_2x_1,\ J(x_1,x_2,x_3,x_4)=0,$$ where $$\begin{gathered} J(x_1,x_2,x_3,x_4)=x_1(x_2(x_3x_4))+(x_2(x_1x_3))x_4+x_3(x_2(x_1x_4))\\ -(x_1x_2)(x_3x_4)-(x_1x_3)(x_2x_4)-(x_3x_2)(x_1x_4)\end{gathered}$$ is the Jordan identity in a multilinear form [@Zhevl:78]. Hence using the general definition of a variety of dialgebras [@Kol:08] we obtain that the class of Jordan dialgebras is defined by two 0-identities (\[eq:0-DialgebraDef\]) and the following identities $$\label{eq:IdOfJordanDialgebras} \begin{gathered} x_1{\mathbin\vdash}x_2=x_2{\mathbin\dashv}x_1, \\ J(\dot{x}_1,x_2,x_3,x_4)=0,\quad J(x_1,\dot{x}_2,x_3,x_4)=0, \\ J(x_1,x_2,\dot{x}_3,x_4)=0,\quad J(x_1,x_2,x_3,\dot{x}_4)=0. \end{gathered}$$ The variety of Jordan dialgebras is denoted ${\mathrm{Di}}{\mathrm{Jord}}$. We can express both operations in a Jordan dialgebra through one operation: $a{\mathbin\vdash}b=ab$, $a{\mathbin\dashv}b=ba$. Then an ordinary algebra arises that is a noncommutative analogue of a Jordan algebra. The corresponding variety is defined by the system of identities $$[x_1 x_2]x_3= 0, \quad (x_1^2,x_2,x_3)=2(x_1,x_2,x_1x_3), \quad x_1(x_1^2 x_2)=x_1^2(x_1 x_2),$$ that is equivalent to identities (\[eq:IdOfJordanDialgebras\]). Such algebras are investigated in [@Br:08; @Br:09; @GubKol:09]. Conformal algebras ------------------ The notion of a conformal algebra over a field of zero characteristic was introduced by V. G. Kac [@Kac:96] as a tool of the conformal field theory in mathematical physics. Over a field of an arbitrary characteristic, it is reasonable to use the following equivalent definition [@Kol:08]: a *conformal algebra* is a linear space $C$ endowed with a linear mapping $T\colon C\to C$ and a set of bilinear operations ($n$-products) $(\cdot{\mathbin{{}_{(n)}}}\cdot)\colon C\times C\to C$. For all $a,b\in C$ there exist just a finite number of elements $n\in \mathbb{Z}^+$ such that $a{\mathbin{{}_{(n)}}} b\not=0$ (locality property). In addition, these operations satisfy the following properties: $$\begin{gathered} Ta{\mathbin{{}_{(n)}}}b=a{\mathbin{{}_{(n-1)}}}b,\ n\ge 1,\quad Ta{\mathbin{{}_{(0)}}}b=0,\\ T(a{\mathbin{{}_{(n)}}}b)=a{\mathbin{{}_{(n)}}}Tb+Ta{\mathbin{{}_{(n)}}}b,\ n\ge 0,\end{gathered}$$ for all $a,b\in C$. Let ${\mathrm{Var}}$ be a variety of ordinary algebras. It was defined in [@Roitman:99] what is the corresponding variety of conformal algebras when ${\mathop{\mathrm{char}}\nolimits}\Bbbk = 0$. Namely, given a conformal algebra $C$ one may consider ${\mathop{\mathrm{Coeff}}\nolimits}C =\Bbbk[t,t^{-1}]\otimes _{\Bbbk[T]} C$, where $\Bbbk[t,t^{-1}]$ is a right $\Bbbk[T]$-module defined by $f(t)\,T = -f'(t)$, $f(t)\in \Bbbk[t,t^{-1}]$. Denote elements of ${\mathop{\mathrm{Coeff}}\nolimits}C$ by $a(n):=t^n\otimes _{\Bbbk[T]}a$, where $n\in \mathbb{Z}$, $a\in C$. The multiplication on ${\mathop{\mathrm{Coeff}}\nolimits}C$ is given by the formula $$a(n)\,b(m)=\sum_{s\ge 0} (-1)^s (n+m-s) \frac{n!}{(n-s)!} a{\mathbin{{}_{(s)}}}b.$$ In [@Roitman:99] the definition was given: $C$ is conformal algebra corresponding to a variety ${\mathrm{Var}}$ (${\mathrm{Var}}$-conformal algebra) iff ${\mathop{\mathrm{Coeff}}\nolimits}C \in {\mathrm{Var}}$. In [@Kol:06] the notion of ${\mathrm{Var}}$-conformal algebra was rephrased in terms of pseudo-algebras, that works for nonzero characteristic of $\Bbbk$. Since we use the term “conformal algebra” for a pseudo-algebra over $\Bbbk[T]$ in this paper, it is possible to define the class of these objects corresponding to the variety ${\mathrm{Var}}$ of ordinary algebras. The class of all ${\mathrm{Var}}$-conformal algebras is closed under subalgebras and homomorphic images, but it is not closed under (infinite) direct products. Therefore, this is not a real variety of algebraic system. We will denote it by ${\mathrm{Var}}_\mathrm{C}$. It was also observed in [@Kol:08] that if $C \in {\mathrm{Var}}_\mathrm{C}$, then the space $C$ can be endowed with a structure of a dialgebra by means of $$a{\mathbin\vdash}b=a{\mathbin{{}_{(0)}}}b,\quad a{\mathbin\dashv}b=\sum_{s\ge 0} T^s(a{\mathbin{{}_{(s)}}}b).$$ The dialgebra obtained is denoted by $C^{(0)}$, it belongs to the variety ${\mathrm{Di}}{\mathrm{Var}}$. The simplest example of a conformal algebra can be constructed as follows. Let $A$ be an ordinary algebra, then a conformal product is uniquely defined on $\Bbbk[T]\otimes A$ by the following formulas for $a,b\in A$: $$a{\mathbin{{}_{(n)}}}b=\begin{cases}ab, & n=0,\\ 0, & n>0.\end{cases}$$ The conformal algebra obtained is denoted by ${\mathop{\mathrm{Cur}}\nolimits}A$ and is called a current conformal algebra. If an algebra $A \in {\mathrm{Var}}$, then ${\mathop{\mathrm{Cur}}\nolimits}A \in {\mathrm{Var}}_\mathrm{C}$. In the language of category theory, we can say that ${\mathop{\mathrm{Cur}}\nolimits}$ is a functor from the category of algebras to the category of conformal algebras. If $\phi\colon A\to B$ is a homomorphism of algebras, then the mapping ${\mathop{\mathrm{Cur}}\nolimits}\phi\colon{\mathop{\mathrm{Cur}}\nolimits}A\to{\mathop{\mathrm{Cur}}\nolimits}B$ acting by the rule ${\mathop{\mathrm{Cur}}\nolimits}\phi(f(T)\otimes a) = f(T)\otimes \phi(a)$ is a morphism of conformal algebras. In [@GubKol:09] it was proved that an arbitrary dialgebra $D$ is embedded into the dialgebra $({\mathop{\mathrm{Cur}}\nolimits}\widehat D)^{(0)}$. Notation for varieties of algebras and dialgebras ------------------------------------------------- An arbitrary variety of ordinary algebras we denote ${\mathrm{Var}}$, the free algebra in this variety generated by a set $X$ is denoted by ${\mathrm{Var}}\,\langle X\rangle$. The corresponding variety of dialgebras is denoted by ${\mathrm{Di}}{\mathrm{Var}}$, the free dialgebra is denoted by ${\mathrm{Di}}{\mathrm{Var}}\,\langle X\rangle$. The denotation for concrete varieties is analogous, for example, ${\mathrm{Jord}}$ is the variety of Jordan algebras, ${\mathrm{Di}}{\mathrm{Jord}}\,\langle X\rangle$ is the free Jordan dialgebra. SPECIAL JORDAN DIALGEBRAS ========================= In this section ${\mathop{\mathrm{char}}\nolimits}\Bbbk\not=2$. This is necessary to define the Jordan product correctly. Special and exceptional Jordan dialgebras ----------------------------------------- Let $D$ be an associative dialgebra. If we define on the set $D$ new operations $$\label{eq:QuasiJordanProduct} a{\mathbin{{}_{(\vdash)}}}b=\frac{1}{2}(a{\mathbin\vdash}b+b{\mathbin\dashv}a),\ a{\mathbin{{}_{(\dashv)}}}b=\frac{1}{2}(a{\mathbin\dashv}b+b{\mathbin\vdash}a)$$ then we obtain a new dialgebra which is denoted by $D^{(+)}$. It is easy to check that this dialgebra is Jordan [@Br:08]. A dialgebra $J$ is called *special*, if $J\hookrightarrow D^{(+)}$ for some associative dialgebra $D$. Jordan dialgebras that are not special we call *exceptional*. Further, we will denote the operations in a special Jordan dialgebra through ${\mathbin{{}_{(\vdash)}}}$ and ${\mathbin{{}_{(\dashv)}}}$. These operations are expressed through associative operations by the formula (\[eq:QuasiJordanProduct\]). The definition of special Jordan dialgebras has been introduced by the analogy with ordinary algebras, where a Jordan algebra $J$ is called special, if $J\hookrightarrow A^{(+)}$ for some associative algebra $A$ and the product in $A^{(+)}$ is given by the formula $$\label{eq:JordanProduct} a\circ b=\frac{1}{2}(ab+ba).$$ Let now $D$ be an associative dialgebra. The mapping $*\colon D\to D$ is called an *involution* (involution of the second type [@Pozh:09]) of the dialgebra $D$, if $*$ is linear and $$\label{eq:DefOfInvolution} (a^*)^*=a,\quad (a{\mathbin\vdash}b)^*=b^*{\mathbin\dashv}a^*,\quad (a{\mathbin\dashv}b)^*=b^*{\mathbin\vdash}a^*$$ for all $a$, $b\in D$. The set $H(D,*)=\{x\in D\mid x=x^*\}$ of symmetric elements with respect to $*$ is closed relative to operations (\[eq:QuasiJordanProduct\]). This set is a subalgebra of the algebra $D^{(+)}$. So, $H(D,*)$ is a special Jordan dialgebra. We now construct an example of an exceptional Jordan dialgebra. \[prop:ExampleExceptionalDialgebra\] Let $(J,\circ)$ be an exceptional Jordan algebra and suppose the condition $x\circ J=0$, $x\in J$, implies $x=0$. Then $J$ as a dialgebra with equal operations $x{\mathbin{{}_{(\vdash)}}}y:=x\circ y$ and $x{\mathbin{{}_{(\dashv)}}}y:=x\circ y$ is an exceptional Jordan dialgebra. Assume the opposite. Let $J\hookrightarrow D^{(+)}$ where $(D,\vdash,\dashv)$ is an associative dialgebra and the product in $D^{(+)}$ is given by the formula (\[eq:QuasiJordanProduct\]). Consider $I={\mathop{\mathrm{Span}}\nolimits}\{a{\mathbin\vdash}b-a{\mathbin\dashv}b\mid a,\,b\in D\}$ that is an ideal of $D$. Then $\bar D=D/I$ is an ordinary associative algebra and $\phi\colon D^{(+)}\to\bar D^{(+)}$ is the natural homomorphism of a Jordan dialgebra on its quotient algebra. The composition of the embedding $\hookrightarrow$ and $\phi$ is a homomorphism too, we denote this homomorphism through $\psi$. It is clear that $K:=\ker\psi$ is an ideal of $J$. Since $\psi$ is a restriction $\phi$ on $J$ so $K=\ker\psi\subseteq\ker\phi=I$. We have $I{\mathbin\vdash}J=J{\mathbin\dashv}I=0$, this is a consequence of the 0-identity. Hence $I\circ J=I{\mathbin{{}_{(\vdash)}}}J=\frac{1}{2}(I{\mathbin\vdash}J+J{\mathbin\dashv}I)=0$, from conditions of the proposition we obtain $I=0$ therefore and $K=0$. So $\psi$ is an embedding and $J\hookrightarrow\bar D^{(+)}$, i. e., $J$ is exceptional. Let $\mathbf{C}$ be the Cayley-Dickson algebra over the field $\Bbbk$, ${\mathop{\mathrm{char}}\nolimits}\Bbbk\not=2$. Consider an algebra $H(\mathbf{C}_3)$ of those $3\times 3$ matrices over $\mathbf{C}$ that are symmetric relative the involution in $\mathbf{C}$. This is so called Albert algebra. It is well-known that $J=H(\mathbf{C}_3)$ is a simple exceptional Jordan algebra, so $J$ satisfies the conditions of Proposition \[prop:ExampleExceptionalDialgebra\]. Therefore, The Albert algebra is exceptional as a Jordan dialgebra. Symmetric and Jordan polynomials -------------------------------- Let ${\mathrm{Alg}}\,\langle X\rangle$ be a free non-associative algebra generated by $X$, ${\mathrm{As}}\,\langle X\rangle$ be a free associative algebra, ${\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle$ be a free non-associative dialgebra, ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ be a free associative dialgebra [@Loday:01]. Products in ${\mathrm{Alg}}\,\langle X\rangle$ and ${\mathrm{As}}\,\langle X\rangle$, also in ${\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle$ and ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ are denoted identically. There is no confusion because by the origin of elements it is clear which the product we mean. Fix $z\in X$ and introduce the following mappings. A mapping $\mathcal{J}\colon{\mathrm{Alg}}\,\langle X\rangle\to {\mathrm{As}}\,\langle X\rangle$ is defined by linearity, on non-associative words it is defined by induction on a length: if $x\in X$ then $\mathcal{J}(x)=x$; if $uv\in {\mathrm{Alg}}\,\langle X\rangle$ then $\mathcal{J}(uv)=\frac{1}{2}(\mathcal{J}(u)\mathcal{J}(v)+\mathcal{J}(v)\mathcal{J}(u))$. So, the value of $\mathcal{J}$ on a non-associative polynomial $f$ is equal to an associative polynomial obtained from $f$ by means of rewriting all products in $f$ as Jordan ones by the formula (\[eq:JordanProduct\]). By analogy, in the case of dialgebras a mapping $\mathcal{J}_{{\mathrm{Di}}}\colon{\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle\to {\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ is defined. It is linear, it acts identically on $x\in X$ and $$\begin{gathered} \mathcal{J}_{{\mathrm{Di}}}(u{\mathbin\vdash}v)=\frac{1}{2}(\mathcal{J}_{{\mathrm{Di}}}(u){\mathbin\vdash}\mathcal{J}_{{\mathrm{Di}}}(v)+\mathcal{J}_{{\mathrm{Di}}}(v){\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(u)),\\ \mathcal{J}_{{\mathrm{Di}}}(u{\mathbin\dashv}v)=\frac{1}{2}(\mathcal{J}_{{\mathrm{Di}}}(u){\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v)+\mathcal{J}_{{\mathrm{Di}}}(v){\mathbin\vdash}\mathcal{J}_{{\mathrm{Di}}}(u)). \end{gathered}$$ Introduce the following notation $${\mathrm{Alg}}_z\,\langle X\rangle=\{\Phi\in {\mathrm{Alg}}\,\langle X\rangle \mid \Phi=\sum f_i,\ f_i\text{~--- monomials, }\deg_z f_i = 1\},$$ $${\mathrm{Di}}{\mathrm{Alg}}_z\,\langle X\rangle=\{\Phi\in {\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle \mid \Phi=\sum f_i,\ f_i\text{~--- dimonomials, }\deg_z f_i = 1,\ c(f_i)=z\},$$ where $c(f_i)$ stands for the central letter of a dimonomial $f_i$. A mapping $\Psi^z_{{\mathrm{Alg}}}\colon{\mathrm{Alg}}_z\,\langle X\rangle\to {\mathrm{Di}}{\mathrm{Alg}}_z\,\langle X\rangle$ places signs of dialgebraic operations in a non-associative polynomial in such a way that every sign points to $z$. By induction it can be defined as follows: $\Psi^z_{{\mathrm{Alg}}}(z)=z$; if $z$ is contained by $u$ then $\Psi^z_{{\mathrm{Alg}}}(uv)=\Psi^z_{{\mathrm{Alg}}}(u){\mathbin\dashv}v^\dashv$; if $z$ is contained by $v$ then $\Psi^z_{{\mathrm{Alg}}}(uv)=u^\vdash{\mathbin\vdash}\Psi^z_{{\mathrm{Alg}}}(v)$. There we introduce two mappings ${}^\vdash,{}^\dashv\colon{\mathrm{Alg}}\,\langle X\rangle\to{\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle$. The first mapping maps a word $u$ to $u^\vdash$ where the word $u^\vdash$ has the same multipliers as $u$ and all signs of operations point to the right. In $v^\dashv$ all signs of operations point to the left respectively. Further in Lemmas \[lemma2\] and \[lemma:CommutOperJPhi\] we use mappings ${}^\vdash,{}^\dashv\colon{\mathrm{As}}\,\langle X\rangle\to{\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ which are defined and denoted in a similar way. Analogously, we may define the sets ${\mathrm{As}}_z\,\langle X\rangle$, ${\mathrm{Di}}{\mathrm{As}}_z\,\langle X\rangle$ and a mapping $\Psi^z_{{\mathrm{As}}}\colon{\mathrm{As}}_z\,\langle X\rangle\to {\mathrm{Di}}{\mathrm{As}}_z\,\langle X\rangle$. Define the following sets: $${\mathrm{SJ}}\,\langle X\rangle=\mathcal{J}({\mathrm{Alg}}\,\langle X\rangle),$$ $${\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle=\mathcal{J}_{{\mathrm{Di}}}({\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle).$$ It is well-known that ${\mathrm{SJ}}\,\langle X\rangle$ is the free special Jordan dialgebra. In fact, ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle$ is the free special Jordan dialgebra (see Lemma \[lemma:FreeSpecJordDialgebra\] below). From the definition of the mapping $\mathcal{J}$ it is clear that ${\mathrm{SJ}}\,\langle X\rangle$ is a subalgebra in ${{\mathrm{As}}\,\langle X\rangle}^{(+)}$ generated by the set $X$. Similarly, ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle\hookrightarrow {{\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle}^{(+)}$. An element from ${\mathrm{As}}\,\langle X\rangle$ is called a *Jordan polynomial* if it belongs to ${\mathrm{SJ}}\,\langle X\rangle$. By analogy, an element from ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ is called a *Jordan dipolynomial* if it belongs to ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle$. \[lemma2\] For arbitrary $u\in{\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$, $v\in{\mathrm{Alg}}\,\langle X\rangle$ we have $$u{\mathbin\dashv}\mathcal{J}(v)^\dashv=u{\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v^\dashv)=u{\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v^\vdash),$$ $$\mathcal{J}(v)^\vdash{\mathbin\vdash}u=\mathcal{J}_{{\mathrm{Di}}}(v^\vdash){\mathbin\vdash}u=\mathcal{J}_{{\mathrm{Di}}}(v^\dashv){\mathbin\vdash}u.$$ Use an induction on the length of the word $v$. A base is evident. Let $v=v_1 v_2$. Then $$\begin{gathered} u{\mathbin\dashv}\mathcal{J}(v)^{\dashv} =u{\mathbin\dashv}\mathcal{J}(v_1 v_2)^{\dashv}=\frac{1}{2}u{\mathbin\dashv}(\mathcal{J}(v_1)^{\dashv}{\mathbin\dashv}\mathcal{J}(v_2)^{\dashv} +\mathcal{J}(v_2)^{\dashv}{\mathbin\dashv}\mathcal{J}(v_1)^{\dashv}) \\ =\frac{1}{2}u{\mathbin\dashv}(\mathcal{J}_{{\mathrm{Di}}}(v_1^{\dashv}) {\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v_2^{\dashv}) +\mathcal{J}_{{\mathrm{Di}}}(v_2^{\dashv}){\mathbin\vdash}\mathcal{J}_{{\mathrm{Di}}}(v_1^{\dashv}))=u{\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v_1^{\dashv}{\mathbin\dashv}v_2^{\dashv})=u{\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v^{\dashv}).\end{gathered}$$ All remaining equalities are proved in the same way. \[lemma:CommutOperJPhi\] For all $\Phi\in {\mathrm{Alg}}_z\,\langle X\rangle$ we have $$\Psi^z_{{\mathrm{As}}}(\mathcal{J}(\Phi))=\mathcal{J}_{{\mathrm{Di}}}(\Psi^z_{{\mathrm{Alg}}}(\Phi)).$$ Since all mappings are linear, it is enough to prove the statement for the case when $\Phi$ is a word. If $\Phi=z$ then the claim is evident. If $\Phi=uv$ then $z$ can belong to either $u$ or $v$. Let $z$ belongs to $u$. Then using Lemma \[lemma2\] we obtain $$\begin{gathered} \Psi^z_{{\mathrm{As}}}(\mathcal{J}(\Phi))=\Psi^z_{{\mathrm{As}}}(\frac{1}{2}[\mathcal{J}(u)\mathcal{J}(v) +\mathcal{J}(v)\mathcal{J}(u)]) \\ =\frac{1}{2}[\Psi^z_{{\mathrm{As}}}(\mathcal{J}(u)){\mathbin\dashv}\mathcal{J}(v)^{\dashv} +\mathcal{J}(v)^{\vdash}{\mathbin\vdash}\Psi^z_{{\mathrm{As}}}(\mathcal{J}(u))] \\ =\frac{1}{2}[\mathcal{J}_{{\mathrm{Di}}}(\Psi^z_{{\mathrm{Alg}}}(u)){\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v^{\dashv}) +\mathcal{J}_{{\mathrm{Di}}}(v^{\dashv}){\mathbin\vdash}\mathcal{J}_{{\mathrm{Di}}}(\Psi^z_{{\mathrm{Alg}}}(u))] \\ =\mathcal{J}_{{\mathrm{Di}}}(\Psi^z_{{\mathrm{Alg}}}(u){\mathbin\dashv}v^{\dashv}) =\mathcal{J}_{{\mathrm{Di}}}(\Psi^z_{{\mathrm{Alg}}}(\Phi)).\end{gathered}$$ The case when $z$ belongs to $v$ is proved similarly. We recall about the quotient $\bar D=D/D_0$ that has been defined in Section \[subsec:DefDialg\]. This quotient compares every 0-dialgebra with an ordinary algebra. The quotient $\bar D$ of a dialgebra $D$ generated by a set $X=\{x_i \mid i\in I\}$ is an algebra generated by the set $\bar X =\{\bar x_i \mid i\in I\}$. In order to simplify notation, we will further denote $\bar x\in \bar X$ by $x$. Following this convention we obtain, for example, $\overline{{\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle}={\mathrm{As}}\,\langle X\rangle$. \[prop:SJDiSJ\] Let $f\in {\mathrm{Di}}{\mathrm{As}}_z\,\langle X\rangle$. Then $$f\in {\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle \Leftrightarrow \bar f\in {\mathrm{SJ}}\,\langle X\rangle.$$ “$\Rightarrow$”. Let $f\in {\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle$ that is $f=\mathcal{J}_{{\mathrm{Di}}}(\Phi)$ for some $\Phi\in{\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle$. Then $\bar f=\overline{\mathcal{J}_{{\mathrm{Di}}}(\Phi)}=\mathcal{J}(\bar\Phi)$, so $\bar f\in {\mathrm{SJ}}\,\langle X\rangle$. There we have used the equality $\overline{\mathcal{J}_{{\mathrm{Di}}}(\Phi)}=\mathcal{J}(\bar\Phi)$ which is easy to prove by induction on the length of $\Phi$. “$\Leftarrow$”. Let $\bar f\in {\mathrm{SJ}}\,\langle X\rangle$ that is $\bar f=\mathcal{J}(\Phi)$ for some $\Phi\in{\mathrm{Alg}}\,\langle X\rangle$. Since the degrees on variables do not change when we apply $\mathcal{J}$, we obtain $\Phi\in{\mathrm{Alg}}_z\,\langle X\rangle$. Thereby, $\Phi\in{\mathrm{Alg}}_z\,\langle X\rangle$. By Lemma \[lemma:CommutOperJPhi\] we obtain $\mathcal{J}_{{\mathrm{Di}}}(\Psi^z_{{\mathrm{Alg}}}(\Phi))=\Psi^z_{{\mathrm{As}}}(\mathcal{J}(\Phi)) =\Psi^z_{{\mathrm{As}}}(\bar f)=f$, the last equality in the sequence is true because $f\in{\mathrm{Di}}{\mathrm{As}}_z\,\langle X\rangle$. So, $f\in {\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle$. Consider the dialgebra $$\Lambda_X = {\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle /I,$$ where $I$ is the ideal of ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ generated by the set $\{ f_{x,y} = x{\mathbin\dashv}y + y{\mathbin\vdash}x \mid x,y \in X\}$. This dialgebra is the analogue of the exterior algebra (Grassmann algebra). Further we will identify the set $X$ and its image $\bar X\subseteq\Lambda_X$. Following this agreement we suppose that $\Lambda_X$ is generated by the set $X$. \[thm:BasisGrassmanDialgebra\] Let $X$ be a linearly ordered set. Then the basis of the algebra $\Lambda_X$ consists of words $\dot x_1x_2\dots x_k$, $k\ge 1$, $x_i\in X$, $x_2<x_3<\dots < x_k$. Use the theory of Gröbner-Shirshov bases for associative dialgebras developed in [@BokutChenLiu:08]. Let $S_0 = \{f_{x,y}\mid x,y\in X \}$ be the initial set of defining relations. Compositions of left product $z{\mathbin\dashv}f_{x,y}$ belong to the ideal $I$ as well as compositions of right product $f_{x,y}{\mathbin\vdash}z$, $x,y,z\in X$. The set of defining relations obtained $$x{\mathbin\dashv}y + y{\mathbin\vdash}x; \quad x{\mathbin\dashv}y{\mathbin\dashv}z + x{\mathbin\dashv}z{\mathbin\dashv}y,\ y>z; \quad x{\mathbin\vdash}y{\mathbin\vdash}z + y{\mathbin\vdash}x{\mathbin\vdash}z,\ x>y; \quad x{\mathbin\vdash}x{\mathbin\vdash}y; \quad x{\mathbin\dashv}y{\mathbin\dashv}y$$ is a Gröbner-Shirshov basis. Reduced words are of the form $$\dot x_1x_2\dots x_k,\ k\ge 1,\ x_2<x_3<\dots < x_k,$$ and the set of all reduced words by [@BokutChenLiu:08] is a linear basis of the algebra $\Lambda_X$. Define an involution $*$ on ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ in the following way: $$(x_{i_1}\ldots \dot{x}_{i_k}\ldots x_{i_n})^*=x_{i_n}\ldots \dot{x}_{i_k}\ldots x_{i_1},$$ and extend to dipolynomials by linearity. This mapping reverses words and signs of dialgebraic operations. It is easy to check that the mapping $*$ satisfies properties of an involution (\[eq:DefOfInvolution\]). Through ${\mathrm{Di}}{\mathrm{H}}\,\langle X\rangle$ we denote the Jordan dialgebra $H({\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle,*)$ of symmetric elements from ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ with respect to $*$ with the product (\[eq:QuasiJordanProduct\]). Further $\{u\}$ denotes $ u+u^*$ where $u$ is a basic word from ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$. Note that $\{u\}=\{u^*\}$. An analogous involution on ${\mathrm{As}}\,\langle X\rangle$ we denote by $*$ too. It acts like as $$(x_{i_1}\ldots x_{i_k}\ldots x_{i_n})^*=x_{i_n}\ldots x_{i_k}\ldots x_{i_1},$$ on monomials and extends to polynomials by linearity. The next theorem is an analogue of the classical Cohn’s Theorem [@Cohn:54 Theorems 4.1 and 4.2] that describes Jordan polynomials from $\le 3$ variables as symmetric elements of the free associative algebra. \[thm:CohnForDialgebra\] For any set $X$ we have ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle\subseteq{\mathrm{Di}}{\mathrm{H}}\,\langle X\rangle$. If $|X|\le 2$ then there is an equality, if $|X|>2$ then there is a strict inclusion. Also, for any $X$ we have that ${\mathrm{Di}}{\mathrm{H}}\,\langle X\rangle$ is generated by $X$ and dotted tetrads $\{\dot xyzt\}$, $\{\dot xxyz\}$, where $x,y,z,t\in X$ are distinct. “$\subseteq$” follows from the equality $\mathcal{J}_{{\mathrm{Di}}}(\Phi)^*=\mathcal{J}_{{\mathrm{Di}}}(\Phi)$ which holds for all $\Phi\in{\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle$. As before, this equality can be proved by induction on the length of $\Phi$ considering cases $\Phi=u{\mathbin\vdash}v$ and $\Phi=u{\mathbin\dashv}v$. Let $|X|=2$. In order to prove the equality, consider an arbitrary $f\in{\mathrm{Di}}{\mathrm{H}}\,\langle x,y\rangle$, i. e., $f\in{\mathrm{Di}}{\mathrm{As}}\,\langle x,y\rangle$ and $f=f^*$. We need to show that $f\in{\mathrm{Di}}{\mathrm{SJ}}\,\langle x,y\rangle$. The dipolynomial $f$ is equal to a sum of dimonomials $f=\sum f_i$. Further, $f=\frac{1}{2}(f+f^*)=\frac{1}{2}\sum{(f_i+f_i^*)}$. Without loss of generality we may assume $f=a+a^*$ where $a$ is a dimonomial. Suppose $x$ is the central letter of $a$. So $f$ can be written in a form $f=u\dot xv+v^*\dot xu^*$ where $u,\,v\in{\mathrm{Di}}{\mathrm{As}}\,\langle x,y\rangle$ or equal to empty words. Consider $g(x,y,z)=u\dot zv+v^*\dot zu^*\in{\mathrm{Di}}{\mathrm{As}}\,\langle x,y,z\rangle$. Since $\bar{g}=\bar{g}^*$ then $\bar{g}\in{\mathrm{SJ}}\,\langle x,y,z\rangle$ by the classical Cohn’s Theorem. In addition, $g\in{\mathrm{Di}}{\mathrm{As}}_z\,\langle x,y,z\rangle$ hence Proposition \[prop:SJDiSJ\] implies $g\in{\mathrm{Di}}{\mathrm{SJ}}\,\langle x,y,z\rangle$. It means that there exists a dipolynomial $\Phi(x,y,z)$ such that $g=\mathcal{J}_{{\mathrm{Di}}}(\Phi(x,y,z))$. Substituting $z:=x$ into the last equality we obtain $f=\mathcal{J}_{{\mathrm{Di}}}(\Phi(x,y,x))$. Therefore, $f\in{\mathrm{Di}}{\mathrm{SJ}}\,\langle x,y\rangle$. We have proved the equality for $|X|=2$ and thus for $|X|=1$. Let $|X|>2$. In order to prove the strict inclusion consider the dotted tetrad $\{\dot xxyz\}=\dot xxyz+zyx\dot x\in{\mathrm{Di}}{\mathrm{H}}\,\langle X\rangle$ where $x,y,z\in X$. There exists a homomorphism $\sigma\colon{\mathrm{Di}}{\mathrm{As}}\,\langle x,y,z\rangle \to {\mathrm{Di}}\Lambda\,\langle x,y,z\rangle$ such that $\sigma(x)=x$, $\sigma(y)=y$, $\sigma(z)=z$. All Jordan dipolynomials of degree greater that 1 map to zero by this homomorphism. Using the basis of ${\mathrm{Di}}\Lambda\,\langle x,y,z\rangle$ from Theorem \[thm:BasisGrassmanDialgebra\] we obtain $$\sigma(\{\dot xxyz\})=2\dot xxyz\neq 0.$$ (When we use Theorem \[thm:BasisGrassmanDialgebra\] we suppose that $x<y<z$.) So, the dipolynomial $\{\dot xxyz\}$ does not belong to ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle$. It is well-known that in Jordan algebras we can permute variables in a tetrad modulo ${\mathrm{SJ}}\,\langle X\rangle$. It follows from the fact that $xy = -yx + x\circ y$, hence $\{xyzt\} = -\{yxzt\} + \{(x\circ y) zt\}$ and $\{xyzt\} \in -\{yxzt\} + {\mathrm{SJ}}\,\langle X\rangle$. Placing a dot in the last equality we get that $\{\dot xyzt\} \in -\{y\dot xzt\} + {\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle$. Therefore, we can permute the variables in a dotted tetrad together with a dot modulo ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle$. To find the generators of ${\mathrm{Di}}{\mathrm{H}}\,\langle X\rangle$, we consider $g(x_1,\ldots,x_n) \in {\mathrm{Di}}{\mathrm{H}}\,\langle X\rangle$. We can write $g=g_1+\ldots+g_n$, where each $f_i$ collects all dimonomials with central letter $x_i$. It suffices to find generators for $g_1$. There exists $f(s,x_1,\ldots,x_n) \in {\mathrm{H}}\,\langle X,s \rangle$ such that $g_1 = \Psi^s_{\mathrm{As}}(f) |_{s:=x_1}$. Theorem 4.1 [@Cohn:54] holds for $f$, hence $f$ is Jordan polynomial $\Phi$ from $X$, $s$ and tetrads, i.e., $f=\mathcal{J}(\Phi)$. Further, $g_1 = \Psi^s_{\mathrm{As}}(\mathcal{J}(\Phi)) |_{s:=x_1} \stackrel{\mathrm{L.4}}{=} \mathcal{J}_{\mathrm{Di}}(\Psi^s_{\mathrm{Alg}}(\Phi) |_{s:=x_1} )$. Therefore, $g_1$ is generated by $X$, by dotted tetrads with all the different variables and by dotted tetrads with two equal variables and with the dot over one of the equal variables. By permutation of variables these dotted tetrads can be reduced to $\{\dot xyzt\}$ and $\{\dot xxyz\}$. Homomorphic images of special Jordan dialgebras ----------------------------------------------- In this section we construct the example of an exceptional two-generated Jordan dialgebra which is a homomorphic image of a special Jordan dialgebra. Denote by $\widehat I$ the ideal of ${\mathrm{Di}}{\mathrm{As}}\,\langle x,y \rangle$ generated by the set $I$. \[lemma:CriterionOfQuotientSpeciality\] Let $I$ be an ideal of ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X \rangle$. Then ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X \rangle/I$ is special iff $\widehat I \cap {\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle \subseteq I$. The proof of this lemma is completely analogous to the proof of Theorem 2.2 [@Cohn:54]. \[prop:CriterionOfQuotientSpecialityTwoGenerated\] Let $I$ be an ideal of ${\mathrm{Di}}{\mathrm{SJ}}\,\langle x,y \rangle$ is generated by elements $u_i$. Then ${\mathrm{Di}}{\mathrm{SJ}}\,\langle x,y \rangle/I$ is special iff $\{u_i \dot xxy\} \in I$ and $\{u_i \dot yyx\} \in I$ for all $i$. By Theorem \[thm:CohnForDialgebra\], ${\mathrm{Di}}{\mathrm{SJ}}\,\langle x,y \rangle = {\mathrm{Di}}{\mathrm{H}}\,\langle x,y \rangle$. Lemma \[lemma:CriterionOfQuotientSpeciality\] implies that ${\mathrm{Di}}{\mathrm{SJ}}\,\langle x,y \rangle/I$ is special iff $\widehat I \cap {\mathrm{Di}}{\mathrm{H}}\,\langle x,y \rangle \subseteq I$. “$\Rightarrow$”. It is clear that $\{u_i \dot xxy\} \in \widehat I \cap {\mathrm{Di}}{\mathrm{H}}\,\langle x,y \rangle$, hence the condition of proposition is necessary. “$\Leftarrow$”. Suppose that $\{u_i \dot xxy\} \in I$ and $\{u_i \dot yyx\} \in I$ for all $i$ and let $w \in \widehat I \cap {\mathrm{Di}}{\mathrm{H}}\,\langle x,y \rangle$. It is clear (as in Lemma 3.2 [@Cohn:54]) that $w$ can be written as a symmetric polynomial $f=f^*$ in the $u$’s and $x$, $y$ which is linear homogenious in the $u$’s. We now regard $x$, $y$, $u_i$ as independent. Because $f\in{\mathrm{Di}}{\mathrm{H}}\,\langle x,y,u_i \rangle$, it can by Theorem \[thm:CohnForDialgebra\] be expressed as Jordan dipolynomial $\Phi$ in $x$, $y$, $u_i$ and dotted tetrads involving this variables. Since $f$ is linear in the $u$’s so is $\Phi$ and therefore no dotted tetrad can involve more than one $u$, but it must involve at least one. By permutation of variables any such tetrad can be reduced to the form $\{u_i \dot xxy\}$ or $\{u_i \dot yyx\}$ plus Jordan dipolynomial. By hypothesis any such tetrad belong to $I$, hence every term of $\Phi$ contains at least one factor from $I$, so $\Phi\in I$. This shows that $w=f=\Phi\in I$ and this completes the proof. \[thm:Example2GeneratedExceptionalDialgebra\] Consider the special Jordan dialgebra ${\mathrm{Di}}{\mathrm{SJ}}\,\langle x,y\rangle$, and let $I$ be its ideal generated by the element $k = \frac{1}{2} (\dot x x + x\dot x) - \frac{1}{2}(\dot yy+y\dot y)$. Then the quotient dialgebra $J ={\mathrm{Di}}{\mathrm{SJ}}\, \langle x,y\rangle / I$ is exceptional. Consider $f = \{kx\dot xy\}$. By Proposition \[prop:CriterionOfQuotientSpecialityTwoGenerated\] it suffices to show that $f\notin I$. Assume $f\in I$. Then there exists a dipolynomial $$\phi (x,y,z) \in {\mathrm{Di}}{\mathrm{SJ}}\, \langle x,y,z\rangle \subset {\mathrm{Di}}{\mathrm{H}}\, \langle x,y,z\rangle$$ such that $\phi (x,y,k) = f$. In addition, every summand from $\phi$ contains at most one entry of $z$. Write $$\phi(x,y,z) = \phi_1 (x,y,z) + \phi_2(x,y,z) + \dots, \quad \deg_z \phi_n = n.$$ The total degree of $f$ (with respect to all variables) is equal to 5, hence $\phi_n = 0$ when $n\ge 3$. Therefore $\phi(x,y,z) = \phi_1 (x,y,z) + \phi_2(x,y,z)$. Suppose $\phi_1 := \phi_{1,0} + \phi_{1,1}+\phi_{1,2}+\phi_{1,3},$ where $\deg_x \phi_{1,0}=0$, $\deg_x \phi_{1,1}=1$, $\deg_x\phi_{1,2}=2$, $\deg_x \phi_{1,3}=3$; $\phi_2 := \phi_{2,0} + \phi_{2,1}$, where $\deg_x \phi_{2,0}=0$, $\deg_x \phi_{2,1}=1$. After the substitution $z=k$ all summands in $\phi_{1,1}$, $\phi_{1,3}$ and $\phi_{2,1}$ have degree 1, 3 or 5 in $x$. All summands from $\phi_{1,0}$, $\phi_{1,2}$ and $\phi_{2,0}$ have degree 0, 2 or 4 in $x$. Since $f$ contains summands of only 2-nd and 4-th degree in $x$, we have $\phi_{1,1}+\phi_{1,3}+\phi_{2,1}=0$. Therefore, $\phi=\phi_{1,0}+\phi_{1,2}+\phi_{2,0}$. Since $x$ is the central letter of the dipolynomial $f$, central letters of dimonomials from $\phi$ can be variables $x$ and $z$. Every dipolynomial from ${\mathrm{Di}}{\mathrm{H}}\,\langle x,y,z \rangle$ with this property is equal to a linear combination of the next dipolynomials: $$\begin{gathered} \{ \dot xyxz \}, \ \{ xy\dot xz \}, \ \{ xyx\dot z \},\quad \{ y\dot xxz \}, \ \{ yx \dot xz \}, \ \{ yxx\dot z \},\\ \{ \dot xxyz \}, \ \{ x\dot xyz \}, \ \{ xxy\dot z \},\quad \{ \dot xyzx \}, \ \{ xyz \dot x \}, \ \{ xy\dot zx \},\\ \{ yz\dot xx \}, \ \{ yzx\dot x \}, \ \{ y\dot zxx \} ,\quad \{ y\dot xzx \}, \ \{ yxz\dot x \}, \ \{ yx\dot zx \},\\ \{\dot zyyy\},\ \{y\dot zyy\},\quad \{\dot zzy\},\ \{z\dot zy\}, \{\dot zyz\}. \end{gathered}$$ Consequently $\phi (x,y,z)$ has the form $$\begin{gathered} \alpha_1 \{ \dot xyxz \} +\alpha_2 \{ y\dot xxz \} +\alpha_3 \{ \dot xxyz \} +\alpha_4 \{ \dot xyzx \} +\alpha_5 \{ yz\dot xx \} +\alpha_6 \{ y\dot xzx \} \\ +\beta_1 \{ xy\dot xz \} +\beta_2 \{ yx \dot xz \} +\beta_3 \{ x\dot xyz \} +\beta_4 \{ xyz \dot x \} +\beta_5 \{ yzx\dot x \} +\beta_6 \{ yxz\dot x \} \\ +2\gamma_1 \{ xyx\dot z \} +2\gamma_2 \{ yxx\dot z \} +2\gamma_3 \{ xxy\dot z \} +2\gamma_4 \{ xy\dot zx \} +2\gamma_5 \{ y\dot zxx \} +2\gamma_6 \{ yx\dot zx \} \\ +2\delta_1\{\dot zyyy\}+2\delta_2\{y\dot zyy\}+2\delta_3\{\dot zzy\}+2\delta_4\{z\dot zy\}+2\delta_5\{\dot zyz\}. \end{gathered}$$ Substituting $z=k$ and using the equalities $$\begin{gathered} 2\dot zz=(\dot xx+x\dot x-\dot yy-y\dot y)\mathbin\dashv(xx-yy) \\ =\dot xx^3+x\dot xx^2-\dot yyx^2-y\dot yx^2-\dot xxy^2-x\dot xy^2+\dot y y^3+y\dot yy^2, \\ 2z\dot z=(xx-yy)\mathbin\vdash(\dot xx+x\dot x-\dot yy-y\dot y) \\ =x^2\dot xx+x^3\dot x-x^2\dot yy-x^2y\dot y-y^2\dot xx-y^2x\dot x +y^2\dot yy+y^3\dot y,\end{gathered}$$ we obtain $\phi(x,y,k)$ is equal to $$\begin{gathered} \alpha_1 \{ \dot xyx^3 \} +\alpha_2 \{ y\dot x x^3 \} +\alpha_3 \{ \dot xxyx^2 \} +\alpha_4 \{ \dot xyx^3 \} +\alpha_5 \{ yx^2\dot xx \} +\alpha_6 \{ y\dot xx^3 \} \\ - \alpha_1 \{ \dot xyxy^2 \} - \alpha_2 \{ y\dot xxy^2 \} - \alpha_3 \{ \dot x xy^3 \} - \alpha_4 \{ \dot xy^3x \} - \alpha_5 \{ y^3\dot xx \} - \alpha_6 \{ y\dot xy^2x \} \\ +\beta_1 \{ xy\dot x x^2 \} +\beta_2 \{ yx \dot x x^2 \} +\beta_3 \{ x\dot xyx^2 \} +\beta_4 \{ xyx^2 \dot x \} +\beta_5 \{ yx^3\dot x \} +\beta_6 \{ yx^3\dot x \} \\ - \beta_1 \{ xy\dot xy^2 \} - \beta_2 \{ yx \dot xy^2 \} - \beta_3 \{ x\dot xy^3 \} - \beta_4 \{ xy^3 \dot x \} - \beta_5 \{ y^3x\dot x \} - \beta_6 \{ yxy^2\dot x \} \\ +\gamma_1 \{ xyx\dot x x \} +\gamma_2 \{ yx^2\dot xx \} +\gamma_3 \{ x^2y\dot xx \} +\gamma_4 \{ xy\dot xx^2 \} +\gamma_5 \{ y\dot xx^3 \} +\gamma_6 \{ yx\dot xx^2 \} \\ +\gamma_1 \{ xyx^2\dot x \} +\gamma_2 \{ yx^3\dot x \} +\gamma_3 \{ x^2yx\dot x\} +\gamma_4 \{ xyx\dot xx \} +\gamma_5 \{ yx\dot xx^2 \} +\gamma_6 \{ yx^2\dot xx \} \\ -\gamma_1 \{ xyx\dot yy \} -\gamma_2 \{ yx^2\dot yy \} -\gamma_3 \{ x^2y\dot yy \} -\gamma_4 \{ xy\dot yyx \} -\gamma_5 \{ y\dot yyx^2 \} -\gamma_6 \{ yx\dot yyx \} \\ -\gamma_1 \{ xyxy\dot y \} -\gamma_2 \{ yx^2y\dot y \} -\gamma_3 \{ x^2y^2\dot y \} -\gamma_4 \{ xy^2\dot yx \} -\gamma_5 \{ y^2\dot yx^2\} -\gamma_6 \{ yxy\dot yx \} \\ \end{gathered}$$ $$\begin{gathered} +\delta_1\{\dot xxy^3\}+\delta_1\{x\dot xy^3\}-\delta_1\{\dot yy^4\}-\delta_1\{y\dot yy^3\} \\ +\delta_2\{y\dot xxy^2\}+\delta_2\{yx\dot xy^2\}-\delta_2\{y\dot y y^3\}-\delta_2\{y^2\dot y y^2\} \\ +\delta_3\{\dot xx^3y\}+\delta_3\{x\dot xx^2y\}-\delta_3\{\dot yyx^2y\} -\delta_3\{y\dot yx^2y\} \\ -\delta_3\{\dot xxy^3\}-\delta_3\{x\dot xy^3\}+\delta_3\{\dot yy^4\}+\delta_3\{y\dot yy^3\} \\ +\delta_4\{x^2\dot xxy\}+\delta_4\{x^3\dot xy\}-\delta_4\{x^2\dot yy^2\}-\delta_4\{x^2y\dot yy\} \\ -\delta_4\{y^2\dot xxy\}-\delta_4\{y^2x\dot xy\}+\delta_4\{y^2\dot yy^2\}+\delta_4\{y^3\dot yy\} \\ +\delta_5\{\dot xxyx^2\}+\delta_5\{x\dot xyx^2\}-\delta_5\{\dot yy^2x^2\}-\delta_5\{y\dot yyx^2\} \\ -\delta_5\{\dot xxy^3\}-\delta_5\{x\dot xy^3\}+\delta_5\{\dot yy^4\}+\delta_5\{y\dot yy^3\}. \end{gathered}$$ This expression must coincide with $f = \{x^3\dot x y \} - \{y^2x\dot x y\} $. In particular, a sum of all dimonomials with the central letter $y$ must be equal to zero: $$\begin{gathered} 0= \gamma_1\{y\dot yxyx\} +(\gamma_2+\delta_3)\{y\dot yx^2y\} +(\gamma_3+\gamma_5+\delta_4+\delta_5)\{y\dot yyx^2\} \\ +\gamma_4\{xy\dot yyx\} +\gamma_6\{xy\dot yxy\} +\gamma_1\{\dot yyxyx\} +(\gamma_2+\delta_3)\{\dot y yx^2y\} +(\gamma_3+\delta_5)\{\dot y y^2x^2\} \\ +\gamma_4\{x\dot yy^2x\} +(\gamma_5+\delta_4)\{x^2\dot yy^2\} +\gamma_6 \{x\dot yyxy\} +(\delta_1-\delta_3-\delta_5)\{\dot yy^4\} \\ +(\delta_1+\delta_2-\delta_3-\delta_4-\delta_5)\{y\dot yy^3\} +(\delta_2-\delta_4)\{y^2\dot yy^2\}. \end{gathered}$$ All coefficients in this sum have to be zero. Solving the obtained system we have $\gamma_2= -\delta_3$, $\gamma_3= -\delta_5$, $\gamma_5= -\delta_4$, $\delta_1= \delta_3+\delta_5$, $\delta_2=\delta_4$, $\gamma_1= \gamma_4= \gamma _6= 0$. Substitute the obtained relations to $\phi(x,y,k)$ we get that all summands with coefficients $\gamma$ and $\delta$ are eliminated. Further, consider the remaining summands (we divide them into two groups by $\deg_y$): $$\begin{gathered} (\alpha_1 + \alpha_4) \{ \dot xyx^3 \} +(\alpha_2 + \alpha_6) \{ y\dot x x^3 \} +\alpha_3 \{ \dot xxyx^2 \} +\alpha_5 \{ yx^2\dot xx \} \\ +\beta_1 \{ xy\dot x x^2 \} +\beta_2 \{ yx \dot x x^2 \} +\beta_3 \{ x\dot xyx^2 \} +\beta_4 \{ xyx^2 \dot x \} +(\beta_5 +\beta_6)\{ yx^3\dot x \} \\ = \{x^3\dot x y\}, \end{gathered}$$ $$\begin{gathered} \alpha_1 \{ \dot xyxy^2 \}+\alpha_2 \{ y\dot xxy^2 \} +\alpha_3\{\dot x xy^3 \} + \alpha_5 \{ y^3\dot xx \} + \alpha_6 \{ y\dot xy^2x \} \\ +\beta_1 \{ xy\dot xy^2 \}+\beta_2 \{ yx \dot xy^2 \}+ \beta_3\{x\dot xy^3 \}+ (\alpha_4 +\beta_4) \{ \dot xy^3x \} +\beta_5 \{ y^3x\dot x \}+\beta_6 \{ yxy^2\dot x \} \\ = \{ y^2x\dot x y\}. \end{gathered}$$ The last two equalities imply $\alpha _2 =1$ and other coefficients are equal to zero. Therefore, $$\begin{gathered} \phi (x,y,z) = \{y\dot xxz\}-2\delta_3\{yxx\dot z\} -2\delta_5\{xxy\dot z\}-2\delta_4\{y\dot zxx\} \\ +2(\delta_3+\delta_5)\{\dot zyyy\}+2\delta_4\{y\dot zyy\} +2\delta_3\{\dot zzy\}+2\delta_4\{z\dot zy\}+2\delta_5\{\dot zyz\}.\end{gathered}$$ By assumption this dipolynomial is Jordan. When we expand Jordan products then the central letter is preserved, hence the dipolynomials consisting of dimonomials from $\phi (x,y,z)$ with the fixed central letter must be Jordan. In particular, if we choose the central letter $x$ then the dipolynomial $\{y\dot xxz\}$ must be Jordan, but this is not true by the proof of Theorem \[thm:CohnForDialgebra\]. The contradiction obtained proves that $f\notin I$. S-IDENTITIES ============ In this section ${\mathop{\mathrm{char}}\nolimits}\Bbbk=0$, so we can perform the process of full linearization of identities and varieties of algebras are always defined by multilinear identities. Equality of varieties ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$ and ${\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$ ------------------------------------------------------------------------------------------------------------- Consider a class of special Jordan dialgebras ${\mathrm{SJ}}$. The class ${\mathrm{SJ}}$ is not a variety because it is not close relative the taking of homomorphic images. Consider the operator ${\mathcal{H}}$ acting on classes of algebraic systems $${\mathcal{H}}(K)=\{A\mid A=\phi(B)\text{ for }B\in K,\phi\colon B\to A \text{ is an epimorphism}\}.$$ It is well-known that ${\mathcal{H}}({\mathrm{SJ}})$ is a variety of algebras which we denote ${\mathcal{H}}{\mathrm{SJ}}$. We recall (see Section \[subsec:DefDialg\]) that if $D\in{\mathrm{Di}}{\mathrm{Alg}}0$ then $D$ can be endowed with left and right actions of the algebra $\bar D$ by the rules $\bar xy=x{\mathbin\vdash}y$, $y\bar x=y{\mathbin\dashv}x$, where $x$, $y\in D$. Let ${\mathrm{Var}}$ be a variety of ordinary algebras. In the paper [@Pozh:09] it is shown that $D\in{\mathrm{Di}}{\mathrm{Var}}$ if and only if $\bar D\in{\mathrm{Var}}$ and $D$ is a ${\mathrm{Var}}$-bimodule over $\bar D$ in the sense of Eilenberg, i. e., the split null extension $\widehat D=\bar D\oplus D$ belongs to the variety ${\mathrm{Var}}$. In this way we can define a variety of dialgebras ${\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$ by a variety ${\mathcal{H}}{\mathrm{SJ}}$. Let ${\mathrm{Di}}{\mathrm{SJ}}$ be the class of special Jordan dialgebras. Consider the closure ${\mathcal{H}}({\mathrm{Di}}{\mathrm{SJ}})$ of this class relative to the operator ${\mathcal{H}}$. The variety obtained we denote by ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$. The purpose of this section is to show that ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}={\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$. \[lemma:FreeSpecJordDialgebra\] ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle$ is a free algebra in the variety ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$. Let $J'\in{\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$ be a homomorphic image of $J\in{\mathrm{Di}}{\mathrm{SJ}}$, $D$ be an associative dialgebra such that $J\hookrightarrow D^{(+)}$. We have the following commutative diagram $$\begin{CD} J' @<\text{на}<< J @>\subseteq>> D \\ @AAA @AAA @AAA \\ X @>\subseteq>> {\mathrm{Di}}{\mathrm{SJ}}\langle X\rangle @>\subseteq>> {\mathrm{Di}}{\mathrm{As}}\langle X\rangle \end{CD}$$ We have $X\subseteq J'$. Consider some preimages of elements of the set $X$ with respect to the mapping $J\to J'$. Since $J\subseteq D$, we obtain the embedding of $X$ into $D$. By the universal property of ${\mathrm{Di}}{\mathrm{As}}\langle X\rangle$ there exists an unique homomorphism ${\mathrm{Di}}{\mathrm{As}}\langle X\rangle\to D$ such that its restriction to ${\mathrm{Di}}{\mathrm{SJ}}\langle X\rangle$ is the homomorphism ${\mathrm{Di}}{\mathrm{SJ}}\langle X\rangle\to J$. The last homomorphism in a composition with the mapping $J\to J'$ gives the required homomorphism ${\mathrm{Di}}{\mathrm{SJ}}\langle X\rangle\to J'$. A *bar-unit* of a 0-dialgebra $D$ is an element $e\in D$ such that $x{\mathbin\dashv}e=e{\mathbin\vdash}x=x$ for every $x\in D$ and $e$ belongs to the associative center of $D$ that is $$(x,e,y)_\times=(e,x,y)_\dashv=(x,y,e)_\vdash=0$$ for all $x$, $y\in D$. \[prop:EmbWithBarUnit\] For every associative dialgebra $D$ there exists an associative dialgebra $D_e$ with the bar-unit $e$ such that $D\hookrightarrow D_e$. \[lemma:SpecUnitEmb\] Let $J$ be a special Jordan dialgebra. Then there exists a special Jordan dialgebra $J_e$ such that $J\hookrightarrow J_e$ and $\bar e$ is a unit in the algebra $\bar J_e$. By the defintion of a special Jordan dialgebra it follows that $J=(J,{{}_{(\vdash)}},{{}_{(\dashv)}})$ is embedded into $D^{(+)}$ for some associative dialgebra $D=(D,\vdash,\dashv)$. By Proposition \[prop:EmbWithBarUnit\] we have an embedding $D^{(+)}\hookrightarrow D_e^{(+)}$ where $e$ is a bar-unit in $D_e$. Therefore, $J_e=D_e^{(+)}$ is the required dialgebra. Further, $e{\mathbin\vdash}x=x{\mathbin\dashv}e=x$ holds for every $x\in D_e$, so in $J_e$ we have $e{\mathbin{{}_{(\vdash)}}}x=\frac{1}{2}(e{\mathbin\vdash}x+x{\mathbin\dashv}e)=x$, $x{\mathbin{{}_{(\dashv)}}}e=x$. Hence $\bar e\bar x=\bar x\bar e=\bar x$ in the quotient algebra $\bar J_e$, so $\bar e$ is a unit in $\bar J_e$. \[lemma:SpecFact\] Let $J$ be a special Jordan dialgebra and such that $\bar{J}$ contains a unit. Then $\bar{J}$ is special. Let $D$ be an associative dialgebra such that $J\hookrightarrow D^{(+)}$. Denote $\langle D,D\rangle:={\mathop{\mathrm{Span}}\nolimits}\{a{\mathbin\vdash}b-a{\mathbin\dashv}b\mid a,\,b\in D\}$, $[J,J]:={\mathop{\mathrm{Span}}\nolimits}\{a{\mathbin{{}_{(\vdash)}}}b-a{\mathbin{{}_{(\dashv)}}}b\mid a,\,b\in J\}$. As before $\bar D=D/\langle D,D\rangle$ is an associative algebra. Since $J\subseteq D$ we have $[J,J]\subseteq\langle D,D\rangle$. Then the homomorphism $\phi\colon \bar J\to\bar D^{(+)}$ is well-defined by the rule $x+[J,J]\mapsto x+\langle D,D\rangle$. $$\begin{CD} J @>\subseteq>> D \\ @VVV @VVV \\ \bar J @>\phi>> \bar D \end{CD}$$ It is evident that $\phi$ is injective if and only if $\langle D,D\rangle\cap J=[J,J]$. Let $x\in\langle D,D\rangle\cap J$. Then $x{\mathbin\vdash}y=y{\mathbin\dashv}x=0$ for every $y\in D$, hence $x{\mathbin{{}_{(\vdash)}}}y=\frac{1}{2}(x{\mathbin\vdash}y+y{\mathbin\dashv}x)=0$ in $J$ and $\bar x\bar y=\bar 0$ in $\bar J$. Take $\bar y=1\in\bar J$ and obtain $\bar x=\bar 0$, i. e., $x\in [J,J]$. So, $\phi$ is injective and $\bar J$ is special. Let $J$ be a Jordan algebra, $A$ be an associative algebra with a unit, then a homomorphism from $J$ to $A^{(+)}$ is called an *associative specialization* $\sigma\colon J\to A$. This is a linear mapping such that $$\sigma(ab)=\frac{1}{2}(\sigma(a)\sigma(b)+\sigma(b)\sigma(a))$$ for all $a,b\in J$. Two associative specializations are called *commuting* if $[\sigma_1(a),\sigma_2(b)]=0$ for all $a,b\in J$. A bimodule $M$ over $J$ is *special* if there exists an embedding of $M$ into a bimodule $N$ such that if $v\in N$, $a\in J$ then $$\label{eq:SpecBiModCond} a\cdot v=\frac{1}{2}(\sigma_1(a)v+\sigma_2(a)v),$$ where $\sigma_1$, $\sigma_2$ are commuting associative specializations of $J$ into $\mathrm{Hom}(N,N)$. \[thm:SpecSplitNullExtCrit\] Let $J$ be a special Jordan algebra, $M$ be a bimodule over $J$. Then the bimodule $M$ is special if and only if the split null extension $J\oplus M$ is a special Jordan algebra. \[lemma:SpecSplitNullExt\] Let $J$ be a special Jordan dialgebra and $\bar{J}$ be a special Jordan algebra. Then $\widehat{J}=\bar{J}\oplus J$ is special too. Since $J=(J,{{}_{(\vdash)}},{{}_{(\dashv)}})$ is special, we have $J\hookrightarrow D^{(+)}$ where $D={(D,\vdash,\dashv)}$ is an associative dialgebra. The dialgebra $J$ is a $\bar J$-bimodule: $\bar a\cdot v=a{\mathbin{{}_{(\vdash)}}}v =v{\mathbin{{}_{(\dashv)}}}a=v\cdot\bar a$, where $\bar a\in\bar J$, $v\in J$. We prove that the bimodule $J$ over the special Jordan algebra $\bar J$ is special. The bimodule $J$ is embedded into $D$ and $D$ is a $\bar J$-bimodule too. Consider mappings $\sigma_1,\,\sigma_2\colon\bar J\to\mathrm{Hom}(D,D)$ defined by the rule $$\sigma_1(\bar a)\colon d\mapsto a{\mathbin\vdash}d\in D,\, \sigma_2(\bar a)\colon d\mapsto d{\mathbin\dashv}a\in D,\quad d\in D,\,a\in J\subseteq D.$$ These mappings are well-defined. We show that they are associative specializations. Indeed for every $\bar{a\vphantom b},\,\bar b\in\bar J$, $d\in D$ $$\begin{gathered} \sigma_1(\bar{a\vphantom b}\bar b)(d) =\sigma_1(\overline{a{\mathbin{{}_{(\vdash)}}}b})(d)=\frac{1}{2}(a{\mathbin\vdash}b+b{\mathbin\dashv}a){\mathbin\vdash}d=\\ \frac{1}{2}(b{\mathbin\vdash}a{\mathbin\vdash}d+a{\mathbin\vdash}b{\mathbin\vdash}d)=\frac{1}{2}(\sigma_1(\bar{a\vphantom b})\sigma_1(\bar b)+\sigma_1(\bar b)\sigma_1(\bar{a\vphantom b}))(d).\end{gathered}$$ (We write a composition of mappings as $fg(x)=g(f(x))$.) Analogously, one may check that $\sigma_2$ is an associative specialization. The relation (\[eq:SpecBiModCond\]) follows from the definition of the operation in our bimodule. Moreover, $\sigma_1$ and $\sigma_2$ are commuting because $$[\sigma_1(\bar{a\vphantom b}),\sigma_2(\bar b)](d) =(\sigma_1(\bar{a\vphantom b})\sigma_2(\bar b) +\sigma_2(\bar b)\sigma_1(\bar{a\vphantom b}))(d)=(a{\mathbin\vdash}d){\mathbin\dashv}b-a{\mathbin\vdash}(d{\mathbin\dashv}b)=0.$$ So, $J$ is a special $\bar J$-bimodule and by Theorem \[thm:SpecSplitNullExtCrit\] we obtain that $\widehat J$ is special. In papers [@Kol:08; @Kol:06] conformal algebras were investigated and the following fact was proved. \[prop:VarCur0DiVar\] If an algebra $A$ belongs to a variety ${\mathrm{Var}}$ then a dialgebra $({\mathop{\mathrm{Cur}}\nolimits}A)^{(0)}$ belongs to a variety ${\mathrm{Di}}{\mathrm{Var}}$. We first prove an auxiliary statement. \[lemma:ifHomSJthenHomDiSJ\] If $\widehat J\in{\mathcal{H}}{\mathrm{SJ}}$ then $J\in{\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$. Use conformal algebras. Let the algebra $\widehat J$ generated by a set $X$ belong to the variety ${\mathcal{H}}{\mathrm{SJ}}$. Since ${\mathrm{SJ}}\,\langle X\rangle$ is a free algebra of the variety ${\mathcal{H}}{\mathrm{SJ}}$, there exists a surjective homomorphism $\phi\colon{\mathrm{SJ}}\,\langle X\rangle\to \widehat J$. Then ${\mathop{\mathrm{Cur}}\nolimits}\phi\colon{\mathop{\mathrm{Cur}}\nolimits}{\mathrm{SJ}}\,\langle X\rangle\to{\mathop{\mathrm{Cur}}\nolimits}\widehat J$ is a morphism of conformal algebras and particularly dialgebras. It is known [@GubKol:09] that $J\hookrightarrow({\mathop{\mathrm{Cur}}\nolimits}\widehat J)^{(0)}$. So $({\mathop{\mathrm{Cur}}\nolimits}\phi)^{-1}[J]$ is a subdialgebra in $({\mathop{\mathrm{Cur}}\nolimits}{\mathrm{SJ}}\,\langle X\rangle)^{(0)}$. The algebra ${\mathrm{SJ}}\,\langle X\rangle\in{\mathrm{SJ}}$ so by the definition of ${\mathrm{SJ}}$ there exists an associative algebra $A$ such that ${\mathrm{SJ}}\,\langle X\rangle\hookrightarrow A^{(+)}$, hence ${\mathop{\mathrm{Cur}}\nolimits}{\mathrm{SJ}}\,\langle X\rangle\hookrightarrow{\mathop{\mathrm{Cur}}\nolimits}A^{(+)}$ and $({\mathop{\mathrm{Cur}}\nolimits}{\mathrm{SJ}}\,\langle X\rangle)^{(0)}\in{\mathrm{Di}}{\mathrm{SJ}}$. To complete the proof we need to note that $J={\mathop{\mathrm{Cur}}\nolimits}\phi(({\mathop{\mathrm{Cur}}\nolimits}\phi)^{-1}[J])$, where $({\mathop{\mathrm{Cur}}\nolimits}\phi)^{-1}[J]\hookrightarrow({\mathop{\mathrm{Cur}}\nolimits}{\mathrm{SJ}}\,\langle X\rangle)^{0}\in{\mathrm{Di}}{\mathrm{SJ}}$ and so $J\in{\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$. Now we can prove the following theorem. \[thm:EqOfVarDialg\] ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}={\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$. To prove the inclusion “$\subseteq$” consider a free algebra ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle$ in the variety ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$. By Lemma \[lemma:SpecUnitEmb\] we have ${\mathrm{Di}}{\mathrm{SJ}}\langle X\rangle\hookrightarrow J_e$, $J_e$ is special and $1\in \bar{J_e}$. Then by Lemma \[lemma:SpecFact\] $\bar J_e$ is special, hence by Lemma \[lemma:SpecSplitNullExt\] $\widehat{J_e}$ is a special Jordan algebra and $J_e\in {\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$. Therefore, ${\mathrm{Di}}{\mathrm{SJ}}\langle X\rangle\in{\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$. Since the free algebra of the variety ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$ belongs to the variety ${\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$, the variety ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$ is embedded into ${\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$. We prove the inclusion “$\supseteq$”. Let $J\in{\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$. By the definition of a variety of dialgebras in the sense of Eilenberg it means that $\widehat J\in{\mathcal{H}}{\mathrm{SJ}}$, hence by Lemma \[lemma:ifHomSJthenHomDiSJ\] we obtain $J\in{\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$. s-identities in dialgebras -------------------------- Let ${\mathrm{Var}}$ be a variety of algebras, $X=\{x_1, x_2, \ldots \}$ be a countable set. Consider a mapping $\phi_{\mathrm{Var}}\colon{\mathrm{Alg}}\,\langle X \rangle \to {\mathrm{Var}}\,\langle X \rangle$ which maps $x_i\mapsto x_i$. Let $T_0({\mathrm{Var}})$ be a set of multilinear polynomials from $\ker\phi_{\mathrm{Var}}$, these are exactly all multilinear identities of ${\mathrm{Var}}$. We suppose that the variety is defined by multilinear identities that is ${\mathrm{Var}}=\{A\mid A\vDash T_0({\mathrm{Var}})\}$. There we use the denotation $A\vDash f$ which means that the identity $f(x_1, \ldots, x_n)=0$ holds on the algebra $A$. Further, let ${\mathrm{Di}}{\mathrm{Alg}}0\,\langle X\rangle$ be a free 0-dialgebra, $\phi_{{\mathrm{Di}}{\mathrm{Var}}}\colon{\mathrm{Di}}{\mathrm{Alg}}0\,\langle X \rangle \to {\mathrm{Di}}{\mathrm{Var}}\,\langle X \rangle$, $T_0({\mathrm{Di}}{\mathrm{Var}})$ be a set of multilinear dipolynomials from $\ker \phi_{{\mathrm{Di}}{\mathrm{Var}}}$, i. e., all multilinear identities from ${\mathrm{Di}}{\mathrm{Var}}$. In paper [@Pozh:09] the following theorem was proved. \[thm:DefDiVar\] Let $D\in{\mathrm{Di}}{\mathrm{Alg}}0$. Then the following conditions are equivalent: 1. $D\in{\mathrm{Di}}{\mathrm{Var}}$ 2. $\widehat D=\bar D\oplus D\in{\mathrm{Var}}$ the definition in the sense of Eilenberg 3. $D\vDash \Psi^{x_i}_{\mathrm{Alg}}\,f$ for every $f\in T_0({\mathrm{Var}})$, $\deg f=n$, $i=1,\,\ldots,\,n$ the definition in the sense of We prove the following \[prop:PsiDiVarVar\] Let $f=f(x_1,\ldots,x_n)\in {\mathrm{Di}}{\mathrm{Alg}}0\,\langle X\rangle$ be multilinear, $f=\Psi^{x_j}_{\mathrm{Alg}}\,\bar f$ for some $j$. Then $$f\in T_0({\mathrm{Di}}{\mathrm{Var}})\Leftrightarrow \bar f\in T_0({\mathrm{Var}}).$$ Since evidently ${\mathrm{Var}}\subseteq{\mathrm{Di}}{\mathrm{Var}}$, the statement “$\Rightarrow$” is trivial. To prove “$\Leftarrow$” consider an identity $\bar f\in T_0({\mathrm{Var}})$. By Theorem \[thm:DefDiVar\] for arbitrary $D\in{\mathrm{Di}}{\mathrm{Var}}$ we have $D\vDash \Psi^{x_i}_{\mathrm{Alg}}\,\bar{f}$ for all $i=1,\,\ldots,\,n$, but $\Psi^{x_j}_{\mathrm{Alg}}\,\bar{f}=f$ and so $f\in T_0({\mathrm{Di}}{\mathrm{Var}})$. \[prop:f1fnDiVarDiVar\] Let $f=f(x_1,\ldots,x_n)\in{\mathrm{Di}}{\mathrm{Alg}}0\,\langle X\rangle$ be multilinear, $f=f_1+\ldots+f_n$ where $f_i$ consists of all dimonomials in $f$ with a central letter $x_i$. Then $$f\in T_0({\mathrm{Di}}{\mathrm{Var}})\Leftrightarrow f_i\in T_0({\mathrm{Di}}{\mathrm{Var}})\text{ for all } i=1,\,\ldots,\,n.$$ “$\Leftarrow$” is evident. We prove “$\Rightarrow$”. Let $f\in T_0({\mathrm{Di}}{\mathrm{Var}})$, consider an arbitrary algebra $A\in{\mathrm{Var}}$. Then by Proposition \[prop:VarCur0DiVar\] we obtain ${({\mathop{\mathrm{Cur}}\nolimits}A)}^{(0)}\in {\mathrm{Di}}{\mathrm{Var}}$, hence ${({\mathop{\mathrm{Cur}}\nolimits}A)}^{(0)}\vDash f$, where $f=f(x_1,\ldots,x_n)$. Fix $i\in\{1,\,\ldots,\,n\}$ and assign the following values to variables: $x_i:=Ta_i$, $a_i\in A$, $x_j:=a_j$ for all $j\not=i$, $a_j\in A$. The properties of a conformal product imply $$0=f(a_1,\ldots,Ta_i,\ldots,a_n)=T\bar{f_i}(a_1,\ldots,a_n).$$ From the last equality we obtain $\bar{f_i}(a_1,\ldots,a_n)=0$, so $A\vDash\bar{f_i}$ and $\bar{f_i}\in T_0({\mathrm{Var}})$. By the previous proposition $f_i\in T_0({\mathrm{Di}}{\mathrm{Var}})$. We recall that $f$ is called a multilinear s-identity (in the case of ordinary algebras) if $$f\in T_0({\mathcal{H}}{\mathrm{SJ}})\setminus T_0({\mathrm{Jord}}):={\mathrm{SId}}.$$ A similar notion can be introduced for dialgebras [@Br:09] $$f\in T_0({\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}})\setminus T_0({\mathrm{Di}}{\mathrm{Jord}}):={\mathrm{Di}}{\mathrm{SId}}.$$ \[thm:CorrespSId\] 1. Let $g = g(x_1,\ldots,x_n)\in{\mathrm{SId}}$. Then $\Psi^{x_i}_{\mathrm{Alg}}\,g\in{\mathrm{Di}}{\mathrm{SId}}$ for all $i=1,\ldots,n$. 2. Let $f=f(x_1,\ldots,x_n)\in{\mathrm{Di}}{\mathrm{SId}}$, $f=f_1+\ldots+f_n$ by a central letter. Then there exists $j\in\{1,\,\ldots,\,n\}$ such that $\bar{f_j}\in{\mathrm{SId}}$. We prove the statement 1. Let $g\in{\mathrm{SId}}$, hence by the definition ${\mathrm{SId}}$ we have $g\in T_0({\mathcal{H}}{\mathrm{SJ}})$ and $g\not\in T_0({\mathrm{Jord}})$. Proposition \[prop:PsiDiVarVar\] implies $\Psi^{x_i}_{\mathrm{Alg}}\,g\in T_0({\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}})$, $\Psi^{x_i}_{\mathrm{Alg}}\,g\not\in T_0({\mathrm{Di}}{\mathrm{Jord}})$. It follows from the equality of varieties ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}={\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$ that $\Psi^{x_i}_{\mathrm{Alg}}\,g\in {\mathrm{Di}}{\mathrm{SId}}$. For proving the statement 2 consider $f\in{\mathrm{Di}}{\mathrm{SId}}$. By the definition of ${\mathrm{Di}}{\mathrm{SId}}$ and Theorem \[thm:EqOfVarDialg\] we have $f\in T_0({\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}})=T_0({\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}})$ and $f\not\in T_0({\mathrm{Di}}{\mathrm{Jord}})$. It follows from $f\in T_0({\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}})$ by Proposition \[prop:f1fnDiVarDiVar\] that $f_i\in T_0({\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}})$ for all $i$. It follows from $f\not\in T_0({\mathrm{Di}}{\mathrm{Jord}})$ that $j$ exists such that $f_j\not\in T_0({\mathrm{Di}}{\mathrm{Jord}})$. Further, by Proposition \[prop:PsiDiVarVar\], $\bar f_i\in T_0({\mathcal{H}}{\mathrm{SJ}})$ and $\bar f_j\not\in T_0({\mathrm{Di}}{\mathrm{Jord}})$, hence by the definition ${\mathrm{SId}}$ we obtain $\bar f_j\in{\mathrm{SId}}$. Now we can easily obtain the following corollary which was proved in [@Br:09] by computer algebra methods. There are no s-identities for Jordan dialgebras of degree $\le 7$ and there exists a multilinear s-identity of a degree 8. Let $f$ be a s-identity for Jordan dialgebras, $\deg f=k\le 7$. After a full linearization of $f$ we can suppose that $f$ is multilinear that is $f\in{\mathrm{Di}}{\mathrm{SId}}$ and $f=f_1+\ldots+f_k$ by central letters. It follows from Theorem \[thm:CorrespSId\] about the corresponding of multilinear s-identities that $\bar f_i\in{\mathrm{SId}}$ for some $i$, $\deg \bar f_i\le k$, but Glennie proved [@Glennie:70] that such an identity does not exist. It is known [@Glennie:66] that there exists $f$ which is a s-identity for Jordan algebras, $\deg f=8$. Again we can suppose that $f$ is multilinear. Then Theorem \[thm:CorrespSId\] implies $\Psi^{x_i}_{\mathrm{Alg}}f$ is a required multilinear s-identity for all $i=1,\,\ldots,\,8$. Analogues for dialgebras of Shirshov’s and Macdonald’s Theorems --------------------------------------------------------------- Since we get the generalization of the Cohn’s Theorem to the case of dialgebras, a question appears about a generalization of the Shirshov’s Theorem for special Jordan algebras ([@Zhevl:78], the simplification of Shirshov’s original proof is contained in [@JacPage:57]): Whenever every Jordan dialgebra with two generators is special? The answer to this questions is negative, it follows from Theorem \[thm:Example2GeneratedExceptionalDialgebra\]. However, the following analogue of the Shirshov’s Theorem holds for dialgebras. Let $J$ be a one-generated dialgebra. Then $J$ is special. We have $J\in{\mathrm{Di}}{\mathrm{Jord}}$. Then by the definition a variety of dialgebras in the sense of Eilenberg $\bar J\in{\mathrm{Jord}}$, $\widehat J=\bar J\oplus J\in{\mathrm{Jord}}$. Let $x$ be the generator of $J$. Then $\widehat J=\langle \bar x,x\rangle$, so $\widehat J$ is a two-generated Jordan algebra. By the Shirshov’s Theorem we obtain that $\widehat J$ is special. We have $J\hookrightarrow({\mathop{\mathrm{Cur}}\nolimits}\widehat J)^{(0)}$ and so $J$ is special too. Consider the particular case when two-generated dialgebra is free. Let $J={\mathrm{Di}}{\mathrm{Jord}}\,\langle x,y\rangle$ be the free Jordan dialgebra generated by $x$, $y$. Then $J$ is special. We need to show that $J\in{\mathrm{Di}}{\mathrm{SJ}}$. First, prove $J\in{\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$. Assume the converse, i. e., $J\not\in{\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$. By Lemma \[lemma:ifHomSJthenHomDiSJ\] we obtain $\widehat J=\bar J\oplus J\not\in{\mathcal{H}}{\mathrm{SJ}}$. Since $\widehat J\in{\mathrm{Jord}}\setminus{\mathcal{H}}{\mathrm{SJ}}$, there exists a multilinear s-identity $f(x_1,\ldots,x_n)$ of Jordan algebras such that ${\mathrm{SJ}}\vDash f$ but $\widehat J\nvDash f$. Therefore, we may find $u_1,\ldots,u_n\in\widehat J$ such that $f(u_1,\ldots,u_n)\not =0$. Since the polynomial $f$ is multilinear, we can suppose that either $u_i\in\bar J$ or $u_i\in J$ for all $i$. The number of elements $u_i\in J$ does not exceed one otherwise, $f(u_1,\ldots,u_n)=0$ because $J\cdot J=0$. Consider two possible cases. The first case is when all $u_i\in\bar J$. Then $\bar J\nvDash f$, which is impossible since $\bar J\in{\mathrm{SJ}}$ and $f$ is an s-identity. The second case is when $u_1,\,\ldots,\,u_{n-1}\in\bar J$, $u_n\in J$. The algebra $\bar J$ is generated by $\bar x$ and $\bar y$, so $u_i=u_i(\bar x,\bar y)$, $i=1,\ldots,n$. Then denote $g(\bar x,\bar y,u_n):=f(u_1(\bar x,\bar y),\ldots,u_{n-1}(\bar x,\bar y),u_n)\not=0$. Note that $g$ does not hold on $\widehat J$. The polynomial $g(x,y,z)$ vanishes in ${\mathrm{SJ}}$, $\deg_z g=1$, hence by the Macdonald’s Theorem we obtain $g=0$ in ${\mathrm{Jord}}$. The contradiction obtained proves that $J\in{\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$. We prove that $J\in{\mathrm{Di}}{\mathrm{SJ}}$. We know that $J$ is a homomorphic image of some special Jordan algebra $J_0$ under some mapping $\phi\colon J_0\to J$. Let $x_0$ and $y_0$ are preimages of $x$ and $y$ with respect to $\phi$. Consider a subdialgebra $U$ in $J_0$ generated by $x_0$ and $y_0$. Since the dialgebra $J_0$ is special, subdialgebra $U$ is special too. The dialgebra $J={\mathrm{Di}}{\mathrm{Jord}}\,\langle x,y\rangle$ is free in the variety of Jordan dialgebras, hence every mapping of $x$ and $y$ to $U$ extends to a homomorphism. Map $x$ and $y$ to $x_0$ and $y_0$ respectively. Since $x_0$ and $y_0$ generate $U$, we obtain a surjective homomorphism inverse to a homomorphism $\phi|_U$. Therefore, $J\backsimeq U$ is a special Jordan dialgebra. If an identity $f(x,y)$ in two variables holds in all special Jordan dialgebras then it holds in all Jordan dialgebras. Consider $f(x,y)$ as an element of the free Jordan dialgebra ${\mathrm{Di}}{\mathrm{Jord}}\,\langle x,y\rangle$. By the previous theorem ${\mathrm{Di}}{\mathrm{Jord}}\,\langle x,y\rangle$ is a special Jordan dialgebra, therefore ${{\mathrm{Di}}{\mathrm{Jord}}\vDash f}$. In the paper [@Br:09] the s-identity of dialgebras was found which depends on three variables and is linear in one of variables. So the naive generalization of the Macdonald’s Theorem to the case of dialgebras is not true. But if an identity is linear in the central letter then the following theorem is true which is an analogue of the Macdonald’s Theorem. Let $f=f(x,y,\dot z)$ be a dipolynomial which is linear in $z$. If ${\mathrm{Di}}{\mathrm{SJ}}\vDash f$ then ${\mathrm{Di}}{\mathrm{Jord}}\vDash f$. Let ${\mathrm{Di}}{\mathrm{SJ}}\vDash f$, then ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}\vDash f$. Consider a Jordan algebra $\bar J\in{\mathcal{H}}{\mathrm{SJ}}$ as a dialgebra $J$ with equal left and right products. Then $\bar J\in{\mathcal{H}}{\mathrm{SJ}}$ and $\widehat J=\bar J\oplus J=\bar J\oplus\bar J\in{\mathcal{H}}{\mathrm{SJ}}$, so $J\in{\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}={\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$. We obtain $J\vDash f$, hence $\bar J\vDash \bar f$. Therefore, ${\mathcal{H}}{\mathrm{SJ}}\vDash\bar f=f(x,y,z)$, so by the classical Macdonald’s Theorem we have ${\mathrm{Jord}}\vDash\bar f$. It remains to note that if $f= f(x,y,\dot z)$ is a multilinear dipolynomial such that ${\mathrm{Jord}}\vDash\bar f=f(x,y,z)$ then ${\mathrm{Di}}{\mathrm{Jord}}\vDash f$: It follows immediately from the definition [@Kol:08] of what is a variety of dialgebras. The polynomial $f(x,y,z)$ can be nonlinear in $x$ and $y$. Suppose $\deg_x f=n$, $\deg_y f=m$. Consider the full linearization $$g(x_1,\dots, x_n, y_1,\dots, y_m, z)= L_x^n L_y^m f(x,y,z)$$ of the identity $f(x,y,z)$ (notations from [@Zhevl:78 ch. 1]). Then ${\mathrm{Jord}}\vDash g(x_1,\dots, x_n, y_1,\dots, y_m, z)$ and so ${\mathrm{Di}}{\mathrm{Jord}}\vDash g(x_1,\dots, x_n,y_1,\dots, y_m, \dot z)$. If we now identify variables, then $$g(x,\dots, x, y,\dots, y,\dot z)=n!m!f(x,y,\dot z).$$ In this section the characteristic of the basic field is equal to zero, so we can divide by $n!m!$ and hence $f(x,y,\dot z)$ is an identity on ${\mathrm{Di}}{\mathrm{Jord}}$. P. M. Cohn in [@Cohn:59] proposed an axiomatic characterization of Jordan algebras $J_1(A)=A^{(+)}$ and $J_2(A,*)=H(A,*)$, where $A$ is an associative algebra and $*$ is an involution on $A$, in terms of $n$-ary operations. This is an interesting task to generalize these results to Jordan dialgebras. Acknowledgements {#acknowledgements .unnumbered} ---------------- In the end of paper the author thanks P. S. Kolesnikov, A. P. Pozhidaev and V. Yu. Gubarev for helpful discussions and valuable comments. 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[^1]: The author was partially supported by ADTP (Grant 2.1.1.10726), RFBR (Grant 09-01-00157-A), SSc-3669.2010.1, and Federal Aim Program (contracts N. 02.740.11.5191, N. 14.740.11.0346).

--- abstract: | The improvements of the observations of the solar system allowed by the use of probes and big instruments let appear several problems: The frequencies of the radio signals received from the probes sent over 5 UA from the Sun are too high; the explanation by spicules or siphon-flows of the frequency shifts of UV emissions observed on the surface of the sun by SOHO is not satisfactory; the anisotropy of the CMB seems bound to the ecliptic. This problems are solved using a coherent optical effect, deduced from standard spectroscopy and easily observed with lasers. In a gas containing atomic hydrogen in states 2S and (or) 2P , transfers of energy between light beams, allowed by thermodynamics, produce the required frequency shifts or amplifications. author: - 'Jacques Moret-Bailly' title: Anomalous frequency shifts in the solar system --- jacques.moret-bailly@u-bourgogne.fr Introduction ============ The remarkable precision of relastivistic mechanics and electrodynamics allows, for instance the good localizations by the the Global Positioning System. However, a discrepancy appears in the observation of the probes (in particular Pioneers 10 and 11) when their distance from the Sun becomes larger than about five astronomical units. The the frequencies of the received radio signals are too high, so that it seems that the attraction by the Sun increases over Newton’s law. Several prudent explanations are proposed, in particular new physics or an acceleration by an anisotropic radiation of the energy provided by the disintegration of the plutonium which feeds the probes in energy. In section \[description\], we show why the previous explanations cannot work, showing that the problem occurs during the propagation of the radio waves. Other discrepancies are found in observations of the Solar system: - the explanations of the redshifts of the UV emission spectra of the Sun observed by SOHO, by spicules or siphon-flows, appear weak; - the anisotropy of the microwave background appears bound to the Solar system. This section defines the properties required for an optical effect able to solve these three problems. In section \[creil\], we study this effect, deduced from standard rules of spectroscopy, but observable only in conditions which allow to qualify the light pulses ”ultrashort”. This effect appears, in particular while light is refracted by a low pressure gas containing atomic hydrogen in states 2S or 2P. Description of the anomalies and their explanations. {#description} ==================================================== Anomalies of the speeds of the Pioneer 10 and 11 probes. {#Pioneer} -------------------------------------------------------- The original description of the Pioneer probes, and of the detection of anomalies in the radio-signals of several probes was given by Anderson et al. [@Anderson1; @Anderson2]. Anderson et al. deduce the radial speed of the probes through an assumed Doppler shift of radio waves: An electromagnetic wave is sent from the Earth to the probes at a frequency deduced from the frequency of an hydrogen maser by a multiplication such that the frequency received by the probe is close to 2.11 GHz. This frequency is multiplied by 240/221 to avoid an interference with the received frequency, amplified and sent back to the Earth where it is detected by an heterodyne system, producing a frequency close to 1 MHz. The weakness of the received signal requires a track more and more difficult with an increase of the distance. Taking into account the main computed frequency shifts, Doppler and gravitational, less important perturbations such as the pressures of radiation of the Solar light, and the pressure of the Solar wind, the gravity of the Kuiper belt ..., the received frequency has the computed value until the distance of the probe is lower than 5AU; at a longer distance the received frequency becomes more and more too high, until the extra acceleration stabilises over 15 AU at the value $(8.6 \pm 1.34)\times 10^{-8} $ cm s$^{-2}$. See figure \[F1\]. ![ Apparent acceleration corresponding to the residual frequency shift, as a function of distance of the Sun, from Anderson et al. [@Anderson2].[]{data-label="F1"}](correlate.eps){height="10cm"} If the origin of the acceleration were a leakage of the valves of the thrusters allowing the maneuvers, the probability that both Pioneers have leaks producing the same acceleration, and that a leak reproduces after a maneuver, is low. Therefore, the main hypothesis is an anisotropy of the radiation of the 2 kW produced by the decay of the plutonium on board the aircraft. The decrease of this energy with the time is not observed, but it may correspond to the uncertainty of the measure of the acceleration[@Markwardt; @Scheffer]. We think that the origin of the anomalous accelerations does not lie in the apparatus for the following reasons : i)The identities of the accelerations of both Pioneers show that they do not probably result from an accidental disworking such as a leakage of a valve. ![ Doppler residuals as a function of time. The top panel shows all of the data. The bottom shows the residuals, excluding the regions perturbed by the solar corona, designated by an horizontal bar “C” and the noisy regions, designated “N”. From Markwardt[@Markwardt].[]{data-label="F2"}](fig-bestfit-gif.eps){height="16cm"} ii)On figure \[F2\], the interferences with the corona produce large perturbations of the observed frequencies (“C” regions), but , after, the linear increase of speed is restored.. If the “N” regions were produced in the apparatus, the large anomalous speeds which would appear should translate the following segments; thus something similar to a path through the corona, happens on the path of the light, and the properties of this path are more easily restored than the properties of a complex apparatus. What can happen on the path of light in the Solar wind ? Maybe the proximity of a planet, maybe an increase of the Solar activity. How do this change in the Solar wind may be transferred to the radio signal ? The redshifts of the UV emission lines of the quiet Sun. {#sun} -------------------------------------------------------- The chromosphere of the quiet Sun was studied by Peter and Judge [@Peter] using data acquired by the Solar Ultraviolet Measurement of Emitted Radiation (SUMER), on the SOHO spacecraft. We consider here only residual frequency shifts obtained by subtraction from the observed shifts of a "main correction”: a) the the Doppler shift produced by the rotation of the Sun and the relative movement of the Sun and the probe; b) the relativistic shift. After a description of spectra, Peter & Judge present the current state of their interpretation, founded on an attribution of the (residual) frequency shifts of the spectral lines to a Doppler effect produced by vertical movements of the gas in the chromosphere. To explain that lines emitted at the same, or at very close places have different redshifts, an hypothesis is that gas is ejected in vertical spicules, then cools and flows down; an other hypothesis is siphon flows through loops. But Peter & Judge write : ”As for the spicule idea, the existing siphon-flow pictures are either non valid or only part of the story”. Other hypothesis are tried, but "still more work is needed”. With the hypothesis of Doppler effects and vertical movements, for all lines, there is no (residual) frequency shift at the limb of the Sun. This hypothesis implies that the frequencies measured at the limb are, after subtraction of the "main correction”, the absolute frequencies. Comparing the absolute frequencies deduced from SUMER measures at the limb to older measured or computed frequencies, discrepancies appear, attributed to a lack of precision of the old results. For instance, a computed value of the wavelength of Mg X is 62495.2 pm, while the value deduced from the observation of the limb is 62496.8 $\pm$ .7 pm. The wavelength of the Ne VIII line was measured in the laboratory by Bockasten et al [@Bockasten] who found 77040.9 $\pm$ .5 pm. From SUMER measures, at the limb, Peter & Judge obtained 77042.8 $\pm$ .7 pm. Considering that this value is a rest wavelength, there is a discrepancy attributed to a too short error bar in the laboratory measure. Peter & Judge write : "If one would take the Bockasten et al. value for granted, this would imply that the Ne VIII is indeed redshifted at disk center and would beg the question of how the redshift of a line seen at disk center $C$ can even increase toward the limb - - we would not be able to explain such a variation with our current understanding of the solar atmosphere.” Peter & Judge do not rely much on the theory they use: ”Neither the nature of the driving motions nor the response of the plasma can be reliably constrained by currently available observations or by numerical simulations ... It might be that the blueshifts we observe are not caused by the out-flowing solar wind but by some other processes.” An other process, a new understanding is supposing that the shifts occurs during the propagation of the light through a shell of the chromosphere: the path through this shell for the rays emitted at $C$ is the thickness of the shell, while it is larger for other rays. Writing $S(M)$ the frequency shift obtained by Peter & Judge at a point $M$, $s(M)$ the newly defined shift, $L$ being a point at the limb, the relation between the shifts is: $$S(M) = s(M)-s(L) \label{frsh}$$ As $|s(L)|$ is larger than $|s(M)|$, the signs of $S(M)$ and $s(M)$ are opposite, the variations of the frequency shifts along a radius are opposite. {height="12cm"} with formation temperature of the line. Error bars for the data of Brekke et al (1997) were typically 2 km s$^{-1}$. The solid line is a by-eye fit of the Doppler Shifts in Peter & Judge study. From Peter & Judge [@Peter]. \[pj\] Figure \[pj\] shows the shifts of various lines $S(C)$ as a function of the temperature of the emitting gas. Suppose that the column density is sufficient to reach nearly a saturation, that is an equilibrium between the temperature of the emitting gas and the temperature of the light [^1] at the centre of the lines. Thermodynamics says that energy flows from hot to cold, so that the three high energy lines Ne VIII, Mg X and Fe XII are allowed by thermodynamics to transfer energy to the other lines provided that the light is refracted by a convenient medium playing the role of a catalyst. This transfer redshifts the three hot lines, and blueshifts the other in conformity with the definition $s(M)$ of the redshifts. Why do the lines emitted at the lowest temperatures (He I, C II, Si IV and C IV) are less blueshifted than the other cold lines? A simple explanation is that the catalytic power of the refracting medium is nearly zero at temperatures lower than 30 000 K (temperature of emission of He I) and gets a maximal mean value if the temperature along the path varies from about 170 000 K (maximum of the curve on figure \[pj\]) to 30 000 K, leading to an optimal temperature of the effect, very roughly, of the order of 100 000 K. The anisotropy of the cosmic microwave background. -------------------------------------------------- In subsection \[Pioneer\], we explained the anomalous increase of frequency of the Pioneer probes by an interaction in the solar wind. If this interaction is similar to the interaction whose characteristics were found in subsection \[sun\], it is a transfer of energy from the solar light to radio waves. This transfer applies to all radio waves propagating in the solar wind over 5 UA, in particular to the cosmic microwave background. The solar wind is generated in the holes of the corona, so that it is anisotropic. Its structure may be modified by the magnetic fields of the planets. Thus, the blueshift of the radio frequencies by the solar wind is anisotropic. For the CMB, a thermal radiation, this shift is an amplification which adds a contribution to the anisotropy due to the movement of the Sun in the galaxy. The analysis of the observed CMB leads to a similar result [@Schwarz; @Land; @Naselsky]. The Coherent Raman Effects on Incoherent Light (CREIL) {#creil} ====================================================== A simultaneous explanation of the anomalies studied in section \[description\] uses an optical effect having the following properties: i)The images and the spectra are not blurred; else the signals from the Pioneers would be too much weakened; ii)The energy transferred from hot beams to cold beams shifts the frequencies; iii)The interacting beams must be refracted by a gas whose optimal temperature is of the order of 100 000 K; observed from the solar frequency shifts and the cooling of the solar wind. This section explains the required effect, which appears very similar to the refraction, but which requires very particular media, or ultra-short laser pulses. Conditions for Doppler-like frequency shifts by interaction with matter. {#con} ------------------------------------------------------------------------ - A Doppler-like redshift must avoid a blur of the images. Therefore, it must be space-coherent, so that the wave surfaces are not disturbed: For an involved molecule, it exists relations between the local phases of all involved electromagnetic fields, and the phases of all molecular oscillators; space coherence” means that these relations are identical for all involved molecules. Consequently, supposing that the number of involved molecules is large, Huygens’ construction shows that the radiated fields generate clean wave surfaces related with the wave surfaces of the exciting fields. - For a time-coherent source (continuous wave laser), frequency shift” means that while the source emits $n$ cycles, the detector receives a different number $m$. Thus, the number of cycles between the source and the receiver is increased of $n-m$; it is an increase of the number of wavelengths, thus an increase of the distance, therefore a Doppler effect. Consequently, a Doppler-like redshift is only possible with time-incoherent light; a parameter measuring this incoherence must appear in the theory to forbid an application to time-coherent light. - The energy absorbed by the redshifting process must not be quantised to avoid a blur of the spectra: If a light beam exchanges a quantified energy with a molecule, a fraction of the intensity of the beam gets a finite shift. In a parametric process, the molecules leave their stationary state only temporarily, their states becoming dressed” during their interactions with the light; the light beams exchange not-quantified energy, the matter plays the role of a catalyst[^2]. Reminding the semi-classical theory of refraction. -------------------------------------------------- To simplify the explanations, suppose that the refracting medium is perfectly transparent. A sheet of matter between two close wave surfaces distant of $\epsilon$ is excited at a pulsation $\Omega$. The sheet radiates a Rayleigh coherent wave late of $\pi/2$ whose amplitude is a small fraction $K\epsilon$ of the exciting amplitude $E_0$. From Huygens’ construction it generates the same wave surfaces, so that the fields add into $$\begin{aligned} E=E_0[\sin(\Omega t)+K\epsilon \cos(\Omega t)]\nonumber\\ \approx E_0[\sin(\Omega t)\cos(K\epsilon)+\sin(K\epsilon )\cos(\Omega t)]=E_0\sin(\Omega t -K\epsilon).\end{aligned}$$ This result defines the index of refraction $n$ by the identification $$K=2\pi n/\lambda=\Omega n/c.\label{refr}$$ Suppose that the light interacts with free identical molecules, initially in the same non-degenerate stationary state $\phi_0$. The perturbation of a molecule by an electromagnetic wave mixes $\phi_0$ with other states $\phi_i$ , producing a non-stationary state $\Phi = C_0\phi_0+\sum_iC_i\phi_i$, where the $C_i$ are very small. We must consider the set of all interacting molecules, adding an upper index $k$ to distinguish the molecules. Without a field, the total, stationary state is $\Psi_0=\prod_k\phi_0^k$. Its degeneracy is the number of molecules. Perturbed by an external field, the refracting medium radiates a scattered, coherent field late of $\pi/2$, generating the same wave surfaces than the exciting field; therefore, the dynamically excited, non-stationary, dressed ” ( or polarisation” ) state $\psi^m$ which emits this field is characterised by an index $m$ representing the exciting mode. Considering other refracted modes, $\Psi$ splits as $\prod_m\psi^m$. Remark that the coherent interactions are much stronger than the incoherent: A refraction by $\approx 0.25 \mu m$ of water delays the light of $\pi/2$, that is the light is fully scattered by the coherent Rayleigh scattering. In a swimming pool, we see well through 25 metres of water, only a fraction of the light is scattered by the incoherent Rayleigh scattering; the factor is $10^8$. Principle of the CREIL. ------------------------ The CREIL results from an interaction between dressed states $\psi^m$; as these states have the same parity, the interaction must be of Raman type, for instance quadrupolar electric. Thermodynamics says that the entropy must increase, so that the floods of energy are from the modes which have a high Planck’s temperature to the colder ones. For an astrophysical application we consider a purely parametric effect: the matter, a low pressure gas in low fields, returns to its initial state after an interaction. The dressed state $\psi^m$ radiates a mixture of the coherent Rayleigh scattering which produces the refraction and coherent Raman scatterings. These locally weak scatterings may be studied independently, so that the CREIL may be considered as a set of [*simultaneous*]{} Stokes and anti-Stokes coherent Raman scatterings with a zero balance of energy for the molecules [^3]. The scattered beams have the same wave surfaces than the exciting beams, so that these beams may interfere, as in the coherent Rayleigh scattering making the refraction; as the scattered fields are much weaker than the exciting field, they may be added independently to it. The pulsations of the Raman beams are shifted by $\pm\omega$, and, at the beginning of a pulse, in phase because the resonance introduces a $-\pi/2$ phaseshift. The sum of the exciting wave and the coherent anti-Stokes scattered wave is: $$\begin{aligned} E=E_0[\sin(\Omega t)+K'\epsilon \sin((\Omega+\omega)t)] \hskip 3mm ({\rm with } (K'>0) \nonumber\\ E=E_0[\sin(\Omega t)+K'\epsilon[\sin(\Omega t)\cos(\omega t)+\sin(\omega t)\cos(\Omega t)]].\end{aligned}$$ Supposing that $\omega t$ and $K'\epsilon$ are small, the second term, product of two small quantities, may be neglected, and the last one transformed: $$\begin{aligned} E\approx E_0[\sin\Omega t+\sin(K'\epsilon\omega t)\cos(\Omega t)]\nonumber\\ E\approx E_0[\sin(\Omega t)\cos(K'\epsilon\omega t)+ \sin(K'\epsilon\omega t)\cos(\Omega t)=E_0\sin[(\Omega+K'\epsilon\omega)t].\label{eq4}\end{aligned}$$ $K'\epsilon$ is an infinitesimal term, but the hypothesis $\omega t$ small requires that the Raman period $2\pi/\omega$ is large in comparison with the duration of the experiment $t$. This condition was set by G. L. Lamb Jr. for the definition of ultrashort pulses” : shorter than all relevant time constants” [@Lamb]. With ordinary light, the time coherence plays the role of length of the pulses: thus, the time-coherence, some nanoseconds, must be shorter than all relevant time constants”. We have found a first relevant time constant. A second is the collisional time constant, because the collisions destroy the space-coherence, producing an ordinary, weak, incoherent Raman scattering; a low pressure gas is needed. The same computation, replacing $K'$ by a negative $K''$ gives the Stokes contribution, so that we replace $K'$ by $K'+K''$ in formula \[eq4\]. $K'+K''$ depends on the difference of population in both levels, that is on $\exp(-h\omega/2\pi kT)-1\propto \omega/T$, where $T$ is the temperature of the gas. The theory of the refraction shows that the index of refraction is nearly constant in the absence of resonance close to $\Omega$, so that, using for the polarisability a formula equivalent to formula \[refr\], [ ]{} $K'+K"$ [ ]{}appears nearly proportional to $\Omega \omega/T$, and the frequency shift is : $$\Delta\Omega=(K'+K")\epsilon\omega\propto \epsilon\Omega\omega^2/T.$$ The relative frequency shift $\Delta\Omega/\Omega$ is nearly independent on $\Omega$. All required properties are obtained: space coherence, limitation of the time-coherence, no excitation of the gas, nearly constant relative frequency shift. As the shift is proportional to $\omega^2$, a strong effect requires a Raman pulsation $\omega$ as large as allowed by the preservation of the coherence. As the time-coherence of ordinary light is some nanoseconds, an Raman frequency is of the order of 100 Mhz. Laboratory observation of the CREIL effect ------------------------------------------ Usually, it is not necessary to take into account the radiations which receive energy because we are surrounded by thermal radiations whose blueshift is simply a heating. In a convenient medium, the CREIL effect transfers also energy between the radio frequencies which make the thermal radiation as long as the thermal equilibrium, including the isotropy, is not reached; this CREIL effect is strong because, all involved frequencies being low, it is nearly resonant, so that the radio frequencies get quickly a thermal equilibrium. The CREIL in optical fibres is so easily obtained that it makes problems for the use of short pulses in telecommunications. With the high peak power of femtosecond lasers, the index of refraction and the components of the tensor of polarisability become increasing functions of the intensity, allowing a study of the effect in small cells. This nonlinear effect named “Impulsive Stimulated Raman Scattering” (ISRS) allows an easy study of the properties of the coherent Raman effect on incoherent light: transfer of energy from a laser beam to another producing frequency shifts, verification of Lamb’s conditions (Yan et al. [@Yan]). While the lengths of the laser pulses increase, the experiments become more and more difficult: To increase the collisional time, it becomes impossible to use dense matter, a gas less and less dense must be used. While it is easy to find strong Raman resonances at the rotational and vibrational frequencies of molecules, resonances close to 100 MHz appear generally in highly excited states, almost unpopulated. Therefore, an observation of a CREIL effect, using ordinary incoherent light would require an expansive experiment while it is well verified in the whole easily accessible domain of frequencies. Propagation of incoherent light in atomic hydrogen {#H} -------------------------------------------------- As atomic hydrogen has a simple spectrum, its levels of energy may be well populated. Its electric quadrupole spin recoupling transition ($\Delta F=1$) in the ground state has the frequency 1420 MHz, too high. But, in the first excited state, the frequencies 178 MHz in the 2S$_{1/2}$ state, 59 MHz in 2P$_{1/2}$ state, and 24 MHz in 2P$_{3/2}$ are very convenient; in these states, the gas will be named H\*. It is more difficult to populate higher states, and the resonance frequencies are low, so that, in these states, the CREIL effect is negligible. Excited atomic hydrogen which redshifts the light may be generated by various processes: The ionisation energy equals $kT$ for a temperature $T=156 000 K$; as the energy needed for a pumping to the states of principal quantum number $n=2$ (H\* states) is the three fourth of the ionisation energy, it equals $kT$ for $T=117 000 K$. Using Boltzman law, these temperatures may be considered as indicating roughly where these particular states of hydrogen are abundant, remarking however that by a thermal excitation, the proportion of hydrogen in the H\* states is limited by the excitation to higher values of $n$, and by the ionisation at low pressures. Remark that, from figure \[pj\], we found in \[sun\] an approximate optimal value $T=100 000 K$ : H\* is clearly the source of the anomalous frequency shifts on the Sun. Over a temperature $T=10 000 K$, the molecules of hydrogen are dissociated. The strong absorption of the Lyman alpha line produces H\*. The effective decay of H\* is very slow at low pressures because this decay can only re-emit the Ly$_\alpha$ line which is strongly, immediately re-absorbed. The surface of the Sun is too cold to provide much energy at the Ly$_\alpha$ frequency. But H\* may be produced close to very hot objects such as quasars, accreting neutron stars. A feed-back may appear in unexcited atomic hydrogen illuminated by a far UV continuous spectrum: The excitation at the Lyman $\alpha$ frequency produces H\*, therefore a redshift which renews the intensity of the light at the Lyman $\alpha$ frequency until a previously absorbed line almost stops the redshift, so that the other Lyman lines are strongly absorbed and will nearly stop the following fast redshift. The combination of the protons and electrons of a plasma produces atomic hydrogen in various states of excitation. The 2S state is stable at a low pressure. The optical transitions from the 2P states generate a Ly$_\alpha$ line which may be reabsorbed. The cooling of the solar wind beyond 5 UA produces H\* and explains the blueshift of the radio-frequencies of the Pioneers 10 and 11, at least a part of the anisotropy of the CMB bound to the ecliptic. Conclusion ========== Introducing coherent optical interactions other than the refraction seems the key of a lot of explanations of up to now difficult to understand astrophysical observations. In particular, the Coherent Raman Effects on Incoherent Light (CREIL) is the true origin of frequency shifts usually considered as produced by a Doppler effect. The use of the CREIL is very simple: light beams refracted simultaneously by a gas containing atomic hydrogen in states 2S or 2P exchange energy to increase the entropy of their set, producing frequency shifts. Where the physical conditions allow the production of H\*, anomalous frequency shifts appear. [9]{} Anderson J. D. , P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, & S. G. Turyshev, [*Phys. Rev. Lett.* ]{}[**81**]{}, 2858-2861 (1998) . Anderson J. D. , P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, & S. G. Turyshev, [*Phys. Rev.* ]{}[**D 65**]{}, 082004 (2002). Markwardt C. B., arxiv:gr-cq/0208046(2002). Scheffer L. K. , [*Phys.Rev.* ]{}[**D67**]{} (2003) 08402. Peter H. & P. G. Judge [*ApJ,* ]{} [**522**]{}, 1148-1166 (1999). Bockasten, K., R. Hallin & T. Hughes, [*Proc. Phys. Soc.* ]{}[**81**]{} 522 (1963). Schwarz, D. J., G. D. Starkman, D. Huterer, C. J. Copi, [*Phys. Rev. Lett.* ]{}**93**, 221301, arxiv:astro-ph/0403353 (2004). Land K., & J. Magueijo, [*Mon. Not. R. Astron. Soc.* ]{}**357**, 994 (2005). Naselsky P., L.-Y Chiang., P. Olesen & I. Novikov, arxiv:astro-ph/0505011 (2005). Lamb G. L. Jr., [*Rev. Mod. Phys.*]{}, **43**, 99-124 (1971). Yan Y.-X., E. B. Gamble Jr. & K. A. Nelson, [*J. Chem Phys.*]{}, **83**, 5391 (1985). [^1]: Temperature deduced from the intensity in a mode, using Planck’s formula for the radiation of a black body. [^2]: We do not follow an extended definition of parametric” interactions in which the matter may be (des)excited during the interaction (for instance in a He-Ne laser medium), parametric” becoming synonymous of coherent”. [^3]: Coherent Raman Scatterings on Incoherent Light” (CREIL) is ambiguous, relative either to a single Raman interaction ( ignoring the quasi-resonant, easy transfer of the Raman energy to the thermal radiation ), or to the whole set of interactions.

--- abstract: 'This paper describes algorithms for the exact symbolic computation of period integrals on moduli spaces ${\mathcal{M}}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points, and applications to the computation of Feynman integrals.' author: - Christian Bogner and Francis Brown title: Feynman integrals and iterated integrals on moduli spaces of curves of genus zero --- MaPhy-AvH/2014-10 Introduction ============ Let $n\geq 0$ and let ${\mathcal{M}}_{0,n}$ denote the moduli space of Riemann spheres with $n$ ordered marked points. The main examples of periods of ${\mathcal{M}}_{0,n+3}$ consist of integrals [@Bro2; @Bro3; @Ter] $$\label{introM0nint} \int_{0\leq t_1 \leq \ldots \leq t_n \leq 1} {\prod_{i=1}^n t_i^{a_{i}} (1-t_i)^{b_i} \over \prod_{1 \leq i< j \leq n} (t_i-t_j)^{c_{ij}} } dt_1 \ldots dt_n$$ for suitable choices of integers $a_i,b_i,c_{ij} \in {\mathbb{Z}}$ such that the integral converges. These integrals have a variety of applications ranging from superstring theory [@Sch1; @Sch2] to irrationality proofs [@AperyVar; @Fisch1]. In [@Bro2] it was shown that such integrals are linear combinations of multiple zeta values $$\label{introMZVdef} \zeta(n_1,\ldots, n_r) = \sum_{1\leq k_1 < \ldots < k_r} {1 \over k_1^{n_1} \ldots k_r^{n_r}} \qquad \hbox{ where } n_i \in {\mathbb{N}}, n_r \geq 2$$ with rational coefficients. One of the goals of this paper is to provide effective algorithms, based on [@Bro2], for computing such integrals $(\ref{introM0nint})$ symbolically. The idea is to integrate out one variable at a time by working in a suitable algebra of iterated integrals (or rather, their symbols) which is closed under the two operations of taking primitives and taking limits along boundary divisors. The second main application is for the calculation of a large class of Feynman amplitudes, based on the universal property of the spaces ${\mathcal{M}}_{0,n}$. The general idea goes as follows. Suppose that $X \rightarrow S$ is a stable curve of genus zero. Then the universal property of moduli spaces yields an $n\geq3$ and a commutative diagram: $$\label{Square} \begin{array}{ccc} X & \longrightarrow & \overline{{\mathcal{M}}}_{0,n+1} \\ \downarrow & & \downarrow \\ S & \longrightarrow & \overline{{\mathcal{M}}}_{0,n} \end{array}$$ The idea is that, for a specific class of (multi-valued) forms on $X$, we can integrate in the fibers of $X$ over $S$ by passing to the right-hand side of the diagram and computing the integral on the moduli space ${\mathcal{M}}_{0,n+1}$. In this way, it only suffices to describe algorithms to integrate on the universal curve ${\mathcal{M}}_{0,n+1}$ over ${\mathcal{M}}_{0,n}$. In practice, this involves computing a change of variables to pass from $X$ to a set of convenient coordinates on the moduli space ${\mathcal{M}}_{0,n+1}$, applying the algorithm of [@Bro2] to integrate out one of these coordinates, and finally changing variables back to $S$. This process can be repeated for certain varieties which can be fibered in curves of genus $0$ and yields an effective algorithm for computing a large class of integrals. Necessary conditions for such fibrations to exist (‘linear reducibility’) were described in [@Bro4] and apply to many families of Feynman integrals, as we discuss in more detail presently. Feynman integrals ----------------- Any Feynman integral in even-dimensional space-time can always be expressed as an integral in Schwinger parameters $\alpha_j$: $$\label{introFeynI} I = \int_{0\leq \alpha_j \leq \infty } {P(\alpha_j) \over Q(\alpha_j)} \, d\alpha_1 \ldots d{\alpha_N}$$ where $P$ and $Q$ are polynomials with (typically) rational coefficients and which perhaps depend on other parameters such as masses or momenta. Cohomologial considerations tell us that the types of numbers occurring as such integrals only depend on the denominator $Q$, and not on the numerator $P$. A basic idea of [@Bro5] is to compute the integral $(\ref{introFeynI})$ by integrating out the Schwinger parameters $\alpha_i$ one at a time in some well-chosen order. After $i$ integrations, we require that the partial integral $$\label{Ikpartial} I(\alpha_1, \ldots, \alpha_{N-i}) = \int_{0\leq \alpha_j \leq \infty} {P \over Q}\, d\alpha_{N-i+1} \ldots d \alpha_{N}$$ be expressed as a certain kind of generalised polylogarithm function, or iterated integral. Under certain conditions on the singularities of the integrand, the next variable can be integrated out. A ‘linear reduction’ algorithm [@Bro4; @Bro5] yields an upper bound for the set of singularities of $(\ref{Ikpartial})$ and can tell us in advance whether $(\ref{introFeynI})$ can be computed by this method. It takes the form of a sequence of sets of polynomials (or rather, their associated hypersurfaces): $$S_1 \ , \ S_2 \ , \ \ldots$$ where $S_1 = \{Q\}$, and $S_{i+1}$ is derived from $S_{i}$ by taking certain resultants of polynomials in $S_i$ with respect to $\alpha_{N-i+1}$. When $Q$ is linearly reducible, we obtain a sequence of spaces for $i\geq 1$: $$\begin{aligned} X_i & =& ({\mathbb{P}}^1 \backslash \{0,\infty\})^{N-i+1} \ \backslash \ V(S_i) \nonumber \\ & =& \{ (\alpha_{1}, \ldots, \alpha_{N-i+1}): \alpha_k \neq 0, \infty \hbox{ and } P(\alpha_{1},\ldots, \alpha_{N-i+1}) \neq 0 \hbox{ for all } P \in S_i \} \nonumber \end{aligned}$$ and maps $\pi_i: X_i \rightarrow X_{i+1}$ which correspond to projecting out the variable $\alpha_{N-i+1}$. The linear reducibility assumption guarantees that $X_i$ fibers over $X_{i+1}$ in curves of genus $0$. Thus setting $(X,S) = (X_i, X_{i+1})$ in the discussion above, we can explicitly find changes of variables in the $\alpha_i$ to write $(\ref{Ikpartial})$ as an iterated integral on a moduli space ${\mathcal{M}}_{0,n}$ and do the next integration. It is perhaps surprising that such a method should ever work for any non-trivial Feynman integrals. The fundamental reason it does, however, is that the polynomial $Q$ can be expressed in terms of determinants of matrices whose entries are linear in the $\alpha_i$ parameters. In the case when $Q$ is the first Symanzik polynomial, and to a lesser extent when $Q$ also depends on masses and external momenta, it satisfies many ‘resultant identities’, which only break down at a certain loop order. A method of hyperlogarithms versus a method of moduli spaces {#secthypversusmod} ------------------------------------------------------------ There are two possible approaches to implementing the above algorithm: one which is now referred to as the ‘method of hyperlogarithms’ [@Bro5], which stays firmly on the left-hand side of the diagram $(\ref{Square})$; the other, which is the algorithm described here [@Bro2], which makes more systematic use of the geometry of the moduli spaces ${\mathcal{M}}_{0,n}$ and works on the right-hand side of the diagram $(\ref{Square})$. The first involves working directly in Schwinger parameters, and expressing all partial integrals as hyperlogarithms (iterated integrals of one variable) whose arguments are certain rational functions in Schwinger parameters. It has been fully implemented by Panzer [@Pan1; @Pan2; @Pan3] and various parts of the algorithm have found applications in different contexts, as described below. A conceptual disadvantage of this method is that the underlying geometry of every Feynman diagram is different. The second method, espoused here, is to compute all integrals on the moduli spaces ${\mathcal{M}}_{0,n}$ (which, by no accident, are the universal domain of definition for hyperlogarithms). Thus the underlying geometry is always the same and is well-understood; all the information about the particular integral $(\ref{introFeynI})$ is contained in the changes of variables $(\ref{Square})$. Another key difference is the systematic use of generalised symbols of functions in several complex variables, as opposed to functions of a single variable (hyperlogarithms). That these two points of view are equivalent is theorem \[thmVdecomp\] below, but leads, in practice, to a rather different algorithmic approach. We nonetheless provide algorithms (the symbol and unshuffle maps) to pass between both points of view. Applicability ------------- The above method can be applied to a range of Feynman integrals provided that the initial integral $(\ref{introFeynI})$ is convergent. The case of massless, single-scale, primitively overall-divergent Feynman diagrams in a scalar field theory was detailed in [@Bro4]. Since then, the method was applied to the computation of integrals of hexagonal Feynman graphs, arising in $\mathcal{N}=4$ supersymmetric Yang-Mills theory [@Del1; @Del2; @Del3], integrals with operator insertions contributing to massive matrix elements of quantum chromodynamics (QCD) [@Abl1; @Abl2; @Abl3], one- and two-loop triangular Feynman graphs with off-shell legs [@Cha], phase-space contributions [@Ana1; @Ana2] to the cross-section for threshold production of the Higgs boson from gluon-fusion at N3LO QCD [@Higgs], coefficients in the expansion of certain hypergeometric functions, contributing to superstring amplitudes [@Sch1; @Sch2], massless multi-loop propagator-type integrals [@Pan1], and a variety of three- and four-point Feynman integrals depending on several kinematical scales [@Pan2]. These applications arise from very different contexts and the method is combined with various other computational techniques. Focussing on Feynman integrals, we can summarize by stating that the method can be extended to the following situations: - To Feynman graphs with several masses or kinematic scales. - To gauge theories, or more generally, integrals with arbitrary numerator structures. - To graphs with ultra-violet subdivergences. In particular, it is compatible with the renormalisation procedure due to Bogoliubov, Parasiuk, Hepp and Zimmermann (BPHZ) in a momentum scheme [@BroKre]. - Finally, it can also be combined with dimensional regularisation to treat UV and IR divergences by the method of [@Pan2]. The method is suited for automatization on a computer. For the special case of harmonic polylogarithms, the programs [@Mai1; @Mai2] support direct integration using these functions. For the general approach, using hyperlogarithms, a first implementation of the method was presented in ref. [@Pan3]. A program for the numerical evaluation of these functions is given in ref. [@Vol]. There appear to be other classes of integrals which are not strictly Feynman diagrams, but for which the method of iterated fibration in curves of genus zero $(\ref{Square})$ still applies. A basic example are periods of arbitrary hyperplane complements [@Bro2], and as a consequence, various families of integrals occurring in deformation quantization, for example. Plan of the paper ----------------- In section $\ref{Sect2}$ we review some of the mathematics of iterated integrals on moduli spaces ${\mathcal{M}}_{0,n}$, based on [@Bro2]. The geometric ideas behind the main algorithms are outlined here. In §\[sec:Computing-on-the\], these algorithms are spelled out in complete detail together with some illustrative examples. In §\[sec:Feynman-type-integrals\], it is explained how to pass between Feynman integral representations and moduli space representations. In §\[sec:Applications\] we discuss some applications, before presenting the conclusions. Some introductory background can be found in the survey papers [@Bro6; @Bro7].\ The methods of $\S\ref{sec:Computing-on-the}$ should in principle generalise to genus $1$, using multiple elliptic polylogarithms defined in [@BrLe], but there remains a considerable amount of theoretical groundwork to be done. A different direction for generalisation is to introduce roots of unity, by replacing ${\mathbb{P}}^1\backslash \{0,1,\infty\}$ with ${\mathbb{P}}^1 \backslash \{0,\mu_N, \infty\}$ where $\mu_N$ is the group of $N^\mathrm{th}$ roots of unity. This should be rather similar to the framework discussed here.\ *Acknowledgements*: The second named author is a beneficiary of ERC grant 257638. The first named author thanks Erik Panzer for very useful discussions and especially helpful suggestions regarding the contents of section \[sec:Feynman-type-integrals\]. We thank Humboldt University for hospitality and support. Our Feynman graphs were drawn using [@Hah]. Iterated integrals on the moduli spaces ${\mathcal{M}}_{0,n}$ {#Sect2} ============================================================= Coordinates ----------- Let $n\geq 3$ and let ${\mathbb{C}}_{\infty}= {\mathbb{C}}\cup \{\infty\}$ denote the Riemann sphere. The complex moduli space ${\mathcal{M}}_{0,n}({\mathbb{C}})$ is the space of $n$ distinct ordered points on ${\mathbb{C}}_{\infty}$ modulo automorphisms $${\mathcal{M}}_{0,n}({\mathbb{C}}) = \{ (z_1,\ldots, z_n)\in {\mathbb{C}}_{\infty}^n \hbox{ distinct} \} /\mathrm{PGL_2}({\mathbb{C}}) \ .$$ There are two sets of coordinates, called simplicial and cubical, which are useful for the sequel. By applying an element of $\mathrm{PGL_2}({\mathbb{C}})$, we can assume that $z_1 =0, z_{n-1} =1$ and $z_n = \infty$ and define $$t_1 = z_2 \ , \ t_2 = z_3 \ , \ldots\ , \ t_{n-3} = z_{n-2} \ .$$ The $(t_1,\ldots, t_{n-3})$ are called simplicial coordinates and define an isomorphism $${\mathcal{M}}_{0,n}({\mathbb{C}}) \cong \{(t_1,\ldots, t_{n-3}) \in {\mathbb{C}}^{n-3} \hbox { such that the } t_i \hbox{ are distinct and } t_i \neq 0,1\}\ .$$ Cubical coordinates, on the other hand, are defined by $$\label{simplicialtocube} x_1 = {t_1 \over t_2} \ , \ x_2 = {t_2 \over t_3 } \ , \ \ldots\ , \ x_{n-4} = {t_{n-4} \over t_{n-3}} \ , \ x_{n-3}= t_{n-3}$$ Cubical coordinates define an isomorphism $${\mathcal{M}}_{0,n}({\mathbb{C}}) \cong \{(x_1,\ldots, x_{n-3}) \in {\mathbb{C}}^{n-3} \hbox { such that } x_{i} x_{i+1} \ldots x_j \neq \{0,1\} \hbox{ for all } 1 \leq i \leq j \leq n-3\}\ .$$ Note that the divisors above only involve products of cubical coordinates with consecutive indices. The main advantage of cubical coordinates is that the divisors corresponding to $$x_i=0 \quad \hbox{for } i=1,\ldots, n-3$$ are strict normal crossing in a neighbourhood of the origin $(0,\ldots, 0)$. The reason for the nomenclature is that the standard cell (a connected component of the set of real points ${\mathcal{M}}_{0,n}({\mathbb{R}})$) is either a simplex: $$X_n \cong \{(t_1,\ldots, t_{n-3}) \in {\mathbb{R}}^{n-3}: 0 < t_1 < \ldots < t_{n-3} < 1\}$$ or a cube: $$X_n \cong \{(x_1,\ldots, x_{n-3}) \in {\mathbb{R}}^{n-3}: 0 < x_i < 1 \hbox{ for all } 1\leq i \leq n-3 \}\ ,$$ depending on the choice of coordinate system. Differential forms ------------------ Let $\Omega^k({\mathcal{M}}_{0,n})$ denote the space of global regular differential $k$-forms on ${\mathcal{M}}_{0,n}$ which are defined over ${\mathbb{Q}}$. Consider the following elements of $\Omega^1({\mathcal{M}}_{0,n})$: $$\omega_{ij} = { {dt_i - dt_j} \over t_i -t_j } \ \hbox{ for } \ 0\leq i,j\leq n-2$$ where we set $t_0=0$ and $t_{n-2}= 1$. Clearly $\omega_{ij} = \omega_{ji}$ and $\omega_{ii}=0$. There are no other linear relations between the $\omega_{ij}$ besides these. Define $${\mathcal{A}}^1({\mathcal{M}}_{0,n}) = \langle \omega_{ij}: \hbox{ for } i < j \ , \ (i,j) \neq (0,n-2)\rangle_{{\mathbb{Q}}}$$ Thus ${\mathcal{A}}^1({\mathcal{M}}_{0,4})$ has the basis ${dt_1 \over t_1}, {dt_1 \over t_1 -1}$. The $\omega_{ij}$ satisfy the following quadratic relation: $$\label{quadrel} \omega_{ij} \wedge \omega_{jk}+ \omega_{jk} \wedge \omega_{ki} + \omega_{ki} \wedge \omega_{ij} =0$$ for all indices $i,j,k$. Define ${\mathcal{A}}^{\bullet}({\mathcal{M}}_{0,n})$ to be the differential graded algebra which is the quotient of the exterior algebra generated by ${\mathcal{A}}^1({\mathcal{M}}_{0,n})$ by the quadratic relations $(\ref{quadrel})$. A theorem due to Arnold states that $${\mathcal{A}}^{\bullet}({\mathcal{M}}_{0,n}) \longrightarrow H_{dR}^{\bullet}({\mathcal{M}}_{0,n};{\mathbb{Q}})$$ is an isomorphism of algebras. Thus ${\mathcal{A}}^{\bullet}({\mathcal{M}}_{0,n})$ is an explicit model for the de Rham cohomology of ${\mathcal{M}}_{0,n}$. In cubical coordinates, it is convenient to take a different basis for ${\mathcal{A}}^1({\mathcal{M}}_{0,n})$ formed by $${dx_i \over x_i} \quad \hbox{ and } \quad { { d (x_i\ldots x_j) \over x_i x_{i+1} \ldots x_j -1 } } \ \hbox{ for } 1\leq i \leq j \leq n-3 \ .$$ We will consider iterated integrals in these one-forms. Iterated integrals and symbols ------------------------------ Recall the definition of iterated integrals from [@Che]. Let $M$ be a smooth complex manifold and let $\omega_1, \ldots, \omega_n$ denote smooth 1-forms. Let $\gamma:[0,1] \rightarrow M$ be a smooth path. The iterated integral of these forms along $\gamma$ is defined by $$\int_{\gamma} \omega_1 \ldots \omega_n = \int_{0\leq t_1 \leq t_2 \leq \ldots \leq t_n \leq 1} \gamma^{*}(\omega_n) (t_1) \ldots \gamma^{*}(\omega_1) (t_n)\ .$$ There are different conventions for iterated integrals: here we integrate starting from the right. The argument of the left-hand integral is ${\mathbb{C}}$-multilinear in the forms $\omega_i$ and can be viewed as a functional on the tensor product $\Omega^1(M)^{\otimes n}$. Elements of this space are customarily written using the bar notation $[\omega_1 | \ldots | \omega_n]$ to denote a tensor product $\omega_1 \otimes \ldots \otimes \omega_n$. Chen’s theorem states that iterated integration defines an isomorphism from the zeroth cohomology of the reduced bar construction on the $C^{\infty}$ de Rham complex of $M$ to the space of iterated integrals on $M$ which only depend on the homotopy class of $\gamma$ relative to its endpoints. The reduced bar construction on ${\mathcal{M}}_{0,n}$ can be written down explicitly using the model ${\mathcal{A}}$ defined above, in terms of a certain algebra of symbols. For $n\geq 3$, define a graded ${\mathbb{Q}}$ vector space $$V({\mathcal{M}}_{0,n}) \subset \bigoplus_{m\geq 0} {\mathcal{A}}^1({\mathcal{M}}_{0,n})^{\otimes m}$$ by linear combinations of bar elements $$\sum_{I=(i_1,\ldots, i_m)} c_I [\omega_{i_1} | \ldots | \omega_{i_m}]$$ which satisfy the integrability condition $$\label{intcond} \sum_I c_I [\omega_{i_1} | \ldots | \omega_{i_{j-1}} | \omega_{i_j} \wedge \omega_{i_{j+1}} |\omega_{i_{j+2}}| \ldots | \omega_{i_m}] = 0 \qquad \hbox{ for all } 1\leq j \leq m-1\ .$$ Then $V({\mathcal{M}}_{0,n})$ is an algebra for the shuffle product ${\, \hbox{\rus x} \,}$ and is equipped with the deconcatenation coproduct $\Delta$, which is defined by: $$\Delta [\omega_{i_1} | \ldots | \omega_{i_m}] = \sum_{k=0}^m [\omega_{i_1} | \ldots | \omega_{i_k}] \otimes [\omega_{i_{k+1}} | \ldots | \omega_{i_m}]$$ Thus $V({\mathcal{M}}_{0,n})$ is a graded Hopf algebra over ${\mathbb{Q}}$. Iterated integration defines a homomorphism $$\begin{aligned} V({\mathcal{M}}_{0,n}) &\longrightarrow& \{\hbox{Multivalued functions on } {\mathcal{M}}_{0,n}({\mathbb{C}}) \} \label{itintonMod0n} \\ \sum_{I=(i_1,\ldots, i_m)} c_I [ \omega_{i_1} | \ldots | \omega_{i_m}] & \mapsto & \sum_I c_I \int_{\gamma_z} \omega_{i_1} \ldots \omega_{i_m} \nonumber \end{aligned}$$ where $\gamma_z$ is a homotopy equivalence class of paths from a fixed (tangential) base-point to $z \in {\mathcal{M}}_{0,n}({\mathbb{C}})$. By a version of Chen’s theorem, this map gives an isomorphism between homotopy invariant iterated integrals (viewed as multi-valued functions of their endpoint) on ${\mathcal{M}}_{0,n}$ and symbols. Equivalently, this means that the map $(\ref{itintonMod0n})$ is a homomorphism of differential algebras (for a certain differential to be defined in $(\ref{BdRdifferential})$) and the constants of integration are fixed as follows. One can show that, in cubical coordinates $(x_1,\ldots, x_{n-3})$, every iterated integral $(\ref{itintonMod0n})$ admits a finite expansion of the form $$\sum_{I=(i_1,\ldots, i_{n-3})} f_I(x_1,\ldots, x_{n-3}) \log(x_1)^{i_1} \ldots \log(x_{n-3})^{i_{n-3}}$$ where $f_I(x_1,\ldots, x_{n-3})$ is a formal power series in the $x_i$ which converges in the neighbourhood of the origin. The normalisation condition is that the regularised value at zero vanishes: $$f_{0,\ldots, 0} (0,\ldots, 0) = 0\ .$$ This gives a bijection between symbols and certain multivalued functions (whose branch is fixed, for example, on the standard cell $X_n)$, and in this way we can work entirely with symbols. Various operations on functions can be expressed algebraically in terms of $V({\mathcal{M}}_{0,n})$. For example, the monodromy of functions around loops can be expressed in terms of the coproduct $\Delta$. The bar-de Rham complex {#sectbardeRham} ----------------------- Differentiation of iterated integrals with respect to their endpoint corresponds to the following left-truncation operator $$\begin{aligned} \label{BdRdifferential} d: V({\mathcal{M}}_{0,n}) & \longrightarrow & \Omega^1({\mathcal{M}}_{0,n}) \otimes V({\mathcal{M}}_{0,n})\\ \sum_I c_I [ \omega_{i_1} | \ldots | \omega_{i_m}] & \mapsto & \sum_I c_I \omega_{i_1} \otimes [ \omega_{i_2} | \ldots | \omega_{i_m}] \nonumber \end{aligned}$$ where $I= (i_1,\ldots, i_m)$. The bar-de Rham complex is defined to be $$B({\mathcal{M}}_{0,n}) = \Omega^{\bullet}({\mathcal{M}}_{0,n}) \otimes V({\mathcal{M}}_{0,n})$$ equipped with the differential induced by $d$. In [@Bro2] it was shown that The cohomology of the bar-de Rham complex of ${\mathcal{M}}_{0,n}$ is trivial: $$H^i ( B({\mathcal{M}}_{0,n})) = \begin{cases} {\mathbb{Q}}\quad \hbox{ if } i = 0 \nonumber \\ 0 \quad \hbox{ if } i > 0 \end{cases}$$ In particular, $B({\mathcal{M}}_{0,n})$ is closed under the operation of taking primitives, which is one ingredient for computing integrals symbolically. The next ingredient states that one can compute regularised limits along irreducible boundary divisors $D \subset \overline{{\mathcal{M}}}_{0,n} \backslash {\mathcal{M}}_{0,n}$ with respect to certain local canonical sections $v$ of the normal bundle of $D$. Let $\mathcal{Z}$ denote the ${\mathbb{Q}}$-vector space generated by multiple zeta values $(\ref{introMZVdef}).$ \[thmreglimmap\] There exist canonical ‘regularised limit’ maps $$\mathrm{Reg}^v_D : V({\mathcal{M}}_{0,n}) \longrightarrow V({\mathcal{M}}_{0,r}) \otimes V({\mathcal{M}}_{0,n+2-r}) \otimes \mathcal{Z}$$ for every irreducible boundary divisor $D$ of $\overline{{\mathcal{M}}}_{0,n}$ which is isomorphic to $\overline{{\mathcal{M}}}_{0,r} \times \overline{{\mathcal{M}}}_{0,n+2-r}$. This states that the regularised limits of iterated integrals on moduli spaces are products of such iterated integrals with coefficients in the ring $\mathcal{Z}$ of multiple zeta values. By applying these two operations of primitives and limits, one can compute period integrals on ${\mathcal{M}}_{0,n}$. In more detail: ### Total primitives {#sectTotalPrimitives} Taking primitives of differential one-forms is a trivial matter. Let $\eta $ be a $1$-form in $B^1({\mathcal{M}}_{0,n})$ such that $d\eta=0$. We can write it as a finite sum $$\eta = \sum_k \omega^{k}_{0} \otimes [ \omega_1^k| \ldots | \omega_n^k]$$ A primitive is given explicitly by $$\int \eta = \sum_k [ \omega^{k}_{0}| \omega_1^k| \ldots | \omega_n^k] \ .$$ The constant of integration is uniquely (and automatically) determined by the property $$\varepsilon(\int \eta ) =0$$ where $\varepsilon: V({\mathcal{M}}_{0,n}) \rightarrow {\mathbb{Q}}$ is the augmentation map (projection onto terms of weight $0$). The fact that $\int \eta $ satisfies the integrability condition $(\ref{intcond})$ follows from the integrability of $\eta$ and the equation $d\eta=0$. In practice, the algorithm we actually use for taking primitives on the universal curve needs to be more sophisticated and is described below. ### Limits When taking limits, one must bear in mind the fact that the elements of $V({\mathcal{M}}_{0,n})$ represent multivalued functions, and hence depend on the (homotopy class) of the path $\gamma_z$ of analytic continuation $(\ref{itintonMod0n})$. When computing period integrals by the method described above, however, all iterated integrals which occur will be single-valued on the domain of integration ([@Bro4], theorem 58). In cubical coordinates, the domain of integration is the unit cube $X_n = [0,1]^{n-3}$, and so it suffices in this case to define limits along the divisors in $\overline{{\mathcal{M}}}_{0,n}$ defined by $x_i=0$ and $x_i=1$, where $x_i$ are cubical coordinates. Recall that the integration map from $V({\mathcal{M}}_{0,n})$ to multivalued functions is normalised at the point $(0,\ldots, 0)$ with respect to unit tangent vectors in cubical coordinates $x_i$, and it follows that the limits at $x_i=0$ are trivial to compute. Any function $f$ in the image of $(\ref{itintonMod0n})$ is uniquely determined on the simply connected domain $X_n=[0,1]^{n-3}$, and admits a unique expansion for some $N$ $$\label{fexpansion} f(x_1,\ldots, 1-\epsilon_i,\ldots, x_{n-3}) = \sum_{k=0}^N \log(\epsilon)^k p_k(\epsilon) f_k(x_1,\ldots, x_{i-1}, x_{i+1}, \ldots,x_{n-3})$$ where $p_k(\epsilon)$ is holomorphic at $\epsilon=0$ and where $f_k$ is in the image of $V({\mathcal{M}}_{0,i+2}) \otimes V({\mathcal{M}}_{0,n-i})$. The ‘regularised limit’ of $f$ along $x_i=1$ (with respect to the normal vector $-{\partial \over \partial x_i}$) is the function $$\mathrm{Reg}_{x_i=1}\, f = p_0(0) f_0(x_1,\ldots, x_{i-1}, x_{i+1}, \ldots,x_{n-3}) \ .$$ It is the composition of the realisation map $(\ref{itintonMod0n})$ with a certain map (theorem $\ref{thmreglimmap}$) $$V({\mathcal{M}}_{0,n}) \longrightarrow \mathcal{Z} \otimes V({\mathcal{M}}_{0,i+2}) \otimes V({\mathcal{M}}_{0,n-i})$$ where $\mathcal{Z}$ is the ring of multiple zeta values. This map can be computed explicitly as follows. Recall first of all the general formula for the behaviour of iterated integrals with respect to composition of paths, where $\gamma_1 \gamma_2$ denotes the path $\gamma_2$ followed by the path $\gamma_1$: $$\int_{\gamma_1 \gamma_2} \omega_1 \ldots \omega_n = \sum_{i=0}^n \int_{\gamma_1} \omega_1 \ldots \omega_i \int_{\gamma_2} \omega_{i+1} \ldots \omega_n\ .\label{pathconcat}$$ If $E_{\gamma}$ is the function on $V({\mathcal{M}}_{0,n})$ which denotes evaluation of a (regularised) iterated integral along a path $\gamma$, then the previous equation can be interpreted as a convolution product: $$\label{convolution} E_{\gamma_1 \gamma_2} = m ( E_{\gamma_1} \otimes E_{\gamma_2}) \circ \Delta$$ Ignoring, for the time being, issues to do with tangential base points and regularisation, a path from the origin $0$ to a point $z=(x_1,\ldots,x_{i-1}, 1, x_{i+1}, \ldots, x_{n-3})$ which lies inside the cube $X_n=[0,1]^{n-3}$ is homotopic to a composition of paths $\gamma_1 \gamma_2$ (‘up the $i^{\mathrm{th}}$ axis and then along to the point $z$’), where $$\gamma_2 =\hbox{straight line from } \ 0\ \hbox{ to } \ 1_i =( \underbrace{0,\ldots, 0}_{i-1}, 1, \underbrace{0,\ldots 0}_{n-i-4})$$ and $\gamma_1$ is a path from $1_i$ to $z$ which lies inside $x_i=1$. The segment of path $\gamma_2$ can be interpreted as a straight line from $0$ to $1$ in ${\mathcal{M}}_{0,4} = {\mathbb{P}}^1\backslash \{0,1,\infty\}$ (with coordinate $x_i$). Iterated integrals along this path give rise to coefficients of the Drinfeld associator, which are multiple zeta values. Iterated integrals along $\gamma_1$ can be identified with our class of multivalued functions on the boundary divisor $D$ of $\overline{{\mathcal{M}}}_{0,n}$ defined by $x_i=1$, which is canonically isomorphic to $\overline{{\mathcal{M}}}_{0,i+2} \times \overline{{\mathcal{M}}}_{0,n-i}$. One can check that the above argument makes sense for regularised (divergent) iterated integrals, and putting the pieces together yields the regularisation algorithm which is described below. For the computation of period integrals, one needs slightly more. We actually require an expansion of the function $(\ref{fexpansion})$ as a polynomial in $\log(\epsilon)$ and a Taylor expansion of $p_k(\epsilon)$ up to some order $K$ in $\epsilon$. This is because $f$ can occur with a rational prefactor which may have poles in $\epsilon$ of order $K$. This Taylor expansion is straightforward to compute recursively by expanding ${\partial \over \partial x_i} f$ and integrating (we know the constant terms by the previous discussion). The partial derivative ${\partial \over \partial x_i} f$ is simply a component of the total differential $d$ defined in $(\ref{BdRdifferential})$, which decreases the length and hence this gives an algorithm which terminates after finitely many steps, also described below. Note that in order to compute period integrals $(\ref{introM0nint})$, one only requires taking limits with respect to the final cubical variable $x_i$ for $i=n$. ### More general limits It can happen, for example when computing Feynman integrals, that one wants to take limits at more general divisors on $\overline{{\mathcal{M}}}_{0,n}$. The compactification of the standard cell $X_n$ (the closure of $X_n$ in $\overline{{\mathcal{M}}}_{0,n}$ for the analytic topology) is a closed polytope $$\overline{X}_n \subset \overline{{\mathcal{M}}}_{0,n}$$ which has the combinatorial structure of a Stasheff polytope. It can happen that one wants to compute limits at a (tangential) base point on the boundary of $\overline{X}_n$. An example is illustrated in the figure below in the case $n=5$, and where $\overline{X}_5$ is a pentagon. (-330,-10)[$x_1=0$]{} (-230,-10)[$x_1=1$]{} (-380,22)[$x_2=0$]{} (-380,120)[$x_2=1$]{} (-265,150)[$x_1x_2=1$]{} (5,80)[$\mathcal{E}$]{} (-50,128)[$z_1$]{} (-25,102)[$z_2$]{} The case of such limits can be dealt with using explicit local normal crossing coordinates on the boundary of $\overline{X}_n$ such as the dihedral coordinates $u_{ij}$ defined in [@Bro2]. One can show that any such limit is in fact a composition of regularised limits along divisors $x_{i_k} =1$ and $x_{i_k}=0$ in some specified (but not necessarily unique) order. This order can be determined from the combinatorics of the dihedral coordinates, and gives an algorithm to compute limits in this more general sense. For example, in the figure, the point $z_1$ is reached by taking the limit first as $x_2$ goes to $1$ and then $x_1$ goes to $1$; the point $z_2$ corresponds to the opposite order. The regularised limits of iterated integrals (such as $\mathrm{Li}_{1,1}(x,y)$) at $(1,1)$ along each path are different. Note that a path which approaches $(1,1)$ with a gradient which is strictly in between $0$ and $\infty$ corresponds to a limit point which is not equal to either $z_1$ or $z_2$ on $\mathcal{E}$ and could take us outside the realm of multiple zeta values. Finally, it is worth noting that one can imagine situations when one needs to take limits at points ‘at infinity’ corresponding to the case when, for example, some cubical coordinates $x_i$ go to infinity. This will not be discussed here. Fibrations ---------- The space $V({\mathcal{M}}_{0,n})$ is defined by a system of quadratic equations $(\ref{intcond})$ and its structure is hard to understand from this point of view. We will never need to actually solve the integrability equations $(\ref{intcond})$. A different description of $V({\mathcal{M}}_{0,n})$ comes from considering the morphism $$\begin{aligned} {\mathcal{M}}_{0,n} & \longrightarrow & {\mathcal{M}}_{0,n-1} \label{fibration} \\ (x_1,\ldots, x_{n-3}) & \mapsto & (x_1,\ldots, x_{n-4}) \nonumber\end{aligned}$$ which is obtained by forgetting the last cubical coordinate. It is a fibration, whose fiber over the point $(x_1,\ldots, x_{n-4})$ is isomorphic to the punctured projective line $$C_{n} = {\mathbb{P}}^1 \backslash \{ 0, (x_1\ldots x_{n-4})^{-1} , \ldots, x_{n-4}^{-1}, 1, \infty\}$$ with coordinate $x_{n-3}$. Let ${\mathcal{A}}_n={\mathcal{A}}({\mathcal{M}}_{0,n})$ denote the model for the de Rham complex on ${\mathcal{M}}_{0,n}$ defined earlier, and let ${\,{}^F \!\!\bar{\mathcal{A}}}_n= {\mathcal{A}}_{n}/{\mathcal{A}}_{n-1}$ denote the ${\mathbb{Q}}$-vector space of relative differentials. Denote the natural projection by $$\label{projectontoAf} \omega \mapsto \overline{\omega}: {\mathcal{A}}_{n} \rightarrow {\,{}^F \!\!\bar{\mathcal{A}}}_n$$ Using the representation of forms in cubical coordinates, we can choose a splitting $$\label{lambdadefn} \lambda_n \quad : \quad {\,{}^F \!\!\bar{\mathcal{A}}}_n \overset{\sim}{\rightarrow} {\,{}^F \!\!\mathcal{A}}_n \subseteq {\mathcal{A}}_{n}$$ which is defined explicitly in $(\ref{lambdaexplicit})$, and obtain a decomposition of ${\mathcal{A}}_{n-1}$-modules: $$\label{Aosproductdecomp} {\mathcal{A}}_{n} \cong {\mathcal{A}}_{n-1} \otimes {\,{}^F \!\!\bar{\mathcal{A}}}_n\ .$$ Armed with this decomposition, the quadratic relation $(\ref{quadrel})$ can be reinterpreted as a multiplication law on $1$-forms on the fiber: $$\begin{aligned} \label{Wedgedecomp} \mu_n \quad : \quad {\,{}^F \!\!\mathcal{A}}_n^1 \wedge {\,{}^F \!\!\mathcal{A}}_n^1 \longrightarrow {\mathcal{A}}^1_{n-1} \otimes {\,{}^F \!\!\mathcal{A}}^1_n \end{aligned}$$ which is used intensively in all computations. The product of two elements in ${\,{}^F \!\!\mathcal{A}}^1_n$ lies in ${\mathcal{A}}_n^2 \cong {\mathcal{A}}^2_{n-1} \oplus ({\mathcal{A}}^1_{n-1} \otimes {\,{}^F \!\!\bar{\mathcal{A}}}^1_n)$ since ${\,{}^F \!\!\bar{\mathcal{A}}}_n^2=0$. In fact, our choice of splitting $\lambda_n$ is such that the component of the previous isomorphism in ${\mathcal{A}}_{n-1}^2$ vanishes, which defines the map $(\ref{Wedgedecomp})$. [@Bro2] \[thmVdecomp\] The choice of map $\lambda_n$ gives a canonical isomorphism of algebras $$\label{Vdecomp} V({\mathcal{M}}_{0,n}) \cong V({\mathcal{M}}_{0,{n-1}}) \otimes V(C_n)\ ,$$ (which does not respect the coproducts on both sides) where $$V(C_n) = \bigoplus_{k\geq 0} ({\,{}^F \!\!\bar{\mathcal{A}}}^1_n)^{\otimes k}$$ is the ${\mathbb{Q}}$-vector space spanned by all words in ${\,{}^F \!\!\bar{\mathcal{A}}}^1_n$, equipped with the shuffle product. This gives a very precise description of the algebraic structure on $V({\mathcal{M}}_{0,n})$: by applying this theorem iteratively, every element of $V({\mathcal{M}}_{0,n})$ can be uniquely represented by a sum of tensors of words in prescribed alphabets. In order to go back and forth between the two representations on the left and right hand sides of $(\ref{Vdecomp})$ we have the symbol and unshuffle maps, defined as follows. 1. The *symbol map* is a homomorphism, which depends on the choice $(\ref{lambdadefn})$, $$\Psi: V(C_n) \longrightarrow V({\mathcal{M}}_{0,n})$$ which can be thought of as the map which takes a function defined on a fiber of the universal curve $C_n$ and extends it to a function on the entire moduli space ${\mathcal{M}}_{0,n}$. It is constructed as follows. One can define a Gauss-Manin connection, corresponding to ‘differentiation under an iterated integral’ which is a linear map $$\nabla : V(C_n) \longrightarrow {\mathcal{A}}^1_{n-1} \otimes V(C_n)$$ by the following recipe: lift words in ${\,{}^F \!\!\bar{\mathcal{A}}}_n$ to words in ${\mathcal{A}}_{n}$ via the map $\lambda_n$; then apply the usual internal differential of the bar construction in degree $0$ (all signs simplify since the $\omega_i$ are $1$-forms): $$\label{integrability} D [\omega_1 | \ldots | \omega_n] = (-1)^n\big( \sum_{i=1}^n [\omega_1 | \ldots | d\omega_i| \ldots | \omega_n] + \sum_{i=1}^{n-1} [\omega_1 | \ldots | \omega_i \wedge \omega_{i+1} | \ldots | \omega_n] \big)$$ and finally project all one-forms on the right-hand side to ${\,{}^F \!\!\bar{\mathcal{A}}}^1$ via the map $(\ref{projectontoAf})$ and project all two forms (namely, $d \omega_i$ and $\omega_i \wedge \omega_{i+1}$) onto ${\mathcal{A}}^1_{n-1} \otimes {\,{}^F \!\!\bar{\mathcal{A}}}^1$ via the decomposition $(\ref{Wedgedecomp})$. Pulling out all factors in ${\mathcal{A}}^1_{n-1}$ to the left gives the required formula for $\nabla$. The connection $\nabla$ can be promoted to a total connection $$\begin{aligned} \label{TotalConnection} \nabla_T: V(C_n) \longrightarrow {\mathcal{A}}^1_{n} \otimes V(C_n)\end{aligned}$$ by setting $\nabla_T= d- \nabla $, and identifying ${\mathcal{A}}^1_{n-1} \oplus {\,{}^F \!\!\bar{\mathcal{A}}}_n^1 \cong {\mathcal{A}}^1_{n}$ via the decomposition $(\ref{Aosproductdecomp})$. It is straightforward to show that in this context the total connection is flat $(\nabla_T^2=0)$. Finally, the symbol map is the unique linear map (necessarily a homomorphism) $$\label{SymbolMap} \Psi: V(C_n) \longrightarrow V({\mathcal{M}}_{0,n})$$ which satisfies the equation $$(id \otimes \Psi) \circ \nabla_T = d \circ \Psi.$$ This can be viewed as a recursive formula to compute the symbol map $\Psi$ since $\nabla_T$ strictly decreases the length of bar elements. Explicitly, it can be rewritten $$\Psi= \int (id \otimes \Psi) \circ \nabla_T$$ where the total primitive operator $\int$ was defined in §\[sectTotalPrimitives\]. 2. In the other direction, there is the *unshuffle* map which is a homomorphism of graded algebras $$\Phi: V({\mathcal{M}}_{0,n}) \overset{\sim}{\longrightarrow} V({\mathcal{M}}_{0,n-1}) \otimes V(C_n)$$ which is the inverse of the map $m({\mathrm{id}}\otimes \Psi) : V({\mathcal{M}}_{0,n-1}) \otimes V(C_n) \rightarrow V({\mathcal{M}}_{0,n}) $ (which we abusively denote simply by $\Psi$), where $m$ denotes multiplication. It can be computed as follows. Denote the natural map $$\begin{aligned} r: V({\mathcal{M}}_{0,n}) & \longrightarrow & V(C_n) \nonumber \\ {[}\omega_1 | \ldots | \omega_r] & \mapsto& [\overline{\omega}_1| \ldots | \overline{\omega}_r] \nonumber \end{aligned}$$ given by restriction of iterated integrals to the fiber induced by $(\ref{projectontoAf})$ component-wise on bar elements. Note that the map $\Psi$ has the property that $r\circ \Psi$ is the identity on $V(C_n)$. Recall the morphism $(\ref{fibration})$ from ${\mathcal{M}}_{0,n}$ to ${\mathcal{M}}_{0,n-1}$ defined in terms of cubical coordinates. The projection map $\pi : {\mathcal{A}}_n \rightarrow {\mathcal{A}}_{n-1}$ implied by the section $\lambda_n$ is given by sending first $dx_{n-3}$ to zero and then $x_{n-3}$ to zero. One can see that it is a homomorphism by inspection of the explicit equations in §\[sub:Arnold-relations\]: the product of two elements in ${}^F\!{\mathcal{A}}_n^1$ have no component in ${\mathcal{A}}^2_{n-1}$. It defines a homomorphism $$\pi: V({\mathcal{M}}_{0,n})\rightarrow V({\mathcal{M}}_{0,n-1})$$ and one easily verifies that the homomorphism $\Phi$ defined by $$\Phi( \xi) = (r \otimes \pi ) \circ \Delta$$ is an inverse to the symbol map $\Psi$. Alternatively, we can view ${\mathcal{M}}_{0,n-1}$ as being embedded in $\overline{{\mathcal{M}}}_{0,n}$ by identifying it with the divisor defined by $x_{n-3}=0$. An element of $V({\mathcal{M}}_{0,n})$ can be thought of as an iterated integral along a path from the unit tangential base point at the origin $0$ in cubical coordinates to a point $x=(x_1,\ldots, x_{n-3})$. It is the composition of a path from the unit tangential base point at $0$ to $(x_1,\ldots, x_{n-4})$ in the base ${\mathcal{M}}_{0,n-1}$, followed by a path in $C_n$ from the unit tangential base point at $x_{n-3}=0$ to $x$. Since composition of paths is dual to deconcatenation in $V({\mathcal{M}}_{0,n})$, this yields a geometric interpretation of the above formula for $\Phi$. Thus it is possible, via the symbol and unshuffle maps, to pass back and forth between a representation of an iterated integral on ${\mathcal{M}}_{0,n}$ as a symbol in $V({\mathcal{M}}_{0,n})$ or a product of words in $V(C_i)$’s. This gives a precise algorithmic equivalence between the two approaches described in §\[secthypversusmod\]. Representation as functions --------------------------- In order to represent elements of $V({\mathcal{M}}_{0,n})$ as functions (although in principle one never needs to do this) the simplest method is to apply the unshuffle map $\Phi$ defined above, which reduces to the problem of representing elements of $V(C_{k})$, for $4\leq k \leq n$ as functions. This is simply the case of computing iterated integrals in a single variable $x_{n-3}$, i.e. hyperlogarithms. $$\begin{aligned} V(C_n) &\longrightarrow & \hbox{Iterated integrals on } C_n \\ \nonumber [ \omega_1 | \ldots | \omega_k] & \mapsto & \int \omega_1 \ldots \omega_k \nonumber\end{aligned}$$ The iterated integrals on $C_n$ are normalised with respect to the tangential base point ${\partial \over \partial x_{n-3}}$ at $x_{n-3}=0$. They can be written as polynomials in $\log(x_{n-3})$ and explicit power series which were studied in the work of Lappo-Danilevsky [@Lap]. In this way, the iterated use of the unshuffle map reduces the expression of elements of $V({\mathcal{M}}_{0,n})$ as functions to a product of hyperlogarithms. These are well understood, and can be expressed in terms of multiple polylogarithms $$\mathrm{Li}_{n_1,\ldots, n_r}(x_1,\ldots, x_r) = \sum_{0<k_1<\ldots< k_r} {x_1^{k_1} \ldots x_r^{k_r} \over k_1^{n_1} \ldots k_r^{n_r}}$$ which can be computed to arbitrary accuracy by standard techniques [@Vol]. ‘Mixed’ primitives {#subMixedPrimitives} ------------------ Suppose that we have an element $\xi \in V({\mathcal{M}}_{0,n})$, and a one form $\omega \in {\,{}^F \!\!\mathcal{A}}_{n}^1$ which is only defined on the fiber. The mixed primitive is defined to be $$\omega \star \xi := \Psi \big( \int \omega \,\Phi(\xi)\big) \qquad \in V({\mathcal{M}}_{0,n})\ .$$ In other words, $\xi$ is viewed as an element of $V({\mathcal{M}}_{0,n-1})\otimes V(C_n)$ via the unshuffle map, then multiplied by $1\otimes \omega$ before computing its primitive $\int$ in $V(C_n)$ (which is simply given by left concatenation of forms in ${\,{}^F \!\!\mathcal{A}}^1$, as in §\[sectTotalPrimitives\]). Clearly, the map $\star$ is bilinear over ${\mathbb{Q}}$ and satisfies $$\label{starproperty1} \omega_0 \star \Psi([\omega_1 | \ldots | \omega_k]) = \Psi( [\omega_0 | \ldots | \omega_k])$$ for all $\omega_i \in {\,{}^F \!\!\bar{\mathcal{A}}}_{n}^1$. Furthermore, $\star$ is right-linear over $V({\mathcal{M}}_{0,n-1})$: $$\label{starproperty2} \omega \star ( b {\, \hbox{\rus x} \,}\xi) = b {\, \hbox{\rus x} \,}(\omega \star \xi)$$ for all $b\in V({\mathcal{M}}_{0,n-1})$, and $\xi \in V({\mathcal{M}}_{0,n})$, and $\star$ is uniquely determined by $(\ref{starproperty1})$, $(\ref{starproperty2})$ and $(\ref{Vdecomp})$. Evidently, one does not want to have to compute $\star$ by applying the unshuffling and symbol maps $\Phi$ and $\Psi$ which would be highly inefficient (and largely redundant). The approach we have adopted is more direct. Suppose that $\xi =\sum_I c_I [\omega_{i_1} | \ldots | \omega_{i_m}]$. As a first approximation to the mixed primitive $ \omega \star \xi $ take the element $$\xi_0=\sum_{I=(i_1,\ldots, i_m)} c_I [\lambda_n(\omega) | \omega_{i_1} | \ldots | \omega_{i_m}]$$ The projection of $\xi_0$ onto $V(C_n)$ coincides with that of $\omega \star \xi$, but $\xi_0$ does not satisfy the integrability condition $(\ref{intcond})$. The idea is to add correction terms $\xi_1, \ldots, \xi_k$ to $\xi_0$ so that the sum $\xi_0 + \ldots + \xi_k = \sum_J c'_J [ \eta_{j_1}| \ldots | \eta_{j_{m+1}}]$ satisfies the first $k$ integrability equations (with the notation of $(\ref{intcond})$) $$\sum_J c'_J [\eta_{i_1} | \ldots | \eta_{j_r} \wedge \eta_{j_{r+1}} | \ldots | \eta_{j_{m+1}}]=0 \qquad \hbox{ for } 1 \leq r \leq k$$ The correction term $\xi_{k+1}$ is obtained using the quadratic relations $\mu_n$ to expand out each wedge product $\omega_i \wedge \omega_j$ in the $k+1$th integrability equation, applied to $\xi_0+\ldots+ \xi_k$. The mixed primitive $\omega \star \xi$ is equal to the sum $\xi_0+ \ldots + \xi_{m}$ if $\xi$ is of length $m$. The precise details are described below. Feasibility and orders of magnitude ----------------------------------- By iterating theorem $(\ref{thmVdecomp})$ one obtains a formula for the dimension of all symbols on ${\mathcal{M}}_{0,n+3}$ in weight $N$: $$\label{dimformula} \sum_{N\geq 0}\, (\dim_{{\mathbb{Q}}} V({\mathcal{M}}_{0,n+3})_{N}) t^N = {1 \over (1-2t) (1-3t) \ldots (1-(n+1)t)}$$ This gives a coarse upper bound for the possible size of expressions which can occur during the integration process. At the initial step of integration, the integrand is of weight $0$ on a moduli space of high dimension ${\mathcal{M}}_{0,n+3}$, and at the final step, the integrand is of high weight on a moduli space of low dimension ${\mathcal{M}}_{0,4}$. The dimensions $(\ref{dimformula})$ peak somewhere in the middle of the computation. For example, for (the maximal weight part) of a period integral $(\ref{introM0nint})$ in five variables, one works in a sequence of vector spaces of dimension $20,125,285, 211,32$ (these are the dimensions of the spaces of functions after taking each primitive and before taking each limit). In the case of Feynman diagrams, one can estimate in advance (using the linear reduction algorithm) the number of marked points $n$ which will be required at each step of the integration to get a bound on the size of the computation. In practice, it seems that the limit of what is reasonable with current levels of computing power should be adequate to reach the ‘non-polylogarithmic’ boundary where amplitudes which are not periods of mixed Tate motives first start to appear. Computing on the moduli space {#sec:Computing-on-the} ============================= In this section we spell out the details of the above algorithms and present them in a version which is ready for implementation on a computer. As a proof of concept we implemented these algorithms in a Maple-based computer program. With this program we computed all examples below and all applications of section \[sec:Applications\]. For notational convenience let $m=n-3$ denote the number of cubical coordinates $x_i$ on ${\mathcal{M}}_{0,n}$. As bases for ${\mathcal{A}}^1_{n}$, ${\,{}^F \!\!\bar{\mathcal{A}}}^1_n$ and ${\,{}^F \!\!\mathcal{A}}^1_n$ we choose the sets of closed 1-forms $$\begin{aligned} \Omega_{m} & = & \left\{ \frac{dx_{1}}{x_{1}},\,...,\,\frac{dx_{m}}{x_{m}},\,\frac{d\left(\prod_{a\leq i\leq b}x_{i}\right)} {\prod_{a\leq i\leq b}x_{i}-1}\textrm{ for }1\leq a\leq b\leq m\right\} ,\\ \bar{\Omega}^F_{m} & = & \left\{ \frac{dx_{m}}{x_{m}},\,\frac{\left(\prod_{a\leq i\leq m-1}x_{i}\right)dx_{m}} {\prod_{a\leq i\leq m}x_{i}-1}\textrm{ for }1\leq a\leq m\right\} ,\\ \Omega_{m}^{F} & = & \left\{ \frac{dx_{m}}{x_{m}},\,\frac{d\left(\prod_{a\leq i\leq m}x_{i}\right)}{\prod_{a\leq i\leq m}x_{i}-1} \textrm{ for }1\leq a\leq m\right\} ,\\\end{aligned}$$ respectively. The isomorphism $\bar{{\,{}^F \!\!\mathcal{A}}_n}\overset{\lambda_{n}}{\cong}{\,{}^F \!\!\mathcal{A}}_n \subseteq {\mathcal{A}}_n$ of $(\ref{lambdadefn})$ is defined explicitly by $$\begin{aligned} \label{lambdaexplicit} \lambda_{n}\frac{dx_{m}}{x_{m}} & = & \frac{dx_{m}}{x_{m}},\\ \lambda_{n}\frac{\left(\prod_{a\leq i\leq m-1}x_{i}\right)dx_{m}}{\prod_{a\leq i\leq m}x_{i}-1} & = & \frac{d\left(\prod_{a\leq i\leq m}x_{i}\right)}{\prod_{a\leq i\leq m}x_{i}-1}\textrm{ for }1\leq a \leq m. \nonumber \end{aligned}$$ According to these chosen bases, we refer to the vector-spaces $V(C_{n}),\, V(\mathcal{M}_{0,n})$ by $V(\Omega^F_{m}),\, V(\Omega_{m})$ respectively. Iterated integrals are written as linear combinations of words $[\omega_{1}|...|\omega_{k}]$, whose letters are 1-forms in these sets. Note that $\Omega_m $ is the disjoint union of $\Omega_{m-1}$ and $\Omega_{m}^F$. Arnold relations {#sub:Arnold-relations} ---------------- With the above choices, the Arnold relations of $(\ref{Wedgedecomp})$ read explicitly: $$\begin{aligned} \frac{dx_{m}}{x_{m}}\wedge\frac{d\left(x_{i}...x_{m}\right)}{x_{i}...x_{m}-1} & = & -\sum_{k=i}^{m-1}\frac{dx_{k}}{x_{k}}\wedge\frac{d\left(x_{i}...x_{m}\right)}{x_{i}...x_{m}-1},\\ \frac{d\left(x_{j}...x_{m}\right)}{x_{j}...x_{m}-1}\wedge\frac{d\left(x_{i}...x_{m}\right)}{x_{i}...x_{m}-1} & = & \frac{d\left(x_{i}...x_{j-1}\right)}{x_{i}...x_{j-1}-1}\wedge\left(\frac{d\left(x_{i}...x_{m}\right)}{x_{i}...x_{m}-1}-\frac{d\left(x_{j}...x_{m}\right)}{x_{j}...x_{m}-1}\right)-\sum_{k=i}^{j-1}\frac{dx_{k}}{x_{k}}\wedge\frac{d\left(x_{i}...x_{m}\right)}{x_{i}...x_{m}-1}\end{aligned}$$ for $1\leq i\leq j\leq m.$ For the implementation on a computer, it is efficient to generate these equations to a desired number of variables once and for all, and to store them as a look-up table since they are used very frequently by the algorithms below. The splitting of theorem \[thmVdecomp\] is realised by a certain application of the Arnold relations. We define an auxiliary map $\rho_{i}$ by the following operations. For a word $\xi=[\omega_{1}|...|\omega_{k}]$ with letters in $\bar{\Omega}^F_{m}$ and some $1\leq i<k$ we consider the neighbouring letters $\omega_i | \omega_{i+1}$ and consider the wedge-product of their images in $\Omega^F_{m}$. By the corresponding Arnold relation, we express this product as a $\mathbb{Q}$-linear combination of wedge-products, with one factor in the base $\Omega_{m-1}$ and one in the fiber $\Omega^F_{m}$. We replace the letters $\omega_i | \omega_{i+1}$ in $\xi$ by the factor in $\Omega^F_{m}$ and pull the base-term in $\Omega_{m-1}$ and rational pre-factors out of the word. In summary, this defines the auxiliary map $$\rho_{i}:\, V\left(\bar{\Omega}^F_{m}\right)\rightarrow\Omega_{m-1}\otimes V\left(\bar{\Omega}^F_{m}\right)$$ by $$\rho_{i}\left[a_{1}|...|a_{k}\right]=\sum_{j}c_{j}\eta_{j}\otimes\left[a_{1}|...|a_{i-1}| \overline{\alpha}_{j}|a_{i+2}|...|a_{k}\right]$$ where $\eta_{j}\in\Omega_{m-1},\,\alpha_{j}\in\Omega_{m}^{F},\, c_{j}\in\mathbb{Q}$ are determined by the Arnold relation $$\lambda_{n}a_{i}\wedge\lambda_{n}a_{i+1}=\sum_{j}c_{j}\eta_{j}\wedge\alpha_{j}.$$ Note that these are the same operations as in our definition of the Gauss-Manin connection $\nabla$ above, which we obtain by summing the $\rho_i$ over $i$. This is because the first sum on the right-hand side of $(\ref{integrability})$ vanishes in our set-up, as all our 1-forms are closed, and the operations on the terms of the second sum correspond to the definition of $\rho_i$. \[exampleArnoldn=5\] For $n=5, m=2$ we have the Arnold relations $$\begin{aligned} \frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}\wedge\frac{dx_{2}}{x_{2}} & = & \frac{dx_{1}}{x_{1}}\wedge\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1},\\ \frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}\wedge\frac{dx_{2}}{x_{2}-1} & = & \left(\frac{dx_{1}}{x_{1}}-\frac{dx_{1}}{x_{1}-1}\right)\wedge\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}+\frac{dx_{1}}{x_{1}-1}\wedge\frac{dx_{2}}{x_{2}-1}.\end{aligned}$$ For the words $\kappa=\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}-1}\right],$ $\xi=\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}}|\frac{dx_{2}}{x_{2}-1}\right]$ in $V(\bar{\Omega}^F_{2})$ we compute $$\begin{aligned} \rho_{1}\kappa & = & \left[\frac{dx_{1}}{x_{1}}\right]\otimes\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}\right] -\left[\frac{dx_{1}}{x_{1}-1}\right]\otimes\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}\right]+\left[\frac{dx_{1}}{x_{1}-1}\right] \otimes\left[\frac{dx_{2}}{x_{2}-1}\right],\\ \rho_{1}\xi & = & \left[\frac{dx_{1}}{x_{1}}\right]\otimes\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}-1}\right],\\ \rho_{2}\xi & = & 0.\end{aligned}$$ The symbol map {#sub:The-symbol-map} -------------- Both the total connection and the symbol map can be computed conveniently by use of the maps $\rho_{i}$. The total connection (see $(\ref{TotalConnection})$) is computed as $$\nabla_{T}\left[a_{1}|...|a_{k}\right]=d\left[a_{1}|...|a_{k}\right]-\sum_{1\leq i<k}\rho_{i}\left[a_{1}|...|a_{k}\right]$$ where (by $(\ref{BdRdifferential})$) $$d\left[a_{1}|...|a_{k}\right]=a_{1}\otimes\left[a_{2}|...|a_{k}\right].$$ The symbol map $\Psi$ (see $(\ref{SymbolMap})$) is applied to a word in $V(\bar{\Omega}^F_{m})$ by the recursive algorithm $$\begin{aligned} \Psi\left(\left[a_{i}\right]\right) & = & \left[\lambda_{n}\left(a_{i}\right)\right],\nonumber \\ \Psi\left(\left[a_{i_{1}}|a_{i_{2}}|...|a_{i_{k}}\right]\right) & = & \lambda_{n}\left(a_{i_{1}}\right)\sqcup\Psi\left(\left[a_{i_{2}}|...|a_{i_{k}}\right]\right)-\sum_{1\leq i<k}\sqcup\left(\left(\textrm{id}\otimes\Psi\right)\rho_{i}\left[a_{i_{1}}|...|a_{i_{k}}\right]\right),\;1<k,\label{eq:symbol map-1}\end{aligned}$$ where $\xi_1\sqcup \xi_2\equiv\sqcup(\xi_1\otimes \xi_2)$ denotes the concatenation of two words $\xi_1,\, \xi_2.$ Note that on the right hand side of $(\ref{eq:symbol map-1})$ the map $\Psi$ acts on words of length $k-1.$ Making use of the relations derived in example \[exampleArnoldn=5\], we compute $$\begin{aligned} \Psi\left(\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}}|\frac{dx_{2}}{x_{2}-1}\right]\right) & = & \left[\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}}|\frac{dx_{2}}{x_{2}-1}\right]-\left[\frac{dx_{1}}{x_{1}}\right]\sqcup\Psi\left(\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}-1}\right]\right)\\ & = & \left[\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}}|\frac{dx_{2}}{x_{2}-1}\right]-\left[\frac{dx_{1}}{x_{1}}|\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}-1}\right]\\ & & -\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}-1}|\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}\right]+\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}-1}|\frac{dx_{2}}{x_{2}-1}\right]\\ & & +\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}}|\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}\right].\end{aligned}$$ The map $\Psi$ is defined such that for any $\xi\in V\left(\bar{\Omega}^F_{m}\right)$ we have $D\Psi\left(\xi\right)=0$ and therefore $\Psi\left(\xi\right)\in V\left(\Omega_{m}\right).$ The vector space $V\left(\Omega_{m}\right)$ is generated, over $V(\Omega_{m-1})$, by the image of $V\left(\bar{\Omega}^F_{m}\right)$ under $\Psi.$ We furthermore note the property $$\Psi\left(\xi_{1}{\, \hbox{\rus x} \,}\xi_{2}\right)=\Psi(\xi_{1}){\, \hbox{\rus x} \,}\Psi(\xi_{2})$$ for any $\xi_{1},\,\xi_{2}\in V\left(\bar{\Omega}^F_{n}\right).$ A slightly different algorithm for $\Psi$ in terms of differentiation under the integral was already given in [@Bog1]. For related constructions, also see references [@Duh1; @Gon1; @Gon2]. In section \[sec:Feynman-type-integrals\] we will make use of $\Psi$ as a part of a procedure to map hyperlogarithms in Schwinger parameters to multiple polylogarithms of cubical variables. We expect the map $\Psi$ also to be useful in different contexts such as [@Druetal]. Primitives {#sub:Primitives} ---------- Let $\omega\in\bar{\Omega}^F_m$ and let $\xi=\sum_{I}c_{I}\left[\omega_{i_{1}}|...|\omega_{i_{k}}\right]$ be an iterated integral in $V(\Omega_{m})$. In subsection \[subMixedPrimitives\], we discussed the strategy of building up the mixed primitive $\omega\star\xi$ by naive left-concatenation of the form $\omega$ to the word $\xi$, yielding $$\label{xi0ofmixedprimitives} \sum_{I}c_{I}\left[\lambda_{n}(\omega)|\omega_{i_{1}}|...|\omega_{i_{k}}\right],$$ and the addition of correction terms until the resulting combination satisfies the integrability condition of $(\ref{intcond})$. For the explicit computation of the correction terms, let us introduce some auxiliary notation. For all $0\leq i < k$ let $C_{i}\left(\Omega_{m}\right)_k= \Omega_{m-1}^{\otimes i} \otimes \Omega_m^F \otimes \Omega_m^{\otimes(k-i-1)}$ be the $\mathbb{Q}$-vector space of words of length $k$ with letters in $\Omega_{m}$, whose first $i$ letters, counted from the left, are in the base $\Omega_{m-1}$, and whose $(i+1)^{\mathrm{th}}$ letter is in the fiber $\Omega_m^F$. The members of these auxiliary sets of words do not necessarily stand for homotopy invariant iterated integrals. We define the auxiliary maps $$\star_{i}:\, C_{i-1}\left(\Omega_{m}\right)_k\rightarrow C_{i}\left(\Omega_{m}\right)_k$$ for $i<k$ by the following recipe $$\begin{aligned} \star_{i}[a_{1}|...|a_{i-1}|a_{i}|a_{i+1}|...|a_{k}] & = & [a_{1}|...|a_{i-1}|a_{i+1}|a_{i}|...|a_{k}]\qquad \textrm{ if } a_{i+1}\in\Omega_{m-1} , \nonumber \\ \star_{i}[a_{1}|...|a_{i-1}|a_{i}|a_{i+1}|...|a_{k}] & = & -\sum_{j}c_{j}[a_{1}|...|a_{i-1}|\eta_{j}|\alpha_{j}|a_{i+2}|...|a_{k}]\qquad \textrm{ if }a_{i+1}\in\Omega_{m}^{F},\, \label{eq:star i}\end{aligned}$$ where the forms $\eta_{j}\in\Omega_{m-1},\,\alpha_{j}\in\Omega_{m}^{F}$ and constants $\, c_{j}\in\mathbb{Q}$ are determined by an Arnold relation $$a_{i}\wedge a_{i+1}=\sum_{j}c_{j}\eta_{j}\wedge\alpha_{j}.$$ Note that indeed, in each word on the right-hand side of $(\ref{eq:star i})$ the 1-forms in the first $i$ positions are in $\Omega_{m-1}$ and the form in the $(i+1)$-th position is in $\Omega_{m}^{F}$. This procedure can be iterated. Since $\ref{xi0ofmixedprimitives}$ lies in $C_{0}\left(\Omega_{m}\right)_ {k+1}$, we repeatedly apply $\star_{\bullet}$ to obtain the following formula for the mixed primitive $$\begin{aligned} \label{eq:Primitive} \omega\star[a_{1}|...|a_{k}]=(1+\star_{1}+\star_{2}\star_{1}+...+\star_{k}...\star_{1})[\lambda_{m}(\omega)|a_{1}|...|a_{k}].\end{aligned}$$ The construction satisfies the relations $(\ref{starproperty1})$ and $(\ref{starproperty2})$. We consider the 1-form $\omega = \frac{dx_2}{x_2}$, the iterated integral $$\begin{aligned} \xi = \Psi \left(\left[ \frac{x_1 d(x_2)}{x_1 x_2 -1} | \frac{dx_2}{x_2} \right] \right) = \left[ \frac{d(x_1 x_2)}{x_1 x_2 -1} | \frac{dx_2}{x_2} \right] - \left[ \frac{dx_1}{x_1} | \frac{d(x_1 x_2)}{x_1 x_2 -1} \right],\end{aligned}$$ and the concatenation $$\begin{aligned} \xi_0 & = & \lambda_2 (\omega) \sqcup \xi = \left[ \frac{dx_2}{x_2} | \frac{d(x_1 x_2)}{x_1 x_2 -1} | \frac{dx_2}{x_2} \right] - \left[ \frac{dx_2}{x_2} | \frac{dx_1}{x_1} | \frac{d(x_1 x_2)}{x_1 x_2 -1} \right].\end{aligned}$$ Following $(\ref{eq:Primitive})$, we compute the primitive $$\begin{aligned} \omega \star \xi = \xi_0 + \xi_1 + \xi_2\end{aligned}$$ where $\xi_1 = \star_1 \xi_0$ and $\xi_2 = \star_2 \star_1 \xi_0$. We obtain $$\begin{aligned} \\ \xi_1 & = & \left[ \frac{dx_1}{x_1} | \frac{d(x_1 x_2)}{x_1 x_2 -1} | \frac{dx_2}{x_2} \right] - \left[ \frac{dx_1}{x_1} | \frac{dx_2}{x_2} | \frac{d(x_1 x_2)}{x_1 x_2 -1} \right], \\ \xi_2 & = & -2\left[ \frac{dx_1}{x_1} | \frac{dx_1}{x_1} | \frac{d(x_1 x_2)}{x_1 x_2 -1} \right]\end{aligned}$$ by use of the Arnold relations given in the example of section \[sub:Arnold-relations\]. Limits {#sub:Limits} ------ We consider limits at $x_{l}=u$, $l\in\{1,\,...,\, m\}$ where $u\in\{0,\,1\}.$ By definition, any $\xi\in V\left(\Omega_{m}\right)$ vanishes along $x_{l}=0.$ Limits at $0$ and $1$ are computed as follows. As in the previous sections, let $\mathcal{Z}$ be the $\mathbb{Q}$-vector space of multiple zeta values. It was shown in [@Bro2] that for any $\xi\in V\left(\Omega_{m}\right)$ the limits $\lim_{x_{l}\rightarrow 1}\xi$ are $\mathcal{Z}$-linear combinations of elements of $V\left(\Omega_{m-1}\right) $ (after a possible renumbering of the cubical coordinates: $(x_{l+1},\ldots, x_m) \mapsto (x_l,\ldots, x_{m-1}$).) Our algorithm for the computation of limits proceeds in two steps: - Expand the function $\xi$ at $x_{l}=u$ as a polynomial in $\log(x_{l}-u)$, whose coefficients are power series in $x_{l}-u$, and - Evaluate the constant term (coefficient of $\log(x_l-u)^0$) at $x_{l}=u.$ The series expansion is the non-trivial part in this computation while the evaluation of the series is immediate. Let $\textrm{Exp}_{x_{l}=u}\xi(x_{l})$ denote the expansion of the function $\xi(x_{l})$ at $x_{l}=u.$ We compute the expansion recursively as $$\label{eq:Expansion} \textrm{Exp}_{x_{l}=u}\xi(x_{l})=\textrm{Reg}_{x_{l}=u}\xi(x_{l})+\int dx'_{l}\,\textrm{Exp}_{x'_{l}=u}\frac{\partial}{\partial x'_l}\xi(x'_{l}).$$ where the integral on the right is the regularised integral from the tangential base point ${\partial \over \partial x_l}$ at $x_l=u$ to $x_l$, or equivalently, is an indefinite integral in $x_l$ whose constant of integration is fixed by declaring that its regularised limit at $x_l=u$ vanishes. Note that if $\xi(x_{l})$ is a linear combination of words of length $k$, then in the integrand on the right-hand side of $(\ref{eq:Expansion})$, $\textrm{Exp}_{x'_{l}=u}$ is computed on words of length $k-1$. Rational prefactors are trivially expanded as power series in $x_l=u$ also. The notation $\textrm{Reg}_{x_{l}=u}\xi(x_{l})$ stands for the operation of taking the regularised limit of $\xi$ at $x_{l}=u.$ For $u=0$ we define $\textrm{Reg}_{x_{l}=0}$ to be the identity-map on terms of weight $0$ and $$\textrm{Reg}_{x_{l}=0}\xi(x_{l})=0$$ for $\xi(x_{l})$ with all terms of weight greater than $0$. For $u=1$ regularised limits are defined and computed in the remainder of this subsection. Let us start by computing regularised limits of iterated integrals in only one variable and then extend to the $n$-variable case. We consider $\Omega_{1}=\left\{ \frac{dx_{1}}{x_{1}},\,\frac{dx_{1}}{x_{1}-1}\right\} $ and for $\xi\in V\left(\Omega_{1}\right)$ we use a simplified notation where in each word we symbolically replace $\frac{dx_{1}}{x_{1}}\rightarrow0$ and $\frac{dx_{1}}{x_{1}-1}\rightarrow1$ and multiply the word with $(-1)^{s}$ where $s$ is the number of 1-forms $\frac{dx_{1}}{x_{1}-1}.$ Following [@Bro1] we define the map $$\textrm{Reg}_{x_{1}=1}:\, V\left(\Omega_{1}\right)\rightarrow\mathcal{Z}$$ by the following relations for different cases of words $\xi=\left[a_{1}|...|a_{k}\right],\, a_{i}\in\{0,\,1\},\, i=1,\,...,\, k$: - Case 1: If all letters are equal, $a_{1}=a_{2}=...=a_{k}$, we have $$\textrm{Reg}_{x_{1}=1}\left[a_{1}|...|a_{k}\right]=0.$$ - Case 2: If the word begins with 0 and ends with 1 (from left to right), we have $$\textrm{Reg}_{x_{1}=1}[\underbrace{0|...|0|1|}_{n_{r}}...|1|\underbrace{0|...|0|1}_{n_{1}}]=\zeta(n_1,\,...,\, n_{r})\textrm{ for }n_{r}\geq2,\, n_{i}\geq1,\, n_{1}+...+n_{r}=k.$$ - Case 3: If the word begins in 1 we apply the relation $$\textrm{Reg}_{x_{1}=1}\left[a_{1}|...|a_{k}\right]=\textrm{Reg}_{x_{1}=1}\left[1-a_{k}|...|1-a_{1}\right]$$ which is also true in all other cases. - Case 4: If the word ends with 0 we use the relation $$\textrm{Reg}_{x_{1}=1}[\underbrace{0|...|0|1|}_{n_{1}}...|1|\underbrace{0|...|0|1}_{n_{r}}|\underbrace{0|...|0}_{q}]=$$ $$(-1)^{q}\sum_{i_{1}+...+i_{r}=q}\left(\begin{array}{c} n_{1}+i_{1}-1\\ i_{1} \end{array}\right)...\left(\begin{array}{c} n_{r}+i_{r}-1\\ i_{r} \end{array}\right)\textrm{Reg}_{x_{1}=1}[\underbrace{0|...|0|1|}_{n_{1}+i_{1}}...|1|\underbrace{0|...|0|1}_{n_{r}+i_{r}}],\label{eq:case 4}$$ where $q,\, n_{1},\,...,\, n_{r}\geq1.$ By these relations, implementing the well-known shuffle-regularization, the regularized value of any $\xi\in V\left(\Omega_{1}\right)$ can be expressed as a $\mathbb{Q}$-linear combination of expressions as in case 2, which are multiple zeta values. We consider $\xi=\left[\frac{dx_{1}}{x_{1}-1}|\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}}\right]$ which in short-hand notation reads $\xi=-[1|0|0]$ and falls into the above case 4. By $(\ref{eq:case 4})$ we have $\textrm{Reg}_{x_{1}=1}(-[1|0|0])=\textrm{Reg}_{x_{1}=1}(-[0|0|1])$ and obtain by case 2: $$\textrm{Reg}_{x_{1}=1}\xi=-\zeta(3).$$ Now we extend the definition of regularized limits to $V\left(\Omega_{m}\right).$ Let us first define the auxiliary restriction maps $$\begin{aligned} R_{x_{l}}:\, V\left(\Omega_{m}\right) & \rightarrow & V\left(\Omega_{1}\right)\end{aligned}$$ by $$R_{x_{l}}\xi=\xi|_{dx_{i}=0,\, x_{i}=0\textrm{ for all }i\in\{1,\,...,\, m\},\, i\neq l}\label{eq:restriction R}$$ and $$L_{x_{l}}:\, V\left(\Omega_{m}\right)\rightarrow V\left(\Omega_{m-1}\right)$$ by $$L_{x_{l}}\xi=\xi|_{dx_{l}=0,\, x_{l}=1}.\label{eq:restriction L}$$ and relabelling cubical coordinates as mentioned above. Note that the map $R_{x_{l}}$ is the projection onto words all of whose 1-forms are $\frac{dx_l}{x_l}$ and $\frac{dx_l}{x_l-1}$. These maps play a similar role as the restrictions $E_\gamma$ in section \[sectbardeRham\]. The map $R_{x_{l}}$ restricts the iterated integral to the straight line from the origin to $1_l$ (called $\gamma_2$ in section \[sectbardeRham\]) and $L_{x_{l}}$ restricts to the divisor of $\overline{{\mathcal{M}}}_{0,n}$ defined by $x_l=1$ (in which $\gamma_1$ of section \[sectbardeRham\] lives). According to $(\ref{convolution})$, we take the deconcatenation co-product $\Delta$ of $\xi\in V\left(\Omega_{m}\right)$ and apply $L_{x_{l}}$ and $R_{x_{l}}$ to the left and right part of the tensor product respectively. The right-hand side of the tensor product is then in $V\left(\Omega_{1}\right)$ and we apply the above map of regularized values to this part. In summary, we extend the definition of regularized values to $$\textrm{Reg}_{x_{l}=1}:\, V\left(\Omega_{m}\right)\rightarrow\mathcal{Z}\otimes V\left(\Omega_{m-1}\right)$$ by $$\label{eq:Reg} \textrm{Reg}_{x_{l}=1}\xi=m\left(L_{x_{l}}\otimes \textrm{Reg}_{x_{l}=1} R_{x_{l}}\right)\circ \Delta\xi.$$ This completes our algorithm for computing limits of $\xi\in V\left(\Omega_{m}\right)$ at $x_{l}=0,\,1.$ We consider the iterated integral $$\begin{aligned} \xi & = & \Psi\left(\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}}|\frac{dx_{2}}{x_{2}-1}\right]\right)\\ & = & \left[\frac{d(x_{1}x_{2})}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}}|\frac{dx_{2}}{x_{2}-1}\right]-\left[\frac{dx_{1}}{x_{1}}|\frac{d(x_{1}x_{2})}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}-1}\right]-\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}-1}|\frac{d(x_{1}x_{2})}{x_{1}x_{2}-1}\right]\\ & & +\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}-1}|\frac{dx_{2}}{x_{2}-1}\right]+\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}}|\frac{d(x_{1}x_{2})}{x_{1}x_{2}-1}\right]\\ & \in & V\left(\Omega_{2}\right).\end{aligned}$$ In this case, the only contibutions to the limit at $x_2=1$ are given by the term $\textrm{Reg}_{x_2 =1} \xi (x_2)$ of $(\ref{eq:Expansion})$, which we compute by use of $(\ref{eq:Reg})$. The coproduct of $\xi$ involves 20 terms, most of which vanish after applying $L_{x_2}$ to the left and $R_{x_2}$ to the right part. From the non-vanishing terms we obtain $$\begin{aligned} \textrm{Reg}_{x_2 =1} \xi (x_2) & = & m \left( \left[ \frac{dx_1}{x_1 -1} \right] \otimes \textrm{Reg}_{x_2 =1} \left[ \frac{dx_2}{x_2} | \frac{dx_2}{x_2 -1} \right] - \left[ \frac{dx_1}{x_1} | \frac{dx_1}{x_1 -1} \right] \otimes \textrm{Reg}_{x_2 =1} \left[ \frac{dx_2}{x_2 -1} \right] \right. \\ & & + \left[ \frac{dx_1}{x_1} | \frac{dx_1}{x_1} | \frac{dx_1}{x_1-1} \right] \otimes 1 - \left[ \frac{dx_1}{x_1} | \frac{dx_1}{x_1-1} | \frac{dx_1}{x_1-1} \right] \otimes 1 \\ & & \left.+\left[ \frac{dx_1}{x_1} | \frac{dx_1}{x_1-1} \right] \otimes \textrm{Reg}_{x_2 =1} \left[ \frac{dx_2}{x_2 -1} \right] \right).\end{aligned}$$ Due to $$\begin{aligned} \textrm{Reg}_{x_2 =1} \left[ \frac{dx_2}{x_2 -1} \right]=0 \textrm{ \ \ and \ \ } \textrm{Reg}_{x_2 =1} \left[ \frac{dx_2}{x_2} | \frac{dx_2}{x_2 -1} \right] = -\zeta(2)\end{aligned}$$ or by cancellation of the second and fifth terms, we obtain the limit $$\begin{aligned} \lim_{x_{2}\rightarrow1}\xi=\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}-1}\right]-\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}-1}|\frac{dx_{1}}{x_{1}-1}\right]-\zeta(2)\left[\frac{dx_{1}}{x_{1}-1}\right].\end{aligned}$$ Feynman-type integrals {#sec:Feynman-type-integrals} ====================== In this section we consider finite integrals derived from (linearly reducible, unramified) Feynman integrals. We present an algorithm to map such integrals to hyperlogarithms in cubical variables (corresponding to the morphism $X\rightarrow \overline{{\mathcal{M}}}_{0,n+1}$ in the diagram \[Square\]). The integration over one chosen Schwinger parameter maps to the integration over one cubical variable. Then this integration can be computed by the algorithms of section \[sec:Computing-on-the\]. After the integration, as a preparation for the integration over a next Schwinger parameter, we map back to iterated integrals in Schwinger parameters. Schwinger parameters -------------------- In dimensional regularization, scalar Feynman integrals of Feynman graphs $G$ with $N$ edges and loop-number $L,$ can be written in the Feynman parametric form $$I_{G}(D)=\frac{\Gamma(\nu-LD/2)}{\prod_{j=1}^{n}\Gamma(\nu_{j})}\int_{\alpha_{j}\geq0}\delta\left(1-\alpha_{N}\right)\left(\prod_{j=1}^{N}d\alpha_{j}\alpha_{j}^{\nu_{j}-1}\right)\frac{\mathcal{U}_{G}^{\nu-(L+1)D/2}}{\mathcal{F}_{G}{}^{\nu-LD/2}},$$ where $\nu=\sum_{i=1}^{N}\nu_{i}$ is the sum of powers of Feynman propagators and $D\in\mathbb{C}$. We refer to the variables $\alpha_{1},\,...,\,\alpha_{N}$ as Schwinger parameters and the above integration is over the positive range of each of these variables. The functions $\mathcal{U}_{G}$ and $\mathcal{F}_{G}$ are the first and second Symanzik polynomials respectively. They are polynomials in the Schwinger parameters and $\mathcal{F}_{G}$ is furthermore a polynomial of kinematical invariants, which are quadratic functions of particle masses and external momenta of $G.$ For more details we refer to [@Itz; @Nak; @Bog2]. Assume that we want to compute $I_{G}(2n)$ for some $n\in\mathbb{N}.$ There are different scenarios in which our algorithms may be useful. In the simplest case, the integral $I_{G}(2n)$ is finite and we may attempt to compute it without further preparative steps. If $I_{G}(2n)$ is divergent there may be a $n\neq m\in\mathbb{N}$ such that $I_{G}(2m)$ is finite and the method of [@Tar1; @Tar2] may provide useful relations between $I_{G}(2n)$ and $I_{G}(2m).$ These relations however may involve further integrals to be computed. The method of [@Bro3] allows for a subtraction of UV divergent contributions by a renormalization procedure on the level of the integrand. Alternatively, for a possibly UV and IR divergent integral, we may attempt to expand $I_{G}$ as $$I_{G}=\sum_{j=-2L}^{\infty}c_{j}\epsilon^{j}$$ where $\epsilon=(2n-D)/2$ and the $c_{j}$ are finite integrals. In principle such an expansion can be computed by sector decomposition [@Bin], however in this case, a use of our algorithms may be prohibited by the type of polynomials appearing in the integrands of the resulting $c_{j}.$ Recently, an alternative approach, where the latter polynomials are given by Symanzik polynomials of $G$ and its minors was suggested in [@Pan2]. Let us assume that these or alternative methods have led us to an integral over the positive range of Schwinger parameters where the integrand is of the form $$\label{hypIntegrand} f(\alpha_{1},\,...,\,\alpha_{N})=\frac{\prod_{Q_{i}\in\mathcal{Q}}Q_{i}^{\delta_{i}}\textrm{ hyperlogarithms}({P_{i}})}{\prod_{P_{i}\in\mathcal{P}}P_{i}^{\beta_{i}}}$$ where all $\delta_{i},\,\beta_{i}\in\mathbb{N}_{0}$ and where $\mathcal{P}=\{P_{1},\,...,\, P_{r}\}$ and $\mathcal{Q}=\{Q_{1},\,...,\, Q_{q}\}$ are finite sets of irreducible polynomials in Schwinger parameters. We assume furthermore that all $P_i$ are homogeneous and positive or negative definite. This is the case for all Symanzik polynomials in the Euclidean momentum region and in the massless case, and also for the polynomials arising from their linear reduction in a large class of situations. This simplifying assumption allows us to apply the particular change of variables constructed below. However, the general method is not restricted to this case. A more precise description of the numerator is given below. For our algorithms to be applicable we furthermore have to assume that there is an ordering on the Schwinger parameters such that the set $\mathcal{P}$ is linearly reducible and unramified [@Bro5; @Bro4]. In the following let $\alpha_N, \alpha_{N-1},...,\alpha_1$ be such a fixed ordering. From Schwinger parameters to cubical variables\[sub:From-Feynman-parameters\] ----------------------------------------------------------------------------- In the following, we transform a given integrand $f$ of the type given by $(\ref{hypIntegrand})$ to an integrand in cubical variables. According to our fixed ordering on the Schwinger parameters, let $\alpha_N$ be the parameter to be integrated out in the present step. Linear reducibility implies that the polynomials in $\mathcal{P}$ are of degree at most 1 in $\alpha_{N},$ while there are no implications for $\mathcal{Q}.$ We write $\mathcal{P}=\mathcal{P}_{N}\cup\mathcal{P}_{\backslash N}$ where $\mathcal{P}_{N}\subset\mathcal{P}$ is the subset of polynomials linear in $\alpha_{N}$ and $\mathcal{P}_{\backslash N}\subset\mathcal{P}$ is the set of polynomials independent of $\alpha_{N}.$ Let us fix the numbering on the $P_{i}$ such that $\mathcal{P}_{N}=\left\{ P_{1},\,...,\, P_{n}\right\} $ with $n\leq r.$ We also write the set of all polynomials $Q_{i}$ as $\mathcal{Q}=\mathcal{Q}_{N}\cup\mathcal{Q}_{\backslash N}$ where the polynomials in $\mathcal{Q}_{N}$ depend on $\alpha_{N}$ and the ones in $\mathcal{Q}_{\backslash N}$ do not. Now let us be more specific about the functions occurring in the numerator of $(\ref{hypIntegrand})$. We write $L_{w}(\alpha_{N})$ for a hyperlogarithm in $\alpha_{N},$ given by a word $w$ in differential 1-forms in the alphabet $$\Omega_{N}^{\textrm{Feynman}}=\left\{ \frac{d\alpha_{N}}{\alpha_{N}},\,\frac{d\alpha_{N}}{\alpha_{N}-\rho_{i}}\textrm{ where }\rho_{i}= -\frac{ P_{i}|_{\alpha_{N}=0} }{\frac{\partial P_{i}}{\partial \alpha_{N}}}\textrm{ for }i=1,\,...,\, n\right\} .\label{eq: Omega Feyn}$$ Here $\rho_{i}$ is a rational function such that $P_{i}$ vanishes for $\alpha_N=\rho_i$. Throughout this section, we shall assume the Feynman integral we are considering is linearly reducible and unramified. The condition for being unramified was defined in [@Bro5], definition 16, and discussed in [@Bro4], §9.3. It implies in particular that if $\rho_i$ is a constant independent of all $\alpha_i$, then it must be equal to $0$ or $-1$. We assume as an induction hypothesis that the functions in the numerator of the integrand are of a certain type. We will see in section \[sub:Back-to-Feynman\] that this assumption will be satisfied after integration and will be the starting point for the next integration. The numerator of the integrand $f$ is assumed to be a linear combination of hyperlogarithms in $\alpha_{N}:$ $$\textrm{numerator}(f)=\sum_{k}a_{k}b_{k}(\alpha_{N})L_{w_{k}}(\alpha_{N}),\label{eq:Feyn integrand hyperlog}$$ where the $w_{k}$ are words in the alphabet $\Omega_{N}^{\textrm{Feynman}}$ and where we denote the $\alpha_{N}$-dependent and $\alpha_{N}$-independent factor of the $k$-th coefficient by $b_{k}(\alpha_{N})$ and $a_{k}$ respectively. The $\alpha_{N}$-dependent factor $b_{k}(\alpha_{N})$ is a product of $Q_{i}\in\mathcal{Q}_{N}$ while the $\alpha_{N}$-independent factor $a_{k}$ is allowed to be a product of $Q_{i}\in\mathcal{Q}_{\backslash N}$ and of hyperlogarithms which do not depend on $\alpha_{N}.$ As $\alpha_{N}$-independent factors of the numerator remain unchanged in the integration procedure, we restrict our attention to integrals of the type $$\label{eq:LIntegral} \int_{0}^{\infty}d\alpha_{N}f(\alpha_{1},\,...,\,\alpha_{N})=\int_{0}^{\infty}d\alpha_{N}\frac{\prod_{Q_{i}\in\mathcal{Q}_{N}} Q_{i}^{\delta_{i}}L_{w}(\alpha_{N})}{\prod_{P_{i}\in\mathcal{P}}P_{i}^{\beta_{i}}}.$$ Let us now express the integral of $(\ref{eq:LIntegral})$ as an integral over cubical coordinates such that the algorithms of section \[sec:Computing-on-the\] apply. Let $\mathbb{R}_{+}^{N}$ be the subspace of $\mathbb{R}^{N}$ where all Schwinger parameters are greater than or equal to zero and let $\mathbb{R}_{\textrm{cube}}^{n}$ be the unit cube in $n$ cubical variables, i.e. $$\begin{aligned} \mathbb{R}_{+}^{N} & = & \left\{ (\alpha_{1},...,\,\alpha_{N})\in\mathbb{R}^{N}|0\leq\alpha_{i},\, i=1,\,...,\, N\right\} ,\\ \mathbb{R}_{\textrm{cube}}^{n} & = & \left\{ (x_{1},...,\, x_{n})\in\mathbb{R}_{n}|0\leq x_{i}\leq1,\, i=1,\,...,\, n\right\} .\end{aligned}$$ Consider the $\alpha_{N}$-dependent polynomials $\mathcal{P}_{N}=\{P_{1},\,...,\, P_{n}\}$ and the corresponding $\rho_{i}=-\frac{P_{i}|_{\alpha_{N}=0}}{ \frac{\partial P_{i}}{\partial\alpha_{N}} }$ for $i=1,\,...,\, n.$ We introduce an ordering on the set $\mathcal{P}_{N}$ as follows. A sufficiently small open region of the form $0\leq \alpha_{N-1} \ll \alpha_{N-2} \cdots \ll \alpha_1 \ll \epsilon$ (where $x\ll y $ denotes $x<y^M$ for some large $M$) does not intersect the hypersurfaces $\rho_i-\rho_j=0$. Therefore number the polynomials in $\mathcal{P}_{N}=\{P_{1},\,...,\, P_{n}\}$ such that everywhere in this region we have $$0 > \rho_{n}>\rho_{1}>\rho_{2}>...>\rho_{n-2}>\rho_{n-1}.\label{eq:ordered zeroes}$$ For the given, ordered set $(P_{1},\,...,\, P_{n})$, consider the rational map between affine spaces $$\phi:\,\mathbb{A}^{N}\rightarrow\mathbb{A}^{n},$$ (equivalently a homomorphism $\phi^*: \mathbb{Q}(x_1,\ldots, x_n) \rightarrow \mathbb{Q}(\alpha_1,\ldots, \alpha_N)$) given by $$\begin{aligned} \phi^*(x_{n}) & = & \frac{\alpha_{N}}{\alpha_{N}-\rho_{n}}\nonumber, \\ \phi^*(x_{n-1}) & = & 1-\frac{\rho_{n}}{\rho_{n-1}},\nonumber \\ \phi^*(x_{k}) & = & \frac{1-\frac{\rho_{n}}{\rho_{k}}}{1-\frac{\rho_{n}}{\rho_{k+1}}}\textrm{ for }1\leq k\leq n-2.\label{eq: change variables}\end{aligned}$$ These variables $x_i$ will be our cubical coordinates and we construct the set of 1-forms $\bar{\Omega}^F_n$ as above. Note that the restriction of $\phi$ to the first $N-1$ coordinates defines a rational map $\phi: \mathbb{A}^{N-1} \rightarrow \mathbb{A}^{n-1}$, since $\rho_1,\ldots, \rho_{n}$ do not depend on $\alpha_N$. For fixed $\alpha_1,\ldots, \alpha_{N-1}$, the curve ${\mathbb{P}}^1$ with coordinate $\alpha_N$ and punctures at $\{0,\rho_1,\ldots, \rho_{n},\infty \}$ (i.e., the fiber of $\mathbb{A}^N \rightarrow \mathbb{A}^{N-1}$), is isomorphic, via $\ref{eq: change variables}$, to the curve with coordinate $x_n$ and punctures at $\{0,(x_1\ldots x_{n-1})^{-1}, (x_2\ldots x_{n-1})^{-1} , \ldots , x_{n-1}^{-1},\infty, 1\}$, in that order. Via such a (family of) isomorphisms, we can explicitly express all 1-forms in $\Omega_{N}^{\textrm{Feynman}}$ as $\mathbb{Q}$-linear combinations of 1-forms in $\bar{\Omega}^F_n$ in cubical coordinates. We obtain $$\begin{aligned} \label{dalphas} \frac{d\alpha_N}{\alpha_N} & = & \frac{dx_n}{x_n} - \frac{dx_n}{x_n-1} ,\\ \frac{d\alpha_N}{\alpha_N-\rho_n} & = & -\frac{dx_n}{x_n-1} \ , \nonumber \\ \frac{d\alpha_N}{\alpha_N-\rho_i} & = & \frac{x_i ... x_{n-1}dx_n}{x_i... x_n-1} - \frac{dx_n}{x_n-1} , \nonumber \end{aligned}$$ since the $\rho_i$ are constant, for $i=1,...,n-1$. As a consequence, we can express each hyperlogarithm $L_{w}(\alpha_{N})$ as a $\mathbb{Q}$-linear combination of hyperlogarithms in cubical variables $\xi\in V\left(\bar{\Omega}^F_{n}\right).$ For simplicity, we make the following assumption (which is slightly stronger than assuming that the linear reduction of the Feynman integral is unramified): $$\label{eq:xlimits} \lim_{\alpha_{1}\rightarrow0}...\lim_{\alpha_{N}\rightarrow0}x_{k}(\alpha_{1},\,...,\,\alpha_{N})\in\{0,\,1\},\, k=1,\,...,\, n,$$ where these limits are approached from inside the cube $\mathbb{R}_{\textrm{cube}}^{n}.$ The domain of the $\alpha_{N}$-integration is mapped to the domain $0\leq x_{n}\leq1.$ The Jacobian is $J=-\frac{\rho_{n}}{(x_{n}-1)^{2}}.$ Up to rational functions which do not depend on $x_{n},$ we can now express integrals of the type of $(\ref{eq:LIntegral})$ as integrals of the type $$\label{eq:IntCubical} \int_{0}^{1}dx_{n}\frac{\prod q_{i}^{\gamma_{i}}}{\prod f_{i}^{\delta_i}} \xi$$ where $\gamma_{i},\delta_{i}\in\mathbb{N},$ and where each $q_{i}$ is a polynomial in Schwinger parameters without $\alpha_{N}$ or in cubical variables, and the integrand involves functions $f_{i}\in\{x_{n},\, x_{n}-1,\, x_{n-1}x_{n}-1,\,...,\, x_{1}\cdot\cdot\cdot x_{n}-1\}$ and hyperlogarithms $\xi\in V\left(\bar{\Omega}^F_{n}\right).$ Before we can apply our algorithm of subsection \[sub:Primitives\] for the computation of primitives, we use a standard procedure of applying finitely many successive partial fraction decompositions and partial integrations until all powers $\delta_i$ are equal to 1. As a last step of preparation, we apply the symbol map $\Psi$ of subsection \[sub:The-symbol-map\] to $\xi$. We obtain an integral as in $(\ref{eq:IntCubical})$ where now $\xi\in V\left(\Omega_{n}\right).$ Now we compute the definite integral $(\ref{eq:IntCubical})$ by use of the algorithms of subsections \[sub:Primitives\] and \[sub:Limits\]. Up to rational prefactors, we obtain a $\mathcal{Z}$-linear combination of functions in $V\left(\Omega_{n-1}\right).$ Back to Schwinger parameters\[sub:Back-to-Feynman\] ---------------------------------------------------- Note that after the integration, we have a function in terms of both types of variables, the Schwinger parameters and the cubical coordinates. In order to proceed with the integration over a next Schwinger parameter and apply the same steps again, we firstly have to express the integrand only in terms of Schwinger parameters again. Let $I$ be the result of the $\alpha_{N}$-integration, expressed as a linear combination $$I=\sum a_{i}\xi_{i}$$ of multiple polylogarithms $\xi_{i}\in V\left(\Omega_{n-1}\right)$. The coefficients $a_{i}$ are trivially expressed by Schwinger parameters by application of $\phi^*.$ However, expressing the multiple polylogarithms $\xi_{i}$ in terms of Schwinger parameters is more subtle, as we have to respect the limiting conditions of iterated integrals in both sets of variables. For any function $f$ of variables $y_{1},\,...,\, y_{n}$ and numbers $c_{1},\,...,\, c_{n}$ let us introduce the notation $$\lim_{(y_{1},\,...,\, y_{n})\rightarrow(c_{1},\,...,\, c_{n})}f=\mathrm{Reg}_{y_{n}\rightarrow c_{n}}...\mathrm{Reg}_{y_{1}\rightarrow c_{1}}f.$$ where in the right-hand side, $\mathrm{Reg}$ denotes the regularised limits with respect to unit tangent vectors in either cubical coordinates $x_i$ (or $1-x_i$), or Schwinger parameters $\alpha_i$. In the following let us write $0_{n}$ for the vector $(0,\,...,\,0)$ with $n$ components. We consider the vector $x_{p}=(x{}_{p(1)},\,...,\, x_{p(n-1)})$ of the remaining cubical coordinates, where the ordering is given by a permutation $p$ on the set $\{1,\,...,\, n-1\}$. We furthermore consider the vector of remaining Schwinger parameters $\alpha=(\alpha{}_{N-1},\,...,\,\alpha_{1})$ in the ordering in which we integrate over them, as fixed above. Consider a multiple polylogarithm $\xi\in V\left(\Omega_{n-1}\right).$ By definition, it satisfies $$\lim_{x_{\sigma}\rightarrow0_{n-1}}\xi= \epsilon ( \xi) \label{eq:limit condition cubical}$$ for every permutation $\sigma$ on $\{1,\,...,\, n-1\}$, where $\epsilon$ is the augmentation map (projection onto components of length $0$). We want to express each $\xi$ as iterated integrals $\eta$ in Schwinger parameters, for which we impose the condition $$\lim_{\alpha\rightarrow0_{N-1}}\eta=\epsilon ( \eta) . \label{eq:limit condition Feynman}$$ Condition $\ref{eq:limit condition Feynman}$ corresponds to a vanishing condition for the iterated integral $\xi \in V\left(\Omega_{n-1}\right) $ at a tangential base point on ${\mathcal{M}}_{0,n+2}$ (strictly speaking, on a related space ${\mathcal{M}}_{0,n+2}^{\dag}$ ([@Bro4], §8.2) which can be read off from the linear reduction algorithm and involves removing from $\mathbb{A}^{n-1}$ only those hypersurfaces $x_i=0$, $x_ix_{i+1}\ldots x_j =1$ which correspond to singularities actually occurring in the integrand), which is on the boundary of the connected component of ${\mathcal{M}}_{0,n+2}(\mathbb{R})$ defined by the unit hypercube $0\leq x_1,\ldots, x_{n-1} \leq 1$. One can verify that such a point can always be represented by a permutation $p$ on $\{1,\,...,\, n-1\}$ (non-uniquely) and a vector $c=(c_{1},\,...,\, c_{n-1})$ (uniquely) with all $c_{i}\in\{0,\,1\}$ such that for any rational function $g$ in the $x_i$ which is regular on ${\mathcal{M}}^{\dag}_{0,n+2}$, we have $$\lim_{x_{p}\rightarrow c}g=\lim_{\alpha\rightarrow0_{N-1}}\phi^\star g,\label{eq:limits}$$ where on the left-hand side $c$ is approached inside $\mathbb{R}_{\textrm{cube}}^{n-1}$ and on the right-hand side $(0,\,...,\,0)$ is approached inside $\mathbb{R}_{+}^{N-1}.$ Such a point $c$ and permutation $p$ determine the procedure to express $\xi$ in terms of Schwinger parameters. The components of $c$ are computed by $$c_{i}=\lim_{\alpha \rightarrow0_{N-1}}x_{i},$$ where $i\in\{1,\,...,\, n-1\}$, and lies in $\{0,1\}$, by assumption $(\ref{eq:xlimits})$. In the case when ${\mathcal{M}}_{0,n+2}^{\dag}= {\mathcal{M}}_{0,n+2}$ (i.e., all possible singularities which can occur actually do occur), a permutation $p$ satisfying $(\ref{eq:limits})$ can easily be computed with the help of dihedral coordinates $u_{ij}$, which are related to the cubical coordinates as discussed in [@Bro2]. A permutation $p$ satisfies $(\ref{eq:limits})$ for any regular function $g$ on ${\mathcal{M}}_{0,n+2}$ (expressed as a rational function of cubical coordinates) if it satisfies $$\lim_{x_{p}\rightarrow c}u_{ij}=\lim_{\alpha \rightarrow0_{N-1}}\phi^\star u_{ij}$$ for all dihedral coordinates $u_{ij}$. This condition determines $p$ in this case. Suppose ${\mathcal{M}}_{0,5}^{\dag}={\mathcal{M}}_{0,5}$. Let $x_1,x_2$ be cubical coordinates. Suppose that $x_1=1-\alpha_2, x_2=1-{\alpha_2 \over \alpha_1}$. Then the five dihedral coordinates $(x_1,x_2, 1-x_1x_2, {1-x_1 \over 1-x_1x_2}, {1-x_2 \over 1-x_1x_2})$ in the limits $\alpha_2\rightarrow 0$ then $\alpha_1 \rightarrow 0$ tend to $(1,1,0,0,1)$ respectively. This corresponds to taking first the limit as $x_1\rightarrow 1$ and then $x_2 \rightarrow 1$. On the other hand, suppose that $x_1 = 1-\alpha_1, x_2 = 1-\alpha_1$. Then the limit of the five dihedral coordinates above as $\alpha_1 \rightarrow 0$ are $(1,1,0,{1\over 2}, {1\over 2})$, which could potentially produce a $\log 2$ in the iterated integrals (ramification at prime $2$). In such a case, the condition of being unramified will ensure that $1-x_1x_2=\alpha_1(2-\alpha_1)$ does not occur as a singularity of the integrand. Thus ${\mathcal{M}}_{0,5}^{\dag} = \mathbb{A}^2 \backslash \{x_1,x_2=0,1\} = {\mathcal{M}}_{0,4} \times {\mathcal{M}}_{0,4}$ strictly contains ${\mathcal{M}}_{0,5}$. The limit as $\alpha_1\rightarrow 0$ can be obtained as the limit as $x_1\rightarrow 1, x_2\rightarrow 1$ in either order. Now let $x_{p}$ and $c$ be vectors satisfying $(\ref{eq:limits})$. We define $\eta$ by the following equation, where $\xi\in V\left(\Omega_{n-1}\right)$ is the result of the integration of $(\ref{eq:IntCubical})$, $$\eta=m\left(\phi^{\star}\otimes\phi^{\star}\lim_{x_{p}\rightarrow c}\right)\Delta\xi \label{eq:limit procedure}$$ and $m$ is multiplication. Note that this is an application of $(\ref{pathconcat})$. Then $\eta $ is the desired expression in terms of Schwinger parameters. As a last step, we express each iterated integral in terms of hyperlogarithms, such that we arrive at the starting point for the next integration over the variable $\alpha_{N-1}$. As a consequence of linear reducibility, all iterated integrals are now given by differential forms of the type $\omega=dP/P$ where $P$ are polynomials in the Schwinger parameters which are of degree $\leq1$ in the variable $\alpha_{N-1}$. In analogy to the construction of the unshuffle map we define the auxiliary restriction operations $$\pi_{\alpha_{i}}\omega=\omega|_{d\alpha_{i}=0,\,\alpha_{i}=0}$$ and $$r_{\alpha_{i}}\omega=\omega|_{d\alpha_{j}=0\textrm{ for all }j\neq i.}$$ By $$\eta'=m\left(r_{\alpha_{N-1}}\otimes \pi_{\alpha_{N-1}}\right)\Delta \eta \label{eq:write hyperlog}$$ we finally arrive at a linear combination of hyperlogarithms $L_{w}(\alpha_{N-1})$ whose coefficients are products of rational functions in Schwinger parameters, multiple zeta values and iterated integrals independent of $\alpha_{N-1}.$ Iterating the computation of $(\ref{eq:write hyperlog})$ for the remaining Schwinger parameters we can express all iterated integrals as hyperlogarithms. With this expression we can repeat the above steps to integrate out $\alpha_{N-1}$, and so on. Summary of the integration algorithm ------------------------------------ Let us summarize the above steps for integrating over one Schwinger parameter $\alpha_{N}.$ We start from a finite integral $I=\int_{0}^{\infty}d\alpha_{N}f$ whose integrand $f$, as in $(\ref{eq:Feyn integrand hyperlog})$, is a linear combination of hyperlogarithms $L_{w}(\alpha_{N})$ as functions of $\alpha_{N}$, and whose coefficients are products of rational functions $b(\alpha_{N})$ of the Schwinger parameters including $\alpha_{N},$ and further functions (possibly hyperlogarithms) not depending on $\alpha_{N}.$ As above, we write $\mathcal{P}_{N}$ for the set of $n$ polynomials depending linearly on $\alpha_{N},$ which are in the denominators of $b(\alpha_{N})$ and define the differential forms of $L_{w}(\alpha_{N})$ by $(\ref{eq: Omega Feyn})$. The set $\mathcal{P}_{N}$ is linearly reducible with respect to an ordered set $(\alpha_N,...,\alpha_1)$ of all Schwinger parameters, and unramified. The main steps of the algorithm are combined as follows: - Define the $n$ cubical variables $x_{1},\,...,\, x_{n}$, and express the integrand $f$ via the map $\ref{dalphas}$ as a linear combination of hyperlogarithms in $V\left(\bar{\Omega}^F_{n}\right)$. The integration over $\alpha_{N}$ is mapped via \[eq: change variables\] to the integration over $x_{n}$ from 0 to 1. - Apply the symbol map $\Psi$ of subsection \[sub:The-symbol-map\] to lift each function in $V\left(\bar{\Omega}^F_{n}\right)$ to multiple polylogarithms in $V\left(\Omega_{n}\right).$ - Use iterated partial integration and partial fraction decomposition to bring the integrand into the appropriate form. Then use the map $\star$ of subsection \[sub:Primitives\] to compute the primitive of $f.$ - Take the limits of the primitive at $x_{n}=0$ and $x_{n}=1$ to obtain the definite integral from 0 to 1, using the algorithm of subsection \[sub:Limits\]. The result is a linear combination of multiple polylogarithms in $V\left(\Omega_{n-1}\right)$ with coefficients possibly involving multiple zeta values. - Apply the change of variables to obtain an expression only in Schwinger parameters again. For iterated integrals, apply $(\ref{eq:limit procedure})$ such that the regularised limit at $\alpha\rightarrow0_{N-1}$ is preserved. - Write the result as a combination of hyperlogarithms in the next integration variable by $(\ref{eq:write hyperlog})$. Examples for the application of this algorithm by use of our computer program are given below. Applications {#sec:Applications} ============ Cellular integrals ------------------ A particular instance of period integrals on moduli spaces are given by the cellular integrals defined in [@AperyVar] in relation to irrationality proofs. The basic construction is to consider a permutation $\sigma $ of $\{1,\ldots, n\}$ and define a rational function and differential form $$\widetilde{f}_{\sigma} = \prod_{i} {z_i -z_{i+1} \over z_{\sigma(i)} - z_{\sigma(i+1)} } \qquad \hbox{ and } \qquad \widetilde{\omega}_{\sigma} = \prod_{i} {dz_i\over z_{\sigma(i)} - z_{\sigma(i+1)} } \nonumber\ ,$$ on the configuration space $C^n ({\mathbb{P}}^1)$ of $n$ distinct points $z_1,\ldots, z_n$ in ${\mathbb{P}}^1$, where the product is over all indices $i$ modulo $n$. Now $\mathrm{PGL}_2$ acts diagonally on $C^n ({\mathbb{P}}^1)$, and the quotient is $$\mathcal{M}_{0,n} \cong C^n( {\mathbb{P}}^1) / \mathrm{PGL}_2\ .$$ The rational function $\widetilde{f}_\sigma$ and the form $\widetilde{\omega}_{\sigma}$ are $\mathrm{PGL}_2$-invariant, and therefore descend in the standard way to a rational function and form $f_{\sigma}, \omega_{\sigma}$ on $\mathcal{M}_{0,n}$. Because $\mathrm{PGL}_2$ is triply-transitive, we can put $z_1=0, z_{n-1}=1, z_n = \infty$, and replace $z_{i+1}$ by $x_ix_{i+1}\ldots x_{n-3}$ for $i=1,\ldots, n-3$, where $x_1,\ldots, x_{n-3}$ are cubical coordinates on ${\mathcal{M}}_{0,n}$. Therefore we can formally write $$f_{\sigma} = \prod_{i} {z_i -z_{i+1} \over z_{\sigma(i)} - z_{\sigma(i+1)} } \qquad \hbox{ and } \qquad \omega_{\sigma} = { dx_1\ldots dx_{n-3} \over \prod_i z_{\sigma(i)} - z_{\sigma(i+1)} } \nonumber\ ,$$ where the product is over all indices $i$ modulo $n$, and all factors involving $z_n=\infty$ are simply omitted. For all $N\geq 0$, consider the family of basic cellular integrals $$I_N^{\sigma} = \int_{[0,1]^{n-3}} f_{\sigma}^N \omega_{\sigma}$$ where the domain of integration is the unit hypercube in the cubical coordinates $x_i$. Conditions for the convergence of the integral are discussed in [@AperyVar]. When it converges, this integral is a rational linear combination of multiple zeta values of weights $\leq n-3$ and can be computed with our program. In the case $n=5,6$ and $\sigma(1,2,3,4,5) = (1,3,5,2,4)$, and $\sigma(1,2,3,4,5,6)=(1,3,6,4,2,5) $ it gives back exactly the linear forms involved in Apéry’s proofs of the irrationality of $\zeta(2)$ and $\zeta(3)$. A systematic study of examples for higher $n$ (described in [@AperyVar]) was undertaken using the algorithms described in this paper. Expansion of generalized hypergeometric functions ------------------------------------------------- Many Feynman integrals can be expressed in terms of generalized hypergeometric functions $$_{p}F_{q}(a_{1},\,...,\, a_{p};\, b_{1},\,...,\, b_{q};\, z)=\sum_{k=0}^{\infty}\frac{\prod_{j=1}^{p}\left(a_{j}\right)_{k}z^{k}}{\prod_{j=1}^{q}\left(b_{j}\right)_{k}k!},$$ converging everywhere in the $z$-plane if $q\geq p,$ and in the case $q=p-1$ for $|z|<1$ or at $|z|=1$ if the real part of $\sum_{j=1}^{p-1}b_{j}-\sum_{j=1}^{p}a_{j}$ is positive. Here we used the Pochhammer symbol $$(a)_n=\frac{\Gamma (a+n)}{\Gamma (a)}.$$ Multi-variable generalizations, such as Appell and Lauricella functions, play a role in Feynman integral computations as well. If the Feynman integral is considered in $D=4-2\epsilon$ dimensions, the parameters take the form $$\label{eq:HyperParameters} a_{i}=A_{i}+\epsilon\alpha_{i},\, b_{i}=B_{i}+\epsilon\beta_{i}\;\textrm{where }\alpha_{i},\,\beta_{i}\in\mathbb{R}.$$ In the case of massless integrals, the numbers $A_{i},\, B_{i}$ are integers while in the case of non-vanishing masses, some of them are half-integers. In order to arrive at a result for the Feynman integral where pole-terms in $\epsilon$ can be separated, one has to expand these functions near $\epsilon=0.$ Several computer programs are available for this task. The programs of [@MocUwe; @Wei] use algorithms for the expansion of very general types of nested sums [@MocUweWei] while the program [@HubMai2] writes an Ansatz in harmonic polylogarithms and determines the coefficients from differential equations. A method using systems of differential equations was presented in [@Kal; @KalWarYos1; @KalWarYos2]. We also refer to [@Kal2; @Grey; @AblX] for recent progress in the field. Alternatively, we can start from an integral representation of the function, expand the integrand and compute the resulting integrals explicitly. This approach was applied in [@HubMai1] for the expansion of $_{2}F_{1}.$ The algorithms presented above are very well suited for this method and can be used to extend it to more general functions. As a first example we still consider $_{2}F_{1}.$ We have the integral representation $$_{2}F_{1}(a_{1},\, a_{2};\, b;\, z_{1})=\frac{\Gamma(b)}{\Gamma(a_{2})\Gamma(b-a_{2})}\int_{0}^{1}z_{2}^{a_{2}-1}(1-z_{2})^{b-a_{2}-1}(1-z_{1}z_{2})^{-a_{1}}dz_{2}$$ for $\textrm{Re}(b)>\textrm{Re}(a_{2})>0$ and $z_1\notin [1,\infty).$ The parameters $a_i$ and $b$ may depend on $\epsilon$ as in $(\ref{eq:HyperParameters})$. If we exclude the case of half-integers mentioned above, the expansion at $\epsilon=0$ leads to integrands whose denominators are products of $z_2, (1-z_2), (1-z_1 z_2)$ and whose numerators may involve powers of logarithms of these functions. We can view the variables $z_1, z_2$ as cubical coordinates and apply the algorithms of section \[sec:Computing-on-the\] to integrate over $z_{2}$ analytically. Example: $$\begin{aligned} _{2}F_{1}(1,\,1+\epsilon;\,3+\epsilon;\, z_{1}) & = & \frac{\Gamma(3+\epsilon)}{\Gamma(1+\epsilon)}\int_{0}^{1}\frac{z_{2}^{\epsilon}(1-z_{2})}{1-z_{1}z_{2}}dz_{2}\\ & = & \int_{0}^{1}\frac{2(z_{2}-1)}{z_{1}z_{2}-1}dz_{2}+\epsilon\int_{0}^{1}\frac{\left(2\ln(z_{2})+3\right)(z_{2}-1)}{z_{1}z_{2}-1}\\ & & +\epsilon^{2}\int_{0}^{1}\frac{\left(\ln(z_{2})^{2}+3\ln(z_{2})+1\right)(z_{2}-1)}{z_{1}z_{2}-1}dz_{2}+\mathcal{O}\left(\epsilon^{3}\right)\\ & = & \frac{2}{z_{1}^{2}}\left(z_{1}+(1-z_{1})\ln(1-z_{1})\right)+\frac{\epsilon}{z_{1}^{2}}\left(z_{1}+3(1-z_{1})\ln(1-z_{1})\right.\\ & & \left.+2(1-z_{1})\textrm{Li}_{2}(z_{1})\right)+\frac{\epsilon^{2}}{z_{1}^{2}}\left((1-z_{1})\ln(1-z_{1})\right.\\ & & \left.+3(1-z_{1})\textrm{Li}_{2}(z_{1})-2(1-z_{1})\textrm{Li}_{3}(z_{1})\right)+\mathcal{O}\left(\epsilon^{3}\right)\end{aligned}$$ We extend the approach to generalized hypergeometric functions, starting from the integral representation $$_{p}F_{q}(a_{1},\,...,\, a_{p};\, b_{1},\,...,\, b_{q};\, z)$$ $$=\frac{\Gamma(b_{q})}{\Gamma(a_{p})\Gamma(b_{q}-a_{p})}\int_{0}^{1}t^{a_{p}-1}(1-t)^{b_{q}-a_{p}-1}\,_{p-1}F_{q-1}(a_{1},\,...,\, a_{p-1};\, b_{1},\,...,\, b_{q-1};\, zt)dt$$ in the region where it converges. Here again the expansion of the integrand leads to integrals over cubical coordinates which can be computed by the algorithms of section \[sec:Computing-on-the\]. Example: $$\begin{aligned} _{3}F_{2}(2,\,1+\epsilon,\,1+\epsilon;\,3+\epsilon,\,2+\epsilon;\, z_{1}) & = & \frac{\Gamma(3+\epsilon)\Gamma(2+\epsilon)}{\Gamma(1+\epsilon)^{2}}\int_{0}^{1}\int_{0}^{1}\frac{z_{2}z_{3}^{\epsilon}(1-z_{2})^{\epsilon}}{(1-z_{1}z_{2}z_{3})^{1+\epsilon}}dz_{2}dz_{3}\\ & = & \frac{2}{z_{1}^{2}}\left((1-z_{1})\ln(1-z_{1})+z_{1}\right)+\frac{\epsilon}{z_{1}^{2}}\left(7(1-z_{1})\ln(1-z_{1})\right.\\ & & \left.+5z_{1}+(2-4z_{1})\textrm{Li}_{2}(z_{1})\right)+\frac{\epsilon^{2}}{z_{1}^{2}}\left(9(1-z_{1})\ln(1-z_{1})\right.\\ & & \left.+(7-12z_{1})\textrm{Li}_{2}(z_{1})+(6z_{1}-2)\textrm{Li}_{3}(z_{1})+4z_{1}\right)+\mathcal{O}\left(\epsilon^{3}\right)\end{aligned}$$ While for these functions the integral representations are readily given in cubical coordinates, an extension to further cases may require a change of variables. For example the first Appell function $$F_{1}(a;\, b_{1},\, b_{2};\, c;\, x,\, y)=\sum_{m\geq0}\sum_{n\geq0}\frac{\left(a\right)_{m+n}\left(b_{1}\right)_{m}\left(b_{2}\right)_{n}}{m!n!\left(c\right)_{m+n}}x^{m}y^{n}\;\textrm{where }|x|,\,|y|<1$$ with the integral representation [@Pic] $$F_{1}(a;\, b_{1},\, b_{2};\, c;\, x,\, y)=\frac{\Gamma\left(c\right)}{\Gamma(a)\Gamma(c-a)}\int_{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-tx)^{-b_{1}}(1-ty)^{-b_{2}}dt$$ for $\textrm{Re}(c)>\textrm{Re}(a)>0$ can be expressed in the appropriate form after introducing cubical coordinates $z_{3}=t,$ $z_{2}=y,$ $z_{1}=x/y.$ Example: $$\begin{aligned} F_{1}(1;\, 1,\, 1;\, 2+\epsilon;\, x,\, y) & = & \frac{\Gamma(2+\epsilon)}{\Gamma(1+\epsilon)}\int_{0}^{1}\frac{(1-z_{3})^{\epsilon}}{(1-z_{1}z_{2}z_{3})(1-z_{2}z_{3})}dz_{3}\\ & = & \frac{1}{x-y}\left(\ln(1-y)-\ln(1-x)\right)\\ & & +\frac{\epsilon}{x-y}\left(\ln(1-y)-\ln(1-x)+\frac{1}{2}\ln(1-y)^{2}-\frac{1}{2}\ln(1-x)^{2}\right.\\ & & -\left.\textrm{Li}_{2}(x)+\textrm{Li}_{2}(y)\right)+\mathcal{O}\left(\epsilon^{2}\right)\end{aligned}$$ We checked the examples with $_{2}F_{1}$ and $_{3}F_{2}$ analytically with the program of [@HubMai2] and the example with $F_{1}$ numerically with the built-in first Appell function in Mathematica. Feynman integrals ----------------- As a third application we turn to the computation of Feynman integrals by direct integration over their Schwinger parameters. As a first example we consider the period integral (in the sense of [@BEK]) of the four-loop vacuum-type graph of figure 5.1 a). The integral is finite in $D=4$ dimensions and is given in terms of Schwinger parameters as $$I_{1}=\int_{\alpha_{i}\geq0}\prod_{i=1}^{8}d\alpha_{i}\delta(1-\alpha_{8})\frac{1}{\mathcal{U}^{2}}.$$ The first Symanzik polynomial $\mathcal{U}$ is linearly reducible in this case. We use our implementation of the algorithms of sections \[sec:Computing-on-the\] and \[sec:Feynman-type-integrals\] to integrate over $\alpha_{1},\,...,\, \alpha_{7}$ in an appropriate ordering and to compute the limit at $\alpha_{8}=1$ in the last step. The computation time per integration grows at first due to the increasing weight and complexity of the functions involved, but decreases in the end as fewer variables remain. Here we compute with multiple polylogarithms of weight 2, 3, 4 and 5 in the fourth, fifth, sixth and seventh integration respectively. We obtain the result $I_{1}=20\zeta(5)$ which is well-known [@CheTka]. Period integrals of this type appear as coefficients of two-point integrals corresponding to graphs obtained from breaking open one edge in the vacuum-graph (see [@Bro5; @CheTka]). As an example for a Feynman integral with non-trivial dependence on masses and external momenta, we consider the hexagon-shaped one-loop graph of figure 5.1 b) with incoming external momenta $p_{1},\,...,\, p_{6}.$ Introducing one particle mass with $m^{2}<0$ we impose the on-shell condition $p_{1}^{2}=m^{2},\, p_{i}^{2}=0,\, i=2,\,...,\,6.$ In $D=6$ dimensions the Feynman integral reads $$I_{2}=\int_{\alpha_{i}\geq0}\prod_{i=1}^{6}d\alpha_{i}\delta(1-\alpha_{6})\frac{2}{\mathcal{F}^{3}}$$ with the second Symanzik polynomial $$\mathcal{F}=\sum_{1\leq i<j\leq 6}\alpha_{i}\alpha_{j}\left(-s_{ij}^{2}\right)$$ and kinematical invariants $$s_{ij}=\left(\sum_{k=i+1}^{j}p_{k}\right)^2.$$ This integral is computed in [@Del3]. In a first step in this computation, the integral is expressed in terms of the cross-ratios $$u_{1}=\frac{s_{26}^{2}s_{35}^{2}}{s_{25}^{2}s_{36}^{2}},\, u_{2}=\frac{s_{13}^{2}s_{46}^{2}}{s_{36}^{2}s_{14}^{2}},\, u_{3}=\frac{s_{15}^{2}s_{24}^{2}}{s_{14}^{2}s_{25}^{2}},\, u_{4}=\frac{s_{12}^{2}s_{36}^{2}}{s_{13}^{2}s_{26}^{2}}$$ as $$I_{2}=\frac{1}{s_{14}^{2}s_{25}^{2}s_{36}^{2}}\int_{\alpha_{i}\geq0}\prod_{i=1}^{3}d\alpha_{i}\frac{1}{(u_{2}+\alpha_{1}+\alpha_{2})(u_{3}\alpha_{1}+u_{1}\alpha_{3}+\alpha_{2})(u_{4}\alpha_{1}\alpha_{2}+\alpha_{2}+\alpha_{1}\alpha_{3}+\alpha_{3})}.$$ We choose the parametrization $$u_{1}=\frac{1}{1+y},\, u_{2}=\frac{1+v}{1+v-u},\, u_{3}=\frac{(1-u)(-y-x)}{(1+y)(-1+u-v)},\, u_{4}=\frac{1+v-x}{1+v}$$ which differs from the one in [@Del3]. This parametrisation is not pulled from thin air: it is constructed recursively out of the polynomials occurring in the linear reduction algorithm, applied to the integrand. With this choice each $u_{i}$ tends to either 0 or 1 at the tangential base-point which we choose by the ordering $(\alpha_{2},\, \alpha_{3},\, \alpha_{1},\, u,\, v,\, x,\, y)$ and furthermore the polynomials in the denominator of the re-written integrand of $I_{2}$ are linearly reducible for the ordering $(\alpha_{2},\, \alpha_{3},\, \alpha_{1})$. Therefore we can apply our implementation to integrate over the $\alpha_{i}$ in this order and we obtain a result for positive $u,\, v,\, x,\, y$. We checked the result analytically with the program of [@Pan3]. \[fig:Graphs\] (220, 120)(2, 1) (5.,10.)(10.,15.)(-0.4,)[Straight]{}[0]{} (10.,15.)(15.,10.)(-0.4,)[Straight]{}[0]{} (15.,10.)(10.,5.)(-0.4,)[Straight]{}[0]{} (10.,5.)(5.,10.)(-0.4,)[Straight]{}[0]{} (5.,10.)(10,10.)(0.,)[Straight]{}[0]{} (10.,15.)(10.,10.)(0.,)[Straight]{}[0]{} (15.,10.)(10.,10.)(0.,)[Straight]{}[0]{} (10.,5.)(10.,10.)(0.,)[Straight]{}[0]{} (7.5,15.)(12.5,15.)(0.,)[Straight]{}[0]{} (12.5,15.)(15.,10.)(0.,)[Straight]{}[0]{} (15.,10.)(12.5,5.)(0.,)[Straight]{}[0]{} (12.5,5.)(7.5,5.)(0.,)[Straight]{}[0]{} (7.5,5.)(5.,10.)(0.,)[Straight]{}[0]{} (5.,10.)(7.5,15.)(0.,)[Straight]{}[0]{} (15.,10.)(18.,10.)(0.,)[Straight]{}[0]{} (12.5,15.)(14.,18.)(0.,)[Straight]{}[0]{} (7.5,15.)(6.,18.)(0.,)[Straight]{}[0]{} (5.,9.95)(2.,9.95)(0.,)[Straight]{}[0]{} (5.,10.05)(2.,10.05)(0.,)[Straight]{}[0]{} (5.,9.9)(2.,9.9)(0.,)[Straight]{}[0]{} (5.,10.1)(2.,10.1)(0.,)[Straight]{}[0]{} (5.,9.85)(2.,9.85)(0.,)[Straight]{}[0]{} (5.,10.15)(2.,10.15)(0.,)[Straight]{}[0]{} (5.,9.8)(2.,9.8)(0.,)[Straight]{}[0]{} (5.,10.2)(2.,10.2)(0.,)[Straight]{}[0]{} (7.5,5.)(6.,2.)(0.,)[Straight]{}[0]{} (12.5,5.)(14.,2.)(0.,)[Straight]{}[0]{} Conclusions =========== In this article we have presented explicit algorithms for symbolic computation of iterated integrals on moduli spaces ${\mathcal{M}}_{0,n+3}$ of curves of genus $0$ with $n+3$ ordered marked points, based on [@Bro2]. These algorithms include the total differential of these functions, computation of primitives and the exact computation of limits at arguments equal to 0 and 1. The algorithms are formulated by use of operations on an explicit model for the reduced bar construction on ${\mathcal{M}}_{0,n+3}$ in terms of cubical coordinates $x_i$. In this formulation, the algorithms are well suited for an implementation on a computer. We have furthermore presented an algorithm for the symbol map, out of which the vector space of homotopy invariant iterated integrals on ${\mathcal{M}}_{0,n+3}$ can be constructed. We expect the algorithms to apply to a variety of problems in theoretical physics and pure mathematics. Here we have concentrated on two main applications. As a first application, we have considered the computation of periods on ${\mathcal{M}}_{0,n+3}$, for which our algorithms are readily applicable. Secondly, we have discussed the computation of a class of Feynman integrals by the method of [@Bro4; @Bog1]. In this approach, the Feynman integral is mapped to an integral on the moduli space, where our algorithms are applied to compute a single integration. We have presented an explicit procedure for the required change of variables from Schwinger parameters to cubical coordinates. A further procedure maps the result of the integration back to iterated integrals in terms of Schwinger parameters, and this process can be iterated. Using an implementation of our algorithms based on Maple, we have computed examples of such applications. As a third type of application, we have briefly discussed an approach for the expansion of generalized hypergeometric functions. [10]{} V. I. Arnold, *The cohomology ring of the coloured braid group*, Mat. Zametki 5 (1969), 227-231; Math Notes 5 (1969), 138-140. 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--- abstract: 'Let $X,S$ be two smooth projective varieties and $X\rightarrow S$ be a smooth family of projective curves of genus $\geq 2$ over an algebraically closed field $k$ and let $E$ be a vector bundle of rank $r\geq 3$ over $X$ and $\mathbb{P}(E)$ be its projectivization. Fix $d\geq 1$. Let $\mathcal{Q}(E,d)$ be the relative quot scheme of torsion quotients of $E$ of degree $d$. Then we show that the identity component of the group of automorphisms of $\mathcal{Q}(E,d)$ over $S$ is isomorphic to the identity component of the group of automorphisms of $\mathbb{P}(E)$ over $S$. As a corollary, the identity component of the automorphism group of flag scheme of filtrations of torsion quotients of $\mathcal{O}^{r}_{C}$, where $r\geq 3$ and $C$ a smooth projective curve of genus $\geq 2$ is computed.' address: 'School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India' author: - Chandranandan Gangopadhyay title: Automorphisms of relative Quot schemes --- Introduction ============ Let $p_{S}:X\rightarrow S$ be a smooth family of projective curves over an algebraically closed field $k$. Assume $X$ and $S$ are smooth projective varieties. Let $E$ be a vector bundle over $X$ of rank $r\geq 3$.\ Fix $d \geq 1$. Let $\pi_{S}:\mathcal{Q}(E,d)\rightarrow S$ be the relative Quot scheme \[1,Th.2.2.4\] whose closed points correspond to quotients $E|_{X_{s}} \rightarrow B_{d}$, $\forall s\in S$ where $X_{s}$ is the fibre of $p_{S}$ over $S$, and $B_{d}$ is a torsion sheaf over the smooth projective curve $X_{s}$ of degree $d$.\ \ **Notation.** If $f:Y\rightarrow Z$ is a smooth morphism between two projective varieties, let us denote the automorphism group of $Y$ over $Z$ by $\text{Aut}(Y/Z)$. It is known that $\text{Aut}(Y/Z)$ is a group scheme and let its identity component be denoted by $\text{Aut}^{o}(Y/Z)$. Let $\mathcal{T}_{Y/Z}$ be the relative tangent bundle of $f$. Then Lie($\text{Aut}^{o}(Y/Z))=H^{0}(Y,\mathcal{T}_{Y/Z})$ \[3,(1)\].\ \ Let us denote the projective bundle associated to $E$ by $\mathbb{P}(E)$. Note that $\mathcal{Q}(E,1)\cong \mathbb{P}(E)$.\ Then, we will be proving the following theorem:\ \ **Theorem 2.1** *(i)$ \text{Aut}^{o}(\mathcal{Q}(E,d))\cong \text{Aut}^{o}(\mathbb{P}(E)/S)$\ (ii) $H^{0}(X,\mathcal{T}_{\mathbb{P}(E)/S})\cong H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)/S})$*\ In the third section, we will use Theorem 2.1 to compute the automorphism group in certain specific cases. Main Theorem ============ (i)$ \text{Aut}^{o}(\mathcal{Q}(E,d))\cong \text{Aut}^{o}(\mathbb{P}(E)/S)$\ (ii) $H^{0}(X,\mathcal{T}_{\mathbb{P}(E)/S})\cong H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)/S})$ Note that by \[3, Cor 2.2\], any automorphism $g\in \text{Aut}^{o}(\mathbb{P}(E)/S)$ descends to an automorphism $g'\in \text{Aut}^{o}(X/S)$. Therefore we have the following diagram:\ (E)((g’)\^[\*]{}E) & (E)\ X & X \ Then, $E\cong (g')^{*} E\bigotimes p^{*}\mathcal{L}$ for some line bundle $\mathcal{L}$ on $X$. Let us denote this isomorphism of bundles by $\Psi_{g}$.\ It is clear that $\Psi_{g}$ induces an automorphism of $\mathcal{Q}(E,d)$ by sending $[E|_{p^{-1}_{S}(s)} \rightarrow B_{d}\rightarrow 0]$ to $[E_{p^{-1}_{S}(s)} \xrightarrow{\Psi|_{p^{-1}_{S}(s)}} (g')^{*}(E|_{p^{-1}_{S}}(s)\rightarrow B_{d}\rightarrow 0)\bigotimes \mathcal{L}|_{p^{-1}_{S}(s)}]$.\ Hence, we have a homomorphism $\text{Aut}^{o}(\mathbb{P}(E)/S) \rightarrow \text{Aut}^{o}(\mathcal{Q}(E,d)/S)$ and clearly it is injective.\ Hence, we have morphism of lie algebras: $H^{0}(X,\mathcal{T}_{\mathbb{P}(E)/S})\hookrightarrow H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)/S})$.\ Now, by Theorem 2.2(iii), both are of same dimension as vector spaces. Therefore,\ $H^{0}(X,\mathcal{T}_{\mathbb{P}(E)/S})\cong H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)/S})$. Hence, $ \text{Aut}^{o}(\mathcal{Q}(E,d))\cong \text{Aut}^{o}(\mathbb{P}(E)/S)$. Let $\mathcal{Z}$ be the fibered product of $d$ copies of $\mathbb{P}(E)$ over $S$. Then (i)There exists an open subset $\mathcal{U}\subseteq \mathcal{Z}$ and a dominant map $\Phi :\mathcal{U}\rightarrow \mathcal{Q}(E,d)$ over $S$ such that codim$_{\mathcal{Z}}(\mathcal{Z}\setminus \mathcal{U})\geq 2$\ (ii)$H^{0}(\mathcal{U},\Phi^{*}\mathcal{T}_{\mathcal{Q}(E,d)/S})=\bigoplus\limits_{i=1}^{d}H^{0}(\mathbb{P}(E),\mathcal{T}_{\mathbb{P}(E)/S})$\ (iii)The natural map $H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)/S})\rightarrow H^{0}(\mathcal{U},\Phi^{*}\mathcal{T}_{\mathcal{Q}(E,d)/S})=\bigoplus\limits_{i=1}^{d}H^{0}(\mathbb{P}(E),\mathcal{T}_{\mathbb{P}(E)/S})$ is an injection and is invariant under permutation of the components of $\bigoplus\limits_{i=1}^{d}H^{0}(\mathbb{P}(E),\mathcal{T}_{\mathbb{P}(E)/S})$ i.e. we have an injection $H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)/S})\hookrightarrow H^{0}(X,\mathcal{T}_{\mathbb{P}(E)/S})$. \(i) Let $p:\mathbb{P}(E)\rightarrow X$ be the projection. Let $p_{i}$ be the $i$-th projection of $\mathcal{Z}$ to $\mathbb{P}(E)$. For, $i\neq j$, let $\Delta_{i,j}$ be the closed subscheme of $\mathcal{Z}$ given by the equation $p_{i}=p_{j}$, and for $i,j,k$ all distinct, let $\Delta_{i,j,k,X}$ be the subscheme given by the equation $p\circ p_{i}=p\circ p_{j}=p \circ p_{k}$. Let $\mathcal{U}=\mathcal{Z} \setminus (\bigcup \limits_{i,j} \Delta_{i,j} \bigcup\bigcup\limits_{i,j,k } \Delta_{i,j,k,X} )$.\ Now, let $\pi_{1}$ and $\pi_{2}$ are the first and secong projections from $X\times_{S} \mathcal{Z}$ to $X$ and $\mathcal{Z}$ respectively, and let $p_{i}\circ \pi_{2}$ be denoted by $\pi_{2,i}$. Let $\Delta_{i}$ be the closed subscheme of $X\times_{S} \mathcal{Z}$ given by the equation $\pi_{1}=p\circ \pi_{2,i}$.\ Then, consider the following morphism of sheaves over $X\times_{S}\mathcal{Z}$:\ \^[\*]{}\_[1]{}E & \_[i=1]{}\^[d]{} \^[\*]{}\_[2,i]{}(1)|\_[\_[i]{}]{} \ Here, $\pi^{*}_{1}E \rightarrow \pi^{*}_{2,i}\mathcal{O}(1)|_{\Delta_{i}}$ is given by the composition of morphisms $\pi^{*}_{1}E \rightarrow \pi^{*}_{1}E|_{\Delta_{i}}$ followed by $\pi^{*}_{1}E|_{\Delta_{i}} \cong \pi^{*}_{2,i}E|_{\Delta_{i}} \rightarrow \pi^{*}_{2,i} \mathcal{O}(1)|_{\Delta_{i}}$.\ Let $u \in \mathcal{U}$. Then, $(\bigoplus\limits_{i=1}^{d} \pi^{*}_{2,i}\mathcal{O}(1)|_{\Delta_{i}})|_{X\times_{S}\{u\}}=\bigoplus\limits_{i=1}^{d} k_{p\circ p_{i}(u)}$, and the surjection $E|_{X\times_{S}\{u\}}\rightarrow k_{p\circ p_{i}(u)}$ is given by the element $p_{i}(u)$.\ By definition of $\mathcal{U}$, for any $1\leq i \leq d$, there can exist atmost one $j\neq i$ such that $p\circ p_{i}(u)=p\circ p_{j}(u)$, and for such a pair $(i,j)$, $p_{i}(u)\neq p_{j}(u)$, i.e. the homomorphism $E|_{X\times_{S}\{u\}}\rightarrow k_{p\circ p_{i}(u)}\bigoplus k_{p\circ p_{j}(u)}$ is a surjection.\ Therefore, $q|_{X\times_{S} \{u\}}$ is surjective $\forall u\in \mathcal{U}$, and clearly $(\bigoplus\limits_{i=1}^{d} \pi^{*}_{2,i}\mathcal{O}(1)|_{\Delta_{i}})|_{X\times_{S}\{u\}}=\bigoplus\limits_{i=1}^{d} k_{p\circ p_{i}(u)}$ has length $d$. Hence, by universal property of $\mathcal{Q}(E,d)$, $q$ induces a map $\Phi:\mathcal{U} \rightarrow \mathcal{Q}(E,d)$, which is dominant \[6,Th.2.1\].\ (ii)Let us denote the projection $X\times_{S} \mathcal{Q}(E,d) \rightarrow X$ by $\pi_{X}$ and the projection $X\times_{S} \mathcal{Q}(E,d) \rightarrow \mathcal{Q}(E,d)$ by $p_{\mathcal{Q}}$ i.e. we have the following diagram:\ X\_[S]{} (E,d) & X\ (E,d) & S \ Over $X\times_{S} \mathcal{Q}(E,d)$, we have the universal exact sequence:\ 0 & (E,d) & \^[\*]{}\_[X]{} E & (E,d) & 0 \ \ Then, it is known that $\mathcal{A}(E,d)$ is a vector bundle of rank $r$ \[4, Lemma 2.2\] and\ $\mathcal{T}_{\mathcal{Q}(E,d)/S}=(p_{\mathcal{Q}})_{*}\mathcal{H}om(\mathcal{A}(E,d),\mathcal{B}(E,d))$. \[1, Prop.2.2.7\]\ Now, we have the following diagram:\ X\_[S]{} & X\_[S]{} (E,d)\ & (E,d) \ By Grauert’s theorem \[7, Cor.12.9\], We get that\ $\Phi^{*}\mathcal{T}_{\mathcal{Q}(E,d)/S}=(\Phi)^{*}(p_{\mathcal{Q}})_{*}\mathcal{H}om(\mathcal{A}(E,d),\mathcal{B}(E,d)) \cong(\pi_{1})_{*}(id_{X}\times \Phi)^{*}\mathcal{H}om(\mathcal{A}(E,d),\mathcal{B}(E,d))$\ Now, since $\mathcal{A}(E,d)$ is a vector bundle, we have\ $(id_{X}\times_{S}\Phi)^{*}\mathcal{H}om(\mathcal{A}(E,d),\mathcal{B}(E,d))=\mathcal{H}om(\Phi^{*}\mathcal{A}(E,d),\Phi^{*}\mathcal{B}(E,d))$\ Let us denote the kernel of $q$ by $\mathcal{F}(E,d)$. Then, by the definition of the map $\Phi$, we have $\Phi^{*}\mathcal{B}(E,d)\cong (\bigoplus\limits_{i=1}^{d} \pi^{*}_{2,i}\mathcal{O}(1)|_{\Delta_{i}})|_{X\times_{S}\mathcal{U}}$. Also, $\Phi^{*}\mathcal{A}(E,d)\cong \mathcal{F}(E,d)|_{X\times_{S} \mathcal{U}}$, since by \[4, Lemma 2.2\] $\mathcal{F}(E,d)|_{X\times_{S}\mathcal{U}}$ is again a vector bundle of rank $r$, and there exists a surjection $\Phi^{*}\mathcal{A}(E,d)\twoheadrightarrow \mathcal{F}(E,d)|_{X\times_{S} \mathcal{U}}$.\ Therefore, $\Phi^{*}\mathcal{T}_{\mathcal{Q}(E,d)/S}=\mathcal{H}om(\mathcal{F}(E,d),\bigoplus\limits_{i=1}^{d} \pi^{*}_{2,i}\mathcal{O}(1)|_{\Delta_{i}} )|_{X\times_{S} \mathcal{U}}$.\ Now (ii) follows from Lemma 2.3.\ (iii)Note that since $\Phi:\mathcal{U}\rightarrow \mathcal{Q}(E,d)$ is dominant, then induced map\ $H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)/S})\rightarrow H^{0}(\mathcal{U},\Phi^{*}\mathcal{T}_{\mathcal{Q}(E,d)/S})=\bigoplus\limits_{i=1}^{d}H^{0}(\mathbb{P}(E),\mathcal{T}_{\mathbb{P}(E)/S})$ is an injection.\ Now, since $\Phi : \mathcal{U}\rightarrow \mathcal{Q}(E,d)$ is invariant under any permutation of the various $\mathbb{P}(E)$ factors of $\mathcal{U}$, hence, the map $H^{0}(\mathcal{Q}(E,d), \mathcal{T}_{\mathcal{Q}(E,d)/S})\hookrightarrow \bigoplus\limits_{i=1}^{d }H^{0}(\mathbb{P}(E),\mathcal{T}_{\mathbb{P}(E)/S})$ factors through the $H^{0}(\mathbb{P}(E),\mathcal{T}_{\mathbb{P}(E)/S}) \hookrightarrow \bigoplus\limits_{i=1}^{d }H^{0}(\mathbb{P}(E),\mathcal{T}_{\mathbb{P}(E)/S})$. $H^{0}(X\times_{S} \mathcal{U}, \mathcal{H}om(\mathcal{F}(E,d), \pi^{*}_{2,j}\mathcal{O}(1)|_{\Delta_{j}} )|_{X\times_{S} \mathcal{U}})= H^{0}(\mathbb{P}(E), \mathcal{T}_{\mathbb{P}(E)/S})$ $\forall j$ Over $X\times_{S} \mathcal{U}$, we have the following commutative diagram:\ & & & 0 &\ & 0 & & \_[i=1,ij]{}\^[d]{} \^[\*]{}\_[2,i]{}(1)|\_[\_[i]{}]{} &\ 0 & (E,d) & \^[\*]{}\_[1]{}E & \_[i=1]{}\^[d]{} \^[\*]{}\_[2,i]{} (1)|\_[\_[i]{}]{} & 0\ 0 & (\_[1]{}\_[2,j]{})\^[\*]{}(E,1) & \^[\*]{}\_[1]{}E & \^[\*]{}\_[2,j]{}(1)|\_[\_[j]{}]{} & 0\ & & & 0 &\ \ Now, using snake lemma for the above diagram, we get the following exact sequence over $X\times_{S} \mathcal{U}$:\ $$\begin{tikzcd} 0 \arrow[r] & \mathcal{F}(E,d) \arrow[r] & (\pi_{1} \times \pi_{2,j})^{*}\mathcal{F}(E,1) \arrow[r] & \bigoplus\limits_{i=1,i\neq j}^{d} \pi^{*}_{2,i}\mathcal{O}(1)|_{\Delta_{i}} \arrow[r] & 0 \end{tikzcd}\\$$ Applying $\mathcal{H}om($ $ ,\pi^{*}_{2,j}\mathcal{O}(1)|_{\Delta_{j}} )$ and Lemma 2.4(i) to the exact sequence $(2.3)$, we get that over $X\times_{S} \mathcal{U}$, we have the following exact sequence:\ 0 & om((\_[1]{}\_[2,j]{})\^[\*]{}(E,1),\_[2,j]{}\^[\*]{}(1)|\_[\_[j]{}]{}) & om((E,d),\_[2,j]{}\^[\*]{}(1)|\_[\_[j]{}]{})\ & 0 & xt\^[1]{}(\_[i=1,ij]{}\^[d]{}\^[\*]{}\_[2,i]{}(1)|\_[\_[i]{}]{},\_[2,j]{}\^[\*]{}(1)|\_[\_[j]{}]{}) \ Applying $H^{0}$ to the above exact sequence and using Lemma 2.4(i) and (ii), we get that\ $H^{0}(X\times_{S} \mathcal{U}, \mathcal{H}om(\mathcal{F}(E,d),\pi_{2,j}^{*}\mathcal{O}(1)|_{\Delta_{j}})=H^{0}(\mathbb{P}(E),\mathcal{T}_{\mathbb{P}(E)/S})$. \(i) $\mathcal{H}om(\bigoplus\limits_{i=1,i\neq j}^{d}\pi^{*}_{2,i}\mathcal{O}(1)|_{\Delta_{i}}, \pi^{*}_{2,j}\mathcal{O}(1)|_{\Delta_{j}})=0$\ (ii) $H^{0}(X\times_{S}\mathcal{U}, \mathcal{E}xt^{1}(\bigoplus\limits_{i=1,i\neq j}^{d}\pi^{*}_{2,i}\mathcal{O}(1)|_{\Delta_{i}}, \pi^{*}_{2,j}\mathcal{O}(1)|_{\Delta_{j}}))=0$\ (iii) $H^{0}(X\times_{S} \mathcal{U},\mathcal{H}om ((\pi_{1}\times \pi_{2,j})^{*}\mathcal{F}(E,1),\pi_{2,j}^{*}\mathcal{O}(1)|_{\Delta_{j}}))= H^{0}(\mathbb{P}(E), \mathcal{T}_{\mathbb{P}(E)/S})$ \(i) $\mathcal{H}om(\bigoplus\limits_{i=1,i\neq j}^{d}\pi^{*}_{2,i}\mathcal{O}(1)|_{\Delta_{i}}, \pi^{*}_{2,j}\mathcal{O}(1)|_{\Delta_{j}})=\bigoplus\limits_{i=1,i\neq j}^{d} \mathcal{H}om(\pi^{*}_{2,i}\mathcal{O}(1)|_{\Delta_{i}},\pi^{*}_{2,j} \mathcal{O}(1)|_{\Delta_{j}})= \\ \bigoplus\limits_{i=1,i\neq j}^{d} (\pi^{*}_{2,i}\mathcal{O}(-1) \bigotimes \pi^{*}_{2,j}\mathcal{O}(1)) \mathcal{H}om(\mathcal{O}_{\Delta_{i}}, \mathcal{O}_{\Delta_{j}})$.\ Now, by adjunction, $\mathcal{H}om_{\mathcal{O}_{X\times_{S} \mathcal{Z}}}(\mathcal{O}_{\Delta_{i}},\mathcal{O}_{\Delta_{j}})=\mathcal{H}om_{\mathcal{O}_{\Delta_{j}}}(\mathcal{O}_{\Delta_{i}\cap \Delta_{j}}, \mathcal{O}_{\Delta_{j}})$.\ Now, since $\Delta_{j}$ is an integral scheme, and $\Delta_{i}\cap \Delta_{j}$ is a proper subset of $\Delta_{j}$, so\ $\mathcal{H}om_{\mathcal{O}_{\Delta_{j}}}(\mathcal{O}_{\Delta_{i}\cap \Delta_{j}}, \mathcal{O}_{\Delta_{j}})=0$.\ (ii) $\mathcal{E}xt^{1}(\bigoplus\limits_{i=1,i\neq j}^{d}\pi^{*}_{2,i}\mathcal{O}(1)|_{\Delta_{i}}, \pi^{*}_{2,j}\mathcal{O}(1)|_{\Delta_{j}})=\bigoplus\limits_{i=1,i\neq j}^{d} \mathcal{E}xt^{1}(\pi^{*}_{2,i}\mathcal{O}(1)|_{\Delta_{i}}, \mathcal{O}(1)|_{\Delta_{j}})= \\ \bigoplus\limits_{i=1,i\neq j}^{d} (\pi^{*}_{2,i}\mathcal{O}(-1) \bigotimes \pi^{*}_{2,j}\mathcal{O}(1)) \mathcal{E}xt^{1}(\mathcal{O}_{\Delta_{i}}, \mathcal{O}_{\Delta_{j}})$.\ Now, consider the exact sequence:\ 0 & (-\_[i]{}) & \_[X\_[S]{}]{} & \_[\_[i]{}]{} & 0 \ \ Applying $\mathcal{H}om(\text{ }, \mathcal{O}_{\Delta_{j}})$ to the above exact sequence, we get:\ 0 & \_[\_[j]{}]{} & (\_[i]{})|\_[\_[j]{}]{} & xt\^[1]{}(\_[\_[i]{}]{},\_[\_[j]{}]{}) & 0 \ \ i.e. $\mathcal{E}xt^{1}(\mathcal{O}_{\Delta_{i}},\mathcal{O}_{\Delta_{j}})\cong \pi^{*}_{1}\mathcal{T}_{X/S}|_{\Delta_{i}\cap \Delta_{j}} $.\ \ Therefore, $H^{0}(X\times_{S}\mathcal{U}, (\pi^{*}_{2,i}\mathcal{O}(-1) \bigotimes \pi^{*}_{2,j}\mathcal{O}(1)) \mathcal{E}xt^{1}(\mathcal{O}_{\Delta_{i}}, \mathcal{O}_{\Delta_{j}}))=\\ H^{0}(X\times_{S}\mathcal{U}, (\pi^{*}_{2,i}\mathcal{O}(-1) \bigotimes \pi^{*}_{2,j}\mathcal{O}(1))\bigotimes \pi^{*}_{1}\mathcal{T}_{X/S}|_{\Delta_{i}\cap \Delta_{j}})\\ = H^{0}((X\times_{S}\mathcal{U}) \cap \Delta_{i} \cap \Delta_{j}, (\pi^{*}_{2,i}\mathcal{O}(-1) \bigotimes \pi^{*}_{2,j}\mathcal{O}(1))\bigotimes \pi^{*}_{1}\mathcal{T}_{X/S}|_{\Delta_{i}\cap \Delta_{j}})$.\ \ Without loss of generality, assume $i=1, j=2$. Then, $\Delta_{1} \cong \mathcal{Z}$, where this isomorphism is given by $\Delta_{1}\hookrightarrow X\times_{S}Z \rightarrow \mathcal{Z}$. Under this ismorphism $(X\times_{S} \mathcal{U})\cap \Delta_{1}\cong \mathcal{U}$ and $\Delta_{1}\cap \Delta_{2} \cong (\mathbb{P}(E)\times_{X} \mathbb{P}(E))\times_{S}(\mathbb{P}(E))^{d-2}$, and the line bundle $(\pi^{*}_{2,i}\mathcal{O}(-1) \bigotimes \pi^{*}_{2,j}\mathcal{O}(1))\bigotimes \pi^{*}_{1}\mathcal{T}_{X/S}|_{\Delta_{1}} \cong p^{*}_{1}\mathcal{O}(-1)\bigotimes p^{*}_{2}\mathcal{O}(1)\times (p\circ p_{1})^{*}\mathcal{T}_{X/S}$.\ Let us denote the subscheme $(\mathbb{P}(E)\times_{X} \mathbb{P}(E))\times_{S}(\mathbb{P}(E))^{d-2}\hookrightarrow \mathcal{Z}$ by $Y$, and the line bundle $p^{*}_{1}\mathcal{O}(-1)\bigotimes p^{*}_{2}\mathcal{O}(1)\times (p\circ p_{1})^{*}\mathcal{T}_{X/S}|_{Y}$ by $\mathcal{L}$.\ \ We need to show that $H^{0}(Y\cap \mathcal{U},\mathcal{L})=0$.\ \ Now, $Y \setminus \mathcal{U}=\bigcup\limits_{i,j} (\Delta_{i,j}\cap Y) \bigcup\limits_{i,j,k} (\Delta_{i,j,k,X}\cap Y)$.\ **Claim.** (a) codim$_{Y}(\Delta_{i,j}\cap Y)\geq 2$\ (b)If $\{1,2\}\nsubseteq \{i,j,k\}$, then codim$_{Y}(\Delta_{i,j,k,X})\geq 2$.\ (c) $\Delta_{1,2,k,X}$ has codimension $1$ in $Y$.\ *Proof of Claim.*(a) If $\{i,j\}\neq \{1,2\}$, then $\Delta_{i,j}\cap Y\cong (\mathbb{P}(E)\times_{X}\mathbb{P}(E))\times_{S}(\mathbb{P}(E))^{d-3}$, and hence has codimension $\geq 2$ subset in $Y$.\ If $(i,j)=(1,2)$, then $\Delta_{1,2}\cong\mathbb{P}(E)\times_{S}(\mathbb{P}(E))^{d-2}$, and since $\text{rank }E\geq 3$, it has codimension $\geq 2$ in $Y$.\ (b) If $\{1,2\}\cap \{i,j,k\}=\phi$, then\ $\Delta_{i,j,k,X}\cap Y\cong (\mathbb{P}(E)\times_{X}\mathbb{P}(E))\times_{S}(\mathbb{P}(E) \times_{X}\mathbb{P}(E)\times_{X}\mathbb{P}(E))\times_{S}(\mathbb{P}(E))^{d-5}$\ If $i=1$ and $\{j,k\}\cap \{1,2\}=\phi$, then\ $\Delta_{i,j,k,X}\cap Y\cong(\mathbb{P}(E)\times_{X}\mathbb{P}(E)\times_{X}\mathbb{P}(E)\times_{X}\mathbb{P}(E))\times_{S}(\mathbb{P}(E))^{d-4}$. Hence, both in these two cases we have codimension of $\Delta_{i,j,k,X}\cap Y \geq 2$ in $Y$.\ (c) $\Delta_{1,2,k,X}\cong \mathbb{P}(E)\times_{X} \mathbb{P}(E) \times_{X} \mathbb{P}(E)\times_{S} (\mathbb{P}(E))^{d-3}$. Hence it has codimension $1$ in $Y$. Let us denote the open set $Y\setminus ( \bigcup\limits_{k\geq 3} \Delta_{1,2,k,X})$ by $V$. Then, by the above claim\ $H^{0}(Y\cap \mathcal{U}, \mathcal{L})=H^{0}(V,\mathcal{L})$.\ Now, if $s \in H^{0}(V,\mathcal{L}|_{V})$, then for some $t\in H^{0}(Y,\mathcal{O}(\sum\limits_{k=3}^{d}\Delta_{1,2,k,X}))$, $st^{n}$ extends to a global section of $\mathcal{L}\bigotimes \mathcal{O}(\sum\limits_{k=3}^{d}n\Delta_{1,2,k,X})$, for some $n\in \mathbb{N}$. So, it is enough to show that $H^{0}(Y, \mathcal{L}\bigotimes \mathcal{O}(\sum\limits_{k=3}^{d}n\Delta_{1,2,k,X}))=0$ $\forall n\in \mathbb{N}$. Now, this follows from the next claim.\ **Claim:** $(p_{2}\times p_{3}\times ..\times p_{d})_{*}(\mathcal{L}\bigotimes \mathcal{O}(\sum\limits_{k=3}^{d}\Delta_{1,2,k,X}))=0$\ *Proof of claim.* Let us denote the $\mathbb{P}(E)\times_{S} (\mathbb{P}(E))^{d-2} \rightarrow \mathbb{P}(E)$ the $k$-th projection to $\mathbb{P}(E)$ by $p'_{k}$, where $3\leq k \leq d$.\ Now, consider the following diagram:\ ((E) \_[X]{} (E))\_[S]{} ((E))\^[d-2]{}\ ((E))\_[S]{} ((E))\^[d-2]{}\ X\_[S]{} X\^[d-2]{} \ Let us denote the map $p_{2}\times p_{3}\times ..\times p_{d}$ by $f$, and $(p\circ p'_{2})\times (p\circ p'_{3})\times ... \times (p\circ p'_{d}))$ by $g$. Then, if we denote by $X_{k}\subseteq X\times_{S} X^{d-2}$ the closed subscheme whose points are of the form $(x_{1},x_{2},..,x_{d-1})$ with $x_{1}=x_{k}$, then $\mathcal{O}(\sum\limits_{k=3}^{d}n\Delta_{1,2,k,X}))=(g\circ f)^{*}\mathcal{O}(\sum\limits^{d}_{k=2}nX_{k})$.\ Hence, by projection formula we have\ $$\begin{split} f_{*}(\mathcal{L}\bigotimes \mathcal{O}(\sum\limits_{k=3}^{d} n\Delta_{1,2,k,X}))=f_{*}((p_{1})_{*}\mathcal{O}(-1)\bigotimes(p_{2})^{*}\mathcal{O}(1)\bigotimes (p\circ p_{2})^{*}\mathcal{T}_{X/S}\bigotimes \mathcal{O}(\sum\limits_{k=3}^{d} n\Delta_{1,2,k,X})) \\ =(f_{*}((p_{1})^{*}\mathcal{O}(-1)))\bigotimes ((p'_{2})^{*}\mathcal{O}(1))\bigotimes (p\circ p'_{2})^{*}\mathcal{T}_{X/S}\bigotimes g^{*}\mathcal{O}(\sum\limits^{d}_{k=2}nX_{k})) \end{split}$$\ Hence, it is enough to show that $f_{*}(p_{1})^{*}\mathcal{O}(-1)=0$.\ Now consider the following fibered diagram:\ ((E)\_[X]{}(E))\_[S]{}((E))\^[d-2]{} & (E)\ (E)\_[S]{} ((E))\^[d-2]{} & X \ Since $p\circ p'_{2}$ is flat, we have by \[7, Prop. 9.3\]\ $f_{*}p^{*}_{1}\mathcal{O}(1)\cong (p\circ p'_{2})^{*}p_{*}\mathcal{O}(-1)=0$.\ (iii) As before, we identify $\Delta_{j}$ with $\mathcal{Z}$. Then, we have,\ $H^{0}(X\times_{S} \mathcal{U}, \mathcal{H}om((\pi_{1}\times \pi_{2,j})^{*}\mathcal{F}(E,1),\pi^{*}_{2,j}\mathcal{O}(1)|_{\Delta_{j}}))\\ =H^{0}(\mathcal{U},\mathcal{H}om (p^{*}_{j}\mathcal{F}(E,1)|_{\Delta_{1}},p^{*}_{j}\mathcal{O}(1) ))=H^{0}(\mathcal{U},p^{*}_{j}(\mathcal{F}(E,1)^{\vee}|_{\Delta_{1}}\bigotimes \mathcal{O}(1)))$.\ Now, since $p^{*}_{j}(\mathcal{F}(E,1)^{\vee}|_{\Delta_{1}}\bigotimes \mathcal{O}(1))$ is vector bundle over $\mathcal{Z}$ and codimension of $\mathcal{Z} \setminus \mathcal{U}\geq 2$, hence,\ $H^{0}(\mathcal{U},p^{*}_{j}(\mathcal{F}(E,1)^{\vee}|_{\Delta_{1}}\bigotimes \mathcal{O}(1)))=H^{0}(\mathcal{Z},p^{*}_{j}(\mathcal{F}(E,1)^{\vee}|_{\Delta_{1}}\bigotimes \mathcal{O}(1)))$.\ Using projection formula for the morphism $p_{j}$, we get that\ $H^{0}(\mathcal{U},p^{*}_{j}(\mathcal{F}(E,1)^{\vee}|_{\Delta_{1}}\bigotimes \mathcal{O}(1)))=H^{0}(\mathbb{P}(E),(\mathcal{F}(E,1)^{\vee}|_{\Delta_{1}}\bigotimes \mathcal{O}(1)))$.\ Now, over $\mathbb{P}(E)$, $\mathcal{F}(E,d)^{\vee}|_{\Delta_{1}}\bigotimes \mathcal{O}(1)\cong \mathcal{T}_{\mathbb{P}(E)/S}$. Hence, we have the result. Applications ============ We have the following left exact sequence of algebraic groups: $$\begin{tikzcd} 0 \arrow[r] & \text{GL}(E)/k^{*} \arrow[r] & \text{Aut}^{o}(\mathcal{Q}(E,d)/S) \arrow[r] & \text{Aut}^{o}(X/S) \end{tikzcd}$$ The corresponding sequence of lie algebras is given by:\ $$\begin{tikzcd} 0 \arrow[r] & H^{0}(X,\text{ad }E) \arrow[r] & H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q})(E,d)/S}) \arrow[r] & H^{0}(X,\mathcal{T}_{X/S}) \end{tikzcd}$$ The left exactness of the above sequences follow from Theorem 2.1 and from the fact that $\text{Aut}^{o}(\mathbb{P}(E)/S)$ and its lie algebra fits into the above exact sequences.\ If genus of the fibres of $X\rightarrow S$ is $\geq 2$, then\ (i)Aut$^{o}(\mathcal{Q}(E,d)/S)=\text{GL}(E)/k^{*}$\ (ii)$H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)/S})=H^{0}(X,\text{ad }E)$\ If genus of each fibre is $\geq 2$, then $(p_{S})_{*}\mathcal{T}_{X/S}=0$. In particular, $H^{0}(X,\mathcal{T}_{X/S})=0$. Hence, $\text{Aut}^{o}(X/S)=0$. Now the corollary follows from Theorem 2.1 Let $C$ is a smooth projective curve of genus $\geq 2$ over an algebraically closed field $k$. Fix $\bar{d}=(d_{1},d_{2},..,d_{k})\in \mathbb{N}^{k}$ with $d_{1}>d_{2}>..>d_{k}$ and $r\geq 1$. Let $\mathcal{D}(r,\bar{d})$ be the flag scheme of filtration of quotients of $\mathcal{O}^{r}_{C}\rightarrow B_{1}\rightarrow B_{2}\rightarrow..\rightarrow B_{d}$, where $\mathcal{O}^{r}_{C}\rightarrow B_{i}$ is a torsion quotient of degree $d_{i}$\[1,2.A.1\]. Then,\ (i)$H^{0}(\mathcal{D}(r,\bar{d}), \mathcal{T}_{D(r,\bar{d})})=sl(r)$\ (ii)Aut$^{o}(\mathcal{D}(r,\bar{d}))=$ PGL$(r)$ Over $C\times \mathcal{D}(r,(d_{2},d_{3},..,d_{k}))$ we have the universal chain of filtrations:\ (r,d\_[2]{})(r,d\_[3]{})..(r,d\_[k]{})\^[r]{}\_[C(r,(d\_[1]{},d-[2]{},..,d\_[k]{}))]{} Then $D(r,\bar{d})$ is the relative quot scheme of torsion quotients of degree $d_{1}-d_{2}$ of the vector bundle $\mathcal{A}(r,d_{1})$ for the map $C\times \mathcal{D}(r,(d_{2},d_{3},..,d_{k}))\rightarrow \mathcal{D}(r,(d_{2},d_{3},..,d_{k}))$. Then, by Corollary 3.1, we get that\ $H^{0}(\mathcal{D}(r,\bar{d}),\mathcal{T}_{\mathcal{D}(r,\bar{d})/\mathcal{D}(r,(d_{2},d_{3,..,d_{k}))}})=H^{0}(C\times \mathcal{D}(r,(d_{2},d_{3},..,d_{k})), \text{ad }\mathcal{A}(r,d_{2}))$. Now, we know that for $k\geq 1$, $\mathcal{A}(r,d_{2})$ is stable with respect to certain polarisations on\ $C\times \mathcal{D}(r,(d_{2},d_{3},..,d_{k}))$ \[4, Th.3.2.4, Th. 5.1\], so $H^{0}(C\times \mathcal{D}(r,(d_{2},d_{3},..,d_{k})), \text{ad }\mathcal{A}(r,d_{2}))=0$. Now, by induction on k, we get that $H^{0}(\mathcal{D}(r,\bar{d}),\mathcal{T}_{D}(r,\bar{d}))=H^{0}(C,\text{ad }\mathcal{O}^{r}_{C})=sl(r)$.(ii) follows immediately from this fact. Let $C$ be a smooth projective curve over an algebraically closed field $k$. Let $E$ be a vector bundle of rank $\geq 3$ over $C$. Fix $d\geq 1$. Let $\mathcal{Q}(E,d)$ be the quot scheme of torsion quotients of $E$ of degree $d$. Then,\ (i)(a)If genus of $C=0$,i.e. $C\cong \mathbb{P}^{1}$, then\ Aut$^{o}(\mathcal{Q}(E,d))=$PGL$(2,k)\ltimes$GL$(E)/k^{*}$.\ (b) If genus of $C=1$ and if $E$ is semistable, then\ We have the following sequence of algebraic groups:\ 0 & (E)/k\^[\*]{} & \^[o]{}((E,d)) & \^[o]{}(C) & 0 \ In both of these cases, we have the exact sequence of lie algebras:\ 0 & H\^[0]{}(C, E) & H\^[0]{}((E,d),\_[(E,d)]{}) & H\^[0]{}(C,\_[C]{}) & 0 \ (ii) If $E$ is not semistable, then $\text{Aut}^{o}(\mathcal{Q}(E,d))=\text{GL}(E)/k^{*}$ and $H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)})=H^{0}(C,\text{ad }E)$ (i)(a) If $C\cong \mathbb{P}^{1}$, then any vector bundle $E$ admits a $\text{GL}(2)$ linearisation, in paricular we have a homomorphism GL$(2)\rightarrow \text{Aut}^{o}(\mathcal{Q}(E,d))$. This homomorphism factors through $\text{PGL}(2,k)$ and gives a section to the map $\text{Aut}^{o}(\mathcal{Q}(E,d))\rightarrow \text{PGL}(2,k)$. Therefore, the left exact sequence (3.1) is exact in this case and it splits.\ (b) We show that for any $g\in \text{Aut}^{o}(C)$ $g^{*}E\cong E$. This will show that the sequence (3.1) is also right exact in this case.\ We know that $E\cong \bigoplus E_{i}$, where $E_{i}$’s are indecomposable vector bundles. Since, $E$ is semistable, hence $\mu(E_{i})=\mu(E_{j})$ $\forall i,j$. Since the $\mathcal{Q}(E,d)\cong \mathcal{Q}(E\bigotimes \mathcal{L},d)$ canonically for any line bundle $\mathcal{L}$, after twisting $E$ by a line bundle of appropriate degree, we can assume $\mu(E)=\mu(E_{i})=0$ $\forall i$. By \[5,Th.5(i)\], $E_{i}\cong F_{r_{i}}\bigotimes \mathcal{L}_{i}$, where $F_{r_{i}}$ is the unique indecomposable vector bundle of rank $r_{i}$ with $H^{0}(C,E_{r_{i}})\neq 0$ and $\mathcal{L}$ is a line bundle of degree $0$.\ It follows that for any $g\in \text{Aut}^{o}(C)$, $g^{*}F_{r}\cong F_{r}$, since $g^{*}F_{r}$ is also indecomposable bundle of degree $0$ and rank $r$ with $H^{0}(C,g^{*}F_{r})\neq 0$.\ So, we need to show that $g^{*}\mathcal{L}\cong \mathcal{L}$ for any $\mathcal{L}\in \text{Pic}^{o}(C)$.\ Fix a base point $x_{0}\in C$. Then, under the group structure of $C$ with $x_{0}$ as the identity, $\text{Aut}^{o}(C)\cong C$ i.e. any element of $\text{Aut}^{o}(C)$ is given by $y \mapsto y+_{C}x$ for a fixed $x\in C$.(Here we denote the group addition of $C$ by $+_{C}$) Fix such an automorphism $x\in C\cong \text{Aut}^{0}(C)$\ Now, $(C,x_{0},+_{C})\cong \text{Pic}^{o}(C)$, with the morphism given by $z \mapsto \mathcal{O}(z-x_{0})$.\ Let us assume $\mathcal{L}=\mathcal{O}(z-x_{0})$ for $z\in C$.\ Then $x^{*}\mathcal{L}=\mathcal{O}((z-_{C}x)-(x_{0}-_{C}x))$.\ Since, $(C,x_{o},+_{C})\cong \text{Pic}^{0}(C)$ is a homomorphism, it follows that\ $x^{*}\mathcal{L}\cong \mathcal{O}((z-_{C}x)-(x_{0}-_{C}x))=\mathcal{O}(((z-_{C}x)-x_{0})-((x_{0}-_{C}x)-x_{0}))\\ =\mathcal{O}((z-x_{0})-(x-x_{0})-(x_{0}-x_{0})+(x-x_{0}))=\mathcal{O}(z-x_{0})=\mathcal{L}$.\ Hence, it follows that for any $g\in \text{Aut}^{o}(C)$, $g^{*}E\cong E$ for any $E$ semistable.\ (ii) By \[6, Proposition 6.13\], every semi-homogenous vector bundle\[6, Definition 5.2\] is semistable. In particular, if $E$ is not semistable, then the map $H^{0}(\mathbb{P}(E),\mathcal{T}_{\mathbb{P}(E)})\rightarrow H^{0}(C,\mathcal{T}_{C})$ is zero. Hence, using sequence (3.2), we get $H^{0}(C,\text{ad }E)\rightarrow H^{0}(\mathcal{Q}(E,d),\mathcal{T}_{\mathcal{Q}(E,d)})$ is an isomorphism. From this, it follows that $\text{Aut}^{o}(\mathcal{Q}(E,d))=\text{GL}(E)/k^{*}$. [AAA]{} D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Cambridge University Press, Second Edition(2010). I.Biswas, A.Dhillon and J.Hurtubise, Automorphisms of the Quot Schemes Associated to Compact Riemann Surfaces, International Mathematics Research Notices, Vol. 2015, No. 6, pp.1445-1460 M.Brion, On automorphism groups of fiber bundles, arXiv: https://arxiv.org/pdf/1012.4606 (2011) C.Gangopadhyay, Stability of sheaves over Quot Schemes, Bulletin des Sciences Mathématiques, Volume 149, (November 2018), pgs. 66-85 M. F. Atiyah, Vector bundles over an elliptic curve. Proc. London Math. Soc. (3), 414-452, 1957 S.Mukai, Semi-homogeneous vector bundles on an abelian variety, J.Math. Kyoto Univ. (JMKYAZ) 18-2(1978) 239-272 R.Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, NewYork–Heidelberg, 1977.

--- abstract: 'For any two configurations of ordered points ${{\mathbf p}}=({{\mathbf p}}_{1},\cdots,{{\mathbf p}}_{N})$ and ${{\mathbf q}}=({{\mathbf q}}_{1},\cdots,{{\mathbf q}}_{N})$ in Euclidean space ${{\mathbb E}}^d$ such that ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$, there exists a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in dimension $2d$; Bezdek and Connelly used this to prove the Kneser-Poulsen conjecture for the planar case. In this paper, we show that this construction is optimal in the sense that for any $d \ge 2$ there exists configurations of $(d+1)^2$ points ${{\mathbf p}}$ and ${{\mathbf q}}$ in ${{\mathbb E}}^d$ such that ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$ but there is no continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in dimension less than $2d$. The techniques used in our proof are completely elementary.' address: - | Department of Computer Science\ National University of Singapore\ Singapore - | Department of Mathematics\ National University of Singapore\ Singapore 119076 - | Department of Mathematics\ National University of Singapore\ Singapore 119076 author: - 'Holun Cheng, Ser Peow Tan and Yidan Zheng' title: On continuous expansions of configurations of points in Euclidean space --- \[section\] \[thm\][Lemma]{} \[thm\][Conjecture]{} \[thm\][Corollary]{} \[thm\][Addendum]{} \[thm\][Proposition]{} \[thm\][Definition]{} \[thm\][Remark]{} \[thm\][[**Example**]{}]{} \[thm\][[ **Question**]{}]{} [^1] Introduction and statement of results. {#s:intro} ====================================== Let ${{\mathbb E}}^d$ be the Euclidean space of dimension $d \ge 2$, where we identify and represent the points of ${{\mathbb E}}^d$ by their position vectors. ${{\mathbb E}}^d$ is endowed with the standard inner product ${{\mathbf u}}\cdot {{\mathbf v}}$ and norm $|{{\mathbf u}}|=\sqrt{{{\mathbf u}}.{{\mathbf u}}}$. Suppose that $d<f$, then ${{\mathbb E}}^f \cong {{\mathbb E}}^d \times {{\mathbb E}}^{f-d}$ and we have the standard projections $\pi_1: {{\mathbb E}}^f \rightarrow {{\mathbb E}}^d$ and $\pi_2: {{\mathbb E}}^f \rightarrow {{\mathbb E}}^{f-d}$ given by $$\pi_1(u_1,\ldots,u_f)=(u_1, \ldots,u_d), \qquad \pi_2(u_1,\ldots,u_f)=(u_{d+1}, \ldots,u_{f}),$$ and the standard inclusion $\iota: {{\mathbb E}}^d \rightarrow {{\mathbb E}}^f$ given by $$\iota({{\mathbf u}})=\iota(u_1,\ldots,u_d)=(u_1,\ldots,u_d,0\ldots,0).$$ Note that $\pi_1 \circ \iota=id$ on ${{\mathbb E}}^d$, and for ${{\mathbf u}}, {{\mathbf v}}\in {{\mathbb E}}^f$, $$\begin{aligned} {{\mathbf u}}&=& (\pi_1({{\mathbf u}}), \pi_2({{\mathbf u}})), \\ {{\mathbf u}}\cdot{{\mathbf v}}&=& \pi_1({{\mathbf u}})\cdot \pi_1({{\mathbf v}})+\pi_2({{\mathbf u}})\cdot \pi_2({{\mathbf v}}), \label{eqn:dotprod}\\ | {{\mathbf u}}|^2 &=& | \pi_1({{\mathbf u}})|^2+| \pi_2({{\mathbf u}}) |^2. \label{eqn:norms}\end{aligned}$$ Let ${{\mathbf p}}=({{\mathbf p}}_{1},\cdots,{{\mathbf p}}_{N})$ and ${{\mathbf q}}=({{\mathbf q}}_{1},\cdots,{{\mathbf q}}_{N})$ be two configurations of $N$ ordered points in ${{\mathbb E}}^{d}$, where ${{\mathbf p}}_{i},{{\mathbf q}}_{i}\in{{\mathbb E}}^{d}$ for $i=1,\ldots,N$, and suppose $f > d$. (Expansions in ${{\mathbb E}}^d$) ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$ if $$| {{\mathbf p}}_i-{{\mathbf p}}_j|\le| {{\mathbf q}}_i-{{\mathbf q}}_j|, \qquad 1 \le i<j \le N.$$ (Continuous expansions in ${{\mathbb E}}^f$) We say that there is a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^f$ if there exists a family of continuous functions (*continuous motions*) $${{\mathbf{f}}}_i:[0,1]\longrightarrow\mathbb{E}^{f}, \quad i=1, \ldots N$$ such that for $1 \le i <j \le N$ and $0 \le t_1<t_2 \le 1$, 1. ${{\mathbf{f}}}_i(0)=\iota({{\mathbf p}}_i)$, ${{\mathbf{f}}}_i(1)=\iota({{\mathbf q}}_i)$; 2. $| {{\mathbf{f}}}_{i}(t_1)-{{\mathbf{f}}}_{j}(t_1)| \le | {{\mathbf{f}}}_{i}(t_2)-{{\mathbf{f}}}_{j}(t_2)|$. Note that if there is a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^f$, then ${{\mathbf q}}$ is necessarily an expansion of ${{\mathbf p}}$ in ${{\mathbb E}}^d$, but an expansion may not admit a continuous expansion in the same or a higher dimension. The following result by R. Alexander [@Alex] shows that any expansion admits a continuous expansion in twice the dimension. \[thm:continuous\][@Alex], see also [@BezCon]. Suppose that ${{\mathbf q}}=({{\mathbf q}}_1, \ldots {{\mathbf q}}_N)$ is an expansion of ${{\mathbf p}}=({{\mathbf p}}_1, \ldots {{\mathbf p}}_N)$ in ${{\mathbb E}}^d$. Then the family of functions ${{\mathbf{f}}}_i:[0,1] \longrightarrow {{\mathbb E}}^{2d}, \quad i=1, \ldots, N$, given by $${{\mathbf{f}}}_{i}(t)=\left(\frac{{{\mathbf p}}_{i}+{{\mathbf q}}_{i}}{2}+(cos\pi t)\frac{{{\mathbf p}}_{i}-{{\mathbf q}}_{i}}{2}, \,(sin\pi t)\frac{{{\mathbf p}}_{i}-{{\mathbf q}}_{i}}{2}\right)$$ is a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^{2d}$. We reproduce the proof here for completeness. Clearly, ${{\mathbf{f}}}_i$ is continuous for $i=1, \ldots, N$ and ${{\mathbf{f}}}_i(0)=\iota({{\mathbf p}}_i)$, ${{\mathbf{f}}}_i(1)=\iota({{\mathbf q}}_i)$. Expanding, $4| {{\mathbf{f}}}_{i}(t)-{{\mathbf{f}}}_{j}(t)|^{2}$ $$=|({{\mathbf p}}_{i}-{{\mathbf p}}_{j})-({{\mathbf q}}_{i}-{{\mathbf q}}_{j})|^{2}+|({{\mathbf p}}_{i}-{{\mathbf p}}_{j})+({{\mathbf q}}_{i}-{{\mathbf q}}_{j})|^{2}+2(\cos\pi t)(| {{\mathbf p}}_{i}-{{\mathbf p}}_{j}|^{2}-| {{\mathbf q}}_{i}-{{\mathbf q}}_{j}|^{2}).$$ Since ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$, $| {{\mathbf p}}_{i}-{{\mathbf p}}_{j}|^{2}-| {{\mathbf q}}_{i}-{{\mathbf q}}_{j}|^{2}\le0$ for all $ i \neq j$. Therefore $| {{\mathbf{f}}}_{i}(t)-{{\mathbf{f}}}_{j}(t)|$ is non-decreasing on $[0,1]$. Bezdek and Connelly used the above in [@BezCon], together with results of Csikós [@Csi] to prove the Kneser-Poulsen conjecture [@Kne] for the plane. More specifically, they showed that if there is a piecewise analytic expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in dimension $d+2$, then the Kneser-Poulsen conjecture holds for balls centered at ${{\mathbf p}}$ and ${{\mathbf q}}$, that is, the volume of the union of the balls $B({{\mathbf p}}_i,r_i)$ is less than or equal to the volume of the union of the balls $B({{\mathbf q}}_i,r_i)$, where $r_i>0$. Similarly, the same method shows that the conjecture holds if the number of balls $N \le d+3$, generalizing a result of Gromov in [@Gro]. This raises the question, as pointed out in [@BezCon], of whether it is possible to find continuous expansions in dimensions less than $2d$ for all expansions ${{\mathbf q}}$ of ${{\mathbf p}}$ in dimension $d$. If so, then the approach of Bezdek and Connelly can be applied to prove the Kneser-Poulsen conjecture in more general settings. Our main result is a negative answer to this question, specifically, we have: \[thm:main\](Main Theorem) There exists configurations ${{\mathbf p}}=({{\mathbf p}}_1, \ldots {{\mathbf p}}_N)$ in ${{\mathbb E}}^d$ with expansions ${{\mathbf q}}=({{\mathbf q}}_1, \ldots {{\mathbf q}}_N)$, where $N=(d+1)^2$, which do not admit continuous expansions in dimensions less than $2d$. [*Remark:*]{} The example we construct is in fact the same as that constructed independently by Belk and Connelly in [@BelCon], and in both cases, based on the example constructed in [@BezCon] for the planar case. However, our proof is more elementary and uses only basic linear algebra and some simple rigidity results. Indeed, our proof shows that away from the endpoints, any continuous expansion cannot be embedded into dimension less than $2d$ at any time $t \in (0,1)$. The configurations ${{\mathbf p}}$ and ${{\mathbf q}}$ are built from the $(d+1)$ vertices ${{\mathbf v}}_0, \ldots, {{\mathbf v}}_d$ of the regular $d$-simplex $\sigma_d \subset {{\mathbb E}}^d$, together with the vertices of the inward and outward flaps associated to the faces of $\sigma_d$, specifically, each face $F^i$ ($i=0. \ldots,d$) of $\sigma_d$ may be pushed orthogonally towards or away from the center of $\sigma_d$ by a distance $s>0$, to obtain flaps $F^i_{inw}$ and $F^i_{out}$ respectively (note that Belk and Connelly had a slightly different definition for flaps in [@BelCon]). The configuration ${{\mathbf p}}$ consists of the vertices of $\sigma_d$ and of the inward flaps $F^i_{inw}$ and the configuration ${{\mathbf q}}$ consists of the corresponding vertices of $\sigma_d$ and of the outward flaps $F^i_{out}$. Note that each flap has $d$ vertices so that ${{\mathbf p}}$ and ${{\mathbf q}}$ consists of $(d+1)^2$ points. The rest of the paper will be devoted to explaining this construction (§\[s:simplex\]), showing that ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$ (§\[s:expansion\]), and proving that there is no continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^f$ for $f <2d$ (§\[s:proof\]). [*Acknowledgements.*]{} This work arose from an undergraduate honors project of the third author under the supervision of the first and second authors. The authors are grateful to Jean-Marc Schlenker for helpful conversations, and also for bringing their attention to [@BelCon] arising from his correspondence with R. Connelly. Regular simplices with flaps {#s:simplex} ============================ Let $\sigma :=\sigma_d \subset {{\mathbb E}}^d$ be the regular simplex with vertices ${{\mathbf u}}_i$, $i=0, \ldots, d$ and center at the origin $O$ such that $| {{\mathbf u}}_i| =1$ for all $i$ (see figure 1). Then $$\label{eqn:innerproductofnorms} {{\mathbf u}}_i \cdot {{\mathbf u}}_j=-\frac{1}{d}, \quad i \neq j$$ {width="70.00000%"} see for example Coxeter [@Cox], or Parks and Wills [@ParWil] for an elementary proof. Denote by $F^i$ the face of $\sigma$ which does not contain the vertex ${{\mathbf u}}_i$. Then the norm of $F^i$, the outward facing unit normal ${{\mathbf n}}_i$ to $F^i$ is the vector $-{{\mathbf u}}_i$. Fix $s>0$. For each face $F^i$, $i=0, \ldots, d$, define the outward $i$th flap of depth $s$ to be $F^i$ translated by $s{{\mathbf n}}_i=-{{\mathbf u}}_i$, that is, $$F^i_{out}:=F^i -s{{\mathbf u}}_i.$$ Similarly, the inward $i$th flap of depth $s$ is given by $$F^i_{inw}:=F^i +s{{\mathbf u}}_i.$$ Each flap has $d$ vertices and if we denote the vertices of $F^i_{out}$ by ${{\mathbf{c}}}^i_j$ and those of $F^i_{inw}$ by ${{\mathbf{b}}}^i_j$, where $j\neq i$ (see figure 2 for the case when $d=2$ and $3$), then we have, for $i,j\in\{0, \ldots, d\}$, $i \neq j$, $$\begin{aligned} {{\mathbf{c}}}^i_j &=& {{\mathbf u}}_j+s{{\mathbf n}}_i = {{\mathbf u}}_j-s{{\mathbf u}}_i \\ {{\mathbf{b}}}^i_j &=& {{\mathbf u}}_j-s{{\mathbf n}}_i = {{\mathbf u}}_j+s{{\mathbf u}}_i\end{aligned}$$ {width="80.00000%"} The configurations ${{\mathbf p}}$ and ${{\mathbf q}}$ we are interested in consists of the vertices of the regular simplex with inward and outward flaps respectively, defined by $$\label{eqn:pandq} {{\mathbf p}}=\{{{\mathbf u}}_i\} \cup \{{{\mathbf{b}}}^i_j\}, \quad {{\mathbf q}}=\{{{\mathbf u}}_i\} \cup \{{{\mathbf{c}}}^i_j\}, \quad i,j \in \{0,1,\ldots,d\},\quad i \neq j$$ where ${{\mathbf p}}$ and ${{\mathbf q}}$ are ordered so that the correspondence between the elements from the indexing is preserved. We have: \[thm:main2\] Suppose that ${{\mathbf p}}$ and ${{\mathbf q}}$ are configurations in ${{\mathbb E}}^d$ consisting of the vertices of the regular simplex with inward and outward flaps defined as in (\[eqn:pandq\]). Then 1. ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$ in ${{\mathbb E}}^d$; 2. there does not exist a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^f$ for $f <2d$. We will prove (a) in the next section and (b) in the following section. We note that although (a) was claimed in [@BelCon], no proof was given, we give a proof here for completeness. Also our proof of (b) is independent of, and more elementary than that given in [@BelCon]. Proof that ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$ {#s:expansion} ============================================================= We only need to consider the distances between vertices on $\sigma$ and vertices on the flaps, or between vertices on the flaps. In the first case, we have $$| {{\mathbf u}}_k- {{\mathbf{b}}}^i_j| =| {{\mathbf u}}_k- {{\mathbf{c}}}^i_j|, \quad \hbox{if}\quad k \neq i,$$ since ${{\mathbf u}}_k \subset F^i$, and $$| {{\mathbf u}}_i- {{\mathbf{b}}}^i_j| < | {{\mathbf u}}_i- {{\mathbf{c}}}^i_j|$$ since by reflecting on the face $F^i$, we see there is a broken path from ${{\mathbf u}}_i$ to ${{\mathbf{b}}}^i_j$ of length $| {{\mathbf u}}_k- {{\mathbf{c}}}^i_j|$. The argument works if we replace $\sigma$ by any simplex. In the second case, we have, for $i \neq j$, $k \neq l$, $$\begin{aligned} | {{\mathbf{b}}}_j^i-{{\mathbf{b}}}_l^k|^2 &=& | ({{\mathbf u}}_j+s{{\mathbf u}}_i)-({{\mathbf u}}_l+s{{\mathbf u}}_k) |^2 \\ ~ &=& | {{\mathbf u}}_j-{{\mathbf u}}_l |^2+2s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf u}}_i-{{\mathbf u}}_k)+s^2| {{\mathbf u}}_i-{{\mathbf u}}_k |^2 \\ | {{\mathbf{c}}}_j^i-{{\mathbf{c}}}_l^k|^2 &=& | {{\mathbf u}}_j-{{\mathbf u}}_l |^2-2s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf u}}_i-{{\mathbf u}}_k)+s^2| {{\mathbf u}}_i-{{\mathbf u}}_k |^2 \\ \Longrightarrow \quad | {{\mathbf{b}}}_j^i-{{\mathbf{b}}}_l^k|^2 &-& | {{\mathbf{c}}}_j^i-{{\mathbf{c}}}_l^k|^2= 4s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf u}}_i-{{\mathbf u}}_k)\end{aligned}$$ If $i=k$, or $j=l$, or $i,j,k,l$ are all distinct, then $4s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf u}}_i-{{\mathbf u}}_k)=0$ by (\[eqn:innerproductofnorms\]) so that $$| {{\mathbf{b}}}_j^i-{{\mathbf{b}}}_l^k|= | {{\mathbf{c}}}_j^i-{{\mathbf{c}}}_l^k|.$$ If $i=l$ or $j=k$, then $$4s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf u}}_i-{{\mathbf u}}_k)=4s(\frac{1}{d}-1)<0$$ by (\[eqn:innerproductofnorms\]), hence in all cases, $$| {{\mathbf{b}}}_j^i-{{\mathbf{b}}}_l^k|\leq | {{\mathbf{c}}}_j^i-{{\mathbf{c}}}_l^k|.$$ [*Remark:*]{} In the case where we start with any simplex instead of $\sigma_d$, then $$| {{\mathbf{b}}}_j^i-{{\mathbf{b}}}_l^k|^2 - | {{\mathbf{c}}}_j^i-{{\mathbf{c}}}_l^k|^2= 4s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf n}}_k-{{\mathbf n}}_i).$$ Again, if $i=k$, or $j=l$, or $i,j,k,l$ are all distinct, then $4s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf n}}_k-{{\mathbf n}}_i)=0$, and if $i=l$ or $j=k$, then $4s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf n}}_k-{{\mathbf n}}_i)<0$, so Theorem \[thm:main2\](a) holds if we replace the regular simplex with any simplex. Proof that there is no continuous expansion in dimension $<2d$ {#s:proof} =============================================================== The main tools we use are some basic linear algebra as described in §\[s:intro\], and the fact that the configurations ${{\mathbf p}}$ and ${{\mathbf q}}$ contain several sub-configurations which are rigid under continuous expansion since the pair-wise distances are preserved in the sub-configurations. We first outline the strategy of our proof, note that it suffices to show that there is no continuous expansion in dimension $2d-1$. 1. We will assume for a contradiction that there exists a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^{2d-1}$; 2. we construct for each face $F^k$ a displacement vector function $${{\mathbf{d}}}_k:[0,1] \longrightarrow {{\mathbb E}}^{2d-1} \cong {{\mathbb E}}^d \times {{\mathbb E}}^{d-1};$$ such that ${{\mathbf{d}}}_k(t)$ is orthogonal to $F^k$ and $| {{\mathbf{d}}}_k(t)| =s$ for all $t \in [0,1]$; 3. show that there is some $t_0 \in [0,1]$ such that the projection $\pi_2({{\mathbf{d}}}_k(t_0))$ to ${{\mathbb E}}^{d-1}$ is non-zero for all $k\in \{0,1, \ldots,d\}$; 4. show that the set $\{{{\mathbf w}}_k=\pi_2({{\mathbf{d}}}_k(t_0))\}\subset {{\mathbb E}}^{d-1}$ consists of pairwise obtuse vectors; 5. show that this is not possible to give the required contradiction. \(I) Consider ${{\mathbb E}}^{2d-1}\cong {{\mathbb E}}^d \times {{\mathbb E}}^{d-1}$ and define the projections $\pi_1:{{\mathbb E}}^{2d-1} \rightarrow {{\mathbb E}}^d$ and $\pi_2:{{\mathbb E}}^{2d-1} \rightarrow {{\mathbb E}}^{d-1}$ and the inclusion $\iota: {{\mathbb E}}^d \rightarrow {{\mathbb E}}^{2d-1}$ as in §\[s:intro\]. Suppose that there is a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^{2d-1} \cong {{\mathbb E}}^d \times {{\mathbb E}}^{d-1}$. Let ${{\mathbf{f}}}_k,~~ {{\mathbf{g}}}_j^i:[0,1]\rightarrow {{\mathbb E}}^{2d-1}$, $i,j,k \in \{0, \ldots, d\}$, $i \neq j$, be the continuous motions of ${{\mathbf u}}_k$ and ${{\mathbf{b}}}_j^i$ respectively which define the continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$. Since $\sigma_d$ is rigid, we may assume without loss of generality that ${{\mathbf u}}_k$ remains stationary throughout the motion, that is $$\label{eqn:ffkt} {{\mathbf{f}}}_k(t) \equiv \iota({{\mathbf u}}_k), \quad k=0, \ldots, d.$$ We also have $$\label{eqn:ggji} {{\mathbf{g}}}_j^i(0)=\iota({{\mathbf{b}}}_j^i)=\iota({{\mathbf u}}_j+s{{\mathbf u}}_i), ~~ {{\mathbf{g}}}_j^i(1)=\iota({{\mathbf{c}}}_j^i)=\iota({{\mathbf u}}_j-s{{\mathbf u}}_i).$$ (II) We will need the following: \[prop:parallelogram\] Suppose that $({{\mathbf u}}_1,{{\mathbf u}}_2, {{\mathbf u}}_3,{{\mathbf u}}_4), ({{\mathbf v}}_1,{{\mathbf v}}_2,{{\mathbf v}}_3,{{\mathbf v}}_4) \subset {{\mathbb E}}^n$ are configurations such that $$\label{eqn:parallelogram} |{{\mathbf u}}_i-{{\mathbf u}}_j|=|{{\mathbf v}}_i-{{\mathbf v}}_j| \quad \hbox{ for all} \quad i \neq j.$$ If $({{\mathbf u}}_1,{{\mathbf u}}_2, {{\mathbf u}}_3,{{\mathbf u}}_4)$ is a parallelogram, then $({{\mathbf v}}_1,{{\mathbf v}}_2,{{\mathbf v}}_3,{{\mathbf v}}_4)$ is also a parallelogram and $({{\mathbf v}}_1, {{\mathbf v}}_2, {{\mathbf v}}_3, {{\mathbf v}}_4)\cong ({{\mathbf u}}_1,{{\mathbf u}}_2, {{\mathbf u}}_3,{{\mathbf u}}_4)$. Let ${{\mathbf w}}$ and ${{\mathbf w}}'$ be the midpoints of $({{\mathbf u}}_2,{{\mathbf u}}_4)$ and $({{\mathbf v}}_2,{{\mathbf v}}_4)$ respectively. We have $\triangle({{\mathbf u}}_1,{{\mathbf u}}_2, {{\mathbf u}}_4)\cong \triangle({{\mathbf v}}_1,{{\mathbf v}}_2, {{\mathbf v}}_4)$, hence $|{{\mathbf w}}-{{\mathbf u}}_1|=|{{\mathbf w}}'-{{\mathbf v}}_1|$ (see figure 3). Similarly, $\triangle({{\mathbf u}}_2,{{\mathbf u}}_3, {{\mathbf u}}_4)\cong \triangle({{\mathbf v}}_2,{{\mathbf v}}_3, {{\mathbf v}}_4)$, so $|{{\mathbf u}}_3-{{\mathbf w}}|=|{{\mathbf v}}_3-{{\mathbf w}}'|$. Also, by (\[eqn:parallelogram\]) $|{{\mathbf v}}_3-{{\mathbf v}}_1|=|{{\mathbf u}}_3-{{\mathbf u}}_1|$ and since $({{\mathbf u}}_1,{{\mathbf u}}_2, {{\mathbf u}}_3,{{\mathbf u}}_4)$ is a parallelogram, $|{{\mathbf u}}_3-{{\mathbf u}}_1|=|{{\mathbf u}}_3-{{\mathbf w}}|+|{{\mathbf w}}-{{\mathbf u}}_1|$. Hence $$|{{\mathbf v}}_3-{{\mathbf v}}_1|=|{{\mathbf u}}_3-{{\mathbf u}}_1|=|{{\mathbf u}}_3-{{\mathbf w}}|+|{{\mathbf w}}-{{\mathbf u}}_1|=|{{\mathbf v}}_3-{{\mathbf w}}'|+|{{\mathbf w}}'-{{\mathbf v}}_1|.$$ ![The configurations (${{\mathbf u}}_1$,${{\mathbf u}}_2$,${{\mathbf u}}_3$,${{\mathbf u}}_4$) and (${{\mathbf v}}_1$,${{\mathbf v}}_2$,${{\mathbf v}}_3$,${{\mathbf v}}_4$)[]{data-label="fig:figure_3"}](./figure_3.png){width="80.00000%"} Hence, ${{\mathbf v}}_1, {{\mathbf w}}'$ and ${{\mathbf v}}_3$ are collinear, and $({{\mathbf v}}_1, {{\mathbf v}}_2, {{\mathbf v}}_3, {{\mathbf v}}_4)$ lies on a plane with the diagonal from ${{\mathbf v}}_1$ to ${{\mathbf v}}_3$ bisecting the diagonal from ${{\mathbf v}}_2$ to ${{\mathbf v}}_4$. A similar argument shows that the diagonal from ${{\mathbf v}}_2$ to ${{\mathbf v}}_4$ bisects the diagonal from ${{\mathbf v}}_1$ to ${{\mathbf v}}_3$, so that $({{\mathbf v}}_1, {{\mathbf v}}_2, {{\mathbf v}}_3, {{\mathbf v}}_4)$ is a parallelogram. Now (\[eqn:parallelogram\]) implies $({{\mathbf v}}_1, {{\mathbf v}}_2, {{\mathbf v}}_3, {{\mathbf v}}_4)\cong ({{\mathbf u}}_1,{{\mathbf u}}_2, {{\mathbf u}}_3,{{\mathbf u}}_4)$. Now, for distinct $i,j,k \in \{0, \ldots, d\}$, consider the continuous family of configurations $({{\mathbf{f}}}_i(t), {{\mathbf{f}}}_j(t), {{\mathbf{g}}}_j^k(t), {{\mathbf{g}}}_i^k(t))$, $t \in [0,1]$. By assumption, this is a continuous expansion, but the pairwise distances between points in the initial configuration $$({{\mathbf{f}}}_i(0), {{\mathbf{f}}}_j(0), {{\mathbf{g}}}_j^k(0), {{\mathbf{g}}}_i^k(0))=((\iota({{\mathbf u}}_i), \iota({{\mathbf u}}_j), \iota({{\mathbf u}}_j+s{{\mathbf u}}_k), \iota({{\mathbf u}}_i+s{{\mathbf u}}_k))$$ and those of the final configuration $$({{\mathbf{f}}}_i(1), {{\mathbf{f}}}_j(1), {{\mathbf{g}}}_j^k(1), {{\mathbf{g}}}_i^k(1))=((\iota({{\mathbf u}}_i), \iota({{\mathbf u}}_j), \iota({{\mathbf u}}_j-s{{\mathbf u}}_k), \iota({{\mathbf u}}_i-s{{\mathbf u}}_k))$$ are equal since they form congruent rectangles. Since the initial configuration describes a rectangle, it follows from proposition \[prop:parallelogram\] that all intermediate configurations are congruent rectangles. Hence, $${{\mathbf{g}}}_j^k(t)-\iota({{\mathbf u}}_j)={{\mathbf{g}}}_i^k(t)-\iota({{\mathbf u}}_i), \quad \forall~~ i \neq j \neq k \neq i.$$ We can define ${{\mathbf{d}}}_k(t): [0,1] \rightarrow {{\mathbb E}}^{2d-1}$ by $${{\mathbf{d}}}_k(t):={{\mathbf{g}}}_j^k(t)-\iota({{\mathbf u}}_j), \quad \hbox{for any} \quad j \neq k,$$ then $$|{{\mathbf{d}}}_k(t)|=|{{\mathbf{g}}}_j^k(t)-\iota({{\mathbf u}}_j)|=|{{\mathbf{g}}}_j^k(0)-\iota({{\mathbf u}}_j)|=s$$ and ${{\mathbf{d}}}_k(t) \cdot (\iota({{\mathbf u}}_j-{{\mathbf u}}_i))=0$ for all $i \neq j \neq k \neq i$, hence, ${{\mathbf{d}}}_k(t)$ is orthogonal to $\iota(F^k)$, since $\{(\iota({{\mathbf u}}_j-{{\mathbf u}}_i))\}$, $i \neq j \neq k \neq i$ spans $\iota(F^k)$. \(III) For $k=0, \ldots, d$, let $$\pi_1({{\mathbf{d}}}_k(t)):={{\mathbf v}}_k(t)\in {{\mathbb E}}^d, \quad \pi_2({{\mathbf{d}}}_k(t)):={{\mathbf w}}_k(t) \in {{\mathbb E}}^{d-1}$$ so that ${{\mathbf{d}}}_k(t)=({{\mathbf v}}_k(t), {{\mathbf w}}_k(t))$. Since ${{\mathbf{d}}}_k(t).\iota({{\mathbf u}}_i-{{\mathbf u}}_j)={{\mathbf v}}_k(t).({{\mathbf u}}_i-{{\mathbf u}}_j)=0$ for all $i \neq j \neq k \neq i$, ${{\mathbf v}}_k(t)$ is orthogonal to $F^k \subset {{\mathbb E}}^d$, so ${{\mathbf v}}_k(t)=a_k(t){{\mathbf u}}_k$, $a_k(t) \in {{\mathbb R}}$, and furthermore, $|a_k(t)| \le s$ since $|{{\mathbf v}}_k(t)|^2+|{{\mathbf w}}_k(t)|^2=|{{\mathbf{d}}}_k(t)|^2=s^2$ by (\[eqn:norms\]). By the intermediate value theorem, since $a_k(0)=s$ and $a_k(1)=-s$, $a_k(t)$ takes all values in $[-s,s]$, so in particular, there exists some $t_0 \in [0,1]$ such that $a_k(t_0)=0$ , so that ${{\mathbf v}}_k(t_0)={\mathbf 0}$. Hence $|{{\mathbf w}}_k(t_0)|^2=s^2$, in particular, ${{\mathbf w}}_k(t_0) \neq {\mathbf 0}$ (in fact, we only need that $|a_k(t_0)|<s$ to get ${{\mathbf w}}_k(t_0) \neq {\mathbf 0}$). Now for $i \neq j \neq k \neq i$, we have $\triangle(\iota({{\mathbf u}}_i),{{\mathbf{g}}}_i^j(0),{{\mathbf{g}}}_i^k(0)) \cong \triangle(\iota({{\mathbf u}}_i),{{\mathbf{g}}}_i^j(1),{{\mathbf{g}}}_i^k(1))$ since $${{\mathbf{g}}}_i^j(0)-\iota({{\mathbf u}}_i)=\iota(s{{\mathbf u}}_j), \quad {{\mathbf{g}}}_i^k(0)-\iota({{\mathbf u}}_i)=\iota(s{{\mathbf u}}_k),$$ $${{\mathbf{g}}}_i^j(1)-\iota({{\mathbf u}}_i)=\iota(-s{{\mathbf u}}_j), \quad {{\mathbf{g}}}_i^k(1)-\iota({{\mathbf u}}_i)=\iota(-s{{\mathbf u}}_k),$$ so all the triangles $ \triangle (\iota({{\mathbf u}}_i),{{\mathbf{g}}}_i^j(t),{{\mathbf{g}}}_i^k(t))$, $t \in [0,1]$ are congruent. In particular, $$\label{eqn:dkdotdj} ({{\mathbf{g}}}_i^k(t)-\iota({{\mathbf u}}_i))\cdot ({{\mathbf{g}}}_i^j(t)-\iota({{\mathbf u}}_i))={{\mathbf{d}}}_k(t) \cdot {{\mathbf{d}}}_j(t)={{\mathbf{d}}}_k(0) \cdot {{\mathbf{d}}}_j(0)=s{{\mathbf u}}_k \cdot s{{\mathbf u}}_j=-\frac{s^2}{d}$$ for all $t \in [0,1]$ by (\[eqn:innerproductofnorms\]). Now using ${{\mathbf v}}_k(t_0)={\mathbf 0}$ and applying (\[eqn:dotprod\]) to (\[eqn:dkdotdj\]) gives, $$\label{eqn:wkdotwj} -\frac{s^2}{d}={{\mathbf{d}}}_k(t_0)\cdot {{\mathbf{d}}}_j(t_0)={{\mathbf v}}_k(t_0) \cdot {{\mathbf v}}_j(t_0)+{{\mathbf w}}_k(t_0)\cdot {{\mathbf w}}_j(t_0)={{\mathbf w}}_k(t_0)\cdot {{\mathbf w}}_j(t_0)$$ for all $j \neq k$. In particular, we see that ${{\mathbf w}}_j(t_0) \neq {\mathbf 0}$ for all $j=0, \ldots, d$ (again, we really only need that $|a_k(t_0)|<s$ to obtain this conclusion). \(IV) We need to show that ${{\mathbf w}}_i(t_0)\cdot {{\mathbf w}}_j(t_0)<0$ for all distinct $i, j \in \{0, \ldots, d\}$. Recall that ${{\mathbf{d}}}_i(t)=({{\mathbf v}}_i(t), {{\mathbf w}}_i(t))=(a_i(t){{\mathbf u}}_i, {{\mathbf w}}_i(t))$. Since ${{\mathbf w}}_i(t_0) \neq {\mathbf 0}$ and by (\[eqn:norms\]) $$s^2=|{{\mathbf{d}}}_i(t_0)|^2=|{{\mathbf v}}_i(t_0)|^2+|{{\mathbf w}}_i(t_0)|^2=|a_i(t_0)|^2+|{{\mathbf w}}_i(t_0)|^2$$ we have $$\label{eqn:aitlessthans} -s< a_i(t_0)<s, \quad \hbox{for all} \quad i=0, \ldots,d.$$ Now by (\[eqn:innerproductofnorms\]), for $i \neq j$, $${{\mathbf{d}}}_i(t_0) \cdot {{\mathbf{d}}}_j(t_0)={{\mathbf v}}_i(t_0) \cdot {{\mathbf v}}_j(t_0) +{{\mathbf w}}_i(t_0) \cdot {{\mathbf w}}_j(t_0).$$ ${{\mathbf{d}}}_i(t_0) \cdot {{\mathbf{d}}}_j(t_0)= {{\mathbf{d}}}_i(0)\cdot {{\mathbf{d}}}_j(0)=-\frac{s^2}{d}$ and $${{\mathbf v}}_i(t_0) \cdot {{\mathbf v}}_j(t_0)=a_i(t_0)a_j(t_0){{\mathbf u}}_i\cdot {{\mathbf u}}_j=-\frac{a_i(t_0)a_j(t_0)}{d},$$ where by (\[eqn:aitlessthans\]), $|{{\mathbf v}}_i(t_0) \cdot {{\mathbf v}}_j(t_0)|< \frac{s^2}{d}$. It follows that ${{\mathbf w}}_i(t_0)\cdot {{\mathbf w}}_j(t_0)<0$ for all distinct $i, j \in \{0,\ldots, d\}$. [*Remark:*]{} In proving the conclusion in (IV) holds, we only really require that the outward normals ${{\mathbf n}}_i$, $i=0, \ldots, d$ of $\sigma_d$ are pairwise obtuse, that is, ${{\mathbf n}}_i \cdot {{\mathbf n}}_j<0$ for all distinct $i,j \in \{0, \ldots, d\}$. Hence we may replace the regular simplex with one for which the above holds. \(V) Recall that ${{\mathbf u}}_1, {{\mathbf u}}_2 \in {{\mathbb E}}^n$ are obtuse if ${{\mathbf u}}_1 \cdot {{\mathbf u}}_2 <0$. The lemma below states that we cannot have a collection of $n+2$ pairwise obtuse vectors in ${{\mathbb E}}^n$. \[lem:obtuse\] For any set $\{{{\mathbf u}}_1, \ldots, {{\mathbf u}}_{n+2}\}$ of $n+2$ vectors in ${{\mathbb E}}^n$, ${{\mathbf u}}_i\cdot {{\mathbf u}}_j \ge 0$ for some $i \neq j$, that is, the vectors cannot be all pairwise obtuse. We prove by induction on the dimension $n$. The result is clearly true when $n=1$ since for any 3 vectors ${{\mathbf u}}_1, {{\mathbf u}}_2, {{\mathbf u}}_3 \in {{\mathbb E}}^1$, either at least one of the vectors is $\mathbf 0$, or two are in the same direction so have positive dot product. Assume the lemma is true for $n$ and suppose for a contradiction that there exists ${{\mathbf u}}_1, \ldots, {{\mathbf u}}_{n+3} \in {{\mathbb E}}^{n+1}$ that are all pairwise obtuse. Without loss of generality, we may assume that none of ${{\mathbf u}}_i$ are zero, and that ${{\mathbf u}}_{n+3}=(-1,0, \ldots, 0)$. Write ${{\mathbb E}}^{n+1}\cong {{\mathbb E}}^1 \times {{\mathbb E}}^{n}$ and consider the projections $\pi_1:{{\mathbb E}}^{n+1} \rightarrow {{\mathbb E}}^1$ and $\pi_2:{{\mathbb E}}^{n+1} \rightarrow {{\mathbb E}}^n$ respectively as in §\[s:intro\]. For $i=1, \ldots, n+2$, let ${{\mathbf v}}_i:=\pi_1({{\mathbf u}}_i) \in {{\mathbb E}}^1 \cong {{\mathbb R}}$, ${{\mathbf w}}_i:=\pi_2({{\mathbf u}}_i) \in {{\mathbb E}}^n$, see figure 4. Note that ${{\mathbf v}}_i >0$ since ${{\mathbf u}}_i \cdot {{\mathbf u}}_{n+3}<0$, so ${{\mathbf v}}_i \cdot {{\mathbf v}}_j>0$ for $i,j \in \{1, \ldots, n+2\}$. Then we have, from (\[eqn:dotprod\]), for distinct $i, j \in \{1, \ldots, n+2\}$, $${{\mathbf u}}_i \cdot {{\mathbf u}}_j={{\mathbf v}}_i\cdot {{\mathbf v}}_j +{{\mathbf w}}_i \cdot {{\mathbf w}}_j.$$ ![Projection of ${{\mathbb E}}^{n+1}$ vectors into ${{\mathbb E}}^{n}$ space []{data-label="fig:figure_4"}](./figure_4.png){width="100.00000%"} By assumption, ${{\mathbf u}}_i \cdot {{\mathbf u}}_j <0$, and ${{\mathbf v}}_i\cdot {{\mathbf v}}_j >0$ from the above, so $${{\mathbf w}}_i\cdot {{\mathbf w}}_j <0.$$ Hence $\{{{\mathbf w}}_1, \ldots, {{\mathbf w}}_{n+2}\}$ is a collection of pairwise obtuse vectors in ${{\mathbb E}}^{n}$ contradicting the induction hypothesis. Applying lemma \[lem:obtuse\] to the set $\{{{\mathbf w}}_0, {{\mathbf w}}_1, \ldots, {{\mathbf w}}_{d}\} \subset {{\mathbb E}}^{d-1}$ in (IV) we get the required contradiction which concludes the proof of Theorem \[thm:main2\] from which Theorem \[thm:main\] follows. [**Concluding remarks.**]{} The method of proof above works if we construct ${{\mathbf p}}$ and ${{\mathbf q}}$ from any simplex in ${{\mathbb E}}^d$ whose pairwise norms are obtuse. It also shows that any intermediate configuration in a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ cannot be embedded in a space of dimension less than $2d$. An interesting open question is, for each $d$, what is the smallest number of points in the configurations ${{\mathbf p}}$ and ${{\mathbf q}}$ for which there is no continuous expansion in ${{\mathbb E}}^{2d-1}$. We have shown that $N=(d+1)^2$ suffices, but this may not be optimal. Finally, it is also interesting to ask if we can find configurations ${{\mathbf p}}$, and expansions ${{\mathbf q}}$ of ${{\mathbf p}}$ such that the continuous expansion given by Theorem \[thm:continuous\] is essentially, up to some trivial motions, the only continuous expansion in dimension $2d$. [1]{} R. Alexander, [*Lipschitzian mappings and total mean curvature of polyhedral surfaces. I*]{}. Trans. Amer. Math. Soc. 288 (1985), no. 2, 661–678. M. Belk, R. Connelly, [*Making contractions continuous: a problem related to the Kneser-Poulsen conjecture*]{}. Preprint (2007). K. Bezdek, R. Connelly, [*Pushing disks apartthe Kneser-Poulsen conjecture in the plane.*]{} J. Reine Angew. Math. 553 (2002), 221236. H.S.M. Coxeter, [*Regular polytopes*]{}. Second edition The Macmillan Co., New York; Collier-Macmillan Ltd., London 1963 xx+321 pp. B. Csikós, [*On the volume of the union of balls*]{}. Discrete Comput. Geom. 20 (1998), no. 4, 449–461. M. Gromov, [*Monotonicity of the volume of intersection of balls*]{}. Geometrical aspects of functional analysis (1985/86), 1–4, Lecture Notes in Math., 1267, Springer, Berlin, 1987. M. Kneser, [*Einige Bemerkungen über das Minkowskische Flächenmass*]{}. Arch. Math. (Basel) 6 (1955), 382–390. H. R.Parks, D.C. Wills, [*An elementary calculation of the dihedral angle of the regular $n$-simplex*]{}. Amer. Math. Monthly 109 (2002), no. 8, 756–758. [^1]: The second author is partially supported by the National University of Singapore academic research grant R-146-000-133-112

--- abstract: 'We prove geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension.' address: | Einstein Institute of Mathematics\ Edmond J. Safra Campus, Givat Ram\ The Hebrew University of Jerusalem\ Jerusalem, 91904, Israel author: - Bruno Duchesne bibliography: - 'biblio.bib' title: | Superrigidity in infinite dimension and finite rank\ via harmonic maps --- [^1] Introduction ============ Geometric Superrigity --------------------- In the nineteen seventies, Margulis proved his famous *superrigidity theorem* to show that irreducible lattices in higher rank semisimple Lie groups and algebraic groups are arithmetic. \[Margulis\]Let $G,H$ be semisimple algebraic groups over local fields without compact factors. Assume that the real rank of $G$ is at least 2 and let $\Gamma$ be an irreducible lattice of $G$. Any homomorphism $\Gamma\to H$ with unbounded and Zariski dense image extends to a homomorphism $G\to H$. Using the dictionary between semisimple algebraic groups over local fields and symmetric spaces of noncompact type (in the Archimedean case) and Euclidean buildings (in the non-Archimedean case), Theorem \[Margulis\] can be interpreted in a geometric way. This is the subject of the so-called *geometric superrigidity*, see [@MR2655318] for a survey in french or the older [@MR1168043]. Using this geometric interpretation, Corlette [@MR1147961] (in the Archimedean case) and later Gromov and Schoen [@MR1215595] extended Margulis superridigity theorem in the case where $G$ is a simple Lie group of rank 1 that is the isometry group of a quaternionic hyperbolic space or the isometry group of the Cayley hyperbolic plane. The main tool in these two former results are harmonic maps. Some time later, Mok, Siu and Yeung gave a very general statement [@MR1223224] of geometric superrigidity in the Archimedean case.\ The framework of geometric superrigidity was extended [@MR2219304; @MR2377496] to nonpositively curved metric spaces, which may be not locally compact, in the particular higher rank case where $\Gamma$ is a lattice in a product.\ In [@MR1253544 6.A], Gromov invited geometers to study some “cute and sexy" infinite dimensional symmetric spaces of nonpositive curvature and finite rank. The geometry of these spaces $X_p({\mathbb{R}})=$O$(p,\infty)/($O$(p)\times$O$(\infty))$ and their analogs, $X_p({\mathbb{K}})$ over the field ${\mathbb{K}}$ of complex or quaternionic numbers, were studied in [@Duchesne:2011fk]. Gromov also conjectured that actions of lattices in semisimple Lie groups on some $X_p({\mathbb{R}})$ should be subject to geometric superrigidity.\ In this article, a Riemannian manifold will be a connected smooth manifold modeled on a separable Hilbert space and endowed with a smooth Riemannian metric. In particular, such a manifold *may have infinite dimension*. See [@MR1330918] or [@MR1666820] for an accurate definition.\ The main result of this paper is the following theorem. \[theorem\]Let $\Gamma$ be an irreducible torsion free uniform lattice in a connected higher rank semisimple Lie group with finite center and no compact factor $G$. Let $N$ be a simply connected complete Riemannian manifold of nonpositive sectional curvature and finite telescopic dimension. If $\Gamma$ acts by isometries on $N$ without fixed point in $N\cup{\partial}N$ then there exists a $\Gamma$-equivariant isometric totally geodesic embedding of a product of irreducible factors of the symmetric space of $G$ in $N$. In the unpublished paper [@ks99], Korevaar and Schoen introduced the notion of *FR-spaces* (Finite Rank spaces). Later Caprace and Lytchak introduced the notion of spaces of *finite telescopic dimension* in [@MR2558883], without knowing [@ks99]. The two notions are the same for complete CAT(0) spaces and can be defined by an inequality at large scale inspired by Jung Inequality. For any bounded subset $Y\subset{\mathbb{R}}^n$, Jung proved that $$\mathrm{rad}(Y)\leq\sqrt{\frac{n}{2(n+1)}}\mathrm{diam}(Y),$$ see [@MR1456512] and references therein. A complete CAT(0) space $X$ has *telescopic dimension less than* $n$ if for any $\delta>0$ there exists $D>0$ such that for any bounded subset $Y\subseteq X$ of diameter larger than $D$, we have $$\mathrm{rad}(Y)\leq\left(\delta+\sqrt{\frac{n}{2(n+1)}}\right)\mathrm{diam}(Y).$$\ Theorem \[theorem\] applies to the particular case where $N$ is a symmetric space of noncompact type and finite rank (see [@sym] for the meaning of noncompact type in infinite dimension). The fact that symmetric spaces of noncompact type and finite rank have finite telescopic dimension was expected in [@ks99] and proved in Corollary 1.8 of [@sym]. Actually, it is proved that a symmetric space of noncompact type is a finite product of irreducible symmetric spaces of noncompact type and that irreducible factors of infinite dimension are some $X_p(\mathbb{K})$.\ This theorem implies that there is no *geometrically Zariski-dense* (see [@Monod:2012fk 5.B]) action of a uniform lattice as in Theorem \[theorem\], on a symmetric space of noncompact type, infinite dimension and finite rank. In rank 1, it was shown that the isometry group of the real hyperbolic space $\mathbf{H}^n$ has geometrically Zariski-dense actions on the infinite dimensional hyperbolic $\mathbf{H}^\infty$, see [@Monod:2012fk].\ The strategy to prove Theorem \[theorem\] goes as follows. We first use a result of existence of harmonic maps due to Korevaar and Schoen (see Theorem \[existence\]) for CAT(0) spaces targets. When the target is moreover a Riemannian manifold, the unique harmonic map is smooth, as it was proved for $\mathbf{H}^\infty$ in Proposition 7 of [@Delzant:2010fk]. We conclude that the harmonic map is totally geodesic, thanks to an argument from [@MR1223224]. \[complex\]Theorem \[theorem\] extends to the case where $G$ is the connected component of the isometry group of the quaternionic hyperbolic space or of the Cayley hyperbolic plane and $N$ has nonpositive complexified sectional curvature. This last condition is satisfied when $N$ is a symmetric space of noncompact type. A flat torus theorem for parabolic isometries --------------------------------------------- In the last section, we include an extension for parabolic isometries of the well-known flat torus theorem [@MR1744486 Chapter II.7]. This extension allows us to obtain a rigidity statement easily.\ Let $\gamma$ be an isometry of a CAT(0) space $X$. The *translation length* of $\gamma$ is the number $|\gamma|=\inf_{x\in X}d(\gamma x,x)$; $\gamma$ is said to be *ballistic* if $|\gamma|>0$ and *neutral* otherwise. Since the infimum in the definition of $|\gamma|$ may or may not be achieved, it is usual to distinguish between *semisimple* isometries, for which the infimum is a minimum, and *parabolic* isometries, for which the infimum is not a minimum. Let $\varphi\colon G\to$Isom$(X)$ be a homomorphism. We say that $G$ *acts by ballistic isometries* on $X$ if $\varphi(g)$ is a ballistic isometry for any $g\neq e$.\ A CAT(0) space $X$ is said to be $\pi$-*visible* if any points $\xi,\eta\in{\partial}X$ that satisfy $\angle(\xi,\eta)=\pi$ are extremities of a geodesic line. For example, Hilbert spaces, Euclidean buildings and symmetric spaces of noncompact type are $\pi$-visible. Let $H$ be a subset of Isom$(X)$, we denote by $\mathcal{Z}_{\mathrm{Isom}(X)}(H)$ the centralizer of $H$, that is the set of elements in Isom$(X)$ that commute with all elements in $H$. \[ftt\]Let $X$ be a complete $\pi$-visible CAT(0) space and let $A$ an abelian free group of rank $n$ acting by ballistic isometries on $X$. Then there exists a $A$-invariant closed convex subspace $Y\subseteq X$. The space $Y$ decomposes as $Z\times{\mathbb{R}}^n$, $\mathcal{Z}_{\textrm{Isom}(X)}(A)$ preserves this decomposition and the action $\mathcal{Z}_{\textrm{Isom}(X)}(A){\curvearrowright}Y$ is diagonal. Moreover, the action $A{\curvearrowright}{\mathbb{R}}^n$ is given by a lattice of ${\mathbb{R}}^n$ acting by translations and for any $a\in A$, the action of $a$ on $Z$ is neutral. \[rr\]Let $\Gamma$ be a lattice in a semi-simple Lie group of real rank $r$. If $\Gamma$ acts by ballistic isometries on a symmetric space of nonpositive curvature $X$ then $r\leq$rank$(X)$. The author thanks Pierre Pansu for suggesting this approach to superrigidity in infinite dimension and thanks Pierre Py for pleasant and useful discussions about regularity of harmonic maps in infinite dimension. Harmonic maps ============= In this section, we recall the standard notions of totally geodesic maps and harmonic maps between Riemannian manifolds (maybe of infinite dimension). We refer to [@MR1896863], among others, for an introduction to these notions in finite dimension.\ Let $(M,g)$ be a smooth Riemannian manifold with Levi-Civita connection $\nabla$. Let $u$ be a chart form an open subset $U\subset M$ to an open subset $V$ of a Hilbert space $\mathcal{H}$. The restriction to $U$ of any vector field $X\in\Gamma(TM)$ can be thought as a smooth map $V\to\mathcal{H}$ and thus we can consider the differential $DX$ of $X$ as a linear map from $\mathcal{H}$ to itself. The *Christoffel symbol* of $\nabla$, $\Gamma(u)$ with respect to $u$, is defined by the relation $$\nabla_YX=D_YX+\Gamma(u)(X,Y),$$ see [@MR1330918 1.5]. Let $f$ be a smooth map between Riemannian manifolds $(M,g)$ and $(N,h)$ with Levi-Civita connections $^M\nabla$ and $^N\nabla$. The vector bundle $f^{-1}TN$, which is the vector bundle over $M$ with fibers $T_{f(x)}N$ for $x\in N$, is endowed with the connection induced from $^N\nabla$, which we denote also by $^N\nabla$. In charts $(u,U),(v,V)$ of $M$ such that $f(U)\subset V$, this connection is given by the formula $$^N\nabla_XY=D_{dfX}Y+\Gamma(v)(dfX,Y)$$ for $X\in\Gamma(TM)$ and $Y\in\Gamma(f^{-1}TN)$. The vector bundle $TM^*$ is also endowed with a connection $^M\nabla^*$ induced from $^M\nabla$. For $\omega\in\Gamma(TM^*)$ and $X,Y\in\Gamma(TM)$, $$^M\nabla^*_X\omega(Y)=X\cdot\omega(Y)-\omega(\nabla_XY).$$ The vector bundle $f^{-1}TN\otimes TM$ is endowed with the connection $\nabla\colon\Gamma(TM^*\otimes f^{-1}TN)\to\Gamma(TM^*\otimes TM^*\otimes f^{-1}TN)$ induced by $^M\nabla^*$ and $^N\nabla$. This connection is defined by the formula $$\nabla_X(\omega\otimes V)= ^M\negthinspace\nabla^*_X\omega\otimes V+\omega\otimes ^N\negthinspace\nabla_XV$$ for $X\in\Gamma(TM)$, $\omega\in\Gamma(TM^*)$ and $V\in\Gamma(f^{-1}TN)$.\ The differential $df$ of a smooth map $f$ is a section of $TM^*\otimes f^{-1}TN$ and $f$ is called *totally geodesic* if $\nabla df=0$. One can think of this property in two equivalent ways. A map $f$ is totally geodesic if and only if it preserves the connections, that is $^N\nabla_XdfY=df(^M\nabla_XY)$ for $X,Y\in\Gamma(TM)$. And $f$ is totally geodesic if and only if it maps geodesics to geodesics.\ When $M$ is finite dimensional, there is a more general notion. Let $\tau(f)$ be the trace of $\nabla df$. It is a section of the vector bundle $TM^*\otimes TM^*\otimes f^{-1}TN$ defined by $$\tau(f)=\sum_{i}\nabla df(e_i,e_i)$$ for any orthonormal base $(e_i)$ of $T_xM$. The map $f$ is *harmonic* if $\tau(f)=0$. Harmonic maps are important because they are solutions of a variational problem. Let $||\ ||$ be the norm associated to the Riemannian metric $g\otimes h$ on $TM^*\otimes f^{-1}TN$. Actually for $x\in M$, $||d_xf||$ is the Hilbert-Schmidt norm of the linear map $d_xf\colon T_xN\to T_{f(x)}N$. This norm is well defined because $T_xM$ is finite dimensional. If $M$ is complete and has finite Riemannian volume, the *energy* of $f$ is $$E(f)=\int_M||df||^2.$$ Harmonic maps are exactly critical points of the energy. There exists an equivariant version of this variational problem. Let $\Gamma$ the fundamental group of a compact Riemann manifold $M$ acting by isometries on $N$ and let $f$ be a $\Gamma$-equivariant map $f\colon \tilde{M}\to N$. Since $||df||^2$ is $\Gamma$-invariant, one can define the energy of $f$ by $E(f)=\int_M||df||^2.$ In the case where $N$ is finite dimensional and non positively curved, the existence of equivariant harmonic maps was considered in [@MR965220; @MR1049845]. Harmonic maps for metric spaces targets ======================================= In [@MR1266480; @MR1483983], Korevaar and Schoen developed a theory of harmonic maps with metric spaces targets (Jost developed also a similar theory, see [@MR1451625] or [@MR2829653 8.2]). We recall the definitions (not in full generality but in a framework convenient to our purpose) and refer to the original papers for details.\ Let $(\Omega,\mu)$ be a standard measure space with finite measure and let $(X,d)$ be a complete separable metric space with base point $x_0$. The space $L^p(\Omega,X)$ for $1\leq p\leq \infty$ is the space of measurable maps $u\colon \Omega\to X$ such that $\int_\Omega d\left(u(\omega),x_0\right)^pd\mu(\omega)$. This space is a complete metric space with distance satisfying $d(u,v)^p=\int_\Omega d\left(u(\omega),v(\omega)\right)^pd\mu(\omega)$ and if $(X,d)$ is CAT(0) then so is $L^2\left(\Omega,X\right)$.\ Let $\Gamma$ be the fundamental group of a compact Riemannian manifold $(M,g)$ and let $\rho\colon \Gamma\to$Isom$(X)$ be a representation of $\Gamma$. The group $\Gamma$ acts by deck transformations on the universal covering $\tilde{M}$ of $M$. We denote by $L^p_\rho(\tilde{M},X)$ the space of measurable equivariant map $u\colon\tilde{M}\to X$ such that the restriction of $u$ to a compact fundamental domain $K\subset \tilde{M}$ is in $L^p(K,X)$. For two maps $u,v\in L^p_\rho(\tilde{M},X)$, the function $x\mapsto d(u(x),v(x))$ is $\Gamma$-invariant and thus can be seen as a function on $M$. The distance on $ L^p_\rho(\tilde{M},X)$ is given by the relation $d(u,v)^p=\int_Md(u(x),v(x))^pd\mu(x)$ where $\mu$ is the measure associated to the Riemannian metric $g$.\ For $u\in L^p_\rho(\tilde{M},X)$ and $\varepsilon>0$ the $\varepsilon$-approximate energy at $x\in \tilde{M}$ is defined by $$e_\varepsilon(x)=\int_{S(x,\varepsilon)}\frac{d(u(x),u(y))^p}{\varepsilon^p}d\sigma(y)$$ where $S(x,{\varepsilon})$ is the $\varepsilon$-sphere around $x$ and $d\sigma$ is the measure induced by $g$ on $S(x,\varepsilon)$ divided by $\varepsilon^{(\mathrm{dim}(M)-1)}$. Now $u$ is said to have *finite energy* if $e_\varepsilon$ converges weakly to a density energy $e$, which is absolutely continuous with respect to $d\mu$ and has finite $L^1$-norm, when $\varepsilon$ goes to $0$. In this case, the energy of $u$ is $E(u)=\int_Me(x)d\mu(x)$. A minimizer of the energy functional is called a *harmonic map*. In [@MR1483983 Theorem 2.3.1], Korevaar and Schoen proved the existence of an equivariant harmonic map when the target is a CAT(-1) space under the assumption there is no fixed point at infinity. Actually, a Gromov-hyperbolic metric space, for example a CAT(-1) space, is nothing else than a metric space of telescopic dimension, or rank, at most 1 [@MR2558883 Introduction]. In the unpublished paper [@ks99], the analog in the higher rank (but finite !) case is proved. We include the original argument. \[existence\] Let $\Gamma$ be the fundamental group of a compact Riemannian manifold $M$ with universal covering $\tilde{M}$ and let $X$ be a complete CAT(0) space of finite telescopic dimension. If $\Gamma$ acts by isometries on $X$ without fixed point at infinity then there exists a unique equivariant harmonic map $f\colon\tilde{M}\to X$. Moreover, this harmonic map is Lipschitz. For $L>0$, let $\mathcal{C}_L$ be the set of $\Gamma$-equivariant maps from $\tilde{M}$ to $X$ that are $L$-Lipschitz and have finite energy. Thanks to Theorem 2.6.4 in [@MR1266480], we fix $L>0$ such that $\mathcal{C}_L$ is not empty. We claim that $\mathcal{C}_L$ is a closed convex subset of $L^2_\rho(\tilde{M},X)$. Let $u,v\in L^2_\rho(\tilde{M},X)$ and let $t\mapsto u_t$ be the geodesic segment with endpoints $u$ and $v$. If $u,v$ are $L$-Lipschitz, then the convexity of distance function on $X$ [@MR1744486 Proposition II.2.2] shows that $u_t$ is also convex for any $t$. Now, the L$^2$-convergence of a sequence with a common Lipschitz bound implies the uniform convergence of this sequence and since a uniform limit of a sequence of $L$-Lipschitz maps is also $L$-Lipschitz, we obtain that $\mathcal{C}_L$ is a closed convex subset of $L^2_\rho(\tilde{M},X)$.\ Let $x_0\in \tilde{M}$ and let $X'=\{x\in X|\ u(x_0)=x,\ u\in\mathcal{C}_L\}$. The convexity of $\mathcal{C}_L$ implies that $X'$ is a convex subset of $X$. We want to show that for any $x\in X'$, there exists a unique map $u\in\mathcal{C}_L$ that minimizes the energy among maps in $\mathcal{C}_L$ such that $u(x_0)=x$. Let $u,v\in\mathcal{C}_L$ such that $u(x_0)=v(x_0)=x\in X_0$ then we have $$\label{ineq} \int_Md(u,v)^2\underset{(\mathrm{PI})}{\leq}C\int_M||\nabla d(u,v)||^2\underset{(\mathrm{CI})}{\leq}C\left[\frac{1}{2}\left(E(u)+E(v)\right)-E(m)\right]$$ where $C$ is some positive number and $m$ is the midpoint of the segment $[u,v]$. Actually, Inequality (PI) is a Poincaré inequality (Lemma \[poincaré\]) for the function $d(u,v)$, which is $2L$-Lipschitz and vanishes at $x_0$, and Inequality (CI) is [@MR1266480 Inequality (2.6ii)]. Inequality (\[ineq\]) shows that an energy minimizing sequence $(u_n)$ with $u_n(x_0)=x$ for any $n$ is Cauchy and thus an energy minimizing map in $\{u\in\mathcal{C}_l\ |\ u(x_0)=x\}$ exists and is unique. Let us denote by $f_x$ this map.\ We define $I(x)$ to be $E(f_x)$. Now, we aim to show that $I\colon X'\to{\mathbb{R}}^+$ is a convex lower semicontinuous function. Assume this is the case, since $I$ is $\Gamma$-invariant and lower semicontinuous, its lower levelsets $X_r:=\{x\in X'|\ I(x)\leq r\}$ are $\Gamma$-invariant closed convex subsets of $X$. Proposition 4.4 in [@Duchesne:2011fk] implies that the intersection $\cap_{r>\inf I}I_r$ is non empty otherwise the center of directions associated to $\{I_r\}_{r>\inf I}$ would be a $\Gamma$-fixed point at infinity. Since $\cap_{r>\inf I}I_r\neq\emptyset$, there is an energy minimizing $\Gamma$-equivariant map, which is unique thanks to Inequality (\[ineq\]).\ From the convexity of $E$, it is clear that $I$ is also convex. Let $r>\inf I$ let $x\in X$ be a limit point of a sequence $(x_n)$ in $X_r$. It suffices to show that $f_n:=f_{x_n}$ is a Cauchy sequence in $\mathcal{C}_L$ to obtain that $I_r$ is closed. Let $I_n=\inf_{X_0\cap B(x,1/n)} I$. We may assume that $x_n\in X_0\cap B(x,1/n)$ and $E(f_n)\leq I_n+1/n$. Now, Inequality (CI) in (\[ineq\]) implies that $$\int_M|\nabla d(f_n,f_m)|^2\underset{n,m\to \infty}{\longrightarrow}0$$ and Lemma \[poincaré\] applied to the function $d(f_n,f_m)-d(f_n(x_0),f_m(x_0))$ allows us conclude that $(f_n)$ is a Cauchy sequence. \[poincaré\]If $f\colon M\to{\mathbb{R}}$ if a $L$-Lipschitz function that vanishes at some point $x_0\in {M}$ then there exists $C>0$ which depends only on ${M}$ and $L$, such that $$\int_Mf^2\leq C\int_M ||\nabla f||^2.$$ Let $R$ be the diameter of $M$. By an abuse of notation, we also denote by $f$ the function $f\circ\exp\colon T_{x_0}M\to{\mathbb{R}}$ and we denote by $\mu$ the pull-back, by the exponential map, of the measure associated to the Riemannian metric on $M$ and we denote by $dx$ the Lebesgue measure on $T_{x_0}M$. The measure $\mu$ is absolutely continuous with respect to $dx$ and there are positive numbers $c$ and $C_1$ such that the density $\upsilon$ of $\mu$ satisfies $c<\upsilon(x)<C_1$ for any $x$ in the $R$-ball around the origin in $T_{x_0}M$. We have $$\int_Mf^2\leq\int_{B(0,R)}f^2(x)d\mu(x)\leq C_1\int_{B(0,R)}f(x)^2dx.$$ Moreover, $$\begin{aligned} \int_{B(0,R)}f(x)^2dx&=\int_{B(0,R)}\int_0^{||x||}\left.\frac{d}{du}\right|_{u=t}f(ux/||x||)^2\ dt\ dx\\ &\leq \int_{B(0,R)}\int_0^R2f(tx/||x||)\nabla_{\frac{tx}{||x||}}f\cdot\frac{x}{||x||}\ dt\ dx.\end{aligned}$$ Now, let $n$ be the dimension of $M$ and let $\sigma$ be the Lebesgue measure on $S^{n-1}$. Using polar coordinates, the fact that $||\nabla f||\leq L$ and Hölder inequality, we have for some $C_2>0$ $$\begin{aligned} \int_Mf^2&\leq 2C_1 L\int_0^R\int_{S^{n-1}}\int_0^Rt\ ||\nabla_{tv}f||\ dt\ d\sigma(v)r^{n-1}dr\\ &\leq C_2\int_{S^{n-1}}\int_0^R ||\nabla_{tv}f||^{n-1} t^{n-1}dt\ d\sigma(v)=C_2\int_{B(0,R)}||\nabla_{x}f||^{n-1}dx.\end{aligned}$$ Once again, using Hölder inequality and the fact that the exponential map is finite to one on $B(x_0,R)$, we have for some $C_3,C>0$, $$\int_Mf^2\leq C_3\int_{B(x_0,R)}||\nabla_xf||^2dx\leq C\int_M||\nabla f||^2.$$ Smoothness ========== It is a standard fact that the most difficult part to obtain smoothness of weak harmonic maps is the first regularity step, which is the continuity of the harmonic map (see for example [@MR2829653 8.4]). In our situation, we already know that the harmonic map is Lipschitz and we can easily adapt the argument given in [@Delzant:2010fk], where the target is the infinite dimensional hyperbolic space. \[smooth\]Let $\Gamma$ be the fundamental group of a compact Riemannian manifold $M$ with universal covering $\tilde{M}$ and let $N$ be a simply connected complete Riemannian manifold of nonpositive sectional curvature. If $\Gamma$ acts by isometries on $N$ and $f\colon M\to N$ is a $\Gamma$-equivariant harmonic map in the sense of Korevaar and Schoen then $f$ is a smooth harmonic map. We only sketch the proof with the slight modifications to adapt [@Delzant:2010fk Proposition 7]. We already know that $f$ is Lipschitz. Choose a point $x\in N$. Since $N$ is a simply connected Riemannian manifold of nonpositive curvature, the Cartan-Hadamard Theorem [@MR1666820 IX.3.8] implies that the exponential map at $x$ is a diffeomorphism from the tangent space $T_xN$ to $N$. This gives us a global chart and we can think of $N$ as a Hilbert space $(\mathcal{H},<\ ,\ >):=(T_xN,h_x)$ with a non constant Riemannian metric $h$. Moreover, since $N$ has nonpositive sectional curvature, for any $v\in\mathcal{H}$ and any point $y\in N$, $$\label{Rauch} h_y(v,v)\geq<v,v>,$$ see [@MR1666820 Theorem IX.3.6]. In this chart, the covariant derivative can be expressed by $$\nabla_YX=D_YX+\Gamma(\exp)(X,Y)$$ where $\Gamma(\exp)$ is the Christoffel symbol of this chart. Let $B$ be a ball in $\tilde{M}$ of radius less than the injectivity radius of $M$. This way, the projection $\tilde{M}\to M$ identifies $B$ with a ball in $M$. Consider $f$ as a map from $\tilde{M}$ to $\mathcal{H}$. Inequality (\[Rauch\]) shows that $f|_B$, which has finite energy for the distance induced by $h$, has finite energy for the one induced by $<\ ,\ >$, too. Thus, $f|_B$ is in the usual Sobolev space (for vector valued maps) $W^{1,2}(B,\mathcal{H})$. Since $f$ is harmonic, it satisfies the equation $$\Delta_hf+\sum_{i,j=1}^{\mathrm{dim}(M)}h^{ij}\Gamma(exp)\left(\frac{{\partial}f}{{\partial}x_i},\frac{{\partial}f}{{\partial}x_j}\right)=0$$ weakly. An induction on $k$ shows that $f|_B$ is in $W^{k,p}(B,\mathcal{H})$ for any $k\in{\mathbb{N}}$ and $p>1$. This shows that $f$ is actually smooth. A vanishing theorem =================== Let $\tilde{M}$ be an irreducible symmetric space of noncompact type that is not the real or complex hyperbolic space and let $\Gamma$ be a uniform torsion free lattice of Isom$(\tilde{M})$. In order to prove a geometric statement of superrigidity in the cocompact Archimedean case, Mok, Siu and Yeung proved the existence [@MR1223224] of a 4-tensor $Q$ on $\tilde{M}$ that satisfies strong conditions. They also proved the following formula [@MR1223224 Theorem 3] for an equivariant map $f\colon \tilde{M}\to N$ where $N$ is a smooth Riemannian manifold of finite dimension $$\int_M\left<\left(Q\circ\sigma_{2\, 4}\right),\nabla df\otimes\nabla df\right>=1/2\int_M\left<Q,f^*R^N\right>.$$ In this formula, $Q\circ\sigma_{2\, 4}(X,Y,Z,T)=Q(X,T,Z,Y)$ and the scalar products are those induced by the Riemannian metrics of $M$ and $N$ on $(T^*M)^{\otimes4}\otimes(f^{-1}TN)^{\otimes2}$ and $(T^*M)^{\otimes4}$. Actually, the proof of this formula goes through in the case where $N$ has infinite dimension, without modification. This formula, conditions satisfied by $Q$ and the harmonicity of $f$ imply that $\nabla df$ vanishes, that is $f$ is totally geodesic. Let $\tilde{M}$ be the symmetric space associated to $G$. Since $\Gamma$ is a torsion free uniform lattice, the quotient space $\Gamma\backslash \tilde{M}$ is a compact manifold. Since $\tilde{M}$ has no fixed point at infinity of $N$, there exists a equivariant harmonic map $f\colon \tilde{M}\to N$ by Theorem \[existence\]. Thanks to Proposition \[smooth\], we know that $f$ is a smooth equivariant harmonic map.\ Assume first that $G$ is simple, that is to say $\tilde{M}$ is irreducible. Now, Mok-Siu-Yeung above argument implies that $f$ is totally geodesic. Since $\Gamma$ does not fix a point in $N$, $f(N)$ is not reduced to a point. Now, since $M$ is irreducible, $f$ is an isometry up to rescaling the metric on $M$ (see for example [@MR0262984]).\ Now, if $\tilde{M}\simeq\tilde{M}_1\times\dots\times\tilde{M}_n$ with $n\geq2$ then thanks to a Bochner formula [@MR1223224 11], it is proved that the restriction of $f$ to any fiber $x_1\times\dots\times x_{i-1}\times \tilde{M_i}\times x_{i+1}\times \dots\times x_n$ is harmonic. The irreducibility of $\Gamma$ allows the authors of [@MR1223224] to prove that $f$ is actually totally geodesic and thus $f$ factorizes through $$\tilde{M}\overset{\pi}{\longrightarrow}\prod_{i\in I}\tilde{M_i}\overset{f'}{\longrightarrow}N$$ where $I$ is a non empty subset of $\{1,\dots,n\}$, $\pi$ is the canonic projection and $f'$ is an isometry (after renormalization of the metric on each factor $\tilde{M_i}$ for $i\in I$). We now explain Remark \[complex\]. Let $(N,h)$ be a Riemannian manifold with Riemann tensor $R$. Let $X,Y$ be vectors in the complexified tangent space $T_xN\otimes{\mathbb{C}}$ at $x\in N$. We also denote by $R$ and $h$ the linear extensions of the Riemann tensor and the metric to the complexification of $T_xN$. The *complexified sectional curvature* between $X$ and $Y$ is $$\mathrm{Sec}_{\mathbb{C}}(X,Y)=\frac{R(X,Y,\overline{X},\overline{Y})}{||X\wedge Y||^2_{\mathbb{C}}}$$ where $||\ ||_{\mathbb{C}}$ is the Hermitian norm on $\wedge^2(T_xN\otimes{\mathbb{C}})$ induced by $h$, which is the norm associated to the scalar product $$<X\wedge Y,Z\wedge T>_h=\det\left[\begin{array}{cc} h(X,Z)&h(X,T)\\ h(Y,Z)&h(Y,T) \end{array}\right]$$ on $\wedge^2T_xN$. The Riemannian manifold $N$ is said to have *nonpositive complexified sectional curvature* if Sec$_{\mathbb{C}}(X,Y)\leq0$ for any $X,Y\in T_xN\otimes{\mathbb{C}}$.\ The result of [@MR1223224], which is the existence of a tensor $Q$ that implies the vanishing of $\nabla df$ for a harmonic map $f$, is true when $N$ has nonpositive complexified sectional curvature and $G$ is the connected component of the isometry group of the quaternionic hyperbolic space or the Cayley hyperbolic plane. Thus Theorem \[theorem\] is also true in this case.\ Let $C$ be the curvature operator as introduced in [@sym 3.2]. We also denote by $C$ its ${\mathbb{C}}$-linear extension to $T_xN\otimes{\mathbb{C}}$. For $X,Y\in T_xN\otimes{\mathbb{C}}$, $$\mathrm{Sec}_{\mathbb{C}}(X,Y)=\frac{<C(X\wedge Y),\overline{X\wedge Y}>_h}{||X\wedge Y||^2_{\mathbb{C}}}.$$ In the case where $N$ is a symmetric space of noncompact type, $C$ is nonpositive and thus, the complexified sectional curvature is nonpositive. A flat torus theorem for parabolic isometries ============================================= We start with some preliminary results. \[lem\]Let $X$ be a $\pi$-visible complete CAT(0) space. If $Y\subseteq X$ is closed and convex then it is also $\pi$-visible. Let $\xi,\eta\in{\partial}Y$ such that $\angle(\xi,\eta)=\pi$. There exists a geodesic $c\colon{\mathbb{R}}\to X$ such that $c(\infty)=\xi$ and $c(-\infty)=\eta$. Let $x$ be the projection of $c(0)$ on $Y$. We define $c_+$ (respectively $c_-$) to be the geodesic ray from $x$ toward $\xi$ (respectively $\eta$). By definition of the boundary, $d(c(t),c_+(t))$ and $d(c(-t),c_-(t))$ are bounded for $t\geq0$. The real function $t\mapsto d(c(t),Y)$ is convex bounded and thus constant equal to some $d_0\geq0$. Now let $c'(t)$ be the projection of $c(t)$ on $Y$ for $t\in{\mathbb{R}}$. For $s,t\in{\mathbb{R}}$ and $x\in[c(t),c(s)]$, $d(x,[c'(t),c'(s)])=d_0$. By the same argument as above, for all $x\in[c'(t),c'(s)]$, $d(x,[c(t),c(s)])=d_0$ and we are in position to apply the Sandwich Lemma [@MR1744486 II.2.12.(2)], which shows that the convex hull of $c(t),c(s),c'(s),c'(t)$ is a Euclidean rectangle. In particular, $c'\colon{\mathbb{R}}\to Y$ is a geodesic with $c'(\infty)=\xi$ and $c'(-\infty)=\eta$. We recall that a ballistic isometry $\gamma$ of a complete CAT(0) space $X$ has two canonical fixed points at infinity, which we denote by $\omega_\gamma$ and $\omega_{\gamma^{-1}}$. They are limit points at infinity of $\gamma^nx$ and $\gamma^{-n}x$ for any $x\in X$ (see [@MR2574740 3.C], for example). \[prop\] Let $X$ be a $\pi$-visible complete CAT(0) space and let $\gamma$ be a ballistic isometry of $X$. Then there exists a closed convex subspace $Y\subseteq X$ that splits as $Z\times{\mathbb{R}}$. Moreover, $\mathcal{Z}_{\mathrm{Isom}(X)}(\gamma)$ preserves $Y$ and acts diagonally. In particular, $\gamma|_Y$ acts as a translation of length $|\gamma|$ along the factor ${\mathbb{R}}$. Let $Y$ be the union of geodesics with endpoints $\omega_\gamma$ and $\omega_{\gamma^{-1}}$. Since $X$ is $\pi$-visible, $Y$ is nonempty and $\gamma$-invariant. Moreover, $Y$ is a closed subspace of $X$. Let $x\in X$ be a limit point of a sequence $(x_n)$ of points in $Y$. Let $c_n$ be the geodesic such that $c_n(0)=x_n$, $c_n(-\infty)=\omega_{\gamma^{-1}}$ and $c_n(\infty)=\omega_{\gamma}$. Thanks to [@MR1744486 Proposition II.9.22], $c_n$ converges to a geodesic $c$ such that $c(0)=x$, $c(-\infty)=\omega_{\gamma^{-1}}$ and $c(\infty)=\omega_{\gamma}$.\ Since $Y$ is closed, convex and $\gamma$-invariant, $\left|\gamma|_Y\right|=|\gamma|$. The subspace $Y$ decomposes as a product $Y\simeq Z\times{\mathbb{R}}$ (see [@MR1744486 Theorem II.2.14]) and $\gamma$ preserves this decomposition. Thus $\gamma|_Y$ can be written $\gamma_0\times\gamma_1$. A simple computation shows that $|\gamma|^2=|\gamma_0|^2+|\gamma_1|^2$. Assume for contradiction that $|\gamma_0|>0$ then there exists $\omega_{\gamma_0}\in{\partial}Z$ such that $\gamma_0^nx_0\to \omega_{\gamma_0}$ for any $x_0\in Z$. Thus, for any $x\in Y$, $\gamma^nx\to(\arccos(|\gamma_0|/|\gamma_1|),\omega_{\gamma_0},\omega_\gamma)$ in the spherical join ${\partial}Z\ast{\partial}{\mathbb{R}}={\partial}Y$. \[add\]Let $A$ be an abelian group acting by isometries on a CAT(0) space $X$. The set $N$ of neutral elements in $A$ is a subgroup of $A$. Moreover, if $A\simeq{\mathbb{R}}^n$ and the action is continuous then it is a linear subspace of $A$. We recall that $|a|=\lim_{n\to\infty}\frac{d(nax,x)}{n}$ for any $x\in X$. Thus $$|a+b|=\lim_n\frac{d(n(a+b)x,x)}{n} \leq\lim_n\frac{d(na(nbx),nax)+d(nax,x)}{n} \leq|a|+|b|.$$ Moreover, $|a|=|-a|$ and $|na|=n|a|$ for any $n\in{\mathbb{N}}$. Thus, if $A\simeq{\mathbb{R}}^n$, by continuity $|\lambda a|=|\lambda||a|$ for any $\lambda\in{\mathbb{R}}$ and $a\in A$. We are ready to prove Theorem \[ftt\]. We prove this theorem by induction on $n$. For $n=1$, this is Proposition \[prop\]. Now suppose $n\geq2$ and choose a primitive element $a\in A$ and apply Proposition \[prop\]. We obtain an $\mathcal{Z}_{\mathrm{Isom}(X)}(a)$-invariant closed convex subspace $Y_1\simeq Z_1\times{\mathbb{R}}$ and $a$ acts as a translation of length $|a|$ on the ${\mathbb{R}}$-factor. The group $\mathcal{Z}_{\mathrm{Isom}(X)}(a)$ acts also diagonally on $Y_1$. Let $N$ be the subset of $A$ formed by elements $b=(b_1,b_2)$ of $A$ such that $|b_1|=0$. Lemma \[add\] shows $N$ is a subgroup of $A$. The subgroup $N$ acts properly on ${\mathbb{R}}$ and thus is cyclic. It contains $a$ and since $a$ is primitive in $A$, $N=a{\mathbb{Z}}$. Now let $B$ be a free abelian group of rank $n-1$ such that $A=N\oplus B$. Observe that $B$ acts by ballistic isometries on $Z_1$. We can now apply an induction for $B{\curvearrowright}Z_1$ and we obtain $Y_2\simeq Z\times{\mathbb{R}}^{n-1}\subseteq Z_1$. By induction $Y_2$ is $\mathcal{Z}_{\mathrm{Isom}(Z_1)}(B)$-invariant and $\mathcal{Z}_{\mathrm{Isom}(Z_1)}(B)$ preserves this decomposition. Moreover, for any $\gamma\in\mathcal{Z}_{\mathrm{Isom}(X)}(A)$, $\gamma_{Z_1}\in\mathcal{Z}_{\mathrm{Isom}(Z_1)}(B)$ and thus $\gamma_{Z_1}$ preserves $Y_2$ and acts diagonally on it. In particular, $a_{Z_1}$ (which is neutral) has a trivial part on ${\mathbb{R}}^{n-1}$. Now if we set $Y=Y_2\times{\mathbb{R}}\simeq Z\times{\mathbb{R}}^{n}\subseteq X$, $Y$ has the desired properties. Thanks to [@MR0302822 Corollary 2.9], $\Gamma$ contains an Abelian free group of rank $r$. Since this Abelian free group acts also by ballistic isometries, it suffices to apply Theorem \[ftt\] to find a Euclidean subspace of $X_p(\mathbb{K})$ of dimension $r$ and thus $p\geq r$. [^1]: The author is supported by a postdoctoral fellowship of the Swiss National Science Foundation.

--- abstract: 'We modified the modal expansion, which is the traditional method used to calculate thermal noise. This advanced modal expansion provides physical insight about the discrepancy between the actual thermal noise caused by inhomogeneously distributed loss and the traditional modal expansion. This discrepancy comes from correlations between the thermal fluctuations of the resonant modes. The thermal noise spectra estimated by the advanced modal expansion are consistent with the results of measurements of thermal fluctuations caused by inhomogeneous losses.' author: - Kazuhiro Yamamoto - Masaki Ando - Keita Kawabe - Kimio Tsubono title: | A theoretical approach to thermal noise caused by an inhomogeneously distributed loss\ — Physical insight by the advanced modal expansion --- Introduction ============ Thermal fluctuation is one of the fundamental noise sources in precise measurements. For example, the sensitivity of interferometric gravitational wave detectors [@LIGO; @VIRGO; @GEO; @TAMA] is limited by the thermal noise of the mechanical components. The calculated thermal fluctuations of rigid cavities have coincided with the highest laser frequency stabilization results ever obtained [@Numata5; @Notcutt]. It is important to evaluate the thermal motion for studying the noise property. The (traditional) modal expansion [@Saulson] has been commonly used to calculate the thermal noise of elastic systems. However, recent experiments [@Yamamoto1; @Harry; @Conti; @Numata3; @Yamamoto3; @Black] have revealed that modal expansion is not correct when the mechanical dissipation is distributed inhomogeneously. In some theoretical studies [@Levin; @Nakagawa1; @Tsubono; @Yamamoto-D], calculation methods that are completely different from modal expansion have been developed. These methods are supported by the experimental results of inhomogeneous loss [@Harry; @Numata3; @Yamamoto3; @Black]. However, even when these method were used, the physics of the discrepancy between the actual thermal noise and the traditional modal expansion was not fully understood. In this paper, another method to calculate the thermal noise is introduced [@Yamamoto-D]. This method, advanced modal expansion, is a modification of the traditional modal expansion (this improvement is a general extension of a discussion in Ref. [@Majorana]). The thermal noise spectra estimated by this method are consistent with the results of experiments concerning inhomogeneous loss [@Yamamoto1; @Yamamoto3]. It provides information about the disagreement between the thermal noise and the traditional modal expansion. We present the details of these topics in the following sections. Outline of advanced modal expansion =================================== Review of the traditional modal expansion ----------------------------------------- The thermal fluctuation of the observed coordinate, $X$, of a linear mechanical system is derived from the fluctuation-dissipation theorem [@Callen; @Greene; @Landau2], $$\begin{aligned} G_{X}(f)&=&-\frac{4 k_{\rm B} T}{\omega} {\rm Im}[H_{X}(\omega)], \label{FDT}\\ H_{X}(\omega)&=&\frac{\tilde{X}(\omega)}{\tilde{F}(\omega)}, \label{transfer function}\\ \tilde{X}(\omega)&=&\frac1{2\pi} \int^{\infty}_{-\infty}X(t)\exp(-{\rm i}\omega t)dt, \label{Fourier transform}\end{aligned}$$ where $f(=\omega/2\pi)$, $t$, $k_{\rm B}$ and $T$, are the frequency, time, Boltzmann constant and temperature, respectively. The functions ($G_{X}$, $H_{X}$, and $F$) are the (single-sided) power spectrum density of the thermal fluctuation of $X$, the transfer function, and the generalized force, which corresponds to $X$. In the traditional modal expansion [@Saulson], in order to evaluate this transfer function, the equation of motion of the mechanical system without any loss is decomposed into those of the resonant modes. The details are as follows: ![\[defX\]Example of the definition of the observed coordinate, $X$, in Eq. (\[observed coordinate\]). The mirror motion is observed using a Michelson interferometer. The coordinate $X$ is the output of the interferometer. The vector $\boldsymbol{u}$ represents the displacement of the mirror surface. The field $\boldsymbol{P}$ is parallel to the beam axis. Its norm is the beam-intensity profile [@Levin].](modefig1){width="8.6cm"} The definition of the observed coordinate, $X$, is described as $$X(t) = \int \boldsymbol{u}(\boldsymbol{r},t) \cdot \boldsymbol{P}(\boldsymbol{r}) dS, \label{observed coordinate}$$ where $\boldsymbol{u}$ is the displacement of the system and $\boldsymbol{P}$ is a weighting function that describes where the displacement is measured. For example, when mirror motion is observed using a Michelson interferometer, as in Fig. \[defX\], $X$ and $\boldsymbol{u}$ represent the interferometer output and the displacement of the mirror surface, respectively. The vector $\boldsymbol{P}$ is parallel to the beam axis. Its norm is the beam-intensity profile [@Levin]. The equation of motion of the mechanical system without dissipation is expressed as $$\rho\frac{\partial^2 \boldsymbol{u}}{\partial t^2} -{\cal L}[\boldsymbol{u}]=F(t)\boldsymbol{P}(\boldsymbol{r}), \label{eq_mo_continuous}$$ where $\rho$ is the density and ${\cal L}$ is a linear operator. The first and second terms on the left-hand side of Eq. (\[eq\_mo\_continuous\]) represent the inertia and the restoring force of the small elements in the mechanical oscillator, respectively. The solution of Eq. (\[eq\_mo\_continuous\]) is the superposition of the basis functions, $$\boldsymbol{u}(\boldsymbol{r},t) =\sum_{n}\boldsymbol{w}_n(\boldsymbol{r})q_n(t). \label{mode decomposition}$$ The functions, $\boldsymbol{w}_n$ and $q_n$, represent the displacement and time development of the $n$-th resonant mode, respectively. The basis functions, $\boldsymbol{w}_n$, are solutions of the eigenvalue problem, written as $${\cal L}[\boldsymbol{w}_n(\boldsymbol{r})] =-\rho {\omega_n}^2 \boldsymbol{w}_n(\boldsymbol{r}), \label{eigenvalue problem}$$ where $\omega_n$ is the angular resonant frequency of the $n$-th mode. The displacement, $\boldsymbol{w}_n$, is the component of an orthogonal complete system, and is normalized to satisfy the condition $$\int \boldsymbol{w}_n(\boldsymbol{r}) \cdot \boldsymbol{P}(\boldsymbol{r}) dS = 1. \label{normalized condition}$$ The formula of the orthonormality is described as $$\int \rho \boldsymbol{w}_n(\boldsymbol{r}) \cdot \boldsymbol{w}_k(\boldsymbol{r}) dV = m_n \delta_{nk}. \label{effective mass}$$ The parameter $m_n$ is called the effective mass of the mode [@Yamamoto1; @Gillespie; @Bondu; @Yamamoto2]. The tensor $\delta_{nk}$ is the Kronecker’s $\delta$-symbol. Putting Eq. (\[mode decomposition\]) into Eq. (\[observed coordinate\]), we obtain a relationship between $X$ and $q_n$ using Eq. (\[normalized condition\]), $$X(t) = \sum_{n} q_n(t). \label{observed coordinate decomposition}$$ In short, coordinate $X$ is a superposition of those of the modes, $q_n$. In order to decompose the equation of motion, Eq. (\[eq\_mo\_continuous\]), Eq. (\[mode decomposition\]) is substituted for $\boldsymbol{u}$ in Eq. (\[eq\_mo\_continuous\]). Equation (\[eq\_mo\_continuous\]) is multiplied by $\boldsymbol{w}_n$ and then integrated over all of the volume using Eqs. (\[eigenvalue problem\]), (\[normalized condition\]) and (\[effective mass\]). The result is that the equation of motion of the $n$-th mode, $q_n$, is the same as that of a harmonic oscillator on which force $F(t)$ is applied. After modal decomposition, the dissipation term is added to the equation of each mode. The equation of the $n$-th mode is written as $$-m_n \omega^2 \tilde{q}_n + m_n {\omega_n}^2 [1+{\rm i}\phi_n(\omega)] \tilde{q}_n=\tilde{F}, \label{traditional1}$$ in the frequency domain. The function $\phi_n$ is the loss angle, which represents the dissipation of the $n$-th mode [@Saulson]. The transfer function, $H_X$, derived from Eqs. (\[transfer function\]), (\[observed coordinate decomposition\]) and (\[traditional1\]) is the summation of those of the modes, $H_n$, $$\begin{aligned} H_{X}(\omega) &=& \frac{\tilde{X}}{\tilde{F}} = \sum_n \frac{\tilde{q}_n}{\tilde{F}} \left(= \sum_n H_n \right)\nonumber\\ &=& \sum_{n} \frac1{-m_n \omega^2 + m_n {\omega_n}^2 [1+{\rm i}\phi_n(\omega)]}. \label{traditional3}\end{aligned}$$ According to Eqs. (\[FDT\]) and (\[traditional3\]), the power spectrum density, $G_{X}$, is the summation of the power spectrum, $G_{q_n}$, of $q_n$, $$\begin{aligned} &&G_{X}(f) = \sum_{n} G_{q_n}\nonumber\\ &&= \sum_{n} \frac{4k_{\rm B}T}{m_n \omega} \frac{{\omega_n}^2\phi_n(\omega)} {(\omega^2-{\omega_n}^2)^2+{\omega_n}^4{\phi_n}^2(\omega)}. \label{traditional2} \end{aligned}$$ Equation of motion in an advanced modal expansion ------------------------------------------------- In the traditional modal expansion, the dissipation term is introduced after decomposition of the equation of motion without any loss. On the contrary, in an advanced modal expansion, the equation with the loss is decomposed [@Yamamoto-D; @optics]. If the loss is sufficiently small, the expansion process is similar to that in the perturbation theory of quantum mechanics [@Sakurai]. The equation of $q_n$ is expressed as $$\begin{aligned} -m_n \omega^2 \tilde{q}_n + m_n {\omega_n}^2 [1&+&{\rm i}\phi_n(\omega)] \tilde{q}_n\nonumber\\ &+& \sum_{k \neq n} {\rm i} \alpha_{nk}(\omega) \tilde{q}_k = \tilde{F},\label{advanced1}\\ \phi_n(\omega) &=& \frac{\alpha_{nn}}{m_n {\omega_n}^2}\label{phi}. \label{phi_n}\end{aligned}$$ The third term in Eq. (\[advanced1\]) is the difference between the advanced, Eq. (\[advanced1\]), and traditional, Eq. (\[traditional1\]), modal expansions. Since this term is a linear combination of the motions of the other modes, it represents the couplings between the modes. The magnitude of the coupling, $\alpha_{nk}$, depends on the property and the distribution of the loss (described below). Details of coupling ------------------- Let us consider the formulae of the couplings caused by the typical inhomogeneous losses, the origins of which exist outside and inside the material (viscous damping and structure damping, respectively) [@Yamamoto-D]. Regarding most of the external losses, for example, the eddy-current damping and residual gas damping are of the viscous type [@Saulson]. The friction force of this damping is proportional to the velocity. Inhomogeneous viscous damping introduces a friction force, ${\rm i} \omega \rho \Gamma(\boldsymbol{r}) \tilde{\boldsymbol{u}}(\boldsymbol{r})$, into the left-hand side of the equation of motion, Eq. (\[eq\_mo\_continuous\]), in the frequency domain. The function $\Gamma (\geq 0)$ represents the strength of the damping. The equation of motion with the dissipation term, ${\rm i} \omega \rho \Gamma(\boldsymbol{r}) \tilde{\boldsymbol{u}}(\boldsymbol{r})$, is decomposed. Since the loss is small, the basis functions of the equation without loss are available [@Sakurai]. Equation (\[mode decomposition\]) is put into the equation of motion along with the inhomogeneous viscous damping. This equation multiplied by $\boldsymbol{w}_n$ is integrated. The coupling of this dissipation is written in the form $$\alpha_{nk} = \omega \int \rho \Gamma(\boldsymbol{r}) \boldsymbol{w}_n(\boldsymbol{r}) \cdot \boldsymbol{w}_k(\boldsymbol{r}) dV = \alpha_{kn}. \label{coupling_viscous}$$ In most cases, the internal loss in the material is expressed using the phase lag, $\phi (\geq 0)$, between the strain and the stress [@Saulson]. The magnitude of the dissipation is proportional to this lag. The phase lag is almost constant against the frequency [@Saulson] in many kinds of materials (structure damping). In the frequency domain, the relationship between the strain and the stress (the generalized Hooke’s law) in an isotropic elastic body is written as [@Saulson; @Levin; @Yamamoto-D; @Landau] $$\begin{aligned} \tilde{\sigma}_{ij} &=& \frac{E_0[1+{\rm i}\phi(\boldsymbol{r})]}{1+\sigma} \left(\tilde{u}_{ij} + \frac{\sigma}{1-2\sigma}\sum_{l}\tilde{u}_{ll} \delta_{ij}\right)\nonumber\\ &=& [1+{\rm i}\phi(\boldsymbol{r})]\tilde{\sigma}'_{ij}, \label{structure_stress}\\ u_{ij} &=& \frac1{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right), \label{strain}\end{aligned}$$ where $E_0$ is Young’s modulus and $\sigma$ is the Poisson ratio; $\sigma_{ij}$ and $u_{ij}$ are the stress and strain tensors, respectively. The tensor, $\tilde{\sigma}'_{ij}$, is the real part of the stress, $\tilde{\sigma}_{ij}$. It represents the stress when the structure damping vanishes. The value, $u_i$, is the $i$-th component of $\boldsymbol{u}$. The equation of motion of an elastic body [@Landau] in the frequency domain is expressed as $$-\rho \omega^2 \tilde{u}_i -\sum_j \frac{\partial \tilde{\sigma}_{ij}}{\partial x_{j}} = \tilde{F}P_i(\boldsymbol{r}), \label{eq_mo_elastic_withloss}$$ where $P_i$ is the $i$-th component of $\boldsymbol{P}$. From Eqs. (\[structure\_stress\]) and (\[eq\_mo\_elastic\_withloss\]), an inhomogeneous structure damping term is obtained, $-{\rm i} \sum_j \partial \phi(\boldsymbol{r}) \tilde{\sigma}'_{ij}/\partial x_{j}$. The equation of motion with the inhomogeneous structure damping is decomposed in the same manner as that of the inhomogeneous viscous damping. The coupling is calculated using integration by parts and Gauss’ theorem [@Landau], $$\begin{aligned} \alpha_{nk} &=& - \int \sum_{i,j} w_{n,i} \frac{\partial \phi(\boldsymbol{r}) \sigma_{k,ij}}{\partial x_j} dV \nonumber\\ &=& - \int \sum_{i,j} \frac{\partial w_{n,i} \phi(\boldsymbol{r}) \sigma_{k,ij}} {\partial x_j} dV + \int \sum_{i,j} \frac{\partial w_{n,i}}{\partial x_j} \phi(\boldsymbol{r}) \sigma_{k,ij} dV \nonumber \\ &=& - \int \sum_{i,j} w_{n,i} \phi(\boldsymbol{r}) \sigma_{k,ij} n_j dS + \int \sum_{i,j} \frac{\partial w_{n,i}}{\partial x_j} \phi(\boldsymbol{r}) \sigma_{k,ij} dV \nonumber \\ &=& \int \frac{E_0 \phi(\boldsymbol{r})}{1+\sigma} \left[\sum_{i,j} \frac{\partial w_{n,i}}{\partial x_j} \left(w_{k,ij} + \frac{\sigma}{1-2\sigma} \sum_l w_{k,ll} \delta_{ij}\right)\right] dV\nonumber\\ &=& \int \frac{E_0 \phi(\boldsymbol{r})}{1+\sigma} \left(\sum_{i,j}w_{n,ij}w_{k,ij} + \frac{\sigma}{1-2\sigma}\sum_{l}w_{n,ll}\sum_{l}w_{k,ll} \right) dV =\alpha_{kn}, \label{coupling_structure}\end{aligned}$$ where $w_{n,i}$ and $n_{i}$ are the $i$-th components of $\boldsymbol{w}_n$ and the normal unit vector on the surface. The tensors, $w_{n,ij}$ and $\sigma_{n,ij}$, are the strain and stress tensors of the $n$-th mode, respectively. In order to calculate these tensors, $w_{n,i}$ is substituted for $u_i$ in Eqs. (\[structure\_stress\]) and (\[strain\]) with $\phi=0$. Equation (\[coupling\_structure\]) is valid when the integral of the function, $\sum_{i,j} w_{n,i} \phi \sigma_{k,ij} n_{j}$, on the surface of the elastic body vanishes. For example, the surface is fixed ($w_{n,i}=0$) or free ($\sum_{j}\sigma_{k,ij}n_{j}=0$) [@Landau]. The equation of motion in the advanced modal expansion coincides with that in the traditional modal expansion when all of the couplings vanish. A comparison between Eqs. (\[effective mass\]) and (\[coupling\_viscous\]) shows that in viscous damping all $\alpha_{nk} (n \neq k)$ are zero when the dissipation strength, $\Gamma(\boldsymbol{r})$, does not depend on the position, $\boldsymbol{r}$. In the case of structure damping, from Eqs. (\[eq\_mo\_continuous\]), (\[mode decomposition\]), (\[eigenvalue problem\]) and (\[eq\_mo\_elastic\_withloss\]), the stress, $\sigma'_{ij}$, without dissipation satisfies $$\sum_j \frac{\partial \tilde{\sigma}'_{ij}}{\partial x_{j}} = - \sum_n \rho {\omega_n}^2 w_{n,i} \tilde{q}_n. \label{stress decomposition}$$ According to Eq. (\[effective mass\]), Eq. (\[stress decomposition\]) is decomposed without any couplings. From Eq. (\[stress decomposition\]) and the structure damping term, $-{\rm i} \sum_j \partial \phi(\boldsymbol{r}) \tilde{\sigma}'_{ij}/\partial x_{j}$, the conclusion is derived; all of the couplings in the structure damping vanish when the loss amplitude, $\phi$, is homogeneous. In summary, the inhomogeneous viscous and structure dampings produce mode couplings and destroy the traditional modal expansion. The reason why the inhomogeneity of the loss causes the couplings is as follows. Let us consider the decay motion after only one resonant mode is excited. If the loss is uniform, the shape of the displacement of the system does not change while the resonant motion decays. On the other hand, if the dissipation is inhomogeneous, the motion near the concentrated loss decays more rapidly than the other parts. The shape of the displacement becomes different from that of the original resonant mode. This implies that the other modes are excited, i.e. the energy of the original mode is leaked to the other modes. This energy leakage represents the couplings in the equation of motion. It must be noticed that some kinds of “homogeneous” loss cause the couplings. For example, in thermoelastic damping [@Zener; @Braginsky-thermo; @Liu; @Cerdonio], which is a kind of internal loss, the energy components of the shear strains, $w_{n,ij}(i \neq j)$, are not dissipated. The couplings, $\alpha_{nk}$, do not have any terms that consist of the shear strain tensors. The coupling formula of the homogeneous thermoelastic damping is different from Eq. (\[coupling\_structure\]) with the constant $\phi$. The couplings are not generally zero, even if the thermoelastic damping is uniform. The advanced, not traditional, modal expansion provides a correct evaluation of the “homogeneous” thermoelastic damping. In this paper, however, only coupling caused by inhomogeneous loss is discussed. Thermal-noise formula of advanced modal expansion {#thermal noise of advanced} ------------------------------------------------- In the advanced modal expansion, the transfer function, $H_{X}$, is derived from Eqs. (\[transfer function\]), (\[observed coordinate decomposition\]), and (\[advanced1\]) (since the dissipation is small, only the first-order of $\alpha_{nk}$ is considered [@alpha2]), $$H_{X}(\omega) = \sum_n \frac1{-m_n {\omega}^2 + m_n {\omega_n}^2 (1 + {\rm i}\phi_n)} - \sum_{k \neq n} \frac{{\rm i} \alpha_{nk}} {[-m_n \omega^2+m_n {\omega_n}^2 (1+{\rm i}\phi_n)] [-m_k \omega^2 +m_k {\omega_k}^2 (1+{\rm i}\phi_k)]}. \label{advanced3}$$ Putting Eq. (\[advanced3\]) into Eq. (\[FDT\]), the formula for the thermal noise is obtained. In the off-resonance region, where $|-\omega^2+{\omega_n}^2| \gg {\omega_n}^2 \phi_n(\omega)$ for all $n$, this formula approximates the expression $$\begin{aligned} G_{X}(f)&=&\sum_{n} \frac{4 k_{\rm B} T}{m_n\omega} \frac{{\omega_n}^2\phi_n(\omega)} {(\omega^2-{\omega_n}^2)^2}\nonumber\\ &+&\sum_{k \neq n}\frac{4k_{\rm B}T}{m_n m_k \omega} \frac{\alpha_{nk}}{(\omega^2-{\omega_n}^2)(\omega^2-{\omega_k}^2)}. \label{advanced2}\end{aligned}$$ The first term is the same as the formula of the traditional modal expansion, Eq. (\[traditional2\]). The interpretation of Eq. (\[advanced2\]) is as follows. The power spectrum density of the thermal fluctuation force of the $n$-th mode, $G_{F_n}$, and the cross-spectrum density between $F_n$ and $F_k$, $G_{F_n F_k}$, are evaluated from Eq. (\[advanced1\]) and the fluctuation-dissipation theorem [@Greene; @Landau2], $$\begin{aligned} G_{F_n}(f) &=& 4 k_{\rm B} T \frac{m_n {\omega_n}^2 \phi_n(\omega)}{\omega}, \label{G_F_n}\\ G_{F_n F_k}(f) &=& 4 k_{\rm B} T \frac{\alpha_{nk}(\omega)}{\omega}. \label{G_F_n_F_k}\end{aligned}$$ The power spectrum density, $G_{F_n}$, is independent of $\alpha_{nk}$. On the other hand, $G_{F_n F_k}$ depends on $\alpha_{nk}$. Having the correlations between the fluctuation forces of the modes, correlations between the motion of the modes must also exist. The power spectrum density of the fluctuation of $q_n$, $G_{q_n}$, and the cross-spectrum density between the fluctuations of $q_n$ and $q_k$, $G_{q_nq_k}$, are described as [@Greene; @Landau2] $$\begin{aligned} G_{q_n}(f) &=& \frac{4 k_{\rm B} T}{m_n\omega}\frac{{\omega_n}^2\phi_n(\omega)} {(\omega^2-{\omega_n}^2)^2}, \label{G_q_n}\\ G_{q_n q_k}(f) &=& \frac{4k_{\rm B}T}{m_nm_k\omega} \frac{\alpha_{nk}}{(\omega^2-{\omega_n}^2)(\omega^2-{\omega_k}^2)}, \label{G_q_n_q_k}\end{aligned}$$ under the same approximation of Eq. (\[advanced2\]). The first and second terms in Eq. (\[advanced2\]) are summations of the fluctuation motion of each mode, Eq. (\[G\_q\_n\]), and the correlations, Eq. (\[G\_q\_n\_q\_k\]), respectively. In conclusion, inhomogeneous mechanical dissipation causes mode couplings and correlations of the thermal motion between the modes. In order to check wheather the formula of the thermal motion in the advanced modal expansion is consistent with the the equipartition principle, the mean square of the thermal fluctuation, $\overline{X^2}$, which is an integral of the power spectrum density over the whole frequency region, is evaluated. This mean square is derived from Eq. (\[FDT\]) using the Kramers-Kronig relation [@Landau2; @KKcomment], $${\rm Re}[H_X(\omega)] = -\frac1{\pi} \int_{-\infty}^{\infty} \frac{{\rm Im}[H_X(\xi)]}{\xi-\omega}d\xi. \label{Kramers-Kronig}$$ The calculation used to evaluate the mean square is written as [@Landau2] $$\begin{aligned} \overline{X^2} &=& \int_0^{\infty} G_{X}(f) df \nonumber\\ &=& \frac1{4\pi} \int_{-\infty}^{\infty} G_{X}(\omega) d\omega \nonumber\\ &=& -\frac{k_{\rm B}T}{\pi} \int_{-\infty}^{\infty} \frac{{\rm Im}[H_{X}(\omega)]}{\omega} d\omega \nonumber\\ &=& k_{\rm B}T {\rm Re}[H_{X}(0)]. \label{X2}\end{aligned}$$ Since the transfer function, $H_{X}$, is the ratio of the Fourier components of the real functions, the value $H_X(0)$ is a real number. The functions $\phi_n$ and $\alpha_{nk}$, which cause the imaginary part of $H_{X}$, must vanish when $\omega$ is zero [@Landau2]. The correlations do not affect the mean square of the thermal fluctuation. Equation (\[X2\]) is rewritten using Eq. (\[advanced3\]) as $$\overline{X^2}=\sum_n \frac{k_{\rm B}T}{m_n {\omega_n}^2}. \label{X2_2}$$ Equation (\[X2\_2\]) is equivalent to the prediction of the equipartition principle. The calculation of the formula of the advanced modal expansion, Eq. (\[advanced2\]), is more troublesome than that of the other methods [@Levin; @Nakagawa1; @Tsubono; @Yamamoto-D], which are completely different from the modal expansion, when many modes contribute to the thermal motion. However, the advanced modal expansion gives clear physical insight about the discrepancy between the thermal motion and the traditional modal expansion, as shown in Sec. \[new insight\]. It is difficult to find this insight using other methods. Experimental check ================== {width="3cm"} {width="8.6cm"} {width="3cm"} {width="8.6cm"} {width="3cm"} {width="8.6cm"} {width="3cm"} {width="8.6cm"} In order to test the advanced modal expansion experimentally, our previous experimental results concerning oscillators with inhomogeneous losses [@Yamamoto1; @Yamamoto3] are compared with an evaluation of the advanced modal expansion [@Yamamoto-D]. In an experiment involving a drum (a hollow cylinder made from aluminum alloy as the prototype of the mirror in the interferometer) with inhomogeneous eddy-current damping by magnets [@Yamamoto3], the measured values agreed with the formula of the direct approach [@Levin], Eq. (6) in Ref. [@Yamamoto3]. This expression is the same as that of the advanced modal expansion [@Yamamoto-D]. Figure \[experiment\] presents the measured spectra of an aluminum alloy leaf spring with inhomogeneous eddy-current damping [@Yamamoto1]. The position of the magnets for the eddy-current damping and the observation point are indicated above each graph. In the figures above each graph, the left side of the leaf spring is fixed. The right side is free. The open circles in the graphs represent the power spectra of the thermal motion derived from the measured transfer functions using the fluctuation-dissipation theorem. These values coincide with the directly measured thermal-motion spectra [@Yamamoto1]. The solid lines are estimations using the advanced modal expansion (the correlations derived from Eqs. (\[coupling\_viscous\]) and (\[G\_q\_n\_q\_k\]) are almost perfect [@Yamamoto-D]). As a reference, an evaluation of the traditional modal expansion is also given (dashed lines). The results of a leaf-spring experiment are consistent with the advanced modal expansion. Therefore, our two experiments support the advanced modal expansion. Physical insight given by the advanced modal expansion {#new insight} ====================================================== The advanced modal expansion provides physical insight about the disagreement between the real thermal motion and the traditional modal expansion. Here, let us discuss the three factors that affect this discrepancy: the number of the modes, the absolute value and the sign of the correlation. Number of modes --------------- Since the difference between the advanced and traditional modal expansions is the correlations between the multiple modes, the number of the modes affects the magnitude of the discrepancy. If the thermal fluctuation is dominated by the contribution of only one mode, this difference is negligible, even when there are strong correlations. On the other hand, if the thermal motion consists of many modes, the difference is larger when the correlations are stronger. Examples of the one-mode oscillator are given in Fig. \[experiment\]. The measured thermal motion spectra of the leaf spring with inhomogeneous losses below 100 Hz were the same as the estimated values of the “traditional” modal expansion. This is because these fluctuations were dominated by only the first mode (about 60 Hz). As another example, let us consider a single-stage suspension for a mirror in an interferometric gravitational-wave detector. The sensitivity of the interferometer is limited by the thermal noise of the suspensions between 10 Hz and 100 Hz. Since, in this frequency region, this thermal noise is dominated by only the pendulum mode [@Saulson], the thermal noise generated by the inhomogeneous loss agrees with the traditional modal expansion. It must be noticed that the above discussion is valid only when the other suspension modes are negligible. For example, when the laser beam spot on the mirror surface is shifted, the two modes (pendulum mode and mirror rotation mode) must be taken into account. In such cases, the inhomogeneous loss causes a disagreement between the real thermal noise of the single-stage suspension and the traditional modal expansion [@Braginsky]. The discrepancy between the actual thermal motion and the traditional modal expansion in the elastic modes of the mirror [@Yamamoto2] is larger than that of the drum, the prototype of the real mirror in our previous experiment [@Yamamoto3]. One of the reasons is that the thermal motion of the mirror (rigid cylinder) consists of many modes [@Gillespie; @Bondu]. The drum (hollow cylinder) had only two modes [@Yamamoto3]. Since the number of modes that contribute to the thermal noise of the mirror in the interferometer increases when the laser beam radius becomes smaller [@Gillespie; @Bondu], the discrepancy is larger with a narrower beam. This consideration is consistent with our previous calculation [@Yamamoto2]. Absolute value of the correlation --------------------------------- ![\[abs ex\]Example for considering the absolute value of the coupling. There are the $n$-th and $k$-th modes, $\boldsymbol{w}_n$ and $\boldsymbol{w}_k$, of a bar with both free ends. The vertical axis is the displacement. The dashed horizontal lines show the bar that does not vibrate. When only the grey part (A), which is narrower than the wavelengths on the left-hand side, has viscous damping, the absolute value of the coupling, Eq. (\[coupling\_viscous\]), is large. Because the signs of $\boldsymbol{w}_n$ and $\boldsymbol{w}_k$ do not change in this region. If viscous damping exits only in the hatching part (B), which is wider than the wavelengths on the right-hand side, the coupling is about zero, because, in this wide region, the sign of the integrated function in Eq. (\[coupling\_viscous\]), which is proportional to the product of $\boldsymbol{w}_n$ and $\boldsymbol{w}_k$, changes.](modefig3){width="8.6cm"} In Eq. (\[G\_q\_n\_q\_k\]), the absolute value of the cross-spectrum density, $G_{q_n q_k}$, is proportional to that of the coupling, $\alpha_{nk}$. Equations (\[coupling\_viscous\]) and (\[coupling\_structure\]) show that the coupling depends on the scale of the dissipation distribution. A simple example of viscous damping is shown in Fig. \[abs ex\]. Let us consider the absolute value of $\alpha_{nk}$ when the viscous damping is concentrated (at around $\boldsymbol{r}_{\rm vis}$) in a smaller volume ($\Delta V$) than the wavelengths of the $n$-th and $k$-th modes. An example of this case is (A) in Fig. \[abs ex\]. It is assumed that the vector $\boldsymbol{w}_n(\boldsymbol{r}_{\rm vis})$ is nearly parallel to $\boldsymbol{w}_k(\boldsymbol{r}_{\rm vis})$. The absolute value of the coupling is derived from Eqs. (\[phi\_n\]) and (\[coupling\_viscous\]) as $$\begin{aligned} |\alpha_{nk}| &\sim& |\omega \rho \Gamma(\boldsymbol{r}_{\rm vis}) \boldsymbol{w}_n(\boldsymbol{r}_{\rm vis}) \cdot \boldsymbol{w}_k(\boldsymbol{r}_{\rm vis}) \Delta V| \nonumber\\ &\sim& \sqrt{\omega \rho \Gamma(\boldsymbol{r}_{\rm vis}) |\boldsymbol{w}_n(\boldsymbol{r}_{\rm vis})|^2 \Delta V} \nonumber\\ &&\hspace{0.5cm}\times \sqrt{\omega \rho \Gamma(\boldsymbol{r}_{\rm vis}) |\boldsymbol{w}_k(\boldsymbol{r}_{\rm vis})|^2 \Delta V}\nonumber\\ &\sim& \sqrt{\alpha_{nn}\alpha_{kk}} = \sqrt{m_n {\omega_n}^2 \phi_n m_k {\omega_k}^2 \phi_k}. \label{coupling_narrow}\end{aligned}$$ The absolute value of the cross-spectrum is derived from Eqs. (\[G\_q\_n\]), (\[G\_q\_n\_q\_k\]), and (\[coupling\_narrow\]) as $$|G_{q_n q_k}| \sim \sqrt{G_{q_n}G_{q_k}}. \label{maxcorrelation}$$ In short, the correlation is almost perfect [@maxcoupling]. On the other hand, if the loss is distributed more broadly than the wavelengths, the coupling, i.e. the correlation, is about zero, $$|G_{q_n q_k}| \sim 0.$$ The dissipation in the case where the size is larger than the wavelengths is equivalent to the homogeneous loss. An example of this case is (B) in Fig. \[abs ex\]. Although the above discussion is for the case of viscous damping, the conclusion is also valid for other kinds of dissipation. When the loss is localized in a small region, the correlations among many modes are strong. The loss in a narrower volume causes a larger discrepancy between the actual thermal motion and the traditional modal expansion. This conclusion coincides with our previous calculation of a mirror with inhomogeneous loss [@Yamamoto2]. Sign of correlation ------------------- The sign of the correlation depends on the frequency, the loss distribution, and the position of the observation area. The position dependence provides a solution to the inverse problem: an evaluation of the distribution and frequency dependence of the loss from measurements of the thermal motion. ### Frequency dependence According to Eq. (\[G\_q\_n\_q\_k\]), the sign of the correlation reverses at the resonant frequencies. For example, in calculating the double pendulum [@Majorana], experiments involving the drum [@Yamamoto3] and a resonant gravitational wave detector with optomechanical readout [@Conti], this change of the sign was found. In some cases, the thermal-fluctuation spectrum changes drastically around the resonant frequencies. A careful evaluation is necessary when the observation band includes the resonant frequencies. Examples are when using wide-band resonant gravitational-wave detectors [@wide1; @wide2; @wide3; @wide4], and thermal-noise interferometers [@Numata3; @Black]. The reason for the reverse at the resonance is that the sign of the transfer function of the mode with a small loss from the force ($F_n$) to the motion ($q_n$), $H_n$ in Eq. (\[traditional3\]) \[$\propto (-\omega^2+{\omega_n}^2)^{-1}$\], below the resonance is opposite to that above it. Since the sign of the correlation changes at the resonant frequencies, the cross-spectrum densities, the second term of Eq. (\[advanced2\]), make no contribution to the integral of the power spectrum density over the whole frequency region, i.e. the mean square of the thermal fluctuation, $\overline{X^2}$, as shown in Sec. \[thermal noise of advanced\]. Therefore, the consideration in Sec. \[thermal noise of advanced\] indicates that a reverse of the sign of the correlation conserves the equipartition principle, a fundamental principle in statistical mechanics. ### Loss and observation area position dependence ![\[signex\]Example for considering the sign of the coupling. There are the lowest three modes, $\boldsymbol{w}_n$, of a bar with both free ends. The vertical axis is the displacement. The dashed horizontal lines show the bar that does not vibrate. The observation point is at the right-hand side end. The normalization condition is Eq. (\[normalized condition\]) [@normalized; @condition]. The sign and shape of the displacement of all the modes around the observation point are positive and similar, respectively. On the contrary, at the left-hand side end, the sign and shape of the $n$-th mode are different from each other in many cases. From Eqs. (\[normalized condition\]), (\[coupling\_viscous\]), (\[coupling\_structure\]), when the loss is concentrated near to the observation area, most of the couplings are positive. On the other hand, when the loss is localized far from the observation area, the number of the negative couplings is about the same as the positive one. In such a case, most of the couplings between the $n$-th and $(n \pm 1)$-th modes are negative.](modefig4){width="8.6cm"} According to Eqs. (\[coupling\_viscous\]) and (\[coupling\_structure\]), and the normalization condition, Eq. (\[normalized condition\]) [@normalized; @condition], the sign of the coupling, $\alpha_{nk}$, depends on the loss distribution and the position of the observation area. A simple example is shown in Fig. \[signex\]. Owing to this normalization condition, near the observation area, the basis functions, $\boldsymbol{w}_n$, are similar in most cases. On the contrary, in a volume far from the observation area, $\boldsymbol{w}_n$ is different from each other in many cases. From Eqs. (\[normalized condition\]), (\[coupling\_viscous\]), (\[coupling\_structure\]) and (\[G\_F\_n\_F\_k\]), when the loss is concentrated near to the observation area, most of the couplings (and the correlations between the fluctuation forces of the modes, $G_{F_nF_k}$) are positive. On the other hand, when the loss is localized far from the observation area, the numbers of the negative couplings and $G_{F_nF_k}$ are about the same as the positive ones. In such a case, most of the couplings between the $n$-th and $(n \pm 1)$-th modes (and $G_{F_nF_{n \pm 1}}$) are negative. These are because the localized loss tends to apply to the fluctuation force on all of the modes to the same direction around itself. Equation (\[G\_q\_n\_q\_k\]) indicates that the sign of the correlation, $G_{q_nq_k}$, is the same as that of the coupling, $\alpha_{nk}$, below the first resonance. In this frequency band, the thermal motion is larger and smaller than the evaluation of the traditional modal expansion if the dissipation is near and far from the observation area, respectively. This conclusion is consistent with the qualitative discussion of Levin [@Levin], our previous calculation of the mirror [@Yamamoto2], and the drum experiment [@Yamamoto3]. ### Inverse problem The above consideration about the sign of the coupling gives a clue to solving the inverse problem: estimations of the distribution and frequency dependences of the loss from the measurement of the thermal motion. Since the sign of the coupling depends on the position of the observation area and the loss distribution, a measurement of the thermal vibrations at multiple points provides information about the couplings, i.e. the loss distribution. Moreover, multiple-point measurements reveal the loss frequency dependence. Even if the loss is uniform, the difference between the actual thermal motion and the traditional modal expansion exits when the expected frequency dependence of the loss angles, $\phi_n(\omega)$, is not correct [@Majorana]. The measurement at the multiple points shows whether the observed difference is due to an inhomogeneous loss or an invalid loss angle. This is because the sign of the difference is independent of the position of the observation area if the expected loss angles are not valid. As an example, our leaf-spring experiment [@Yamamoto1] is discussed. The two graphs on the right (or left) side of Fig. \[experiment\] show thermal fluctuations at different positions in the same mechanical system. The spectrum is smaller than the traditional modal expansion. The other one is larger. Thus, the disagreement in the leaf-spring experiment was due to inhomogeneous loss, not invalid loss angles. When the power spectrum had a dip between the first (60 Hz) and second (360 Hz) modes, the sign of the correlation, $G_{q_1q_2}$, was negative. According to Eq. (\[G\_q\_n\_q\_k\]), the sign of the coupling, $\alpha_{12}$, was positive. The loss was concentrated near to the observation point when a spectrum dip was found. The above conclusion agrees with the actual loss shown in Fig. \[experiment\]. Conclusion ========== The traditional modal expansion has frequently been used to evaluate the thermal noise of mechanical systems [@Saulson]. However, recent experimental research [@Yamamoto1; @Harry; @Conti; @Numata3; @Yamamoto3; @Black] has proved that this method is invalid when the mechanical dissipation is distributed inhomogeneously. In this paper, we introduced a modification of the modal expansion [@Yamamoto-D; @Majorana]. According to this method (the advanced modal expansion), inhomogeneous loss causes correlations between the thermal fluctuations of the modes. The fault of the traditional modal expansion is that these correlations are not taken into account. Our previous experiments [@Yamamoto1; @Yamamoto3] concerning the thermal noise of the inhomogeneous loss support the advanced modal expansion. The advanced modal expansion gives interesting physical insight about the difference between the actual thermal noise and the traditional modal expansion. When the thermal noise consists of the contributions of many modes, the loss is localized in a narrower area, which makes a larger difference. When the thermal noise is dominated by only one mode, this difference is small, even if the loss is extremely inhomogeneous. The sign of this difference depends on the frequency, the distribution of the loss, and the position of the observation area. It is possible to derive the distribution and frequency dependence of the loss from measurements of the thermal vibrations at multiple points. There were many problems concerning the thermal noise caused by inhomogeneous loss. 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Sakurai, [*Modern Quantum Mechanics*]{} (Benjamin/Cummings, California, 1985). Chapters 1 and 3 of L.D. Landau and E.M. Lifshitz, [*Theory of Elasticity*]{} (Pergamon, New York, 1986). C. Zener, Phys. Rev. [**52**]{}, 230 (1937); [**53**]{}, 90 (1938). V.B. Braginsky, M.L. Gorodetsky, and S.P. Vyatchanin, Phys. Lett. A [**264**]{}, 1 (1999). Y.T. Liu and K.S. Thorne, Phys. Rev. D [**62**]{}, 122002 (2000). M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, Phys. Rev. D [**63**]{}, 082003 (2001). In the off-resonance region, where $|-\omega^2+{\omega_n}^2| \gg {\omega_n}^2 \phi_n(\omega)$ for all $n$, this approximation is always appropriate because the maximum of ${\alpha_{nk}}^2$ is $m_n {\omega_n}^2 \phi_n m_k {\omega_k}^2 \phi_k$ [@maxcoupling]. In the calculation to derive Eq. (\[advanced3\]), Cramer’s rule is useful. The sign on the right-hand side of the Kramers-Kronig relation in Ref. [@Landau2] is positive. The definition of the Fourier transformation in Ref. [@Landau2] is conjugate to that of this paper, Eq. (\[Fourier transform\]). V.B. Braginsky, Yu. Levin, and S.P. Vyatchanin, Meas. Sci. Technol. [**10**]{}, 598 (1999). This discussion suggests that $|\alpha_{nk}|$ is never larger than $\sqrt{m_n {\omega_n}^2 \phi_n m_k {\omega_k}^2 \phi_k}$. The validity of this conjecture is supported by the Cauchy-Schwarz inequality [@Yamamoto-D]. M. Cerdonio [*et al.*]{}, Phys. Rev. Lett. [**87**]{}, 031101 (2001). T. Briant [*et al.*]{}, Phys. Rev. D [**67**]{}, 102005 (2003). M. Bonaldi [*et al.*]{}, Phys. Rev. D [**68**]{}, 102004 (2003). M. Bonaldi [*et al.*]{}, Phys. Rev. D [**74**]{}, 022003 (2006). The results of the discussion presented in this section are valid under arbitrary normalization conditions. If a normalization condition other than Eq. (\[normalized condition\]) is adopted, the signs of some couplings change. The signs of the right-hand sides of Eqs. (\[observed coordinate decomposition\]) and (\[advanced1\]) also change. Equations (\[observed coordinate\]), (\[eq\_mo\_continuous\]), and (\[mode decomposition\]) say that the right-hand sides of Eqs. (\[observed coordinate decomposition\]) and (\[advanced1\]) in the general style are $\sum_{n} q_n(t) \int \boldsymbol{w}_n(\boldsymbol{r}) \cdot \boldsymbol{P}(\boldsymbol{r}) dS$ and $\tilde{F} \int \boldsymbol{w}_n(\boldsymbol{r}) \cdot \boldsymbol{P}(\boldsymbol{r}) dS$, respectively. The change in the sign cancels each other in the process of the calculation for the transfer function, $H_X$, and the power spectrum of the thermal motion, $G_{X}$.

--- abstract: 'We show that Pb and Bi adatoms and dimers have a large tunneling anisotropic magnetoresistance (TAMR) of up to 60% when adsorbed on a magnetic transition-metal surface due to strong spin-orbit coupling and the hybridization of $6p$ orbitals with $3d$ states of the magnetic layer. Using density functional theory, we have explored the TAMR effect of Pb and Bi adatoms and dimers adsorbed on a Mn monolayer on W(110). This surface exhibits a noncollinear cycloidal spin spiral ground state with an angle of 173$^\circ$ between neighboring spins which allows to rotate the spin quantization axis of an adatom or dimer quasi-continuously and is ideally suited to explore the angular dependence of TAMR using scanning tunneling microscopy (STM). We find that the induced magnetic moments of Pb and Bi adatoms and dimers are small, however, the spin-polarization of the local density of states (LDOS) is still very large. The TAMR obtained from the anisotropy of the vacuum LDOS is up to 50-60 % for adatoms. For dimers the TAMR depends sensitively on the dimer orientation with respect to the crystallographic directions of the surface due to the formation of bonds between the adatoms with the Mn surface atoms and the symmetry of the spin-orbit coupling induced mixing. Dimers oriented along the spin spiral direction of the Mn monolayer display the largest TAMR of 60 % which is due to hybrid $6p-3d$ states of the dimers and the Mn layer.' author: - Soumyajyoti Haldar - Mara Gutzeit - Stefan Heinze title: 'Tunneling anisotropic magnetoresistance of Pb and Bi adatoms and dimers on Mn/W(110): A first-principles study' --- Introduction ============ The tunneling magnetoresistance (TMR), in which the flow of current depends on the relative magnetization directions of two magnetic layers, has a significant impact on modern day applications ranging from spintronics to magnetic data storage. Using spin-polarized scanning tunneling microscopy (STM), it is even possible to detect the TMR effect for single magnetic adatoms on surfaces [@Yayon2007; @Meier82; @Tao2009; @Loth2010; @Ziegler2011; @Lazo2012; @Khajetoorians55]. The resistance can also depend on the magnetization direction relative to the current direction because of spin-orbit coupling (SOC), which is known as the tunneling anisotropic magnetoresistance (TAMR) [@Bode2002; @Gould2004]. The TAMR is driven by SOC which couples spin and orbital momentum degrees of freedom by the Hamiltonian $H_{SOC} = \xi \, \mathbf{L} \cdot \mathbf{S}$, where $\xi$, $\mathbf{L}$, and $\mathbf{S}$ are the SOC constant, orbital momentum operator and spin operator, respectively. SOC and magnetocrystalline anisotropy effects depend on the environment of an adatom and hence can be tuned by adatom adsorption which have been studied quite extensively  [@Gambardella1130; @Hirjibehedin1199; @Loth1628; @Khajetoorians2011; @Rau988]. The TAMR can be observed with only one ferromagnetic electrode and it does not require any coherent spin-dependent transport. Hence, the TAMR is very attractive for spintronics applications [@Fert2008; @Sinova2012]. The TAMR was first observed for a double layer of Fe on W(110) [@Bode2002]. Subsequently, the TAMR has been observed in various systems, [[e.g.,]{}]{} planar ferromagnetic surfaces [@Shick2006; @Chantis2007], tunnel junctions [@Gould2004; @Matos2009a; @Matos2009b; @Gao2007], mechanically controlled break junctions [@Viret2006; @Bolotin2006]. The observed values of TAMR in the above cases are $\approx$ 10%. Attempts have been made to increase the value of TAMR by using $3d$ or $5d$ elements, [[e.g.,]{}]{} using isolated adatoms  [@Neel2013; @Schoneberg2016], bimetallic alloys [@Shick2010] and with antiferromagnetic electrodes [@Park2011]. Recently, Hervé [[*et al.*]{}]{} have reported a TAMR of up to 30% for Co films on Ru(0001) mediated by surface states [@Herve2018b]. Another approach to tune SOC is to use single atoms and dimers of $6p$ elements. The strength of SOC scales with atomic number ($Z$), principal quantum number ($n$), and orbital quantum number ($l$) as $\xi \propto Z^{4}n^{-3}l^{-2}$. Hence, $6p$ elements such as Pb and Bi have a higher SOC strength as compared to that of the $3d$ or $5d$ elements studied before. Further tuning of SOC can be achieved by reducing the high rotational symmetry of single atom, [[i.e.,]{}]{} by using dimers of these elements. The effect of strong SOC on unsupported $6p$ dimers has been discussed recently [@Borisova2016a]. In an experimental and theoretical study, Schöneberg [[*et al.*]{}]{} [@Schoneberg2018] have achieved TAMR values of $\approx$ 20% by using suitably oriented Pb dimers on the Fe bilayer on W(110) substrate where magnetic domains with out-of-plane magnetization and domain walls with in-plane magnetization can be observed [@Bode2002]. In recent years noncollinear magnetic structures at transition-metal interfaces have gained popularity as promising candidates for spintronic applications due to their interesting dynamical and transport properties [@Fert2013; @Nagaosa2013]. A monolayer Mn grown on W(110) surface (Mn/W(110)) is a prominent example which exhibits a noncollinear magnetic structure with a cycloidal 173$^{\circ}$ spin-spiral ground state along the $[1\overline{1}0]$ direction [@Bode2007] that is driven by the Dzyaloshinskii-Moriya interaction. Using this magnetic surface with a noncollinear spin structure, it is possible to control the spin direction of adsorbed Co adatoms due to local exchange coupling which has been demonstrated in recent experiments using scanning tunneling microscopy (STM) by Serrate [[*et al.*]{}]{} [@Serrate2010; @Serrate2016]. The noncollinear spin state of the Mn monolayer is reflected due to hybridization even in the orbitals of the adsorbed Co adatom [@Haldar2018]. The possibility of controlling the magnetization direction of an adatom on this surface without the presence of external magnetic field makes this system very promising for TAMR studies. Compared to the domain walls of Fe/W(110) used in previous studies [@Schoneberg2018] the spin structure of this surface is known on the atomic scale and allows a quasi continuous rotation of the local spin quantization axis. Recently, Caffrey [[*et al.*]{}]{} have predicted TAMR values up to 50% for Ir adatoms, i.e. a $5d$ transition metal, on Mn/W(110) [@Caffrey2014a], however, experimental evidence is missing. Here we have explored Pb and Bi adatoms and dimers on Mn/W(110) in order to explore the magnitude of TAMR and its dependence on the $6p$ element and atomic arrangement on the surface. We have used first-principles density functional theory (DFT) calculations to investigate the adsorption of Pb and Bi adatoms and dimers on Mn/W(110) and studied their electronic and magnetic properties. The spin structure of Mn/W(110) is locally well approximated as a two-dimensional antiferromagnet [@Heinze2000]. We considered two limiting cases of spin directions which are possible due to the cycloidal nature and propagation direction of the spin spiral in the Mn layer: (i) a magnetization direction perpendicular to the surface (out-of-plane) and (ii) a magnetization direction pointing along the $[1\overline{1}0]$ direction (in-plane). Our results indicate that the adsorption of these adatoms facilitates local enhancement of SOC above the surface leading to very large values of the TAMR of 50% to 60% for adatoms. The orientation of Pb and Bi dimers is shown to be crucial in order to achieve even larger TAMR values. This can be understood based on the symmetry of the matrix elements of the SOC Hamiltonian as well as the hybridization of $6p$ adsorbate with $3d$ substrate states. This paper is organized as follows. First, we briefly discuss the computational methods used in our calculations. Then we proceed to discuss the structural, electronic, and magnetic properties, as well as the TAMR of adatoms and the same for dimers in different orientations. The TAMR effects are discussed focusing on the local density of states at the adsorbate atoms and the Mn layer and the vacuum density of states and interpreted based on a simplified model. We summarize our main conclusions in the final section. Computational details {#sec:compdet} ===================== In this work we used first-principles calculations using a plane wave based DFT code <span style="font-variant:small-caps;">vasp</span> [@vasp1; @vasp2] within the projector augmented wave method (PAW) [@blo; @blo1]. For the exchange-correlation, we have used the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [@PBE; @PBEerr]. For SOC, we followed the methods described by Hobbs [[*et al.*]{}]{} [@Hobbs2000]. We used a 450 eV energy cutoff for the plane wave basis set convergence. Structural relaxations are performed using a $6\times6\times1$ k-point Monkhorst-Pack mesh [@Monkhorst1976]. The vacuum local density of states (LDOS) was calculated by placing an empty sphere at a specific height of 5.3 [Å]{} above the adatoms onto which the LDOS was projected. For the calculation of electronic properties, magnetic properties and LDOS, we have used $20\times20\times1$ k-point Monkhorst-Pack mesh. Structural details {#subsec:structure} ------------------ ![(a) Top view and (b) perspective view of a $c(4\times4)$ supercell used for the adatom on Mn/W(110) calculation. Gray spheres represent W atoms while Mn atoms are depicted as green (light blue) spheres with arrows showing the ferromagnetic (antiferromagnetic) magnetic moments with respect to the $6p$ adatom (dark blue sphere). ‘nn’ and ‘nnn’ are the nearest neighbor and next nearest neighbor Mn atoms to the adatom. $x$ and $y$ refer to the direction of coordinates for the supercell. (c) Top view of a $c(6\times6)$ supercell used for dimer adsorption on Mn/W(110) along with the three dimer orientations considered in our calculations. $m_\perp$ and $m_\parallel$ denote the direction of a perpendicular magnetization and a parallel magnetization with respect to the surface, respectively.[]{data-label="fig:geom"}](figure1.png) We modeled Mn/W(110) using a symmetric slab consisting of five atomic layers of W with a pseudomorphic Mn layer on each side. We have approximated the local magnetic order of the system as antiferromagnetic, [[i.e.,]{}]{} collinear due to the long periodicity of the spin spiral ground state [@Heinze2000; @Bode2007; @Serrate2010]. The effect of the noncollinearity of the spin structure on the electronic states of adatoms has been studied before [@Haldar2018]. We used a $c(4\times4)$ AFM surface unit cell, as shown in Fig. \[fig:geom\](a)-(b). The GGA calculated lattice constant of W, [[i.e.,]{}]{} 3.17 [Å]{} is used for our calculations as it is in good agreement with the experimental value of 3.165 [Å]{}. A thick vacuum layer of $\approx$ 25 [Å]{} is included in the $z$ direction normal to the surface to remove interactions between repeating slabs. We added Pb or Bi adatoms at the hollow-site position on each Mn monolayer. The $c(4\times4)$ unit cell is large enough to keep the interactions between the periodic images of the adatoms small. For the adsorption of dimers, we have used a larger $c(6\times6)$ AFM surface unit cell (see Fig. \[fig:geom\](c)) to keep the interactions coming from the periodic images of the dimers negligible. In the case of Pb or Bi dimers, we have considered three possible dimer orientations on the surface: (i) along the \[001\] direction, (ii) along the $[1\overline{1}0]$ direction, and (iii) along the $[1\overline{1}1]$ direction as shown in Fig. \[fig:geom\](c). The magnetization direction in calculations including SOC has been chosen normal to the surface, $\perp$, and along the $[1\bar{1}0]$ in-plane direction, $\parallel$, as enforced by the cycloidal nature of the underlying spin spiral structure of Mn/W(110) [@Bode2007]. The position of the adatoms, dimers and the Mn layers are relaxed with 0.01 eV/[Å]{} force tolerance. We have kept the coordinates of W atoms fixed in all our calculations. Tunneling anisotropic magnetoresistance {#subsec:tamr} --------------------------------------- Using the spectroscopic mode of an STM, the TAMR can be obtained by measuring the differential conductance (d$I$/d$V$) above an adatom or a dimer for two different magnetization directions. The TAMR is obtained from $$\begin{aligned} \mathrm{TAMR} &= \frac{[(\mathrm{d}I/\mathrm{d}V)_\perp-(\mathrm{d}I/\mathrm{d}V)_\parallel]}{(\mathrm{d}I/\mathrm{d}V)_\perp}\; ,\end{aligned}$$ where $\perp$ and $\parallel$ denote a perpendicular magnetization and a parallel magnetization with respect to the surface, respectively. Within the Tersoff-Hamann model [@Tersoff1983; @Tersoff1985], the d$I$/d$V$ signal is directly proportional to the local density of states (LDOS), $n(z, \epsilon)$, at the tip position in the vacuum, $z$, a few [Å]{}ngströms above the surface. Hence, the TAMR can be calculated theoretically from the anisotropy of the LDOS arising due to SOC [@Bode2002; @Neel2013]. Then the TAMR can be calculated as: $$\begin{aligned} \mathrm{TAMR} &= \frac{n_\perp(z, \epsilon)-n_\parallel(z, \epsilon)}{n_\perp(z, \epsilon)}\; . \label{eq:TAMR}\end{aligned}$$ Results and Discussion {#sec:results} ====================== Pb and Bi adatoms on Mn/W(110) {#sbsec:adatoms} ------------------------------ ### Structural and magnetic properties {#sub2sec:str_mag} ---- ----------------- ------------------ ------------------------ ------------------------- ------------------------ ------------------------- $d_{\text{nn}}$ $d_{\text{nnn}}$ $\Delta z_{\text{nn}}$ $\Delta z_{\text{nnn}}$ $\Delta x_{\text{nn}}$ $\Delta y_{\text{nnn}}$ Pb 2.76 3.17 $-0.13$ +0.02 $-0.01$ +0.02 Bi 2.70 2.97 $-0.10$ +0.06 $-0.05$ +0.03 ---- ----------------- ------------------ ------------------------ ------------------------- ------------------------ ------------------------- : Relaxed distances (in [Å]{}) of Pb and Bi adatoms from the Mn atoms of the Mn/W(110) surface. $d_{\text{nn}}$ and $d_{\text{nnn}}$ denotes the nearest neighbor (nn) and the next-nearest neighbor (nnn) Mn atoms, respectively. $\Delta x$, $\Delta y$, and $\Delta z$ are the displacements with respect to the clean surface of Mn atoms after the adsorption of the adatoms. Positive (negative) values imply that the Mn atoms move towards (away from) the adatom.[]{data-label="tab:distance"} We begin our discussion with the local structural relaxations upon adsorption of Pb and Bi adatoms on Mn/W(110) which are tabulated in Table \[tab:distance\]. Our calculations indicate that the hollow site (see Fig. \[fig:geom\](a)) is the most stable adsorption site for both Pb and Bi adatoms. The other sites, [[e.g.,]{}]{} bridge and top sites are unstable for both adatoms in our calculations and collapse to the hollow site position. The adsorption of the adatoms creates a buckling in the underlying Mn layer in the vicinity of the adsorption sites (see Table \[tab:distance\]). Significant changes can be observed for the nearest neighbor (nn) Mn atoms, which move away from the adatoms, while the next nearest neighbor (nnn) Mn adatoms move slightly towards the adatoms. ---- -------- ------------------ ------------------- --------------------- Adatom Mn$_{\text{nn}}$ Mn$_{\text{nnn}}$ Mn$_{\text{clean}}$ Pb +0.00 +2.36 $-3.34$ $\pm$3.41 Bi +0.08 +2.60 $-3.35$ $\pm$3.41 ---- -------- ------------------ ------------------- --------------------- : Magnetic moments (in [[$\mu_{\text{B}}$]{}]{}) of the adsorbed Pb and Bi adatoms and the nearest neighbor (nn) and next nearest neighbor (nnn) Mn atoms of the Mn monolayer on W(110). For comparison the value of the clean Mn/W(110) surface is given.[]{data-label="tab:magmom"} The magnetic properties of these systems are affected by the hybridization between the $6p$ adatoms and underlying Mn atoms of Mn monolayer (see Table \[tab:magmom\]). The clean Mn surface of Mn/W(110) have magnetic moments $\pm$ 3.41 [[$\mu_{\text{B}}$]{}]{}. The $p_z$ orbitals of the adatoms mainly hybridize with the $d_{z^2}$ orbitals of nn Mn adatoms. The magnetic moments of the nn Mn adatoms drop quite significantly for both atom types. They are reduced by 1.05 [[$\mu_{\text{B}}$]{}]{} and 0.81 [[$\mu_{\text{B}}$]{}]{} for Pb and Bi adsorption, respectively. Due to the hybridization, the induced magnetic moment on Bi adatom is 0.08 [[$\mu_{\text{B}}$]{}]{}, whereas the Pb adatom is non-magnetic. The effect of hybridization is less prominent for the nnn Mn adatoms where a slight reduction of magnetic moment $\sim$ 0.06 [[$\mu_{\text{B}}$]{}]{} occurs for both adatoms. ### Electronic properties  Next, we discuss the electronic properties of the $6p$ adatoms adsorbed on the Mn/W(110) surface. Fig. \[fig:spinpol\_adatom\] shows the spin-resolved LDOS of the Pb and Bi adatom adsorbed on Mn/W(110), the LDOS of the neighboring Mn atoms, and $m_l$ decomposed $p$ states of Pb and Bi adatom. These calculations have been performed in the scalar relativistic approximation, [[i.e.,]{}]{} neglecting SOC. A possible hybridization can be observed by calculating and comparing the spin-resolved LDOS of the adatoms with the neighboring Mn states as shown in Fig. \[fig:spinpol\_adatom\]. This hybridization effect is clearly observed just below [[$E_{\text{F}}$]{}]{} where minority Mn peaks are located at the same position as $p_x$ and $p_z$ states of the adatoms. Further interactions are observed for Pb around [[$E_{\text{F}}$]{}]{}$-$0.50 eV, [[$E_{\text{F}}$]{}]{}$-$0.34 eV, where the states from the adatoms interact with the states from the nn Mn atoms. In this energy range one also sees reduced exchange splitting of the nn Mn states as compared to the nnn Mn states which affects the magnetic moment of the nn Mn atoms as mentioned in section \[sub2sec:str\_mag\]. The magnetic moment of nn Mn atoms drops to 2.36 [[$\mu_{\text{B}}$]{}]{} and 2.6 [[$\mu_{\text{B}}$]{}]{} upon adsorption of the Pb and Bi adatom, respectively. Despite the small spin splitting observed for both adatoms, the spin polarization of the adatoms is quite large. The spin polarization of the adatoms varies in-between $\pm 40$% which is mainly arising from the $p_z$ and $p_y$ states of the adatoms. ![(a, b) Spin-resolved partial charge density plots at 3 [Å]{} above the Pb adatom on Mn/W(110) in the energy range \[$E_\text{F}-0.065$, $E_\text{F}-0.045$ eV\]. (c, d) cross-sectional plots through the Pb adatom parallel to the \[001\] direction for the charge densities of the top panel. []{data-label="fig:chargedensity_Pb"}](figure3.png) Previously, it has been shown that the spin direction of adsorbed Co adatoms on Mn/W(110) can be detected in spin-polarized STM images at small bias voltages due to the different orbital symmetry of $d$ states in majority and minority spin channel [@Serrate2010]. We find a similar effect for the $p$ orbitals of Pb close to the Fermi energy, $E_F$. Figure \[fig:chargedensity\_Pb\] shows top and cross-sectional spin-resolved partial charge density plots in a small energy window \[[[$E_{\text{F}}$]{}]{}$-$0.065, [[$E_{\text{F}}$]{}]{}$-$0.045 eV\] for Pb adatom adsorbed on Mn/W(110). A strong interaction between the minority $p_x$ states of the adatom and the minority $d_{z^2}$ orbitals of the neighboring Mn atoms is clearly visible in the cross-sectional plot along the $[001]$ direction shown in Figs. \[fig:chargedensity\_Pb\](d). Here, the axes of the $d_{z^2}$ Mn orbitals are distorted pointing towards the Pb atom and a large part of the charge density is concentrated at the interface between adsorbate and substrate. However, such hybridization is less prominent in the majority channel which displays the rotationally symmetric shape of a $p_z$ orbital \[Fig. \[fig:chargedensity\_Pb\](c)\]. The partial charge density calculated at a height of 3 [Å]{} in the vacuum \[Fig. \[fig:chargedensity\_Pb\](a-b)\] shows that the both spin channels are clearly distinguishable from each other due to the shape of their orbitals. For the majority channel one can clearly observe the $p_z$ states of the adatom in the vacuum. In the minority channel, the double-lobed structure of the $p_x$ state protrudes rotationally symmetric states such as $s$, $p_z$ and $d_{z^2}$ orbitals which usually extend further into the vacuum. Similar behavior has been reported previously by Serrate [[*et al.*]{}]{} for different $d$-states of a Co adatom adsorbed on Mn/W(110) [@Serrate2010]. Hence, we can conclude that in an STM experiment with a magnetic tip it will be possible to identify the spin direction of the Pb adatom by means of the respective orbitals dominating near [[$E_{\text{F}}$]{}]{} yielding similar effects observed in spin-polarized STM [@Serrate2010; @Serrate2016; @Haldar2018]. However, for the Bi adatom the above mentioned feature is not present in the vicinity of $E_F$ which is accessible for STM. In this case, the majority $p_y$ states of Bi are completely covered by the rotationally symmetric orbitals in the vacuum (not shown). Therefore, the orbital shapes for majority and minority states for the charge densities calculated in the vicinity of [[$E_{\text{F}}$]{}]{} do not differ from one another. ### TAMR of Pb and Bi adatoms on Mn/W(110) {#sub2sec:TAMR_adatom} ![(a) Total (black lines) and spin-resolved (Majority: blue, Minority: red) vacuum LDOS including SOC above the Pb adatom on Mn/W(110) for out-of-plane ($\perp$, solid lines) and in-plane (parallel to the $[1\overline{1}0]$ direction) magnetizations ($\parallel$, dashed lines). (b) TAMR obtained from the spin-averaged vacuum LDOS according to Eq. (\[eq:TAMR\]). (c) Orbital decomposition of the LDOS of the Pb adatom in terms of the majority (up) and minority (down) states. Solid (dashed) lines correspond to the magnetization direction perpendicular (parallel) to the surface plane. The orange up and down arrow indicates majority and minority spin channels, respectively.[]{data-label="fig:PbAdatom"}](figure4.pdf) ![(a) Total (black lines) and spin-resolved (Majority: blue, Minority: red) vacuum LDOS including SOC above the Bi adatom on Mn/W(110) for out-of-plane ($\perp$, solid lines) and in-plane (parallel to the $[1\overline{1}0]$ direction) magnetizations ($\parallel$, dashed lines). (b) TAMR obtained from the spin-averaged vacuum LDOS according to Eq. (\[eq:TAMR\]). (c) Orbital decomposition of the LDOS of the Bi adatom in terms of the majority (up) and minority (down) states. Solid (dashed) lines correspond to the magnetization direction perpendicular (parallel) to the surface plane. The orange up and down arrow indicates majority and minority spin channels, respectively.[]{data-label="fig:BiAdatom"}](figure5.pdf) In this section we will focus on the description of the electronic structure of $6p$ adatoms adsorbed on the Mn monolayer of W(110). Especially the anisotropy of the LDOS due to SOC and the subsequent TAMR effect will be discussed in detail. Fig. \[fig:PbAdatom\](a) shows both the total (spin-averaged) and spin-resolved vacuum LDOS above the Pb adatom – in an energy range around $E_F$ typically accessible to STM – calculated for the two magnetization directions including SOC: (i) perpendicular to the surface (out-of-plane) denoted as $n_\perp(z, \epsilon)$ and parallel to the $[1\overline{1}0]$ direction (in-plane) denoted as $n_\parallel(z, \epsilon)$. Differences between both magnetization components are clearly discernible in the energy range below the Fermi level ([[$E_{\text{F}}$]{}]{}). The most significant feature is located at $-0.37$ eV in $n_\parallel(z, \epsilon)$ and corresponds to a peak of majority $p_z$ states being split and shifted towards lower energies as the magnetization rotates from the film plane ($\parallel$) to the perpendicular ($\perp$) direction of the surface. The same effect, although much less prominent, is also visible for the minority states. This behavior leads to a maximum value in the TAMR of $-49$% (see Fig. \[fig:PbAdatom\](b)). Around [[$E_{\text{F}}$]{}]{} this effect is considerably smaller and of opposite sign with TAMR values up to +22%. Similar observations can be seen in the vacuum LDOS of the Bi adatom on Mn/W(110) shown in Fig. \[fig:BiAdatom\](a). Here, the dominant peak of majority $p_z$ states which splits likewise upon rotation of the magnetization direction is shifted by 0.2 eV towards lower energies compared to Pb. Linked to this state, the value of the TAMR first takes a local maximum of +42% at $-0.66$ eV before dropping abruptly to a minimum of $-61$% at $-0.57$ eV below [[$E_{\text{F}}$]{}]{}. Similar to Pb, differences concerning $n_\perp$ and $n_\parallel$ for the minority channel are small in this energy range and the main part of the TAMR originates from majority states. In contrast, states with minority character are causing a modest TAMR of +20% just below [[$E_{\text{F}}$]{}]{}. For both adatoms the anisotropy of the vacuum LDOS shows only little magnetization-direction dependent differences in the unoccupied regions and giant values in the TAMR effect are restricted to areas below [[$E_{\text{F}}$]{}]{}. A closer look at the orbitally resolved LDOS of the adatoms in Fig. \[fig:PbAdatom\](c) and Fig. \[fig:BiAdatom\](c) reveals that the above-mentioned changes between both magnetization components in the vacuum can be attributed to $p_z$ states of the adatoms which are mostly below [[$E_{\text{F}}$]{}]{}. The curves in the vacuum almost coincide with the ones for states of this character calculated directly at the respective adatom. As the $p_z$ states are oriented along the surface normal, they preponderate in the vacuum compared to the $p_x$ and $p_y$ states. In contrast, the prominent peak of majority $p_y$ states dominating the LDOS in the vicinity of [[$E_{\text{F}}$]{}]{} of both Pb and Bi is not visible in the vacuum LDOS because they are aligned parallel to the film plane. The shift of this peak, from a position of 0.2 eV above [[$E_{\text{F}}$]{}]{} for Pb, towards occupied regions for Bi can be explained by the increasing number of electrons in the $p$ shell. On the other hand, the shift of the majority states with $p_z$ character which are identified to generate the large anisotropy of the vacuum LDOS and hence the shift of the position of the maximum TAMR can be ascribed to the different strength of the attractive potential acting between valence electrons and nucleus. Due to the larger nuclear charge these potentials lead to a stronger binding of the $p_z$ states to the nucleus for Bi. Further reasons for the majority $p_z$ states of the Bi adatom being shifted towards lower energies is the higher spin polarization compared to Pb as well as the smaller distance from its nearest neighbor Mn atom in the Mn monolayer. Hereby the orbital overlap increases resulting in a larger splitting of the states. ### Modeling of the TAMR In order to explain the large TAMR found for $6p$ adatoms adsorbed on Mn/W(110), we revert to the Hamiltonian of SOC mentioned in the introduction. As shown in Ref. [@Abate1965], the SOC operator can be written as a matrix in the following way: $$\mathcal{H}_{SOC}= \frac{\xi}{2} \begin{pmatrix} M&N\\ -N^{*}&M^{*} \end{pmatrix}\; .$$ Here, the diagonal matrices $M$ describe the coupling of two states with equal spin direction, whereas the secondary diagonal matrices $N$ denote the interaction of states with different spin character via SOC. Both can be calculated for an arbitrary orientation of the spin quantization axis by applying ladder operators of spin and angular momentum to linear combinations of complex spherical harmonics which represent both $p$ and $d$ orbitals. This approach yields the matrix element describing a spin-orbit induced hybridization between states with $p_z$ and $p_x$ symmetry in the same spin channel as [@Schoeneberg2016Diss]: $$\langle \uparrow,p_z|\mathcal{H}_{SOC}|p_x, \uparrow \rangle= i\sin\theta\sin\phi \;, \label{eq:H_SOC_up_up}$$ and the element for coupling states of the same symmetry, but with opposite spin direction as $$\langle \uparrow,p_z|\mathcal{H}_{SOC}|p_x, \downarrow \rangle=\cos\phi+i\sin\phi\cos\theta \;. \label{eq:H_SOC_up_down}$$ In the first case (cf. Eq. (\[eq:H\_SOC\_up\_up\])) the matrix element vanishes for the perpendicular magnetization direction ($\phi$=0$^{\circ}$, $\theta$=0$^{\circ}$) and becomes maximal for its magnetization pointing along the $[1\overline{1}0]$ direction ($\phi$=90$^{\circ}$, $\theta$=90$^{\circ}$), i.e. we expect a mixing of the two states only for a spin-quantization axis chosen along the film plane. The reverse is true if both states have opposite spin direction (cf. Eq. (\[eq:H\_SOC\_up\_down\])). Evaluating the matrix elements given in Ref. [@Schoeneberg2016Diss] for a potential hybridization mediated by SOC for states with $p_z$ and $p_y$ character shows that such interaction can not be realized on the Mn/W(110) surface for the two above mentioned magnetization directions, which are possible on the substrate due to the spin spiral ground state. For this reason the discussion concerning the anisotropy of the vacuum LDOS is restricted to $p_x$ and $p_z$ states for both $6p$ adatoms and dimers in this paper. Applying the above considerations first to the case of a Bi adatom on Mn/W(110) \[Fig. \[fig:BiAdatom\]\], one can explain the maximum value of the TAMR at $-0.57$ eV below [[$E_{\text{F}}$]{}]{} by a magnetization-direction dependent mixing of $p_x$ and $p_z$ orbitals of opposite spin channels. At this energy the prominent peak of majority $p_z$ states whose in-plane magnetization component resembles a single peak is split and shifted towards lower energies upon rotation of the spin-quantization axis (see Fig. \[fig:BiAdatom\](c)). According to the matrix elements, this behavior hints at a SOC-mediated hybridization with a minority $p_x$ state which can be found at $-0.82$ eV. The TAMR of the Pb adatom can also be understood based on the matrix elements of $H_{\rm SOC}$. E.g. the vacuum LDOS of the minority spin channel \[Fig. \[fig:PbAdatom\](a)\] just below $E_F$ is reduced upon rotating the magnetization direction from in-plane to out-of-plane. This is due to mixing by SOC in the minority spin channel \[Fig. \[fig:PbAdatom\](c)\] between a $p_z$ state located at $-0.12$ eV and a peak at $-0.05$ eV of $p_x$ orbital character. For an in-plane magnetization direction, which allows mixing within the same spin channel by SOC according to Eq. (\[eq:H\_SOC\_up\_up\]), the $p_x$ minority state peak at $-0.05$ eV splits into two peaks which coincide with the positions of two minority states $p_z$ peaks. This creates a large negative TAMR within the minority spin channel of $-56$% (not shown here). However, the TAMR is obtained from the total, spin-averaged LDOS. Just below the Fermi energy it is positive with a value of $+22$% due to a majority $p_z$ peak whose height is reduced due to SOC for an in-plane magnetization \[Fig. \[fig:PbAdatom\](c)\]. The maximum TAMR effect of the Pb adatom of $-49$% occurs at $0.37$ eV below [[$E_{\text{F}}$]{}]{}. It originates from the majority spin channel \[Fig. \[fig:PbAdatom\](a)\] and it is due to the splitting of a majority $p_z$ state as can be seen from the orbital decomposition at the Pb atom \[Fig. \[fig:PbAdatom\](c)\]. Since the mixing occurs for a magnetization direction perpendicular to the surface it can be explained by a SOC induced mixing with $p_x$ states of the opposite spin channel according to Eq. (\[eq:H\_SOC\_up\_down\]). While changes in the minority $p_x$ LDOS can be noted within the relevant energy interval it is not possible to unambiguously propose a single peak which is responsible for the mixing. As will be discussed in detail for the Pb dimers at the end of this manuscript, there is also an impact of the Mn $3d$ states which are also subject to SOC and with which the Pb $p$ states are hybridizing. Pb and Bi dimers on Mn/W(110) {#subsec:dimers} ----------------------------- ### Structural and magnetic properties {#sub2sec:str_mag_dimers} -------------------- ------ -------------------------- ------ -------------------------- $d$ [[$\mu_{\text{B}}$]{}]{} $d$ [[$\mu_{\text{B}}$]{}]{} $[001]$ 3.23 +0.08 3.22 +0.12 $[1\overline{1}0]$ 3.35 +0.18 3.93 +0.02 $[1\overline{1}1]$ 3.11 $\pm$0.02 3.12 $\pm$0.02 Free 2.96 0.67 2.68 0.0 -------------------- ------ -------------------------- ------ -------------------------- : The dimer bond lengths $d$ (in Å) and the individual magnetic moments (in [[$\mu_{\text{B}}$]{}]{}) of the adsorbed Pb and Bi dimers on the Mn monolayer on W(110). For comparison, the bond lengths and the magnetic moments of the free dimers (calculated with SOC) are given.[]{data-label="tab:dimer_d_m"} Since the Pb and Bi adatoms adsorb in the hollow-site position of the Mn layer, the dimers can be oriented along the $[001]$, $[1\overline{1}0]$, and $[1\overline{1}1]$ directions (see Fig. \[fig:geom\](c)). The relaxed dimer bond lengths and the magnetic moments for the three orientations along with the values for free dimers are given in Table \[tab:dimer\_d\_m\]. The dimer bond lengths increase after the adsorption due to structural relaxation from the bond length values of free dimers. For Pb dimers an increase of $\approx$ 10% in bond length can be observed. For Bi dimers, a larger increase of $\approx$ 20% in bond length has been observed except along the $[1\overline{1}0]$ orientation. In this case, we find an increase of bond length values by $\approx$ 45 %. In Pb dimers, the individual atoms carry small induced magnetic moments for all three orientations due to the hybridization with the Mn monolayer. Among the three orientations, the largest individual magnetic moment of +0.18 [[$\mu_{\text{B}}$]{}]{} is observed for the $[1\overline{1}0]$ orientation. These induced moments for Pb dimers are in contrast with the single atom adsorption where Pb remains nonmagnetic. Similar to the single adatom adsorption, Bi dimers also pick up small induced magnetic moment for all orientations with the largest value of +0.12 [[$\mu_{\text{B}}$]{}]{} along the $[001]$ direction. Similar to the single adatom adsorption, reduction of magnetic moments for both nn and nnn Mn adatoms have been observed here as well. ### Electronic properties  We proceed by describing and comparing the electronic structure of the dimers with those presented for the single adatoms before explaining the anisotropy of the LDOS. It should be pointed out here that the notation of the $p$ orbitals of the dimers refers to the global coordinate axes of the Mn/W(110) surface as shown in Fig. \[fig:geom\], i.e. no local system for the adsorbates rotated for different orientations has been used. Therefore, the $p_x$ and $p_y$ orbitals of both \[001\] and $[1\overline{1}0]$ dimers are aligned along the \[001\] and $[1\overline{1}0]$ direction, respectively. As a result, the orbitals responsible for the covalent bond are changing. Fig. \[fig:Pb\_Bi\_dimer\_dos\_full\] shows the $m_l$ decomposed $p$ states of both Pb and Bi dimers on a large energy scale around [[$E_{\text{F}}$]{}]{}. We will exemplify the differences in the LDOS compared to the single adatoms by means of the Bi dimers; similar observations can be made for the respective Pb adsorbates. Compared with the Bi adatom (cf. Fig. \[fig:spinpol\_adatom\](f)), the Bi dimer along the $[001]$ orientation exhibits the largest modifications in its $p_x$ orbitals which are responsible for the covalent bond in this case hereby forming $\sigma$ orbitals (cf. Fig. \[fig:Pb\_Bi\_dimer\_dos\_full\](b)). This becomes most evident in the minority channel just below [[$E_{\text{F}}$]{}]{} where the corresponding states of the single adatom are shifted by 1 eV to the left due to the orbital overlap of the two atoms of the dimer. In contrast, the $p_y$ and $p_z$ states only show minor differences compared to the Bi adatom; especially the large peak of majority $p_y$ states dominating close to [[$E_{\text{F}}$]{}]{} of the adatom is also found for the $[001]$ dimer. Owing to the large distance of 3.93 [Å]{} between the two Bi atoms of the $[1\overline{1}0]$ dimer the overlap of their orbitals is small resulting in similar features as for the adatom (cf. Fig. \[fig:Pb\_Bi\_dimer\_dos\_full\](d)). The main change in its $m_l$ resolved LDOS is the disappearance of the dominant majority $p_y$ peak at [[$E_{\text{F}}$]{}]{} which can be attributed to the fact that these orbitals are forming $\sigma$ bonds in this case. For the Bi $[1\overline{1}1]$ dimer, differences in the $m_l$ decomposed LDOS (cf. Fig. \[fig:Pb\_Bi\_dimer\_dos\_full\](f)) are clearly discernible compared to the single adatom (cf. Fig. \[fig:spinpol\_adatom\](f)). This observation can partly be ascribed to the small bond length of 3.12 [Å]{} and hence a large orbital overlap and partly to the combination of $p_x$ and $p_y$ states forming $\sigma$ bonds. Both orbitals are tilted with respect to the global coordinate axes of the substrate leading to a covalent bond of a mixture of the two states with different symmetry. The $p_z$ states which are crucial for STM are relatively weakly affected for all dimers. ### TAMR effect of the Pb dimers on Mn/W(110) {#sub2sec:TAMR_dimers_Pb}  In the following section we study the TAMR of Pb dimers on Mn/W(110) for the three different dimer orientation discussed before. In Fig. \[fig:Pb\_dimer\_dos\](a) both the total and spin-resolved components of the vacuum LDOS of the Pb dimer oriented along the \[001\] direction are plotted for the two different magnetization directions which can occur due to the spin spiral states of the Mn monolayer on W(110). Note, that the in-plane magnetization direction is perpendicular to the dimer axis in this case. Compared to the respective adatom (cf. Fig. \[fig:PbAdatom\]), only small differences between $n_{\perp}$ and $n_{\parallel}$ can be observed in the occupied regions below [[$E_{\text{F}}$]{}]{}. The most striking features occur now at energies $-0.7$ eV, $-0.42$ eV, $-0.2$ eV, and $0.05$ eV leading to maximum values in the TAMR of $\pm 28$% (see Fig. \[fig:Pb\_dimer\_dos\](b)). Unlike the Pb adatom, the anisotropy of the vacuum LDOS takes another local maximum of +29% just above [[$E_{\text{F}}$]{}]{}. As one can see from the spin-resolved curves in Fig. \[fig:Pb\_dimer\_dos\](a), the TAMR at $-0.7$ eV stems from a modification of both spin channels upon rotation of the magnetization direction, whereas at [[$E_{\text{F}}$]{}]{} only majority states contribute whose perpendicular magnetization components are clearly enhanced compared to the parallel ones. Similar to the Pb adatom, differences between $n_{\perp}$ and $n_{\parallel}$ in the unoccupied regions are barely noticeable for both spin directions of the \[001\] dimer. The orbitally decomposed LDOS of this dimer plotted in Fig. \[fig:Pb\_dimer\_dos\](c), shows further similarities with the adatom. It is dominated by a prominent peak of majority $p_y$ states at 0.15 eV above [[$E_{\text{F}}$]{}]{} which is not reflected in the vacuum LDOS and exhibits discernible changes in the $p_z$ states with respect to both magnetization directions below [[$E_{\text{F}}$]{}]{}. The $p_z$ states predominate the vacuum LDOS due to their double-lobed orbitals pointing along the surface normal. However, they experience a small shift towards lower energies as well as a splitting which is a consequence of the interaction between both Pb atoms composing the dimer. Compared to the $[001]$ Pb dimer, the anisotropy of the vacuum LDOS is much larger for the dimer oriented along the $[1\overline{1}0]$ direction representing the propagation direction of the spin spiral on Mn/W(110) (see Fig. \[fig:Pb\_dimer\_dos\](d)-(e)). For this dimer orientation the in-plane magnetization direction is along the dimer axis (cf. Fig. \[fig:geom\]). As for the single adatom, the appearance of the LDOS in the vacuum below [[$E_{\text{F}}$]{}]{} is characterized by magnetization-direction dependent differences of the majority $p_z$ states where the main contribution comes from a dominant peak of the in-plane magnetized dimer at $-0.45$ eV below [[$E_{\text{F}}$]{}]{}. Being split multiple times upon reorientation of the spin-quantization axis, it creates a steep descent in the TAMR up to $-64$% thereby even exceeding the maximum value of the Pb adatom by 15%. Just above [[$E_{\text{F}}$]{}]{}, the anisotropy of the vacuum LDOS takes a local minimum of $-38$% which is due to the shift of a minority $p_z$ state as the magnetization rotates from the perpendicular direction to the film plane along the $[1\overline{1}0]$ direction. The substantial similarity of the LDOS of this dimer in the $p_z$ orbitals compared to the single adatom (cf. Fig. \[fig:PbAdatom\]) can be explained by means of the relatively large distance of both Pb atoms (see Table \[tab:dimer\_d\_m\]). If they are further apart, their interaction, i.e. the overlap of their orbitals, will be small thereby causing a similar behavior as for a single atom (cf. Fig. \[fig:PbAdatom\](c) and Fig. \[fig:Pb\_dimer\_dos\](f)). The $p_y$ orbitals, on the other hand, which are forming $\sigma$ bonds in the case of the $[1\overline{1}0]$ dimer are expected to show more remarkable differences in comparison with the adatom. This becomes mostly evident in the unoccupied regions where the prominent peak of majority $p_y$ states is completely absent as seen in Fig.  \[fig:Pb\_dimer\_dos\](f) with respect to the adatom (cf. Fig. \[fig:PbAdatom\](c)).  The electronic structure of the Pb dimer along $[1\overline{1}1]$ direction is shown in Fig. \[fig:Pb\_dimer\_dos\](g)-(i) for both magnetization directions. As for the \[001\] Pb dimer, only minor differences between $n_{\perp}$ and $n_{\parallel}$ are visible in the vacuum LDOS above the $[1\overline{1}1]$ dimer along the diagonal of the unit cell. At $-0.6$ eV below [[$E_{\text{F}}$]{}]{}, a peak of majority states for parallel magnetization direction is enhanced compared to the perpendicular magnetization direction resulting in a negative TAMR of $-25$%. The vacuum LDOS of the unoccupied spectrum on the other hand is mainly characterized by magnetization-direction dependent changes of the minority states causing maximum values of +28% in the TAMR at +0.62 eV. The magnitude of TAMR is much smaller for $[1\overline{1}1]$ dimer orientation as compared to the $[1\overline{1}0]$, which is oriented along the natural magnetization direction of the Mn/W(110) substrate. However, it is very similar as for the dimer oriented along $[001]$, which is perpendicular to the magnetization of the surface. We will discuss these behavior in more detail in Section \[TAMR\_origin\_Dimers\]. ### TAMR of Bi dimers on Mn/W(110) {#sub2sec:TAMR_dimers_Bi} Similar changes of the TAMR with the dimer orientation can be observed for Bi dimers on Mn/W(110). Considering first the electronic structure of the \[001\] Bi dimer which is shown in Fig. \[fig:Bi\_dimer\_dos\](a)-(c), one notices that as for the \[001\] Pb dimer the curves for both magnetization directions of the spin-averaged vacuum LDOS do not differ significantly from each other. The largest differences are now located in the energy range between $-0.9$ and $-0.3$ eV leading to a local maximum in the TAMR of $-37$% at $-0.75$ eV. The negative TAMR value is much smaller than for the Bi adatom (cf. Fig. \[fig:BiAdatom\](b)). Consistent with the single Bi adatom, the energy range just below [[$E_{\text{F}}$]{}]{} is dominated by a large peak of minority states with $p_z$ character (see Fig. \[fig:Bi\_dimer\_dos\](c)), whereas the prominent peak of majority $p_y$ orbitals is absent in the vacuum LDOS due to its orientation within the film plane. In contrast to the anisotropy of the vacuum LDOS of the corresponding Pb dimer, changes between $n_{\perp}$ and $n_{\parallel}$ vanish in the case of the \[001\] Bi dimer directly at [[$E_{\text{F}}$]{}]{}. A closer look at the orbitally resolved LDOS reveals that the $p_z$ states which dominate the LDOS above the surface are affected by the mutual interaction of both Bi atoms composing the dimer. In comparison with the single Bi adatom, they are shifted towards lower energies and experience a larger splitting which is mostly apparent in the majority channel. The dominant peak of majority $p_z$ states causing the large TAMR of $-61$% at $-0.57$ eV for the single Bi adatom (cf. Fig. \[fig:BiAdatom\](b)) is therefore located outside of the presented energy range for the dimers. However, owing to the interaction of the orbitals it is split as well and less pronounced than for the single Bi adatom (not shown). The hybridization of majority $p_z$ orbitals is less prominent in the case of a Bi dimer oriented along the $[1\overline{1}0]$ direction (Fig. \[fig:Bi\_dimer\_dos\](d)-(f)) of the Mn/W(110) surface due to the large distance of nearly 4 [Å]{} between both Bi atoms. As one can see from its orbital decomposition in Fig. \[fig:Bi\_dimer\_dos\](f), $p_z$ states move closer to [[$E_{\text{F}}$]{}]{} showing similar characteristics as in the case of the single Bi adatom (cf. Fig. \[fig:BiAdatom\]). At $E_{\rm F}-0.55$ eV a dominant peak of majority $p_z$ states for the in-plane magnetization orientation becomes visible which both decreases in height and shifts towards lower energies upon rotation of the spin-quantization axis. This is also the largest observable change between $n_{\perp}$ and $n_{\parallel}$ of the total (spin-averaged) vacuum LDOS for the $[1\overline{1}0]$ Bi dimer that is plotted in Fig. \[fig:Bi\_dimer\_dos\](d). The just mentioned magnetization-direction dependent changes in the majority $p_z$ states thereby correspond to a huge TAMR effect of $-64$% at $-0.55$ eV. Hence, the anisotropy of the LDOS is in the same order of magnitude as for the single Bi adatom and takes its largest value at the same energetic position as well. The TAMR is considerably smaller for the rest of the presented energy range, especially at [[$E_{\text{F}}$]{}]{} where differences for parallel and perpendicular magnetizations of both spin channels only create a modest TAMR of approximately 15%. As observed for the corresponding Pb dimer, only minor differences between both magnetization directions occur in the spin-averaged vacuum LDOS of the Bi dimer placed along the diagonal of the unit cell, i.e. the $[1\overline{1}1]$ direction. Whereas the spin-resolved curves are indeed clearly characterized by changes upon rotation of the magnetization in the occupied regions (see Fig. \[fig:Bi\_dimer\_dos\](g)), changes in the sum of both spin channels only lead to small values in the TAMR ranging from $-16$% at $-0.8$ eV up to $-10$% at [[$E_{\text{F}}$]{}]{}. As shown in the orbital decomposed LDOS of the dimer in Fig. \[fig:Bi\_dimer\_dos\](i), the main part of the anisotropy is created by the $p_z$ orbitals. Additionally, this dimer orientation exhibits the smallest TAMR of all studied Bi dimer geometries and consistent with the previously presented results for $[1\overline{1}1]$ Pb dimers on Mn/W(110). ### Origin of the TAMR for $6p$ dimers on Mn/W(110) {#TAMR_origin_Dimers} The variation of the TAMR magnitude depending on the orientation of both Pb and Bi dimers can partially be explained by means of a physical model considering two atomic states coupled via SOC proposed in Ref. . For a possible SOC induced hybridization of the dimer $p$ states, we refer to the matrix elements presented in section \[sub2sec:TAMR\_adatom\]. The orbitals are defined with respect to the global coordination axes of the unit cell. Within this simplified model, our expectations match quite well with the $6p$ dimers oriented along the $[1\overline{1}0]$ direction showing the largest TAMR of all studied configurations. However, the SOC induced hybridization is not so clearly visible between their $p_z$ and $p_x$ states forming $\pi_z$ and $\pi_x$ molecular orbitals, respectively. The reduction of the TAMR effect for the $[1\overline{1}1]$ $6p$ dimers can be understood as well using the simplified model of two atomic states with differing orbital symmetry. For the case of a dimer orientation along the diagonal of the supercell the magnetization direction of the substrate is rotated with respect to the bonding axis of the atoms leading to a reduction of the respective matrix elements. For instance the hybridization between the $p$ states, $ \langle \uparrow,p_z|\mathcal{H}_{SOC}|p_x, \uparrow \rangle \propto \sin\theta\sin\phi$, is reduced for an azimuth angle of 45$^{\circ}$. Since changes in the LDOS scale with the square of the matrix elements [@Bode2002], for both $6p$ dimers, the TAMR in $[1\overline{1}1]$ orientation is diminished by a factor of 4 compared to the $[1\overline{1}0]$ dimer ($\sim$ 15% vs. $\sim$ 60%; cf. Fig. \[fig:Pb\_dimer\_dos\](e), Fig. \[fig:Pb\_dimer\_dos\](h) and Fig. \[fig:Bi\_dimer\_dos\](e), Fig. \[fig:Bi\_dimer\_dos\](h)). The same behavior has recently been observed for Pb dimers on a Fe bilayer on W(110) [@Schoneberg2018]. In addition, the $p_x$ states are partially involved in the formation of molecular $\sigma$ bonds along the dimer axis and thereby not available for the mixing with the $p_z$ states which further reduces the possible value of the TAMR. The reduction of the LDOS of $p_x$ states in the shown energy range due to hybridization is even more apparent in the $[001]$ and $[1\bar{1}1]$ Bi dimers (cf. Fig. \[fig:Bi\_dimer\_dos\](c), Fig. \[fig:Bi\_dimer\_dos\](i)) while it is similar to that of the Bi adatom (cf. Fig. \[fig:BiAdatom\]) for the $[1\bar{1}0]$ dimer (cf. Fig. \[fig:Bi\_dimer\_dos\](f)). For the $[001]$ Pb dimer, within the simplified model, one can interpret the changes in the curves of the minority $p_x$ and $p_z$ states upon rotation of the spin-quantization axis at $-0.52$ eV due to hybridization mediated by SOC (see Fig. \[fig:Pb\_dimer\_dos\](c)). The same effect could already be observed for the single Pb adatom directly at [[$E_{\text{F}}$]{}]{} (cf. Fig. \[fig:PbAdatom\](c)). If the dimer orientation ($[001]$) is perpendicular to the magnetization direction ($[1\overline{1}0]$) of the substrate, one would expect SOC to mix molecular $\pi_z$ and antibonding $\sigma^{*}$ orbitals which are composed of $p_x$ states here [@Schoneberg2018]. Molecular orbitals of this symmetry are located further apart in the energy spectrum than $\pi_z$ and $\pi_x$ molecular orbitals that can easily hybridize via SOC for a dimer axis along the magnetization of the Mn/W(110) surface, [[i.e.,]{}]{} the $[1\overline{1}0]$ direction. Hence, within this simple model, the $[001]$ $6p$ dimers were expected to exhibit a much smaller variation of their electronic structure under the influence of SOC. However, our DFT calculations show that the anisotropy of the LDOS actually takes an unexpected high value of $-28$% and $-37$% for the case of the $[001]$ Pb and Bi dimer adsorbed on Mn/W(110), respectively. Hence, this model based on only two atomic/molecular states is not sufficient to quantitatively understand the TAMR for dimers along the $[001]$ axis. ![(a) and (b): top view (cross section) of the spin-resolved partial charge density plots of the Pb \[001\] dimer on Mn/W(110) in the energy range \[[[$E_{\text{F}}$]{}]{}$-0.51$, [[$E_{\text{F}}$]{}]{}$-0.49$ eV\]. (c) and (d): cross-sectional plots through the Pb dimer parallel to the \[001\] direction for the charge densities shown in (a) and (b).[]{data-label="fig:pb_001_chg"}](figure9.png) ![(a) and (b): top view (cross section) of the spin-resolved partial charge density plots of the Pb $[1\overline{1}0]$ dimer on Mn/W(110) in the energy range \[[[$E_{\text{F}}$]{}]{}$-0.46$, [[$E_{\text{F}}$]{}]{}$-0.44$ eV\]. (c) and (d): cross-sectional plots through the Pb dimer parallel to the $[1\overline{1}0]$ direction for the charge densities shown in (a) and (b). []{data-label="fig:pb_110_chg"}](figure10.png) In order to achieve a deeper understanding of the effect of $p-d$ hybridization on TAMR for the Pb and Bi dimers on Mn/W(110), we have calculated the partial charge densities within the scalar-relativistic approximation, [[i.e.,]{}]{} neglecting SOC, for a small energy range where the TAMR appears most prominent. The inclusion of SOC will not affect the hybridization as evident from the LDOS in the scalar relativistic approximation \[Fig. \[fig:Pb\_Bi\_dimer\_dos\_full\]\] and including SOC \[Figs. \[fig:Pb\_dimer\_dos\], \[fig:Bi\_dimer\_dos\]\]. In the following we exemplify the influence of the substrate by means of both $[001]$ and $[1\overline{1}0]$ Pb dimers only as we observe similar characteristic behaviors for the Bi dimers. The spin-resolved partial charge density of the $[001]$ Pb dimer at approximately [[$E_{\text{F}}$]{}]{}$-0.5$ eV at which the large TAMR of $\sim$ 28% occurs (cf. Fig. \[fig:Pb\_dimer\_dos\](b)) is shown in Fig. \[fig:pb\_001\_chg\]. From the top view \[Fig. \[fig:pb\_001\_chg\](a,b)\] which represents a cross section through the dimer one can clearly see the molecular $\pi_z$ and $\sigma_x$ character of the adsorbate for both spin channels, whereas from the side view \[Fig. \[fig:pb\_001\_chg\](c,d)\] a strong hybridization with the $d$ orbitals of the Mn atoms of the surface becomes visible. In the majority spin channel the axes of the $p_z$ orbitals at the Pb dimer deviate from the $z$ direction of the unit cell and the $d$ orbitals of the Mn atoms are twisted towards the adsorbate \[Fig. \[fig:pb\_001\_chg\](c)\]. This leads to a tilt of the upper lobes of the $p_z$ orbitals towards the center of the dimer and an overlap of the lower lobes with the Mn $d$ states in the case of the majority channel. For the minority channel on the other hand \[Fig. \[fig:pb\_001\_chg\](d)\] a clear differentiation between the Pb and Mn states is not possible anymore due to the strong hybridization which becomes manifest in an accumulation of the charge density at the interface. These observations already indicate that for the explanation of the TAMR effect of the $6p$ dimers on Mn/W(110) more than two atomic states have to be taken into account. Hybrid Pb-Mn interface states are also present in the majority channel of the $[1\overline{1}0]$ Pb dimer at the position of the maximum TAMR around 0.45 eV below [[$E_{\text{F}}$]{}]{} (see Fig. \[fig:pb\_110\_chg\]). A closer look at the calculated charge density reveals that its majority $p_z$ orbitals only interact with the Mn atom below the dimer axis, but not with the other atoms of the Mn monolayer \[Fig. \[fig:pb\_110\_chg\](c)\]. Exactly the same behavior can also be observed for the corresponding $[1\overline{1}0]$ Bi dimer on Mn/W(110) (not shown). The reason for this hybridization can be explained by means of the different distances of the $6p$ atoms and their neighboring Mn atoms. While the central Mn atom and one atom of the Pb dimer are separated by just 2.84 [Å]{}, the respective distance towards the next Mn atoms is 3.15 [Å]{} and hence significantly larger. However, the interaction with the central Mn atom described above cannot be realized for the $[1\overline{1}1]$ dimer since the respective atom of the substrate is missing below a bonding axis along the diagonal (see Fig. \[fig:geom\](c)). ![Orbital decomposed LDOS including SOC of the central Mn atoms below the $[001]$ and $[1\overline{1}0]$ Pb dimer in terms of the majority (up) and minority (down) states. Solid (dashed) lines correspond to the magnetization direction perpendicular (parallel) to the surface plane. The orange up and down arrow indicates majority and minority spin channels, respectively.[]{data-label="fig:Mn_Soc_Dos"}](figure11.pdf) Keeping in mind the studies of the partial charge densities, we did a further investigation of the LDOS of the central Mn atoms below the Pb dimer axes for the case of the $[001]$ and $[1\overline{1}0]$ direction. Fig. \[fig:Mn\_Soc\_Dos\] shows their orbitally decomposed $d$ states in an energy interval of $\pm$1 eV around [[$E_{\text{F}}$]{}]{} along with the $p$ states of the adsorbates for the two different magnetization directions discussed before. It is evident that the Mn atoms below both Pb dimers are likewise affected by the rotation of the magnetization direction and bear resemblance to the changes in the states of the Pb atoms at same energetic positions. For the $[001]$ direction this becomes mostly apparent at 0.15 eV where a large peak of majority $d_{yz}$ orbitals of the central Mn atom shows the same behavior upon a change of the spin-quantization axis as the dominant $p_y$ state of the adsorbed dimer (cf. Fig. \[fig:Pb\_dimer\_dos\](c)). Moreover one can observe an enhancement of the parallel magnetization component of majority states with $d_{z^2}$ and $d_{x^2-y^2}$ character at $-0.41$ eV which corresponds with a small peak of $n_{\parallel}$ for the majority $p_z$ states at the same energy. Further resemblance regarding magnetization-direction dependent differences in the LDOS are found between $-0.70$ eV and $-0.50$ eV for the minority $d_{xz}$ orbitals of Mn and $p_z$ and $p_x$ orbitals of the $[001]$ Pb dimer. For the $[1\overline{1}0]$ Pb dimer this SOC-dependent hybridization is most prominent for the $d_{yz}$ states both at [[$E_{\text{F}}$]{}]{} in the minority channel and at [[$E_{\text{F}}$]{}]{}$-0.45$ eV in the majority channel. Especially at the last-mentioned position, for the in-plane magnetization, it becomes clear that the majority Mn $d_{yz}$ states and the majority Pb $p_z$ states interacts quite strongly \[Fig. \[fig:Mn\_Soc\_Dos\]\] and produce a large TAMR value (see Fig. \[fig:Pb\_dimer\_dos\](e)). Conclusion ========== In conclusion, we have presented a detailed study of the spin-resolved electronic structure of single Pb and Bi adatoms and dimers adsorbed on the Mn monolayer on W(110) including the effect of spin-orbit coupling. Using density functional theory, we calculated the tunneling anisotropic magnetoresistance effect from two magnetization directions, imposed due to the cycloidal spin spiral ground state in the Mn layer, for the respective $6p$ adsorbate: perpendicular to the surface (out-of-plane) and parallel to the $[1\overline{1}0]$ direction representing the propagation direction of the spin spiral ground state (in-plane). Our calculations for the $6p$ adatoms which are characterized by large spin-orbit coupling constants predict an enhancement of the TAMR up to 49% for Pb and 61% for Bi adatoms. In both cases it can mainly be attributed to magnetization-direction dependent changes of majority $p_z$ states of the adatom. The anisotropy of the LDOS of both adatoms can generally be explained by means of a simplified physical model which considers the coupling of two atomic states with different orbital symmetry ($p_z$ and $p_x$ in the present case) via spin-orbit coupling. Although Pb and Bi adatoms carry almost no magnetic moment, they exhibit a large spin polarization directly at the surface and also in the vacuum due to the hybridization with the substrate. The spin polarization becomes maximal with values up to 60% around [[$E_{\text{F}}$]{}]{} for the Pb adatom. We have also investigated the TAMR for three different dimer orientations adsorbed on the Mn/W(110) surface. Consistent with the expectations both Pb and Bi dimers with their bonding axis along the magnetization direction of the substrate, i.e. the $[1\bar{1}0]$ direction, show the maximum anisotropy of the vacuum LDOS with values of 64% in the occupied regions. The origin of this large effect is a molecular $\pi_z$ orbital with majority spin character which strongly interacts with the central Mn atom below the dimer axis. Similar interactions are also found for a dimer orientation perpendicular to the magnetization direction of Mn/W(110), but with much smaller TAMR values of 37% for Bi and 28% for Pb, respectively. The TAMR becomes minimal for $6p$ dimers along the diagonal $[1\overline{1}1]$ direction (16% in the case of Bi, 27% for Pb) due to reduced SOC induced mixing of the $p$ states on the one hand and due to missing Mn atoms for hybridization below their bonding axes on the other hand. A further exploration of the central Mn atoms below the $[001]$ and $[1\overline{1}0]$ dimers has shown that their $d$ orbitals are likewise affected by changes upon rotation of the magnetization direction which has to be taken into account for the comprehension of the TAMR effect apart from the simple model of only two atomic states interacting by SOC. Acknowledgments {#acknowledgments .unnumbered} =============== We acknowledge the DFG via SFB677 for financial support. We gratefully acknowledge the computing time at the supercomputer of the North-German Supercomputing Alliance (HLRN). We thank N. M. Caffrey for valuable discussions. [50]{} ifxundefined \[1\][ ifx[\#1]{} ]{} ifnum \[1\][ \#1firstoftwo secondoftwo ]{} ifx \[1\][ \#1firstoftwo secondoftwo ]{} ““\#1”” @noop \[0\][secondoftwo]{} sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{} @startlink\[1\] @endlink\[0\] @bib@innerbibempty [****,  ()](https://doi.org/10.1103/PhysRevLett.99.067202) [****,  ()](https://doi.org/10.1126/science.1154415) [****,  ()](https://doi.org/10.1103/PhysRevLett.103.057202) [****, ()](https://doi.org/10.1038/nphys1616) [****,  ()](http://stacks.iop.org/1367-2630/13/i=8/a=085011) [****,  ()](https://doi.org/10.1103/PhysRevB.86.180406) [****,  ()](https://doi.org/10.1126/science.1228519) [****, ()](https://doi.org/10.1103/PhysRevLett.89.237205) [****, ()](https://doi.org/10.1103/PhysRevLett.93.117203) [****,  ()](https://doi.org/10.1126/science.1082857) [****,  ()](https://doi.org/10.1126/science.1146110) [****,  ()](https://doi.org/10.1126/science.1191688) [****, ()](https://doi.org/10.1103/PhysRevLett.106.037205) [****,  ()](https://doi.org/10.1126/science.1252841) [****,  ()](https://doi.org/https://doi.org/10.1016/j.tsf.2008.08.172) [****,  ()](https://doi.org/10.1038/nmat3304) [****,  ()](https://doi.org/10.1103/PhysRevB.73.024418) [****, ()](https://doi.org/10.1103/PhysRevLett.98.046601) [****,  ()](https://doi.org/10.1103/PhysRevB.79.155303) [****, ()](https://doi.org/10.1103/PhysRevB.80.045312) [****, ()](https://doi.org/10.1103/PhysRevLett.99.226602) [****,  ()](https://doi.org/10.1140/epjb/e2006-00201-3) [****, ()](https://doi.org/10.1103/PhysRevLett.97.127202) [****, ()](https://doi.org/10.1103/PhysRevLett.110.037202) [****, ()](https://doi.org/10.1021/acs.nanolett.5b05071) [****, ()](https://doi.org/10.1103/PhysRevB.81.212409) [****, ()](https://doi.org/10.1038/nmat2983) [****,  ()](https://doi.org/10.1103/PhysRevB.97.220406) [****,  ()](https://doi.org/10.1134/S0021364016070043) [****,  ()](https://doi.org/10.1103/PhysRevB.97.041114) @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****, ()]{} [****,  ()](https://doi.org/10.1103/PhysRevB.98.220401) [****,  ()](https://doi.org/10.1088/0953-8984/26/39/394010) [****,  ()](https://doi.org/10.1126/science.288.5472.1805) [****,  ()](https://doi.org/10.1103/PhysRevB.54.11169) @noop [****,  ()](https://doi.org/10.1103/PhysRevB.50.17953) [****,  ()](https://doi.org/10.1103/PhysRevB.59.1758) [****, ()](https://doi.org/10.1103/PhysRevLett.77.3865) [****,  ()](https://doi.org/10.1103/PhysRevLett.78.1396) [****,  ()](https://doi.org/10.1103/PhysRevB.62.11556) [****,  ()](https://doi.org/10.1103/PhysRevB.13.5188) [****,  ()](https://doi.org/10.1103/PhysRevLett.50.1998) [****,  ()](https://doi.org/10.1103/PhysRevB.31.805) @noop [****,  ()]{} **, @noop [Ph.D. thesis]{},  ()

--- abstract: 'Published in 1999, Christodoulou proved that the naked singularities of a self-gravitating scalar field are not stable in spherical symmetry and therefore the cosmic censorship conjecture is true in this context. The original proof is by contradiction and sharp estimates are obtained strictly depending on spherical symmetry. In this paper, appropriate a priori estimates for the solution are obtained. These estimates are more relaxed but sufficient for giving another robust argument in proving the instability, in particular not by contradiction. In another related paper, we are able to prove instability theorems of the spherical symmetric naked singularities under certain isotropic gravitational perturbations without symmetries. The argument given in this paper plays a central role.' address: - | Department of Mathematics, Sun Yat-sen University\ Guangzhou, China - | Department of Mathematics, Sun Yat-sen University\ Guangzhou, China author: - Jue Liu - Junbin Li title: A robust proof of the instability of naked singularities of a scalar field in spherical symmetry --- [^1] Introduction ============ In the paper [@Chr99], Christodoulou proved both the *weak cosmic censorship conjecture* and the *strong cosmic censorship conjecture* for spherically symmetric solutions of the Einstein equations coupled with a massless scalar field. The coupled system reads $$\begin{aligned} \mathbf{Ric}_{\alpha\beta}-\frac{1}{2}\mathbf{R}g_{\alpha\beta}=\mathbf{T}_{\alpha\beta}=\nabla_\alpha\phi\nabla_\beta\phi-\frac{1}{2}g_{\alpha\beta}g^{\mu\nu}\nabla_\mu\phi\nabla_\nu\phi,\end{aligned}$$ which we call the Einstein-scalar field equations. The proof, which is by contradiction, contains sharp estimates which may not be easily obtained beyond spherical symmetry. In this paper, we will provide a robust proof which is not by contradiction and contains only relaxed estimates. The main advantage of this proof is that it has the potential to be extended beyond spherical symmetry. Consider the characteristic initial value problem of the Einstein-scalar field equations in spherical symmetry. The initial data is given on a null cone $C_o$ issuing from a fixed point $o$ of the symmetry group $SO(3)$, and consists of a function $\alpha_0=\frac{\partial}{\partial r}(r\phi)\big|_{C_o}$ defining on $[0,+\infty)$, where $r$ is area radius of the orbit spherical sections of $C_o$, and $\phi$ is the scalar field function. Then what was exactly proved by Christodoulou is the following theorem. \[Chr99\] Let $\mathcal{E}$ be the complement of the collection of functions $\alpha_0\in BV$ whose maximal future development is either complete, or possesses a complete future null infinity and a strictly spacelike singular future boundary. Then if $\alpha_0\in\mathcal{E}$, then there exists some $f\in BV$ such that $\alpha_0+\lambda f\notin\mathcal{E}$ and has non-complete maximal future development for all $\lambda\ne0$. Moreover, if $\alpha_0+\lambda f\equiv\alpha_0'+\lambda'f'$, then $\alpha_0\equiv \alpha_0'$, $f\equiv f'$ and $\lambda=\lambda'$. We may therefore say that the exceptional set $\mathcal{E}$ is of codimension at least $1$. The proof can roughly be divided into three steps. Consider an arbitrary initial data $\alpha_0\in BV$. The first step is, which was shown in [@Chr93], that the maximal future development of such data is complete, unless there exists a singular endpoint $e$ of the central timelike geodesic $\Gamma$ from $o$. A sharp criterion of the appearance of $e$ was also found: $\frac{2m}{r}\nrightarrow0$ when approaching $e$, where $m$ is the mass function. The second step is to understand how an apparent horizon forms. We have the following theorem also by Christodoulou. \[Chr91\] Consider the spherically symmetric solution of the Einstein-scalar field equations with initial data given on a null cone $C_o$. Let $S_1$ and $S_2$ be two sphercal sections with area radii $r_1$, $r_2$ and mass contents $m_1$, $m_2$, and $S_2$ is in the exterior to $S_1$. Denote $$\delta=\frac{r_2}{r_1}-1.$$ Then there exists positive constants $c_0$, $c_1$ such that if $\delta\le c_0$ and $$\begin{aligned} 2(m_2-m_1)>c_1r_2\delta\log\left(\frac{1}{\delta}\right),\end{aligned}$$ then the incoming null cone through $S_2$ intersects the apparent horizon and enters the trapped region, the region of trapped surfaces. Moreover, there exists an event horizon and the trapped region terminates at a spacelike singular boundary. Inspired by this theorem, we then consider the future of $\Cb_e\bigcup C_o$, where $\Cb_e$ is the boundary of the causal past of $e$ and intersects $C_o$ at $s=s_e$. Then the last step, what was really proved in [@Chr99] is that, allowing a perturbation on $\alpha_0$, there exists a sequence $p_n\in\Gamma$ where $p_n\to e$ such that the null cone $C_{p_n}$ issuing from $p_n$ satisfies the assumptions of Theorem \[Chr91\] at two spheres $S_{1,n}$ and $S_{2,n}$ on $C_{p_n}$, between which the distance tends to zero. The corresponding spacetimes have an event horizon and therefore possess a complete future null infinity, which verifies the *weak cosmic censorship conjecture*. Moreover, the distance of $S_{1,n}$ and $S_{2,n}$ tending to zero implies that the apparent horizon issues from $e$ and therefore the future boundary of the maximal development is spacelike and singular. This verifies the *strong cosmic censorship conjecture*. In this paper, we are going to give a new argument of this last step. We would like to give a statement of this single step. First of all, we introduce a double null coordinate $(\ub,u)$ of the quotient spacetime relative to the singular endpoint $e$ of $\Gamma$ as follows. Let $\ub$ and $u$ be optical functions, i.e., their level sets, which we denote by $\Cb_{\ub}$ and $C_u$, are incoming and outgoing null cones respectively. We take $\ub=0, u=-r$ on $\Cb_e$, and take $u=u_0$ and $\ub$ increasing towards the future on $C_o$ where $-u_0=r_0$ is the area radius of the sphere $\Cb_e\bigcap C_o$. Using this notation, we will write $\Cb_e=\Cb_0$, $C_o=C_{u_0}$. In terms of the double null coordinate $(\ub,u)$ relative to $e$, what we are going to reprove can be stated as follows. \[main\] Let $\mathcal{E}$ be the complement of the collection of functions $\alpha_0\in BV$ whose maximal future development is either complete or has the property, that if $e$ is the singular endpoint of $\Gamma$ and $(\ub,u)$ is the double null coordinate relative to $e$, then there exists two sequences $\delta_n\to0^+$ and $u_n\to0^-$ such that $$\begin{aligned} \label{maintheoremcondition} 2(m-m_n)>\frac{c_1r(r-r_n)}{r_n}\log\frac{r_n}{r-r_n},\ \text{with}\ \frac{r-r_n}{r_n}\le c_0\end{aligned}$$ where $m$ and $r$ take values at $(\ub,u)=(\delta_n,u_n)$, $m_n=m(0,u_n)$, $r_n=|u_n|$ and $c_0,c_1$ are the constants given in Theorem \[Chr91\]. Then if $\alpha_0\in\mathcal{E}$, then there exists two functions $f_1,f_2\in BV$, such that $\alpha_0+\lambda_1f_1+\lambda_2f_2\notin\mathcal{E}$ and has non-complete maximal future development for all $\lambda_1,\lambda_2$ with $\lambda_1\ne0$ or $\lambda_2\ne0$. Moreover, if $\alpha_0+\lambda_1f_1+\lambda_2f_2\equiv\alpha_0'+\lambda_1'f'_1+\lambda_2'f'_2$, then $\alpha_0\equiv \alpha_0'$, $f_1\equiv f'_1$, $f_2\equiv f'_2$ and $\lambda_1=\lambda_1'$, $\lambda_2=\lambda_2'$. As in [@Chr99], we will also reprove this theorem where the exceptional set $\mathcal{E}$ is of codimension at least $2$, which is stronger than what we state in Theorem \[Chr99\]. Both in Christodoulou’s proof and the proof in this paper, $f_2$ is shown to be absolutely continuous, and therefore the conclusions of Theorem \[Chr99\] hold for $\alpha_0$ being absolutely continuous, which is of course more regular than being of bounded variation. In proving this theorem, instead of using double null coordinate, Christodoulou worked in a dimensionless coordinate $(s,t)$ relative to the singular endpoint $e$: $$\begin{aligned} u=u_0\mathrm{e}^{-t},\ -2r=u_0\mathrm{e}^{s-t},\end{aligned}$$ and the first step of the proof by contradiction is to assume that given any $\varepsilon>0$, the opposite of , i.e., $$\begin{aligned} \label{contradictionassumption} 2(m(s,t)-m(0,t))\le c_1r(s,t)s\log\left(\frac{1}{s}\right)\end{aligned}$$ holds in $\{0\le s\le c_0\}\bigcap\{0\le\ub\le\varepsilon\}$. For the Einstein-scalar field system, the mass $m$ governs the whole system and has good monotone properties. Christodoulou was able to estimate all related geometric quantities in a sharp way in terms of $m(s,t)-m(0,t)$, which is bounded from , and use these estimates to find some $(s_\varepsilon, t_\varepsilon)\in\{0\le s\le c_0\}\bigcap\{0\le\ub\le\varepsilon\}$ such that the opposite of holds for some particular $(s,t)=(s_\varepsilon,t_\varepsilon)$, which is a contradiction. However, in order to extend this result beyond spherical symmetry, much more things need to be concerned. First of all, we need to derive suitable a priori estimates in order to establish the *existence* of the solution. Second, we may not benefit from the assumptions like which is from proof by contradiction because the mass, which is essentially the $L^2$ integral of $L\phi$ (together with the outgoing shear) over $C_u$, can no longer govern the whole system without symmetries. In addition, the estimates derived by Christodoulou is so sharp that it is not easy to extend them beyond spherical symmetry. In this paper, we are able to derive a priori $L^\infty$ bounds of the geometric quantities, including $\frac{\partial}{\partial\ub}\phi$, $\frac{\partial}{\partial u}\phi$, and the derivatives of $r$ and $\Omega$ defined by $-2\Omega^2=g\left(\frac{\partial}{\partial\ub},\frac{\partial}{\partial u}\right)$. These a priori estimates are proved by bootstrap argument to hold in a region deep enough to the future. It then follows easily that the condition eventually holds before these estimates fail. These estimates are robust and analogues of them may hold when no symmetries are imposed. The generalization of these estimates without symmetries will also be used in proving the *existence*. There is a simple way to understand the difference between two arguments, that is, the a priori estimates we derive will in particular imply that some analogue of the assumption really holds with $c_1$ replaced by a larger constant depending on the $L^\infty$ bounds of the data on $C_{o}$ and this condition with a larger constant still implies that holds by Christodoulou’s argument. Nevertheless, we do not need to repeat Christodoulou’s argument because from the bootstrap argument, all estimates are obtained simultaneously and is then simply a direct conclusion. The new argument presented in this paper can possibly be extended to the case when no symmetries are assumed. In an another paper [@Li-Liu] by the authors, we consider the characteristic initial value problem of the Einstein-scalar field equations, with the initial data given on two null cones intersecting at a sphere. The incoming null cone is assumed to be spherically symmetric and singular at its vertex, in the sense that $\frac{2m}{r}\nrightarrow0$ when approaching it. No symmetries are imposed on the outgoing null cone. Then we will show that the argument presented in this paper can be directly generalized and we can also prove an instability theorem like Theorem \[Chr99\]. We suggest the readers refer to [@Li-Liu] for the precise statement. Finally, we should also mention that the estimates derived in this paper, and also in [@Li-Liu], share many common features with those in the work of An-Luk [@An-Luk] where they worked with the spacetime region deep near the vertex which is regular. Readers may also refer to [@Li-Liu] for some discussions about this. Double null coordinates and equations ===================================== Double null coordinate ---------------------- The spherically symmetric spacetime can be studied through its 2-dimensional quotient spacetime manifold with boundary $\Gamma$, the fixed point set of the $SO(3)$ action, being a timelike geodesic, which we call the central line. We use a double null coordinate $(\ub,u)$, where $\ub$, $u$ are optical functions, which means that their level sets $\Cb_{\ub}$ and $C_u$ are incoming and outgoing null cones invariant under the $SO(3)$ action respectively. In the quotient spacetime, $\Cb_{\ub}$ and $C_u$ are then incoming and outgoing null lines respectively. We then denote $$\begin{aligned} L=\frac{\partial}{\partial \ub},\ \Lb=\frac{\partial}{\partial u},\end{aligned}$$ and define the lapse function $\Omega$ by $$\begin{aligned} -2\Omega^2=g(L,\Lb).\end{aligned}$$ Then the metric has the form $$\begin{aligned} -2\Omega^2({\mathrm{d}}\ub\otimes{\mathrm{d}}u+{\mathrm{d}}u\otimes{\mathrm{d}}\ub)+r^2{\mathrm{d}}\sigma_{\mathbb{S}^2}\end{aligned}$$ where the area radius function $r=r(\ub,u)$ is defined by $$\begin{aligned} \text{Area}(S_{\ub,u})=4\pi r^2,\end{aligned}$$ and ${\mathrm{d}}\sigma_{\mathbb{S}^2}$ is the standard metric of the unit sphere. Equations --------- From the form of the metric, the unknowns of the Einstein-scalar field equations are $r$, $\Omega$ and the scalar field function $\phi$. What we really concern are their derivatives. We define the null expansions relative to the normalized pair of null vectors $\Omega^{-2}L$, $\Lb$ and the mass function $m$ by $$\begin{aligned} h=\Omega^{-2}D r,\ {\underline{h}}={\underline{D}}r,\ m=\frac{r}{2}(1+h{\underline{h}}),\end{aligned}$$ where $D$ and ${\underline{D}}$ are the restrictions on the orbit spheres of the Lie derivatives along $L$ and $\Lb$. When applying on functions, $D$ and ${\underline{D}}$ are simply the ordinary derivatives. We then define the $D$ derivative of the lapse $\Omega$$$\begin{aligned} \omega=D\log\Omega,\end{aligned}$$ while its ${\underline{D}}$ derivative is not needed in this paper. Finally, we also need the derivatives of the scalar field function $\phi$: $$\begin{aligned} L\phi=\frac{\partial}{\partial\ub}\phi,\ \Lb\phi=\frac{\partial}{\partial u}\phi.\end{aligned}$$ We then list below all the equations which are satisfied by the above quantities above and are needed in this paper. First of all, we have the following five null structure equations:[^2] $$\begin{aligned} \label{Dh}Dh=&-r\Omega^{-2}(L\phi)^2,\\ \label{Dbh}{\underline{D}}(\Omega^2h)=&-\frac{\Omega^2(1+h{\underline{h}})}{r},\\ \label{Dhb}D{\underline{h}}=&-\frac{\Omega^2(1+h{\underline{h}})}{r},\\ \label{Dbhb}{\underline{D}}(\Omega^{-2}{\underline{h}})=&-r\Omega^{-2}(\Lb\phi)^2,\\ \label{Dbomega}{\underline{D}}\omega=&\frac{\Omega^2(1+h{\underline{h}})}{r^2}-L\phi\Lb\phi.\end{aligned}$$ The following two equations, which are equivalent, are the wave equation: $$\begin{aligned} \label{DbLphi}{\underline{D}}(rL\phi)=&-\Omega^2h\Lb\phi,\\ \label{DLbphi}D(r\Lb\phi)=&-{\underline{h}}L\phi.\end{aligned}$$ Finally, we have the following equation about the mass function $m$: $$\begin{aligned} \label{Dm}Dm=&-\frac{1}{2}{\underline{h}}\Omega^{-2}(rL\phi)^2,\\ \label{Dbm}{\underline{D}}m=&-\frac{1}{2}h(r\Lb\phi)^2.\end{aligned}$$ A priori bounds for the solution ================================ We begin the proof of Theorem \[main\]. Recall that we start from an arbitrary initial data $\alpha_0\in BV$ and the central line has a singular endpoint $e$, approaching which $\frac{2m}{r}\nrightarrow0$. The double null coordinate $(\ub,u)$ is chosen such that $\ub=0, u=-r$ on the boundary of the causal past of $e$, and $u=u_0=-r_0$ and $\ub$ increases towards the future on the initial null cone $C_o$ where $r_0$ is the area radius of $\Cb_e\bigcap C_o$. Geometry on $\Cb_0$ ------------------- First of all we would like derive some identities on $\Cb_0$, the boundary of the causal past of $e$. We denote the restrictions on $\Cb_0$ of some geometric quantities, which are considered as functions of $u$: $$\psi=\psi(u)=r\Lb\phi\Big|_{\Cb_0},\ \varphi=\varphi(u)=rL\phi\Big|_{\Cb_0},\ \Omega_0=\Omega_0(u)=\Omega\Big|_{\Cb_0},\ h_0=h_0(u)=h\big|_{\Cb_0}.$$ From $u=-r$ on $\Cb_0$, we must have ${\underline{h}}\big|_{\Cb_0}\equiv-1$. Substituting this into , we find $$\begin{aligned} \label{Omega_0} \frac{\partial}{\partial u}\log\Omega_0=-\frac{1}{2}\frac{\psi^2}{|u|},\ \text{and hence}\ -\log\frac{\Omega_0^2(u)}{\Omega_0^2(u_0)}=\int_{u_0}^u\frac{\psi^2(u')}{|u'|}{\mathrm{d}}u'.\end{aligned}$$ From , we have $$\begin{aligned} \label{Omega_0^2h_0} \frac{\partial}{\partial u}(\Omega_0^2h_0)=-\frac{\Omega_0^2(1-h_0)}{|u|},\ \text{and hence}\ -\log\frac{\Omega_0^2(u)h_0(u)}{\Omega_0^2(u_0)h_0(u_0)}=\int_{u_0}^u\frac{1}{|u'|}\left(\frac{1}{h_0(u')}-1\right){\mathrm{d}}u'.\end{aligned}$$ Because $m\big|_{\Cb_0}\ge0$ and the apparent horizon does not intersects $\Cb_0$, then $0<h_0\le 1$. Then we are going to prove an important lemma. Both $\Omega_0^2h_0$ and $\Omega_0$ are monotonically decreasing and converge to $0$ as $u\to0^-$. The monotonicity follows from the fact that the integrands in and are positive. From Lemma 2 in [@Chr99] the integral in tends to infinity as $u\to0^-$. We rewrite this proof using the notations in this paper. Indeed, on $\Cb_0$, it holds $\frac{2m}{r}\big|_{\Cb_0}=1-h_0$, then using the fact that $m(0,u)$ is decreasing which follows from , we have $$\begin{aligned} &\int_{3u}^u\frac{1}{|u'|}\left(\frac{1}{h_0(u')}-1\right){\mathrm{d}}u'=\int_{3u}^u\frac{1}{|u'|}\frac{\frac{2m(0,u')}{|u'|}}{1-\frac{2m(0,u')}{|u'|}}{\mathrm{d}}u'\\ \ge&\int_{3u}^u\frac{1}{|u'|}\frac{\frac{|u|}{|u'|}\frac{2m(0,u)}{|u|}}{1-\frac{|u|}{|u'|}\frac{2m(0,u)}{|u|}}{\mathrm{d}}u'=\log\left(\frac{1-\frac{1}{3}\frac{2m(0,u)}{|u|}}{1-\frac{2m(0,u)}{|u|}}\right).\end{aligned}$$ If the integral in is bounded for all $u\in[u_0,0)$, then the first integral above should tend to zero when $u\to0^-$. However, this implies that $\frac{2m}{r}\to0$ from the above inequality, a contradiction. Therefore $\Omega_0^2h_0\to0$ as $u\to0^-$. If $\Omega_0\nrightarrow0$, then $\Omega_0$ has a positive lower bound because it is decreasing. Therefore $\Omega^2h_0\to0$ implies that $h_0\to0$. Substitute this to the equation in , we find as $u\to0^-$, $\Omega_0^2(u_0)h_0(u_0)-\Omega_0^2(u)h_0(u)=\int_{u_0}^u\frac{\Omega_0^2(1-h_0)}{|u'|}{\mathrm{d}}u'\to+\infty$ and hence $\Omega_0^2(u)h_0(u)\to-\infty$, a contradiction. We conclude that $\Omega_0\to0$ as $u\to0^-$ and the proof is completed. The proof in the following then depends only on the fact that $\Omega_0\to0$ monotonically. The infiniteness of the integral in depends strictly on the monotonicity of mass $m$ along $\Cb_0$. We do not expect a robust argument of this proof since the criteria $\frac{2m}{r}\nrightarrow0$ may not make sense beyond spherical symmetry. A robust version of this lemma, which is beyond reach right now, should include another suitable criteria in general case which is still an active area of research. The a priori estimates ---------------------- We are going to derive the a priori estimates for the geometric quantities. We fix a small constant $\delta>0$ and a constant $u_1\in(u_0,0)$ and denote $$\begin{aligned} \mathscr{F}=\mathscr{F}(u_0,u_1)=\max\{1,\sup_{u_0\le u\le u_1}|\varphi(u)|\}.\end{aligned}$$ Without loss of generality, we also assume that $\Omega(u_0)\le1$. By the monotonicity of $\Omega_0$, we have $\Omega_0(u)\le1$ for all $u\in[u_0,0)$. Then we are going to prove \[estimate\] There exists a universal large constant $C_0\ge1$ such that the following statement is true. Suppose that $$\begin{aligned} \mathcal{A}=\mathcal{A}(\delta,u_0,u_1)=\max\{1,\sup_{0\le\ub\le\delta}\mathscr{F}^{-1}(|rL\phi(\ub,u_0)|+|u_0||\omega(\ub,u_0)|)\}<+\infty,\end{aligned}$$ and for some $C\ge C_0$ we have $$\begin{aligned} \label{smallness} C^2\delta|u_1|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\le1,\ \text{where}\ \mathscr{W}=\mathscr{W}(u_0,u_1)=\max\left\{1,\left|\log\frac{\Omega_0(u_1)}{\Omega_0(u_0)}\right|\right\}.\end{aligned}$$ Then we have the following estimates for $0\le\ub\le\delta$, $u_0\le u\le u_1$:[^3] $$\begin{aligned} \label{geometryestimate}\frac{1}{2}\Omega_0\le\Omega\le 2\Omega_0,&\ \frac{1}{2}|u|\le r\le 2|u|,\\ \label{estimate-Lphi}|rL\phi|\lesssim&\mathscr{F}\mathcal{A},\\ \label{estimate-Lbphi}|r\Lb\phi-\psi|\lesssim&\delta|u|^{-1}\mathscr{F}\mathcal{A},\\ \label{estimate-h}|h-h_0|\lesssim&\delta|u|^{-1}\Omega_0^{-2}\mathscr{F}^2\mathcal{A}^2,\\ \label{estimate-hb}|{\underline{h}}+1|\lesssim&\delta|u|^{-1}\mathscr{F}\mathcal{A},\\ \label{estimate-omega}|u||\omega|\lesssim&\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}.\end{aligned}$$ Moreover, we have the improved estimate $$\begin{aligned} \label{estimate-Lphiimp} |rL\phi(\ub,u)-\varphi(u)|\lesssim|rL\phi(\ub,u_0)-\varphi(u_0)|+\delta|u|^{-1}\mathscr{F}^2\mathscr{W}^{\frac{1}{2}}\mathcal{A}^2.\end{aligned}$$ We begin the proof by the following *bootstrap assumptions*: $$\begin{aligned} \label{bootstrapLphi}|rL\phi|\lesssim&C^{\frac{1}{4}}\mathscr{F}\mathcal{A},\\ \label{bootstrapLbphi}|r\Lb\phi-\psi|\lesssim&C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathcal{A},\\ \label{bootstraph}|h-h_0|\lesssim&C^{\frac{1}{2}}\delta|u|^{-1}\Omega_0^{-2}\mathscr{F}^2\mathcal{A}^2,\\ \label{bootstraphb}|{\underline{h}}+1|\lesssim&C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathcal{A},\\ \label{bootstrapomega}|u||\omega|\lesssim&C^{\frac{1}{4}}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}.\end{aligned}$$ From the equation and , we have $$\begin{aligned} |\log\Omega-\log\Omega_0|\le\int_0^\delta|\omega|{\mathrm{d}}\ub\lesssim C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\lesssim C^{-1}.\end{aligned}$$ The last inequality is because of and $|u|\ge|u_1|$[^4]. By choosing $C_0$ (and hence $C$) sufficiently large, we have $$\begin{aligned} |\log\Omega-\log\Omega_0|\le \log 2\end{aligned}$$ and therefore holds for $\Omega$. Moreover, we have $$\begin{aligned} \label{estimate-Omega-Omega0} |\Omega-\Omega_0|\le\int_0^\delta\left|\Omega\omega\right|{\mathrm{d}}\ub\lesssim C^{\frac{1}{4}}\delta|u|^{-1}\Omega_0\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}.\end{aligned}$$ For $r$, we note that, from , $$\begin{aligned} \label{estimate-h1} |\Omega^2h|\lesssim\Omega_0^2\left(h_0+C^{\frac{1}{2}}\delta|u|^{-1}\Omega_0^{-2}\mathscr{F}^2\mathcal{A}^2\right)\lesssim\mathscr{F}\mathcal{A}.\end{aligned}$$ The second inequality holds because $\Omega_0\le1\le \mathscr{F}\mathcal{A}$ by definition. We then use the equation $Dr=\Omega^2h$ to obtain$$\begin{aligned} \label{estimate-r-r0} |r-|u||\le\int_0^\delta|\Omega^2h|{\mathrm{d}}\ub\lesssim\delta\mathscr{F}\mathcal{A}.\end{aligned}$$ We then deduce that $|r-|u||\lesssim C^{-1}|u|$ and holds for $r$ if $C_0$ is sufficiently large[^5]. For $L\phi$, we consider the equation . We write $$\begin{aligned} \label{DbrLphi-varphi} \frac{\partial}{\partial u}(rL\phi-\varphi)=-\left(\Omega^2hr^{-1}(r\Lb\phi)-\Omega_0^2h_0|u|^{-1}\psi\right).\end{aligned}$$ Using , , , and , the right hand side can be estimated by $$\begin{aligned} &|\Omega^2hr^{-1}(r\Lb\phi)-\Omega_0^2h_0|u|^{-1}\psi|\\ \lesssim&|\Omega^2-\Omega_0^2||h_0|u|^{-1}\psi|+|\Omega^2||h-h_0|||u|^{-1}\psi|+|\Omega^2h||r^{-1}-|u|^{-1}||\psi|+|\Omega^2hr^{-1}||r\Lb\phi-\psi|\\ \lesssim&C^{\frac{1}{4}}\delta|u|^{-1}\Omega_0^2\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\cdot|u|^{-1}|\psi|+C^{\frac{1}{2}}\delta|u|^{-1}\mathscr{F}^2\mathcal{A}^2\cdot|u|^{-1}|\psi|\\ &+\mathscr{F}\mathcal{A}\cdot\delta|u|^{-2}\mathscr{F}\mathcal{A}\cdot|\psi|+|u|^{-1}\mathscr{F}\mathcal{A}\cdot C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathcal{A}\\ \lesssim& C^{\frac{1}{2}}\delta|u|^{-1}\mathscr{F}^2\mathcal{A}^2\cdot|u|^{-1}\left(1+(1+\Omega_0^2\mathscr{W}^{\frac{1}{2}})|\psi|\right).\end{aligned}$$ Integrating the equation , we have $$\label{proof-estimate-Lphiimp} \begin{split} |rL\phi-\varphi| \lesssim&|rL\phi-\varphi|\big|_{C_{u_0}}+C^{\frac{1}{2}}\delta|u|^{-1}\mathscr{F}^2\mathcal{A}^2\\ &+C^{\frac{1}{2}}\delta\mathscr{F}^2\mathcal{A}^2\left(\int_{u_0}^u\frac{(1+\Omega_0^2\mathscr{W}^{\frac{1}{2}})^2|\psi|^2}{|u'|}{\mathrm{d}}u'\right)^{\frac{1}{2}}\left(\int_{u_0}^u\frac{1}{|u'|^3}{\mathrm{d}}u'\right)^{\frac{1}{2}}\\ \lesssim&|rL\phi-\varphi|\big|_{C_{u_0}}+ C^{\frac{1}{2}}\delta|u|^{-1}\mathscr{F}^2\mathscr{W}^{\frac{1}{2}}\mathcal{A}^2 \end{split}$$ where the second inequality is because of . Then the estimate follows by . For $\Lb\phi$, we note that, from $$\begin{aligned} \label{estimate-hb1} |{\underline{h}}|\lesssim1+C^{\frac{1}{4}}\delta|u|^{-1}\mathscr{F}\mathcal{A}\lesssim1+C^{-1}\lesssim1.\end{aligned}$$ We then simply integrate the equation and obtain, by we have proved above, $$\begin{aligned} |r\Lb\phi-\psi|\lesssim\int_0^\delta|{\underline{h}}L\phi|{\mathrm{d}}\ub\lesssim\delta|u|^{-1}\mathscr{F}\mathcal{A}.\end{aligned}$$ For $h$ and ${\underline{h}}$, we use the equations and . From , using , we have $$\begin{aligned} |h-h_0|\lesssim\int_0^\delta |r\Omega^{-2}(L\phi)^2|{\mathrm{d}}\ub\lesssim\delta\Omega_0^{-2}|u|^{-1}\mathscr{F}^2\mathcal{A}^2,\end{aligned}$$ which is the desired estimate . Using , using and , we have $$\begin{aligned} |{\underline{h}}+1|\lesssim\int_0^\delta\left|\frac{\Omega^2(1+h{\underline{h}})}{r}\right|{\mathrm{d}}\ub\lesssim\delta(1+\mathscr{F}\mathcal{A})\cdot|u|^{-1}\lesssim\delta|u|^{-1}\mathscr{F}\mathcal{A},\end{aligned}$$ which is the desired estimate . For $\omega$, we use the equation . The right hand side of can be estimated by, using , , and , $$\begin{aligned} \left|\frac{\Omega^2(1+h{\underline{h}})}{r^2}-L\phi\Lb\phi\right|\lesssim|u|^{-2}\mathscr{F}\mathcal{A}+|u|^{-2}\mathscr{F}\mathcal{A}(|\psi|+\delta|u|^{-1}\mathscr{F}\mathcal{A}).\end{aligned}$$ Integrating and using , we then have $$\begin{aligned} |\omega|\lesssim|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A},\end{aligned}$$ which is the desired estimate . Finally, we find estimates - we have proved improve the bootstrap assumptions - if $C_0$ is sufficiently large. Therefore the estimates - hold without assuming -. Using these estimates to repeat the derivation as in , with the constant $C$ dropped, we know that also holds. Instability theorems ==================== We then turn to the proof of the instability theorems. We divide the proof in two cases according to the behavior of $\varphi(u)$ as $u\to0^-$. The first case is the following. \[instability1\] If $\varphi(u)$ is unbounded as $u\to0^-$, then there exists two sequences $\delta_n\to0^+$ and $u_n\to0^-$ such that holds for $\ub=\delta_n,u=u_n$. Because $\varphi(u)$ is unbounded, we can find a sequence $u_n\to0^-$ such that $$\begin{aligned} \varphi_n=\varphi(u_n)=\sup_{u_0\le u\le u_n}|\varphi(u)|\to\infty\ \text{as}\ n\to\infty.\end{aligned}$$ Define $\delta_n$ in terms of $u_n=-r_n$ by $$\begin{aligned} \label{deltan} \varphi_n^2=2^8c_1\Omega_n^4\log\frac{r_n}{4\Omega_n^2\delta_n}\end{aligned}$$ where $\Omega_n=\Omega_0(u_n)$. It is obvious that $\delta_n\to0^+$ because $\Omega_n\to0$. We are going to prove such $\delta_n, u_n$ are two sequences we need. We hope to apply Theorem \[estimate\], so we compute, for each $n$, $$\begin{aligned} C^2\delta_n|u_n|^{-1}|\varphi_n|\mathscr{W}_n^{\frac{1}{2}}=\frac{1}{4}C^2\Omega_n^{-2}\exp\left(-\frac{\varphi_n^2}{2^8c_1\Omega_n^4}\right)|\varphi_n|\mathscr{W}_n\end{aligned}$$ where $\mathscr{W}_n=\mathscr{W}(u_0,u_n)=\left\{1,\left|\log\frac{\Omega_n}{\Omega_0(u_0)}\right|\right\}$. We can see the right hand side tends to zero and therefore holds for $\delta=\delta_n$, $u=u_n$ for sufficiently large $n$ depending on $C$ and the initial bound of $L\phi$ on $C_{u_0}$. As a consequence, we have the following estimates for a sufficiently large $C\ge C_0$ and $(\ub,u)\in[0,\delta_n]\times\{u_n\}$: - $\displaystyle |{\underline{h}}+1|\le C^{-1},\ \text{which implies}\ -{\underline{h}}\ge\frac{1}{2}$. - $\displaystyle \Omega^{-2}\ge\frac{1}{4}\Omega_n^{-2}, 1\ge\frac{r}{2r_n}$. - $|rL\phi-\varphi_n|\le c|rL\phi(\ub,u_0)-\varphi(u_0)|+cC^{-1}|\varphi_n|$ for some $c$ depending on the initial bound of $L\phi$ on $C_{u_0}$ which follows from and implies that $\displaystyle |rL\phi|>\frac{1}{2}|\varphi_n|$ for $n$ sufficiently large. - $\displaystyle \Omega_n^2\delta_n\ge\Omega_n^2h_n\delta_n=\int_0^{\delta_n}\Omega_n^2h_n{\mathrm{d}}\ub\ge\frac{1}{4}\int_0^{\delta_n}\Omega^2h{\mathrm{d}}\ub=\frac{1}{4}(r-r_n)$, where we use $h_n=h(0,u_n)\ge h$ because of $Dh\le0$ from equation . From , and the above all estaimtes, we have, for $n$ sufficiently large, $$\begin{aligned} m-m_n=&\frac{1}{2}\int_0^{\delta_n}(-{\underline{h}})\Omega^{-2}(rL\phi)^2{\mathrm{d}}\ub\\ >&\frac{1}{2^6}\delta_n\Omega_n^{-2}\varphi_n^2\\ =&\frac{1}{2^6}\delta_n\Omega_n^{-2}\cdot2^8c_1\Omega_n^4\log\frac{r_n}{4\Omega_n^2\delta_n}\\ \ge&\frac{c_1r}{2r_n}\cdot4\Omega_n^2\delta_n\log\frac{r_n}{4\Omega_n^2\delta_n}\\ \ge&\frac{c_1r}{2r_n}(r-r_n)\log\frac{r_n}{r-r_n}\end{aligned}$$ which is the inequality in . The last inequality above is because the function $x\log\frac{r_n}{x}$ is monotonically increasing for $x\in(0,4\Omega_n^2\delta_n]\subset(0,\frac{r_n}{\mathrm{e}}]$. Finally, $\frac{r-r_n}{r_n}\le c_0$ follows from $$\begin{aligned} \frac{r-r_n}{r_n}\le\frac{4\Omega_n^2\delta_n}{|u_n|}=\exp\left(-\frac{\varphi_n^2}{2^8c_1\Omega_n^4}\right)\end{aligned}$$ since the right hand side tends to zero and hence not larger than $c_0$ if $n$ is sufficiently large. The proof is then completed. The second case is the following. \[instability2\] Suppose that $\varphi(u)$ is bounded by $\Phi\ge0$, and there exists some $\gamma\in(0,4)$ such that $$\begin{aligned} \label{genericcondition} \limsup_{u\to0^-}\Omega_0^{\gamma-4}(u)f(u;\gamma)>1\end{aligned}$$ where the function $f$ is defined by $$\begin{aligned} f(u;\gamma)=\frac{1}{\delta(u;\gamma)}\int_0^{\delta(u;\gamma)}|rL\phi(\ub,u_0)+(\varphi(u)-\varphi(u_0))|^2{\mathrm{d}}\ub\end{aligned}$$ and $\delta(u;\gamma)$ is defined in terms of $u$ by $$\begin{aligned} \label{deltau} \Omega^{4-\gamma}_0(u)=2^8c_1\Omega_0^4(u)\log\frac{|u|}{4\Omega_0^2(u)\delta(u;\gamma)}.\end{aligned}$$ Then the conclusion of Theorem \[instability1\] also holds. From , there exists a sequence $u_n\to0^-$ such that $$\begin{aligned} \label{genericcondition1} f(u_n;\gamma)>\Omega^{4-\gamma}(u_n).\end{aligned}$$ From , we have $\delta_n=\delta(u_n;\gamma)\to0^+$ and $$\begin{aligned} C^2\delta_n|u_n|^{-1}\Phi\mathscr{W}_n=C^2\frac{1}{4}\Omega_n^{-2}\exp\left(-\frac{1}{2^8c_1\Omega_n^{\gamma}}\right)\mathscr{W}_n.\end{aligned}$$ The right hand side tends to zero and therefore holds for $\delta=\delta_n$, $u=u_n$ for $n$ sufficiently large. Then once we can prove that $$\begin{aligned} \int_0^{\delta_n}|rL\phi(\ub,u_n)|^2{\mathrm{d}}\ub>\frac{1}{4}\delta_n\Omega_n^{4-\gamma}.\end{aligned}$$ the conclusion follows using the argument in the proof of Theorem \[instability1\]. To this end, we go back to equation . Integrating it on $C_{u_n}$ leads to $$\begin{aligned} (rL\phi(\ub,u_n)-\varphi_n)-(rL\phi(\ub,u_0)-\varphi(u_0))=\int_{u_0}^u-\left(\Omega^2hr^{-1}(r\Lb\phi)-\Omega_0^2h_0|u'|^{-1}\psi\right){\mathrm{d}}u'.\end{aligned}$$ The right hand side can be estimated similarly to the estimate of the error terms in . Then we have, for $n$ sufficiently large, $$\begin{aligned} |rL\phi(\ub,u_n)|\ge|rL\phi(\ub,u_0)+(\varphi_n-\varphi(u_0))|-c\delta|u_n|^{-1}\Phi^2\mathscr{W}_n^{\frac{1}{2}}\end{aligned}$$ for some constant $c$ depending on the initial bound of $L\phi$ on $C_{u_0}$. Now from again, $$\begin{aligned} c\delta|u_n|^{-1}\Phi^2\mathscr{W}_n^{\frac{1}{2}}=\frac{1}{4}\Omega_n^{-2}\exp\left(-\frac{1}{2^8c_1\Omega_n^{\gamma}}\right)\mathscr{W}_n^{\frac{1}{2}}\cdot c\Phi^2\le\sqrt{\frac{1}{4} \Omega^{4-\gamma}_n}\end{aligned}$$ if $n$ is sufficiently large, then from , $$\begin{aligned} \int_0^{\delta_n}|rL\phi(\ub,u_n)|^2{\mathrm{d}}\ub>\frac{1}{2}\delta_nf(u_n;\gamma)-\frac{1}{4}\delta_n\Omega_n^{4-\gamma}\ge\frac{1}{4}\delta_n\Omega_n^{4-\gamma}\end{aligned}$$ which is the desired inequality. It is worth mentioning that in Christodoulou’s original proof, when $\varphi(u)$ is bounded but not tends to zero, the conclusions of Theorem \[instability1\] holds without any additional conditions like . Indeed, the condition is slightly different from that in Christodoulou’s proof and we can see when $\varphi(u)$ is bounded but not tends to zero, holds identically because $rL\phi(\ub,u_0)$ is of bounded variation and hence can be made right-continuous. The remaining part of the proof of Theorem \[main\] is then similar to that in the last section in [@Chr99]. We still present the proof here for the sake of completeness. We fix the coordinate $\ub=r-r_0$ on $C_o=C_{u_0}$. Then $$\begin{aligned} \alpha_0=\frac{\partial}{\partial r}(r\phi)=rL\phi\big|_{C_o}+\phi\big|_{C_o}.\end{aligned}$$ We denote $\theta_0=\theta_0(r)=rL\phi\big|_{\ub=r-r_0,u=u_0}$. As in [@Chr99], $\alpha_0$ being of bounded variation is equivalent to $\theta_0$ being bounded variation and $\frac{|\theta_0|}{r}\in L^1(0,+\infty)$. Therefore we consider instead $\theta_0$ in such a space. Suppose that $\theta_0\in\mathcal{E}$, then there exists a singular endpoint $e$ on $\Gamma$ and we have a double null coordinate $(\ub,u)$ relative to $e$ and in particular, $\Cb_e\bigcap C_o$ has area radius $r_0$. According to Theorem \[instability1\] and \[instability2\], we have $\varphi(u)$ is bounded and $$\begin{aligned} \label{nogenericcondition} \limsup_{u\to0^-}\Omega_0^{\gamma-4}(u)f(u;\gamma)\le1\end{aligned}$$ for all $\gamma\in(0,4)$. We then define $f_1=f_1(r)$ such that it vanishes on $[0,r_0)$ and near infinity, and is absolutely continuous on $[r_0,+\infty)$ with $f_1(r_0)=1$. We also define $f_2=f_2(r)$ to be absolutely continuous on $[0,+\infty)$ such that it vanishes on $[0,r_0]$ and near infinity, and $$\begin{aligned} f_2(r)=\sqrt{\frac{{\mathrm{d}}}{{\mathrm{d}}r}\left[(r-r_0)\Omega_0^2(u)\right]},\ r\in[r_0,r_0+1]\end{aligned}$$ where $u$ and $r$ are related through $r-r_0=\delta(u;\gamma=2)$ defined by . Then for all $\gamma\in(0,4)$ and $\lambda_1\ne0$, we have $$\begin{aligned} \label{limf1} \lim_{u\to0^-}\frac{\Omega_0^{\gamma-4}(u)}{\delta(u;\gamma)}\int_0^{\delta(u;\gamma)}\lambda_1^2f_1^2(\ub+r_0){\mathrm{d}}\ub=+\infty.\end{aligned}$$ On the other hand, we define $u_{(\gamma)}$ through $\delta(u;\gamma)=\delta(u_{(\gamma)};2)$. From , $\delta(u;\gamma)$ is increasing relative to $|u|$ and decreasing relative to $\gamma$. If $\gamma\in(0,2)$, we must have $|u|<|u_{(\gamma)}|$ and therefore $\Omega_0(u_{(\gamma)})>\Omega_0(u)$. We then have $$\label{lim1} \begin{split} &\lim_{u\to0^-}\frac{\Omega_0^{\gamma-4}(u)}{\delta(u;\gamma)}\int_0^{\delta(u;\gamma)}\lambda_2^2f_2^2(\ub+r_0){\mathrm{d}}\ub\\ =&\lim_{u\to0^-}\lambda_2^2\Omega_0^{\gamma-4}(u)\Omega_0^2(u_{(\gamma)})\ge\lambda_2^2\lim_{u\to0^-}\Omega_0^{\gamma-2}(u)=+\infty. \end{split}$$ If $\gamma\in(2,4)$, we have $\Omega_0(u_{(\gamma)})<\Omega_0(u)$ and $$\label{lim2} \begin{split} &\lim_{u\to0^-}\frac{\Omega_0^{\gamma-4}(u)}{\delta(u;\gamma)}\int_0^{\delta(u;\gamma)}\lambda_2^2f_2^2(\ub+r_0){\mathrm{d}}\ub\\ =&\lim_{u\to0^-}\lambda_2^2\Omega_0^{\gamma-4}(u)\Omega_0^2(u_{(\gamma)})\le\lambda_2^2\lim_{u\to0^-}\Omega_0^{\gamma-2}(u)=0. \end{split}$$ We then compute, when $\lambda_1\ne0$, for $\gamma\in(2,4)$, from , , and , $$\begin{aligned} &\limsup_{u\to0^-}\frac{\Omega_0^{\gamma-4}(u)}{\delta(u;\gamma)}\int_0^{\delta(u;\gamma)}|rL\phi(\ub,u_0)+\lambda_1f_1(\ub+r_0)+\lambda_2f_2(\ub+r_0)+(\varphi(u)-\varphi(u_0))|^2{\mathrm{d}}\ub\\ \ge&\liminf_{u\to0^-}\frac{\Omega_0^{\gamma-4}(u)}{2\delta(u;\gamma)}\int_0^{\delta(u;\gamma)}|\lambda_1f_1(\ub+r_0)|^2{\mathrm{d}}\ub\\ &-\limsup_{u\to0^-}\frac{\Omega_0^{\gamma-4}(u)}{\delta(u;\gamma)}\int_0^{\delta(u;\gamma)}|\lambda_2f_2(\ub+r_0)|^2{\mathrm{d}}\ub\\ &-\limsup_{u\to0^-}\frac{\Omega_0^{\gamma-4}(u)}{\delta(u;\gamma)}\int_0^{\delta(u;\gamma)}|rL\phi(\ub,u_0)+(\varphi(u)-\varphi(u_0))|^2{\mathrm{d}}\ub\\ =&+\infty.\end{aligned}$$ When $\lambda_1=0,\lambda_2\ne0$, we compute, for $\gamma\in(0,2)$, from and , $$\begin{aligned} &\limsup_{u\to0^-}\frac{\Omega_0^{\gamma-4}(u)}{\delta(u;\gamma)}\int_0^{\delta(u;\gamma)}|rL\phi(\ub,u_0)+\lambda_1f_1(\ub+r_0)+\lambda_2f_2(\ub+r_0)+(\varphi(u)-\varphi(u_0))|^2{\mathrm{d}}\ub\\ \ge&\liminf_{u\to0^-}\frac{\Omega_0^{\gamma-4}(u)}{2\delta(u;\gamma)}\int_0^{\delta(u;\gamma)}|\lambda_2f_2(\ub+r_0)|^2{\mathrm{d}}\ub\\ &-\limsup_{u\to0^-}\frac{\Omega_0^{\gamma-4}(u)}{\delta(u;\gamma)}\int_0^{\delta(u;\gamma)}|rL\phi(\ub,u_0)+(\varphi(u)-\varphi(u_0))|^2{\mathrm{d}}\ub\\ =&+\infty.\end{aligned}$$ This proves that $\theta_0+\lambda_1f_1+\lambda_2f_2\notin\mathcal{E}$ for all $\lambda_1,\lambda_2$ with $\lambda_1\ne0$ or $\lambda_2\ne0$. Now suppose that $\theta,\theta'\in\mathcal{E}$ and $$\theta_{\lambda_1,\lambda_2}:=\theta_0+\lambda_1f_1+\lambda_2f_2\equiv\theta'_{\lambda'_1,\lambda'_2}:=\theta_0'+\lambda_1'f_1'+\lambda_2'f_2'.$$ Assume that $e'$ is the singular endpoint of $\Gamma$ in the maximal future development of $\theta_0'$ (and hence of $\theta'_{\lambda'_1,\lambda'_2}$) and $\Cb_{e'}\bigcap C_o$ has area radius $r_0'$. We then have $e=e'$ and $r_0=r_0'$. Because $f_i,f_i'$ vanish on $[0,r_0)$, we will have $\theta(r)\equiv\theta'(r)$ for $r\in[0,r_0)$ and hence the double coordinate $(\ub,u)$, the functions $\varphi(u)$ and $\Omega_0(u)$, are the same. Therefore $f_1\equiv f_1'$, $f_2\equiv f_2'$. We then write $$\begin{aligned} \theta_{\lambda_1-\lambda_1',\lambda_2-\lambda_2'}\equiv\theta_0'.\end{aligned}$$ From the above argument, when $\lambda_1\ne\lambda_1'$ or $\lambda_2\ne\lambda_2'$, $\theta_{\lambda_1-\lambda_1',\lambda_2-\lambda_2'}\notin\mathcal{E}$ but $\theta_0'\in\mathcal{E}$. Therefore we must have $\lambda_1=\lambda_1'$ and $\lambda_2=\lambda_2'$. Finally, we conclude that $\theta\equiv\theta'$ and the proof is completed. [99]{} X. An and J. Luk, *Trapped surfaces in vacuum arising dynamically from mild incoming radiation*, Advances in Theoretical and Mathematical Physics 21 (2017), 1–120. D. Christodoulou, *The formation of black holes and singularities in spherically symmetric gravitational collapse*, Communications on Pure and Applied Mathematics 44, no. 3 (1991): 339-373. D. Christodoulou, *Bounded variation solutions of the spherically symmetric einstein-scalar field equations*, Communications on Pure and Applied Mathematics 46, no. 8 (1993): 1131-1220. D. Christodoulou, *The instability of naked singularities in the gravitational collapse of a scalar field*, Ann. of Math. 149, 183-217 (1999). D. Christodoulou, *The Formation of Black Holes in General Relativity*, Monographs in Mathematics, European Mathematical Soc. 2009. J. Li and J. Liu, *Instability of spherical naked singularities of a scalar field under isotropic gravitational perturbations*, preprint. [^1]: Both authors are supported by NSFC 11501582, 11521101. [^2]: Readers may refer to [@Chr91] for the derivations of these equations, though the notations have slight differences. These equations can also be directly written down from the general null structure equations without symmetries, which can be found in the authors related paper [@Li-Liu] mentioned above. The derivations of these equations in vacuum can be found in Christodoulou’s work [@Chr] on the formation of black holes. [^3]: The notation $A\lesssim B$ means $A\le cB$ for some universal constant $c$. [^4]: Because is used very frequently in a similar manner, we will not point it out again when we use in the rest of the paper. Because $\mathscr{W}\ge1$, will also be used in the form $C^2\delta|u_1|^{-1}\mathscr{F}\mathcal{A}\le1$. [^5]: Similar to , the estimates are used frequently and we will not point this out in the argument.

--- abstract: 'We critically examine issues associated with determining the fundamental properties of the black hole and the surrounding accretion disk in an X-ray binary based on modeling the disk X-ray continuum of the source. We base our work mainly on two [*XMM-Newton*]{} observations of GX 339$-$4, because they provided high-quality data at low energies (below 1 keV), which are critical for reliably modeling the spectrum of the accretion disk. A key issue examined is the determination of the so-called “color correction factor”, which is often empirically introduced to account for the deviation of the local disk spectrum from a blackbody (due to electron scattering). This factor cannot be predetermined theoretically, because it may vary with, e.g., mass accretion rate, among a number of important factors. We follow up on an earlier suggestion to estimate the color correction observationally by modeling the disk spectrum with saturated Compton scattering. We show that the spectra can be fitted well, and the approach yields reasonable values for the color correction factor. For comparison, we have also attempted to fit the spectra with other models. We show that even the high-soft state continuum (which is dominated by the disk emission) cannot be satisfactorily fitted by state-of-the-art disk models. We discuss the implications of these results.' author: - Gabor Pszota and Wei Cui title: 'Modeling Accretion Disk X-ray Continuum of Black Hole Candidates' --- \[firstpage\] Introduction ============ The X-ray continuum of black hole candidates (BHCs) is roughly composed of two main elements (see review by Liang 1998), an ultra-soft component that is thought to be associated with emission from the accretion disk, and a hard component that is thought to be produced by inverse Compton scattering of soft photons by energetic electrons that can be either thermal or non-thermal in origin. Modeling the disk component could, in principle, allow one to determine the radius of the inner edge of the accretion disk in a BHC (review by Tanaka & Lewin 1995, and references therein). This has been tried and the results have provided evidence that the accretion disk extends all the way in to the last stable orbit under certain circumstances (Tanaka & Lewin 1995). Motivated by this observation, Zhang et al. (1997) suggested that modeling the X-ray continuum of a BHC could lead to a measurement of the spin of the black hole, if the mass of the black hole can be independently derived. In retrospect, we now know that the accretion disk reaches the last stable orbit probably only in the high-soft state (e.g., Narayan 1996) [^1], so the proposed technique may only be applicable to data taken in such a state. Since the X-ray spectrum of BHCs is dominated by the disk component in the high-soft state, the determination of the disk parameters based on spectral modeling should, in principle, be quite accurate, even if one neglects the hard component whose physical origin is less well understood, particularly in the high-soft state. However, there are still serious issues associated with the exercise. First, the local spectrum of the X-ray emitting portion of the accretion disk is not a blackbody, because the opacity is dominated by electron scattering. Saturated Comptonization leads to a “diluted” blackbody spectrum, whose color temperature is given by $T_{col}=f_{col} T_{eff}$, where $f_{col}$ is the color correction factor and $T_{eff}$ is the effective temperature (Ebisuzaki et al. 1984). Much effort has gone into finding the values of $f_{col}$ that are appropriate for BHCs (Shimura & Takahara 1995; Merloni et al. 2000; Davis et al. 2006). The situation is still uncertain, but it is clear that $f_{col}$ depends on a number of important physical parameters, such as mass accretion rate, which can vary even for a given source. It is, therefore, not possible to know what value to use [*a priori*]{}. Cui et al. (2002) proposed an observational approach to derive $f_{col}$ from the data (see also Shrader & Titarchuk 1999). Although the technique showed some promise with limited data, it needs to be tested further. Second, there is observational evidence (Zhang et al. 2000) that the surface layer of the accretion disk in BHCs might deviate from the standard $\alpha$-disk structure (Shakura & Sunyaev 1973). Such an effect is expected from X-ray heating of the disk by a central hard X-ray source (e.g., Nayakshin & Melia 1997; Mistra et al. 1998), but it is not clear why the effect is still significant even for the high-soft state, in which hard X-ray production is expected to be quite weak. The presence of such a “warm” layer would add further complication in modeling the observed X-ray spectrum (Zhang et al. 2000), because Compton scattering in the layer can further modify the spectrum. Third, some of the widely-used disk models (e.g., the multi-color disk; Mitsuda et al. 1984) do not take into account general relativistic effects that can affect the formation of the X-ray spectrum. Attempts have been made to incorporate the effects empirically in the analysis by introducing a number of correction factors (Zhang et al. 1997). Recently, two new disk models have been developed that account for the general relativistic effects (Li et al. 2005; Davis & Hubeny 2006). The models also consider spectral hardening due to scattering, with one treating $f_{col}$ as a free parameter (Li et al. 2005) and the other carrying out radiative transfer in the disk (Davis & Hubeny 2006). The models have been applied to observations of a number of BHCs (Shafee et al. 2006; Davis et al. 2006; McClintock et al. 2006; Middleton et al. 2006). In this work, we examined some of the issues and also assessed the viability of the state-of-the-art disk models, making use of data of much improved quality that have recently become available. Specifically, we analyzed two [*XMM-Newton*]{} observations of GX 339$-$4 and attempted to fit the observed X-ray spectra with different models. With its large effective area and good sensitivity at low energies ($<$ 1 keV), [*XMM-Newton*]{} offers distinct advantages over other X-ray observatories for our purposes. The low-energy sensitivity is often not appreciated as much as it should be; it is critical to reliable modeling of the disk spectrum, because the effective temperature of the disk is typically $\lesssim$ 1 keV for BHCs. Data ==== XMM-Newton Observations ----------------------- We analyzed data from two archival [*XMM-Newton*]{} observations (ObsIDs 0093562701 and 0148220201) of GX 339$-$4 during its 2002–2003 outburst. The first observation was taken near the peak of the outburst (on 2002 August 24), judging from the ASM/RXTE light curve [^2], while the second one was taken at the tail end of the episode (on 2003 March 8). GX 339$-$4 was observed for about 61 and 20 ks during the two observations, respectively. Since we are mainly interested in the X-ray continuum here, we focused on the EPIC data. The pn/EPIC detector was operated in the burst mode, with the thin optical blocking filter, during the first observation, and the MOS/EPIC detectors were not used. In the second observation, the pn and MOS detectors were both run in the timing mode with the medium blocking filter. Even with the timing mode, the MOS data still suffer from severe photon pile-up, due to the high count rate. In contrast, the pile-up effects are minimal in the pn data. This work is, therefore, based on the pn data. The data were reduced with the standard [*SAS*]{} package (version 7.0.0). We followed the procedures described in the [*XMM-Newton*]{} data analysis cookbook [^3] in preparing and filtering the data, making light curves, extracting spectra, and generating the corresponding arf and rmf files for subsequent spectral modeling. We did need to turn off bad-pixel search in processing the first observation because of a bug in the searching routine for the burst mode. The effects should be negligible because the source was very bright then. The events of interest were extracted from a rectangular region, with RAWX 32–40 RAWY 3–179 and RAWX 34–42 RAWY 3–199 for the 2002 and 2003 observations, respectively. Filtering expressions “FLAG = 0” and “PATTERN $\leq$ 4” were applied to select good single and double events. Because the source was bright during both observations, a significant number of source events are present even near the edge of the CCD chip, which makes it impossible to cleanly extract background events. This should only affect the high-energy end of the spectrum (where the background counts may become comparable or exceed the source counts). Our choice of the central 9 columns of the chip was made to minimize the effect on the shape of the spectrum. However, it led to an underestimation of the overall normalization, which is also important here. To determine the normalization more accurately, we also made spectra with events from the whole chip. The difference amounts to roughly 8%. For spectral modeling, we added a 1% systematic error to the data and grouped the channels so that each bin contains at least 500 counts. RXTE Observations ----------------- To complement the soft-band coverage of [*XMM-Newton*]{}, we obtained simultaneous [*RXTE*]{} data from the public archive. GX 339$-$4 was observed with [*RXTE*]{} for about 4 and 16 ks, respectively, during the two [*XMM-Newton*]{} observing periods. The data were reduced with [*FTOOLS 5.2*]{}. We followed the standard steps [^4] in preparing and filtering the data, deriving PCA and HEXTE spectra from data taken in the standard modes, and generating the corresponding response files for spectral modeling. A PCA or HEXTE spectrum consists of separate spectra from the individual detector units that were in operation. In deriving the PCA spectra, we only used data from the first xenon layer of each detector unit (which is best calibrated) and combined spectra from all the live detectors into one, to maximize the signal-to-noise ratio (S/N). To estimate the PCA background, we used the background model for bright sources (). As for the HEXTE data, we extracted a spectrum for each of the two clusters separately. For spectral modeling, we rebinned the HEXTE spectra so that each bin contains at least 5000 counts. We also added a 1% systematic error to both the PCA and HEXTE spectra. Results ======= We carried out spectral modeling in [*XSPEC*]{} (Arnaud 1996). The spectral bands of interest are 0.5–10 keV (pn/EPIC), 3–25 keV (PCA), and $>$ 15 keV (HEXTE). The spectra are always jointly fitted with a common model, except for a normalization factor (fixed at unity for the pn data) that was introduced to account for any residual difference in the calibration of the throughput of the detectors. Strictly speaking, however, the [*XMM-Newton*]{} and [*RXTE*]{} coverages are not always simultaneous, due to the difference not only in the observing time but also in the orbits of the two satellites. To justify joint modeling, we broke each of the [*XMM-Newton*]{} observations into 8 segments and extracted a spectrum for each segment. We compared the individual spectra and observed no apparent variation in the shape of the spectrum in either case. We experimented with several models for the ultra-soft and hard components of the spectrum. The former is often modeled with a non-relativistic, multi-temperature blackbody model (“diskbb” in XSPEC; Mitsuda et al. 1984). For this work, we instead used the two relativistic disk models (“kerrbb” in XSPEC, Li et al. 2005; and “bhspec”, Davis & Hubeny 2006). To test the procedure of deriving the color correction factor from the data, as proposed by Cui et al. (2002), we also modeled the disk component with saturated Compton scattering (“comptt” in XSPEC, in a disk geometry; Titarchuk 1994). In all cases, the hard component of the spectrum was modeled with unsaturated Compton scattering (also “comptt” but in a spherical geometry). Interstellar absorption was taken into account (with “phabs” in XSPEC). The best and only formally acceptable fit to the continuum was obtained with . In this case, the residuals reveal the presence of discrete features, which include absorption edges at 863 eV and 880 eV for the 2002 and 2003 observations, respectively, and emission lines at 569 eV and 562 eV. We suspect that the edges are calibration artifacts, since we were not able to associate them with any elements. On the other hand, the emission features could be real, with the former being associated with O VIII and the latter with O VII (corresponding to transitions at rest-frame energies 569 eV and 561 eV, respectively), which would imply a plasma temperature of 0.1–0.2 keV. The lines are unresolved and are quite weak, with equivalent widths of 26 and 21 eV for the 2002 and 2003 observations, respectively. We will not discuss the discrete spectral features any further, since the main focus here is on the X-ray continuum. The 2002 data also show the presence of an emission feature at 2.2 keV, which is likely an artifact caused by calibration uncertainty around the M-edge of gold (in the mirror coating). However, the feature is not apparent in the 2003 data, which is a bit puzzling, because the statistics are comparable in the two cases. We consulted with the [*XMM-Newton*]{} Helpdesk about it, and were told that it had probably been corrected for by the calibration in the timing mode, but not so well in the burst mode. After accounting for the discrete spectral features (with “edge” and “gaussian” in XSPEC), we still saw, in the residuals, genuine inconsistency between the pn/EPIC and PCA data at low energies, which could be related to known PCA calibration uncertainties around the L-edge of xenon. For this work, we resolved the issue simply by excluding the PCA data below 9 keV in the joint fits. For the 2003 data, the continuum fit also shows significant structures in the residuals roughly in the range of 5–8 keV, which might be similar to those reported by Miller et al. (2004) based on an [*XMM-Newton*]{} observation taken several months earlier. They are most likely associated with the K$\alpha$ emission of the iron and its associated absorption edge. The excess appears broad and asymmetric in shape, as illustrated in Figure 1. Therefore, we modeled it as a gravitationally redshifted disk line (“laor” in XSPEC; Laor 1991). Also, we included a smeared edge (“smedge” in XSPEC) in the fit. The results are: $E_{Laor}=6.48^{+0.07}_{-0.09}$ keV, $i=51$$^{+2}_{-1}$, $q = 5.2\pm 0.2$, and $R_{in} = 1.76^{+0.10}_{-0.06}$ $R_g$ (where $R_g$ is the gravitational radius) for the line; $E_{edge} = 8.5\pm 0.1$ keV, $W = 2.7^{+0.5}_{-0.4}$ keV, and $\tau=0.59^{+0.07}_{-0.05}$ for the edge. Note that we fixed $R_{out}$ at $400$ $R_g$ in the “laor” model. The obtained value for the inclination angle ($i$) is consistent with those estimated for the system (e.g., Zdziarski et al. 2004). If this interpretation is correct, the results would require a very high value ($a^* \gtrsim 0.97$) for the black hole spin (cf. Miller et al. 2004). However, no such broad line (nor the edge) is apparent in the 2002 data. Adding the line (as a Gaussian component) to the model, we found that the data could accommodate it, but its equivalent width would be merely $14^{+12}_{-9}$ eV, compared to $485^{+217}_{-130}$ eV based on the 2003 data. Figure 2 shows the observed X-ray spectra of GX 339$-$4, along with the best-fit models and the associated residuals. The parameters of the continuum fits are summarized in Table 1. The source was clearly in the high-soft state during the 2002 observation, with the disk contributing about 96% of the 0.5–10 keV flux. The spectrum became harder during the 2003 observation, but the disk still contributed about 80% of the 0.5–10 keV flux. Following Cui et al. (2002), we attempted to derive the color correction factor from the continuum fits. Briefly, to account for the effects of scattering in a Shakura-Sunyaev disk (Shakura & Sunyaev 1973), one should, strictly speaking, start with a multitemperature blackbody spectrum for the seed photons. However, [*comptt*]{} assumes a Wien spectrum for the seed photons. Fitting the peak of [*diskbb*]{} with a Wien distribution leads to $T_{diskbb} = 2.7T_{Wien}$. Based on spectral modeling with [*comptt*]{}, therefore, we can approximate the color correction factor as $f_{col} = T_e/2.7T_0$ (Cui et al. 2002; see also Zhang 2005). For the 2002 and 2003 observations, respectively, we have $f_{col} = 1.48^{+0.09}_{-0.08}$ and $1.35^{+0.01}_{-0.01}$, which seem quite reasonable. This lends support to the viability of the observational approach in deriving $f_{col}$. We then replaced the saturated Compton component with a multicolor disk model, but failed to obtain any formally acceptable fits to the observed X-ray continua with either “kerrbb” or “bhspec”. In this case, we fixed the inclination angle at the value from relativistic line modeling (51), the mass of the black hole at $10~M_{\sun}$, and the distance at 8 kpc (Zdziarski et al. 2004). With “kerrbb”, we also adopted the default settings for torque-free inner boundary condition, returning radiation, and limb darkening, and fixed the normalization at unity and the color correction factors at the values that we derived. The best-fit models are shown in Figure 3. Neither one is formally acceptable, with $\chi^2$/dof = 2634/1203 and 2010/1079 for the 2002 and 2003 observations, respectively. The residuals show significant structures in both cases. Taken at its face value, the black hole spin would be about 0.7, after correcting for the loss of flux due to the use of the central nine columns of the pn chip (see § 2.1). The situation is hardly improved when the inclination angle and the color correction factor are allowed to vary. Figure 4 shows the best-fit models with “bhspec”. Again, significant features are noticeable in the residuals. The $\chi^2$ values of the fits are $\chi^2$/dof = 2246/1203 and 2505/1079 for the 2002 and 2003 observations, respectively. As already mentioned, in this model spectral hardening (due to electron scattering) is taken into account in modeling the disk atmosphere. Again, taken at its face value, the black hole spin is about 0.5. Relaxing the inclination angle does not improve the fits. Discussion ========== The importance of accurately modeling the accretion disk X-ray continuum of BHCs goes beyond gaining insights into radiative processes associated with accretion flows. It also lies in the exciting prospect of deriving the spin of black holes from such spectral modeling. The technique is one of many that have been proposed for BHCs (Laor 1991; Bromley et al. 1997; Zhang at al. 1997; Nowak et al. 1997; Cui et al. 1998; Stella et al. 1999; Wagoner et al. 2001; Abramowicz & Kluzniak 2001). Although varying degrees of success have been achieved, it is fair to say that the techniques all have serious issues in their applications to the data. Further investigation, both theoretical and observational, is thus needed to examine the issues. We have demonstrated in this work that the high quality of the data is starting to demand a proper treatment of electron scattering in radiative transfer through the accretion disk around a stellar-mass black hole. Some of the effects that were not appreciated previously in fitting low S/N data are now becoming apparent. At present, this demanding situation fundamentally limits our ability to reliably derive the physical parameters of the accretion disk or the black hole in an X-ray binary, based on modeling the disk X-ray continuum. There are also observational issues that add additional uncertainties to the exercise. For instance, many key parameters (e.g., black hole mass, inclination angle, and distance) that characterize a source are often poorly determined but are needed to determine, e.g., the black hole spin. This is entirely independent of the quality of X-ray data. Also, perhaps less appreciated are the significant uncertainties in the absolute and cross calibrations of the detectors on different X-ray satellites. This issue is relevant, because the determination of the spin of a black hole in an X-ray binary depends critically on the overall normalization of the X-ray continuum. This is the reason why one must be very careful in comparing results based on data from different satellites. We have shown that neither of the two state-of-the-art disk models is capable of satisfactorily fitting the observed ultra-soft component of the spectra of GX 339$-$4. While this is perhaps not totally surprising for “kerrbb”, since it does not actually carry out radiative transfer calculations, it is for “bhspec”. These models have recently been applied to data to derive the spin of black holes in a number of systems, so our finding is somewhat disappointing. If we take the best-fit parameters at their face values, the models would suggest that GX 339$-$4 contains a moderately rotating black hole (with $a^* \sim$ 0.5–0.6). On the other hand, if we attribute the asymmetry in the profile of the observed Fe K$_{\alpha}$ line to gravitational redshift, we would conclude that the source contains a rapidly rotating black hole (with $a^* \approx 0.96$). We should note, however, that the apparent inconsistency can be easily reconciled when we take into account the large uncertainties associated with, e.g., black hole mass, inclination angle, and distance. For example, if we adopt $13.5~M_{\sun}$ for the black hole mass, 51 for the inclination, and $7.5$ kpc for the distance, the “kerrbb’ model yields $a^* \approx 0.93$ and $0.96$ when fitting the 2002 and 2003 data, respectively. We were able to fit the ultrasoft component quite satisfactorily with a simple saturated Compton scattering model. The results allowed us to test a procedure that was previously suggested by Cui et al. (2002) to empirically derive the color correction factor from the same X-ray data. The values obtained are very close to the theoretical expectation (e.g., Shimura & Takahara 1995), which is also often adopted in spectral modeling. Therefore, our results have provided further support for this observational approach. Although the use of a single color correction factor ignores possible radial dependence of spectral hardening in the disk, it does not seem unreasonable given that the X-ray emission from the disk originates from a relatively narrow region (closest to the black hole). Conclusions =========== Based on our joint spectral analysis of two simultaneous XMM-Newton/RXTE observations of GX 339-4, we can draw following conclusions: - The empirical procedure to derive the color correction factor ($f_{col}$) observationally, as proposed by Cui et al. (2002), yields reasonable results. If confirmed by further investigations, this would eliminate a major (theoretical) uncertainty in deriving the parameters of the disk from modeling the X-ray continuum. - The observed X-ray continuum of GX 339-4 in the high-soft state, which is dominated by emission from the optically-thick accretion disk, cannot be satisfactorily fitted by any existing disk model. Therefore, one should excise caution in assessing quantitative results from such spectral modeling. We wish to thank Shuangnan Zhang for suggesting the derivation of the spectral hardening factor from modeling the disk X-ray continuum and for subsequently collaborating on the subject. This work is a follow-up to much of the initial discussions. We also thank Lev Titarchuk for candid discussions on the theoretical aspects of the subject. This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center. It was supported in part by NASA through the LTSA grant NAG5-9998. We also gratefully acknowledge financial support from the Purdue Research Foundation and from a Grodzins Summer Research Award from the Department of Physics at Purdue University (to G.P.). 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--- abstract: 'We derive a residual-based a posteriori error estimator for the conforming $hp$-Adaptive Finite Element Method ($hp$-AFEM) for the steady state Stokes problem describing the slow motion of an incompressible fluid. This error estimator is obtained by extending the idea of a posteriori error estimation for the classical $h$-version of AFEM. We also establish the reliability and efficiency of the error estimator. The proofs are based on the well-known Cl[é]{}ment-type interpolation operator introduced in [@Melenk2005] in the context of the $hp$-AFEM. Numerical experiments show the performance of an adaptive hp-FEM algorithm using the proposed a posteriori error estimator.' author: - 'A. Ghesmati[^1]' - 'W. Bangerth$\phantom{\,}^{\ddag}$' - 'B. Turcksin$\phantom{\,}^{\#}$[^2]' bibliography: - 'hp-Stokes.bib' title: 'Residual-Based a posteriori error estimation for $hp$-adaptive finite element methods for the Stokes equations' --- Introduction {#sec:Intro} ============ $h$-adaptive finite element methods – in which the mesh size is adjusted to resolve features of the solution – have been known to be efficient tools for solving partial differential equations since the late 1970s [@Babuska_1978; @Babuska_1979]. The development of practical and efficient estimators of the local error over the past 25 years [@Verfurth-1996; @Ainsworth1997; @BR03] has made them a standard tool in the finite element analysis of many equations and is now widely used in applications. On the other hand, the $p$ or $hp$ versions of adaptive finite element methods – in which one adjusts either the polynomial degree of the approximation on every cell, or both the polynomial degree and the mesh size – has seen much less practical attention. Originally introduced in [@Babuska_1981; @Babuska-Dorr-1981], it is known both theoretically and practically that the $hp$-adaptive FEM can achieve exponential rates of convergence with respect to the number of degrees of freedom [@Schwab1998; @Melenk-Schwab; @Schotzau-Schwab; @Costabel-Dauge]. However, it is technically much more complicated to derive reliable and efficient estimates of the error for $hp$ approximations. Furthermore, even once estimates for the error on each cell are available, one is faced with the decision whether increasing the polynomial degree $p$ of the approximation or reducing the mesh size $h$ is more likely to reduce the error, measured with regard to the computational cost of the two possible resulting meshes (see, for example, [@Eibner-Melenk; @Wihler; @Demkowicz-2002; @Rachowicz-89; @Ainsworth-98; @Heuveline-2003; @Buerg_Conv]). Finally, the implementation of algorithms and data structures for conforming $hp$ finite element methods is complex in practice [@BK07]. Furthermore, it has proven to be significantly more difficult to extend many results that are well-established for $h$ adaptivity to $hp$ adaptivity for equations that are not as simple as the Laplace equation. Consequently, published theoretical considerations of error estimates and optimality of refinement strategies are still largely confined to the Laplace equation. Despite the known superiority of $hp$ adaptivity in terms of computational efficiency, its practical impact has therefore not been as profound as $h$-adaptive methods. In this contribution, we address one of these difficulties by deriving residual-based a posteriori error estimates for conforming $hp$ discretizations of the Stokes equation. This work is inspired by previous work for the Laplace equation [@Buerg_Conv; @Dorfler2007; @Melenk2001]. However, it has to address the key difficulty of the Stokes equation that the solution is not the unconstrained minimizer of an energy. Therefore, the Stokes operator is not positive definite, so that working with it is not as straightforward as for example with elliptic operators with their implied coercivity condition. In particular, we present the following results: - We derive estimates for the error between the finite-dimensional $hp$ approximation and the continuous solution of the Stokes equation. - As in similar approaches for the Laplace equation, it is not easily possible to show that these estimates are reliable and efficient, i.e., that the true error is bounded from above and below by our estimator up to a constant that does not depend on $h$ or $p$. This is so because the inverse estimates that are used to derive reliability and efficiency statements typically involve the polynomial degree $p$. To overcome this deficiency, we instead introduce a whole *family* of estimates $\eta_{\alpha}$ parameterized by an index $\alpha\in[0,1]$. For a fixed $\alpha$, we can not show that an estimator is both efficient and reliable; on the other hand, we can show that for some members of this family, either one or the other property hold. However, we demonstrate through numerical experiments that our estimator for a given $\alpha$ is, in practice, indeed both reliable and efficient. - We devise a strategy to mark cells for either $h$ or $p$ refinement based on criteria for a systematic reduction of the error. - Although we make no claims about the optimality of this strategy – i.e., we can not prove that among all strategies it leads to the greatest error reduction – we show numerical results that suggest that the strategy can achieve the desired exponential convergence rate for the $hp$-adaptive refinement. To the best of our knowledge, none of these properties have previously been derived or demonstrated for the Stokes equation using continuous $hp$-adaptive finite element methods. (However, some related work for *dis*continuous Galerkin discretizations of the Stokes equations is available in [@Houston2004].) The outline of the remainder of this paper is as follows: In Section \[sec:model problem\], we introduce the Stokes problem, its weak formulation and the conforming discretization with which we intend to solve it computationally. In Section \[sec: Aux\_Results\], we introduce necessary notation and state our assumptions as well as some important theoretical results (such as the Clément interpolation operator and polynomial inverse estimates) on which we rely throughout this work. The main results are derived in Section \[sec: Error Est\], where we develop an $hp$ residual-based a posteriori error estimator for the Stokes problem, followed by the analysis of the reliability and the efficiency of our error estimator. In Section \[sec:hp-Adaptive Ref\] we discuss the details of our $hp$ algorithm, i.e., the criterion upon which we choose either $h$ or $p$ refinement. Finally in Section \[sec: Numerical Results\] we present numerical results and demonstrate the performance of the proposed error estimators using practical examples. The Stokes problem and basic assumptions {#sec:model problem} ======================================== Let $\Omega\in\IR^{\text{2}}$ be an open and connected domain with smooth boundary $\Gamma=\partial\Omega$ such that it satisfies a Lipschitz condition. $u(\text{x})$ is the velocity and $\varrho(\text{x})$ be the pressure of the fluid at some point $\text{x} \in \Omega$, respectively. Given body forces $f \in L^2(\Omega)^{\text{2}}$ and the constant viscosity parameter $\nu>0$, consider stationary incompressible fluid flow as our model problem: For the Stokes equations, we are interested in finding $u:\Omega\to\IR^{\text{2}}$ and $\varrho:\Omega\to\IR$ such that $$\label{stocks} \begin{split} -\nu \Delta u+ \nabla \varrho &= f \qquad \text{in } \enspace \Omega,\\ -\nabla\cdot u &= 0 \qquad\text{in } \enspace \Omega,\\ u &= 0 \qquad\text{on } \enspace \Gamma. \end{split}$$ For ease of presentation, we here assume homogenous *no slip* boundary condition on the velocity field. (However, similar results as the ones shown herein are also valid for other type of boundary conditions.) To ensure uniqueness of solution, we require *vanishing mean* for pressure field, i.e., that $\int_{\Omega} \varrho = 0$. Here and below, we limit ourselves to the two-dimensional case primarily because Lemmas \[inverse\_ineq\] and \[edge\_inverseq\] below are only available for this case; however, we expect that with additional work, all main results herein could also be shown to hold in three space dimensions. We denote the standard Sobolev spaces by $H^m(\Omega)$ for $m\in\IN_0$. In particular, the norm and the scalar product of $L^2(\Omega)= H^0(\Omega)$ are denoted by $\Vert\cdot\Vert_\Omega$ and $(\cdot,\cdot)_\Omega$, respectively. To account for homogeneous Dirichlet boundary conditions, we set $$H_0^1(\Omega):=\lbrace v\in H^1(\Omega): \varphi=0\text{ on }\Gamma \rbrace.$$ Further, we denote the space containing all functions in $L^2(\Omega)$ with zero mean value by $$L_0^2(\Omega):=\lbrace v\in L^2(\Omega): (\varphi, 1)_\Omega=0 \rbrace$$ and define $$\calH (\Omega):= H_0^1(\Omega)^{\text{2}} \times L_0^2(\Omega).$$ Then, we introduce the bilinear form $\mathcal{L}:\calH (\Omega)\times\calH (\Omega)\to\IR$ by $$\label{bilin} \mathcal{L}([u,\varrho];[v,q]):= (\nu \nabla u, \nabla v)_\Omega-(\varrho, \nabla\cdot v)_\Omega- (\nabla\cdot u, q)_\Omega.$$ The weak formulation of problem (\[stocks\]) then seeks $[u, \varrho]\in \calH$ so that $$\label{weak} \mathcal{L}([u,\varrho];[v,q])=(f,v)_\Omega \qquad \forall [v,q]\in \calH (\Omega).$$ Due to the continuous $\inf$-$\sup$ condition $$\inf_{[u,\varrho]\in \calH } \sup_{[v,q]\in \calH}\frac{\mathcal{L}([u,\varrho];[v,q])}{\left(\Vert\nabla u \Vert_\Omega+ \Vert \varrho \Vert_\Omega\right)\left(\Vert\nabla v \Vert_\Omega+ \Vert q \Vert_\Omega\right)}\geq\kappa>0,$$ where $\kappa$ is the $\inf$-$\sup$ constant depending only on $\Omega$, the weak problem is well-posed and has a unique solution, see [@Brezzi1974] and [@Girault1986]. Now, assume $\calT= \lbrace K \rbrace$ is a triangulation of domain $\Omega$. For each element $K$, we associate an element map $T_K:\hat{K}\to K$ where the reference cell is $\hat{K}=[0,1]^{\text{2}}$. Further, we define the mesh size vector $h:=\left(h_K\right)_{K\in\calT},$ where $h_K:=\diam(K)$. With each element $K \in \calT$, we associate a polynomial degree $p_K \in \IN$ and collect them in a polynomial degree vector $p:=\left(p_K\right)_{K\in\calT}$. Throughout this work, we assume that the discretization $(\calT,p)$ of $\Omega$ is $\left(\gamma_h,\gamma_p\right)$-regular [@Schwab1998; @Szab1991]. \[regular\_def\] A discretization $(\calT,p)$ is called $\left(\gamma_h,\gamma_p\right)$-regular if and only if there exist constants $\gamma_h,\gamma_p > 0$ such that for all $K, K' \in \calT$ with $\overline K\cap \overline{K'}\neq \emptyset$ there holds $$\label{hp-regularity} \gamma_h^{-1}h_K\leq h_{K'} \leq \gamma_hh_K, \qquad \text{and} \qquad \gamma_p^{-1}p_K \leq p_K'\leq \gamma_pp_K.$$ In other words, the condition implies that the element sizes and also the polynomial degrees of neighboring elements are comparable. To define the discrete solution space, for an element $K\in\calT$ denote $\mathcal{F}(K)$ the set of all interior faces of cell $K$. Then, define by $h_{f}:=\diam(f)$ the diameter of face $f\in\cal F(K)$ and by $p_{f}:= \max\left\{p_K, p_{K'}\right\}$ its polynomial degree where for $K,K'\in\calT$ are the cells adjacent to $f$. Further, the problem is discretized by the standard $(p_{k}, p_{k-1})$ Taylor-Hood finite element. The corresponding $hp$ spaces for velocity and pressure are then $$\begin{aligned} \label{velocity-FE-space} V^p_u(\calT)^2 &:= \left\{u \in H_0^1(\Omega)^2:\enspace u|_{K} \circ T_K \in \calQ_{p_K}^2\left(\hat{K}\right) \text{ for all }K\in\calT\right\}, \\ \label{pressure-FE-space} V^p_\varrho(\calT)&:= \left\{\varrho \in L_0^2(\Omega):\enspace \varrho|_{K}\circ T_K \in \calQ_{p_K-1}\left(\hat{K}\right)\text{ for all }K\in\calT\right\} \\ \label{FE-space} \calV^p(\calT) &:=V_u^p(\calT)^2\times V_\varrho^p(\calT) \subseteq \mathcal{H}(\Omega).\end{aligned}$$ Here, $\calQ_r$ is the tensor-product polynomial space of complete degree at most $r\in\IN_0$. Then, the discrete approximation to ($\ref{weak}$) consists of seeking $\left[u_{\FE},\varrho_{\FE}\right] \in \calV^p(\calT)$ such that $$\label{discrete_weak} \mathcal{L}\left(\left[u_{\FE},\varrho_{\FE}\right];\left[v_{\FE},q_{\FE}\right]\right)=\left(f,v_{\FE}\right)_\Omega\qquad\forall\left[v_{\FE},q_{\FE}\right]\in\calV^p(\calT).$$ This choice of spaces satisfies the discrete Babuska-Brezzi condition [@Arnold84] $$\inf_{[u_h,\varrho_h]\in \cal{H} } \sup_{[v_h,q_h]\in \cal{H}}\frac{\mathcal{L}([u_h,\varrho_h];[v_h,q_h])}{\left(\Vert\nabla u_h \Vert+ \Vert \varrho_h \Vert \right)\left(\Vert\nabla v_h \Vert+ \Vert q_h \Vert\right)}\geq\kappa_d>0,$$ where the constant $\kappa_d$ is independent of cell size $h$ and polynomial degree $p$. Consequently, problem is well posed. Furthermore, Galerkin orthogonality holds: Let $[u,\varrho]\in\calH$ be the solution of ($\ref{weak}$) and $\left[u_{\FE},\varrho_{\FE}\right]\in\calV^p(\calT)$ be the solution of ($\ref{discrete_weak}$), then $$\label{Galerkin-Orthogonality} \mathcal{L}\left(\left[u-u_{\FE},\varrho-\varrho_{\FE}\right];[v_{\FE},q_{\FE}]\right)=0\qquad\forall\left[v_{\FE},q_{\FE}\right]\in\calV^p(\calT).$$ Auxiliary results {#sec: Aux_Results} ================= We provide some auxiliary results which we use later in our work. This includes an $H^1$-conforming interpolation operator that preserves homogeneous Dirichlet boundary conditions, and some polynomial smoothing estimates. The $H^1$-conforming interpolation operator is a Clément-type interpolation which replaces point evaluation by a local average [@Clement1975]. The procedure does not require the extra regularity of the point evaluation, and is consequently well-defined for functions in $H^1(\Omega)$. In [@Scott1990], this interpolation operator was modified in such a way that it also preserves polynomial boundary conditions. Melenk in [@Melenk2005] extended the aforementioned $H^1$-conforming interpolation to the context of $hp$-adaptive finite element spaces.\ In our definition of $hp$-Clément interpolation operators, consider $\mathcal{T}$ as a $(\gamma_{h}, \gamma_{p})$-regular triangulation of $\IR^{\text{d}}$. (For cases where we would want to impose Dirichlet boundary conditions on only a subset $\Gamma_D\subset\Gamma$, we can require that $\Gamma_D$ can be exactly represented by a collection of faces, i.e., $\bar\Gamma_D=\cup_{K \in \cal{T}} \partial K \cap\bar{\Gamma}_D$.) Then, for a cell $K\in\calT$ and a face $f \in \mathcal{F}(K)$ we define the patch sets $$\begin{aligned} \label{patch} \omega_K&:=K \cup \bigcup\{ L \in\calT: \text{$L$ shares a common edge with $K$}\}, \\ \label{patch-face} \omega_f&:=\bigcup\{ L \in\calT : \text{$f$ is an edge of $L$}\}.\end{aligned}$$ The following result from [@Melenk2001] then provides an estimate for the interpolation error in terms of the gradient of the interpolated function: \[h1interpol\_thm\] Let $\mathcal{T}$ be $\left(\gamma_h,\gamma_p\right)$-regular and $K \in \mathcal{T}$ be arbitrary. Then, there exists a bounded linear operator $\Pi^{hp}:H_0^1(\Omega)^2\to \mathcal{V}^{p} (\mathcal{T})$ – namely, the Clément interpolation operator –, and a constant $C >0$ independent of mesh size $h$ and polynomial degree $p$ such that for all $u\in H_0^1(\Omega)$ and all $f\in\mathcal{F}(K)$ $$\begin{aligned} \label{clem_1} \left\Vert u-\Pi^{hp} u\right\Vert_{L^2(K)}&\leq C\frac{h_K}{p_K}\Vert \nabla u\Vert_{L^2({\omega_K})}, \\ \label{clem_2} \left\Vert u-\Pi^{hp} u\right\Vert_{L^2(f)}&\leq C\sqrt{\frac{h_f}{p_f}}\Vert\nabla u\Vert_{L^2({\omega_f})}.\end{aligned}$$ Following the lines of [@Schwab1998], one can find proofs in [@Melenk2005 Theorem 3.3]. Next, let us present some polynomial smoothing estimates that are widely used in the error analysis of many numerical methods for partial differential equations and integral equations [@Bernardi2001; @Melenk2001]. We will later use them in proving upper and lower bounds of our error estimator. Specifically, define the smoothing weight functions $\Phi_{K}: K\subset \IR^{2} \to\IR^{+}$ and $\Phi_{\omega_f}: \omega_f\subset \IR^{2}\to\IR^{+}$ by $$\begin{aligned} \label{weightfunc} \Phi_{K}(x) &:= \frac{1}{h_K}\operatorname{dist}\left(x,\partial K\right), \\ \label{weightfunc_edge} \Phi_{\omega_{f}}(x) &:= \frac{1}{\text{diam}(\omega_f)} \text{dist} (x, \partial \omega_f).\end{aligned}$$ Then we have: \[inverse\_ineq\] Let $\delta \in [0,1]$, $a,b\in\IR$ such that $-1\le a\le b$. Then, for any $\pi_p \in \calQ_p\left( K \right)$, there exists some constant $C>0$ independent of $h$ and $p$ so that $$\begin{aligned} \label{inv-equ-01} \Vert \pi_p\left(\Phi_{K}\right)^a \Vert_{\text{L}^2(K)} &\leq C (a,b) p^{(b-a)} \Vert \pi_p\left(\Phi_{K}\right)^b \Vert_{\text{L}^2(K)}, \\ \label{inv-equ-02} \Vert \nabla \pi_p \left(\Phi_{K}\right)^{\delta} \Vert_{\text{L}^2(K)} &\leq \frac {C (\delta) p^{(2-\delta)}}{h_K} \Vert \pi_p \left(\Phi_{K}\right)^{\frac{\delta}{2}} \Vert_{\text{L}^2(K)}. \end{aligned}$$ See [@Bernardi2001 Lemmas 4, 5] and [@Melenk2001 Lemma 2.5]. The next lemma provides results for the extension of a polynomial from an edge to a domain. These estimates are used in the efficiency analysis of our error estimator. \[edge\_inverseq\] Let $\hat{f}$ be the edge of unit square $\hat{K}$, and $0 \le \alpha \leq 1$. $\Phi_{\omega_{\hat{f}}}$ defined as in (\[weightfunc\_edge\]) the edge $\hat{f}$ corresponding to the unit cell $\hat{K}$. Then there exists $C_{\alpha}>0$, such that for any polynomial $\pi_{p} \in \calQ_p$ and every $\delta \in (0, 1]$, there exists some extension $v_{\hat{f}} \in H_{0}^1\left( \hat{K} \right)$ so that: $$\begin{aligned} v_{\hat{f}}\vert_{\hat{f}}&=\pi_p\Phi_{\omega_{\hat{f}}}^{\alpha}, \hspace{3pt} v_{\hat{f}}\vert_{\partial \hat{K} \backslash \hat{f}}= 0, \\ \Vert v_{\hat{f}} \Vert_{L^2({\hat{K}})}^2 &\leq C_{\alpha} \delta \Vert \pi_p \Phi^{\frac{\alpha}{2}}_{\omega_{\hat{f}}} \Vert_{L^2({\hat{f}})}^2, \\ \Vert \nabla v_{\hat{f}} \Vert_{L^2(\hat{K})}^2 &\leq C_{\alpha} (\delta p^{2(2-\alpha)}+ \delta^{-1}) \left\Vert \pi_p \Phi^{\frac{\alpha}{2}}_{\omega_{\hat{f}}} \right\Vert^{2}_{L^2(\hat{f})}. \end{aligned}$$ See [@Melenk2001 Lemma 2.6]. A posteriori error estimation {#sec: Error Est} ============================= A posteriori error estimates assess the error between the exact solution $[u,\varrho]\in\calH$ and its finite element approximation $\left[u_{\FE},\varrho_{\FE}\right]\in\calV^p(\calT)$ only in terms of known quantities [@Ern2013; @Babuska1978; @Verfurth1989] – i.e., the problem data and the approximate solution. We call a functional $\eta\left(u_{\FE},\varrho_{\FE},f\right)$ an *a posteriori error estimator for the Stokes equation*, if and only if there exists a constant $C>0$ such that $$\label{apost_ee} \left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_\Omega+\left\Vert\varrho-\varrho_{\FE}\right\Vert_\Omega\le C\eta\left(u_{\FE},\varrho_{\FE},f\right).$$ Furthermore, if $\eta\left(u_{\FE},\varrho_{\FE},f\right)$ can be decomposed into localized quantities $\eta_K\left(u_{\FE},\varrho_{\FE},f\right)$, $K\in\calT$, such that $$\label{global_est} \eta(u_{\FE},\varrho_{\FE},f)^2=\sum_{K\in\calT}\eta_{K}\left(u_{\FE},\varrho_{\FE},f\right)^2,$$ then $\eta_K\left(u_{\FE},\varrho_{\FE},f\right)$ is called a *local error indicator*. Estimate (\[apost\_ee\]) is usually called a “reliability estimate” since it guarantees that the error is controlled by the error estimator $\eta\left(u_{\FE},\varrho_{\FE},f\right)$ up to a constant independent of mesh size $h$ and polynomial degree $p$. Further, the local error indicators $\eta_K\left(u_{\FE},\varrho_{\FE},f\right)$ provides the basis for adaptive mesh refinement by identifying those cells $K\in\calT$ where the error is large and that, consequently, should be refined locally. This procedure is then repeated until $\eta\left(u_{\FE},\varrho_{\FE},f\right)$ is smaller than a prescribed tolerance. Computational efficiency requires that the $\eta_K$ also satisfy some efficiency property guaranteeing that the upper bound (\[apost\_ee\]) is sharp and does not asymptotically overestimate the true error. To this end, we would like to derive a local lower bound for the energy error for every cell $K\in\calT$: $$\eta_K\left(u_{\FE},\varrho_{\FE},f\right)\le C\left(\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert^2_{\omega_K}+\left\Vert\varrho-\varrho_{\FE}\right\Vert^2_{\omega_K}\right)^{1/2}.$$ Residual-based a posteriori error analysis {#sec: estimator-Err Analysis} ------------------------------------------ Let us now define a residual-based a posteriori error estimator for problem (\[stocks\]), and derive upper and lower bounds for it in terms of the energy error of the approximated solution. In the spirit of [@Melenk2001], we define a family of error estimators $\eta_{\alpha}$, $\alpha \in [0,1]$. This estimator is local, i.e., $\eta_\alpha^2:=\sum_{K\in\calT}\eta_{\alpha;K}^2$ and can be decomposed into cell and interface contributions: $$\begin{aligned} \label{residual_boun} \eta_{\alpha;K}^2&:=\eta_{\alpha;K;R}^2+\eta_{\alpha;K;B}^2, \\ \label{residual_term} \eta_{\alpha;K;R}^2&:=\frac{h_K^2}{p_K^2}\left\Vert\left(I_{p_K}^Kf+\nu\Delta u_{\FE}-\nabla\varrho_{\FE}\right)\Phi_K^{\frac\alpha2}\right\Vert^2_K+\left\Vert\left(\nabla\cdot u_{\FE}\right)\Phi_K^{\frac\alpha2}\right\Vert^2_K, \\ \label{bound_term-chap-3} \eta_{\alpha;K;B}^2&:= \sum_{f\in\cal F(K)} \frac{h_f}{2 p_f} \left\Vert \left[ \nu\frac{\partial u_{\FE}}{\partial n_K} \right] \Phi_{\omega_f}^{\frac\alpha2}\right\Vert_f^2.\end{aligned}$$ Here, $I_{p_K}^K f$ denotes the local $L^2$-projection of $f$ into the space of piecewise polynomials of degree $p_K$. Furthermore, $h_f:=\text{diam}(f)$ and $p_f:=\max(p_{K}, p_{K'})$ for a face $f$ that is shared by cells $K$ and $K'$. Finally, $[\cdot]$ denotes the jump of a quantity across a face whose outward normal relative to $K$ is indicates by $n_K$. In the following, we will first derive an upper bound for the energy error in terms of the estimator $\eta_\alpha$, i.e., state a reliability estimate. \[relibi\_error\_est\] Let $[u,\varrho]\in\calH$ and $\left[u_{\FE},\varrho_{\FE}\right]\in\calV^p(\calT)$ be the solutions of (\[weak\]) and (\[discrete\_weak\]), respectively. Further, let $ \alpha\in[0,1]$ and assume that triangulation $\calT$ is $\left(\gamma_h,\gamma_p\right)$-regular. Then there exists a constant $C_{\text{rel}}>0$ independent of mesh size vector $h$ and polynomial degree vector $p$ such that $$\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_\Omega^2+\left\Vert\varrho-\varrho_{\FE}\right\Vert_\Omega^2\leq C_{\text{rel}}\sum_{K\in\calT}\left(p_K^{2\alpha}\eta_{\alpha;K}^2+\frac{h_K^2}{p_K^2}\left\Vert I_{p_K}^{K}f-f\right\Vert^2_K\right).$$ In particular, the statement provides a $p$-independent reliability bound for $\alpha=0$. Set $e_{\FE}:=u-u_{\FE}$ and $\epsilon_{\FE}:=\varrho-\varrho_{\FE}$. From , we have $$\begin{aligned} \mathcal{L}\left(\left[e_{\FE},\epsilon_{\FE}\right];[v,q]\right) &= \left(\nu\nabla e_{\FE},\nabla\left(v-\Pi^{hp}v\right)\right)_\Omega \\ & \qquad -\left(\epsilon_{\FE},\nabla\cdot\left(v-\Pi^{hp}v\right)\right)_\Omega - \left(\nabla\cdot e_{\FE},q\right)_\Omega \\ &= \sum_{K\in\calT}\bigg(\left(\nu\nabla e_{\FE},\nabla\left(v-\Pi^{hp}v\right)\right)_K \\ & \qquad\qquad - \left(\epsilon_{\FE},\nabla\cdot\left(v-\Pi^{hp}v\right)\right)_K - \left(\nabla\cdot e_{\FE},q\right)_K\bigg),\end{aligned}$$ where $\Pi^{hp}:H_0^1(\Omega)^2\to \mathcal{V}^p(\mathcal{T})$ is the $H^1$-conforming interpolation operator from Theorem \[h1interpol\_thm\]. Using integration by parts and the incompressibility condition $\nabla\cdot u=0$ yields $$\begin{gathered} \mathcal{L}\left(\left[e_{\FE},\epsilon_{\FE}\right];[v,q]\right) =\sum_{K\in\calT}\Bigg(\left(f+\nu\Delta u_{\FE}-\nabla\varrho_{\FE},v-\Pi^{hp}v\right)_{K} \\ -\left(\nabla\cdot u_{\FE},q\right)_{K} + \sum_{f\in\cal F(K)}\left(\left[\nu\frac{\partial u_{\FE}}{\partial n}\right],v-\Pi^{hp}v\right)_f\Bigg).\end{gathered}$$ The continuous Cauchy-Schwarz inequality then results in the estimate $$\begin{gathered} \mathcal{L}\left(\left[e_{\FE},\epsilon_{\FE}\right];[v, q]\right) \le\sum_{K\in\calT}\Bigg(\left\Vert I_{p_K}^Kf+\nu\Delta u_{\FE}-\nabla\varrho_{\FE}\right\Vert_K\left\Vert v-\Pi^{hp}v\right\Vert_K \\ \qquad\qquad\qquad +\left\Vert\nabla\cdot u_{\FE}\right\Vert_K\Vert q\Vert_K +\left \Vert f-I_{p_K}^Kf\right\Vert_K\left\Vert v-\Pi^{hp}v\right\Vert_K \\ +\sum_{f\in\cal F(K)}\left\Vert\left[\nu\frac{\partial u_{\FE}}{\partial n_K}\right]\right\Vert_f\left\Vert v-\Pi^{hp}v\right\Vert_f\Bigg).\end{gathered}$$ Theorem \[h1interpol\_thm\] allows us to locally bound the differences $v-\Pi^{hp}v$. This yields $$\begin{split} \mathcal{L}\left(\left[e_{\FE},\epsilon_{\FE}\right];[v, q]\right)& \le C\sum_{K\in\calT}\Bigg(\frac{h_K}{p_K}\left\Vert I_{p_K}^Kf+\nu\Delta u_{\FE}-\nabla\varrho_{\FE}\right\Vert_K \\&+\left\Vert\nabla\cdot u_{\FE}\right\Vert_K +\frac{h_K}{p_K}\left\Vert f-I_{p_K}^Kf\right\Vert_K \quad \\& + \sum_{f\in\cal F(K)}\sqrt{\frac{h_f}{p_f}}\left\Vert\left[\nu\frac{\partial u_{\FE}}{\partial n_K}\right]\right\Vert_f\Bigg)\left(\Vert\nabla v\Vert_{\omega_K}+\Vert q\Vert_K\right), \end{split}$$ which we can further estimate as follows: $$\begin{gathered} \mathcal{L}\left(\left[e_{\FE},\epsilon_{\FE}\right];[v, q]\right) \\ \le C\left(\sum_{K\in\calT}\left(\eta_{0;K}^2+\frac{h_K^2}{p_K^2}\left\Vert f-I_{p_K}^Kf\right\Vert_K^2\right)\right)^{\frac12}\left(\Vert\nabla v\Vert_\Omega^2+\Vert q\Vert_\Omega^2\right)^{\frac12}\end{gathered}$$ for some constant $C>0$ independent of mesh size vector $h$ and polynomial degree vector $p$. Moreover, for $(e_{\FE}, \varepsilon_{\FE}) \in \mathcal{H} $ we have $$\left(\left\Vert\nabla e_{\FE}\right\Vert_\Omega^2+\left\Vert\epsilon_{\FE}\right\Vert_\Omega^2\right)^{\frac12} \leq C \sup_{[v,q]\in\calH}\frac{\mathcal{L}\left(\left[e_{\FE},\epsilon_{\FE}\right];[v,q]\right)}{(\Vert\nabla v\Vert_\Omega^2+\Vert q\Vert_\Omega^2)^{\frac12}},$$ for some constant $C>0$. This implies the claimed result for $\alpha=0$. Using the inverse estimates given in Lemma \[inverse\_ineq\], we can bound $\eta_{0; K}$ in terms of $\eta_{\alpha; K}$ for $\alpha\in(0,1]$ from above. Therefore, setting $a:=0$ and $b:=\alpha$ in Lemma \[inverse\_ineq\] and we get $$\left(\left\Vert\nabla e_{\FE}\right\Vert_\Omega^2+\left\Vert\epsilon_{\FE}\right\Vert_\Omega^2\right)^{\frac12} \le C_{rel}\left(\sum_{K\in\calT}\left(p_K^{2\alpha}\eta_{\alpha;K}^2+\frac{h_K^2}{p_K^2}\left\Vert f-I_{p_K}^Kf\right\Vert_K^2\right)\right)^{\frac12}$$ which concludes the proof. Next, we derive an upper bound for the a posteriori error estimator $\eta_{\alpha; K}$ in terms of the energy error $\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_{\omega_K}^2+\left\Vert\varrho-\varrho_{\FE}\right\Vert_{\omega_K}^2$ defined on the patch $\omega_K$ around cell $K$. Under mild assumptions on the mesh, this then constitutes an efficiency estimate for the error estimator. We will first consider the residual and jump terms $\eta_{\alpha;K;R}, \eta_{\alpha;K;B}$ separately and combine the derived efficiency estimates later to obtain an upper bound for the residual-based a posteriori error estimator from definition (\[residual\_boun\]). \[lemma1\] Let $[u,\varrho]\in\calH$, $\left[u_{\FE},\varrho_{\FE}\right]\in\calV^p(\calT)$, and $\calT$ as in Theorem \[relibi\_error\_est\], and $\alpha\in[0,1]$ be arbitrary. Then, there exists a constant $C>0$ independent of the mesh size vector $h$ and polynomial degree vector $p$ so that $$\begin{gathered} \eta_{\alpha;K;R}^2\leq C \bigg(p_{K}^{2(1-\alpha)}\left(\nu^2\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_K^2+\left\Vert\varrho-\varrho_{\FE}\right\Vert^2_K\right) \\ +\frac{h_K^{2+\frac\alpha2}}{p_K^{1+\alpha}}\left\Vert f-I^K_{p_K}f\right\Vert^2_K \bigg). \end{gathered}$$ In particular, the statement provides a $p$-independent efficiency bound of the cell-residual term for $\alpha=1$. Let us write the residual-based term as $ \eta_{\alpha;K; R}^2= \eta_{\alpha;K;R_1}^2+ \eta_{\alpha;K;R_2}^2, $ with $$\label{eq_01} \begin{split} \eta_{\alpha;K;R_1}^2 & := \frac{h^2_K}{p^2_K}\left\Vert\left(I^K_{p_K}f+\nu\Delta u_{\FE}-\nabla\varrho_{\FE}\right)\Phi_K^{\frac\alpha2}\right\Vert^2_K,\\ \eta_{\alpha;K;R_2}^2 & := \left\Vert\nabla\cdot u_{\FE}\Phi_K^{\frac\alpha2}\right\Vert^2_K. \end{split}$$ Using the idea in [@Melenk2001] to build test functions, for $0<\alpha\le1$, we define the cell residual term $R_K$ as, $R_K:=\left(I^K_{p_K}f+\nu\Delta u_{\FE}-\nabla\varrho_{\FE}\right)\Phi_K^{\alpha}\in H_0^1(K)$ and obtain $$\label{eq_02} \left\Vert R_K\Phi_{K}^{-\frac\alpha2}\right\Vert^2_K=\left(f+\nu\Delta u_{\FE}-\nabla\varrho_{\FE},R_K\right)_K+\left(I^K_{p_K}f-f,R_K\right)_K.$$ With equation (\[weak\]) and applying integration by parts, the first term reads $$\begin{gathered} \left(f+\nu\Delta u_{\FE}-\nabla\varrho_{\FE},R_K\right)_K \notag \\ =\left(\nu\nabla\left(u-u_{\FE}\right),\nabla R_K\right)_K-\left(\varrho-\varrho_{\FE},\nabla\cdot R_K\right)_K -\left(\nabla\cdot u,q\right)_K.\end{gathered}$$ Inserting into (\[eq\_02\]) and using that $\nabla \cdot u=0$ implies $$\begin{aligned} \left\Vert R_K\Phi_{K}^{-\frac\alpha2}\right\Vert^2_K & = \left(\nu\nabla\left(u-u_{\FE}\right),\nabla R_K\right)_K-\left(\varrho-\varrho_{\FE},\nabla\cdot R_K\right)_K \notag\\&\qquad +\left(I^K_{p_K}f-f,R_K\right)_K \notag \\ & \le \bigg(\nu\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_K+\left\Vert\varrho-\varrho_{\FE}\right\Vert_K \bigg) \left\Vert\nabla R_K\right\Vert_K \notag \\ \label{equ 001} &\qquad + \left\Vert\left(I^K_{p_K} f-f\right)\Phi_K^{\frac{\alpha}{2}}\right\Vert_K\left\Vert R_K\Phi_K^{-\frac\alpha2}\right\Vert_K.\end{aligned}$$ Using equations (\[inv-equ-01\]) and (\[inv-equ-02\]) in Lemma \[inverse\_ineq\], we can estimate $$\begin{aligned} \left\Vert\nabla R_K\right\Vert_K^2 &= \left\Vert\nabla\bigg(\left(I^K_{p_K}f+\nu\Delta u_{\FE}-\nabla\varrho_{\FE}\right)\Phi_K^{\alpha}\bigg)\right\Vert_K^2\\ &\le 2 \left\Vert\nabla\left(I^K_{p_K}f+\nu\Delta u_{\FE}-\nabla\varrho_{\FE}\right)\Phi_{K}^{\alpha}\right\Vert^2_K \\&\qquad + 2 \left\Vert\left(I^K_{p_K}f+\nu\Delta u_{\FE}-\nabla\varrho_{\FE}\right)\Phi_K^{\alpha-1}\nabla\Phi_K\right\Vert^2_K \\& \le C\bigg( \frac{p_{K}^{2(2-\alpha)}}{h_K^2}\left\Vert R_K\Phi_K^{-\frac\alpha2}\right\Vert^2_K \\&\qquad + \frac{C}{h^2_K} \left\Vert\left(I^K_{p_K}f+\nu\Delta u_{\FE}-\nabla\varrho_{\FE}\right)^2\Phi_K^{2(\alpha-1)}\right\Vert_K\bigg),\end{aligned}$$ with some $C>0$ independent of $h$ and $p$. For the second of these two terms, we have to distinguish between two cases. Assuming $\alpha > \frac{1}{2}$, we set $a:=2(\alpha-1)$ and $b:=\alpha$ in Lemma \[inverse\_ineq\] to get $$\left\Vert\left(I^K_{p_K}f+\nu\Delta u_{\FE}-\nabla\varrho_{\FE}\right)\Phi_K^{\alpha-1}\right\Vert_K\le Cp_K^{1-\frac\alpha2}\left\Vert R_K\Phi_K^{-\frac\alpha2}\right\Vert_K$$ and inserting into the estimate above yields $$\label{used_in_goal_paper} \left\Vert\nabla R_K\right\Vert_K\le C\frac{p_{K}^{2-\alpha}}{h_K}\left\Vert R_K\Phi_K^{-\frac\alpha2}\right\Vert_K.$$ Inequality (\[equ 001\]) then reads as $$\begin{gathered} \left\Vert R_K\Phi_{K}^{-\frac\alpha2}\right\Vert_K \\ \le C\frac{p_K^{2-\alpha}}{h_K}\bigg(\nu\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_K+\left\Vert\varrho-\varrho_{\FE}\right\Vert_K\bigg)+h_K^{\frac\alpha2}\left\Vert I^K_{p_K} f-f\right\Vert_K,\end{gathered}$$ and, after multiplying both sides by $\frac{h_K}{p_K}$ and using definition (\[eq\_01\]), we have $$\begin{gathered} \label{equ 004} \eta_{\alpha;K;R_1} \leq Cp_K^{1-\alpha}\bigg(\nu\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_K+\left\Vert\varrho-\varrho_{\FE}\right\Vert_K\bigg) \\+\frac{h_K^{1+\frac\alpha2}}{p_K}\left\Vert I^K_{p_K} f-f\right\Vert_K.\end{gathered}$$ Now, let us consider the case $0\le\alpha\le\frac12$. Let $\beta:=\frac{1+\alpha}2$. Again, using the smoothing estimates given in Lemma \[inverse\_ineq\] and considering the fact that $\beta>\alpha$, we find $$\begin{split} \left\Vert R_K\Phi_K^{-\frac\alpha2}\right\Vert_K &\le C p_K^{\beta-\alpha}\left\Vert\left(I_{p}^{K}f+\nu\Delta u_{\FE}-\nabla\varrho_{\FE}\right)\Phi_K^{\frac\beta2}\right\Vert_K\\ & = C\frac{p_K^{1+\beta-\alpha}}{h_K}\eta_{\beta;K;R_1}. \end{split}$$ Estimate (\[equ 004\]) then implies $$\begin{split} \left\Vert R_K\Phi_K^{-\frac\alpha2}\right\Vert_K\le C\bigg( &\frac{p_K^{2-\alpha}}{h_K}\left(\nu\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_K+\left\Vert\varrho-\varrho_{\FE}\right\Vert_K\right)\\& +\frac{h_K^{\frac\beta2}}{p_K^{\alpha-\beta}}\left\Vert I^K_{p_K} f-f\right\Vert_K \bigg). \end{split}$$ Then, the definition of $\beta$ yields $$\begin{split} \eta_{\alpha;K;R_1}\le C\bigg( & p_K^{1-\alpha}\left(\nu\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_K+\left\Vert\varrho-\varrho_{\FE}\right\Vert_K\right)\\& +\frac{h_K^{\frac{5+\alpha}4}}{p_K^{\frac{1+\alpha}2}}\left\Vert I^K_{p_K} f-f\right\Vert_K\bigg). \end{split}$$ To obtain the upper bound for $\eta_{\alpha;K;R_2}^2$, we observe $$\begin{split} \eta_{\alpha;K;R_2} = \left\Vert (\nabla\cdot u_{\FE} )\Phi_K^{\frac\alpha2}\right\Vert_K \le h_K^{\frac\alpha2}\left\Vert\nabla\cdot u_{\FE}\right\Vert_K. \end{split}$$ Since $\nabla\cdot u=0$, we have $\nabla\cdot u_{\FE}=\nabla\cdot\left(u-u_{\FE}\right)$ and, hence, $$\label{resid_02} \eta_{\alpha;K;R_2}\le h_K^{\frac\alpha2}\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_K.$$ Finally, combining estimates (\[equ 004\]) and (\[resid\_02\]) gives the desired result. Similarly, we can derive matching estimates for the jump-based term $\eta_{\alpha ;K;B}$ in equation (\[bound\_term-chap-3\]): \[lemma\_02\] Let $[u,\varrho]\in\calH$, $\left[u_{\FE},\varrho_{\FE}\right]\in\calV^p(\calT)$, and $\calT$ as in Theorem \[relibi\_error\_est\]. Let $\alpha\in[0,1]$. Then, there exists some constant $C>0$ independent of mesh size vector $h$ and polynomial degree vector $p$ such that $$\begin{split} \eta_{\alpha ;K;B}^{2} \leq C\bigg( & p_K^{\frac{3-\alpha}2}\left(\nu^2\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert^2_{\omega_K}+\left\Vert\varrho-\varrho_{\FE}\right\Vert^2_{\omega_K}\right)\\& +\frac{h^2_K}{p_K^{\frac{3+\alpha}2}}\left\Vert I^K_{p_K}f-f\right\Vert_{\omega_K}^2\bigg). \end{split}$$ For a given element $K \in \calT$ and an interior face $f \in \cal F (K)$, there exists some $K_1 \in \calT$ such that $f= \partial K \cap\partial K_1$ and a face patch $\omega_f$ as given in (\[patch-face\]). Moreover, by Lemma \[inverse\_ineq\] there exists an extension function $R_f \in H_0^{1}(\omega_f)$ such that $R_{f} \vert_{f}=\left[ \nu\frac{\partial u_{\FE}}{\partial n}\right]\Phi_{\omega_f}^{\alpha}$ that is continuous on $K$, vanishes on $\partial \omega_{f}$, and can be extended by zero to all of $\Omega$. Thus, we can consider $R_{f} \in H_0^1(\Omega)$. Now, to derive an upper bound for the jump-based term $\eta_{\alpha ;K;B}^{2}$, we use integration by parts to get $$\begin{split} \left\Vert R_f \Phi_{\omega_{f}}^{-\frac{\alpha}{2}}\right\Vert^2_{f} & =(\nu \Delta u_{\FE}, R_{f})_{\omega_f}+(\nu \nabla u_{\FE}, \nabla R_f)_{\omega_{f}}. \end{split}$$ From the weak formulation (\[weak\]) we have $$\begin{split} \left\Vert R_f \Phi_{\omega_{f}}^{-\frac{\alpha}{2}}\right\Vert^2_{f} & = (\nu \Delta u_{\FE}, R_f)_{\omega_f}-\left(\nu \nabla\left(u-u_{\FE}\right), \nabla R_f\right)_{\omega_f}+(f, R_f)_{\omega_f} \\ &\qquad +(\varrho, \nabla \cdot R_f)_{\omega_f} + (\nabla \cdot u, R_f)_{\omega_f}\\ &= (\nu \Delta u_{\FE}, R_f)_{\omega_f}-\left(\nu \nabla\left(u-u_{\FE}\right), \nabla R_f\right)_{\omega_f}+(f, R_f)_{\omega_f} \\&\qquad+ \left(\varrho_{\FE},\nabla\cdot R_f\right)_{\omega_f} +\left(\varrho-\varrho_{\FE}, \nabla \cdot R_f\right)_{\omega_f}, \end{split}$$ using again $\nabla \cdot u =0$. Then, performing integration by parts gives $$\begin{aligned} \left\Vert R_f \Phi_{\omega_{f}}^{-\frac{\alpha}{2}}\right\Vert^2_{f} &= \left(I^K_{p_K}f + \nu \Delta u_{\FE}- \nabla \varrho_{\FE}, R_f\right)_{\omega_f}-(\nu \nabla (u-u_{\FE}), \nabla R_e)_{K_e} \notag \\&\qquad +(\varrho-\varrho_{\FE}, \nabla \cdot R_f)_{\omega_f} + \left(f- I^K_{p_K}f, R_f\right)_{\omega_f} \notag \\ \label{eq_03} & \le \left(\left\Vert I^K_{p_K}f + \nu \Delta u_{\FE}-\nabla\varrho_{\FE}\right\Vert_{\omega_f}+ \left\Vert f- I^K_{p_K}f\right\Vert_{\omega_f}\right)\left\Vert R_e\right\Vert_{\omega_f} \notag \\& \qquad+ \nu \left\Vert \nabla(u-u_{\FE})\right\Vert_{\omega_f} \left\Vert \nabla R_f \right\Vert_{\omega_f} + \left\Vert \varrho-\varrho_{\FE} \right\Vert_{\omega_f} \left\Vert \nabla \cdot R_e\right\Vert_{\omega_f}.\end{aligned}$$ We again distinguish two cases. First, if $\alpha>\frac{1}{2}$, we use Lemma \[edge\_inverseq\] and obtain the following upper bounds for $\Vert R_f \Vert_{\omega_f}$ and $ \Vert \nabla R_f \Vert_{\omega_f}$ on face $f$: $$\begin{aligned} \Vert \nabla R_f \Vert^2_{\omega_f} &\leq C\frac{\delta p_K^{(2(2-\alpha))} +\delta^{-1}}{h_K} \left \Vert \left[ \nu\frac{\partial u_{\FE}}{\partial n}\right]\Phi_{\omega_f}^{\frac{\alpha}{2}} \right \Vert_{f}^{2},\\ \Vert R_f \Vert^2_{\omega_f} &\leq C\delta h_K \left \Vert \left[ \nu\frac{\partial u_{\FE}}{\partial n} \right]\Phi_{\omega_{f}}^{\frac{\alpha}{2}} \right \Vert_{f}^{2}.\end{aligned}$$ Knowing that $\Vert \nabla \cdot R_f \Vert_{\omega_f} \leq \Vert \nabla R_f \Vert_{\omega_f}$, estimate (\[eq\_03\]) yields $$\begin{aligned} \left\Vert\left[\nu\frac{\partial u_{\FE}}{\partial n}\right]\Phi_{\omega_f}^{\frac\alpha2}\right\Vert_f &\le C \bigg(\left(\delta h_K\right)^{\frac12}\bigg(\left\Vert I^K_{p_K}f+\nu \Delta u_{\FE}-\nabla \varrho_{\FE}\right\Vert_{\omega_f} \\&\qquad+\left\Vert f- I^K_{p_K}f\right\Vert_{\omega_f}\bigg)\\ & \qquad + \sqrt{\frac{\delta p_K^{2(2-\alpha)}+\delta^{-1}}{h_K}}\bigg(\nu\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_{\omega_f}\\&\qquad\qquad +\left\Vert\varrho-\varrho_{\FE}\right\Vert_{\omega_f}\bigg) \bigg),\end{aligned}$$ and it follows with Lemma \[lemma1\] that $$\begin{aligned} \left\Vert\left[\nu\frac{\partial u_{\FE}}{\partial n}\right]\Phi_{\omega_f}^{\frac\alpha2}\right\Vert_f &\le C\bigg\{\left(\delta h_K\right)^{\frac12}\bigg[\frac{p_K^2}{h_K}\bigg(\nu\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_{\omega_f}\\\qquad&+\left\Vert\varrho-\varrho_{\FE}\right\Vert_{\omega_f}\bigg)+p_K^{\frac12}\left\Vert f-I^K_{p_K}f\right\Vert_{\omega_f}\bigg] \\\qquad& + \sqrt{\frac{\delta p_K^{2(2-\alpha)}+\delta^{-1}}{h_K}}\bigg(\nu\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_{\omega_f} \\&\qquad\qquad +\left\Vert\varrho-\varrho_{\FE}\right\Vert_{\omega_f}\bigg)\bigg\}.\end{aligned}$$ By squaring both sides and summing over all edges $f\in\cal F(K)$, we get $$\label{equ_02} \begin{split} \eta_{\alpha;K;B}^2 \le C\delta & \bigg[p_K^3\left(\nu^2\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_{\omega_K}^2+\left\Vert\varrho- \varrho_{\FE}\right\Vert_{\omega_K}^2\right)+h_K^2\left\Vert f-I^K_{p_K}f\right\Vert_{\omega_K}^2\\ &\quad+ \frac{p_K^{2(2-\alpha)}+\delta^{-2}}{p_K}\left(\nu^2\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_{\omega_K}^2+\left\Vert\varrho-\varrho_{\FE}\right\Vert_{\omega_K}^2\right)\bigg]. \end{split}$$ Setting $\delta:=p_K^{-2}$ gives the desired result. For $0 \leq \alpha \leq \frac{1}{2}$, similar to the proof of Lemma \[lemma1\], we set $\beta:= \frac{1+\alpha}{2}$ and apply Lemma \[inverse\_ineq\] to get $\eta_{\alpha;K;B}\le p_K^{\beta-\alpha}\eta_{\beta;K;B}$. Then, using (\[equ\_02\]) gives $$\begin{aligned} \eta_{\alpha;K;B}^2 &\le C\delta \bigg[p_K^{\frac{7-\alpha}2}\left(\nu^2\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_{\omega_K}^2+\left\Vert\varrho-\varrho_{\FE}\right\Vert_{\omega_K}^2\right) \\&\qquad +\frac{h_K^2}{p_K^{\frac{\alpha-1}2}}\left\Vert f-I^K_{p_K}f\right\Vert_{\omega_K}^2\\ &\qquad+\frac{p_K^{2(2-\alpha)}+\delta^{-2}}{p_K^{\frac{1+\alpha}2}}\left(\nu^2\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert_{\omega_K}^2+\left\Vert\varrho-\varrho_{\FE}\right\Vert_{\omega_K}^2\right) \bigg].\end{aligned}$$ Again setting $\delta:=p_K^{-2}$ concludes the proof. \ Lemmas \[lemma1\] and \[lemma\_02\] combine to yield the desired “efficiency” upper bound for the error estimator $ \eta $ in terms of the quasi-local energy error. \[reli\_effici\_thm\] Let $[u,\varrho]\in\calH$, $\left[u_{\FE},\varrho_{\FE}\right]\in\calV^p(\calT)$, and $\calT$ as in Theorem \[relibi\_error\_est\], and $\alpha\in[0,1]$ be arbitrary. Then, there exists some constant $C_{\text{eff}}> 0$ independent of mesh size vector $h$ and polynomial degree vector $p$ such that $$\begin{split} \eta_{\alpha ;K}^{2}\leq C_{\text{eff}} & \bigg(p_K^k\left(\nu^2\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert^{2}_{\omega_K}+\left\Vert\varrho-\varrho_{\FE}\right\Vert^2_{\omega_K}\right)\\& +\frac{h^2_K}{p_K^{1+\alpha}}\left\Vert I^K_{p_K}f-f\right\Vert_{\omega_K}^2\bigg) \end{split}$$ for all $K\in\calT$, where $k:=\max\left\{2(1-\alpha),\frac{3-\alpha}2\right\}$. By assuming that each cell is only part of a bounded number of cell patches, the efficiency upper bound also holds for the entire estimator $\eta_\alpha$. $hp$-adaptive refinement {#sec:hp-Adaptive Ref} ======================== To define a fully automatic $hp$-adaptive finite element algorithm, we base our approach on the error estimator introduced in Section \[sec: estimator-Err Analysis\]. It consists of the standard adaptive loop $$\text{SOLVE} \longrightarrow \text{ESTIMATE} \longrightarrow \text{MARK} \longrightarrow \text{REFINE}.$$ Of concern in this section is only the marking strategy for the third step (given an estimate of the error as derived previously), for which we follow the ideas of [@Burg2012; @Buerg_Conv]. We then apply either the usual bisection strategy of marked cells for mesh refinement followed by ensuring that there is only one hanging node per edge ($h$ refinement), or increase the polynomial degree (if $p$ refinement is favored). The question in marking is whether to perform $h$- or $p$-refinement. In both cases, one can also ask how exactly a cell is to be subdivided, or by how much the polynomial degree should be increased. Unfortunately, the size of the estimated error $\eta_K$ by itself is not enough to tell us which option is to be preferred. Rather, we should estimate the error one would “expect” after each of these choices, and balance this information against the cost of each choice. Convergence indicators {#sec: Conv-Indicator} ---------------------- Let $j \in \{1, 2, \cdots, n\}$, where $n$ indicates the number of different $h$ and $p$ refinement patterns, and let $K \in \calT_N$ be a cell during the $N$-th cycle of refinement. Following [@Dorfler2007], we define a “convergence indicator” $k_{K,j}\ge 0$ that estimates the error reduction on cell $K$ (relative to the current estimated error $\eta_K$) if $K$ were refined by refinement pattern $j$. For the Stokes problem, similar to [@Ainsworth1997], we generate this estimate by measuring the residual in a norm equivalent to the norm on the dual of $\cal H(\omega_K)$. Let $e:= u - u_{\FE}$ and $E:= \varrho-\varrho_{\FE}$ such that $(e,E) \in \cal H$. Considering the residual of the Stokes problem on the local patch domain $\omega_K$, and notation from $(\ref{bilin})$, then we have for all $(v,q)\in \cal H$: $$\int_{\omega_K} v f -\int_{\omega_K} \nabla v \cdot \nabla u_{\FE} +\int_{\omega_K} (\nabla \cdot v)\varrho_{\FE}+ \int_{\omega_K} q \nabla \cdot u_{\FE} = \mathcal{L}([v,q];[e, E])_{\omega_K}.$$ Integration by parts gives $$\int_{\omega_K} v\left (f+\nu\Delta u_{\FE}-\nabla\varrho_{\FE}\right)-\int_{\omega_K} q\left( \nabla\cdot u_{\FE} \right)= \mathcal{L}([v,q];[e, E])_{\omega_K}.$$ The pair $(w_u ,w_\varrho)\in \cal H$ is defined to be the Ritz projection of the residual, as follows: $$\label{ritz-rep} (\nabla v ,\nabla(w_u))_{\omega_K}+(q , w_\varrho)_{\omega_K} = \mathcal{L}([v,q];[e, E])_{\omega_K},\hspace{20pt} \forall (v,q)\in \cal{H}.$$ Existence and uniqueness of $(w_u, w_{\varrho})$ follows from the continuity of the operators in the definition of the bilinear form in $(\ref{bilin})$. The energy norm of the error can then be defined as $$\label{norm-equality} {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (e, E) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\omega_K}^2 = \Vert \nabla(w_u) \Vert_{\omega_K}^2+ \Vert w_\varrho \Vert_{\omega_K}^2.$$ Of course, this pair of functions can not be found analytically – we need to approximate it by solving a discrete problem for $(w_u^j,w_\rho^j)$ using either a finer mesh, or a finite element space with a higher polynomial degree – i.e., one of the choices $j$ for refinement. For cell $K$ refined by pattern $j$, we combine the idea of the convergence estimator in [@Dorfler2007] and the above discussion on the Ritz representation of the residual and define $$\label{conv-est} k_{K,j}= \frac{1}{\eta_{K}(u_{\FE}, \varrho_{\FE})} \left(\left\Vert \nabla w_u^{j} \right\Vert^2_{\omega_K}+ \left\Vert w_{\varrho}^{j} \right\Vert^2_{\omega_K} \right)^{\frac{1}{2}}.$$ The convergence estimator $k_{K,j}$ as defined in $(\ref{conv-est})$ indicates which refinement pattern $j$ provides the biggest error reduction on every cell. In order to choose the most efficient refinement pattern, we need to balance this reduction against a workload number $\varpi_{K,j} > 0$ that indicates the work required to achieve the error reduction $k_{K,j}$ on cell $K$. This workload number can be defined in a variety of ways; here, we take it as the number of degrees of freedom in the local finite element space, i.e., $\varpi_{K,j} = \text{dim}\;\calV ^{p}_{K, j}(\calT_N\rvert_{ \omega_K})$. For each cell $K$, we then define $j_K$ to be that refinement strategy that maximizes the expected (normalized) relative error reduction, i.e., $j_K = \arg\max_{j \in \{1, 2, \cdots, n \}} \frac{k_{K,j}}{\varpi_{K,j}}$. For the purpose of this work, we only consider two refinement patterns, $j \in \{1, 2\}$, namely isotropic $h$-refinement, and $p$-refinement by increasing the polynomial degree by one, but the strategy above is clearly applicable also to more general choices. Marking {#sec:mark-primal-hp} ------- We still have to decide *which* cells should be refined using the strategies $j_K$ defined above. To this end, we seek that set $\mathcal{M}\subseteq \mathcal{T}$ of minimal cardinality so that $$\label{constraint} \sum_{K\in \cal M} k^2_{K, j_K} \eta^2_{K} \geq \theta^2 \eta^2.$$ We solve this problem approximately using a greedy strategy, i.e., using Dörfler marking. It is known, see [@Dorfler2007], that such an $\mathcal M$ exists if $\theta$ is chosen small enough. Numerical results {#sec: Numerical Results} ================= Our numerical verification of the algorithms proposed above are implemented using the software library deal.II [@dealii; @dealII85]. In particular, we will keep track of the estimated error and demonstrate that it decreases with the same asymptotic rate as the actual error in the energy norm on a sequence of non-uniform, $hp$-adaptively refined meshes. The *effectivity index* $I_\text{eff}$ then measures the quality of the estimator $\eta$: $$\label{eff-Index} I_\text{eff} := \frac{\text{error estimator}}{\text{energy error}} = \frac{\eta\left(u_{\FE},\varrho_{\FE},f\right)}{\bigg(\left\Vert\nabla\left(u-u_{\FE}\right)\right\Vert^2_\Omega+\left\Vert\varrho-\varrho_{\FE}\right\Vert^2_\Omega\bigg)^{1/2}}.$$ Ideally, one would want to have $I_\text{eff} =1$ as $h \rightarrow 0$; however, the equivalence of $\eta$ and the error in Section \[sec: Error Est\] has only been shown up to unknown constants, and consequently in practice we will be content if $C_1 \leqslant I_\text{eff} \leqslant C_2$ for some $C_1 , C_2 > 0$. Example 1 {#sec: Example-1} --------- Let us consider a domain $ \Omega= (-1,1)^2 \;\backslash\; ([0,1]\times [-1,0]) \subset \IR^2 $ shaped like an “L”, and choose the right hand side $f$ as well as inhomogeneous Dirichlet boundary conditions for $u$ so that the solution of the Stokes equations equals the smooth functions $$u= \left[ \begin{matrix} -e^{x} (y \cos(y) +\sin(y)) \\ e^x y \sin(y) \end{matrix} \right], \qquad \varrho= 2e^x \sin(y) - \frac 23 (1-e) (\cos(1)-1)).$$ In the following experiment, we start with a triangulation $\mathcal{T}_0$ consisting of 12 uniform cells, and initially choose $\calQ_3^2\times \calQ_2$ elements on all cells. We then start the adaptive mesh iteration as discussed previously with $\theta=0.75$. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Example 1. Left: Final mesh after 11 $hp$-adaptive refinement steps, with color indicating the polynomial degrees. Right: Final mesh after 7 $h$-adaptive refinement steps. Both meshes have approximately the same number of degrees of freedom.[]{data-label="h_and-hp-ref"}](Ex1-hp-fedeg "fig:"){width="0.45\linewidth"} ![Example 1. Left: Final mesh after 11 $hp$-adaptive refinement steps, with color indicating the polynomial degrees. Right: Final mesh after 7 $h$-adaptive refinement steps. Both meshes have approximately the same number of degrees of freedom.[]{data-label="h_and-hp-ref"}](Ex1-h-ref "fig:"){width="0.45\linewidth"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Fig. \[h\_and-hp-ref\] shows meshes after a number of cycles if $hp$-refinement is allowed, or if we only do $h$-refinement. Unsurprisingly, and confirming expectations, given the smooth nature of the exact solution, the $hp$-adaptive strategy consistently chooses $p$-refinement. Fig. \[error-errorest-eff-index\] presents the decay of the energy error and the a posteriori error estimator as a function of number of degrees of freedom. The graph both demonstrates the exponential convergence rate, and also that the $hp$-error estimator is a sharp upper bound for the energy error – validating this as an efficient and reliable a posteriori error estimator. We observe from the effectivity index graph in Fig. \[error-errorest-eff-index\] that the $I_\text{eff}$ remains bounded in the range $5.4 \leqslant I_\text{eff} \leqslant 8.1$. The figure’s right panel also shows a comparison of errors for $h$- and $hp$-adaptive refinement strategies. This plot clearly shows the superiority of $hp$-AFEM over the $h$-AFEM. ![Example 1. Left: Comparison of the energy error and the error estimator for an $hp$-adaptive computation. Center: Effectivity indices $I_\text{eff}$ for the same computation. Right: A comparison of the errors for this computation with the errors obtained through pure $h$-refinement.[]{data-label="error-errorest-eff-index"}](Ex1-Err-EssEst "fig:"){width="0.32\linewidth"} ![Example 1. Left: Comparison of the energy error and the error estimator for an $hp$-adaptive computation. Center: Effectivity indices $I_\text{eff}$ for the same computation. Right: A comparison of the errors for this computation with the errors obtained through pure $h$-refinement.[]{data-label="error-errorest-eff-index"}](Ex1-EffIndex-edited "fig:"){width="0.32\linewidth"} ![Example 1. Left: Comparison of the energy error and the error estimator for an $hp$-adaptive computation. Center: Effectivity indices $I_\text{eff}$ for the same computation. Right: A comparison of the errors for this computation with the errors obtained through pure $h$-refinement.[]{data-label="error-errorest-eff-index"}](Ex1-h-hp-comparison "fig:"){width="0.32\linewidth"} Example 2 {#sec: Example-2} --------- On the same $L$-shaped domain, we now choose right hand side and boundary values so that we reproduce the singular solution of [@Dauge1989], which reads in polar coordinates as $$\begin{aligned} u(r, \varphi) &= r^{\alpha} \left[ \begin{matrix} \cos(\varphi)\psi^{'}(\varphi)+(1+\alpha)\sin(\varphi)\psi(\varphi)\\ \sin(\varphi)\psi^{'}(\varphi)-(1-\alpha)\cos(\varphi) \psi(\varphi) \end{matrix} \right ], \\ \varrho(r, \varphi) &= -r^{\alpha-1} \frac{(1+\alpha)^2 \psi^{'}(\varphi)+ \psi^{'''}(\phi) }{1-\alpha},\end{aligned}$$ where $$\begin{gathered} \psi(\varphi) = \frac{\sin((1+\alpha)\varphi) \cos(\alpha \omega)}{1+\alpha} -\cos((1+\alpha)\varphi) \\ -\frac{\sin((1-\alpha)\varphi) \cos(\alpha\omega)}{1-\alpha} + \cos((1-\alpha)\varphi), \end{gathered}$$ and $\omega= \frac{3 \pi}{2}$. Here $\alpha$ is the smallest positive solution of $ \sin(\alpha\omega) + \alpha \sin(\omega)=0 $ and is $\alpha\approx 0.54448373678246$. We choose the same initial triangulation, but this time start with $\calQ_2^2\times \calQ_1$ elements on all cells. We use $\theta=0.85$. Fig. \[Ex-2-h\_and-hp-ref\] again shows $hp$- and $h$-adaptively refined meshes generated by our error estimator. The corner singularity in the solution is apparent. A comparison of error and error estimator, efficiency indices, and a comparison between $hp$- and $h$-adaptively refinement strategies is shown in Fig. \[error-errorest-eff-index-Ex2\]. In particular, the efficiency indices again remain bounded. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Example 2. Left: Final mesh after 10 $hp$-adaptive refinement steps, with color indicating the polynomial degrees. Right: Final mesh after 12 $h$-adaptive refinement steps. Both meshes have approximately the same number of degrees of freedom.[]{data-label="Ex-2-h_and-hp-ref"}](Ex2-hp-ref "fig:"){width="0.45\linewidth"} ![Example 2. Left: Final mesh after 10 $hp$-adaptive refinement steps, with color indicating the polynomial degrees. Right: Final mesh after 12 $h$-adaptive refinement steps. Both meshes have approximately the same number of degrees of freedom.[]{data-label="Ex-2-h_and-hp-ref"}](Ex2-h-ref-2 "fig:"){width="0.45\linewidth"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Example 2. Left: Comparison of the energy error and the error estimator for an $hp$-adaptive computation. Center: Effectivity indices $I_\text{eff}$ for the same computation. Right: A comparison of the errors for this computation with the errors obtained through pure $h$-refinement.[]{data-label="error-errorest-eff-index-Ex2"}](Ex2-hp-Err-ErrEst "fig:"){width="0.32\linewidth"} ![Example 2. Left: Comparison of the energy error and the error estimator for an $hp$-adaptive computation. Center: Effectivity indices $I_\text{eff}$ for the same computation. Right: A comparison of the errors for this computation with the errors obtained through pure $h$-refinement.[]{data-label="error-errorest-eff-index-Ex2"}](Ex2-EffIndex-edited "fig:"){width="0.32\linewidth"} ![Example 2. Left: Comparison of the energy error and the error estimator for an $hp$-adaptive computation. Center: Effectivity indices $I_\text{eff}$ for the same computation. Right: A comparison of the errors for this computation with the errors obtained through pure $h$-refinement.[]{data-label="error-errorest-eff-index-Ex2"}](h-hp-convplot-Ex2 "fig:"){width="0.32\linewidth"} Example 3 {#sec: Example-4} --------- As our last example, we consider a less contrived flow field of a fluid moving through a pipe with a bend. The exact solution is here not known, but the solution on a very fine grid is shown in Fig. \[bend-exact-solu-velocity\]. For this case, we prescribe homogeneous Dirichlet boundary condition on the sides of the pipe; for the inlet and outlet, we prescribe parabolic velocity boundary conditions. The adaptive algorithms uses $\theta=0.75$ and starts with 28 equally sized cells. The meshes generated by $h$-adaptive refinement are shown in Figure \[bend-exact-solu-velocity\]. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![Example 3. Left: Pressure field and velocity vectors on a fine mesh. Right: The mesh after 12 $h$-adaptive refinement steps.[]{data-label="bend-exact-solu-velocity"}](Exact-bend-Solu-Primal-velocity "fig:"){width="0.45\linewidth"} ![Example 3. Left: Pressure field and velocity vectors on a fine mesh. Right: The mesh after 12 $h$-adaptive refinement steps.[]{data-label="bend-exact-solu-velocity"}](bend-h-energyEst-13 "fig:"){width="0.45\linewidth"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ A comparison between the $h$- and $hp$-adaptively generated meshes in Fig.s \[bend-exact-solu-velocity\] and \[bend-hp-mesh\] shows the expected pattern of h-refinement where the solution is not smooth, and $p$ refinement (if allowed) where the solution is smooth. Because the exact solution is not known, it is not possible to compare the exact errors for these two strategies; however, having established the quality of our error estimator in the previous example, we can compare how quickly the error estimates are reduced for both strategies, with results shown in Fig. \[bend-compare-h-hp\] – clearly showing the superiority of $hp$ refinement. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![Example 3. Mesh generated after 16 $hp$-adaptive steps, where the color bar indicates the polynomial degrees[]{data-label="bend-hp-mesh"}](Bend-hp-primal-15 "fig:"){width="0.75\linewidth"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Example 3. Comparison of the energy error estimator with $h$- and $hp$-adaptive mesh refinement.[]{data-label="bend-compare-h-hp"}](Bend-h-hp-comparison "fig:"){width="0.65\linewidth"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Conclusion {#sec: Conclusion} ========== In the spirit of previous work by Melenk on other equations (see [@Melenk2005; @Melenk2001]), we have here introduced a residual-based a posteriori error estimator for the Stokes problem for continuous, $hp$-adaptive finite element methods (AFEM). In particular, we have introduced a family $\eta_{\alpha}, \alpha \in [0,1]$ of residual based error estimators. We then proved upper and lower bounds for the estimators applied to the Stokes problems. We were inspired by Dörfler and Heuveline’s work [@Dorfler2007] for one-dimensional problems and later work on higher space dimensions by Bürg [@Buerg_Conv], and introduced an $hp$-adaptive refinement algorithm for our application. In order to decide which refinement gives the best possible $hp$-refinement, in terms of the largest error reduction, we solve local patch problems in parallel for each individual cell. The numerical examples demonstrate the exponential convergence rate for $hp$-AFEM in comparison with $h$-AFEM. They also show the efficiency and reliability of the estimator with respect to the norm of the exact error. Acknowledgements ================ This material is based upon work supported by the U.S. Department of Energy, Office of Science, under contract number DE-AC05-00OR22725. AG and WB’s work was supported by the National Science Foundation under award OCI-1148116 as part of the Software Infrastructure for Sustained Innovation (SI2) program. WB was also supported by the Computational Infrastructure in Geodynamics initiative (CIG), through the National Science Foundation under Awards No. EAR-0949446 and EAR-1550901, administered by The University of California – Davis. [^1]: Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA. Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1864, USA. \# Computational Engineering and Energy Sciences Group, Computational Sciences and Engineering Division, Oak Ridge National Laboratory, 1 Bethel Valley Rd, TN 37831, USA. Emails: [aghesmati@math.tamu.edu](aghesmati@math.tamu.edu), [bangerth@colostate.edu](bangerth@colostate.edu) [turcksinbr@ornl.gov](turcksinbr@ornl.gov), [^2]: This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

--- abstract: 'We introduce a new family of shift spaces — the subordinate shifts. Using subordinate shifts we prove in an elementary way that for every nonnegative real number $t$ there is a shift space with entropy $t$.' author: - 'Marcin Kulczycki, Dominik Kwietniak, and Jian Li' title: Entropy of subordinate shift spaces --- Introduction. ============= Let ${\mathscr{A}}$ be a finite collection of *symbols* which we call an *alphabet*. Typical choices are ${\mathscr{A}}=\{0,1,\ldots,r-1\}$ for some integer $r$ or a set of Roman letters, for example ${\mathscr{A}}=\{a,b,c,d\}$. The *full shift* over ${\mathscr{A}}$ is the set ${\mathscr{A}}^{\mathbb{N}}$ of all infinite sequences of symbols. A *block* is any finite sequence of symbols. A *shift space* over ${\mathscr{A}}$ is a subset of ${\mathscr{A}}^{\mathbb{N}}$ defined by establishing some constraints on blocks, which are allowed to appear as subsequences. For example, the set of all infinite sequences of $0$’s and $1$’s that do not contain the block $11$ is a shift space over $\{0,1\}$. A shift space becomes a dynamical system when we equip it with the *shift map*, which discards the first element of a sequence and shifts the remaining elements by one place to the left. The branch of mathematics concerning the study of shift spaces is known as *symbolic dynamics*. Shift spaces are mathematical models for digitized information, arising frequently as a result of the discretization of dynamical processes. For example, imagine a point moving along a trajectory in space. Partition the space into finitely many pieces and assign a symbol to each piece. Now write down the sequence of symbols labelling the successive partition elements visited by the point as it follows the trajectory. We have encoded information about the trajectory in a sequence from the shift space over the set of labels of the partition cells. As is often the case in mathematics, there is a notion of isomorphism for shift spaces known as *conjugacy* showing us when two seemingly different shift spaces are practically the same. In other words, conjugate shift spaces can be treated as two instances of the same underlying object (for more details we refer the reader to [@LM95 Def. 1.5.9]). For example, there is neither loss nor gain of information when the full $\{0,1\}$-shift is transformed into the full $\{4,7\}$-shift by switching every $0$ to $4$ and every $1$ to $7$. A common problem in symbolic dynamics is to decide whether two shift spaces are conjugate or not. Various invariants are devised to help us distinguish shift spaces which are *not* conjugate. A property of a shift space is a *conjugacy invariant* if whenever a shift space $X$ possesses that property, then every shift space conjugate to $X$ also possesses that property. Among the most important invariants is *entropy*, which is a measure of the complexity of a shift space. Entropy is a nonnegative real number equal to the asymptotic growth rate of the number of blocks that occur in a shift space. Entropy is a conjugacy invariant [@LM95 Cor. 4.1.10]. Therefore, if two shift spaces have different entropy, then they have different structure — they are not conjugate. For example, the full shift over $\{0,1\}$ is not conjugate to the full shift over $\{a,b,c\}$, as they have different entropies. There is one problem with this useful tool: computing the entropy of an arbitrary shift space is often a difficult or even a hopeless task. For example, it is known that for every nonnegative real number $t$ there is a shift space with entropy $t$, but this result is a part of nontrivial theory (see for example [@Walters pp. 178–9]). We will show, however, that this can be proven in an elementary way. To this end, we will define a new class of shift spaces — the subordinate shifts — for some of which the calculation of entropy is straightforward and requires only basic combinatorics. Notation and definitions. ------------------------- Throughout this paper, the symbol ${\mathbb{N}}$ denotes the set of *positive* integers. We denote the number of elements of a finite set $A$ by $|A|$. Given any real number $x$, by $\lfloor x\rfloor$ we mean the largest integer not greater than $x$. Recall that a sequence of real numbers $\{a_n\}_{n=1}^\infty$ is *subadditive* if $a_{m+n}\leq a_m+a_n$ for all $m,n\in{\mathbb{N}}$. Let $\{a_n\}_{n=1}^\infty$ be a subadditive sequence of nonnegative real numbers. Then the sequence $\{a_n/n\}_{n=1}^\infty$ converges to a limit equal to the infimum of the terms of this sequence, that is $$\lim_{n\to\infty}\frac{a_n}{n} = \inf_{n\in{\mathbb{N}}}\frac{a_n}{n}.$$ We follow the notation of Lind and Marcus [@LM95] as close as possible. One notable exception is that we consider only one-sided shifts, while Lind and Marcus consider two-sided (invertible) shifts throughout most of their book. Let ${\mathscr{A}}$ be a finite set, which we call the *alphabet*. We refer to the elements of ${\mathscr{A}}$ as *symbols*. The *full ${\mathscr{A}}$-shift* is the collection of all infinite sequences of symbols from ${\mathscr{A}}$. The full ${\mathscr{A}}$-shift is denoted by $${\mathscr{A}}^{\mathbb{N}}=\{x=(x_i)_{i=1}^\infty : x_i\in{\mathscr{A}}\text{ for all } i\in{\mathbb{N}}\}.$$ We usually write an element of ${\mathscr{A}}^{\mathbb{N}}$ as $x=(x_i)_{i=1}^\infty=x_1x_2x_3\ldots$. Often we identify a finite set ${\mathscr{A}}$ such that $|{\mathscr{A}}|=r$ with $\{0,1,\ldots,r-1\}$. A *full $r$-shift* is then the full shift over the alphabet $\{0,1,\ldots,r-1\}$ and a *full binary shift* is the full $2$-shift. A *block over ${\mathscr{A}}$* is a finite sequence of symbols from ${\mathscr{A}}$. We write blocks without separating their symbols, so a block over ${\mathscr{A}}=\{0,1,2\}$ might look like $01220120$. The *length of a block $u$* is the number of symbols it contains. An *$n$-block* stands for a block of length $n$. We identify a symbol with the $1$-block consisting of this symbol. An *empty block* is the unique block with no symbols and length zero that we denote ${\varepsilon}$. The set of all blocks over ${\mathscr{A}}$ (including ${\varepsilon}$) is denoted by ${\mathscr{A}}^*$. A *concatenation* of two blocks $u=a_1\ldots a_k$ and $v=b_1\ldots b_l$ is the block $uv$ obtained by writing $u$ first and then $v$, that is, $uv=a_1\ldots a_k b_1\ldots b_l$. The concatenation is an associative operation, because $(uv)w=u(vw)$ for any blocks $u,v,w\in{\mathscr{A}}^*$. For this reason we may write $uvw$, or indeed concatenate any sequence of blocks (finite or not) without ambiguity. If $n\ge 1$, then $u^n$ stands for the concatenation of $n$ copies of $u$. Given a nonempty block $u\in{\mathscr{A}}^*$ we denote by $u^\infty$ the sequence $uuu\ldots\in{\mathscr{A}}^{\mathbb{N}}$. Let $x=(x_i)_{i=1}^\infty\in{\mathscr{A}}^{\mathbb{N}}$ and let $1\le i \le j$ be integers. We write $x_{[i,j]}=x_ix_{i+1}\ldots x_j$ for the block of symbols in $x$ starting from the $i$-th and ending at the $j$-th position. We say that a block $w\in A^*$ *occurs in $x$* and $x$ *contains* $w$ if $w=x_{[i,j]}$ for some integers $1\le i \le j$. Note that ${\varepsilon}$ occurs in every sequence from ${\mathscr{A}}^{\mathbb{N}}$. Similarly, given an $n$-block $w=w_1\ldots w_n \in{\mathscr{A}}^*$ we define $w_{[i,j]}=w_iw_{i+1}\ldots w_j\in{\mathscr{A}}^*$ for each $1\le i\le j\le n$. A *prefix* of a block $z\in{\mathscr{A}}^*$ is a block $u$ such that $z=uv$ for some $v\in{\mathscr{A}}^*$. For every ${\mathscr{A}}$ we define the *shift map* $\sigma\colon{\mathscr{A}}^{\mathbb{N}}\rightarrow{\mathscr{A}}^{\mathbb{N}}$. It maps a sequence $x=(x_i)_{i=1}^\infty$ to the sequence $\sigma(x)=(x_{i+1})_{i=1}^\infty$. Equivalently, $\sigma(x)$ is the sequence obtained by dropping the first symbol of $x$ and moving the remaining symbols by one position to the left. Given any collection ${\mathscr{F}}$ of blocks over ${\mathscr{A}}$ (i.e., a subset of ${\mathscr{A}}^*$) we define a *shift space specified by ${\mathscr{F}}$*, denoted by $X_{\mathscr{F}}$, as the set of all sequences from ${\mathscr{A}}^{\mathbb{N}}$ which do not contain any blocks from ${\mathscr{F}}$. We say that ${\mathscr{F}}$ is a collection of *forbidden blocks for $X_{\mathscr{F}}$* as blocks from ${\mathscr{F}}$ are forbidden to occur in $X_{\mathscr{F}}$. A *shift space* is a set $X\subset{\mathscr{A}}^{\mathbb{N}}$ such that $X=X_{\mathscr{F}}$ for some ${\mathscr{F}}\subset{\mathscr{A}}^*$. A *binary shift space* is a shift space over the alphabet $\{0,1\}$. Show that for every shift space $X$ we have $\sigma(X)\subset X$. Find a shift space $X$ for which $\sigma(X)\neq X$. With each shift space $X$ over ${\mathscr{A}}$, we may associate a set of blocks over ${\mathscr{A}}$ which occur in some sequence $x\in X$. We call this set the *language of $X$* and denote it by ${\mathscr{B}}(X)$. We write ${\mathscr{B}}_n(X)$ for the set of all $n$-blocks contained in ${\mathscr{B}}(X)$. Show that if $\mathcal{L}$ is a language of some shift space over ${\mathscr{A}}$, then $\mathcal{L}$ is: 1. *factorial*, meaning that if $u\in\mathcal{L}$ and $u=vw$ for some blocks $v,w\in {\mathscr{A}}^*$, then both $v$ and $w$ also belong to $\mathcal{L}$, 2. *prolongable*, meaning that for every block $u$ in $\mathcal{L}$ there is a symbol $a\in {\mathscr{A}}$ such that $ua$ also belongs to $\mathcal{L}$. Actually, the converse is also true. Given a factorial and prolongable subset $\mathcal{L}\subset{\mathscr{A}}^*$ there is a shift space $X$ such that $\mathcal{L}$ is the language of $X$. A collection of forbidden blocks defining $X$ is ${\mathscr{F}}={\mathscr{A}}^*\setminus\mathcal{L}$. We can also characterize points in a shift space $X$. \[lem:char\] Let $\mathcal{L}\subset{\mathscr{A}}^*$ be factorial and prolongable. Let $X$ be a shift space such that $\mathcal{L}={\mathscr{B}}(X)$. Then a point $x\in {\mathscr{A}}^{\mathbb{N}}$ is in $X$ if and only if $x_{[i,j]}\in\mathcal{L}$ for all $i,j\in{\mathbb{N}}$ with $ i < j$. Shift spaces are determined by their language. In other words, two shift spaces are equal if and only if they have the same language [@LM95 Proposition 1.3.4]. Hence there is a one-to-one correspondence between shift spaces over ${\mathscr{A}}$ and factorial, prolongable subsets of ${\mathscr{A}}^*$. To define a shift space, one can either specify its set of forbidden blocks or its language. Let $x\in{\mathscr{A}}^{\mathbb{N}}$. Let ${\mathscr{B}}_x$ be the collection of all blocks occurring in $x$. Since ${\mathscr{B}}_x$ is a factorial and prolongable language, it defines a shift space, which we denote by $\Sigma_x$. Let $S\subset{\mathbb{N}}\cup\{0\}$. We define $${\mathscr{F}}_S=\{1\underbrace{0\ldots 0}_p1:p\notin S\}.$$ The binary shift defined by forbidding blocks from ${\mathscr{F}}_S$ is called the *$S$-gap shift* and is denoted by $X_S$. In particular, we call $X_{\mathbb{N}}$ the *golden mean shift* (a sequence belongs to it if and only if it does not contain the block $11$). The following fact merely states that we may construct inductively an *infinite* sequence starting from an infinite collection of *finite* sequences such that each sequence in it coincides with any shorter one as long as both are defined. \[lem:growing-words\] Let $\{w^{(n)}\}_{n=1}^\infty$ be a sequence in ${\mathscr{A}}^*$ and for each $n\in{\mathbb{N}}$ let $l(n)$ be the length of $w^{(n)}$. If for each $k\in{\mathbb{N}}$ the block $w^{(k)}$ is a prefix of $w^{(k+1)}$, then there is a point $x\in{\mathscr{A}}^{\mathbb{N}}$ such that for each $n\in{\mathbb{N}}$ we have $x_{[1,l(n)]}=w^{(n)}$. Moreover, if $\lim_{n\rightarrow\infty} l(n)=\infty$, then $x$ is unique. We use superscripts in brackets as above to denote indices for sequences of blocks. That is, we write $\{w^{(n)}\}_{n=1}^\infty$ to denote a sequence of blocks. This way we may reserve subscripts for enumerating symbols within the block: $w^{(n)}=w^{(n)}_1w^{(n)}_2\ldots w^{(n)}_k$. Finally, we are ready to move to the entropy itself. In full generality, this concept was defined by Adler, Konheim and McAndrew [@AKM] for an arbitrary compact topological space $X$ and a continuous map $f\colon X\to X$. The definition below applies only to shift spaces but the resulting number is equal to the Adler, Konheim and McAndrew entropy of the shift treated as a dynamical system (see [@LM95 Exercise 6.3.8]). Let $\log$ denote the logarithm to base $2$ (choosing a different base would also yield a valid definition; it would change the value of entropy only by a multiplicative constant). Let $X\subset{\mathscr{A}}^{\mathbb{N}}$ be a nonempty shift space and let $m,n\in{\mathbb{N}}$. Observe that every block $w\in{\mathscr{B}}_{m+n}(X)$ can be written in a unique way as a concatenation $w=uv$, where $u\in {\mathscr{B}}_m(X)$ and $v\in {\mathscr{B}}_n(X)$. Therefore $|{\mathscr{B}}_{m+n}(X)|\le|{\mathscr{B}}_{m}(X)|\cdot |{\mathscr{B}}_{n}(X)|$, and hence $$\log |{\mathscr{B}}_{m+n}(X)|\le \log|{\mathscr{B}}_{m}(X)|+\log |{\mathscr{B}}_{n}(X)|.$$ By applying Fekete’s Lemma to the nonnegative sequence $\log|{\mathscr{B}}_n(X)|$ we may now define the *entropy of $X$*, denoted by $h(X)$, as $$h(X)=\lim_{n\to\infty} \frac{1}{n}\log |{\mathscr{B}}_n(X)|=\inf_{n\ge 1} \frac{1}{n}\log |{\mathscr{B}}_n(X) |.$$ Roughly speaking, the entropy measures the complexity of a shift space $X$ in terms of the asymptotic growth rate of the number of $n$-blocks that appear in the language $X$. In other words, the number of $n$-blocks in a shift space of entropy $h\ge 0$ roughly equals $2^{nh}$. Every finite shift space has entropy zero. The full ${\mathscr{A}}$-shift has entropy $\log|{\mathscr{A}}|$. In particular, the full $r$-shift has entropy $\log r$. Observe that if $X,Y\subset{\mathscr{A}}^{\mathbb{N}}$ and $X\subset Y$, then $h(X)\leq h(Y)$. Since a shift space over ${\mathscr{A}}$ is a subset of the full ${\mathscr{A}}$-shift, we may conclude that the entropy of any shift space over ${\mathscr{A}}$ is a nonnegative real number bounded above by $\log|{\mathscr{A}}|$ (for systems that are not shifts it may well be infinite — see [@ALM Example 4.2.6]). As mentioned in the introduction, computing the entropy of a shift space is a hard problem — try, for example to verify straight from the definition that the entropy of the golden mean shift is $\log((1+\sqrt{5})/2)$. There are (relatively rare) families of shift spaces (e.g. shifts of finite type, see [@LM95]) for which we can actually provide a (theoretically) computable formula for entropy. Even in these special cases, one needs to apply some non-trivial tools. For the sake of illustration we recall some results from [@LM95 Exercise 4.3.7]. Let $S\subset{\mathbb{N}}\cup\{0\}$ and let $X_S$ be the associated $S$-gap shift. Then $h(X_S)=\log\lambda$, where $\lambda$ is the unique positive solution of the equation $$\sum_{j\in S} x^{-j-1}=1.$$ For every $t\in[0,1]$ there is a set $S\subset{\mathbb{N}}\cup\{0\}$ ($S$ depends on $t$) such that $h(X_S)=t$. Subordinate shifts. =================== The goal of this section is to introduce a family of shift spaces with easily calculable entropy; a family rich enough to contain a shift space with every possible nonnegative entropy. For the remainder of the paper, we fix ${\mathscr{A}}=\{0,1,\ldots, r-1 \}$. We say that a block $w=w_1\ldots w_k\in{\mathscr{A}}^*$ *dominates* a block $v=v_1\ldots v_k\in{\mathscr{A}}^*$ if $v_i\le w_i$ for $i=1,\ldots,k$. In an analogous way we define when one sequence from ${\mathscr{A}}^{\mathbb{N}}$ dominates another. A *subordinate of $\mathcal{L}\subset {\mathscr{A}}^*$* is the set $\mathcal{L}^\le$ of all blocks over ${\mathscr{A}}$ that are dominated by some block in $\mathcal{L}$. Observe that if $\mathcal{L}$ is factorial and prolongable, then the same holds for $\mathcal{L}^\le$. In particular, given a point $x\in{\mathscr{A}}^{\mathbb{N}}$, we may define a *subordinate shift of $x$*, denoted by $X^{\le x}$, as a shift space given by the language ${\mathscr{B}}^\le_x$, where ${\mathscr{B}}_x$ is the language of blocks occurring in $x$. All binary blocks of length $3$ are dominated by $111$. The blocks $0000$, $0001$, $0100$, and $0101$ are the only blocks dominated by $0101$. Subordinate shifts are *hereditary* (this is a notion introduced in [@KerrLi] and examined in [@Kwietniak]). It can be shown that a hereditary shift is subordinate if and only if it is irreducible in the sense of [@LM95 Definition 1.3.6]. It turns out that a shift space that has been recently extensively studied is an example of a subordinate shift. Recall that a positive integer $n$ is *square-free* if there is no prime number $p$ such that $p^2$ divides $n$. Let $\eta$ be a point in $\{0,1\}^{\mathbb{N}}$ given by $$\eta_n=\begin{cases} 1&\text{ if $n$ is square-free,}\\ 0& \text{ otherwise.} \end{cases}$$ In other words, $\eta_n=(\mu(n))^2$, where $\mu\colon{\mathbb{N}}\to{\mathbb{N}}$ is the famous Möbius function. It can be shown that $S=X^{\le \eta}$ is the *square-free flow*; that is a shift space, whose structure is strongly tied to the statistical properties of square-free numbers. For more details see [@Peckner; @Sarnak]. The study of the square-free flow has been recently extended to the more general context of $\mathcal{B}$-free integers; that is to say integers with no factor in a given family $\mathcal{B}$ of pairwise relatively prime integers, the sum of whose reciprocals is finite, see [@B-free-dynamical; @B-free-measures]. We aim to show that if given $t\in [0,1]$, then we are able to choose a point $x(t)$ from the full binary shift such that $h(X^{\le x(t)})=t$. First we tackle rational entropies. \[lem:rational-case\] If $w\in\{0,1\}^*$ is a block of length $q$ with $p$ occurrences of the symbol $1$ and $x=w^\infty$, then $h(X^{\le x})=p/q$. Since replacing in $w$ any subset of $1$’s with $0$’s leads to a block dominated by $w$ which is in ${\mathscr{B}}(X^{\le x})$, we know that there are at least $2^p$ blocks in ${\mathscr{B}}_q(X^{\le x})$. It follows that $h(X^{\le x})\geq p/q$. On the other hand, the periodicity of $x=w^\infty$ implies that for each $j\in{\mathbb{N}}$ there are at most $q$ different blocks of length $qj$ in ${\mathscr{B}}_{x}$. Each such block dominates exactly $2^{pj}$ blocks from $\{0,1\}^*$. Therefore there are at most $q\cdot 2^{pj}$ blocks in ${\mathscr{B}}_{qj}(X^{\le x})$. Consequently, $$\begin{aligned} h(X^{\le x}) & = \lim_{n\to\infty} \frac{1}{n}\log|{\mathscr{B}}_n(X^{\le x})| = \lim_{j\to\infty} \frac{1}{qj}\log|{\mathscr{B}}_{qj}(X^{\le x})|\\ & \le \lim_{j\to\infty} \frac{1}{qj}\log (q\cdot 2^{pj})=p/q.\qedhere\end{aligned}$$ We now show how to construct a shift space with entropy $\pi/8$. We hope that this example will make the general construction that comes after it much clearer. Let $t=\pi/8=0.3926990816\ldots$. We consider the following sequence of rational approximations of $t$ from the above: $$0.4,\, 0.4,\, 0.393,\, 0.3927,\,0.3927,\,0.3927,\,0.3926991,\,0.39269909,\,\ldots .$$ We obtain our $n$-th approximation by rounding up $t$ to the nearest number with no more than $n$ digits after the decimal point. Let $p_n$ be this $n$-th approximation times $10^n$. We now inductively build a sequence of blocks $\{w^{(n)}\}_{n=1}^\infty$ from $\{0,1\}^*$ such that for every $n\in{\mathbb{N}}$: 1. $w^{(n)}$ has length $10^n$, 2. the symbol $1$ appears exactly $p_n$ times in $w^{(n)}$, 3. $w^{(n)}$ is a prefix of $w^{(n+1)}$, 4. $w^{(n+1)}$ is dominated by $(w^{(n)})^{10}$. We can visualize the construction of $w^{(n+1)}$ as a process with two steps. In the first step we concatenate ten copies of $w^{(n)}$. In the second step we keep the first $p_{n+1}$ occurrences of the symbol $1$ in $w^{(n+1)}$ and replace the rest by $0$’s. In our example, we first put $w^{(1)}=1111000000$. We want the symbol $1$ to appear $p_2=40$ times in $w^{(2)}$, so we define $w^{(2)}=(w^{(1)})^{10}$ — there is no need to remove any $1$’s. Then in $w^{(3)}$ we need to see $393$ appearances of $1$, so we define $w^{(3)}$ as a concatenation of nine copies of $w^{(2)}$ and a block $v$ of length $100$ which agrees with $w^{(2)}$ except that the last seven $1$’s appearing in $w^{(2)}$ are replaced by $0$’s in $v$. The construction continues on inductively. Applying Lemma \[lem:growing-words\] to the sequence of binary blocks $\{w^{(n)}\}_{n=1}^\infty$, we obtain a point $x\in\{0,1\}^{\mathbb{N}}$. The entropy of the subordinate shift $X^{\le x}$ is $t=\pi/8$. The proof of this fact is contained in the general result below. We are now equipped to tackle the main theorem of this paper. \[thm:binary-case\] For every $t\in[0,1]$ there is a binary subordinate shift with entropy $t$. For every $n\in{\mathbb{N}}$ let $s_n$ be the rational approximation of $t$ obtained by rounding up the decimal expansion of $t$ to the nearest number with no more than $n$ digits after the decimal point. Let $p_n=s_n\cdot 10^n$. We now inductively build a sequence of blocks $\{w^{(n)}\}_{n=1}^\infty$ from $\{0,1\}^*$ such that for every $n\in{\mathbb{N}}$: 1. $w^{(n)}$ has length $10^n$, 2. the symbol $1$ appears exactly $p_n$ times in $w^{(n)}$, 3. $w^{(n)}$ is a prefix of $w^{(n+1)}$, 4. $w^{(n+1)}$ is dominated by $(w^{(n)})^{10}$. We start with $$w^{(1)}=\underbrace{1\ldots1}_{p_1}\underbrace{0\ldots 0}_{10-p_1}.$$ Assume now that we have defined $w^{(1)},\ldots,w^{(n)}$ so that the four conditions above are satisfied to the extent to which they apply to $w^{(1)},\ldots,w^{(n)}$. We now concatenate ten copies of $w^{(n)}$. We keep the first $p_{n+1}$ occurrences of the symbol $1$ unaltered in the sequence, replace the rest by $0$, and call the result $w^{(n+1)}$. It is easy to verify that the conditions above now hold to the extent to which they apply to $w^{(1)},\ldots,w^{(n+1)}$, so the inductive construction is complete. Applying Lemma \[lem:growing-words\] to the sequence $\{w^{(n)}\}_{n=1}^\infty$ we obtain a point $x\in\{0,1\}^{\mathbb{N}}$. Observe that for every $n\in{\mathbb{N}}$ the point $x$ is dominated by $(w^{(n)})^\infty$. It follows that $X^{\leq x}\subset X^{\le (w^{(n)})^\infty}$, and so $h(X^{\leq x})\leq h(X^{\le (w^{(n)})^\infty})$. By Lemma \[lem:rational-case\] we have $h(X^{\le (w^{(n)})^\infty})=s_n$, and therefore $$h(X^{\leq x})\leq\lim_{n\to\infty}h(X^{\le (w^{(n)})^\infty})=\lim_{n\to\infty}s_n=t.$$ On the other hand, observe that for every $n\in{\mathbb{N}}$ we have $w^{(n)}\in{\mathscr{B}}_{10^n}(X^{\leq x})$, so all $2^{p_n}$ blocks dominated by $w^{(n)}$ are also in ${\mathscr{B}}_{10^n}(X^{\leq x})$. Therefore $\log |{\mathscr{B}}_{10^n}(X^{\leq x})|\ge \log 2^{p_n} = p_n$, and so $$t=\lim_{n\to\infty}s_n=\lim_{n\to\infty}\frac{p_n}{10^n}\le \lim_{n\to\infty} \frac{1}{10^n}\log |{\mathscr{B}}_{10^n}(X^{\leq x})|=h(X^{\leq x}),$$ which completes the proof of $h(X^{\leq x})=t$. Prove that the shift space $X^{\leq x}$ constructed in the proof of Theorem \[thm:binary-case\] is *irreducible* (see [@LM95 Definition 1.3.6]), meaning that given any pair of blocks $u,v\in{\mathscr{B}}(X^{\leq x})$ there is a block $y\in{\mathscr{B}}(X^{\leq x})$ such that $uyv\in{\mathscr{B}}(X^{\leq x})$. It remains to show that the conclusion of Theorem \[thm:binary-case\] is true for every $t> 1$. For every $t\in (1,\infty)$ there is a shift space with entropy $t$. Let $t\in (1,\infty)$ be given. Pick $k\in{\mathbb{N}}$ and $s\in (0,1)$ such that $ks=t$. Use Theorem \[thm:binary-case\] to obtain a binary subordinate shift $X$ with entropy $s$. Using $X$ we now construct a shift space $Y\subset\{0,1,\ldots,2^k-1\}^{\mathbb{N}}$. We start with $Y=\emptyset$. For every point $w\in X$ we perform the following procedure: 1. express $w$ as a concatenation $w_1w_2\ldots$, where each block $w_n$ has length $k$, 2. for every $w_n$ we determine the number $a_n\in\{0,1,\ldots,2^k-1\}$ such that $w_n$ is the binary notation for $a_n$ (for example, for $k=2$, we get $00\mapsto 0$, $01\mapsto 1$, $10\mapsto 2$, and $11\mapsto 3$), 3. we add the sequence $a_1a_2\ldots$ to $Y$ (note that it is the $a_n$’s that are symbols here, not their digits). It is elementary to check that $Y$ is a shift space. It suffices to analyze how the language of $Y$ is created from the language of $X$. Observe that for every $n\in{\mathbb{N}}$ we have $|{\mathscr{B}}_n(Y)|=|{\mathscr{B}}_{kn}(X)|$. Therefore $$h(Y)=\lim_{n\to\infty}\frac{1}{n}\log |{\mathscr{B}}_n(Y)|=k\cdot\lim_{n\to\infty}\frac{1}{kn}\log |{\mathscr{B}}_{kn}(X)|=k\cdot h(X)=t.\qedhere$$ Acknowledgment. {#acknowledgment. .unnumbered} =============== The authors would like to thank the referees for their thorough and careful work. Preparing this article we asked our students and colleagues to comment on it. We are grateful to: Jakub Byszewski, Vaughn Climenhaga, Jakub Konieczny, Marcin Lara, Simon Lunn, Martha [Ł]{}[a]{}cka, Dariusz Matlak, Samuel Roth, and Maciej Ulas for their remarks and suggestions. The research of Dominik Kwietniak was supported by the National Science Centre (NCN) under grant Maestro 2013/08/A/ST1/00275. The research of Jian Li was supported by Scientific Research Fund of Shantou University (YR13001). [10]{} R. L. Adler, A. G. Konheim, M. H. McAndrew, Topological entropy, [*Trans. Amer. Math. Soc.*]{} [**114**]{} (1965) 309–319. L. Alsedá, J. Llibre, M. Misiurewicz, [*Combinatorial Dynamics and Entropy in Dimension One.*]{} Second edition. World Scientific, River Edge, NJ, 2000. E.-H. El Abdalaoui, M. Lemańczyk, T. De La Rue, A dynamical point of view on the set of $\mathscr{B}$-free integers (2013), available at <http://arxiv.org/abs/1311.3752>. D. Kerr, H. Li, Independence in topological and $C\sp *$-dynamics, [*Math. Ann.*]{} [**338**]{} (2007) 869–926. J. Ku[ł]{}aga-Przymus, M. Lemańczyk, B. Weiss, On invariant measures for $\mathscr{B}$-free systems (2014), available at <http://arxiv.org/abs/1406.3745>. D. Kwietniak, Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts, [*Discrete Contin. Dyn. Syst.*]{} [**33**]{} (2013) 2451–2467. D. Lind, B. Marcus, [*An Introduction to Symbolic Dynamics and Coding.*]{} Cambridge University Press, Cambridge, 1995. R. Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow (2014), available at <http://arxiv.org/abs/1205.2905>. P. Sarnak, Three lectures on the Möbius function, randomness, and dynamics (2011), available at <http://publications.ias.edu/sarnak/paper/512>. P. Walters, [*An Introduction to Ergodic Theory.*]{} Graduate Texts in Mathematics, Vol. 79, Springer-Verlag, New York-Berlin, 1982.

--- abstract: 'The problem of the $\Xi_{c}^{+}$ lifetime is considered in the framework of [*Heavy-Quark Expansion*]{} and $SU(3)_{flavor}$ symmetry. The lifetime of $\Xi_{c}^{+}$ is expressed in terms of measurable inclusive quantities of the other two charmed baryons belonging to the same $SU(3)_{flavor}$ multiplet in a model-independent way. In such a treatment, inclusive decay rates of singly Cabibbo suppressed decay modes have a prominent role. An analogous approach is applied to the multiplet of charmed mesons yielding interesting predictions on $D_{s}^{+}$ properties. The results obtained indicate that a more precise measurement of inclusive decay quantities of some charmed hadrons (such as $\Lambda_{c}^{+}$) that are more amenable to experiment can contribute significantly to our understanding of decay properties of other charmed hadrons (such as $\Xi_{c}^{+}$) where discrepancies or ambiguities exist.' author: - 'B. Guberina' - 'H. Štefančić' title: 'Cabibbo suppressed decays and the $\Xi_{c}^{+}$ lifetime' --- The investigation of inclusive decays and lifetimes of hadrons containing heavy quarks [@review] is already a mature subject with many fruitful applications and numerous significant achievements. The fusion of the [*Operator Product Expansion (OPE)*]{} techniques developed in the nineties [@90] with the phenomenological insights of the eighties [@80] has created a consistent framework known as [*Heavy-Quark Expansion (HQE)*]{}, within which one can systematically treat inclusive decays of heavy quarks and hadrons containing them. A host of experimental data, first on $c$ hadron decays and then, with the advent of $B$ factories, on $b$ decays, have made possible a comparison of experimental and theoretical results and revealed broad agreement with several notable exceptions [^1]. Addressing these discrepancies has become one of the most important tasks in heavy-quark physics, given that data extracted from inclusive weak decays represent an essential input in research of fundamental questions of the Standard Model (such as [*CP*]{} violation) or its extensions. Increasing quantity and quality of experimental data opens new directions in treating inclusive weak decays which may contribute to the resolution of existing problems. Consideration of inclusive weak decay rates of Cabibbo suppressed modes as individual objects (not only as a small correction to inclusive weak decay rates of Cabibbo dominant modes) supported by the application of standard symmetries (such as $SU(3)_{flavor}$ or [*Heavy-Quark Symmetry (HQS)*]{}) traces along one of these directions. As the [*HQE*]{} depends crucially on the heaviness of the decaying heavy quark, the predictions are more reliable in the sector of $b$ hadrons than in the sector of $c$ hadrons. Nevertheless, rather acceptable predictions of lifetime hierarchies and lifetime ratios were obtained in the sector of charmed hadrons too. Furthermore, very reasonable agreement was achieved in the sector of singly-charmed baryons [@GM; @PDG2000]. However, recent measurements of the $\Xi_{c}^{+}$ lifetime by [*FOCUS*]{} [@focus] and [*CLEO*]{} [@cleo] collaborations indicate substantial discrepancy between new experimental data and the presently available theoretical result [@GM; @Bigi]: $$\begin{aligned} \label{eq:exp} \left ( \tau(\Xi_{c}^{+})/\tau(\Lambda_{c}^{+}) \right )_{FOCUS} & = & 2.29 \pm 0.14 \, ,\nonumber \\ \left ( \tau(\Xi_{c}^{+})/\tau(\Lambda_{c}^{+}) \right )_{CLEO} & = & 2.8 \pm 0.3 \, , \nonumber \\ \left ( \tau(\Xi_{c}^{+})/\tau(\Lambda_{c}^{+}) \right )_{th} & \sim & 1.3 \, .\end{aligned}$$ The results displayed above show that there is a difference by a factor of $\sim 2$ between experiment and theory. It is reasonable to pose a question whether the [*HQE*]{} can correctly describe lifetimes of singly-charmed baryons. The new experimental data on the lifetime of $\Xi_{c}^{+}$ are certainly out of reach of the calculations performed so far. However, experimental data for other singly-charmed, weakly-decaying baryons ($\Lambda_{c}^{+}$, $\Xi_{c}^{0}$, and $\Omega_{c}^{0}$) are consistent with theoretical calculations of [@GM]. However, as the data on the lifetimes of $\Xi_{c}^{0}$ and $\Omega_{c}^{0}$ are presently of marginal quality, it is not excluded that future updates of these lifetimes might disturb the agreement in the case of these baryons, too. The theoretical procedure is based on some assumptions (e.g., calculation of four-quark operator matrix elements in a nonrelativistic quark model) that may limit its explanatory power in the case of $\Xi_{c}^{+}$. Therefore, it is justified to investigate if a theoretical procedure based on the [*HQE*]{} with relaxed assumptions of analysis [@GM] can be formulated so that it might explain new experimental results. To this end, one must invoke Cabibbo suppressed modes of decay as a new source of information. Let us begin our analysis with a brief discussion about the inclusive weak decay rate. The principal result of the [*HQE*]{} is the expression for any inclusive weak decay rate of a heavy hadron given as a series of matrix elements of local operators with the inverse mass of the decaying heavy quark as an expansion parameter: $$\begin{aligned} \label{eq:master} \Gamma(H_{Q} \rightarrow f) & = & \frac{G_{F}^{2} m_{Q}^{5}}{192 \pi^{3}} \mid V \mid^{2} \frac{1}{2 M_{H_{Q}}} \\ \nonumber & & \times \left [ \sum_{D=3}^{\infty} c_{D}^{f} \frac{\langle H_{Q} \mid O_{D} \mid H_{Q} \rangle}{m_{Q}^{D-3}} \right ] \, ,\end{aligned}$$ where $D$ denotes the canonical dimension of the local operator $O_{D}$. The coefficients $c_{D}$ are calculated perturbatively (therefore given as a series in $\alpha_{S}$). $V$ stands for a product of [*CKM*]{} matrix elements appearing in a given weak decay mode. For the sake of practical calculations, one has to truncate the series at some point in the series hoping that the disregarded remainder of the series does not contribute significantly to the final result. The quality of such an approximation depends on the magnitude of the expansion parameter, i.e., on the speed of convergence of the series. The underlying hypothesis is that the inclusive hadron decay rates can be described by calculating the inclusive quark decay rates – the [*ansatz*]{} known as quark-hadron duality. The [*ansatz*]{} is not trivially obvious as one can see by inspection of the leading term in (\[eq:master\]). The decay rate is given by $\Gamma^{dec} \sim m_{Q}^{5}$ and this expression has, [*prima facie*]{}, nothing to do with the hadrons in the final states. This is, however, misleading since the summation of hadronic widths of different channels agrees with the widths computed at the quark level [^2]. Another problem stems from the matrix elements appearing in the expansion. They are dominated by nonperturbative dynamics and therefore so far there has been no systematic way of calculating them. The matrix elements of several operators of the lowest dimensions can be determined by applying [*Heavy-Quark Effective Theory (HQET)*]{}, lattice [*QCD*]{}, or, in some cases, extracted from the lepton energy spectra, but the matrix elements of some operators essential for understanding lifetime differences of heavy hadrons (e.g., four-quark operators) are still not generally calculable in such a manner, but one must recourse to quark models, which introduces the undesirable feature of model dependence. A further source of uncertainty is the heavy-quark mass $m_{Q}$. Since in the leading order the inclusive weak decay rate depends on the fifth power of $m_{Q}$, very small uncertainties in the determination of this mass parameter can lead to a significant uncertainty in the inclusive weak decay rate. Finally, using a truncated expression instead of the entire series raises the possibility of violation of quark-hadron duality [@Shifmandual; @BUdual], which emerges as another possible source of contributions beyond the present theoretical control. The [*OPE*]{} was originally formulated for deep Euclidean kinematics and its net effect was to factorize perturbative short-distance physics (Wilson coefficients) from soft, nonperturbative one (matrix elements). On the other hand, the quark-hadron duality is the concept dealing exclusively with Minkowskian dynamics [^3]. It appears that the small corrections that one safely neglects in the Euclidean regime often turn out to be enhanced in the Minkowski regime [@Shifmandual; @BUdual]. The Wilson coefficients themselves are generally not free of nonperturbative (nonlogarithmic) terms. They are generated, e.g., by small-size instantons [@Shifmandual]. Similarly, perturbative corrections appear in the soft physics of matrix elements. Generally, the truncation of the series (\[eq:master\]) in $\alpha_{s}$ and condensate terms is known to be necessary since both series are factorially divergent [@Mueller]. Therefore, a “practical” calculation at any given order $\alpha_{s}^{m}$ and $m_{Q}^{-n}$ will have a “natural uncertainty” coming from the higher-order terms $\alpha_{s}^{m+1}$ and $m_{Q}^{-(n+1)}$. The “natural uncertainty” also includes the ordinary uncertainties like the uncertainties in quark masses, $\Lambda_{QCD}$, etc. The uncertainties beyond this “natural uncertainty” are considered to be violations of quark-hadron duality. Resolutions of the problems stated above presumably lead to the explanation of discrepancies between present experimental and theoretical results. Since the contributions of higher-dimensional operators, uncertainties in matrix elements and $m_{Q}$ as well as effects of duality violation are all intertwined in the full expression for the weak decay rate, it is very difficult to distinguish precisely which of these factors causes the problem and should be improved accordingly. One possible strategy is to eliminate or reduce the importance of all (in practice as many as possible) factors but one in order to test the influence of the remaining factor. In this paper we adopt this strategy and implement it using symmetries in multiplets of heavy hadrons. Investigations along similar lines (connecting the charmed with the beauty sector) were performed in [@Vol; @GMS1; @GMS2]. The standard procedure of truncating the series (\[eq:master\]) is to keep operators of dimensions 3 (decay operator) and 5 (chromomagnetic operator) [^4], which are insensitive to the light-quark content of the heavy hadron (at least on the quark-gluon operator level). Operators of dimension 6, which are sensitive to flavors of light quarks (four-quark operators), also have to be kept in order to describe lifetime differences within multiplets of heavy hadrons. The effects of four-quark operators (clearly presented in [@GM]) are traditionally referred to as W-exchange, positive and negative Pauli interference in baryons, and W-annihilation, W-exchange, and negative Pauli interference in mesons. We shall adopt this procedure along with the assumption of $SU(3)_{flavor}$ symmetry at the level of matrix elements. The validity of this assumption and its influence on the final result will be discussed below. We start by expressing decay rates of individual Cabibbo modes for singly-charmed baryons within the framework that we have set. As already mentioned, operators of dimension 3 and 5 are insensitive to the light-(anti)quark content of a heavy hadron. Nevertheless, their coefficients comprise a phase-space correction coming from the fact that some of the resulting quarks in the decay of a heavy quark have a nonnegligible mass compared with the heavy-quark mass. Thus, contributions of operators of dimensions 3 and 5 have slightly different values in the treatment of various Cabibbo modes of the decay of the heavy quark. In the case of $c$ quark decays, these corrections are generally not large and we shall neglect them in our initial treatment. Their effect will be taken into account in the discussion of our results. The assumption of $SU(3)_{flavor}$ symmetry guarantees that the matrix elements of operators of dimension 3 and 5 are the same for all hadrons in any $SU(3)_{flavor}$ multiplet of heavy hadrons. These approximations allow us to describe the contribution of the aforementioned operators with a single quantity $\Gamma_{35}$ in all Cabibbo modes, for all members of the multiplet, up to the product of the [*CKM*]{} matrix elements specific for every individual Cabibbo mode. The coefficients of four-quark operators also include phase-space corrections owing to the massive particles in the final state of the decay of the heavy quark. In this case, however, these corrections are at the percent level and can be safely disregarded in $c$ quark decays. The contributions of these operators of dimension 6 for the case of baryons can then be expressed in terms of several parameters (under the assumption of $SU(3)_{flavor}$ symmetry) related to the aforementioned types of four-quark effects: W-exchange ($\Gamma_{exch}$), negative Pauli interference ($\Gamma_{negint}$), and positive Pauli interference ($\Gamma_{posint}$), again up to the [*CKM*]{} matrix elements. Analogous claims are valid in the case of charmed meson decays. We should emphasize that $\Gamma$’s are conveniently chosen combinations of products of coefficients and operator matrix elements which appear in expressions for the inclusive weak decay rates of all Cabibbo modes. As we do not engage in a direct calculation of matrix elements, all these matrix elements can be taken as determined at the same scale $\mu$, i.e., there is no need for the hybrid renormalization in the case of four-quark operators. One needs to know nothing else about the matrix elements of the operators. In such a suitably defined theoretical environment one can express inclusive decay rates in a straightforward manner. The decay rates for nonleptonic modes are $$\begin{aligned} \label{eq:nllambda} \frac{\Gamma^{c \rightarrow s \overline{d} u} (\Lambda_{c}^{+})} {|V_{cs}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{exch} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow s \overline{s} u}(\Lambda_{c}^{+})} {|V_{cs}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{d} u}(\Lambda_{c}^{+})} {|V_{cd}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{exch} + \Gamma_{negint} + \Gamma_{posint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{s} u}(\Lambda_{c}^{+})} {|V_{cd}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{posint} + \Gamma_{negint} \end{aligned}$$ for $\Lambda_{c}^{+}$, $$\begin{aligned} \label{eq:nlxiplus} \frac{\Gamma^{c \rightarrow s \overline{d} u}(\Xi_{c}^{+})} {|V_{cs}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{posint} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow s \overline{s} u}(\Xi_{c}^{+})} {|V_{cs}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{negint} + \Gamma_{posint} + \Gamma_{exch} \nonumber \, , \\ \frac{\Gamma^{c \rightarrow d \overline{d} u}(\Xi_{c}^{+})} {|V_{cd}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{s} u}(\Xi_{c}^{+})} {|V_{cd}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{exch} + \Gamma_{negint}\end{aligned}$$ for $\Xi_{c}^{+}$, and $$\begin{aligned} \label{eq:nlxi0} \frac{\Gamma^{c \rightarrow s \overline{d} u}(\Xi_{c}^{0})} {|V_{cs}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{posint} + \Gamma_{exch} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow s \overline{s} u}(\Xi_{c}^{0})} {|V_{cs}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{posint} + \Gamma_{exch} \nonumber \, , \\ \frac{\Gamma^{c \rightarrow d \overline{d} u}(\Xi_{c}^{0})} {|V_{cd}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{posint} + \Gamma_{exch} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{s} u}(\Xi_{c}^{0})} {|V_{cd}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{posint} + \Gamma_{exch} \end{aligned}$$ for $\Xi_{c}^{0}$. For the decay rates of the semileptonic modes one obtains ($l = e, \mu$) $$\begin{aligned} \label{eq:sllambda} \frac{\Gamma^{c \rightarrow s \overline{l} \nu_{l}}(\Lambda_{c}^{+})} {|V_{cs}|^2} & = & \Gamma_{35}^{SL} \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{l} \nu_{l}}(\Lambda_{c}^{+})} {|V_{cd}|^2} & = & \Gamma_{35}^{SL} + \Gamma_{posint}^{SL}\end{aligned}$$ for $\Lambda_{c}^{+}$, $$\begin{aligned} \label{eq:slxiplus} \frac{\Gamma^{c \rightarrow s \overline{l} \nu_{l}}(\Xi_{c}^{+})} {|V_{cs}|^2} & = & \Gamma_{35}^{SL} +\Gamma_{posint}^{SL} \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{l} \nu_{l}}(\Xi_{c}^{+})} {|V_{cd}|^2} & = & \Gamma_{35}^{SL}\end{aligned}$$ for $\Xi_{c}^{+}$, and $$\begin{aligned} \label{eq:slxi0} \frac{\Gamma^{c \rightarrow s \overline{l} \nu_{l}}(\Xi_{c}^{0})} {|V_{cs}|^2} & = & \Gamma_{35}^{SL} +\Gamma_{posint}^{SL} \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{l} \nu_{l}}(\Xi_{c}^{0})} {|V_{cd}|^2} & = & \Gamma_{35}^{SL} + \Gamma_{posint}^{SL}\end{aligned}$$ for $\Xi_{c}^{0}$. One can introduce the following notation for the [*CKM*]{} matrix elements: $|V_{cs}|^2 = |V_{ud}|^2 = (\cos \theta_{c})^2 \equiv c^2$ and $|V_{cd}|^2 = |V_{us}|^2 = (\sin \theta_{c})^2 \equiv s^2$. Combining relations from (\[eq:nllambda\]) and (\[eq:nlxiplus\]), one obtains $$\begin{aligned} \label{eq:nllamxi} \Gamma^{c \rightarrow s \overline{d} u} (\Xi_{c}^{+}) & = & \frac{c^2}{s^2} \left ( \Gamma^{c \rightarrow s \overline{s} u} (\Lambda_{c}^{+}) + \Gamma^{c \rightarrow d \overline{d} u} (\Lambda_{c}^{+}) \right ) \nonumber \\ & - & \Gamma^{c \rightarrow s \overline{d} u} (\Lambda_{c}^{+}) \, , \nonumber \\ \Gamma^{c \rightarrow s \overline{s} u} (\Xi_{c}^{+}) & + & \Gamma^{c \rightarrow d \overline{d} u} (\Xi_{c}^{+}) = \nonumber \\ & & \Gamma^{c \rightarrow s \overline{s} u} (\Lambda_{c}^{+}) + \Gamma^{c \rightarrow d \overline{d} u} (\Lambda_{c}^{+}) \, , \nonumber \\ \Gamma^{c \rightarrow d \overline{s} u} (\Xi_{c}^{+}) & = & \frac{s^4}{c^4} \Gamma^{c \rightarrow s \overline{d} u} (\Lambda_{c}^{+})\end{aligned}$$ for the nonleptonic decay rates and from (\[eq:sllambda\]), (\[eq:slxiplus\]), and (\[eq:slxi0\]) we have $$\label{eq:slbar} \Gamma_{SL}(\Xi_{c}^{+}) = \Gamma_{SL}(\Xi_{c}^{0}) + \frac{s^2}{c^2} ( \Gamma_{SL}(\Lambda_{c}^{+}) - \Gamma_{SL}(\Xi_{c}^{0}))$$ for the semileptonic decay rates, where $\Gamma_{SL} (X) = \Gamma^{c \rightarrow s \overline{l} \nu_{l}} (X) + \Gamma^{c \rightarrow d \overline{l} \nu_{l}} (X)$, $X = \Xi_{c}^{+}, \Xi_{c}^{0}, \Lambda_{c}^{+}$. Expressions (\[eq:nllamxi\]) and (\[eq:slbar\]) show that all contributions to the total inclusive weak decay rate of $\Xi_{c}^{+}$ are expressed in terms of some of the analogous contributions of $\Lambda_{c}^{+}$ and $\Xi_{c}^{0}$. In this way, we have succeeded in expressing the lifetime of a “problematic” baryon $\Xi_{c}^{+}$ in terms of quantities of “nonproblematic” baryons $\Lambda_{c}^{+}$ and $\Xi_{c}^{0}$. If we introduce the notation $$\label{eq:brlambda} BR_{\Delta C =-1, \Delta S = 0} (\Lambda_{c}^{+}) = \frac{\left ( \Gamma^{c \rightarrow s \overline{s} u} (\Lambda_{c}^{+}) + \Gamma^{c \rightarrow d \overline{d} u} (\Lambda_{c}^{+}) \right )} {\Gamma_{TOT}(\Lambda_{c}^{+})} \, ,$$ the final expression (after neglecting all terms $\sim s^{4}$) for the ratio specified in (\[eq:exp\]) becomes $$\begin{aligned} \label{eq:ratioxilam} \frac{\tau(\Xi_{c}^{+})}{\tau(\Lambda_{c}^{+})} & = & \left[ -1 + \left( 2 + \frac{c^2}{s^2} \right) BR_{\Delta C =-1, \Delta S = 0} (\Lambda_{c}^{+}) \right. \nonumber \\ & & + \left. 2 \left( 1 - \frac{s^2}{c^2} \right) \frac{\tau(\Lambda_{c}^{+})} {\tau(\Xi_{c}^{0})} BR_{SL} (\Xi_{c}^{0}) + 2 \left( 1 + \frac{s^2}{c^2} \right) BR_{SL} (\Lambda_{c}^{+}) \right] ^{-1} \, .\end{aligned}$$ This type of analysis can be extended to the sector of charmed mesons. The hierarchy of charmed meson lifetimes is in general well understood in the framework of the [*HQE*]{} [@Bigi95], although some disrepancies exist that motivate alternative approaches [@Nussinov] and raise corresponding controversies [@Bigi2001]. We shall pursue our analysis in the framework of [*HQS*]{} and perform a model-independent analysis. This analysis, apart from its intrinsic value as a contribution to the understanding of charmed meson lifetimes, can also be a testing ground of our approach because of a higher quality of available experimental data for charmed mesons. Therefore, we conduct our analysis on a $SU(3)_{flavor}$ antitriplet of charmed mesons. The inclusive weak decay rates for individual Cabibbo nonleptonic decay modes are ($\Gamma$’s used in the mesonic case are different from those used in the baryonic case although the notation is very similar) $$\begin{aligned} \label{eq:nlDplus} \frac{\Gamma^{c \rightarrow s \overline{d} u} (D^{+})} {|V_{cs}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow s \overline{s} u}(D^{+})} {|V_{cs}|^2 |V_{us}|^2} & = & \Gamma_{35} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{d} u}(D^{+})} {|V_{cd}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{ann} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{s} u}(D^{+})} {|V_{cd}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{ann} \end{aligned}$$ for $D^{+}$, $$\begin{aligned} \label{eq:nlD0} \frac{\Gamma^{c \rightarrow s \overline{d} u}(D^{0})} {|V_{cs}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{exch} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow s \overline{s} u}(D^{0})} {|V_{cs}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{exch} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{d} u}(D^{0})} {|V_{cd}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{exch} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{s} u}(D^{0})} {|V_{cd}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{exch} \end{aligned}$$ for $D^{0}$, and $$\begin{aligned} \label{eq:nlDsplus} \frac{\Gamma^{c \rightarrow s \overline{d} u}(D_{s}^{+})} {|V_{cs}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{ann} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow s \overline{s} u}(D_{s}^{+})} {|V_{cs}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{ann} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{d} u}(D_{s}^{+})} {|V_{cd}|^2 |V_{ud}|^2} & = & \Gamma_{35} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{s} u}(D_{s}^{+})} {|V_{cd}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{negint}\end{aligned}$$ for $D_{s}^{+}$. For the decay rates of the semileptonic modes one obtains ($l = e, \mu$) $$\begin{aligned} \label{eq:slDplus} \frac{\Gamma^{c \rightarrow s \overline{l} \nu_{l}}(D^{+})} {|V_{cs}|^2} & = & \Gamma_{35}^{SL} \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{l} \nu_{l}}(D^{+})} {|V_{cd}|^2} & = & \Gamma_{35}^{SL} + \Gamma_{ann}^{SL}\end{aligned}$$ for $D^{+}$, $$\begin{aligned} \label{eq:slD0} \frac{\Gamma^{c \rightarrow s \overline{l} \nu_{l}}(D^{0})} {|V_{cs}|^2} & = & \Gamma_{35}^{SL} \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{l} \nu_{l}}(D^{0})} {|V_{cd}|^2} & = & \Gamma_{35}^{SL}\end{aligned}$$ for $D^{0}$, and $$\begin{aligned} \label{eq:slDsplus} \frac{\Gamma^{c \rightarrow s \overline{l} \nu_{l}}(D_{s}^{+})} {|V_{cs}|^2} & = & \Gamma_{35}^{SL} +\Gamma_{ann}^{SL} \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{l} \nu_{l}}(D_{s}^{+})} {|V_{cd}|^2} & = & \Gamma_{35}^{SL}\end{aligned}$$ for $D_{s}^{0}$. Combining relations from (\[eq:nlDplus\]) and (\[eq:nlDsplus\]), one obtains $$\begin{aligned} \label{eq:nlDplusDsplus} \Gamma^{c \rightarrow s \overline{d} u} (D_{s}^{+}) & = & \frac{c^2}{s^2} \left ( \Gamma^{c \rightarrow s \overline{s} u} (D^{+}) + \Gamma^{c \rightarrow d \overline{d} u} (D^{+}) \right ) \nonumber \\ & & - \Gamma^{c \rightarrow s \overline{d} u} (D^{+}) \, , \nonumber \\ \Gamma^{c \rightarrow s \overline{s} u} (D_{s}^{+}) & + & \Gamma^{c \rightarrow d \overline{d} u} (D_{s}^{+}) = \nonumber \\ & & \Gamma^{c \rightarrow s \overline{s} u} (D^{+}) + \Gamma^{c \rightarrow d \overline{d} u} (D^{+}) \, , \nonumber \\ \Gamma^{c \rightarrow d \overline{s} u} (D_{s}^{+}) & = & \frac{s^4}{c^4} \Gamma^{c \rightarrow s \overline{d} u} (D^{+})\end{aligned}$$ for the nonleptonic decay rates and from (\[eq:slDplus\]), (\[eq:slD0\]) and (\[eq:slDsplus\]) we have $$\label{eq:slmes} \Gamma_{SL}(D_{s}^{+}) = \Gamma_{SL}(D^{0}) + \frac{c^2}{s^2} (\Gamma_{SL}(D^{+}) - \Gamma_{SL}(D^{0}))$$ for the semileptonic decay rates, where $\Gamma_{SL} (X) = \Gamma^{c \rightarrow s \overline{l} \nu_{l}} (X) + \Gamma^{c \rightarrow d \overline{l} \nu_{l}} (X)$, $X = D^{+}, D^{0}, D_{s}^{+}$. Expressions (\[eq:nlDplusDsplus\]) and (\[eq:slmes\]) show that all contributions to the total inclusive weak decay rate of $D_{s}^{+}$ are expressed in terms of some of the analogous contributions of $D^{+}$ and $D^{0}$. Let us comment briefly on the findings of [@slCher; @slShif; @slVol] which indicate that the [*HQE*]{} could not reproduce semileptonic inclusive widths of charmed mesons. Let us point out that although the expressions for semileptonic inclusive decay widths are calculated using the [*HQE*]{}, the relations among them (such as (\[eq:slmes\])) simply state that inclusive semileptonic widths for all three charmed mesons are very close, which is satisfied very well experimentally [@PDG2000]. Therefore, the possibility that the [*HQE*]{} does not describe semileptonic inclusive widths ideally (although contributions of higher dimensional operators should be investigated before making this statement definite) does not bare a consequence on our final results which depend only on the relations among semileptonic decay widths. If we introduce the notation $$\label{eq:brDplus} BR_{\Delta C =-1, \Delta S = 0} (D^{+}) = \frac{\left ( \Gamma^{c \rightarrow s \overline{s} u} (D^{+}) + \Gamma^{c \rightarrow d \overline{d} u} (D^{+}) \right )} {\Gamma_{TOT}(D^{+})} \, ,$$ we obtain the following final expression (after neglecting terms $\sim s^{4}$) for the ratio of lifetimes of $D^{+}$ and $D^{0}$ mesons $$\begin{aligned} \label{eq:ratioDplusDsplus} \frac{\tau(D^{+})}{\tau(D_{s}^{+})} (1 - BR_{\tau}(D_{s}^{+} )) & = & -1 + \left( 2 + \frac{c^2}{s^2} \right) BR_{\Delta C =-1, \Delta S = 0} (D^{+}) \nonumber \\ & + & 2 \left( 1 - \frac{c^2}{s^2} \right) \frac{\tau(D^{+})} {\tau(D^{0})} BR_{SL} (D^{0}) + 2 \left( 1 + \frac{c^2}{s^2} \right) BR_{SL} (D^{+}) \, , $$ where $BR_{\tau}(D_{s}^{+} )$ denotes the branching ratio of the leptonic $D_{s}^{+} \rightarrow \tau^{+} \nu_{\tau}$ decay [^5]. Once we have obtained the results (\[eq:ratioxilam\]) and (\[eq:ratioDplusDsplus\]), we can clearly see their theoretical and experimental appeal. These relations have an intrinsic value since they express the lifetime of one charmed hadron in terms of measurable quantities of other two charmed hadrons belonging to the same $SU(3)_{flavor}$ multiplet. This result represents exploitation of advantages of the [*HQE*]{} at a new deeper level. The approach that leads to (\[eq:ratioxilam\]) and (\[eq:ratioDplusDsplus\]) also suppresses some of the problems referred to in the introduction. Let us briefly discuss these problems in the light of our approach. The problem of convergence seems rather important in charmed baryon decays. The operators of the lowest dimension in (\[eq:master\]), which are neglected in our approach, are some operators of dimension 6 (which are insensitive to the light content of the heavy hadron) followed by the operators of dimension 7 and higher. In our approach, all operators that are insensitive to the light content of heavy hadrons give the same contribution to the inclusive weak decay rate of each Cabibbo mode (up to the [*CKM*]{} matrix elements) and for every hadron within a given $SU(3)_{flavor}$ multiplet. If we look at the relations (\[eq:nllamxi\]), (\[eq:slbar\]), (\[eq:nlDplusDsplus\]), and (\[eq:slmes\]) as relations between exact inclusive weak decay rates for individual Cabibbo modes (and not only as approximations with several lowest dimensional operators), we can see that contributions of all light-flavor insensitive operators (of any dimension) get cancelled. Thus, these relations are correct up to the contributions of higher light-flavor sensitive operators. Since apart from four-quark operators there are other operators of dimension 6 in (\[eq:master\]) but they are all light-flavor insensitive, the aforementioned relations get the first correction from those operators of dimension 7 (or higher) which are light-flavor sensitive. Therefore, relations (\[eq:nllamxi\]), (\[eq:slbar\]), (\[eq:nlDplusDsplus\]), and (\[eq:slmes\]) are in the form that ameliorates the convergence issue. The phase-space corrections represent corrections which are different in various Cabibbo modes, depending on the number of massive quarks in the final state. Still, relations (\[eq:nllamxi\]), (\[eq:slbar\]), (\[eq:nlDplusDsplus\]), and (\[eq:slmes\]) are in such a form that the effect of phase space is significantly reduced. Let us consider the first equation of (\[eq:nllamxi\]): the sum of decay rates of two modes with one $s$ quark in the final state equals (up to the [*CKM*]{} matrix elements) the sum of decay rates of modes with two and zero $s$ quarks in the final state. Numerical values of the phase-space corrections to operators of dimensions 3 and 5 [@Bigi95] indicate that the sum of corrections for two $s$ quarks and zero $s$ quarks in the final state is very close to the double correction for one $s$ quark in the final state. The effects of phase-space corrections largely cancel. A similar situation appears in all other relations in (\[eq:nllamxi\]), (\[eq:slbar\]), (\[eq:nlDplusDsplus\]), and (\[eq:slmes\]). Therefore, inclusion of phase-space corrections does not notably worsen the accuracy of the aforementioned relations. The problem of calculating matrix elements is in our approach completely avoided. From the span of lifetimes of charmed hadrons [@PDG2000] it is clear that four-quark operators must play a very prominent role. Since, in contradistinction to operators of dimension 3 and 5, the matrix elements of four-quark operators cannot be calculated in a model-independent way, it is clear that even a modest inaccuracy in their determination may lead to significant deviations from the correct result. Moreover, a recent analysis [@Voloshin] indicates that there might exist serious deviations from some standard approximations, such as the valence quark approximation. Evading these pitfalls is one of the greatest advantages of our approach. Another advantage is that all crucial relations in this paper do not depend on the heavy quark mass $m_{Q}$ in the case when the assumed symmetries apply. In the realistic case, the form of relations (\[eq:nllamxi\]), (\[eq:slbar\]), (\[eq:nlDplusDsplus\]), and (\[eq:slmes\]) reduces the dependence of results on $m_{Q}$ significantly (to the level of breaking of underlying symmetries). Finally, there remains the assumption on $SU(3)_{flavor}$ symmetry. The effects of breaking of this symmetry were analyzed in [@GM]. From that analysis one can conclude that the effects of $SU(3)_{flavor}$ breaking are generally less than $30\%$ and probably significantly smaller. Therefore, we expect the same level of accuracy in our treatment, too. After the discussion of theoretical advantages and limitations of our approach there remains an important problem of confrontation of theoretical findings with experimental values. From the final relation for baryons (\[eq:ratioxilam\]) and mesons (\[eq:ratioDplusDsplus\]) it is evident that theoretical predictions depend on the branching ratios of the singly Cabibbo suppressed nonleptonic modes $BR_{\Delta C =-1, \Delta S = 0} (\Lambda_{c}^{+})$ and $BR_{\Delta C =-1, \Delta S = 0} (D^{+})$, respectively. These values are not available from experiment and their determination represents a crucial step in numerical analysis. An estimate of these quantities can be obtained indirectly from exclusive modes and depends on the quality of data for these modes. From the flavor quantum numbers of the final decay products in heavy-hadron decays one can determine which Cabibbo mode governed that particular decay at the quark level. The only exceptions are the modes $c \rightarrow s \overline{s} u$ and $c \rightarrow d \overline{d} u$ which lead to the final hadronic state with the same flavor quantum numbers. However, this fact does not pose a problem since in all expressions the decay rates of these two modes appear in the form of sum and therefore there is no need to make difference between them. From the flavor quantum numbers of the final states of any particular exclusive mode one can determine whether it was governed by the Cabibbo dominant, singly Cabibbo suppressed, or doubly Cabibbo suppressed nonleptonic modes at the quark level. An analogous conclusion follows for semileptonic decays. It is, therefore, possible to obtain a decay rate for any Cabibbo inclusive mode (all decay channels coming from the same Cabibbo mode at the quark level) by summing the decay rates of associated exclusive modes. In performing this procedure one encounters the effect of quantum interference. Namely, different final states originating from the same quark decay mode can mix owing to final state strong interactions. The most notable manifestation of this effect is that summing of the branching ratios of all exclusive modes taken from [@PDG2000] can lead to a result well over $100 \%$ (e.g., for $D^{0}$ or $D^{+}$). To minimize this effect, we invoke the following procedure: we calculate the inclusive decay rate of singly Cabibbo suppressed modes by summing the decay rates of all appropriate exclusive decay modes; then we calculate the [*total decay rate*]{} by summing decay rates of [*all*]{} exclusive modes and then divide the two numbers to obtain the wanted ratio. Using the sum of all exclusive modes instead of the measured lifetime for the total decay rate insures the same treatment of interference effects in both quantities in the ratio. Other quantities appearing in the expressions (\[eq:ratioxilam\]) and (\[eq:ratioDplusDsplus\]) are lifetimes and semileptonic branching ratios, which are a standard part of information on any weakly decaying particle. In general, they are well measured and available in [@PDG2000]. Let us first consider the presently very interesting question of the $\tau(\Xi_{c}^{+})/\tau(\Lambda_{c}^{+})$ ratio. The sum of branching ratios of all measured exclusive decay modes is approximately $50 \%$ which shows that the set of available decay modes is not complete. The branching ratio $BR_{\Delta C =-1, \Delta S = 0} (\Lambda_{c}^{+})$ is obtained at the level of $0.0295 \pm 0.0115$, which is probably an underestimated result since only a few exclusive modes corresponding to singly Cabibbo suppressed modes are available [@PDG2000]. Another problem is the lack of data on the semileptonic branching ratio of the $\Xi_{c}^{0}$ baryon. This value can be taken from [@GM] to be $BR_{SL}(\Xi_{c}^{0}) = (0.092 \pm 0.006)$. As the contribution coming from $BR_{SL}(\Xi_{c}^{0})$ is the nonleading one (the leading one coming from the $BR_{\Delta C =-1, \Delta S = 0} (\Lambda_{c}^{+})$), this mixing of theoretical and experimental results does not introduce a significant model dependence. Still, only the future arrival of experimental data on $BR_{SL}(\Xi_{c}^{0})$ will complete the set of experimental values needed for a fully consistent analysis. The rest of the data is taken to be [@PDG2000]: $BR_{SL}(\Lambda_{c}^{+}) = (0.045 \pm 0.017)$, $\tau(\Lambda_{c}^{+}) = (0.206 \pm 0.012) \, \rm ps$, and $\tau(\Xi_{c}^{0}) = (0.098 \pm 0.019) \, \rm ps$. The analysis using the set of parameters specified above yields a result for the $\tau(\Xi_{c}^{+})/\tau(\Lambda_{c}^{+})$ ratio which is far above the new experimental results and has a very large error. The principal reason for such a result can be seen from (\[eq:ratioxilam\]). The value of $BR_{\Delta C =-1, \Delta S = 0} (\Lambda_{c}^{+})$ is multiplied by a large factor $c^2/s^2$, which makes the final result very sensitive to the value of this branching ratio. The conclusion stemming from this analysis is that the presently available data on $\Lambda_{c}^{+}$ exclusive modes are insufficiently accurate and abundant to insure a reliable result. A more extensive and precise measurement of exclusive decay modes of $\Lambda_{c}^{+}$ (especially Cabibbo suppressed ones) can however lead to interesting new predictions on $\Xi_{c}^{+}$. Numerical analysis in the sector of charmed meson decays is more promising. Addition of the branching ratios of all exclusive modes of $D^{+}$ gives a value of $110 \%$, which shows that the data on exclusive decay modes of $D^{+}$ can be considered complete. The branching ratio $BR_{\Delta C =-1, \Delta S = 0} (D^{+})$ attains the value $0.140 \pm 0.026$. Other values taken from [@PDG2000] are $BR_{SL}(D^{+}) = 0.172 \pm 0.019$, $BR_{SL}(D^{0}) = 0.0675 \pm 0.0029$, $\tau(D^{+}) = (1.051 \pm 0.013) \, \rm ps$, and $\tau(D^{0}) = (0.4126 \pm 0.0028) \, \rm ps$. Using the expression (\[eq:ratioDplusDsplus\]) one obtains the value $(\tau(D^{+})/\tau(D_{s}^{+}))(1-BR_{\tau}(D^{+}_{s}))_{th} = 2.63 \pm 0.98$. This result obtained from theoretical considerations can be compared with the value for the same quantity following from the experiment. To this end, we use the experimental values [@PDG2000]: $\tau(D_{s}^{+}) = (0.496 \pm 0.0095) \rm ps$ and $BR_{\tau}(D_{s}^{+}) = 0.07 \pm 0.04$. This leads to a value $(\tau(D^{+})/\tau(D_{s}^{+}))(1-BR_{\tau}(D^{+}_{s}))_{exp} = 1.971 \pm 0.096$. Comparison of these two results shows that they are consistent within their errors. A relatively large error of the result obtained through relation (\[eq:ratioDplusDsplus\]) originates to a great extent from the expression (\[eq:slmes\]) where the inclusive semileptonic decay rate of $D_{s}^{+}$ is expressed in terms of respective quantities for the other two charmed mesons. In this relation a large factor $c^{2}/s^{2}$ multiplies a small quantity $\Gamma_{SL}(D^{+}) - \Gamma_{SL}(D^{0})$ (the inclusive decay rates for these two charmed mesons are numerically very close). In the final expression, this fact contributes very little to the central value, but gives a significant contribution to the error since $\Gamma_{SL}(D^{+})$ and $\Gamma_{SL}(D^{0})$ are treated as independent quantities and their individual errors are significantly larger than their difference. The consequences of these facts can be better observed if one performs the following analysis. For the sake of error analysis, we take that $\Gamma_{SL}(D^{+})$ and $\Gamma_{SL}(D^{0})$ are identically equal (while in reality they differ by the small Cabibbo suppressed correction). This approximation removes the problematic term of a large factor multiplying a small quantity. This procedure changes the central value at the permille level while the error is almost halved (even with this reduced errors our two results are in a 2$\sigma$ interval). The procedures presented so far are by no means restricted to the calculation of the lifetimes of $\Xi_{c}^{+}$ and $D_{s}^{+}$. Any inclusive quantity (such as semileptonic branching ratios) for these hadrons can be expressed by means of inclusive quantities of the other two charmed hadrons belonging to the same multiplet. Similar relations can also be established in multiplets of $b$ hadrons bearing in mind that, e.g., phase-space corrections in the $b$ case can be substantial. Nevertheless, the full success of this approach is dependent on accumulation of experimental data [^6] and measurement of inclusive decay rates of suppressed decay modes. Considerations displayed in this paper are motivated by recent experimental results on charmed baryon lifetimes and the need to establish whether a standard existing formalism can be brought into agreement with these results by eliminating or reducing some of its uncertainties. Our formalism procures model-inedependent results with the assumption of some symmetries. Apart from these desirable properties, the theoretical appeal of our approach consists in expressing some measurable quantity of a heavy hadron in terms of measurable quantities of other heavy hadrons from the same $SU(3)_{flavor}$ multiplet. This feature enables us to set a new course in testing the formalism of inclusive weak decays. Using relations such as (\[eq:ratioxilam\]) and (\[eq:ratioDplusDsplus\]) one can use the data for those hadrons the decays of which are more amenable to experimental determination to produce predictions for hadrons where experimental data lack or need theoretical verification (like in the $\Xi_{c}^{+}$ case). As any advantage, this one has its price, too. One has to introduce inclusive decay rates of singly Cabibbo suppressed modes which so far have not been measured (as inclusive modes). Use of data on exclusive decay modes can give a reasonable estimate of necessary decay rates. Nevertheless, the full strength of our approach would manifest itself if direct measurements of inclusive decay rates of singly Cabibbo suppressed modes of $\Lambda_{c}^{+}$ should be possible in the near future. Even better and more detailed data on exclusive decay modes of $\Lambda_{c}^{+}$ could improve our understanding of new experimental data on the $\Xi_{c}^{+}$ lifetime. The real challenge now faces the experimental community. There is a clear indication that by measuring the parameters of one heavy hadron ( $\Lambda_{c}^{+}$) we can draw definite conclusions on the other heavy hadron ($\Xi_{c}^{+}$). These conclusions may clarify the question of applicability of the [*HQE*]{} in charmed decays or at least decide whether $\Xi_{c}^{+}$ really fits into the, so far successful, description of charmed baryon lifetime hierarchy. [**Acknowledgments**]{} The authors would like to thank the referee for kind suggestions which have improved the comprehensibility of the paper. This work was supported by the Ministry of Science and Technology of the Republic of Croatia under the contract No. 0098002. [88]{} I. Bigi, The Lifetimes of Heavy Flavor Hadrons - a Case Study in Quark-Hadron Duality, in: Proc. $\rm 3^{rd}$ Intern. Conf. on B Physics and CP Violation, Taipei, Taiwan, December 3-7, 1999, to appear, hep-ph/0001003;\ M. 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Lett. B 484 (2000) 43. I. Bigi, UND-HEP-95-BIG02, hep-ph/9508408. S. Nussinov, M.V. Purohit, Phys. Rev. D 65 (2002) 034018. I. Bigi, UND-HEP-01-BIG08, hep-ph/0112155. V. Chernyak, Nucl. Phys. B 457 (1995) 96. B. Blok, R.D. Dikeman, M.A. Shifman, Phys. Rev. D 51 (1995) 6167. M.B. Voloshin, TPI-MINN-02/01-T, UMN-TH-2041-02, hep-ph/0202028. M.B. Voloshin, Phys. Rev. D 61 (2000) 074026. V. Eiges (BELLE collaboration), private communication with B. Guberina. [^1]: Like the still present problem of the $\tau(\Lambda_{b}^{0})/\tau(B_{d}^{0})$ ratio or the recently escalating problem of the $\tau(\Xi_{c}^{+})/\tau(\Lambda_{c}^{+})$ ratio. [^2]: As nicely demonstrated in $(1 + 1)$-dimensional [*QCD*]{} [@11QCD1; @11QCD2; @Shifmandual]. [^3]: Therefore, it cannot be studied in lattice QCD, which is essentially a numerical Euclidean approach. [^4]: there are no operators of dimension 4 owing to color-gauge invariance [^5]: This mode contributes significantly only to the decays of the $D_{s}^{+}$ meson and therefore cannot be related to the analogous decay rates of other members of the $SU(3)_{flavor}$ multiplet. [^6]: The upcoming high-statistics measurements, especially for charmed baryons [@private], are in this respect very encouraging.

--- abstract: | There are fundamental reasons as to why there should exist a reformulation of quantum mechanics which does not refer to a classical spacetime manifold. It follows as a consequence that quantum mechanics as we know it is a limiting case of a more general nonlinear quantum theory, with the nonlinearity becoming significant at the Planck mass/energy scale. This nonlinearity is responsible for a dynamically induced collapse of the wave-function, during a quantum measurement, and it hence falsifies the many-worlds interpretation of quantum mechanics. We illustrate this conclusion using a mathematical model based on a generalized Doebner-Goldin equation. The non-Hermitian part of the Hamiltonian in this norm-preserving, nonlinear, Schrödinger equation dominates during a quantum measurement, and leads to a breakdown of linear superposition. 1.0 in *This essay received an Honorable Mention in the* *Gravity Research Foundation Essay Competition, 2007* ---  1.0 in [ ]{} 0.2 in [[ [^1] ]{}]{} [*Tata Institute of Fundamental Research,*]{}\ [*Homi Bhabha Road, Mumbai 400 005, India.*]{} 0.5cm 1.0 in There are two fundamental unsolved problems in our understanding of quantum mechanics. The first is the famous problem of quantum measurement, for which one of the possible solutions is the mechanism of decoherence, in conjunction with the many-worlds interpretation of quantum mechanics. An alternative explanation of a quantum measurement is a dynamically induced collapse of the wave-function, which requires modification of the Schrödinger equation in the measurement domain. The second unsolved fundamental problem is the need for a reformulation of quantum mechanics, which does not refer to a classical spacetime manifold [@singh]. In this essay we show that these two unsolved problems have a deep connection, and the resolution of the second problem implies that quantum measurement is explained by dynamically induced collapse of the wave-function. This, in turn, falsifies the many-worlds interpretation of quantum mechanics. The standard formulation of quantum theory depends on an external classical time. The need for a reformulation of quantum mechanics which does not refer to a classical spacetime manifold arises because the geometry (metric and curvature) of the manifold is produced by [*classical*]{} matter fields. One can envisage a Universe in which there are only quantum, and no classical, fields. This will cause the spacetime geometry to undergo quantum fluctuations, which, in accordance with the Einstein hole argument, destroy the underlying classical spacetime manifold. However, one should still be able to describe quantum dynamics; hence the need for the aforementioned reformulation. The new formulation becomes equivalent to standard quantum mechanics as and when an external classical spacetime geometry becomes available. When one tries to construct such a reformulation of quantum mechanics, it follows from very general arguments [@singh] that quantum gravity is effectively a nonlinear theory. What this means is that the ‘quantum gravitational field’ acts as a source for itself. Such a nonlinearity cannot arise in the standard canonical quantization of general relativity, which is inherently based on linear quantum theory, and which leads to the Wheeler-DeWitt equation. It also follows as a consequence that at the Planck mass/energy scale, quantum theory itself becomes an effectively nonlinear theory \[because of self-gravity\], and that the Hamiltonian describing a quantum system depends nonlinearly on the quantum state. The standard linear quantum theory is recovered as an approximation at energy scales much smaller than the Planck mass/energy scale. In [@singh] we have developed a model for the above-mentioned reformulation of quantum mechanics, based on noncommutative differential geometry. One of the outcomes of this model is that the non-relativistic quantum mechanics of a particle of mass $m$ is described by a nonlinear Schrödinger equation, which belongs to the Doebner-Goldin class [@DG] of nonlinear equations. The nonlinear terms depend on the mass of the particle, and are extremely small when the particle’s mass is much smaller than Planck mass $m_{Pl}\sim 10^{-5}$ grams. Thus in the microscopic domain the theory reduces to standard quantum mechanics. The nonlinearity becomes significant in the mesoscopic domain, where the particle’s mass is comparable to Planck mass. This is also the domain where the quantum to classical transition is expected to take place; a nonlinearity in this domain can play a decisive role in explaining quantum measurement. It is pertinent to mention here that current experimental tests of quantum mechanics do not rule out such a nonlinearity, and furthermore, because our model is based on an underlying noncommutative geometry, the usual objections against a nonlinear quantum mechanics do not apply [@singh]. When the particle’s mass is greater than Planck mass, the nonlinear theory reduces to standard classical mechanics. We now demonstrate how the Doebner-Goldin equation can explain quantum measurement as dynamical collapse of the wave-function. The simplest D-G equation is $$i\hbar \frac{\partial\psi}{\partial t} = -\frac{\hbar^{2}}{2m}\nabla^{2}\psi + V\psi + iD(m/m_{Pl})\hbar\left( \nabla^{2}\psi + \frac{|\nabla\psi|^{2}}{|\psi|^{2}}\psi\right). \label{dg}$$ The coefficient $D$ of the nonlinear, imaginary, part of the Hamiltonian is a real constant, which depends on the ratio of the particle’s mass to Planck mass. $D$ goes to zero in the limit $m\ll m_{Pl}$, so that then the D-G equation reduces to the linear Schrödinger equation. As $m$ approaches $m_{Pl}$, $D$ becomes large enough for the imaginary part of the Hamiltonian to dominate over the real part. The equation is norm-preserving, although the probability density obeys not the continuity equation, but a Fokker-Planck equation. The equation is of interest also because it arises in the study of unitary representations of an infinite-dimensional Lie algebra of vector fields $Vect(R^{3})$ and group of diffeomorphisms $Diff(R^{3})$ - these representations provide a way to classify physically distinct quantum systems. Further, the equation is a special case [@gri] of the following class of norm-preserving nonlinear Schrödinger equations $$i\hbar d|\psi>/dt = H|\psi> + (1-P_\psi)U|\psi> \label{gri}$$ where $H$ is the Hermitian part of the Hamiltonian, $(1-P_\psi)U$ is the non-Hermitian part, $P_\psi=|\psi><\psi|$ is the projection operator, and $U$ is an arbitrary nonlinear operator. We will work with a generalization of the $U$ operator for this D-G equation, given by $U=iF(m/m_{Pl})\Sigma_n D_n U_n$, where $$U_n = \left[ \frac{<\psi|\nabla |\chi_n><\chi_n|\nabla|\psi>} {<\psi|\chi_n><\chi_n|\psi>}|\chi_n><\chi_n|+\nabla^2 \right] \label{UR}$$ and where $D_n$ are state-dependent scalars; the real function $F(m/m_{Pl})$ vanishes as $m\rightarrow 0$ and monotonically increases with mass, and $|\chi_n>$ are a complete set of orthonormal vectors. We will use the term ‘initial system’ to refer to the quantum system ${\cal Q}$ on which a measurement is to be made by a classical apparatus ${\cal A}$, and the term ‘final system’ to refer jointly to ${\cal Q}$ and ${\cal A}$ after the initial system has interacted with ${\cal A}$. A quantum measurement will be thought of as an increase in the mass (equivalently, number of degrees of freedom) of the system, from the initial value $m_{\cal Q}\ll m_{Pl}$ to the final value $m_{\cal Q} + m_{\cal A} \gg m_{Pl}$. Clearly then, the non-Hermitian part in (\[gri\]), which is proportional to $U$, and hence to the scalars $D_n$ in (\[UR\]), will play a critical role in the transition from the initial system to the final system. We assume that ${\cal A}$ measures an observable ${\hat O}$ of ${\cal Q}$, having a complete set of eigenstates $|\phi_n>$. Let the quantum state of the initial system be given as $|\psi>=\Sigma_n\ a_n|\phi_n>$. The onset of measurement corresponds to mapping the state $|\psi>$ to the state $|\psi>_F$ of the final system as $$|\psi>\rightarrow |\psi>_F\ \equiv \sum_n a_n|\psi>_{Fn} = \sum_n\ a_n|\phi_n>|A_n> \label{map}$$ where $|A_n>$ is the state the measuring apparatus would be in, had the initial system been in the state $|\phi_n>$, and the $|\chi_n>$ in (\[UR\]) should be understood as the direct product $|\chi_n>=|\phi_n>|A_n>$. During a quantum measurement the non-Hermitian part of the Hamiltonian in (\[gri\]) dominates over the Hermitian part, and governs the evolution of the state $|\psi>_F$ given by (\[map\]). Assuming that the Hermitian operator $U_n$ maps the state $|\psi>_F$ to a state $|\xi>_{nF}$ which can be expanded as $$|\xi>_{nF}= \sum_m\ b_{nm}|\phi_m>|A_m> \label{map2}$$ we substitute the expansion for $|\psi>_F$ from (\[map\]) in (\[gri\]), and neglecting the Hermitian part of the Hamiltonian we get [@gri] $$\frac{da_n}{dt} = \frac{F(m/m_{Pl})}{\hbar}\ a_n(q_n-L)$$ where $q_n=t_n/a_n$, $L=\Sigma_m \ a_m^{*}t_m$, $t_m=\Sigma_s b_{ms} D_s$. If the dependence of the $D_n$’s on the state is such that the $q_n$’s are [*random constants*]{} then it follows that [@gri] $$\frac{d}{dt} \left(\ln \frac{a_i}{a_j}\right) = \frac{F(m/m_{Pl})}{\hbar}\ [q_i -q_j]. \label{evo}$$ It follows that only the state $|\psi>_{Fi}$ having the largest real part of $q_i$ survives at the end of a measurement (since $\Sigma_n |a_n|^2=1)$, and in this manner superposition is broken. It is noteworthy that the time-scale for breakdown of superposition is directly proportional to Planck’s constant, and it decreases with increasing mass. The randomness of the $q_n$’s is needed to ensure that repeated measurements of the observable ${\hat O}$ lead to different outcomes $|\psi>_{Fn}$. In order to reproduce the observed Born probability rule, the measurement should cause the quantum system to collapse to the eigenstate $|\phi_n>$ with the probability $p_n=|<\psi(t_0)|\phi_n>|^2$. The most plausible way to introduce randomness in the $q_n$’s is to propose that they are related to the random phase $\theta_0$ of the initial quantum state. As an example, if the phase is uniformly distributed in the range $[0,2\pi]$ and the $q_n$’s are related to $\theta_0$ by the relations [@gri] $$q_1=-2\pi\theta_0, \ \ q_n=-\frac{1}{n}\left(2\pi\theta_0 - \sum_k^{n-1}|<\psi(t_0)|\phi_k>|^2\right) -\sum_k^{n-1} \frac{|<\psi(t_0)|\phi_k>|^2}{k}$$ and possess the probability distribution $$\omega(q_n) = |<\psi(t_0)|\phi_n>|^2\ \exp(|<\psi(t_0)|\phi_n>|^2)$$ the Born probability rule is reproduced. The detailed assumptions of the above model can only be justified after a better understanding of the relation between quantum mechanics and noncommutative geometry has been achieved. However, it is already clear that the natural requirement of a reformulation of quantum mechanics which does not refer to a classical spacetime manifold compels us to consider a nonlinear modification of the Schrödinger equation at the Planck mass/energy scale. Such a nonlinearity, which explicitly depends on Newton’s gravitational constant (via the Planck mass) is responsible for the breakdown of superposition during a quantum measurement, and provides a dynamical explanation for collapse of the wave-function. Modifications of the Schrödinger equation hitherto investigated in the literature have been [*ad*]{} [*hoc*]{}, and introduced solely for the purpose of explaining quantum measurement. However, the nonlinear modification considered by us has its origin elsewhere, in quantum gravity; yet it has an impact on quantum measurement. The experimentally observed mechanism of decoherence destroys the [*interference*]{} between different possible outcomes of measurement, but as it is based on standard linear quantum mechanics, it preserves [*superposition*]{} amongst the alternatives. In this scheme (assuming that the wave-function describes individual quantum systems, and not merely their statistical ensemble), the only natural way to explain the observed lack of superposition amongst the results of a measurement is to invoke the many-worlds interpretation of quantum mechanics, wherein upon a measurement, the Universe splits into many branches, one for every decohered state. Up until now, no theoretical argument had been presented, to choose between a decoherence based explanation of quantum measurement, and the alternative explanation based on dynamically induced collapse. Our analysis in this essay establishes that the wave-function does collapse during a measurement, and hence the many-worlds interpretation stands falsified. Above all, the proposal that the initial random phase of the quantum state is correlated with the outcome of a quantum measurement is experimentally testable with current generation experiments, and if confirmed, will provide the first experimental evidence for quantum gravity. [99]{} T. P. Singh, in [*Quantum theory and symmetries*]{}, Ed. V. K. Dobrev (Heron Press, 2006) \[gr-qc/0510042\]; T. P. Singh, [*Intl. Jour. Mod. Phys*]{} 15, 2153 (2006) \[hep-th/0605112\]. H.-D. Doebner and G. A. Goldin, Phys. Lett. A162, 397 (1992); G. A. Goldin, quant-ph/0002013 (2003). A. N. Grigorenko, [*J. Phys.A: Math. Gen.*]{} 28, 1459 (1995); V. V. Dodonov and S. S. Mizhary, [*J. Phys. A: Math. Gen.*]{} 26, 7163 (1993). [^1]: Talk given at the Meeting ‘Himalayan Relativity Dialogue’, Mirik, India, 18-20 April, 2007

Introduction ============ The idea of current-carrying edge states[@Halperin-82] is one of the major paradigms in the theory of the quantum Hall (QH) effect. For simple filling fractions $\nu=(2m+1)^{-1}$, Wen has shown[@Wen-90; @Wen-91; @Wen-91A; @Wen-91B; @Wen-92rev] that edge modes can be represented as one-component chiral Luttinger liquids, with the universal coupling determined by $\nu$. Within this simple model, controlled calculations are possible. This lead to many beautiful results, including the universal inter-edge tunneling exponent[@Wen-91B; @Kane-Fisher-Tunnel], exact expressions for tunneling conductance, the non-linear tunneling $I$–$V$ curve[@Weiss-Exact; @Fendley-95B], and tunneling noise[@Kane-94A; @Fendley-95C; @QHpers-book]. Experimentally, however, there are more dimensions to this problem. The results of the first pinch-off tunneling experiment[@Milliken-96], where the scaling appeared to be in agreement with theory[@Wen-91B; @Kane-Fisher-Tunnel], have only recently received a partial confirmation[@Maasilta-97; @Turley-98]. Furthermore, in Ref.  no scaling was observed at all, and in Ref.  the measured tunneling exponent was off by a factor of two. Such discrepancies were attributed in part to edge reconstruction in samples with “soft” confinement[@softedge]. However, the tunneling measurements in cleaved-edge samples[@Chang-96; @Grayson-98], where the confining potential is expected to be sharp, yield tunneling exponents shifted off the predicted values even at the magic filling fractions $\nu=1$, $1/3$. Previously, much effort[@mechanisms] was dedicated to identify mechanisms leading to (non-universal) corrections to tunneling exponents. In particular, the effect of the long-range Coulomb interaction was analyzed[@Zuelicke-96; @Moon-96; @Oreg-96; @Imura-97] in the geometry of two counterpropagating parallel edges ($\alpha\to0$ in Fig. \[fig:angles\]). In exact analogy with its effect in one-dimensional electron gas[@Emery-1DEG], repulsive Coulomb interaction renormalizes the Luttinger liquid coupling parameter. Thus, a weak impurity-associated inter-edge tunneling becomes a relevant perturbation, so that the current flow (from top to bottom in Fig. \[fig:angles\]) is [*enhanced*]{} at low temperature $T$ and applied voltage $V$. However, the same interaction [ *suppresses*]{}[@Imura-97] the tunneling in the dual configuration, of two semi-infinite non-chiral Luttinger liquids connected by a tunneling point ($\alpha\to\pi$ in Fig. \[fig:angles\]), and the system is pushed towards the insulating regime. This indicates that even the [*sign*]{} of the Coulomb interaction effect on the tunneling exponent is not the same in different geometries. 0.75 0.2pc The purpose of this work is to analyze in detail the Coulomb interaction effect on the properties of QH tunneling junctions in different geometries. First, we demonstrate that the well-known duality relating weak tunneling and weak backscattering remains exact in the presence of long-range interactions. Then, we focus on scale-invariant [X]{}-shaped constrictions, and calculate the renormalized Luttinger coupling constant $g_\star^2$ (which, in particular, determines the power law dependence of the conductance on $T$ and $V$) as a function of the opening angle $\alpha$ (Fig. \[fig:angles\]). We show that the unscreened Coulomb interaction drives a zero temperature delocalization transition as a function of $\alpha$ in both integer and fractional QH constrictions. In the integer case the transition occurs precisely at the self-dual value $\alpha_c=\pi/2$, independent of the interaction strength. At the fractions $\nu=(2m+1)^{-1}$, the critical angle $\alpha_c$ is non-universal, but its value is always smaller than $\pi/2$. We also analyze the effect of Coulomb interactions in the geometry of cleaved-edge tunneling experiments. The paper is organized as follows. In Sec. \[sec:effective-tunneling\] we introduce the tunneling action which accounts for the long-range interactions. A general proof of the duality between weak tunneling and weak backscattering is given in Sec. \[sec:duality\]. In Sec. \[sec:self-similar\], we present our results for the renormalized Luttinger coupling $g_\star^2$ in different geometries, and in Sec. \[sec:discussion\] we discuss the implications on tunneling experiments. Related analytic results are collected in Appendices: in App. \[sec:appendix-pi\], the case of $\alpha=\pi$ is solved; in App. \[sec:wiener-hopf\], the Wiener-Hopf technique is used to directly solve the self-dual case $\alpha=\pi/2$, and evaluate the lowest order correction for $|\cos\alpha|\ll1$. The effective tunneling action {#sec:effective-tunneling} ============================== Gapless edge excitations $u\equiv u(x,\tau)$ for Laughlin’s QH states with filling fractions $\nu\!=\!(2m+1)^{-1}$ can be described[@Wen-91A; @Wen-91B; @Wen-92rev] by the imaginary-time quadratic action $$\label{eq:edge-action} {\cal S}_0=\frac{1}{4\pi}\int_0^\beta \!d\tau\int dx\,{\partial_x u\,(i\partial_\tau{u}+ v \,\partial_x u)},$$ where $x$ is the coordinate along the edge, and $v\equiv v(x)$ is the edge wave velocity. The field $u$ is related to the linear charge density at the edge, $\rho=\sqrt\nu\,\partial_x u/(2\pi)$ (note the unconventional normalization). Formally, gauge invariance requires that the field $u(x,\tau)$ be treated as a compact boson of radius $R=\sqrt\nu$, [*i.e.*]{}, the values $u$ and $u+2\pi\sqrt\nu$ must be identified. This, however, is [*not*]{} achieved within the usual path integral formalism[@Oreg-95] in a finite geometry if we assume the field $u(x,\tau)$ continuous everywhere along the circumference. Indeed, the equal-time commutation relationship $$[u(x),\,u(x')]=i\pi\sgn (x-x')$$ on the edge of length $L$ implies that the fields $u_0\equiv u(0,\tau)$ and $u_L\equiv u(L,\tau)$ are canonically conjugated, which contradicts the continuity of the field along the circle. The difference $u_L-u_0$ (proportional to the topological charge associated with the zero mode) is also proportional to the total charge $Q=\sqrt\nu\,(u_L-u_0)/(2\pi)$ accumulated at the edge; only in the absence of tunneling into the edge this charge is a dynamically conserved quantized quantity. The correct zero-mode quantization spectrum can be obtained if we consider the variables $u_0$ and $u_L$ as independent, and write the bare edge action (\[eq:edge-action\]) more explicitly as[@Pryadko-98] $$\begin{aligned} {\cal S}_0& =& \frac{1}{4\pi}\int_{0\strut}^\beta d\tau\int_0^L dx\,{\partial_x u\,(i\partial_\tau{u}+ v \,\partial_x u)}\nonumber \\ & & +\,{1\over8\pi}\int_0^{\beta\strut} d\tau \,(u_L-u_0)\, i\partial_\tau (u_L+u_0). \label{eq:finite-size-edge-action}\end{aligned}$$ The boundary term in the second line is added to fix the canonical quantization of the zero mode, and to decouple it from the edge modes with finite momenta. Since the charge density $\rho$ is expressed linearly in terms of the field $u$, the action remains quadratic[@Wen-92rev; @Moon-96; @Hangmo-96] even in the presence of non-local Coulomb interaction $$\label{eq:coulomb-action} {\cal S}_1= {\nu\,e^2\over 8\pi^2\varepsilon}\int_0^\beta \!d\tau\int dx\,dy \, u'(x)\,V\left(\left|{\bf r}_x-{\bf r}_y\right|\right)\,u'(y),$$ where ${\bf r}_x$ is the actual position of the point $x$ as measured along the edge, and $\varepsilon$ is the dielectric constant of the material. The problem is non-trivial because now both the distance $x$ measured along the edge, and the geometrical distance $\left|{\bf r}_x-{\bf r}_y\right|$ are important. The inter-edge tunneling is introduced by the non-linear term $$\label{eq:tunn-action} {\cal S}_{\rm t}=\int_0^\beta \! d\tau \, \re\,\lambda\,e^{i g \varphi}, \quad \varphi\equiv u(x_1)-u(x_2);$$ here $g\!=\!\sqrt\nu$ for the quasiparticles’ tunneling between the points $x_1$ and $x_2$ through the QH liquid with the filling fraction $\nu$, or $g\!\rightarrow \!\tilde g=1/\sqrt{\nu}$ for tunneling of electrons through the insulating region. The tunneling amplitude $\lambda$ is set by the details[@Jain-88] of the self-consistent potential near the tunneling point and considered as a phenomenological parameter. The non-linear tunneling action (\[eq:tunn-action\]) depends on the values of the field $u(x,\tau)$ in the points $x_1$, $x_2$; the values of this field in all other points can be integrated out. Leaving the argument $\varphi$ of the tunneling term as the only independent variable, we can write the most general form of the effective action $$\label{eq:effective-model} S={T\over4\pi}\sum_{n} % |\omega_n| \,{\cal K}(\omega_n)\,|\varphi_n|^2 +\int_0^\beta \! d\tau \, \re\,\lambda\,e^{i g \varphi(\tau)},$$ where the harmonics $\varphi_n\equiv \int_0^\beta d\tau \varphi(\tau)\,\exp(-i\omega_n\tau)$ and $\bar\varphi_n\equiv \varphi_{-n}$ are evaluated at the Matsubara frequencies $\omega_n=2\pi\,n T$. This effective tunneling model is fully characterized by the frequency-dependent coupling ${\cal K}(\omega_n)$, which contains all relevant information about the form of the interaction potential $V(r)$ and the geometry of the system. Formally, its functional form is defined by the correlator[@Pryadko-98] $$\label{eq:thermal-average} {\cal K}^{-1}(\omega_n) ={|\omega_n|\over2\pi} \bigl\langle\left|\varphi_n\right|^2\bigr\rangle_{\lambda=0}.$$ If the coupling ${\cal K}(\omega)$ is independent of the frequency, the effective action (\[eq:effective-model\]) can be visualized as describing an overdamped particle in a periodic (cosine) potential with Ohmic dissipation $\kappa={\cal K}/g^2$; the transport properties for this problem are known exactly[@Weiss-Exact; @Fendley-95B]. In general, however, the exact solution is not available, and we have to rely on the frequency-shell perturbative renormalization group (RG). The main idea is that the non-linear term is irrelevant for large-frequency modes $\varphi(\omega)$, as long as $|\omega|\gg \lambda$. When such modes are integrated out, the tunneling constant for the remaining slow modes is reduced, $$\label{eq:eff-tunn} \lambda(\Lambda) =\lambda(\Lambda_0)\left\langle e^{ig\varphi} \right\rangle_{\Lambda<\omega<\Lambda_0},$$ or, equivalently, $$-\ln{\lambda(\Lambda)\over \lambda(\Lambda_0)}= {g^2}\int_{\Lambda_0}^\Lambda {d\omega\over2\pi} \left\langle |\varphi(\omega)|^2\right\rangle_{\lambda=0} ={g^2}\int_{\Lambda_0}^\Lambda {d\omega\over\omega{\cal K}(\omega)},$$ where we used the definition (\[eq:thermal-average\]). After the frequencies are rescaled to restore the original upper cutoff, we arrive at the usual RG equation $${d\ln\lambda\over d\ln\Lambda} =1-g^2\,{\cal K}^{-1}(\Lambda) \equiv 1-g_\star^2(\Lambda). \label{eq:define-gstar}$$ The renormalization stops at a lower cutoff scale determined either by the temperature or the applied voltage. Most importantly, for $g_\star^2 >1$, the tunneling amplitude flows to weak coupling as the temperature is lowered, so that the channel along the tunneling current becomes more insulating; for $g_\star^2 <1$ it flows to strong coupling. It should be pointed out that in the case where ${\cal K}(\omega)$ is [*frequency-independent*]{}, the parameter $g_\star^2$ \[defined in Eq. (\[eq:define-gstar\])\] is a constant, and the effective Euclidean action describing the system can be recast in the simpler form $$S={T\over4\pi}\sum_{n} |\omega_n|\,|\varphi_n|^2 +\int_0^\beta \! d\tau \, \re\,\lambda\,e^{i g_\star \varphi(\tau)}. \label{eq:effective-action-disc}$$ Such is indeed the case (for sufficiently small $\omega$) for the scale-invariant models considered in detail in Sec. \[sec:self-similar\]. In this situation, the RG equation leads to the standard result[@Kane-Fisher-Tunnel; @QHpers-book] $$\lambda_{\rm eff}\sim \max(T,\,V)^{g_\star^2-1}, \label{eq:lameff}$$ which can be also obtained by expanding the exact solution[@Weiss-Exact; @Fendley-95B]. Duality between weak tunneling and weak bacscattering {#sec:duality} ===================================================== The partition function corresponding to the effective action (\[eq:effective-model\]) \[which also describes an overdamped particle in a non-Ohmic dissipative environment, $\kappa(\omega)={\cal K}(\omega)/g^2$\] can be also rewritten[@Schmid-83; @Guinea-85] in terms of the dual variable $\Delta\theta$ with the identical action, up to a replacement ${\cal K}(\omega_n)\to 1/{\cal K}(\omega_n)$, $g\to 1/g$, and the modified tunneling coefficient $\lambda\to \tilde\lambda$ (which has the meaning of fugacity for the instanton of the original field $\varphi$). In terms of edge modes, this duality[@Fendley-95B; @Weiss-Exact] represents a freedom to describe the same junction in terms of [*weak*]{} tunneling or [*strong*]{} backscattering, and vice versa. The main advantage of the duality is the ability to substitute a problem at [*strong*]{} tunneling with its dual, which can be then accessed perturbatively. This argument relies heavily on the properties of the effective model (\[eq:effective-model\]), which, in principle, may or may not remain equivalent to the original edge model after the addition of the non-local coupling (\[eq:coulomb-action\]). To illustrate the mutual consistency of the two models, we derive the relationship between the coupling ${\cal K}(\omega)$ in the two tunneling geometries directly, using only the quadratic action ${\cal S}_{\rm q}\equiv {\cal S}_0+{\cal S}_1$. Consider a field configuration with the boundary conditions fixed as in Fig. \[fig:dual-proof\]a, where $u_i=u_i(\tau)$ are given. Everywhere on the composite contour $C\equiv C_1+C_2$ the action is quadratic, and the corresponding Euler-Lagrange equation is linear, $$\partial_x\left[i\partial_\tau u+v(x)\,\partial_x u+ {\nu\,e^2\over 2\pi\varepsilon}\int_{C} dy\, V(|{\bf r}_x-{\bf r}_y|)\,\partial_y u\right]=0.$$ The classical solution is uniquely determined by the given values $u_i(\tau)$ of the fields at the endpoints. The quadratic action (\[eq:finite-size-edge-action\]), (\[eq:coulomb-action\]), evaluated along this classical solution, can be written as $$\begin{aligned} {\cal S}_{\rm q}[u]&=&{\cal G}[u_1\!-\!u_0,\,u_3\!-\!u_2] +\int d\tau{(u_1\!-\!u_0)\,i\partial_\tau (u_1\!+\!u_0)\over 4\pi}\nonumber\\ & & +\int d\tau{(u_3\!-\!u_2)\,i\partial_\tau (u_3\!+\!u_2)\over 4\pi}, \label{eq:formally-evaluated-action}\end{aligned}$$ where ${\cal G}[a,b]$ is a quadratic, non-local in time, and generally very complicated functional of its arguments. =0.9 The conservation of the total charge $$Q={\sqrt\nu \over 2\pi}(u_3-u_2+u_1-u_0) \label{eq:charge-conserv}$$ requires that $\varphi\equiv{}u_1-u_0=u_2-u_3$, up to a time-independent constant. Setting the total charge to zero, we can write Eq. (\[eq:formally-evaluated-action\]) as $$\label{eq:formally-evaluated-action-two} {\cal S}_{\rm q}[\varphi,\Delta\theta]= {\cal G}[\varphi,-\varphi] -{1\over 2\pi}\! \int\! d\tau\,{\varphi\,i\partial_\tau \Delta\theta}.$$ where $\Delta\theta\equiv (u_3\!+\!u_2\!-\!u_1\!-\!u_0)/2$. For the tunneling geometry in Fig. \[fig:dual-proof\]b, Eq. (\[eq:charge-conserv\]) implies that the field $u(x,\tau)$ can be chosen continuous everywhere along the combined edge $C_1+C_2$, $\Delta\theta=0$ and hence the effective quadratic action becomes $$S_{\rm q} ={\cal G}[\varphi,\,-\varphi]\equiv {T\over4\pi}\sum_{\omega_n= 2\pi\,nT} |\omega_n| \,{\cal K}(\omega_n)\,|\varphi_n|^2 ,$$ where we introduced the coupling ${\cal K}(\omega)$ as in Eq. (\[eq:effective-model\]). For the tunneling geometry in Fig. \[fig:dual-proof\]c, the charges in upper and lower areas change with time as a result of the tunneling, and we must keep the field $u(x,\tau)$ discontinuous. The corresponding action becomes $$\tilde S_{\rm q} = {T\over4\pi}\sum_{n} |\omega_n| \,{\cal K}(\omega_n)\,|\varphi_n|^2 +\omega_n (\bar\varphi_{n}\Delta\theta_n-\Delta\bar\theta_{n}\varphi_n).$$ (Note that a different choice of $\Delta\theta$, [*e.g.*]{}, $\Delta\theta=u_3-u_0$ or $\Delta\theta=u_2-u_1$, only changes the Euclidean Lagrangian by a total time derivative, thus leaving the action $\tilde{S}_{\rm q}$ invariant.) The field $\varphi$ can be now trivially integrated out, and we arrive at the final form of quadratic action for this geometry, $$\label{eq:dual-effective-action} \tilde S_{\rm q} = {T\over4\pi}\sum_{\omega_n} |\omega_n| \,\tilde{\cal K}(\omega_n)\,|\Delta\theta_n|^2, \quad \tilde{\cal K}(\omega_n) ={1\over{\cal K}(\omega_n)}.$$ This result can be generalized for systems with several junctions, where the coupling ${\cal K}(\omega)$ is replaced by a matrix, which is inverted when all junctions are replaced by their duals[@Pryadko-98]. This simple calculation shows that even in the presence of long-range interactions the duality between weak tunneling and weak backscattering for the model described by Eqns. (\[eq:finite-size-edge-action\]), (\[eq:coulomb-action\]), (\[eq:tunn-action\]) coincides with the duality between weak and strong coupling for the effective tunneling model (\[eq:effective-model\]), independent of the actual geometry of the system. The only assumption we made is that the geometries in Fig. \[fig:dual-proof\]b and Fig. \[fig:dual-proof\]c should not differ “substantially”, that is, the size of a junction near a saddle point should be sufficiently small ([*e.g.*]{}, compared with a short-distance cut-off length, or, at small enough frequencies, with the wavelength $v/\omega$), so that the Coulomb potential would be the same in the points $u_0,\ldots,u_3$. Scale-invariant models {#sec:self-similar} ====================== In the absence of long-range forces ($e^2=0$), the properties of any system are determined only by the relative location of the tunneling points along the edges. If such a system has only one tunneling point, in the limit where both contours $C_1$ and $C_2$ in Fig. \[fig:dual-proof\] become infinite, the system would not “know” the difference between the geometries in Fig. \[fig:dual-proof\]b and Fig. \[fig:dual-proof\]c, and the duality implies that the coupling has a universal self-dual value ${\cal K}(\omega)=1$, independent of the actual geometry of the edges. Of course, this statement requires that $\omega\,L/v\gg1$, otherwise one can obtain[@Pryadko-98] for Figs. \[fig:dual-proof\]b and \[fig:dual-proof\]c respectively $$%% \label{eq:K-matr-c1a} {\cal K}^{(\ref{fig:dual-proof}b)}= \left[{\cal K}^{(\ref{fig:dual-proof}c)}\right]^{-1} =%{s_1 s_2-1\over(s_1-1)(s_2-1)}\equiv %% {e^{\omega (L_1+L_2)}\!-\!1\over(e^{\omega L_1}\!-\!1) (e^{\omega %% L_2}\!-\!1)}= {1\over2}\left|\coth\Bigl({\omega L_1\over 2v}\Bigr) +\coth\Bigl({\omega L_2\over 2v}\Bigr)\right|,$$ where $L_i$ is the length of the contour $C_i$, and a uniform edge velocity $v(x)=v$ is assumed for simplicity. In the presence of Coulomb interactions, the functional form ${\cal K}(\omega)$ has been previously found[@Moon-96; @Oreg-96; @Imura-97] only for two [*parallel*]{} edges ($\alpha\to 0$ or $\alpha=\pi$ in Fig. \[fig:angles\]), where the translational symmetry of the quadratic part of the action is restored. In any other geometry the distance $|x-y|$ measured along the edges, and the geometrical distance $R_{xy}\equiv |{\bf r}_x-{\bf r}_y|$ in Eq. (\[eq:coulomb-action\]) are no longer equivalent, and an analytic computation of the average (\[eq:thermal-average\]) with “non-interacting” quadratic action ${\cal S}_0+{\cal S}_1$ becomes virtually impossible. Some simplification can be achieved for an idealized [X]{}-shaped geometry (see Fig. \[fig:angles\]), which can be also introduced as the zero-bias limit of the edges in a vicinity of a saddle point with the opening angle $\alpha$. For the special case of unscreened Coulomb potential, $$\label{eq:coulomb} V(R)=\left({R^2+a^2}\right)^{-1/2},$$ the long-range interaction term (\[eq:coulomb-action\]) scales the same way as the local potential (velocity) term in Eq. (\[eq:edge-action\]). Then, if the edge velocity $v(x)=v$ is coordinate-independent the action becomes scale invariant for a sufficiently small short-distance cutoff $a$. This implies that the function ${\cal K}_\alpha(\omega)$, for a given opening angle $\alpha$, can depend on the frequency at most logarithmically. In this regime the geometry of the edges and the tunneling properties of the junction \[[*i.e.*]{}, the function ${\cal K}_\alpha(\omega)$\] are fully determined by the angle $\alpha$ and the dimensionless coupling constant[@Moon-96] $$\chi\equiv {\nu\,e^2/(\pi\hbar v\varepsilon)}\label{eq:coupling-constant}.$$ The duality discussed in the previous section implies that ${\cal K}_{\pi-\alpha}(\omega)={\cal K}^{-1}_{\alpha}(\omega)$, for given values of the coupling constant $\chi$ and the cut-off scale $a$. Therefore, in the self-dual geometry at $\alpha=\pi/2$, we expect $K_{\pi/2}=1$ exactly, independent of the form or the strength of the interaction potential $V(x)$. To rewrite more explicitly the general Coulomb action (\[eq:coulomb-action\]) for the infinite geometry in Fig. \[fig:angles\], let us introduce the coordinate $x$ along each edge, with the origin at the tunneling point and positive direction to the right. Then the charge densities along the top and the bottom boundaries are respectively $\rho_1(x)=\sqrt\nu\partial_x\,u_1(x)/(2\pi)$ and $\rho_2(x)=-\sqrt\nu\partial_x\,u_2(x)/(2\pi)$ (the sign in the second expression differs because the coordinate is now chosen in the direction opposite to the edge velocity). The Coulomb part of the action becomes $${\cal S}_1= {\chi\over8\pi}\!\int\! d\tau\!\int_{-\infty}^\infty\!\! dx\,dy \! \sum_{i,j=1,2}(-1)^{i+j} \partial_x u_iV_{ij}(x,y)\partial_y u_j,$$ where the potential $V_{ij}(x,y)\equiv V\biglb(|{\bf r}_{i}(x)-{\bf r}_{j}(y)|\bigrb)$ denotes the interaction energy between unit charges at the points $x$ and $y$ at the edges $i$ and $j$ respectively, and we changed the units of distance: from now on $v=1$. For symmetric geometries $V_{ij}(x,y)=V_{ij}(y,x)$, $V_{11}(x,y)=V_{22}(x,y)$, and the obtained expression can be diagonalized by introducing the symmetric and antisymmetric combinations $\varphi=u_1-u_2$, $\vartheta=u_1+u_2$. The quadratic part (\[eq:finite-size-edge-action\]), (\[eq:coulomb-action\]) of the Euclidean action becomes $$\begin{aligned} \lefteqn{{\cal S}_{\rm q}={T\over8\pi}\sum_n\biggl\{\int dx\left[ 2\omega_n\bar\varphi(x)\,\vartheta'_x+|\varphi'_x|^2 +|\vartheta'_x|^2\right] \nonumber} & & \\ & &\;\, + {\chi\over2} \int \!dx\!\int \!dy\,\left[\bar\varphi'_x\,V_+(x,y)\,\varphi'_y +\bar\vartheta'_x\,V_-(x,y)\,\vartheta'_y\right]\biggr\}, \label{eq:symmetrized-action}\end{aligned}$$ where $V_\pm(x,y)\equiv V_{11}(x,y)\pm V_{12}(x,y)$ and the coordinate integrations are performed along the entire real axis. Note that the first term of the integrand is not written as $\omega_n(\bar\varphi\,\vartheta'_x-\bar\vartheta\,\varphi'_x)$ as would be expected from the action (\[eq:edge-action\]); the integrand in Eq. (\[eq:symmetrized-action\]) differs by a full spatial derivative, exactly equivalent to the surface term in the second line of Eq. (\[eq:finite-size-edge-action\]). The interaction potential is always an even function with respect to simultaneous reflection of both coordinates, $V_\pm(x,y)=V_\pm(-x,-y)$, and the fields $\varphi=\varphi_s+\varphi_a$ and $\vartheta=\vartheta_s+\vartheta_a$ can be separated into symmetric ($s$) and antisymmetric ($a$) components. The first term of the action (\[eq:symmetrized-action\]) couples only the components of two fields with the opposite symmetry: $\varphi_s$ with $\vartheta_a$, and $\varphi_a$ with $\vartheta_s$. Since the tunneling term depends on the field $\varphi(0)=\varphi_s(x=0)$ only, the components $\varphi_a(x)$ and $\vartheta_s(x)$ decouple and can be integrated out independently of the value $\varphi(0)$. In the following, we shall presume that this symmetrization has been done, and use $$\varphi(x)=\varphi(-x),\quad \vartheta(x)=-\vartheta(-x), \label{eq:field-symmetry}$$ with the indices “$s$” and “$a$” dropped for convenience. Exactly-solvable example ------------------------ To illustrate the properties of the symmetrized action (\[eq:symmetrized-action\]), consider a model problem where the interaction happens only between the points at equal distance from the origin, $$\begin{aligned} (\chi/2)\, V_{11}(x,y)&=&v_0\delta(x-y)+v_1\delta(x+y),\\ (\chi/2)\, V_{12}(x,y)&=&v_2\delta(x-y)+v_3\delta(x+y), \end{aligned}$$ where the velocity $v_0$ (measured in units of bare velocity $v$) denotes the strength of additional interaction at the same edge, $v_1$ and $v_2$ denote the interaction between the neighboring edges (left–right and top–bottom), while $v_3$ denotes the interaction between the points at the opposing edges. (Physically, this set of interactions corresponds to four locally-interacting chiral edges, running along the surface of a semi-infinite cylinder and meeting in the tunneling point at its near end). With interaction of this simple form we can use the symmetry properties (\[eq:field-symmetry\]), and the quadratic action (\[eq:symmetrized-action\]) becomes entirely [*local*]{}, $${\cal S}_{\rm q}={T\over8\pi}\sum_n\int dx\left[ 2\omega_n\bar\varphi(x)\,\partial_x\vartheta+ v_\varphi |\partial_x\varphi|^2+ v_\vartheta |\partial_x\vartheta|^2 \right],$$ with $v_{\varphi,\vartheta}=1+v_0-v_3\pm (v_1-v_2)$. Now the field $\vartheta(x)$ can be trivially integrated out, and Eq. (\[eq:thermal-average\]) gives $${\cal K}^{-1}(\omega_n)={1+v_0-v_3+ (v_1-v_2)\over 1+v_0-v_3- (v_1-v_2)}.$$ Clearly, under interchange $v_1\leftrightarrow v_2$ this expression goes to its inverse according to the duality relation derived in Sec. \[sec:duality\], and ${\cal K}(\omega_n)=1$ for the self-dual case $v_1=v_2$ where all edges are equivalent. Coulomb interactions near a saddle point {#sec:coulomb-wedge} ---------------------------------------- Now let us consider more realistic long-distance interactions in the edge geometry shown in Fig. \[fig:angles\]. We write the intra- and inter-edge interaction potentials $$\begin{aligned} V_{11}(x,y)&=&\tf(xy)V(x-y) +\tf(-xy)V(R_\alpha),\\ V_{12}(x,y)&=&\tf(xy)V(R_\alpha) +\tf(-xy)V(x-y),\end{aligned}$$ where the bulk distance $R_\alpha\equiv ({x^2+y^2-2xy\cos\alpha})^{1/2}$, $\tf(x)$ is the usual step function, $\tf(x)=1$ for $x>0$ and $\tf(x)=0$ otherwise, and $V(x)$ is, [ *e.g.*]{}, the Coulomb potential (\[eq:coulomb\]). The resulting effective action has the form (\[eq:symmetrized-action\]), with $$\begin{aligned} \label{eq:v-plus} V_+&=&V(x-y) +V(R_\alpha),\\ V_-&=&\left[V(x-y) -V(R_\alpha)\right]\sgn (xy). \label{eq:v-minus}\end{aligned}$$ In the limit $\alpha=0$, $R_\alpha=|x-y|$, the antisymmetric part of the potential vanishes, $V_-(x,y)=0$, while $V_+(x,y)=2V(x-y)$, and we obtain the usual translationally-invariant action for two parallel edges. Integrating out the field $\vartheta$ and diagonalizing the remaining part of the action with the help of Fourrier transformation, we use Eq. (\[eq:thermal-average\]) to calculate the coupling, $$\label{eq:coupling-zero-angle} {\cal K}_{\alpha=0}^{-1}(\bar\omega) ={2\bar\omega\over\pi}\int_0^\infty {d\zeta\over \bar\omega^2+\zeta^2\,[1+2\chi\,K_0(\zeta)] },$$ where the Fourrier-transformed Coulomb potential (\[eq:coulomb\]), $V(\zeta)=2K_0(\zeta)$, is expressed in terms of the modified Bessel function $K_0$, and the reduced frequency $\bar\omega=a\omega/v$. Performing the integration with logarithmic accuracy, we obtain, in agreement with Refs.  $${\cal K}_{\alpha=0}= \left[1+2\chi\ln\left({ 2\sqrt{2\chi}\,e^{-\gamma}\over\bar\omega }\right)\right]^{1/2} +{\cal O}\bigl(|\ln \bar\omega|^{-1/2}\bigr), \label{eq:alpha0}$$ where $\gamma\approx 0.577$ is the Euler constant. The case $\alpha=\pi$ corresponds to two semi-infinite non-chiral Luttinger liquids connected by a tunneling point ($\alpha\to\pi$ in Fig. \[fig:angles\]); by duality we expect[@Imura-97] ${\cal K}_{\alpha=\pi} =1/{\cal K}_{\alpha=0}$. This expression is proved again, specifically for this geometry, in Appendix \[sec:appendix-pi\]. We argued that in the self-dual case $\alpha=\pi/2$, ${\cal K}(\omega)=1$ identically, independently of the properties of the potential $V(x)$, as long as it is appropriately regularized at short distances. We have also constructed a direct analytical solution for this case. The major simplification comes from an observation that the potential $V(R_{\pi/2})= V(\sqrt{x^2+y^2})$ is a symmetric function of $x$ and $y$ independently; the corresponding contribution vanishes from the action (\[eq:symmetrized-action\]) by the symmetry (\[eq:field-symmetry\]). As a result, only the potentials $V(x\pm y)$ with the distance measured along the edge enter the extremum equations, and these equations can be solved exactly using the Wiener-Hopf method, as detailed in Appendix \[sec:wiener-hopf\]. This direct solution confirms the universal result ${\cal K}_{\alpha=\pi/2}=1$. In addition, the explicitly found extremum configuration of the fields $\varphi(x)$, $\vartheta(x)$ is used to get a perturbative expression for ${\cal K}_\alpha(\omega)$ near the self-dual point $\alpha_0=\pi/2$. This yields (see Appendix B) $$\label{eq:perturbation} K_\alpha(\omega\to0)\approx 1+{\cal N}(\chi)\,\chi\cos\alpha, \quad |\cos\alpha|\ll 1,$$ where ${\cal N}(\chi)$ is [*independent of*]{} $\omega$. In the limit of weak Coulomb interactions, ${\cal N}(\chi\to0)\approx1.51$, while ${\cal N}(\chi=1.0)\approx0.21$. To get a handle on the dependence of the coupling ${\cal K}_\alpha(\omega)$ on the parameters and the cut-off scales, we have also evaluated the average (\[eq:thermal-average\]) numerically for the quadratic action (\[eq:symmetrized-action\]) with the Coulomb potential (\[eq:coulomb\]) at different frequencies $\omega$, and for different values of the angle $\alpha$ and the dimensionless coupling constant $\chi$. To perform this calculation we wrote a discretized version of the quadratic action (\[eq:symmetrized-action\]) in terms of lattice values $\varphi(x_n)$ and $\vartheta'(x_n)$, $0<n<N-1$, and then integrated out the values of the fields away from the origin, which only required inverting two $N\times N$ matrices. In addition to the cut-off distance $a$ in Eq. (\[eq:coulomb\]), the discretization involved two explicit cut-off scales: the total system size $L$ and the lattice grid size $h=L/N$. The calculations were performed in the regime $h\ll a\ll L$; the results are independent of these cut-off scales in the frequency range $h\ll v/\omega\ll L$. These inequalities substantially limited the dynamical range where the results are accurate. Typical results of the calculations are illustrated in Fig. \[fig:K-frequency\] and Fig. \[fig:K-angle\]. The curves in Fig. \[fig:K-frequency\] with marked values of $\cos(\alpha)$ show superimposed values ${\cal K}_\alpha(\bar\omega)$, ${\cal K}^{-1}_{\pi-\alpha}(\bar\omega)$ calculated with the lattice size $N=1600$, for cut-off parameters $a=0.05$, $0.1$. The deviation betwen the curves shows that our discretization violated the self-duality of the problem at both large and small cutoff scales. Nevertheless, as illustrated in Fig. \[fig:K-angle\], the self-duality holds with a very good numerical accuracy near the middle of the dynamical range, $a\,\omega/v\sim 0.1$. As indicated by finite-size scaling analysis of our data (not shown), at small enough $\omega$, ${\cal K}_\alpha(\omega)$ saturates to a frequency-independent value in the range $0<\alpha<\pi$. This behavior is consistent with the small-angle expansion (\[eq:perturbation\]). In addition, Fig. \[fig:K-angle\] indicates that Eq. (\[eq:perturbation\]) provides a good approximation to ${\cal K}_\alpha(\omega)$ in a rather wide range of $\alpha$. For small $\alpha\ll1$, as the frequency is reduced, the numerical values ${\cal K}_\alpha(\omega)$ seem to closely follow the logarithmically divergent line (\[eq:alpha0\]), but eventually cross over to a constant value ${\cal K}_\alpha(\omega=0,\chi)$, which (logarithmically) depends on the angle and the cut-off scale $a$. = = Coulomb interactions in the cleaved edge geometry {#sec:Grayson} ------------------------------------------------- Here we consider the effect of long-range interactions in the cleaved-edge geometry[@Chang-96; @Grayson-98], where the tunneling happens between a three-dimensional metal and the edge of a 2DEG, located in the plane perpendicular to the surface of the metal. It is believed that the tunneling in these experiments is dominated by localized “hot” spots or impurities. Chamon and Fradkin[@Chamon-97] demontstated that in the absence of interactions, a point contact between a 3D metal and a QH edge with the filling fraction $\nu$ is equivalent to a point tunneling junction between such an edge and an ideal non-interacting $\nu=1$ edge; furthermore, they mapped this latter problem to that of tunneling between two identical edges with filling fractions $\nu_*=2\nu/(1+\nu)$. = The effect of the Coulomb interaction in this setup is limited to the chiral Luttinger liquid, the “real” quantum Hall edge, while the Fermi-liquid nature of quasiparticles in the metal imply that they remain non-interacting for the purposes of tunneling measurements. The metallic surface only provides additional screening charges, which modify the form of the interaction potential $V\left(|{\bf r}_x-{\bf r}_y|\right)$. Assuming characteristic frequencies at the edge are small compared with the plasma frequency of electrons in metal (which is always true for a good metal), the retardation can be neglected, and the modified interaction potential is obtained simply by adding the appropriate image charges. The quadratic part of the action for the translationally-invariant geometry shown in the left part of Fig. \[fig:cleaved\] ([*i.e*]{}., the case $\alpha=0$) is obtained by combining Eq. (\[eq:finite-size-edge-action\]) with the Coulomb energy $$\label{eq:cleaved-action-coulomb} {\cal S}_1= {\chi\over 8\pi}\int_0^\beta d\tau\int_{-\infty}^\infty \!\!dx\,dy \,\partial_x u\,\hat V(x-y)\, \partial_y u ,$$ where $\hat V(x)\equiv V(x)-V(\sqrt{x^2+4a^2})$ is corrected for the image potential, and the units of length are again chosen so that the edge velocity $v=1$. Because we work with the chiral field now, the surface term in the second line of Eq. (\[eq:finite-size-edge-action\]) is absolutely essential even in an infinite geometry. To properly account for this term, we formally separate the field $u=\phi+\theta$ into its symmetric $\phi(x,\tau)=\phi(-x,\tau)$ and antisymmetric $\theta(x,\tau)=-\theta(-x,\tau)$ compontents; then the surface term can be absorbed after an integration by parts, and the action (\[eq:finite-size-edge-action\]) becomes $$\label{eq:cleaved-symmetrized-quadratic} {\cal S}_0\!=\!{1\over 4\pi}\! \int%_0^\beta \! d\tau\!\int_{-\infty}^\infty\!\! dx \left[ 2i\partial_\tau\phi\,\partial_x\theta+ (\partial_x\phi)^2+ (\partial_x\theta)^2\right].$$ This transformation is equivalent to “folding” the chiral edge in half, which produces two non-chiral fields defined on a semiaxis, and simultaneously eliminates the zero mode and associated subtleties. The translationally invariant action can be now diagonalized by a Fourrier transformation, and, after integrating out the fluctuations away from the origin, we obtain the single-edge contribution to the quadratic part of the effective action, $$\label{eq:half-action} {\cal S}^{(1)}_{\rm q}={T\over 2\pi}\sum_n |\omega_n| \,\hat{\cal K}(\omega_n)\,|\phi_1|^2,$$ where $\phi_1\equiv u(0)=\phi(0)$ by definition, and $$\begin{aligned} \label{eq:cleaved-parallel-K} \hat{\cal K}^{-1}(\omega)&=&{2|\omega|\over\pi}\!\int_0^\infty\!\! {dk\,Z(k)\over \omega^2+k^2\,Z^2(k)}, \\ \nonumber %% {\rm where}\;\; Z(k)&=&1+{\chi\over2}\,\hat V(k).\end{aligned}$$ The argument[@Chamon-97] that a point contact with a metal is equivalent to that with a non-interacting $\nu=1$ edge holds independently of the interactions affecting the “real” edge. Therefore, the full effective action can be written as $$\label{eq:full-asymmetric-action} {\cal S}={T\over2\pi}\sum_n |\omega_n| \left({ \hat{\cal K}\,|\phi_1|^2 \! +|\phi_2|^2}\right) \!+\!\int\! d\tau\,\re\,\lambda\,e^{i(g \phi_1-\phi_2)},$$ where we used $\hat{\cal K}=1$ for the auxiliary $\nu=1$ edge. The canonical form (\[eq:effective-model\]) of the tunneling action can be obtained by introducing the tunneling degree of freedom, $\varphi=g \phi_1-\phi_2$, with the corresponding effective coupling ${\cal K}_{\rm eff}$ calculated, [*e.g.*]{}, using the average as in Eq. (\[eq:thermal-average\]). As before, the resulting model describes an overdamped particle in a washboard potential; the corresponding non-Ohmic “friction” coefficient $$\kappa_{\rm eff}(\omega)\equiv{{\cal K}_{\rm eff}\over g_{\rm eff}^2}={2\over g^2/\hat{\cal K}(\omega)+1}. \label{eq:friction-cleaved}$$ In the non-interacting limit $\hat{\cal K}(\omega)=1$ this expression safely goes into the result[@Chamon-97] obtained by a different method. Notice that the long-distance part of the Coulomb potential $\hat V$ in Eq. (\[eq:cleaved-action-coulomb\]) is screened by the metallic surface. Then, at sufficiently small frequencies, $a\omega\ll v_{\rm r}\equiv Z(0)\,v$, the momentum dependence of the coefficient $Z(k)$ can be ignored, and the integral (\[eq:cleaved-parallel-K\]) gives precisely the non-interacting coupling, $\hat{\cal K}=1$. This is not at all surprising, since the interaction happens within a single chiral edge, and its long-range part (most dangerous at small frequencies) is screened. As usual[@Volkov-88], the only effect of the additional interaction in this chiral system is the velocity renormalization, $v\to v_{\rm r}$. The translational symmetry is lost for the “wedge” geometry shown in the right part of Fig. \[fig:cleaved\]. The Coulomb part of the corresponding action can be written in the form (\[eq:cleaved-action-coulomb\]) with the potential $\hat V(x-y)\to%\hat V(x,y)= V_-(x,y)$ given by Eq. (\[eq:v-minus\]). In the limit $\alpha\to0$, the potential $V_-(x,y)$ vanishes identically, and hence $\hat{\cal K}(\omega)=1$ in this case as well. At general values of $\alpha$ we again use the “folding” trick by introducing symmetric and antisymmetric variables $\phi$, $\theta$. Up to an overall coefficient, the resulting action looks like Eq. (\[eq:symmetrized-action\]), with the exception that both components $\phi$ and $\theta$ couple with the [*same*]{} potential $V_-(x,y)$. The most prominent difference is that at $\alpha=\pi/2$ the symmetry no longer leads to a cancellation of the part $V(\sqrt{x^2+y^2})$ of the total potential, and the effect of the long-distance interactions is no longer trivial, $\hat{\cal K}_{\pi/2}(\omega)\neq1$. Again, this comes as no surprise, since there is no self-duality in this geometry. Finally, in the limiting case $\alpha=\pi$, the potential $V_-(x,y)$ becomes an even function of each argument; as a result, the coupling with the symmetric field $\phi$ (antisymmetric derivative $\partial_x\phi$) vanishes by symmetry. Up to an overall coefficient, the resulting action is identical to that considered in Appendix \[sec:appendix-pi\], and we obtain \[note that the extra coefficient was already accounted for in the corresponding effective action, [*cf*]{}. Eqns. (\[eq:half-action\]) and (\[eq:effective-model\])\], $$\hat{\cal K}_{\alpha=\pi}(\omega)={\cal K}_{\alpha=\pi}(\omega) ={2|\omega|\over \pi}\!\int_{0}^\infty\!\! {dk\over \omega^2+k^2 \biglb(1+\chi\,V(k)\bigrb)}.$$ This result is quite intuitive: metallic screening becomes non-effective in the case where a wire is perpendicular to the conducting surface. Our calculations imply that the tunneling exponent is modified by the Coulomb interaction only if the edge is bent near the tunneling point. In an ideal sample, the edge runs along a straight line parallel to the surface of the metal, and long-range interactions do not modify the tunneling exponents. In any real sample, however, imperfections near the tunneling point always reduce the effective coupling ${\cal K}(\omega)$, or, equivalently, systematically [*increase*]{} the tunneling exponent in Eq. (\[eq:lameff\]). Nevertheless, we do not believe this effect would be sufficient to explain a $10\%$ increase of the tunneling exponent observed[@Grayson-98] by Grayson [*et al*]{}. near $\nu=1$: cleaved-edge samples are characterized by sharp confinement and large drift velocities, meaning that the corresponding dimensionless coupling constant $\chi$ \[see Eq. (\[eq:coupling-constant\])\] is small. Discussion {#sec:discussion} ========== We have shown that the effect of long-range interactions on transport through a QH tunneling junction depends crucially on its geometry. In particular, in a self-similar [X]{}-shaped junction (see Fig. \[fig:angles\]) characterized by an opening angle $\alpha$, unscreened Coulomb interactions only renormalize the effective Luttinger-liquid exponent, $$g_\star^2=g^2/{\cal K}_\alpha(\omega=0,\,\chi),$$ where $g^2=1/\nu$ for electron tunneling between the edges of 2DEGs with Laughlin fractions $\nu$. Therefore, the renormalized exponent depends non-universally on the angle $\alpha$ and the dimensionless Coulomb interaction strength $\chi$. This implies that the system should exhibit a zero-temperature delocalization transition at a critical angle characterized by $g_\star^2=1$. This is in contrast with the transport properties expected in the absence of long range interactions, which are exclusively determined by the filling fraction $\nu$. For integer QH systems with $\nu=1$, the transition always corresponds to a self-dual geometry, [*i.e.*]{}, $\alpha_c=\pi/2$, independently of the details of the interaction. In fractional QH constrictions, however, the transition (if any) occurs at a non-universal critical angle $\alpha_c<\pi/2$, such that ${\cal K}_{\alpha_c}(0,\chi)=\nu^{-1}$. Properties of all charge transfer processes through the junction are defined by the parameter $g_\star$ in the effective action (\[eq:formally-evaluated-action-two\]), which determines the tunneling exponents[@Wen-91B; @Kane-Fisher-Tunnel] \[see Eq. (\[eq:lameff\])\], the form of the non-linear $I$–$V$ curve[@Weiss-Exact; @Fendley-95B], as well as the tunneling noise [@Kane-94A; @Fendley-95C; @QHpers-book]. In the limit of weak tunneling, the quantization of transferred charge is ultimately determined by gauge invariance, and a shot noise measurement would show current transferred by unit charges. However, the shot noise measured in the opposite, strongly coupled limit (reached, [*e.g.*]{}, by driving a large tunneling current through the junction), is set[@Sandler-noise] by the instanton charge for the effective tunneling action (\[eq:formally-evaluated-action-two\]). The value of this charge is determined solely by the value of $g_\star$. Hence, in this regime a noise measurement would show a non-universal charge $$e_\star/e=1/g^2_\star=\nu\,{\cal K}_\alpha(0,\,\chi),$$ clearly an interaction effect. The described situation applies to ideal systems without screening. More realistically, Coulomb interactions are screened at some finite length $\xi$. Then, for a junction with finite opening angle, $|\cos\alpha|<1$, the correction to tunneling exponents always vanishes in the static limit, ${\cal K}^{\rm (scr)}_\alpha(0)=1$, even though it may be significant at larger frequencies, $\omega\gtrsim v/\xi$ (this corresponds to a temperature $0.1$ K for $\xi=1\;\,\mu$m and $v=10^{-7}$ cm/s). Consequently, a system at a fractional $\nu$ with originally metallic behavior would eventually localize at small enough temperatures.[@Moon-96; @Imura-97] Contrarily, the interaction-induced flow in an integer junction would gradually stop without changing its direction. For an [X]{}-shaped junction with a given opening angle $\alpha$, the magnitude of the renormalization parameter ${\cal K}_\alpha(0,\chi)$ is determined by the value of the dimensionless Coulomb interaction constant (\[eq:coupling-constant\]), which, in turn, is defined by the edge wave (drift) velocity. For cleaved-edge samples, edge magnetoplasmon velocities have been measured[@velocity-cleaved] by Ashoori [*et al.*]{}, yielding $v\sim 10^8$ cm/s which corresponds to $\chi\sim 0.05$. On the other hand, edge electric fields equivalent to drift velocities as small as $v\sim 10^6$ cm/s have been measured by Maasilta and Goldman[@Maasilta-97], who analyzed discrete energy levels of a quantum antidot. This value of velocity results in a relatively large coupling constant value $\chi\sim5$. We must point out, however, that our discussion of Coulomb interaction effects was based on a single-mode sharp edge, which implies large confining electric fields of order ${\cal E}\sim E_g/(e\ell)$, where $E_g$ is the energy gap associated with the incompressible QH state, and $\ell$ is the magnetic length. Using the drift velocity $v=c {\cal E}/B$, we obtain $$\chi={\nu \,e^2\over \pi\varepsilon\hbar v} \sim\left({\nu\,e^2\over \pi\varepsilon\ell}\right)\,E_g^{-1},$$ which, for a typical QH sample, leads to $\chi\lesssim 1$. Samples with much larger values of the Coulomb coupling are likely to have a tendency to edge reconstruction. This would lead to additional polarization at the edge due to neutral modes, and, consequently, a partial screening of Coulomb interaction. Therefore, to observe the predicted effects, samples with well-defined, but not too sharp edges are necessary. This excludes the cleaved-edge samples (where the drift velocity $v$ is large), as well as the samples with electrostatically-defined geometry (where confinement tends to be soft). The best choice would therefore be a Hall bar with lithographically defined [X]{}-shaped constriction and a narrow local gate to fine-tune the tunneling. For a given base temperature $T$, the linear size of the constriction should be at least of order $\xi\sim\hbar v/T$, [*i.e.*]{}, approaching a millimeter scale for millikelvin temperature range. Tunneling junctions with small opening angles will give larger values of ${\cal K}_\alpha$ \[in principle, limited only by the logarithm (\[eq:alpha0\]), divergent at small frequencies\]. However, as illustrated in Fig. \[fig:K-frequency\], for such junctions the renormalized Luttinger parameter $g_\star^2$ is more likely to retain some frequency (temperature) dependence, which would modify the measured exponents. We gratefully acknowledge useful discussions with C. de C. Chamon, M. Fogler, C. Glattli, E. Gwinn, S. A. Kivelson, D.-H. Lee, Z. Nussinov, S. L. Sondhi, and X.-G. Wen. A.A. acknowledges support of the Israeli Science Foundation and the fund for Promotion of Research in the Technion. E.S. acknowledges support by grant no. 96–00294 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel. L.P.P. was supported in part by DOE Grant No. DE-FG02-90ER40542. Coupling at $\alpha=\pi$. {#sec:appendix-pi} ========================= Here we derive the form of the coupling ${\cal K}(\omega)$ for the saddle-point geometry shown in Fig. \[fig:angles\] in the special limit $\alpha=\pi$, which corresponds to two vertical semi-infinite wires connected by a single tunneling point. In this case the distance $R_\alpha=|x+y|$, and the contribution of the symmetric potential $V_+(x,y)=V(x-y)+V(x+y)$ to Eq. (\[eq:symmetrized-action\]) vanishes by symmetry (\[eq:field-symmetry\]), so that only the part $V_-(x,y)=[V(x-y)-V(x+y)]\sgn (xy)$ remains. The symmetry of the derivative $\partial_x\vartheta$ implies that both parts of the potential $V_-$ give identical contribution, and the quadratic part of the action (\[eq:symmetrized-action\]) can be written as $$\begin{aligned} \lefteqn{{\cal S}_{\rm q} ={T\over8\pi}\sum_n\biggl\{\int_{-\infty}^\infty dx\left[ 2\omega_n\bar\varphi(x)\,\partial_x\vartheta +|\partial_x\varphi|^2 +|\partial_x\vartheta|^2\right] \nonumber} & & \\ & &\quad + {\chi} \int_{-\infty}^\infty dx \,dy\,\left[\partial_x\bar\vartheta\,V(x-y)\,\partial_y\vartheta\,{\rm sgn}(xy) \right]\biggr\}. \label{eq:wedge-action-pi}\end{aligned}$$ Unlike the case $\alpha=0$, the non-local interaction in the second line cannot be diagonalized by a simple Fourrier transformation; we need to get rid of the sign function first. Naively, this could be done by multiplying both $\varphi(x)$ and $\vartheta(x)$ by ${\rm sgn}(x)$. However, since $\varphi(0)\neq0$, the function $\varphi(x)\sgn (x)$ would not be continuous at the origin, so that spurious $\delta$-functions may be generated. Instead, we define auxiliary continuous functions $u(x)$, $g(x)$, so that $$\varphi(x)=\varphi(0)+\sgn (x)\,u(x),\quad g(x)=\vartheta(\infty)-\sgn (x)\,\vartheta(x),$$ and $u(0)=g(\infty)=0$. After integrating out the field $u(x)$, the effective action becomes $$\begin{aligned} {\cal S}_{\rm q} &=&{T\over8\pi}\sum_n\biggl\{ -4\omega_n\bar\varphi(0)\,g(0)\\ & & \quad+\int {dk\over2\pi} |g_k|^2 \left[ \omega_n^2+k^2\biglb(1+\chi \,V(k) \bigrb) \right]\biggr\}. \end{aligned}$$ In the first term here we substitute $g(0)=\int {dk\,g_k/(2\pi)}$ in terms of the Fourrier-transformed field $g_k$, integrate this field out, and obtain the effective action for the the field $\varphi(0)$ alone, $${\cal S}_{\rm q} ={T\over4\pi}\sum_n \omega_n \,|\varphi(0)|^2 \left[ {2\omega_n\over \pi}\!\int_{0}^\infty\!\! {dk\over \omega_n^2+k^2 \biglb(1+\chi\,V(k)\bigrb)}\right];$$ comparing the result with the general form of the effective action (\[eq:effective-model\]), and the result (\[eq:coupling-zero-angle\]) for $\alpha=0$, we conclude that $${\cal K}_{\alpha=0}(\omega_n)\, {\cal K}_{\alpha=\pi}(\omega_n)=1$$ exactly, independent of the form of the potential $V(x)$. Self-dual tunneling junction, $\alpha=\pi/2$ {#sec:wiener-hopf} ============================================ General Wiener-Hopf solution. ----------------------------- Here we give a direct solution of the extremum equations for the self-dual case $\alpha=\pi/2$. This solution gives the coupling ${\cal K}_{\pi/2}=1$ directly, without utilizing the self-duality of the problem. In addition, it allows us to calculate $K_\alpha$ perturbatively for small values of $|\cos(\alpha)|\ll 1$. Begin with the Euclidean action (\[eq:symmetrized-action\]) at $\alpha=\pi/2$, $$\begin{aligned} \label{eq:wedge-action-half-pi} {\cal S}_{\rm q} & =& {T\over8\pi}\sum_n\int dx\biggl\{ 2\omega_n\bar\varphi(x)\,\vartheta'_x +\int dy\,\bar\varphi'_x\,Z(x-y)\,\varphi'_y \nonumber \\ & & \qquad+\int dy\, \bar\vartheta'_x\,Z(x-y)\sgn (xy)\,\vartheta'_y \biggr\},\end{aligned}$$ where the total potential $$Z(x-y)=\delta(x-y)+{\chi\over2}\,V(x-y); \label{eq:total-potential}$$ note that due to the symmetry (\[eq:field-symmetry\]), the contribution from the part of the potential with geometrical distance, $V(R_{\pi/2})=V(\sqrt{x^2+y^2})$, was cancelled. The Euler-Lagrange equations (valid at $x\neq0$, where the non-linear tunneling term gives no contribution) are $$\begin{aligned} \label{eq:EL-phi} \omega\partial_x\vartheta&-&\partial_x\int_{-\infty}^\infty \!dy \, Z(x-y)\,\partial_y\varphi =0,\\ \label{eq:EL-theta} \omega\partial_x\varphi&-&\partial_x \int_{-\infty}^\infty\! dy \, Z(x-y)\sgn (xy)\,\partial_y\vartheta=0.\end{aligned}$$ We assume that both fields are continuous everywhere, and that $\varphi(x)$ and $\partial_x\vartheta(x)$ vanish at infinity. Multiplying the first of the obtained equations by $\bar\varphi(x)$, the second by $\bar\vartheta(x)$, and subtracting the results from the integrand in the action (\[eq:wedge-action-half-pi\]), with the help of the definition (\[eq:total-potential\]) we obtain $$\begin{aligned} \lefteqn{{\cal S}_{\rm q}= {T\over8\pi}\sum_n\!\int\!\! dx\,\partial_x\biggl[ \omega_n\bar\varphi\,\vartheta\!+\!\bar\varphi(x)\!\int\!\! dy \,Z(x-y)\,\partial_y\varphi}\nonumber\\ & & \qquad+ \bar\vartheta(x) \int\! \!dy \,Z(x-y) \sgn(xy)\,\partial_y\vartheta\biggr] \nonumber\\ & =& - {T\over8\pi}\sum_n\bar\varphi(0)\,\Delta\varphi'_0, \quad \Delta\varphi'_0\equiv \varphi'(0_+)-\varphi'(0_-), \label{eq:effective-action-evaluated}\end{aligned}$$ where the integration was performed over the entire axis excluding the point $x=0$. The Euler-Lagrange equations (\[eq:EL-phi\]), (\[eq:EL-theta\]) can be simplified by defining linear combinations (symmetric with respect to $x$) $$A,B(x)=[\varphi(x)\pm\vartheta(x)\sgn(x)]/2, \label{eq:a-b-defined}$$ then, multiplying Eq. (\[eq:EL-theta\]) by $\sgn(x)$ and taking symmetric and antisymmetric combinations of the result with Eq. (\[eq:EL-phi\]), we obtain at $x\neq0$ $$\label{eq:EL-A} \omega\sgn(x)\partial_x A-\partial_x\int_{-\infty}^\infty \!dy \, Z(x-y)\,\partial_y A =0,$$ and an identical equation (up to the substitution $\omega\to-\omega$) for $B(x)$. We integrate, keeping in mind that Eq. (\[eq:EL-A\]) is valid for $x\neq0$, $$\label{eq:integrated-EL-A} \omega A\sgn(x)-\int_{-\infty}^\infty \!dy \, Z(x-y)\,\partial_y A =C_a\sgn(x),$$ where the integration constants in the intervals $x<0$ and $x>0$ were related using the symmetry $A(x)=A(-x)$. The value of the constant $C_a$ is determined by the boundary conditions; using the definition (\[eq:a-b-defined\]) we obtain $$\label{eq:discontinuity-a} 2C_a= \omega\,\varphi(0)-\varphi'(0_+)-\vartheta'(0) =\omega\,\vartheta(\infty).$$ Similarly, the integration of the corresponding equation for the function $B(x)$ yields $$\label{eq:discontinuity-b} 2C_b= -\omega\,\varphi(0)-\varphi'(0_+)+\vartheta'(0) =\omega\,\vartheta(\infty).$$ Together, Eqns. (\[eq:discontinuity-a\]) and (\[eq:discontinuity-b\]) imply that $$\label{eq:CaCb} C_a=C_b={-\varphi'(0_+)/ 2}.$$ Because of the sign function multiplying the first term in the l.h.s., Eq. (\[eq:integrated-EL-A\]) cannot be solved directly by a Fourrier transformation. It is, however, of the form solvable by the Wiener-Hoph technique[@wiener-hopf-book]. Following the standard prescription, we introduce the functions $A_\pm(x)=A(x)\,\tf(\pm x)$, so that, [*e.g.*]{}, $A(x)=A_+(x)+A_-(x)$, $A(x)\sgn(x)=A_+(x)-A_-(x)$. After this substitution we can Fourrier-transform Eq. (\[eq:integrated-EL-A\]), $$\label{eq:fourrier-transformed-A} \omega [A_+-A_-]+ik\, Z(k)\, [A_++A_-] =2i\,C_a{\cal P}{1\over k},$$ where ${\cal P}$ denotes the principal value, and the Fourrier-transformed functions $A_\pm\equiv A_\pm(k)$ have no singularities above and below the real axis respectively (regularization at infinity ensures that they are also analytic everywhere along the real axis). The functions $A_\pm(x)$ are only discontinuous in the origin, and the asymptotic form of their Fourrier transformations at $|k|\to \infty$ is $$\label{eq:asymptotic-ug} A_\pm(k)=\pm{ i\over k}\,A_\pm(0_\pm)+{\cal O}(|k|^{-2}) =\pm i\,{\varphi(0)\over2 k}+\ldots$$ The independent functions in Eq. (\[eq:fourrier-transformed-A\]) can be rearranged as follows, $$\begin{aligned} A_+(k)&=&-{\cal R}(k)\, A_-(k)+{2C_a\over k\,Z-i\omega}\,{\cal P}{1\over k}, \label{eq:function-R}\\ {\cal R}(k)&\equiv& {k\,Z+i\omega\over k\,Z-i\omega}={{\cal R}_-(k)\over{\cal R}_+(k)}\label{eq:fraction-decomposition}\end{aligned}$$ where the function ${\cal R}(k)$ was separated into the ratio of the function ${\cal R}_-(k)$ which has neither singularities nor zeros at and below the real axis, and ${\cal R}_+(k)$, which has the same properties at and above the real axis. This separation is possible because the function ${\cal R}(k)$ is analytic in a vicinity of the real axis (which is correct for any $\omega$, assuming that the interaction potential $V(x)$ is properly regularized at infinity). In the absence of the long-distance interactions, $\chi=0$, the decomposition is trivial, ${\cal R}_\pm^{0}=(k\pm i\omega)^{-1}$, where we assume $\omega>0$. At very large values of $k$ the long-distance part of the potential should not matter. Therefore, to ensure the regularity of the decomposition (\[eq:fraction-decomposition\]) at $\chi>0$, we can use the Cauchy formula $$\ln {r}_\pm(q)=\!\int_{-\infty}^\infty {dk\over2\pi i}\, {\ln {r}(k)\over q-k\pm i0},\;\; {r}_\pm(q)\equiv{ {\cal R}_\pm(q)\over {\cal R}^{0}_\pm(q)} \label{eq:Cauchy-R}$$ for the ratio $r(k)={\cal R}(k)/{\cal R}^{0}(k)$. Since ${r}(k)\to1$ at large $k$, this expression implies that $r_\pm(k)\to1$ (and hence that ${\cal R}_\pm\sim 1/k$) as $|k|\to\infty$. Multiplying Eq. (\[eq:function-R\]) by ${\cal R}_+$, and separating the free term of the obtained expression into a sum of functions analytic above and below the real axis respectively, we obtain $$A_+(k)\,{\cal R}_+-2C_a\,h_+=-A_-(k)\,{\cal R}_-+2C_a\,h_-. \label{eq:eqn-A-decomposed}$$ Here the functions $h_\pm\equiv h_\pm(k)$, analytic in the upper (lower) complex half-plane, are defined so that $h_+(k)+h_-(k)=h(k)$, where $$\label{eq:function-h-decomposed} h(k)\equiv {{\cal R}_+(k)\over k\,Z-i\omega} \,{\cal P} {1\over k} ={{\cal R}_-(k)-{\cal R}_+(k)\over 2i\,\omega}\,{\cal P}{1\over k};$$ these functions can be found using the Cauchy formula $$h_\pm(q)=\mp {1\over 2\pi\, i}\,\int_{-\infty}^\infty\! {dk}\, {h(k)\over q-k\pm i0}. \label{eq:Cauchy-h}$$ We assumed that ${\cal R}_\pm(k)$ are non-singular in the origin (and elsewhere along the real axis), therefore, using the identity ${\cal R}_-(0)={\cal R}(0)\,{\cal R}_+(0)=-{\cal R}_+(0)$, we obtain $$\label{eq:h-plus-evaluated} h_\pm(k)=\pm i\,{{\cal R}_\pm(k)\over 2\omega \,(k\pm i0)}.$$ By construction, the LHS of Eq. (\[eq:eqn-A-decomposed\]) has no singularities at and above the real axis, while its RHS has no singularities at and below the real axis. Therefore, the whole expression is analytic everywhere in the complex plane, and, as long as it is uniformly limited at infinity, it can only be a constant. Moreover, since both sides of Eq. (\[eq:eqn-A-decomposed\]) actually [*vanish*]{} at infinity \[as follows from Eq. (\[eq:asymptotic-ug\]) and the properties of the functions ${\cal R}_\pm$, $h_\pm$\], this implies that the whole expression can only be zero everywhere at the complex plane $k$. We obtain $$\label{eq:solution-trivial} A_\pm(k)=2C_a\,{h_\pm(k)\over {\cal R}_\pm(k)}=\pm {i C_a\over \omega\, (k\pm i0)},$$ and by matching with the asymptotic expansion (\[eq:asymptotic-ug\]), we get $$C_a={\omega\,\varphi(0)\over2}, \quad A_\pm(k)=\pm {i \varphi(0)\over2\, (k\pm i0)}.\label{eq:A-found}$$ Comparing to Eq. (\[eq:CaCb\]), we obtain $$\Delta\varphi'_0=2\varphi'(0_+)=-2\omega\varphi(0)$$ and the contribution at the frequency $\omega>0$ to the effective action (\[eq:effective-action-evaluated\]) becomes $${\cal S}_{\rm q}(\omega)={T\over4\pi} |\omega|\,|\varphi(0)|^2.$$ One can also obtain an identical contribution at $\omega<0$, so that $$\label{eq:K-half-pi-exact} {\cal K}_{\alpha=\pi/2}(\omega)= 1,$$ as expected by the self-duality of the problem. The analogue of Eq. (\[eq:EL-A\]) for the function $B(x)$ differs only by the sign of $\omega$, which leads to a replacement ${\cal R}\to1/{\cal R}$, ${\cal R}_\pm\to 1/{\cal R}_\pm$. Instead of Eq. (\[eq:eqn-A-decomposed\]) we get $$\label{eq:eqn-B-decomposed} B_+(k)\,{\cal R}^{-1}_+-2C_b\,f_+=-B_-(k)\,{\cal R}^{-1}_-+2C_b\,f_-.$$ By analogy with Eq. (\[eq:h-plus-evaluated\]), we obtain $$\label{eq:f-plus-evaluated} f_\pm(k)=\mp i\,{{\cal R}^{-1}_\pm(k)\over 2\omega \,(k\pm i0)}.$$ By the same analyticity argument, both sides of equation (\[eq:eqn-B-decomposed\]) are analytic everywhere in the complex plane; at $|k|\to\infty$ they asymptotically approach a constant value $i\varphi(0)$. Therefore, $$B_\pm(k)= \pm{i \varphi(0)}\,\left[{\cal R}_\pm(k)-{1\over 2(k\pm i0)}\right] ,$$ and, combining with Eq. (\[eq:A-found\]), we can use the definition (\[eq:a-b-defined\]) to restore the original fields in the extremum, $$\begin{aligned} \label{eq:final-phi-half-pi} \varphi(x)&=&\varphi(0)\int {dk\over2\pi i}\left[ {\cal R}_-(k)-{\cal R}_+(k)\right] e^{-ikx},\\ \vartheta(x)&=&\sgn(\omega\,x)\,[\varphi(0)-\varphi(x)], \label{eq:final-theta-half-pi}\end{aligned}$$ where the $\sgn(\omega)$ in the second line is needed because the case $\omega<0$ is equivalent to the interchange of $A$ and $B$, which changes the sign of $\theta(x)$. It is easy to verify that the obtained functions obey the boundary conditions assumed when deriving Eqns. (\[eq:effective-action-evaluated\]), (\[eq:discontinuity-a\]), (\[eq:discontinuity-b\]). This self-consistency check ensures that the obtained expressions give us the exact formal solution of the problem. To understand the structure of this solution, let us introduce the expansion $$\chi V(x)=\sum_{l=1}^N {A_l\over a_l} \,e^{-a_l\,|x|}, \quad \chi V(k)=\sum_{l=1}^N{2A_l\over k^2+a_l^2}, \label{eq:exponent-expansion}$$ which, for sufficiently large $N$, gives an adequate regularized representation of any non-pathological even function $V(x)$. For example, the Coulomb potential $ V(x)=1/|x|$ can be rewritten as follows, $${1\over |x|}=\lim_{a\to0}{a\over 1-\exp({-a |x|})} =\lim_{a\to0}a\sum_{l=0}^\infty e^{-a l\,|x|},$$ so that, given a finite $a$, any partial sum provides a regularization of the form (\[eq:exponent-expansion\]) with $a_l=a\,l$ and $A_l=\chi a^2l$. We obtain $$\begin{aligned} Z&=&1+{\chi\over2} V(k)=1+\sum_{l=1}^N {A_l\over k^2+a_l^2}, \\ k Z-i\omega&=&{P_{2N+1}(k) \prod_{l=1}^N (k^2+a_l^2)^{-1}},\end{aligned}$$ where the polynomial $$P_{2N+1}(k)= \prod_{s=1}^{2N+1}(k-i\kappa_s)$$ has precisely $(2N+1)$ purely imaginary distinct roots $k_s\equiv i\kappa_s\neq0$. One can also show that for $\omega>0$ exactly $N$ of the roots lie below the imaginary axis; we shall assume $\kappa_s<0$ for $1<s<N$. The Cauchy integral (\[eq:Cauchy-R\]) is readily evaluated, and we obtain $$\begin{aligned} {\cal R}_+={(k-i\kappa_1)\ldots (k-i\kappa_N)\over (k+i\kappa_{N+1})\ldots (k+i\kappa_{2N+1})}; \label{eq:exponent-expansion-R}\end{aligned}$$ using the form similar to that in the first part of Eq. (\[eq:function-h-decomposed\]), the extremum solution (\[eq:final-phi-half-pi\]) can be explicitly rewritten as $$\varphi(x)=2|\omega|\,\varphi(0)\int {dk\over 2\pi}\, {(k^2+a_1^2)\ldots (k^2+a_N^2)\,\cos(kx)\over (k^2+\kappa^2_{N+1})\ldots (k^2+\kappa_{2N+1}^2)}.\label{eq:found-explicit-phi}$$ Expansion around the self-dual solution {#sec:perturbation-theory} --------------------------------------- To get an approximate expression for ${\cal K}(\alpha)$ in a vicinity of $\alpha=\pi/2$, we expand $V_\pm(x,y)$ to first order in $\cos\alpha$, and employ perturbation theory. The solution of the extremum equations at $\alpha_0=\pi/2$ is unique, and the lowest order non-degenerate perturbation theory suffices. This amounts to evaluating the Euclidean action (\[eq:symmetrized-action\]) along the non-perturbed solution $\varphi(x)$, $\vartheta(x)$, $$\begin{aligned} \delta{\cal S}_{\rm q}&\equiv&{T\over 4\pi}\sum_n|\omega_n|\,\delta {\cal K}_\alpha\,|\varphi(0)|^2\\ &=&{T\over 4\pi}\sum_n {\chi\over2} \int dx \,dy \left[\bar\varphi'_x\,\delta V_+\,\varphi'_y +\bar\vartheta'_x\,\delta V_-\,\vartheta'_y\right],\end{aligned}$$ where the integration is performed everywhere except the origin, and the potentials $$\begin{aligned} \delta V_+&=&-{ xy\,\cos\alpha\over \sqrt{x^2+y^2}}V'(\sqrt{x^2+y^2}),\\ \delta V_-&=&-\delta V_+\,\sgn(xy). \end{aligned}$$ were found by expanding Eqns. (\[eq:v-plus\]), (\[eq:v-minus\]). According to our solution (\[eq:final-theta-half-pi\]), the functions $\varphi'(x)$, $-\vartheta'(x)\,\sgn(\omega\,x)$ are identical, and the two terms give equal contributions, leading to $$\delta{\cal K}_\alpha=-{\chi\cos\alpha\over|\omega|\,|\varphi(0)|^2} \int_{-\infty}^\infty\! dx\,dy\,\bar\varphi'_x\,\varphi'_y\,{y}\, \partial_x\,V(\sqrt{x^2+y^2}).$$ For the Coulomb potential (\[eq:coulomb\]), this gives $$\delta{\cal K}_\alpha={4\chi\cos\alpha\over|\omega|\,|\varphi(0)|^2} \int_0^\infty\! dx\int_0^\infty\!dy\, {x \,y\,\bar\varphi'_x\,\varphi'_y\over (x^2+y^2+a^2)^{3/2}}.$$ This integral converges at small distances even if we set $a\to0$; in this scale-invariant limit the “wavefunctions” $\varphi(x)$ can depend only on the dimensionless quantities $|\omega|x$ and $\chi$, $\varphi(x)\equiv\varphi(0)\,\phi_\chi(|\omega|\,x)$. Scaling out the frequency leads to a [*frequency-independent*]{} correction, $$\begin{aligned} \nonumber \delta{\cal K}_\alpha(\omega,\chi)&=&{\chi\,{\cal N}(\chi)\,\cos\alpha}+{\cal O}(\chi^2\cos^2\alpha),\quad \omega\,a\ll1,\\ {\cal N}(\chi)&\equiv& 4\!\int_0^\infty\!\! dx\int_0^\infty\!\!dy\, {x \,y\,\bar\phi'_\chi(x)\,\phi'_\chi(y)\over (x^2+y^2)^{3/2}}. \label{eq:app-linear-expansion}\end{aligned}$$ This result supports the numerical data, which indicates that ${\cal K}_\alpha(\omega)$ is [*independent*]{} of $\omega$ at small enough frequencies. This statement is true for all finite angles, $|\cos\alpha|<1$, while ${\cal K}_{\alpha=0}(\omega)$ diverges logarithmically according to Eq. (\[eq:alpha0\]). The specific value of the correction depends on the coupling constant $\chi$. 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--- abstract: 'When dealing with highly accurate modeling of time and frequency transfers into arbitrarily moving dielectrics medium, it may be convenient to work with Gordon’s optical spacetime metric rather than the usual physical spacetime metric. Additionally, an accurate modeling of the geodesic evolution of observable quantities (e.g. the range and the Doppler) requires us to know the reception or the emission time transfer functions. In the physical spacetime, these functions can be derived to any post-Minkowskian orders through a recursive procedure. In this work, we show that the time transfer functions can be determined to any order in Gordon’s optical spacetime as well. The exact integral forms of the gravitational, the refractive, and the coupling contributions are recursively derived. The expression of the time transfer function is given within the post-linear approximation assuming a stationary optical spacetime covered with GCRS coordinates. The light-dragging effect due to the steady rotation of the neutral atmosphere of the Earth is found to be at the threshold of visibility in many experiments involving accurate modeling of the time and frequency transfers.' author: - 'A. Bourgoin' bibliography: - 'TTF\_opt.bib' title: General expansion of time transfer functions in optical spacetime --- Introduction ============ In geometrical optics, the concept of light rays is introduced as curves whose tangents coincide with the direction of propagation of an electromagnetic wave [@1975ctf..book.....L]. In this approximation, refraction operates at two different levels. First, it causes the phase velocity of the electromagnetic wave to slow down or speed up while crossing a region of higher or lower refractivity, respectively. Secondly, light rays tend to bend towards region of higher refractivity. These outcomes produce an excess path delay and a geometric delay in the light time. Depending on the context, these two effects must be either thoroughly modeled or precisely measured while designing highly accurate experiments involving time and frequency transfers in presence of a refractive medium. In many fields of astronomy such like planetary physics, astrometry, metrology, geodesy, fundamental physics, or even cosmology, we can think of situations were refractivity plays a significant role in the time and frequency transfers. For instance, we can mention that ground-based astro-geodetic techniques operating for the realization of the international terrestrial reference frame (ITRF) are currently limited by errors in modeling the group delay during the signal propagation through the Earth’s atmosphere [@2010ITN....36....1P; @2004GeoRL..3114602M; @2002GeoRL..29.1414M; @1997JGR...10220489C; @doi1010292006JB004834]. We can also mention the cases of atmospheric radio occultations [@1965Sci...149.1243K; @1985AJ.....90.1136L; @1987JGR....9214987L; @1992AJ....103..967L; @2012Icar..221.1020S; @2015RaSc...50..712S] and atmospheric stellar ocultation experiments . Indeed, both techniques aim at determining a refractivity profile towards an occulting atmosphere from precise measurements of an *a priori* known frequency (usually given in the frame at rest with the emitter) and from an accurate modeling of the frequency transfer in presence of the occulting refractive medium. To an even higher degree of accuracy, we can cite experiments involving frequency transfers between distant atomic clocks via ground-ground free-space optical (FSO) link [@Djerroud.10; @2013NaPho...7..434G; @doi.10.1063.1.4963130], space-ground FSO link [@Fujieda_2014; @Hachisu.14], and optical fiber links [@2010Sci...329.1630C; @Predehl441; @2015Metro..52..552G]. Finally, let us emphasize that in the context of cosmology, it has been shown that the accumulated effect of an artificial refractivity over the distance-redshift relation fits perfectly Hubble curve of type Ia supernovae data in the framework of a non-accelerating cosmological model [@PhysRevD.78.044040; @*PhysRevD.79.104007; @*PhysRevD.80.044019]. All these examples highlight how important refraction can be in highly accurate experiments involving time and frequency transfers. In the past, two independent theoretical formalism have been introduced, namely Gordon’s optical metric and the time transfer functions. On one side, Gordon’s metric allows one to handle refraction in curved spacetime; on the other side, the time transfer functions formalism handles theoretical problems related to the time and frequency transfers in curved spacetime. In this work, we intend to combine the two formalisms which are discussed in turn in the next. In the early 20’s, Gordon introduced [@doi101002andp19233772202] a useful theoretical tool to study light refraction caused by an arbitrarily moving fluid dielectric medium, namely Gordon’s optical metric. In this work, he showed that in the presence of a fluid whose electromagnetic properties are described by a permittivity $\epsilon(x)$ and a permeability $\mu(x)$, any solutions to the macroscopic Maxwell’s equations can be looked for indifferently either in the usual physical spacetime fitted with the metric tensor, or in an artificial optical spacetime fitted with Gordon’s metric. Conveniently, in the optical spacetime and within the geometric optics approximation, by means of a slightly different set of Maxwell’s equations, the electromagnetic properties of the fluid medium are reduced to their vacuum values, that is to say $\epsilon(x)=\epsilon_0$ and $\mu(x)=\mu_0$. In other words, in the physical spacetime, the interaction between the electromagnetic field and the dielectric fluid medium must be carefully modeled, whereas in the optical spacetime this interaction is implicitly accounted for in the vacuum limit of the macroscopic version of Maxwell’s equations. Consequently, within the geometric optics approximation, light rays propagate into dielectrics medium along null geodesic lines of the optical spacetime. At the same time, theoretical problems dealing with the deflection of light rays or the frequency transfer, require us to know the function relating the (coordinate) time transfer to the coordinate time at the reception and to the spatial coordinates of the reception and the emission points. Such function is called a reception time transfer function. Obliviously, an emission time transfer function can be introduced as well. The formalism which aims at determining the time transfer functions was first introduced by [@2002PhRvD..66b4045L] relying on the theory of the world function developed by Synge [@SyngeBookGR]. General expansions of the world function and the time transfer functions was first proposed by [@2004CQGra..21.4463L], and then, a simplified recursive approach, based on the determination of time delay functions instead of Synge’s world function, was presented in [@2008CQGra..25n5020T]. The usefulness of the time transfer function formalism lies in the fact that it spares the trouble of solving explicitly the null geodesic equation which usually leads to heavy calculations beyond the post-Minkowskian regime (see e.g. for explicit resolution of the null geodesic equation in the linearized weak field limit, and see e.g. [@1983PhRvD..28.3007R; @1987KFNT....3....8B; @Linet_2013] for resolution in the post-post Minkowskian approximation). Indeed, assuming that the emission and reception points-events are linked by a null geodesic path (quasi-Minkowskian path approximation), the time transfer functions formalism achieves a complete resolution of the time and frequency transfers to any post-Minkowskian order by means of an algorithmic resolution method [@2008CQGra..25n5020T]. For this reason, this formalism is currently one of the most powerful theoretical tool to derive the time and frequency transfers along null geodesics of the curved physical spacetime. The scope of this paper is to generalize the formalism of the time transfer functions to optical spacetime. The aim is to provide a recursive method allowing one to solve theoretical problems related to the propagation of light in presence of arbitrarily moving refractive medium. The present work is organized as follows. In Sec. \[sec:not\], we present the notations and conventions used throughout this paper. Sec. \[sec:geO\] is a short reminder about the use of Gordon’s metric in relativistic geometrical optics. In this section, we derive the optical counterpart of the scalar Eikonal equation (fundamental equation of geometrical optics) which is at the basis of the demonstration which follows. Sec. \[sec:TF\] is a recall about the time transfer functions formalism. In Sec. \[sec:intTF\], by applying a method initially proposed by [@2008CQGra..25n5020T], we show that working in optical spacetime induces that the time transfer functions can be decomposed in three components, that we call the gravitational, the refractive, and the coupling time transfer functions. In Sec. \[sec:exp\], we present the general expansion of the three contributions. In Sec. \[sec:app\], we illustrate the method by computing the time transfer function of an optical spacetime describing Earth’s rotating atmosphere in the geocentric celestial reference system (GCRS) within the post-linear approximation. Finally, we discuss about the importance of taking into account the light-dragging effect in future generation of data reduction softwares. Notations and conventions {#sec:not} ========================= In this work, the metric of spacetime is denoted $g$ and its signature is $(+,-,-,-)$. The optical metric (also called Gordon’s metric) is denoted $\bar g$. We suppose that spacetime is covered with some global coordinates system $\bm x=(x^0,\mathbf{x})$. We put $x^0=ct$ with $c$ being the speed of light in a vacuum and $t$ being the coordinate time. Greek indices run from 0 to 3 and Latin indices run from 1 to 3. Straight bold letters denote ordered triples such as $\mathbf x=(x^i)$ and italic bold letters denote ordered quadruples such like $\bm x=(x^\mu)$. Einstein’s summation convention on repeated indices is used for expressions like $a^ib^i$ as well as for expressions like $A^\mu B_\mu$. The ordinary Euclidean norm of $\mathbf{x}$ is denoted $|\mathbf x|$ and is defined as $|\mathbf{x}|=(\delta_{ij}x^ix^j)^{1/2}$ where $\delta_{ij}$ is the Kronecker delta. The maximum absolute value of the component $A_{\mu\nu}$ is denoted $|A_{\mu\nu}|_{\mathrm{max}}$. The 3-dimensional antisymmetric Levi-Civita tensor is denoted $e^{ijk}$. For the sake of legibility, we would employ $(f)_{x}$ or $[f]_{x}$ instead of $f(x)$ whenever necessary. When a quantity $f(x)$ is to be evaluated at two point-events $x_A$ and $x_B$, we would employ $(f)_{A/B}$ to denote $f(x_A)$ and $f(x_B)$, respectively. The partial differentiation with respect to coordinates $x^\mu$ is denoted $\partial_\mu A_\nu$. The physical and the optical covariant differentiation with respect to $x^\mu$ are denoted $\nabla_\mu A_\nu$ and $\bar\nabla_\mu A_\nu$, respectively. Given a scalar function $f(\bm x)$, we have the relation $\bar\nabla_\mu f=\nabla_\mu f=\partial_\mu f$. Throughout the paper, we assume the presence of an arbitrarily moving fluid dielectric medium filling a finite domain $\mathcal{D}$ of spacetime. We call $\bm w(x)$ the unit 4-velocity vector of a point-event $x$ belonging to a fluid element of the optical medium. The expression of $\bm w(x)$ is given by $$\bm w(x)\equiv\frac{{\mathrm{d}}\bm x}{{\mathrm{d}}s} \label{eq:4velel}$$ where the spacetime interval ${\mathrm{d}}s$ is defined by $${\mathrm{d}}s^2=g_{\mu\nu}(x){\mathrm{d}}x^{\mu} {\mathrm{d}}x^{\nu}\text{.}$$ We call $\xi^i(x)$ the coordinate 3-velocity vector of the point-event $x$ belonging to a fluid element of the optical medium. Its expression is given by $$\xi^i(x)\equiv\frac{w^i}{w^0}=\frac{1}{c}\frac{{\mathrm{d}}x^i}{{\mathrm{d}}t}\text{.} \label{eq:3velel}$$ Finally, $G$ is the Newtonian gravitational constant. Relativistic Geometrical optics {#sec:geO} =============================== We assume the presence of an optical medium filling $\mathcal{D}$. Additionally, we consider for simplicity that the fluid’s electromagnetic properties are linear, isotropic, non-dispersive and can be summarized by two scalar functions, namely the permittivity $\epsilon(x)$ and the permeability $\mu(x)$. These two quantities completely determine the refractive properties of the optical medium through the following relationship $$n(x)\equiv c\sqrt{\epsilon(x)\mu(x)} \label{eq:n}$$ where $n$ is the index of refraction of the medium. When $x\notin\mathcal{D}$, the permittivity and the permeability reduce to their vacuum values $\epsilon(x)=\epsilon_0$ and $\mu(x)=\mu_0$, respectively. Thus, considering that $c\equiv(\epsilon_0\mu_0)^{-1/2}$, the index of refraction becomes $n(x)=1$. By subtracting its vacuum value from the index of refraction, we obtain the refractivity $$N(x)\equiv n(x)-1\text{,} \label{eq:N}$$ which is obviously null in a vacuum. In the physical spacetime, the evolution of electromagnetic phenomenon occurring in the presence of an optical medium are usually described by the macroscopic version of Maxwell’s equations. These equations are separated into two distinct sets involving a covariant antisymmetric tensor $F_{\mu\nu}$ called the electromagnetic field tensor (or Faraday tensor), and a contravariant antisymmetric tensor $B^{\mu\nu}$ called the electromagnetic field excitation tensor (or Maxwell tensor), respectively. The macroscopic version of Maxwell’s equations are given by [@1960ecm..book.....L; @doi101002andp200810313] \[eq:Max\] $$\begin{aligned} \partial_{[\sigma} F_{\mu\nu]}&=0\text{,}\label{eq:Max1}\\ \nabla_\mu B^{\mu\nu}&=j^\nu\label{eq:Max2}\end{aligned}$$ where $\bm j(x)$ is a 4-vector denoting the free charge density-current. The square brackets denote the complete antisymmetrization of the enclosed indices. The first set , allows one to postulate the existence of a covector field $A_\mu(x)$, such that the electromagnetic field tensor $F_{\mu\nu}$ can be locally written as the rotational of the covector field, that is $$F_{\mu\nu}=\mathrm{Re}\big\{\partial_{\mu}A_\nu-\partial_{\nu}A_\mu\big\}\text{.} \label{eq:F}$$ The second set cannot be used alone to fully determine the six independent components of the electromagnetic field excitation tensor $B^{\mu\nu}$. In addition, it does not provide a way to determine the components of the electromagnetic field tensor $F_{\mu\nu}$ which yet governs the motion of particles through the Lorentz force. Therefore, Maxwell’s equations must be supplemented with constitutive relations. For an arbitrarily moving medium of permittivity $\epsilon(x)$ and permeability $\mu(x)$ the covariant constitutive relationships are given by [@1960ecm..book.....L] \[eq:HFctv\] $$\begin{aligned} B^{\mu\nu}w_\nu&=\epsilon c^2F^{\mu\nu}w_\nu\text{,}\label{eq:Hctv}\\ \mu B_{[\mu\nu}w_{\sigma]}&=F_{[\mu\nu}w_{\sigma]}\text{.}\label{eq:Fctv}\end{aligned}$$ Eqs. can be written as a single relationship involving $B^{\mu\nu}$, $F_{\mu\nu}$, and $\bm w(x)$. Indeed, as initially shown by Gordon [@doi101002andp19233772202], when dealing with problems of electromagnetic waves propagating into dielectrics, it is convenient to introduce an optical spacetime in which refractivity is considered as spacetime curvature. The Gordon’s metric (or optical metric) is defined by \[eq:Gor\] $$\bar g_{\mu\nu}\equiv g_{\mu\nu}+\gamma_{\mu\nu}\text{,} \qquad \gamma_{\mu\nu}=-\left(1-\frac{1}{n^2}\right)w_\mu w_\nu\text{,}\label{eq:Gorcov}$$ with inverse $$\bar g^{\mu\nu}\equiv g^{\mu\nu}+\kappa^{\mu\nu}\text{,} \qquad \kappa^{\mu\nu}=(n^2-1)w^\mu w^\nu\text{.}\label{eq:Gorcon}$$ Making use of Eq. , one can see that Eqs. are summarized within the single following relation [@doi101002andp19233772202] $$\mu B^{\mu\nu}=\bar F^{\mu\nu} \label{eq:ctv}$$ where the optical metric has been used to raise covariant indices of $F_{\alpha\beta}$, that is $$\bar F^{\mu\nu}\equiv\bar g^{\mu\alpha}\bar g^{\nu\beta} F_{\alpha\beta}\text{.} \label{eq:bF}$$ It is now possible to express Maxwell’s equation in the optical spacetime. Because, the covariant components of the electromagnetic field tensor are equivalents in both spacetimes [^1], the first pair of Maxwell’s equations remains unchanged. The optical form of the second pair is obtained after substituting for $B^{\mu\nu}$ from while introducing the optical covariant derivative [@PhysRevD.78.044040; @PhysRevD.79.104007]. After little algebra, we find $$\bar{\nabla}_\mu\left(\sqrt{\frac{\epsilon}{\mu}}\,\bar F^{\mu\nu}\right)=\sqrt{\epsilon\mu}\,j^{\nu}\text{.} \label{eq:bMax}$$ Eq. is perfectly equivalent to Eq. equipped with the constitutive relations . While working in the optical spacetime, Eq. allows to find $\bar F^{\mu\nu}$ and Eq.  lets to express the components of the electromagnetic field tensor in the physical spacetime. Hereafter, we work in the optical spacetime where the light propagation into the dielectric medium is simply given by the vacuum limit of the macroscopic version of Maxwell’s equations (no free density current i.e. $j^\nu=0$). In this work, we consider geometrical optics approximation, so we assume that the 4-potential covector $A_\mu(x)$ of a traveling quasi-monochromatic wave possesses an expansion of the form [@1975ctf..book.....L] $$A_\mu=\big[a_\mu+\mathcal{O}(\omega^{-1})\big]e^{i\omega\mathscr{S}}\text{.} \label{eq:pot}$$ Here, $\mathscr{S}(x)$ is the usual eikonal function which determines the surfaces of constant phase for the wave, $a_\mu(x)$ is the complex covector amplitude varying slowly in comparison to $\mathscr{S}(x)$, and $\omega$ is a book-keeping parameter that we take to be high during our manipulations [@gravitationBook]. Then, substituting for $A_\mu$ from Eq. into , allows one to infer $$F_{\mu\nu}=\mathrm{Re}\Big\{\big[i\omega f_{\mu\nu}+\mathcal{O}\left(\omega^0\right)\big]e^{i\omega\mathscr{S}}\Big\}\text{,} \label{eq:Fopt}$$ where $f_{\mu\nu}(x)$ represent the electromagnetic field tensor amplitude’s components, that is $$f_{\mu\nu}=k_\mu a_\nu-k_\nu a_\mu\text{,} \label{eq:A}$$ with $k_\mu$ being the wave covector defined by $$k_\mu\equiv\partial_\mu\mathscr{S}\text{.} \label{eq:kdef}$$ We can introduce the contravariant optical wave vector such that $$\bar k^\mu\equiv\bar g^{\mu\nu}k_\nu \label{eq:bkdef}$$ where the low index has been raised with the help of the optical spacetime metric. We can directly check from the inverse conditions $\bar g_{\mu\sigma}\bar g^{\sigma\nu}=\delta_\mu^\nu$, that the covariant components of the wave vector are identical in physical and optical spacetimes, that is to say $$\bar k_\mu =k_\mu\text{.} \label{eq:kbk}$$ Assuming that the 4-potential fulfills the Lorentz gauge in the optical spacetime, that is $$\bar\nabla_\mu \bar A^\mu=0 \label{eq:bLG}$$ where we introduced $\bar A^\mu\equiv\bar g^{\mu\nu}A_\nu$, we find $$\bar g^{\mu\nu}k_\mu a_\nu=0 \label{eq:bLGopt}$$ within the geometrical optics approximation. This relationship states the orthogonality between the optical wave vector $\bar{\bm k}$ and the wave covector amplitude $a_\mu$. Finally, the fundamental equations of geometrical optics can be derived from the vacuum limit of the optical version of Maxwell’s equations. We first determine the optical electromagnetic field tensor by making use of Eqs.  and . Then, by taking the covariant derivative of $\bar F^{\mu\nu}$, we find $$\bar\nabla_\mu\bar F^{\mu\nu}=-\mathrm{Re}\Big\{\big[\omega^2k_\mu\bar f^{\mu\nu}+\mathcal{O}(\omega)\big]e^{i\omega\mathscr{S}}\Big\} \label{eq:CDFopt}$$ where we introduced $\bar f^{\mu\nu}\equiv\bar g^{\mu\alpha}\bar g^{\nu\beta}f_{\alpha\beta}$. By substituting this result into the vacuum limit of Eq. , and by keeping the geometrical optics order only, one deduces $$\mathrm{Re}\big\{k_\mu\bar f^{\mu\nu}\big\}=0\text{.} \label{eq:beikA}$$ Then, substituting for $\bar f^{\mu\nu}$ from Eq. into and considering , we finally deduce $$\bar g^{\mu\nu}k_\mu k_\nu=0\text{.} \label{eq:beik}$$ This is the fundamental equation of geometrical optics expressed in optical spacetime. After inserting Eq. , we infer that the phase $\mathscr{S}(x)$ satisfies the well-known scalar Eikonal equation $$\bar g^{\mu\nu}\partial_\mu\mathscr{S}\partial_\nu\mathscr{S}=0\text{.} \label{eq:beikS}$$ We close this section by showing that $\bar{\bm k}$ is a null vector satisfying the geodesic equation for the optical metric. From Eqs. and , we easily infer $$\bar g_{\mu\nu}\bar k^\mu \bar k^\nu=0\text{.} \label{eq:beikcont}$$ This relation shows that $\bar{\bm k}$ is indeed isotropic for the optical metric $\bar g_{\mu\nu}$. Then, we differentiate Eq. with respect to $x^\sigma$. Considering the symmetry of the components of the optical metric together with Eq. , it comes $$\bar k^\nu(\bar\nabla_\sigma k_\nu)=0\text{.}$$ Making use of the definition , we infer $\bar\nabla_\sigma k_\nu=\bar\nabla_\nu k_\sigma$. Finally, considering Eqs. and , we deduce $$(\bar k^\nu\bar\nabla_\nu) \bar k^\sigma=0\text{,} \label{eq:geod}$$ which states (together with Eq. ) that curves admitting $\bar{\bm k}$ as tangent vector are null geodesic lines of the optical metric. In that respect, a null line which is solution of Eq. can be interpreted as a ray of light whose tangent at any point $x$ is orthogonal to the surface of constant phase $\mathscr{S}(x)$ [@SyngeBookGR]. Time transfer formalism {#sec:TF} ======================= Let us consider a light ray $\Gamma$ propagating in a region of spacetime covered with some coordinate system $(x^\mu)$. Let $\bm x_A=(ct_A,\mathbf{x}_A)$ be the coordinates of a point-event $x_A$. We introduce $\mathcal{C}_A$ as the curve of parametric equations $x=x_A(\tau)$ with $\tau$ being a parametrization of $\mathcal{C}_A$. Let us suppose that $(x^\mu)$ is chosen in such away that $\mathcal{C}_A$ is a time-like worldline for any $x_A$, which means that $\partial/\partial x^0$ is a time-like vector field, that is to say $g_{00}>0$ everywhere. Let $x_A$ be the point-event where $\Gamma$ is emitted and let $x_B$ be the point-event of coordinates $\bm x_B=(ct_B,\mathbf{x}_B)$ where it is observed. The quantity $t_B-t_A$ is the (coordinate) travel time of the light ray connecting the emission point-event $x_A$ and the reception point-event $x_B$. This quantity allows us to introduce the time transfer functions $\mathcal{T}_{r,\Gamma}$ and $\mathcal{T}_{e,\Gamma}$ [@2004CQGra..21.4463L] as $$t_B-t_A\equiv\mathcal{T}_{r,\Gamma}(\mathbf x_A,t_B,\mathbf x_B)\equiv\mathcal{T}_{e,\Gamma}(t_A,\mathbf x_A,\mathbf x_B)\text{.} \label{eq:TTFdefGen}$$ We call $\mathcal{T}_{r,\Gamma}$ the reception time transfer function and $\mathcal{T}_{e,\Gamma}$ the emission time transfer function associated to $\Gamma$. As shown in [@PhysRevD.93.044028], given a point-event $x_B$ and a spatial position $\mathbf x_A$, $\Gamma$ is not unique in general. Let $\Gamma^{[\sigma]}(\mathbf x_A,x_B)$ be a light ray intersecting $x_B$ and flowing from the point-event $x_A^{[\sigma]}\in\mathcal{C}_A$ of coordinates $(ct_A^{[\sigma]},\mathbf x_A)$. For each $\Gamma^{[\sigma]}$, there exists a reception time transfer function, denoted $\mathcal{T}_{r,\Gamma^{[\sigma]}}(\mathbf x_A,t_B,\mathbf{x}_B)$, such that $$t_B-t_A^{[\sigma]}=\mathcal{T}_{r,\Gamma^{[\sigma]}}(\mathbf x_A,t_B,\mathbf{x}_B)\text{,}$$ with $\sigma\in\mathbb{N}_{>0}$ (the same reasoning works for the emission time transfer function as well). This fact shows that, in general, we cannot expect to find a unique reception (or emission) time transfer function. However, for a very particular type of null geodesics, referred to as quasi-Minkowskians [@2012CQGra..29x5010T; @PhysRevD.93.044028], it has been shown that the reception (or the emission) time transfer function, if it exists, can be uniquely determined [@2008CQGra..25n5020T]. Henceforth, we assume that $\Gamma$ is a quasi-Minkowskian light ray so that the corresponding time transfer functions are indeed unique. In agreement with this assumptions, we suppose that the past null cone at $x_B$ intersects $\mathcal{C}_A$ at one and only one point $x_A$. Therefore, we can rewrite Eq.  as $$t_B-t_A\equiv\mathcal{T}_r(\mathbf x_A,t_B,\mathbf x_B)\equiv\mathcal{T}_e(t_A,\mathbf x_A,\mathbf x_B)\text{.} \label{eq:TTFdef}$$ Hereafter, in order to shorten future notations, we introduce the reception and the emission range transfer functions being defined by \[eq:RTFdef\] $$\mathcal{R}_r(\mathbf x_A,x_B)\equiv c\mathcal{T}_r(\mathbf x_A,t_B,\mathbf x_B)\text{,} \label{eq:RTFrdef}$$ and $$\mathcal{R}_e(x_A,\mathbf x_B)\equiv c\mathcal{T}_e(t_A,\mathbf x_A,\mathbf x_B)\text{.} \label{eq:RTFedef}$$ An important theorem (cf. Theorem 1 of [@2004CQGra..21.4463L]) states that the covariant components of the tangent vector are totally known, as soon as one of the time transfer functions (or equivalently, one of the range transfer functions) is explicitly determined. Therefore, if we define $$(l_i)_{A/B}\equiv \left(\frac{k_i}{k_0}\right)_{A/B}\text{,} \label{eq:l}$$ we have the following relationships \[eq:kTFdef\] $$\begin{aligned} (l_i)_A&\equiv\frac{\partial\mathcal{R}_r}{\partial x_A^i}=\frac{\partial\mathcal{R}_e}{\partial x_A^i}\left(1+\frac{\partial\mathcal{R}_e}{\partial x_A^0}\right)^{-1}\text{,}\label{eq:kTFA}\\ (l_i)_B&\equiv-\frac{\partial\mathcal{R}_r}{\partial x_B^i}\left(1-\frac{\partial\mathcal{R}_r}{\partial x_B^0}\right)^{-1}=-\frac{\partial\mathcal{R}_e}{\partial x_B^i}\text{,}\label{eq:kTFB}\end{aligned}$$ and $$\frac{(k_0)_B}{(k_0)_A}\equiv1-\frac{\partial\mathcal{R}_r}{\partial x_B^0}=\left(1+\frac{\partial\mathcal{R}_e}{\partial x_A^0}\right)^{-1}\text{.} \label{eq:kTF0}$$ Consequently, Eqs. completely solve theoretical problems related to frequency transfer. Indeed, it is well-known that the instantaneous expression of the Doppler shift along the null-geodesic path between the emitter and the receiver can be expressed as [@SyngeBookGR] $$\frac{\nu_B}{\nu_A}\equiv\frac{(u^{\mu}k_\mu)_B}{(u^{\mu}k_\mu)_A}=\frac{(u^0k_0)_B}{(u^0k_0)_A}\frac{(1+\beta^il_i)_B}{(1+\beta^il_i)_A} \label{eq:dop}$$ where $(\bm u)_{A/B}$ is the emitter/receiver’s unit 4-velocity vectors being defined as $$(\bm u)_{A/B}\equiv\left(\frac{{\mathrm{d}}\bm x}{{\mathrm{d}}s}\right)_{A/B}\text{,} \label{eq:4velAB}$$\ with $({\mathrm{d}}s^2)_{A/B}=(g_{\mu\nu}{\mathrm{d}}x^{\mu} {\mathrm{d}}x^{\nu})_{A/B}$. By definition, the 4-velocities satisfy the unity condition $(g_{\mu\nu}u^\mu u^\nu)_{A/B}=1$, which implies $$(u^0)_{A/B}=(g_{00}+2g_{0i}\beta^i+g_{ij}\beta^i\beta^j)^{-1/2}_{A/B}\text{.} \label{eq:4vel}$$ The quantities $(\beta^i)_{A/B}$ in Eq. are the components of the emitter/receiver’s coordinate 3-velocity vectors and are defined such that $$(\beta^i)_{A/B}\equiv\left(\frac{u^i}{u^0}\right)_{A/B}=\frac{1}{c}\left(\frac{{\mathrm{d}}x^i}{{\mathrm{d}}t}\right)_{A/B}\text{.} \label{eq:3velAB}$$ It is then straightforward to determine the exact expression of the instantaneous Doppler formulation in terms of the range transfer functions . Indeed, after inserting Eqs.  and within , we infer $$\frac{\nu_B}{\nu_A}=\frac{(u^0)_B}{(u^0)_A}\frac{q_B}{q_A}\text{,} \label{eq:dopTF}$$ with \[eq:qTF\] $$\begin{aligned} q_A&=1+\beta^i_A\frac{\partial\mathcal{R}_r}{\partial x_A^i}=1+\frac{\partial\mathcal{R}_e}{\partial x_A^0}+\beta^i_A\frac{\partial\mathcal{R}_e}{\partial x_A^i}\text{,}\label{eq:qA}\\ q_B&=1-\frac{\partial\mathcal{R}_r}{\partial x_B^0}-\beta^i_B\frac{\partial\mathcal{R}_r}{\partial x_B^i}=1-\beta^i_B\frac{\partial\mathcal{R}_e}{\partial x_B^i}\text{,}\label{eq:qB}\end{aligned}$$ and $$\frac{(u^0)_B}{(u^0)_A}=\frac{(g_{00}+2g_{0i}\beta^i+g_{ij}\beta^i\beta^j)^{1/2}_{A}}{(g_{00}+2g_{0i}\beta^i+g_{ij}\beta^i\beta^j)^{1/2}_{B}}\text{.} \label{eq:uBuA}$$ From the fundamental equation of geometrical optics (see Eq. ), we know that the covariant components of the 4-wave optical vector at point-events $x_A$ or $x_B$ satisfy a relation as follows $$(\bar g^{\mu\nu}k_\mu k_\nu)_{A/B}=0\text{.} \label{eq:beikAB}$$ Then, dividing by $[(k_0)_{A/B}]^2$ and making use of Eqs. -, we infer the following theorem which generalizes theorem 1 of [@2008CQGra..25n5020T] to optical spacetime. Within geometrical optics approximation, the range transfer functions $\mathcal{R}_r$ and $\mathcal{R}_e$ satisfy the following Hamilton-Jacobi-like equations over the optical spacetime, namely \[eq:beikHJ\] $$\bar g^{00}(x_B^0-\mathcal{R}_r,\mathbf{x}_A)+2\bar g^{0i}(x_B^0-\mathcal{R}_r,\mathbf{x}_A)\frac{\partial\mathcal{R}_r}{\partial x_A^i}+\bar g^{ij}(x_B^0-\mathcal{R}_r,\mathbf{x}_A)\frac{\partial\mathcal{R}_r}{\partial x_A^i}\frac{\partial\mathcal{R}_r}{\partial x_A^j}=0\text{,}\label{eq:beikHJA}$$ and $$\bar g^{00}(x_A^0+\mathcal{R}_e,\mathbf{x}_B)-2\bar g^{0i}(x_A^0+\mathcal{R}_e,\mathbf{x}_B)\frac{\partial\mathcal{R}_e}{\partial x_B^i}+\bar g^{ij}(x_A^0+\mathcal{R}_e,\mathbf{x}_B)\frac{\partial\mathcal{R}_e}{\partial x_B^i}\frac{\partial\mathcal{R}_e}{\partial x_B^j}=0\text{,}\label{eq:beikHJB}$$ respectively. \[th:theo1\] This theorem is at the basis of the demonstration for deriving the integral form of the range and then the time transfer functions. Henceforth, in order to avoid repetitions, we pursue the demonstration giving details only for the reception time delay function. However, the same results can be derived for the emission time delay function by applying the exact same reasoning. Integral form of the time delay functions {#sec:intTF} ========================================= Now, let us assume that the physical spacetime metric takes the following form \[eq:gm\] $$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu} \label{eq:gmcov}$$ throughout spacetime, where $\eta_{\mu\nu}$ is the Minkowski metric and $h_{\mu\nu}$ is the gravitational perturbation. In Cartesian coordinates, $\eta_{\mu\nu}=\text{diag}(+1,-1,-1,-1)$. The contravariant components of the physical spacetime metric can be decomposed as $$g^{\mu\nu}=\eta^{\mu\nu}+k^{\mu\nu} \label{eq:gmcon}$$ where the components $k^{\mu\nu}$ satisfy $$k^{\mu\nu}=-\eta^{\mu\alpha}\eta^{\beta\nu}h_{\alpha\beta}-\eta^{\mu\alpha}h_{\alpha\beta}k^{\beta\nu}\text{.} \label{eq:kmcon}$$ Therefore, the optical spacetime metric can be expressed as \[eq:Gordec\] $$\bar g_{\mu\nu}=\eta_{\mu\nu}+H_{\mu\nu}\text{,} \label{eq:Gordeccov}$$ with the contravariant components $$\begin{aligned} \bar g^{\mu\nu}&=\eta^{\mu\nu}+K^{\mu\nu}\text{.} \label{eq:Gordeccon}\end{aligned}$$ Thus, the optical metric reduces to the sum of the flat Minkowski metric plus a spacetime curvature contribution which is given by \[eq:Cur\] $$H_{\mu\nu}=h_{\mu\nu}+\gamma_{\mu\nu}\text{,}\label{eq:Curcov}$$ with the contravariant components $$K^{\mu\nu}=k^{\mu\nu}+\kappa^{\mu\nu}\text{.}\label{eq:Curcon}$$ From here, we suppose that the curvature contribution is small so that spacetime is mainly flat, that is to say $|h_{\mu\nu}|_{\mathrm{max}}\ll|\eta_{\mu\nu}|_{\mathrm{max}}$ and $|\gamma_{\mu\nu}|_{\mathrm{max}}\ll|\eta_{\mu\nu}|_{\mathrm{max}}$. In other words, we focus on the post-Minkowskian approximation. Under this condition, we ensure that the null geodesic path is quasi-Minkowskian. The form of the optical metric in Eqs. implies that the reception and the emission range transfer functions can be looked for according to the following expressions \[eq:RTFdec\] $$\mathcal{R}_r(\mathbf{x}_A,x_B)=|\mathbf{x}_B-\mathbf{x}_A|+\Delta(\mathbf{x}_A,x_B)\text{,}\label{eq:RTFdecR}$$ and $$\mathcal{R}_e(x_A,\mathbf{x}_B)=|\mathbf{x}_B-\mathbf{x}_A|+\Xi(x_A,\mathbf{x}_B)\text{,}\label{eq:RTFdecE}$$ respectively. Following [@2008CQGra..25n5020T], we will call $\Delta/c$ the reception time delay function and $\Xi/c$ the emission time delay function [^2]. Now, if we assume that the reception point-event $x_B$ is perfectly known, then, we can regard its components $t_B$ and $\mathbf x_B$ as fixed parameters. Hence, the reception time delay function becomes a function of the spatial components of the emission point-event $x_A$ [^3]. Thus, if we now substitute $\mathbf x$ to $\mathbf x_A$, the reception time delay function $\Delta(\mathbf x,x_B)/c$ uniquely defines the point-event $x_-(\mathbf x)$ for the given set of spatial components $\mathbf{x}$, that is $$\bm x_-(\mathbf x)=\big(x_B^0-|\mathbf x_B-\mathbf x|-\Delta(\mathbf{x},x_B),\mathbf{x}\big)\text{.} \label{eq:x-}$$ Furthermore, assuming that the point-event $x_-$ lies in the vicinity of $x_B$, we can determine the spatial variation of the reception time delay function. Indeed, after inserting Eqs.  and into taken at $x_-$ instead of $x_A$, we deduce the following relation [@2008CQGra..25n5020T] $$-2N^i\partial_i\Delta(\mathbf{x},x_B)=\Omega_-(x_-,x_B) \label{eq:beikDP}$$ where $\mathbf{N}=(\mathbf{x}_B-\mathbf{x})/|\mathbf{x}_B-\mathbf{x}|$, and $$\begin{aligned} \Omega_-(x_-&,x_B)=\big(K^{00}-2K^{0i}N^i+K^{ij}N^iN^j\big)_{x_-}\nonumber\\ &+2\big(K^{0i}-K^{ij}N^j\big)_{x_-}\partial_i\Delta(\mathbf{x},x_B)\nonumber\\ &+\big(\eta^{ij}+K^{ij}\big)_{x_-}\partial_i\Delta(\mathbf{x},x_B)\partial_j\Delta(\mathbf{x},x_B)\text{.}\label{eq:Om}\end{aligned}$$ Since $\mathbf x$ is a free variable, we follow [@2008CQGra..25n5020T] and choose for convenience to focus on the case where $\mathbf x$ is varying along the straight line segment connecting $\mathbf x_A$ to $\mathbf x_B$, that is to say $\mathbf{x}=\mathbf{z}_-(\lambda)$ where $$\begin{aligned} \mathbf{z}_-(\lambda)&=\mathbf{x}_B-\lambda R_{AB}\mathbf{N}_{AB}\text{,} & 0\leqslant\lambda\leqslant 1\text{,}\label{eq:z-i}\end{aligned}$$ with $R_{AB}=|\mathbf{x}_B-\mathbf{x}_A|$ and $\mathbf N_{AB}=(\mathbf{x}_{B}-\mathbf{x}_{A})/R_{AB}$. In that respect, we also have the relation $$\mathbf{N}=\mathbf N_{AB}\text{.} \label{eq:NAB}$$ We can now determine the integral form of the time delay function by differentiating $\Delta(\mathbf{z}_-(\lambda),x_B)$ with respect to $\lambda$. Using Eq. , we can always write $$\frac{{\mathrm{d}}}{{\mathrm{d}}\lambda}\Delta(\mathbf{z}_-(\lambda),x_B)=-R_{AB}N_{AB}^i\big[\partial_i\Delta\big]_{(\mathbf{z}_-(\lambda),x_B)} \label{eq:DTdel}$$ where $[\partial_i\Delta]_{(\mathbf{z}_-(\lambda),x_B)}$ denotes the partial derivative of $\Delta(\mathbf{x},x_B)$ with respect to $x^i$ taken at $\mathbf x=\mathbf z_-(\lambda)$. Then, after inserting Eqs. and into , we infer $$\frac{{\mathrm{d}}}{{\mathrm{d}}\lambda}\Delta(\mathbf{z}_-(\lambda),x_B)=\frac{R_{AB}}{2}\,\Omega_-(\widetilde{z}_-(\lambda),x_B) \label{eq:DTdeleik}$$ where the coordinates of the point-event $\widetilde{z}_-(\lambda)$ are obtained from Eq. , that is $\widetilde{\bm z}_-(\lambda)=\bm x_-(\mathbf z_-(\lambda))$. The components are given explicitly later on in Eq. . By fixing the following boundary conditions \[eq:bdR\] $$\begin{aligned} \Delta(\mathbf{z}_-(0),x_B)&=0\text{,}\label{eq:bdR0}\\ \Delta(\mathbf{z}_-(1),x_B)&=\Delta(\mathbf{x}_A,x_B)\text{,}\label{eq:bdR1}\end{aligned}$$ which follow from the requirement that $\Delta(\mathbf{x}_B,x_B)=0$ when $\mathbf{z}_-(0)=\mathbf{x}_B$, we find $$\Delta(\mathbf{x}_A,x_B)=\frac{R_{AB}}{2}\int_0^1\Omega_-(\widetilde{z}_-(\lambda),x_B)\,{\mathrm{d}}\lambda\text{.} \label{eq:intdel}$$ Then, insertion of Eq. allows us to recover the theorem 2 of [@2008CQGra..25n5020T] which would be expressed here in terms of the contravariant components $K^{\mu\nu}$ instead of $k^{\mu\nu}$’s. In principle, the all machinery developed in [@2008CQGra..25n5020T] for computing the delay functions could be applied using directly the components $K^{\mu\nu}$. However, such an approach possesses the inconvenience of hiding the role played by the different components $k^{\mu\nu}$ and $\kappa^{\mu\nu}$ during the determination of the total delay functions. Indeed, according to Eqs. the curvature of the optical spacetime is described simultaneously with the help of the components $h_{\mu\nu}$ and $\gamma_{\mu\nu}$ which might act on different characteristic lengths (e.g. $\gamma_{\mu\nu}(x)=0$ for $x\notin\mathcal{D}$) and might possess completely different orders of magnitude *a priori*. Therefore, in order to disentangle the contribution of each perturbation into the determination of the total delay, we must perform a complete separation between the physical quantities in Eq. . As it might be seen from Eqs. and such a separation can be achieved when the total time delay functions take the following forms \[eq:del\] $$\begin{aligned} \Delta(\mathbf{x}_A,x_B)&=\Delta_{\mathrm{g}}(\mathbf{x}_A,x_B)+\Delta_{\mathrm{r}}(\mathbf{x}_A,x_B)\nonumber\\ &+\Delta_{\mathrm{gr}}(\mathbf{x}_A,x_B)\text{,}\label{eq:delR}\end{aligned}$$ and $$\begin{aligned} \Xi(x_A,\mathbf{x}_B)&=\Xi_{\mathrm{g}}(x_A,\mathbf{x}_B)+\Xi_{\mathrm{r}}(x_A,\mathbf{x}_B)\nonumber\\ &+\Xi_{\mathrm{gr}}(x_A,\mathbf{x}_B)\text{.} \label{eq:delE}\end{aligned}$$ The subscripts $\mathrm{g}$, $\mathrm{r}$, and $\mathrm{gr}$ refer to the gravitational, the refractive, and the coupling contributions, respectively. The gravitational and the refractive time delay functions are expected to be driven by gravitational and refractive perturbations, respectively. Instead, the coupling time delay functions is expected to be of the order of the product of both perturbations. By Substituting for $\Delta(\mathbf{x}_A,x_B)$ from into and , and then by making use of the contravariant components of the optical and the physical spacetime metrics (see Eq. ), we deduce the following theorem. In the optical spacetime, the function $\Delta$ introduced in Eq. , can be decomposed as shown in Eq.  where each term in the summation satisfy an integro-differential equation \[eq:intdeltR\] $$\begin{aligned} \Delta_{\mathrm{g}}(\mathbf{x}_A,x_B)&=\frac{R_{AB}}{2}\int_0^1\bigg\{\big(k^{00}-2k^{0i}N^i_{AB}+k^{ij}N^i_{AB}N^j_{AB}\big)_{\widetilde{z}_-(\lambda)}\nonumber\\ &+2\big(k^{0i}-k^{ij}N^j_{AB}\big)_{\widetilde{z}_-(\lambda)}\left[\frac{\partial\Delta_{\mathrm{g}}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}+\big(\eta^{ij}+k^{ij}\big)_{\widetilde{z}_-(\lambda)}\left[\frac{\partial\Delta_{\mathrm{g}}}{\partial x^i}\frac{\partial\Delta_{\mathrm{g}}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\bigg\}{\mathrm{d}}\lambda\text{,}\label{eq:intdeltRg}\\ \Delta_{\mathrm{r}}(\mathbf{x}_A,x_B)&=\frac{R_{AB}}{2}\int_0^1\bigg\{\big(\kappa^{00}-2\kappa^{0i}N^i_{AB}+\kappa^{ij}N^i_{AB}N^j_{AB}\big)_{\widetilde{z}_-(\lambda)}\nonumber\\ &+2\big(\kappa^{0i}-\kappa^{ij}N^j_{AB}\big)_{\widetilde{z}_-(\lambda)}\left[\frac{\partial\Delta_{\mathrm{r}}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}+\big(\eta^{ij}+\kappa^{ij}\big)_{\widetilde{z}_-(\lambda)}\left[\frac{\partial\Delta_{\mathrm{r}}}{\partial x^i}\frac{\partial\Delta_{\mathrm{r}}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\bigg\}{\mathrm{d}}\lambda\text{,}\label{eq:intdeltRr} \end{aligned}$$ and $$\begin{aligned} \Delta_{\mathrm{gr}}(\mathbf{x}_A,x_B)&=\frac{R_{AB}}{2}\int_0^1\bigg\{\big(\eta^{ij}+k^{ij}+\kappa^{ij}\big)_{\widetilde{z}_-(\lambda)}\left[\frac{\partial\Delta_{\mathrm{gr}}}{\partial x^i}\frac{\partial\Delta_{\mathrm{gr}}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+2\big(k^{0i}+\kappa^{0i}-(k^{ij}+\kappa^{ij})N^j_{AB}\big)_{\widetilde{z}_-(\lambda)}\left[\frac{\partial\Delta_{\mathrm{gr}}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\big(k^{ij}\big)_{\widetilde{z}_-(\lambda)}\left[\frac{\partial\Delta_{\mathrm{r}}}{\partial x^i}\frac{\partial\Delta_{\mathrm{r}}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}+2\big(k^{0i}-k^{ij}N^j_{AB}\big)_{\widetilde{z}_-(\lambda)}\left[\frac{\partial\Delta_{\mathrm{r}}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\big(\kappa^{ij}\big)_{\widetilde{z}_-(\lambda)}\left[\frac{\partial\Delta_{\mathrm{g}}}{\partial x^i}\frac{\partial\Delta_{\mathrm{g}}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}+2\big(\kappa^{0i}-\kappa^{ij}N^j_{AB}\big)_{\widetilde{z}_-(\lambda)}\left[\frac{\partial\Delta_{\mathrm{g}}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+2\big(\eta^{ij}+k^{ij}+\kappa^{ij}\big)_{\widetilde{z}_-(\lambda)}\left[\frac{\partial\Delta_{\mathrm{g}}}{\partial x^i}\frac{\partial\Delta_{\mathrm{r}}}{\partial x^j}+\frac{\partial\Delta_{\mathrm{g}}}{\partial x^i}\frac{\partial\Delta_{\mathrm{gr}}}{\partial x^j}+\frac{\partial\Delta_{\mathrm{r}}}{\partial x^i}\frac{\partial\Delta_{\mathrm{gr}}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\bigg\}{\mathrm{d}}\lambda\text{.}\label{eq:intdeltRgr} \end{aligned}$$ The coordinates of the point-event $\widetilde{z}_-(\lambda)$ are given by $$\widetilde{\bm z}_-(\lambda)=\big(x_B^0-\lambda R_{AB}-\Delta(\mathbf z_-(\lambda),x_B),\mathbf z_-(\lambda)\big)\text{,} \label{eq:tz-m}$$ $\mathbf z_-(\lambda)$ being defined in Eq. . \[th:theo2\] Following the exact same reasoning, we state a similar theorem for the emission time delay function. However, for the emission case, the straight line segment connecting the emitter $x_A$ to the receiver $x_B$ is defined by [^4] $$\begin{aligned} \mathbf z_+(\mu)&=\mathbf x_A+\mu R_{AB}\mathbf{N}_{AB}\text{,} & 0\leqslant\mu\leqslant 1\text{.}\label{eq:z+i}\end{aligned}$$ Then, from the requirement that $\Xi(x_A,\mathbf{x}_A)=0$ when $\mathbf{z}_+(0)=\mathbf{x}_A$, we can set the following boundary conditions \[eq:bdE\] $$\begin{aligned} \Xi(x_A,\mathbf{z}_+(0))&=0\text{,}\label{eq:bdE0}\\ \Xi(x_A,\mathbf{z}_+(1))&=\Xi(x_A,\mathbf{x}_B)\text{.}\label{eq:bdE1}\end{aligned}$$ Hence, the theorem for the emission time delay function $\Xi(x_A,\mathbf x_B)/c$ reads as follows. In the optical spacetime, the function $\Xi$ introduced in Eq. , can be decomposed as shown in Eq.  where each term in the summation satisfy an integro-differential equation \[eq:intdeltE\] $$\begin{aligned} \Xi_{\mathrm{g}}(x_A,\mathbf x_B)&=\frac{R_{AB}}{2}\int_0^1\bigg\{\big(k^{00}-2k^{0i}N^i_{AB}+k^{ij}N^i_{AB}N^j_{AB}\big)_{\widetilde{z}_+(\mu)}\nonumber\\ &-2\big(k^{0i}-k^{ij}N^j_{AB}\big)_{\widetilde{z}_+(\mu)}\left[\frac{\partial\Xi_{\mathrm{g}}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\mu))}+\big(\eta^{ij}+k^{ij}\big)_{\widetilde{z}_+(\mu)}\left[\frac{\partial\Xi_{\mathrm{g}}}{\partial x^i}\frac{\partial\Xi_{\mathrm{g}}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\bigg\}{\mathrm{d}}\mu\text{,}\label{eq:intdeltEg}\\ \Xi_{\mathrm{r}}(x_A,\mathbf x_B)&=\frac{R_{AB}}{2}\int_0^1\bigg\{\big(\kappa^{00}-2\kappa^{0i}N^i_{AB}+\kappa^{ij}N^i_{AB}N^j_{AB}\big)_{\widetilde{z}_+(\mu)}\nonumber\\ &-2\big(\kappa^{0i}-\kappa^{ij}N^j_{AB}\big)_{\widetilde{z}_+(\mu)}\left[\frac{\partial\Xi_{\mathrm{r}}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\mu))}+\big(\eta^{ij}+\kappa^{ij}\big)_{\widetilde{z}_+(\mu)}\left[\frac{\partial\Xi_{\mathrm{r}}}{\partial x^i}\frac{\partial\Xi_{\mathrm{r}}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\bigg\}{\mathrm{d}}\mu\text{,}\label{eq:intdeltEr} \end{aligned}$$ and $$\begin{aligned} \Xi_{\mathrm{gr}}(x_A,\mathbf x_B)&=\frac{R_{AB}}{2}\int_0^1\bigg\{\big(\eta^{ij}+k^{ij}+\kappa^{ij}\big)_{\widetilde{z}_+(\mu)}\left[\frac{\partial\Xi_{\mathrm{gr}}}{\partial x^i}\frac{\partial\Xi_{\mathrm{gr}}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &-2\big(k^{0i}+\kappa^{0i}-(k^{ij}+\kappa^{ij})N^j_{AB}\big)_{\widetilde{z}_+(\mu)}\left[\frac{\partial\Xi_{\mathrm{gr}}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\lambda))}\nonumber\\ &+\big(k^{ij}\big)_{\widetilde{z}_+(\lambda)}\left[\frac{\partial\Xi_{\mathrm{r}}}{\partial x^i}\frac{\partial\Xi_{\mathrm{r}}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}-2\big(k^{0i}-k^{ij}N^j_{AB}\big)_{\widetilde{z}_+(\mu)}\left[\frac{\partial\Xi_{\mathrm{r}}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &+\big(\kappa^{ij}\big)_{\widetilde{z}(\lambda)}\left[\frac{\partial\Xi_{\mathrm{g}}}{\partial x^i}\frac{\partial\Xi_{\mathrm{g}}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}-2\big(\kappa^{0i}-\kappa^{ij}N^j_{AB}\big)_{\widetilde{z}_+(\mu)}\left[\frac{\partial\Xi_{\mathrm{g}}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &+2\big(\eta^{ij}+k^{ij}+\kappa^{ij}\big)_{\widetilde{z}_+(\mu)}\left[\frac{\partial\Xi_{\mathrm{g}}}{\partial x^i}\frac{\partial\Xi_{\mathrm{r}}}{\partial x^j}+\frac{\partial\Xi_{\mathrm{g}}}{\partial x^i}\frac{\partial\Xi_{\mathrm{gr}}}{\partial x^j}+\frac{\partial\Xi_{\mathrm{r}}}{\partial x^i}\frac{\partial\Xi_{\mathrm{gr}}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\bigg\}{\mathrm{d}}\mu\text{.}\label{eq:intdeltEgr} \end{aligned}$$ The coordinates of the point-event $\widetilde{z}_+(\mu)$ are given by $$\widetilde{\bm z}_+(\mu)=\big(x_A^0+\mu R_{AB}+\Xi(x_A,\mathbf z_+(\mu)),\mathbf z_+(\mu)\big)\text{,} \label{eq:tz+m}$$ $\mathbf z_+(\mu)$ being defined in Eq. . \[th:theo3\] Theorems \[th:theo2\] and \[th:theo3\] generalize theorems 2 and 3 of [@2008CQGra..25n5020T] for the optical spacetime. Indeed, in the limit where refractivity vanishes, that is to say $\kappa^{\mu\nu}\rightarrow 0$, theorems 2 and 3 of [@2008CQGra..25n5020T] are recovered. From Eqs. , we see that the choice does achieve the separation between the different physical quantities entering the computation of the total time delay. As a matter of fact, the right-hand sides of Eqs.  and contains purely gravitational and purely refractive quantities, respectively. The right-hand side of Eq. regroups all terms being a mixture of both. However, as it may be observed from the presence of the total delay in Eq. , the expressions of the different contributions are not fully independents but remain linked via the path of integration. In the next section, we shall further discuss this point and shall present a recursive resolution method for determining the time delay functions at any order. General expansions of the time delay functions {#sec:exp} ============================================== Because the line integrals in Eqs.  are taken along the path $\widetilde{z}_-(\lambda)$ for $0\leqslant\lambda\leqslant1$, the time delay functions $\Delta_{\mathrm{g}}/c$, $\Delta_{\mathrm{r}}/c$, and $\Delta_{\mathrm{gr}}/c$ cannot be solved independently from each other. Indeed, the total delay appearing in Eq.  depends on the three functions as it can be seen from the decomposition . Therefore, a systematic and recursive resolution of $\Delta/c$ can only be achieved, once the relative contributions of $\Delta_{\mathrm{g}}/c$, $\Delta_{\mathrm{r}}/c$, and $\Delta_{\mathrm{gr}}/c$ to the total time delay are known. In Sec. \[subsec:QM\], we first show how to determine the relative importance between the different contributions. Then, within the approximation of a quasi-Minkowskian path, we show that the interdependence between each functions $\Delta_{\mathrm{g}}/c$, $\Delta_{\mathrm{r}}/c$, and $\Delta_{\mathrm{gr}}/c$ can always be rejected to the following order during the resolution of $\Delta/c$. This fact allows one to sort out the occurrence of the different contributions within the determination of the total delay function (cf. theorems \[th:theo4\] and \[th:theo5\]). In Sec. \[sec:detref\], we assume that the refractive components of the optical metric admit a series expansion in term a parameter $N_0$. Then, we show that the refractive delay functions can be determined to any order through a recursive resolution method presented in theorems \[th:theo6\] and \[th:theo7\]. In Sec. \[sec:detgra\], we assume that the gravitational components of the spacetime metric admit a post-Minkowskian expansion (series expansion in ascending power of $G$). Then, the recursive method allowing one to determine the gravitational delay expressions to any order is presented in theorems \[th:theo8\] and \[th:theo9\]. Finally in Sec. \[sec:detcou\], we determine the coupling delay expressions up to any order within theorems \[th:theo10\] and \[th:theo11\]. Quasi-Minkowskian path regime {#subsec:QM} ----------------------------- As shown in theorems \[th:theo2\] and \[th:theo3\], the relative magnitude between each contribution to the total time delay rely on the line integrals of the gravitational and the refractive perturbations. Generally speaking, if gravity acts all along the light path $\Gamma$ joining $x_A$ to $x_B$, the refractive domain $\mathcal{D}$ is localized in spacetime and it follows that the action of refractivity remains bounded to a certain portion of $\Gamma$. Therefore, in order to determine the relative contributions of each time delay functions, not only the relative magnitude between the gravitational and refractive perturbations must be known, but also the typical length scales over which each perturbation acts. Let $\ell$ be the length of $\Gamma$ passing through $\mathcal{D}$. For a Minkowskian path, we always have $\ell\leqslant R_{AB}$ whatever is the size of $\mathcal{D}$. From Eq.  which has been formulated under the assumption that the light path is quasi-Minkowskian, we deduce $\Delta/R_{AB}\ll 1$. This implies that $\Delta_{\mathrm{g}}/R_{AB}\ll 1$, $\Delta_{\mathrm{r}}/R_{AB}\ll 1$, and $\Delta_{\mathrm{gr}}/R_{AB}\ll 1$. Considering that $\Delta_{\mathrm{gr}}$ represents the coupling contributions, its magnitude is expected to be of the order $$\frac{\Delta_{\mathrm{gr}}}{R_{AB}}\sim\left(\frac{\Delta_{\mathrm{g}}}{R_{AB}}\right)\left(\frac{\Delta_{\mathrm{r}}}{R_{AB}}\right)\text{.}$$ Therefore, we can first focus on the relative importance between the gravitational and refractive contributions. To do so, let us introduce the parameter $s$ defined by $$s=\left\lfloor\frac{\log_{10}(\Delta_{\mathrm{g}}/R_{AB})}{\log_{10}(\Delta_{\mathrm{r}}/R_{AB})}\right\rceil\text{,} \label{eq:s}$$ with $\lfloor i\rceil$ denoting the operation of rounding to the nearest integer of $i$. Hereafter, we intend to show that the expansion pattern of the delay functions can totally be determined once $s$ is known. Indeed, it allows one to sort out the occurrences of the gravitational and refractive terms in the determination of the total delay functions. Because we are only focusing on the main integer value of $s$ in Eq. , it is sufficient to get the first-order expressions of $\Delta_{\mathrm{g}}$ and $\Delta_{\mathrm{r}}$. Therefore, in Eqs. , line integrals can be changed into line integrals along the Minkowskian path between $x_A$ and $x_B$ by performing a Taylor series expansion of $\kappa^{\mu\nu}(\widetilde{z}_-(\lambda))$ and $k^{\mu\nu}(\widetilde{z}_-(\lambda))$ about the point-event $z_-(\lambda)$ whose coordinates are given by $$\bm z_-(\lambda)=\big(x_B^0-\lambda R_{AB},\mathbf{z}_-(\lambda)\big)\text{.} \label{eq:z-m}$$ Thus, optical metric components become infinite series in ascending power of the total time delay \[eq:Tay\] $$\begin{aligned} &k^{\mu\nu}\left(\widetilde{z}_-(\lambda),\frac{\Delta(\mathbf{z}_-(\lambda),x_B)}{R_{AB}}\right)=k^{\mu\nu}(z_-(\lambda))\nonumber\\ &+\sum_{l=1}^{\infty}\frac{(-R_{AB})^l}{l!}\left(\frac{\Delta(\mathbf{z}_-(\lambda),x_B)}{R_{AB}}\right)^l\big[\partial_0^lk^{\mu\nu}\big]_{z_-(\lambda)}\text{,} \label{eq:kTay}\end{aligned}$$ and $$\begin{aligned} &\kappa^{\mu\nu}\left(\widetilde{z}_-(\lambda),\frac{\Delta(\mathbf{z}_-(\lambda),x_B)}{R_{AB}}\right)=\kappa^{\mu\nu}(z_-(\lambda))\nonumber\\ &+\sum_{l=1}^{\infty}\frac{(-R_{AB})^l}{l!}\left(\frac{\Delta(\mathbf{z}_-(\lambda),x_B)}{R_{AB}}\right)^{l}\big[\partial_0^l\kappa^{\mu\nu}\big]_{z_-(\lambda)}\text{.} \label{eq:kapTay}\end{aligned}$$ After inserting these expressions into Eqs. and , we infer that the zeroth-order terms in Eqs. , namely $k^{\mu\nu}(z_-(\lambda))$ and $\kappa^{\mu\nu}(z_-(\lambda))$, correspond to the first-order determination of the gravitational and the refractive delays \[eq:dgdrrelmag\] $$\frac{\Delta_{\mathrm{g}}^{(1)}}{R_{AB}}=\frac{1}{2}\int_0^1(k^{00}-2k^{0i}N^i_{AB}+k^{ij}N^i_{AB}N^j_{AB})_{z_-(\lambda)}{\mathrm{d}}\lambda\text{,}$$ and $$\frac{\Delta_{\mathrm{r}}^{(1)}}{R_{AB}}=\frac{1}{2}\int_0^1(\kappa^{00}-2\kappa^{0i}N^i_{AB}+\kappa^{ij}N^i_{AB}N^j_{AB})_{z_-(\lambda)}{\mathrm{d}}\lambda\text{.}$$ (We will see with Eqs. and , that in the context of a quasi-Minkowskian path, these equations can be further simplified. But for now let us pursue the discussion with Eqs. ). These equations can be inserted into in order to determine the value of $s$. We shall discuss now how the expansion pattern of the delay functions can be worked out form $s$. Henceforth, we consider the case $s\in\mathbb{N}_{>1}$ (the result will be still valid for $s\in\mathbb{N}_{>0}$). In other words, we suppose that the refractive perturbation is dominant with respect to the gravitational one [^5]. In order to simplify the next discussion, and without loss of generality, we focus on orders of magnitude only. In addition, we consider that the light path occurs in a sufficiently small region of spacetime where the metric components do not vary significantly. Thus, we deduce from Eqs. that $$\frac{\Delta_{\mathrm{g}}^{(1)}}{R_{AB}}\sim |k^{\mu\nu}|_{\mathrm{max}}\text{,} \qquad \frac{\Delta_{\mathrm{r}}^{(1)}}{R_{AB}}\sim\frac{\ell}{R_{AB}}|\kappa^{\mu\nu}|_{\mathrm{max}}\text{.} \label{eq:del0ord}$$ In order to keep track of the relative magnitude between the gravitational and the refractive terms, we introduce a dimensionless parameter denoted $\varepsilon$ and being of the order of the dominant term, that is $$\varepsilon=\frac{\Delta_{\mathrm{r}}^{(1)}}{R_{AB}}\text{.} \label{eq:varepDr}$$ Thus, from Eqs. and , we immediately infer $$\frac{\Delta_{\mathrm{g}}^{(1)}}{R_{AB}}\sim\mathcal{O}(\varepsilon^s)\text{.} \label{eq:varepDg}$$ Therefore, the first-order expression of the total delay is driven by the refractive term only $$\Delta^{(1)}(\mathbf{x}_A,x_B)=\Delta_{\mathrm{r}}^{(1)}(\mathbf{x}_A,x_B) \label{eq:del1delr}$$ which means that $$\frac{\Delta^{(1)}}{R_{AB}}=\varepsilon\text{,} \label{eq:deltot1}$$ when $s>1$ in Eq. . Let us take a look at the relation between metric components. Eqs. , , and , allow us to deduce $$|k^{\mu\nu}|_{\mathrm{max}}\sim\mathcal{O}(\varepsilon^s)\text{,} \qquad \frac{\ell}{R_{AB}}|\kappa^{\mu\nu}|_{\mathrm{max}}\sim\mathcal{O}(\varepsilon)\text{.} \label{eq:kkap0}$$ At the same time, it might be seen from Eqs. , , and that the second-order refractive delay is driven by terms such like $$\frac{\ell}{R_{AB}}|\kappa^{\mu\nu}|_{\mathrm{max}}\frac{\Delta^{(1)}}{R_{AB}}\text{,} \qquad \frac{\Delta^{(1)}}{R_{AB}}\frac{\Delta^{(1)}}{R_{AB}}\text{,}$$ which, according to Eqs. and , are of the order of $\varepsilon^2$. Therefore, we conclude that the series expansion of the total delay goes on like $$\Delta^{(l)}(\mathbf{x}_A,x_B)=\Delta_{\mathrm{r}}^{(l)}(\mathbf{x}_A,x_B)\text{.}$$ for $1\leqslant l<s$. The first occurrence of the gravitational contribution to the total delay arises for $l=s$ as anticipated in Eq. . Therefore, the $s$th-order expression of the total delay is given by $$\Delta^{(s)}(\mathbf{x}_A,x_B)=\Delta_{\mathrm{r}}^{(s)}(\mathbf{x}_A,x_B)+\Delta_{\mathrm{g}}^{(1)}(\mathbf{x}_A,x_B)\text{.}$$ Then, by looking at the first-order term in Eq. , one might see that the second-order expression of the gravitational delay is proportional to $$\Delta_{\mathrm g}^{(2)}\sim|k^{\mu\nu}|_{\mathrm{max}}\frac{\Delta^{(1)}}{R_{AB}} \label{eq:delg2}$$ which, according to Eqs. and , is of the order of $\varepsilon^{s+1}$. Additionally, after inserting Eqs. into , we infer that the first-order expression of the coupling delay is driven by terms such like $$\frac{\ell}{R_{AB}}|\kappa^{\mu\nu}|_{\mathrm{max}}\frac{\Delta_{\mathrm g}^{(1)}}{R_{AB}}\text{,} \qquad |k^{\mu\nu}|_{\mathrm{max}}\frac{\Delta_{\mathrm{r}}^{(1)}}{R_{AB}}\text{,} \qquad \frac{\Delta_{\mathrm{r}}^{(1)}}{R_{AB}}\frac{\Delta_{\mathrm g}^{(1)}}{R_{AB}}\text{,}$$ which are of the order of $\varepsilon^{s+1}$ too. Therefore, the $(s+1)$th-order expression of the total delay is given by $$\begin{aligned} \Delta^{(s+1)}(\mathbf{x}_A,x_B)&=\Delta_{\mathrm{r}}^{(s+1)}(\mathbf{x}_A,x_B)+\Delta_{\mathrm{g}}^{(2)}(\mathbf{x}_A,x_B)\nonumber\\ &+\Delta_{\mathrm{gr}}^{(1)}(\mathbf{x}_A,x_B)\text{.}\end{aligned}$$ A quick look at the second-order expression of the coupling delay shows that it is driven by terms proportional to $\varepsilon^{s+2}$. Consequently, one deduces that $$\begin{aligned} \Delta^{(l)}(\mathbf{x}_A,x_B)&=\Delta_{\mathrm{r}}^{(l)}(\mathbf{x}_A,x_B)+\Delta_{\mathrm{g}}^{(l-s+1)}(\mathbf{x}_A,x_B)\nonumber\\ &+\Delta_{\mathrm{gr}}^{(l-s)}(\mathbf{x}_A,x_B)\text{,}\end{aligned}$$ for $l\geqslant s+1$. Let us summarize the discussion. Within the quasi-Minkowskian regime, the total delay satisfies $\Delta/R_{AB}\ll 1$, so the line integrals in Eqs. are simplified into line integrals along the Minkowskian path by performing a Taylor series expansion about the point-event $z_-(\lambda)$. Then, by assuming that refractivity is the dominant effect all along the light path $\Gamma$, it results that, in general, the total time delay admits an expansion as follows $$\Delta(\mathbf{x}_A,x_B)=\sum_{l=1}^{\infty}\Delta^{(l)}(\mathbf{x}_A,x_B) \label{eq:delRPM}$$ where the terms $\Delta^{(l)}$ are proportional to $\varepsilon^lR_{AB}$. In that respect, the different contributions to the total delay, namely the refractive, the gravitational, and the coupling delays, all admit series expansion as follows \[eq:delRrgrPM\] $$\begin{aligned} \Delta_{\mathrm{r}}(\mathbf{x}_A,x_B)&=\sum_{l=1}^{\infty}\Delta_{\mathrm{r}}^{(l)}(\mathbf{x}_A,x_B)\text{,}\label{eq:delRrPM}\\ \Delta_{\mathrm{g}}(\mathbf{x}_A,x_B)&=\sum_{l=1}^{\infty}\Delta_{\mathrm{g}}^{(l)}(\mathbf{x}_A,x_B)\text{,}\label{eq:delRgPM}\end{aligned}$$ and $$\Delta_{\mathrm{gr}}(\mathbf{x}_A,x_B)=\sum_{l=1}^{\infty}\Delta_{\mathrm{gr}}^{(l)}(\mathbf{x}_A,x_B) \label{eq:delRgrPM}$$ where the terms $\Delta_{\mathrm{r}}^{(l)}$, $\Delta_{\mathrm{g}}^{(l)}$, and $\Delta_{\mathrm{gr}}^{(l)}$ are of the order of $$\frac{\Delta_{\mathrm{r}}^{(l)}}{R_{AB}}\sim\mathcal{O}(\varepsilon^{l})\text{,} \quad \frac{\Delta_{\mathrm{g}}^{(l)}}{R_{AB}}\sim\mathcal{O}(\varepsilon^{l+s-1})\text{,} \quad \frac{\Delta_{\mathrm{gr}}^{(l)}}{R_{AB}}\sim\mathcal{O}(\varepsilon^{l+s})\text{.}$$ By making use of the Heaviside step function $$\Theta(i)=\left\{ \begin{array}{l l} 1 & \text{for }i\geqslant 0\text{,}\\ 0 & \text{otherwise,} \end{array} \right. \label{eq:stp}$$ we can write the terms $\Delta^{(l)}$ in Eq. as $$\begin{aligned} \Delta^{(l)}(\mathbf{x}_A,x_B)&=\Delta_{\mathrm{r}}^{(l)}(\mathbf{x}_A,x_B)+\Theta(l\!-\!s)\Delta_{\mathrm{g}}^{(l-s+1)}(\mathbf{x}_A,x_B)\nonumber\\ &+\Theta(l\!-\!s\!-\!1)\Delta_{\mathrm{gr}}^{(l-s)}(\mathbf{x}_A,x_B)\text{.}\label{eq:delRPMl}\end{aligned}$$ Let us recall that $\varepsilon$ is of the order of $\ell/R_{AB}|\kappa^{\mu\nu}|_{\mathrm{max}}$ only for a light path occurring in a sufficiently small region of spacetime where the metric components do not vary significantly. In general, it is given by Eq. . In the next two sections, according to the fact that the light ray follows a quasi-Minkowskian path, we will assume that the components $\kappa^{\mu\nu}$ and $k^{\mu\nu}$ admit series expansion in ascending power of parameters $N_0$ and $G$, respectively (see Eqs.  and ). If this fact does not change the pattern of the series expansions and , we should nevertheless, for completeness, specify that the quasi-Minkowskian path is parametrized by the expansion coefficients $N_0$ and $G$. Therefore, by making use of Eq. , we can state a theorem as follows. Within the approximation that the light path is quasi-Minkowskian and is parametrized by $N_0$ and $G$, the function $\Delta$ admits a series expansion as follows $$\Delta(\mathbf{x}_A,x_B,N_0,G)=\sum_{l=1}^{\infty}\Delta^{(l)}(\mathbf{x}_A,x_B)\text{,} \label{eq:delRPMN0G}$$ with $$\Delta^{(l)}(\mathbf{x}_A,x_B)=\Delta_{\mathrm{r}}^{(l)}(\mathbf{x}_A,x_B)+\Theta(l-s)\Delta_{\mathrm{g}}^{(l-s+1)}(\mathbf{x}_A,x_B)+\Theta(l-s-1)\Delta_{\mathrm{gr}}^{(l-s)}(\mathbf{x}_A,x_B)\text{.}$$ The parameter $s\in\mathbb{N}_{>0}$ is determined from Eq. by making use of the first-order expressions and . \[th:theo4\] A similar reasoning works for the emission time delay function as well. Indeed, the line integrals in Eqs. can be expanded in a Taylor series about the point-event $z_+(\mu)$ defined by $$\bm z_+(\mu)=\big(x_B^0+\mu R_{AB},\mathbf{z}_+(\mu)\big)\text{.} \label{eq:z+m}$$ Then, $\kappa^{\mu\nu}(\widetilde{z}_+(\mu))$ and $k^{\mu\nu}(\widetilde{z}_+(\mu))$ become infinite series in ascending power of $\Xi$ similarly to what have been done in Eqs. . Therefore, we end up with similar expansion for $\Xi$ than for $\Delta$ and we state the following theorem. Within the approximation that the light path is quasi-Minkowskian and is parametrized by $N_0$ and $G$, the function $\Xi$ admits a series expansion as follows $$\Xi(x_A,\mathbf{x}_B,N_0,G)=\sum_{l=1}^{\infty}\Xi^{(l)}(x_A,\mathbf{x}_B)\text{,} \label{eq:delEPMN0G}$$ with $$\Xi^{(l)}(x_A,\mathbf{x}_B)=\Xi_{\mathrm{r}}^{(l)}(x_A,\mathbf{x}_B)+\Theta(l-s)\Xi_{\mathrm{g}}^{(l-s+1)}(x_A,\mathbf{x}_B)+\Theta(l-s-1)\Xi_{\mathrm{gr}}^{(l-s)}(x_A,\mathbf{x}_B)\text{.}$$ The parameter $s\in\mathbb{N}_{>0}$ is determined from Eq. by making use of the first-order expressions and . \[th:theo5\] Equipped with theorems \[th:theo2\] to \[th:theo5\], we can now recursively determine the integral form of each component in the time delay function expressions . The refractive time delay functions {#sec:detref} ----------------------------------- We saw in Sec. \[sec:intTF\] that a quasi-Minkowskian path implies the small refractivity approximation, that is to say $N(x)\ll1$. Let $N_0$ be the refractivity at a well-chosen point-event $x_0\in\mathcal{D}$ located on $\Gamma$. $N_0$ is a constant defined by $N_0=N(x_0)$. We can always write $$N(x)=N_0\left(\frac{N(x)}{N_0}\right)\text{.}$$ Hence, considering a quasi-Minkowskian light path, it follows that $N_0\ll1$. Therefore, we can always expand the components $\kappa^{\mu\nu}$ in ascending power of $N_0$ such like $$\kappa^{\mu\nu}(x,N_0)=\sum_{l=1}^{\infty}\kappa^{\mu\nu}_{(l)}(x) \label{eq:kapPM}$$ where $\kappa^{\mu\nu}_{(l)}\propto(N_0)^l$. Considering that the wave-vector is by definition a covector (see Eq. ), the optical metric is intrinsically defined for its contravariant components as seen from Eq. . Therefore, the covariant components of the optical metric are not needed *a priori* to solve the time and frequency transfers. However, we provide their expressions in Sec. \[sec:genexp\] for completeness. As discussed previously, line integrations in Eq.  are taken along the real light path $\widetilde{z}_-(\lambda)$ for $0\leqslant\lambda\leqslant 1$. Within the quasi-Minkowskian path regime, we saw in Eq. that the metric components $\kappa^{\mu\nu}(\widetilde{z}_-(\lambda))$ can be expanded in ascending power of the total delay. In that respect, the right-hand side of Eq.  involves terms such as $\kappa^{\mu\nu}(z_-(\lambda))$. By making use of Eq. , we immediately find $$\kappa^{\mu\nu}(z_-(\lambda),N_0)=\sum_{l=1}^{\infty}\kappa^{\mu\nu}_{(l)}(z_-(\lambda))\text{.} \label{eq:kapPMz-}$$ Therefore, the general expansion of $\kappa^{\mu\nu}(\widetilde{z}_-(\lambda))$ is obtained after substituting for $\Delta$ and $\kappa^{\mu\nu}$ from Eqs.  and into , respectively. After some algebra, we find a relation as follows $$\kappa^{\mu\nu}(\widetilde{z}_-(\lambda),N_0,G)=\sum_{l=1}^{\infty}\widehat{\kappa}^{\mu\nu}_{-(l)}(z_-(\lambda),x_B) \label{eq:kapRPM}$$ where the quantities $\widehat{\kappa}^{\mu\nu}_{-(l)}(z_-(\lambda),x_B)$ are given by \[eq:kapRPMdec\] $$\widehat{\kappa}^{\mu\nu}_{-(1)}(z_-(\lambda),x_B)=\kappa^{\mu\nu}_{(1)}(z_-(\lambda))\text{,}$$ and $$\begin{aligned} &\widehat{\kappa}^{\mu\nu}_{-(l)}(z_-(\lambda),x_B)=\kappa^{\mu\nu}_{(l)}(z_-(\lambda))\nonumber\\ &+\sum_{m=1}^{l-1}\sum_{n=1}^{m}\Phi_-^{(m,n)}(\mathbf z_-(\lambda),x_B)\left[\frac{\partial^n\kappa^{\mu\nu}_{(l-m)}}{(\partial x^0)^n}\right]_{z_-(\lambda)}\end{aligned}$$ for $l\geqslant 2$. The function $\Phi_-^{(l,m)}(\mathbf x,x_B)$ is called a reception function and is defined [@2008CQGra..25n5020T] such that $$\begin{aligned} \Phi_-^{(l,m)}&(\mathbf x,x_B)=\frac{(-1)^m}{m!}\nonumber\\ &\times\sum_{n_1+\cdots+n_m=l-m}\Bigg[\prod_{d=1}^m\Delta^{(n_d+1)}(\mathbf x,x_B)\Bigg]\text{,}\label{eq:recfun}\end{aligned}$$ with $l\geqslant 1$ and $1\leqslant m\leqslant l$, and with $n_1,\ldots,n_m\in\mathbb{N}_{\geqslant 0}$. The summation in is taken over all sequences of $n_1$ through $n_m$ such that the sum of all $n_m$ is $l-m$. Finally, by substituting for $\kappa^{\mu\nu}(\widetilde{z}_-(\lambda))$ from Eq. into , we infer the theorem which follows concerning the refractive time delay function at reception. In the optical spacetime, within the quasi-Minkowskian path regime, $\Delta$ admits the series expansion introduced in theorem \[th:theo4\], so the function $\Delta_{\mathrm{r}}$ is given by $$\Delta_{\mathrm{r}}(\mathbf{x}_A,x_B,N_0,G)=\sum_{l=1}^{\infty}\Delta_{\mathrm{r}}^{(l)}(\mathbf{x}_A,x_B) \label{eq:delRrPMN0G}$$ where \[eq:intdelRrPM\] $$\begin{aligned} \Delta_{\mathrm{r}}^{(1)}(\mathbf{x}_A,x_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\big(\kappa^{00}_{(1)}-2\kappa^{0i}_{(1)}N^i_{AB}+\kappa^{ij}_{(1)}N^i_{AB}N^j_{AB}\big)_{z_-(\lambda)}{\mathrm{d}}\lambda\text{,}\label{eq:intdelRrPM1}\\ \Delta_{\mathrm{r}}^{(2)}(\mathbf{x}_A,x_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\Bigg\{\big(\widehat{\kappa}^{00}_{-(2)}-2\widehat{\kappa}^{0i}_{-(2)}N^i_{AB}+\widehat{\kappa}^{ij}_{-(2)}N^i_{AB}N^j_{AB}\big)_{(z_-(\lambda),x_B)}\nonumber\\ &+2\big(\kappa^{0i}_{(1)}-\kappa^{ij}_{(1)}N_{AB}^j\big)_{z_-(\lambda)}\left[\frac{\partial\Delta_{\mathrm{r}}^{(1)}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}+\eta^{ij}\left[\frac{\partial\Delta_{\mathrm{r}}^{(1)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{r}}^{(1)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\Bigg\}{\mathrm{d}}\lambda\text{,} \end{aligned}$$ and $$\begin{aligned} \Delta_{\mathrm{r}}^{(l)}(\mathbf{x}_A,x_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\Bigg\{\big(\widehat{\kappa}^{00}_{-(l)}-2\widehat{\kappa}^{0i}_{-(l)}N^i_{AB}+\widehat{\kappa}^{ij}_{-(l)}N^i_{AB}N^j_{AB}\big)_{(z_-(\lambda),x_B)}\nonumber\\ &+2\sum_{m=1}^{l-1}\big(\widehat\kappa^{0i}_{-(m)}-\widehat\kappa^{ij}_{-(m)}N_{AB}^j\big)_{(z_-(\lambda),x_B)}\left[\frac{\partial\Delta_{\mathrm{r}}^{(l-m)}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}+\eta^{ij}\sum_{m=1}^{l-1}\left[\frac{\partial\Delta_{\mathrm{r}}^{(m)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{r}}^{(l-m)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\sum_{m=1}^{l-2}\big(\widehat{\kappa}^{ij}_{-(m)}\big)_{(z_-(\lambda),x_B)}\sum_{n=1}^{l-m-1}\left[\frac{\partial\Delta_{\mathrm{r}}^{(n)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{r}}^{(l-m-n)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\Bigg\}{\mathrm{d}}\lambda \end{aligned}$$ for $l \geqslant 3$. The quantities $\widehat{\kappa}^{\mu\nu}_{-(l)}(z_-(\lambda),x_B)$ are defined in Eqs. . \[th:theo6\] Applying the exact same reasoning, the analogous theorem for the refractive time delay function at emission can be stated as well. Line integrations in Eq.  are taken along the light path $\widetilde{z}_+(\mu)$ for $0\leqslant\mu\leqslant 1$. After Taylor expanding the light path about the point-event $z_+(\mu)$, the right-hand side of Eq.  involves terms such as $\kappa^{\mu\nu}(z_+(\mu))$. By making use of Eq. , we find $$\kappa^{\mu\nu}(z_+(\mu),N_0)=\sum_{l=1}^{\infty}\kappa^{\mu\nu}_{(l)}(z_+(\mu))\text{.} \label{eq:kapPMz+}$$ The general expansion of $\kappa^{\mu\nu}(\widetilde{z}_+(\mu))$ is obtained after substituting for $\Xi$ from Eq.  and for $\kappa^{\mu\nu}$ from Eq. , into . After some algebra, we find $$\kappa^{\mu\nu}(\widetilde{z}_+(\mu),N_0,G)=\sum_{l=1}^{\infty}\widehat{\kappa}^{\mu\nu}_{+(l)}(x_A,z_+(\mu)) \label{eq:kapEPM}$$ where the quantities $\widehat{\kappa}^{\mu\nu}_{+(l)}(x_A,z_+(\mu))$ are given by \[eq:kapEPMdec\] $$\widehat{\kappa}^{\mu\nu}_{+(1)}(x_A,z_+(\mu))=\kappa^{\mu\nu}_{(1)}(z_+(\mu))\text{,}$$ and $$\begin{aligned} &\widehat{\kappa}^{\mu\nu}_{+(l)}(x_A,z_+(\mu))=\kappa^{\mu\nu}_{(l)}(z_+(\mu))\nonumber\\ &+\sum_{m=1}^{l-1}\sum_{n=1}^{m}\Phi_+^{(m,n)}(x_A,\mathbf z_+(\mu))\left[\frac{\partial^n\kappa^{\mu\nu}_{(l-m)}}{(\partial x^0)^n}\right]_{z_+(\mu)}\end{aligned}$$ for $l\geqslant 2$. The function $\Phi_+^{(l,m)}(x_A,\mathbf x)$ is called an emission function and is defined such as $$\begin{aligned} \Phi_+^{(l,m)}&(x_A,\mathbf x)=\frac{1}{m!}\nonumber\\ &\times\sum_{n_1+\cdots+n_m=l-m}\Bigg[\prod_{d=1}^m\Xi^{(n_d+1)}(x_A,\mathbf x)\Bigg]\text{,}\label{eq:emifun}\end{aligned}$$ with $l\geqslant 1$ and $1\leqslant m\leqslant l$, and with $n_1,\ldots,n_m\in\mathbb{N}_{\geqslant 0}$. The summation in is taken over all sequences of $n_1$ through $n_m$ such that the sum of all $n_m$ is $l-m$. Finally, the theorem for the refractive time delay function at emission is obtained after substituting for $\kappa^{\mu\nu}(\widetilde{z}_+(\mu))$ from Eq. into . In the optical spacetime, within the quasi-Minkowskian path regime, $\Xi$ admits the series expansion introduced in theorem \[th:theo5\], so the function $\Xi_{\mathrm{r}}$ is given by $$\Xi_{\mathrm{r}}(x_A,\mathbf{x}_B,N_0,G)=\sum_{l=1}^{\infty}\Xi_{\mathrm{r}}^{(l)}(x_A,\mathbf{x}_B) \label{eq:delErPM}$$ where \[eq:intdelErPM\] $$\begin{aligned} \Xi_{\mathrm{r}}^{(1)}(x_A,\mathbf{x}_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\big(\kappa^{00}_{(1)}-2\kappa^{0i}_{(1)}N^i_{AB}+\kappa^{ij}_{(1)}N^i_{AB}N^j_{AB}\big)_{z_+(\mu)}{\mathrm{d}}\mu\text{,}\\ \Xi_{\mathrm{r}}^{(2)}(x_A,\mathbf{x}_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\Bigg\{\big(\widehat{\kappa}^{00}_{+(2)}-2\widehat{\kappa}^{0i}_{+(2)}N^i_{AB}+\widehat{\kappa}^{ij}_{+(2)}N^i_{AB}N^j_{AB}\big)_{(x_A,z_+(\mu))}\nonumber\\ &-2\big(\kappa^{0i}_{(1)}-\kappa^{ij}_{(1)}N_{AB}^j\big)_{z_+(\mu)}\left[\frac{\partial\Xi_{\mathrm{r}}^{(1)}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\mu))}+\eta^{ij}\left[\frac{\partial\Xi_{\mathrm{r}}^{(1)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{r}}^{(1)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\Bigg\}{\mathrm{d}}\mu\text{,} \end{aligned}$$ and $$\begin{aligned} \Xi_{\mathrm{r}}^{(l)}(x_A,\mathbf{x}_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\Bigg\{\big(\widehat{\kappa}^{00}_{+(l)}-2\widehat{\kappa}^{0i}_{+(l)}N^i_{AB}+\widehat{\kappa}^{ij}_{+(l)}N^i_{AB}N^j_{AB}\big)_{(x_A,z_+(\mu))}\nonumber\\ &-2\sum_{m=1}^{l-1}\big(\widehat\kappa^{0i}_{+(m)}-\widehat\kappa^{ij}_{+(m)}N_{AB}^j\big)_{(x_A,z_+(\mu))}\left[\frac{\partial\Xi_{\mathrm{r}}^{(l-m)}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\mu))}+\eta^{ij}\sum_{m=1}^{l-1}\left[\frac{\partial\Xi_{\mathrm{r}}^{(m)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{r}}^{(l-m)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &+\sum_{m=1}^{l-2}\big(\widehat{\kappa}^{ij}_{+(m)}\big)_{(x_A,z_+(\mu))}\sum_{n=1}^{l-m-1}\left[\frac{\partial\Xi_{\mathrm{r}}^{(n)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{r}}^{(l-m-n)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\Bigg\}{\mathrm{d}}\mu \end{aligned}$$ for $l \geqslant 3$. The quantities $\widehat{\kappa}^{\mu\nu}_{+(l)}(x_A,z_+(\mu))$ are defined in Eqs. . \[th:theo7\] In the case where the components $\kappa^{\mu\nu}$ represent the leading perturbation (see Eq. ), it may be seen that the expansion pattern in theorems \[th:theo6\] and \[th:theo7\] is almost the same as the one in theorems 4 and 5 of [@2008CQGra..25n5020T]. Actually, if one assumes that the gravitational components $k^{\mu\nu}$ are the leading perturbations, one obtains quasi similar theorems than theorems 4 and 5 of [@2008CQGra..25n5020T]. The difference would be in the definition of the quantities $\widehat{k}^{\mu\nu}_{+(l)}$. In our case, they would have involved the total time delay containing the gravitational, the refractive, and the coupling components, instead of the unique gravitational contribution. The gravitational time delay functions {#sec:detgra} -------------------------------------- Following [@2008CQGra..25n5020T], we suppose that the gravitational perturbation terms $h_{\mu\nu}$ can be expressed as a post-Minkowskian expansion, such like $$h_{\mu\nu}(x,G)=\sum_{l=1}^{\infty}h_{\mu\nu}^{(l)}(x)\text{.} \label{eq:hPM}$$ The contravariant components are given by $$k^{\mu\nu}(x,G)=\sum_{l=1}^{\infty}k^{\mu\nu}_{(l)}(x) \label{eq:kPM}$$ where the components $k^{\mu\nu}_{(l)}$ can be recursively determined using the following relationships \[eq:kPMord\] $$\begin{aligned} k^{\mu\nu}_{(1)}&=-\eta^{\mu\alpha}\eta^{\beta\nu}h_{\alpha\beta}^{(1)}\text{,}\label{eq:kPMord1}\\ k^{\mu\nu}_{(l)}&=-\eta^{\mu\alpha}\eta^{\beta\nu}h_{\alpha\beta}^{(l)}-\sum_{m=1}^{l-1}\eta^{\mu\alpha}h_{\alpha\beta}^{(m)}k^{\beta\nu}_{(l-m)}\label{eq:kPMord2}\end{aligned}$$ for $l\geqslant 2$. The right-hand side of Eq.  involves terms such as $k^{\mu\nu}(z_-(\lambda))$. Thus, by making use of Eq. , we find $$k^{\mu\nu}(z_-(\lambda),G)=\sum_{l=1}^{\infty}k^{\mu\nu}_{(l)}(z_-(\lambda)) \label{eq:kPMz-}$$ where $k^{\mu\nu}_{(l)}\propto G^l$. Therefore, the general expansion of $k^{\mu\nu}(\widetilde{z}_-(\lambda))$ is obtained after substituting for $\Delta$ and $k^{\mu\nu}$ from Eqs.  and into , respectively. After some algebra, we find the following expression $$k^{\mu\nu}(\widetilde{z}_-(\lambda),N_0,G)=\sum_{l=s}^{\infty}\widehat{k}^{\mu\nu}_{-(l)}(z_-(\lambda),x_B) \label{eq:kRPM}$$ where the quantities $\widehat{k}^{\mu\nu}_{-(l)}(z_-(\lambda),x_B)$ are given for $l\geqslant s$ by the following expressions \[eq:kRPMdec\] $$\widehat{k}^{\mu\nu}_{-(s)}(z_-(\lambda),x_B)=k_{(1)}^{\mu\nu}(z_-(\lambda))\text{,}$$ and $$\begin{aligned} &\widehat{k}^{\mu\nu}_{-(ps)}(z_-(\lambda),x_B)=k^{\mu\nu}_{(p)}(z_-(\lambda))\nonumber\\ &+\sum_{m=1}^{p-1}\sum_{n=1}^{ms}\Phi_-^{(ms,n)}(\mathbf z_-(\lambda),x_B)\left[\frac{\partial^nk_{(p-m)}^{\mu\nu}}{(\partial x^0)^n}\right]_{z_-(\lambda)}\label{eq:kRPMdecl}\end{aligned}$$ for $p\geqslant 2$, and $$\begin{aligned} &\widehat{k}^{\mu\nu}_{-(ps+q)}(z_-(\lambda),x_B)=\nonumber\\ &+\sum_{n=1}^{q}\Phi_-^{(q,n)}(\mathbf z_-(\lambda),x_B)\left[\frac{\partial^nk_{(p)}^{\mu\nu}}{(\partial x^0)^n}\right]_{z_-(\lambda)}\nonumber\\ &+\sum_{m=1}^{p-1}\sum_{n=1}^{ms+q}\Phi_-^{(ms+q,n)}(\mathbf z_-(\lambda),x_B)\left[\frac{\partial^nk_{(p-m)}^{\mu\nu}}{(\partial x^0)^n}\right]_{z_-(\lambda)}\label{eq:kRPMdecl}\end{aligned}$$ for $p\geqslant 1$ and $1\leqslant q\leqslant s-1$, where $p$ and $q$ are determined from $l$ using the following relationships $$\begin{aligned} p&=\lfloor l/s\rfloor\text{,} & q&=l-ps\text{,}\label{eq:pq}\end{aligned}$$ with $\lfloor i\rfloor$ denoting the integer part of $i$. By substituting for $k^{\mu\nu}(\widetilde{z}_-(\lambda))$ from Eq. into , we infer the theorem which follows concerning the gravitational time delay function at reception. In the optical spacetime, within the quasi-Minkowskian path regime, $\Delta$ admits the series expansion introduced in theorem \[th:theo4\], so the function $\Delta_{\mathrm{g}}$ is given by $$\Delta_{\mathrm{g}}(\mathbf{x}_A,x_B,N_0,G)=\sum_{l=1}^{\infty}\Delta_{\mathrm{g}}^{(l)}(\mathbf{x}_A,x_B) \label{eq:delRgPMN0G}$$ where \[eq:intdelRgPM\] $$\begin{aligned} \Delta_{\mathrm{g}}^{(1)}(\mathbf{x}_A,x_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\big(k^{00}_{(1)}-2k^{0i}_{(1)}N^i_{AB}+k^{ij}_{(1)}N^i_{AB}N^j_{AB}\big)_{z_-(\lambda)}{\mathrm{d}}\lambda\text{,}\label{eq:intdelRgPM1}\\ \Delta_{\mathrm{g}}^{(l)}(\mathbf{x}_A,x_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\big(\widehat k^{00}_{-(s+l-1)}-2\widehat k^{0i}_{-(s+l-1)}N^i_{AB}+\widehat k^{ij}_{-(s+l-1)}N^i_{AB}N^j_{AB}\big)_{(z_-(\lambda),x_B)}{\mathrm{d}}\lambda \end{aligned}$$ for $2\leqslant l\leqslant s$, and $$\begin{aligned} \Delta_{\mathrm{g}}^{(l)}(\mathbf{x}_A,x_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\Bigg\{\big(\widehat{k}^{00}_{-(s+l-1)}-2\widehat{k}^{0i}_{-(s+l-1)}N^i_{AB}+\widehat{k}^{ij}_{-(s+l-1)}N^i_{AB}N^j_{AB}\big)_{(z_-(\lambda),x_B)}\nonumber\\ &+2\sum_{m=s}^{l-1}\big(\widehat k^{0i}_{-(m)}-\widehat k^{ij}_{-(m)}N_{AB}^j\big)_{(z_-(\lambda),x_B)}\left[\frac{\partial\Delta_{\mathrm{g}}^{(l-m)}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\eta^{ij}\sum_{m=1}^{l-s}\left[\frac{\partial\Delta_{\mathrm{g}}^{(m)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{g}}^{(l-s-m+1)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\Bigg\}{\mathrm{d}}\lambda \end{aligned}$$ for $s+1\leqslant l\leqslant 2s$, and finally $$\begin{aligned} \Delta_{\mathrm{g}}^{(l)}(\mathbf{x}_A,x_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\Bigg\{\big(\widehat{k}^{00}_{-(s+l-1)}-2\widehat{k}^{0i}_{-(s+l-1)}N^i_{AB}+\widehat{k}^{ij}_{-(s+l-1)}N^i_{AB}N^j_{AB}\big)_{(z_-(\lambda),x_B)}\nonumber\\ &+2\sum_{m=s}^{l-1}\big(\widehat k^{0i}_{-(m)}-\widehat k^{ij}_{-(m)}N_{AB}^j\big)_{(z_-(\lambda),x_B)}\left[\frac{\partial\Delta_{\mathrm{g}}^{(l-m)}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\eta^{ij}\sum_{m=1}^{l-s}\left[\frac{\partial\Delta_{\mathrm{g}}^{(m)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{g}}^{(l-s-m+1)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\sum_{m=s}^{l-s-1}\big(\widehat{k}^{ij}_{-(m)}\big)_{(z_-(\lambda),x_B)}\sum_{n=1}^{l-s-m}\left[\frac{\partial\Delta_{\mathrm{g}}^{(n)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{g}}^{(l-s-m-n+1)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\Bigg\}{\mathrm{d}}\lambda \end{aligned}$$ for $l\geqslant 2s+1$. The quantities $\widehat{k}^{\mu\nu}_{-(l)}(z_-(\lambda),x_B)$ are defined in Eqs. . \[th:theo8\] A similar reasoning allows us to state an analogous theorem for the gravitational time delay function at emission. Indeed, the right-hand side of Eq.  involves line integrals along the light path $\widetilde{z}_+(\mu)$ parametrized by $0\leqslant\mu\leqslant1$. After Taylor expanding the light path about the point-event $z_+(\mu)$, the right-hand side of Eq.  involves terms such as $k^{\mu\nu}(z_+(\mu))$. Thus, by making use of Eq. , we find $$k^{\mu\nu}(z_+(\mu),G)=\sum_{l=1}^{\infty}k^{\mu\nu}_{(l)}(z_+(\mu))\text{.} \label{eq:kPMz+}$$ Then, the general expansion of $k^{\mu\nu}(\widetilde{z}_+(\mu))$ is obtained after substituting for $\Xi$ from Eq. , and for $k^{\mu\nu}$ from Eq. , into . After some algebra, we find $$k^{\mu\nu}(\widetilde{z}_+(\mu),N_0,G)=\sum_{l=s}^{\infty}\widehat{k}^{\mu\nu}_{+(l)}(x_A,z_+(\mu)) \label{eq:kEPM}$$ where the quantities $\widehat{k}^{\mu\nu}_{+(l)}(x_A,z_+(\mu))$ are given for $l\geqslant s$ by the following expressions \[eq:kEPMdec\] $$\widehat{k}^{\mu\nu}_{+(s)}(x_A,z_+(\mu))=k_{(1)}^{\mu\nu}(z_+(\mu))\text{,}$$ and $$\begin{aligned} &\widehat{k}^{\mu\nu}_{+(ps)}(x_A,z_+(\mu))=k^{\mu\nu}_{(p)}(z_+(\mu))\nonumber\\ &+\sum_{m=1}^{p-1}\sum_{n=1}^{ms}\Phi_+^{(ms,n)}(x_A,\mathbf z_+(\mu))\left[\frac{\partial^nk_{(p-m)}^{\mu\nu}}{(\partial x^0)^n}\right]_{z_+(\mu)}\label{eq:kEPMdecl}\end{aligned}$$ for $p\geqslant 2$, and $$\begin{aligned} &\widehat{k}^{\mu\nu}_{+(ps+q)}(x_A,z_+(\mu))=\nonumber\\ &+\sum_{n=1}^{q}\Phi_+^{(q,n)}(x_A,\mathbf z_+(\mu))\left[\frac{\partial^nk_{(p)}^{\mu\nu}}{(\partial x^0)^n}\right]_{z_+(\mu)}\nonumber\\ &+\sum_{m=1}^{p-1}\sum_{n=1}^{ms+q}\Phi_+^{(ms+q,n)}(x_A,\mathbf z_+(\mu))\left[\frac{\partial^nk_{(p-m)}^{\mu\nu}}{(\partial x^0)^n}\right]_{z_+(\mu)}\label{eq:kEPMdecl}\end{aligned}$$ \ for $p\geqslant 1$ and $1\leqslant q\leqslant s-1$ where $p$ and $q$ are determined from $l$ using the relationships in Eq. . By substituting for $k^{\mu\nu}(\widetilde{z}_-(\lambda))$ from Eq. into , we infer the theorem which follows. In the optical spacetime, within the quasi-Minkowskian path regime, $\Xi$ admits the series expansion introduced in theorem \[th:theo5\], so the function $\Xi_{\mathrm{g}}$ is given by $$\Xi_{\mathrm{g}}(x_A,\mathbf{x}_B,N_0,G)=\sum_{l=1}^{\infty}\Xi_{\mathrm{g}}^{(l)}(x_A,\mathbf{x}_B) \label{eq:delEgPM}$$ where \[eq:intdelEgPM\] $$\begin{aligned} \Xi_{\mathrm{g}}^{(1)}(x_A,\mathbf{x}_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\big(k^{00}_{(1)}-2k^{0i}_{(1)}N^i_{AB}+k^{ij}_{(1)}N^i_{AB}N^j_{AB}\big)_{z_+(\mu)}{\mathrm{d}}\mu\text{,}\\ \Xi_{\mathrm{g}}^{(l)}(x_A,\mathbf{x}_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\big(\widehat k^{00}_{+(s+l-1)}-2\widehat k^{0i}_{+(s+l-1)}N^i_{AB}+\widehat k^{ij}_{+(s+l-1)}N^i_{AB}N^j_{AB}\big)_{(x_A,z_+(\mu))}{\mathrm{d}}\mu \end{aligned}$$ for $2\leqslant l\leqslant s$, and $$\begin{aligned} \Xi_{\mathrm{g}}^{(l)}(x_A,\mathbf{x}_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\Bigg\{\big(\widehat{k}^{00}_{+(s+l-1)}-2\widehat{k}^{0i}_{+(s+l-1)}N^i_{AB}+\widehat{k}^{ij}_{+(s+l-1)}N^i_{AB}N^j_{AB}\big)_{(x_A,z_+(\mu))}\nonumber\\ &-2\sum_{m=s}^{l-1}\big(\widehat k^{0i}_{+(m)}-\widehat k^{ij}_{+(m)}N_{AB}^j\big)_{(x_A,z_+(\mu))}\left[\frac{\partial\Xi_{\mathrm{g}}^{(l-m)}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &+\eta^{ij}\sum_{m=1}^{l-s}\left[\frac{\partial\Xi_{\mathrm{g}}^{(m)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{g}}^{(l-s-m+1)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\Bigg\}{\mathrm{d}}\mu \end{aligned}$$ for $s+1\leqslant l\leqslant 2s$, and finally $$\begin{aligned} \Xi_{\mathrm{g}}^{(l)}(x_A,\mathbf{x}_B)&=\frac{R_{AB}}{2}\int^{1}_{0}\Bigg\{\big(\widehat{k}^{00}_{+(s+l-1)}-2\widehat{k}^{0i}_{+(s+l-1)}N^i_{AB}+\widehat{k}^{ij}_{+(s+l-1)}N^i_{AB}N^j_{AB}\big)_{(x_A,z_+(\mu))}\nonumber\\ &-2\sum_{m=s}^{l-1}\big(\widehat k^{0i}_{+(m)}-\widehat k^{ij}_{+(m)}N_{AB}^j\big)_{(x_A,z_+(\mu))}\left[\frac{\partial\Xi_{\mathrm{g}}^{(l-m)}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &+\eta^{ij}\sum_{m=1}^{l-s}\left[\frac{\partial\Xi_{\mathrm{g}}^{(m)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{g}}^{(l-s-m+1)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &+\sum_{m=s}^{l-s-1}\big(\widehat{k}^{ij}_{+(m)}\big)_{(x_A,z_+(\mu))}\sum_{n=1}^{l-s-m}\left[\frac{\partial\Xi_{\mathrm{g}}^{(n)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{g}}^{(l-s-m-n+1)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\lambda))}\Bigg\}{\mathrm{d}}\mu \end{aligned}$$ for $l\geqslant 2s+1$. The quantities $\widehat{k}^{\mu\nu}_{+(l)}(x_A,z_+(\mu))$ are defined in Eqs. . \[th:theo9\] As a final remark, let us emphasize that Eqs.  and are independent of the total delay function. Therefore, as mentioned previously, they can directly be used in Eq. for the determination of $s$. The coupling time delay functions {#sec:detcou} --------------------------------- All the basic ingredients needed for the establishment of the general expansion of the coupling time delay functions, have been introduced in Secs. \[sec:detref\] and \[sec:detgra\]. The general expansions of the refractive and gravitational spacetime perturbations are given in Eqs. and , respectively. Then, the expansions of the reception time delay functions can be found in Eqs. , , and . Therefore, by substituting for $\kappa^{\mu\nu}(\widetilde{z}_-(\lambda))$ and $k^{\mu\nu}(\widetilde{z}_-(\lambda))$ from Eqs.  and into , respectively, we obtain the theorem which follows. In the optical spacetime, within the quasi-Minkowskian path regime, $\Delta$ admits the series expansion introduced in theorem \[th:theo4\], so the function $\Delta_{\mathrm{gr}}$ is given by $$\Delta_{\mathrm{gr}}(\mathbf{x}_A,x_B,N_0,G)=\sum_{l=1}^{\infty}\Delta_{\mathrm{gr}}^{(l)}(\mathbf{x}_A,x_B) \label{eq:delRgrPMN0G}$$ where \[eq:intdelRgrPM\] $$\begin{aligned} \Delta_{\mathrm{gr}}^{(l\geqslant 1)}(\mathbf{x}_A,x_B)&=R_{AB}\int_0^1\Bigg\{\eta^{ij}\sum_{m=1}^{l}\left[\frac{\partial\Delta_{\mathrm{r}}^{(m)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{g}}^{(l-m+1)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\sum_{m=1}^{l}\big(\widehat{\kappa}^{0i}_{-(m)}-\widehat{\kappa}^{ij}_{-(m)}N^j_{AB}\big)_{(z_-(\lambda),x_B)}\left[\frac{\partial\Delta_{\mathrm{g}}^{(l-m+1)}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\sum_{m=s}^{l+s-1}\big(\widehat{k}^{0i}_{-(m)}-\widehat{k}^{ij}_{-(m)}N^j_{AB}\big)_{(z_-(\lambda),x_B)}\left[\frac{\partial\Delta_{\mathrm{r}}^{(l+s-m)}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}\Bigg\}{\mathrm{d}}\lambda \end{aligned}$$ for $l\geqslant 1$, and $$\begin{aligned} \Delta_{\mathrm{gr}}^{(l\geqslant 2)}(\mathbf{x}_A,x_B)&=\Delta_{\mathrm{gr}}^{(l\geqslant 1)}(\mathbf{x}_A,x_B)\nonumber\\ &+R_{AB}\int_0^1\Bigg\{\eta^{ij}\sum_{m=1}^{l-1}\left[\frac{\partial\Delta_{\mathrm{r}}^{(m)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{gr}}^{(l-m)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\sum_{m=1}^{l-1}\big(\widehat{\kappa}^{0i}_{-(m)}-\widehat{\kappa}^{ij}_{-(m)}N^j_{AB}\big)_{(z_-(\lambda),x_B)}\left[\frac{\partial\Delta_{\mathrm{gr}}^{(l-m)}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\sum_{m=1}^{l-1}\big(\widehat{\kappa}^{ij}_{-(m)}\big)_{(z_-(\lambda),x_B)}\sum_{n=1}^{l-m}\left[\frac{\partial\Delta_{\mathrm{r}}^{(n)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{g}}^{(l-m-n+1)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\frac{1}{2}\sum_{m=s}^{l+s-2}\big(\widehat{k}^{ij}_{-(m)}\big)_{(z_-(\lambda),x_B)}\sum_{n=1}^{l+s-m-1}\left[\frac{\partial\Delta_{\mathrm{r}}^{(n)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{r}}^{(l+s-m-n)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\Bigg\}{\mathrm{d}}\lambda \end{aligned}$$ for $l\geqslant 2$, and $$\begin{aligned} \Delta_{\mathrm{gr}}^{(l\geqslant 3)}(\mathbf{x}_A,x_B)&=\Delta_{\mathrm{gr}}^{(l\geqslant 2)}(\mathbf{x}_A,x_B)\nonumber\\ &+R_{AB}\int_0^1\Bigg\{\sum_{m=1}^{l-2}\big(\widehat{\kappa}^{ij}_{-(m)}\big)_{(z_-(\lambda),x_B)}\sum_{n=1}^{l-m-1}\left[\frac{\partial\Delta_{\mathrm{r}}^{(n)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{gr}}^{(l-m-n)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\Bigg\}{\mathrm{d}}\lambda \end{aligned}$$ for $l\geqslant 3$, and $$\begin{aligned} \Delta_{\mathrm{gr}}^{(l\geqslant s+1)}(\mathbf{x}_A,x_B)&=\Delta_{\mathrm{gr}}^{(l\geqslant s)}(\mathbf{x}_A,x_B)\nonumber\\ &+R_{AB}\int_0^1\Bigg\{\eta^{ij}\sum_{m=1}^{l-s}\left[\frac{\partial\Delta_{\mathrm{g}}^{(m)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{gr}}^{(l-s-m+1)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\sum_{m=s}^{l-1}\big(\widehat{k}^{0i}_{-(m)}-\widehat{k}^{ij}_{-(m)}N^j_{AB}\big)_{(z_-(\lambda),x_B)}\left[\frac{\partial\Delta_{\mathrm{gr}}^{(l-m)}}{\partial x^i}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\sum_{m=s}^{l-1}\big(\widehat{k}^{ij}_{-(m)}\big)_{(z_-(\lambda),x_B)}\sum_{n=1}^{l-m}\left[\frac{\partial\Delta_{\mathrm{r}}^{(n)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{g}}^{(l-m-n+1)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+\frac{1}{2}\sum_{m=1}^{l-s}\big(\widehat{\kappa}^{ij}_{-(m)}\big)_{(z_-(\lambda),x_B)}\sum_{n=1}^{l-s-m+1}\left[\frac{\partial\Delta_{\mathrm{g}}^{(n)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{g}}^{(l-s-m-n+2)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\Bigg\}{\mathrm{d}}\lambda \end{aligned}$$ for $l\geqslant s+1$, and $$\begin{aligned} \Delta_{\mathrm{gr}}^{(l\geqslant s+2)}(\mathbf{x}_A,x_B)&=\Delta_{\mathrm{gr}}^{(l\geqslant s+1)}(\mathbf{x}_A,x_B)\nonumber\\ &+\frac{R_{AB}}{2}\int_0^1\Bigg\{\eta^{ij}\sum_{m=1}^{l-s-1}\left[\frac{\partial\Delta_{\mathrm{gr}}^{(m)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{gr}}^{(l-s-m)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+2\sum_{m=s}^{l-2}\big(\widehat{k}^{ij}_{-(m)}\big)_{(z_-(\lambda),x_B)}\sum_{n=1}^{l-m-1}\left[\frac{\partial\Delta_{\mathrm{r}}^{(n)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{gr}}^{(l-m-n)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\nonumber\\ &+2\sum_{m=1}^{l-s-1}\big(\widehat{\kappa}^{ij}_{-(m)}\big)_{(z_-(\lambda),x_B)}\sum_{n=1}^{l-s-m}\left[\frac{\partial\Delta_{\mathrm{g}}^{(n)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{gr}}^{(l-s-m-n+1)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\Bigg\}{\mathrm{d}}\lambda \end{aligned}$$ for $l\geqslant s+2$, and $$\begin{aligned} \Delta_{\mathrm{gr}}^{(l\geqslant s+3)}(\mathbf{x}_A,x_B)&=\Delta_{\mathrm{gr}}^{(l\geqslant s+2)}(\mathbf{x}_A,x_B)\nonumber\\ &+\frac{R_{AB}}{2}\int_0^1\Bigg\{\sum_{m=1}^{l-s-2}\big(\widehat{\kappa}^{ij}_{-(m)}\big)_{(z_-(\lambda),x_B)}\sum_{n=1}^{l-s-m-1}\left[\frac{\partial\Delta_{\mathrm{gr}}^{(n)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{gr}}^{(l-s-m-n)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\Bigg\}{\mathrm{d}}\lambda \end{aligned}$$ for $l\geqslant s+3$, and $$\begin{aligned} \Delta_{\mathrm{gr}}^{(l\geqslant 2s+1)}(\mathbf{x}_A,x_B)&=\Delta_{\mathrm{gr}}^{(l\geqslant 2s)}(\mathbf{x}_A,x_B)\nonumber\\ &+R_{AB}\int_0^1\Bigg\{\sum_{m=s}^{l-s-1}\big(\widehat{k}^{ij}_{-(m)}\big)_{(z_-(\lambda),x_B)}\sum_{n=1}^{l-s-m}\left[\frac{\partial\Delta_{\mathrm{g}}^{(n)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{gr}}^{(l-s-m-n+1)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\Bigg\}{\mathrm{d}}\lambda \end{aligned}$$ for $l\geqslant 2s+1$, and finally $$\begin{aligned} \Delta_{\mathrm{gr}}^{(l\geqslant 2s+2)}(\mathbf{x}_A,x_B)&=\Delta_{\mathrm{gr}}^{(l\geqslant 2s+1)}(\mathbf{x}_A,x_B)\nonumber\\ &+\frac{R_{AB}}{2}\int_0^1\Bigg\{\sum_{m=s}^{l-s-2}\big(\widehat{k}^{ij}_{-(m)}\big)_{(z_-(\lambda),x_B)}\sum_{n=1}^{l-s-m-1}\left[\frac{\partial\Delta_{\mathrm{gr}}^{(n)}}{\partial x^i}\frac{\partial\Delta_{\mathrm{gr}}^{(l-s-m-n)}}{\partial x^j}\right]_{(\mathbf{z}_-(\lambda),x_B)}\Bigg\}{\mathrm{d}}\lambda \end{aligned}$$ for $l\geqslant 2s+2$. The quantities $\widehat{k}^{\mu\nu}_{-(l)}(z_-(\lambda),x_B)$ and $\widehat{\kappa}^{\mu\nu}_{-(l)}(z_-(\lambda),x_B)$ are defined in Eqs. and , respectively. \[th:theo10\] Applying the exact same reasoning, the analogous theorem for the coupling time delay function at emission can be stated. Indeed, substituting for $\kappa^{\mu\nu}(\widetilde{z}_+(\mu))$ from Eq. and for $k^{\mu\nu}(\widetilde{z}_+(\mu))$ from Eq. into , we obtain the theorem which follows. In the optical spacetime, within the quasi-Minkowskian path regime, $\Xi$ admits the series expansion introduced in theorem \[th:theo5\], so the function $\Xi_{\mathrm{gr}}$ is given by $$\Xi_{\mathrm{gr}}(x_A,\mathbf x_B,N_0,G)=\sum_{l=1}^{\infty}\Xi_{\mathrm{gr}}^{(l)}(x_A,\mathbf x_B) \label{eq:delEgrPM}$$ where \[eq:intdelEgrPM\] $$\begin{aligned} \Xi_{\mathrm{gr}}^{(l\geqslant 1)}(x_A,\mathbf x_B)&=R_{AB}\int_0^1\Bigg\{\eta^{ij}\sum_{m=1}^{l}\left[\frac{\partial\Xi_{\mathrm{r}}^{(m)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{g}}^{(l-m+1)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &-\sum_{m=1}^{l}\big(\widehat{\kappa}^{0i}_{+(m)}-\widehat{\kappa}^{ij}_{+(m)}N^j_{AB}\big)_{(x_A,z_+(\mu))}\left[\frac{\partial\Xi_{\mathrm{g}}^{(l-m+1)}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &-\sum_{m=s}^{l+s-1}\big(\widehat{k}^{0i}_{+(m)}-\widehat{k}^{ij}_{+(m)}N^j_{AB}\big)_{(x_A,z_+(\mu))}\left[\frac{\partial\Xi_{\mathrm{r}}^{(l+s-m)}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\mu))}\Bigg\}{\mathrm{d}}\mu \end{aligned}$$ for $l\geqslant 1$, and $$\begin{aligned} \Xi_{\mathrm{gr}}^{(l\geqslant 2)}(x_A,\mathbf x_B)&=\Xi_{\mathrm{gr}}^{(l\geqslant 1)}(x_A,\mathbf x_B)\nonumber\\ &+R_{AB}\int_0^1\Bigg\{\eta^{ij}\sum_{m=1}^{l-1}\left[\frac{\partial\Xi_{\mathrm{r}}^{(m)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{gr}}^{(l-m)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &-\sum_{m=1}^{l-1}\big(\widehat{\kappa}^{0i}_{+(m)}-\widehat{\kappa}^{ij}_{+(m)}N^j_{AB}\big)_{(x_A,z_+(\mu))}\left[\frac{\partial\Xi_{\mathrm{gr}}^{(l-m)}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &+\sum_{m=1}^{l-1}\big(\widehat{\kappa}^{ij}_{+(m)}\big)_{(x_A,z_+(\mu))}\sum_{n=1}^{l-m}\left[\frac{\partial\Xi_{\mathrm{r}}^{(n)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{g}}^{(l-m-n+1)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &+\frac{1}{2}\sum_{m=s}^{l+s-2}\big(\widehat{k}^{ij}_{+(m)}\big)_{(x_A,z_+(\mu))}\sum_{n=1}^{l+s-m-1}\left[\frac{\partial\Xi_{\mathrm{r}}^{(n)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{r}}^{(l+s-m-n)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\Bigg\}{\mathrm{d}}\mu \end{aligned}$$ for $l\geqslant 2$, and $$\begin{aligned} \Xi_{\mathrm{gr}}^{(l\geqslant 3)}(x_A,\mathbf x_B)&=\Xi_{\mathrm{gr}}^{(l\geqslant 2)}(x_A,\mathbf x_B)\nonumber\\ &+R_{AB}\int_0^1\Bigg\{\sum_{m=1}^{l-2}\big(\widehat{\kappa}^{ij}_{+(m)}\big)_{(x_A,z_+(\mu))}\sum_{n=1}^{l-m-1}\left[\frac{\partial\Xi_{\mathrm{r}}^{(n)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{gr}}^{(l-m-n)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\Bigg\}{\mathrm{d}}\mu \end{aligned}$$ for $l\geqslant 3$, and $$\begin{aligned} \Xi_{\mathrm{gr}}^{(l\geqslant s+1)}(x_A,\mathbf x_B)&=\Xi_{\mathrm{gr}}^{(l\geqslant s)}(x_A,\mathbf x_B)\nonumber\\ &+R_{AB}\int_0^1\Bigg\{\eta^{ij}\sum_{m=1}^{l-s}\left[\frac{\partial\Xi_{\mathrm{g}}^{(m)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{gr}}^{(l-s-m+1)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &-\sum_{m=s}^{l-1}\big(\widehat{k}^{0i}_{+(m)}-\widehat{k}^{ij}_{+(m)}N^j_{AB}\big)_{(x_A,z_+(\mu))}\left[\frac{\partial\Xi_{\mathrm{gr}}^{(l-m)}}{\partial x^i}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &+\sum_{m=s}^{l-1}\big(\widehat{k}^{ij}_{+(m)}\big)_{(x_A,z_+(\mu))}\sum_{n=1}^{l-m}\left[\frac{\partial\Xi_{\mathrm{r}}^{(n)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{g}}^{(l-m-n+1)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &+\frac{1}{2}\sum_{m=1}^{l-s}\big(\widehat{\kappa}^{ij}_{+(m)}\big)_{(x_A,z_+(\mu))}\sum_{n=1}^{l-s-m+1}\left[\frac{\partial\Xi_{\mathrm{g}}^{(n)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{g}}^{(l-s-m-n+2)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\Bigg\}{\mathrm{d}}\mu \end{aligned}$$ for $l\geqslant s+1$, and $$\begin{aligned} \Xi_{\mathrm{gr}}^{(l\geqslant s+2)}(x_A,\mathbf x_B)&=\Xi_{\mathrm{gr}}^{(l\geqslant s+1)}(x_A,\mathbf x_B)\nonumber\\ &+\frac{R_{AB}}{2}\int_0^1\Bigg\{\eta^{ij}\sum_{m=1}^{l-s-1}\left[\frac{\partial\Xi_{\mathrm{gr}}^{(m)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{gr}}^{(l-s-m)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &+2\sum_{m=s}^{l-2}\big(\widehat{k}^{ij}_{+(m)}\big)_{(x_A,z_+(\mu))}\sum_{n=1}^{l-m-1}\left[\frac{\partial\Xi_{\mathrm{r}}^{(n)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{gr}}^{(l-m-n)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\nonumber\\ &+2\sum_{m=1}^{l-s-1}\big(\widehat{\kappa}^{ij}_{+(m)}\big)_{(x_A,z_+(\mu))}\sum_{n=1}^{l-s-m}\left[\frac{\partial\Xi_{\mathrm{g}}^{(n)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{gr}}^{(l-s-m-n+1)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\Bigg\}{\mathrm{d}}\mu \end{aligned}$$ for $l\geqslant s+2$, and $$\begin{aligned} \Xi_{\mathrm{gr}}^{(l\geqslant s+3)}(x_A,\mathbf x_B)&=\Xi_{\mathrm{gr}}^{(l\geqslant s+2)}(x_A,\mathbf x_B)\nonumber\\ &+\frac{R_{AB}}{2}\int_0^1\Bigg\{\sum_{m=1}^{l-s-2}\big(\widehat{\kappa}^{ij}_{+(m)}\big)_{(x_A,z_+(\mu))}\sum_{n=1}^{l-s-m-1}\left[\frac{\partial\Xi_{\mathrm{gr}}^{(n)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{gr}}^{(l-s-m-n)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\Bigg\}{\mathrm{d}}\mu\text{,} \end{aligned}$$ for $l\geqslant s+3$, and $$\begin{aligned} \Xi_{\mathrm{gr}}^{(l\geqslant 2s+1)}(x_A,\mathbf x_B)&=\Xi_{\mathrm{gr}}^{(l\geqslant 2s)}(x_A,\mathbf x_B)\nonumber\\ &+R_{AB}\int_0^1\Bigg\{\sum_{m=s}^{l-s-1}\big(\widehat{k}^{ij}_{+(m)}\big)_{(x_A,z_+(\mu))}\sum_{n=1}^{l-s-m}\left[\frac{\partial\Xi_{\mathrm{g}}^{(n)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{gr}}^{(l-s-m-n+1)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\Bigg\}{\mathrm{d}}\mu \end{aligned}$$ for $l\geqslant 2s+1$, and finally $$\begin{aligned} \Xi_{\mathrm{gr}}^{(l\geqslant 2s+2)}(x_A,\mathbf x_B)&=\Xi_{\mathrm{gr}}^{(l\geqslant 2s+1)}(x_A,\mathbf x_B)\nonumber\\ &+\frac{R_{AB}}{2}\int_0^1\Bigg\{\sum_{m=s}^{l-s-2}\big(\widehat{k}^{ij}_{+(m)}\big)_{(x_A,z_+(\mu))}\sum_{n=1}^{l-s-m-1}\left[\frac{\partial\Xi_{\mathrm{gr}}^{(n)}}{\partial x^i}\frac{\partial\Xi_{\mathrm{gr}}^{(l-s-m-n)}}{\partial x^j}\right]_{(x_A,\mathbf{z}_+(\mu))}\Bigg\}{\mathrm{d}}\mu \end{aligned}$$ for $l\geqslant 2s+2$. The quantities $\widehat{k}^{\mu\nu}_{+(l)}(x_A,z_+(\mu))$ and $\widehat{\kappa}^{\mu\nu}_{+(l)}(x_A,z_+(\mu))$ are defined in Eqs. and , respectively. \[th:theo11\] Finally, from the gravitational, the refractive, and the coupling components, we can now determine the time delay expression up to $l$th-order by applying theorem \[th:theo4\]. Then, the expressions for the range and the time transfer functions are determined from Eqs. and , respectively. Let us emphasis that line integrals occurring in Eqs. , , and are now zeroth-order null geodesics with parametric equations $x=z_-(\lambda)$. Similarly, Eqs. , , and are integrated along the zeroth-order null geodesic path with parametric equations $x=z_+(\mu)$. This specificity of the time transfer functions formalism considerably simplifies the integrations and constitutes one of the most important advantage with respect to an explicit resolution of the null geodesic equation. The usefulness of the decomposition which has been performed in Eq.  is really apparent in stationary optical spacetimes. Indeed, when the coordinates $(x^\mu)$ are chosen so that the optical spacetime metric does not depend on $x^0$, it is seen that the series expansion in Eqs.  and reduce to \[eq:kapkRPMsta\] $$\begin{aligned} \kappa^{\mu\nu}(\widetilde{z}_-(\lambda),N_0)&=\sum_{l=1}^{\infty}\widehat{\kappa}^{\mu\nu}_{-(l)}(z_-(\lambda),x_B)\text{,}\label{eq:kapRPMsta}\\ k^{\mu\nu}(\widetilde{z}_-(\lambda),G)&=\sum_{l=1}^{\infty}\widehat{k}^{\mu\nu}_{-(l)}(z_-(\lambda),x_B)\label{eq:kRPMsta}\end{aligned}$$ where \[eq:kapkRPMdecsta\] $$\begin{aligned} \widehat{\kappa}^{\mu\nu}_{(l)}(z_-(\lambda),x_B)&=\kappa^{\mu\nu}_{(l)}(z_-(\lambda))\text{,}\label{eq:kapRPMdecsta}\\ \widehat{k}^{\mu\nu}_{(ps+q)}(z_-(\lambda),x_B)&=\delta(q)\,k^{\mu\nu}_{(p)}(z_-(\lambda))\text{,}\label{eq:kRPMdecsta}\end{aligned}$$ respectively. We recall that $p$ and $q$ are determined from $l$ using . Hence, Eq. does no longer involve the formal expansion parameter $G$, while is now independent of $N_0$. In other words, the different theorems can be solved separately, each one being completely independent from the others. As a matter of fact, theorems involving gravitational perturbation become independent of $\widehat{k}^{\mu\nu}_{(l)}$ for any $l$ which is not a multiple of $s$. Application to stationary optical spacetime in geocentric celestial reference system {#sec:app} ==================================================================================== Let us now illustrate the method by determining the time transfer function up to the post-linear approximation. We investigate, the light-dragging effect experienced by a signal during its propagation inside a flowing media of non-null refractivity. In the GCRS, the effect shows up at the post-linear approximation. In the case where the motion of the Earth’s atmosphere is mainly a steady rotation (e.g. in GCRS), we show that the light-dragging effect reduce to a geometrical factor scaling the static atmospheric contribution. During the computation, we never make use of an index of refraction profile in order to keep the equations under a form which is as general as possible. Notations and definitions ------------------------- We consider that spacetime is covered with some global coordinates $(x^\mu)$. We choose the coordinate system such that the optical metric components are independent of $x^0$. In addition, the coordinate system shall be chosen in such a way that it is convenient to model the outcomes of an experiment taking place in the Earth’s close vicinity. Therefore, we consider that $(x^\mu)$ are the GCRS coordinates. We recall that the GCRS is centered in the Earth’s center of mass and is non-rotating with respect to distant stars. We suppose that the domain $\mathcal{D}$ defines the spacetime boundaries of the Earth’s neutral atmosphere. In that sense, $\mathcal{D}$ draws a time-like tube in spacetime. The Earth’s atmosphere is considered spherically symmetric and we suppose that it is filled with a fluid dielectric medium whose refractive properties are independent of the component $x^0$, that is to say $$n(\mathbf{x})= 1+N(\mathbf{x})\text{.}$$ We consider that the atmosphere is still in the reference system rotating with the Earth, thus we assume that the unit 4-velocity vector is given in GCRS by $$w^{\mu}=w^0(1,\xi^i) \label{eq:4velmed}$$ where $\xi^i$ is the coordinate 3-velocity vector of the fluid dielectric medium. Hereafter, we assume that the 3-velocity vector is given by the following expression $$\xi^i(\mathbf x)=\frac{\omega_\oplus}{c}\,e^{ijk}S_\oplus^jx^k \label{eq:3velmed}$$ where $\omega_\oplus$ is the magnitude of the Earth’s angular velocity of rotation and $\mathbf S_\oplus$ is the direction of the spin axis. Moreover, we consider the case of a one-way transfer, with the transmitter being right outside $\mathcal{D}$ and the receiver being comoving with the fluid dielectrics medium, that is to say at rest in the reference system rotating with the Earth. To fix ideas for future discussion, let us assume that the emitter is transmitting from the International Space Station (ISS) at an altitude of $h\simeq 400\,\mathrm{km}$. Furthermore, let us consider that the emitter is moving along the time-like worldline $\mathcal{C}_A$ with the unit 4-velocity vector $\bm u_{A}$ defined by $$u^{\mu}_A=u^0_A(1,\beta^i_A) \label{eq:4velA}$$ where $\beta^i_A$ is the coordinate 3-velocity vector expressed in GCRS coordinates. Similarly, we assume that the receiver moves along the time-like worldline $\mathcal{C}_B$ with the unit 4-velocity vector $\bm u_{B}$ defined by $$u^{\mu}_B=u^0_B(1,\beta^i_B) \label{eq:4velB}$$ where $\beta^i_B$ is the coordinate 3-velocity vector expressed in GCRS coordinates. For a receiver comoving with the medium, we have $$\beta^i_B=\xi^i(\mathbf x_B)\text{.}$$ Expansion of the delay functions -------------------------------- The components of the physical spacetime metric expressed in GCRS coordinates are given in [@2003AJ....126.2687S]. Let us emphasize that i) our convention for the signature of spacetime is $(+,-,-,-)$, and ii) our $g_{\mu\nu}$ corresponds to $G_{\alpha\beta}$ in [@2003AJ....126.2687S]. By keeping terms in $1/c^2$ only, the first-order gravitational perturbation reads as follows \[eq:gGCRS\] $$h_{00}^{(1)}=-\frac{2U}{c^2}\text{,} \qquad h_{0i}^{(1)}=0\text{,} \qquad h_{ij}^{(1)}=-\frac{2U}{c^2}\delta_{ij} \label{eq:gGCRScov}$$ where the contravariant components are determined from Eq. $$k^{00}_{(1)}=\frac{2U}{c^2}\text{,} \qquad k^{0i}_{(1)}=0\text{,} \qquad k^{ij}_{(1)}=\frac{2U}{c^2}\delta_{ij}\text{.} \label{eq:gGCRScon}$$ In these expressions, we restrict $U$ to the monopole term of the Newtonian gravitational potential of the Earth, that is $$U(\mathbf x)=\frac{Gm_\oplus}{|\mathbf x|} \label{eq:U}$$ where $m_\oplus$ is the mass of the Earth. In that respect, at the level of the surface of the Earth, we find $$|k^{\mu\nu}_{(1)}(R_\oplus)|_{\mathrm{max}}\propto \frac{U(R_\oplus)}{c^2}\sim10^{-10} \label{eq:ordk}$$ where $R_\oplus$ denotes Earth’s equatorial radii. According to [@doi1010292010JD015214], at the sea level an average parcel of air possesses a refractivity $N(R_\oplus)\simeq 3\times10^{-4}$, so we consider $N_0\sim10^{-4}$. Additionally, at the Earth’s surface, the 3-velocity of the refractive medium expressed in GCRS coordinates is $|\xi^i(R_\oplus)|_{\mathrm{max}}\propto\omega_\oplus R_\oplus/c\sim 10^{-6}$. Consequently, we can expand the refractive perturbation in term of the refractivity at the Earth’s surface and in the approximation of small velocities. Therefore, it can be seen that the first-order term of the refractive perturbation is given by (see Eqs. ) $$|\kappa^{\mu\nu}_{(1)}(R_\oplus)|_{\mathrm{max}}\propto N_0\sim 10^{-4} \label{eq:ordkap}$$ once evaluated at the Earth’s surface. At the same time, typical measurement profile for the neutral atmosphere using GPS/MET (Global Positioning System Meteorology) occultations data starts at $\ell\simeq100\,\mathrm{km}$ [@1999AnGeo..17..122S], so that $\ell/h\simeq 0.4$. For observations at lower elevation than the zenith direction, we can roughly take $\ell/R_{AB}\sim0.1$. Then, if we consider that the light path is sufficiently small so that the metric components vary slowly during the integration, we can get a rough estimation of $s$ by making use of Eqs. . We quickly infer that $s$ must satisfy the zeroth-order following relation $$\frac{\ell}{R_{AB}}|\kappa^{\mu\nu}|_{\mathrm{max}}\sim |k^{\mu\nu}|_{\mathrm{max}}^{1/s}\text{.} \label{eq:relmag}$$ Inserting numerical values, we deduce $s=2$. This results can be double-checked by inserting the first-order expressions of the gravitational and refractive delays (see Eqs.  and ) into Eq. . In this application, we exclude third-order terms and beyond, that is to say, all terms of the order of $\varepsilon^3$ with $\varepsilon\sim\ell/R_{AB}|\kappa^{\mu\nu}|_{\mathrm{max}}\sim 10^{-5}$. This means that a post-linear expression of the range transfer function neglects terms of the order $\varepsilon^3R_{AB}$. Therefore, the coupling terms which are of third-order are neglected too. A look at Eq.  and allows one to infer that the time component of the 4-velocity vector of the fluid dielectric medium must be known up to $10^{-5}$ in order to account for all second-order terms. Considering that the four-velocity of the medium must be a unit-vector for the spacetime metric $g_{\mu\nu}$, we have the relation $$w^0=\left(g_{00}+2g_{0i}\xi^i+g_{ij}\xi^i\xi^j\right)^{-1/2}\text{.} \label{eq:u0}$$ Therefore, to sufficient accuracy, we can safely consider for the rest of the application that $w^0=1$. Hence, we end up with the following contravariant components for the refractive perturbation \[eq:kapGCRS\] $$\kappa^{00}_{(1)}=2N\text{,} \qquad \kappa^{0i}_{(1)}=0\text{,} \qquad \kappa^{ij}_{(1)}=0\text{,} \label{eq:kapGCRS1}$$ with the second-order $$\kappa^{00}_{(2)}=N^2\text{,} \qquad \kappa_{(2)}^{0i}=2N\xi^i\text{,} \qquad \kappa_{(2)}^{ij}=0\text{.} \label{eq:kapGCRS2}$$ Let us note that the cross component is non-null at the post-linear approximation. It represents the light-dragging effect due to the motion of the fluid dielectric medium in GCRS coordinates. Additionally, let us mention that the optical spacetime is stationary as seen from Eqs.  and . In that respect, the emission or the reception time transfer functions become identical. As a consequence, the distinction between emission and reception functions is not relevant anymore meaning that the time component at emission or reception is no longer an independent variable [@2008CQGra..25n5020T]. Hence, $\widehat \kappa_{(l)}^{\mu\nu}$ and $\widehat k_{(l)}^{\mu\nu}$ are now given by Eqs. and which are independent of the total time delay. Therefore, the refractive and the gravitational delays may be solved separately. A straightforward application of theorem \[th:theo4\] assuming $s=2$, allows us to infer the following form of the expansion of the total time delay function \[eq:ord2\] $$\begin{aligned} \Delta^{(1)}(\mathbf{x}_A,\mathbf x_B)&=\Delta^{(1)}_{\mathrm r}(\mathbf{x}_A,\mathbf x_B)\text{,}\\ \Delta^{(2)}(\mathbf{x}_A,\mathbf x_B)&=\Delta^{(2)}_{\mathrm r}(\mathbf{x}_A,\mathbf x_B)+\Delta^{(1)}_{\mathrm g}(\mathbf{x}_A,\mathbf x_B)\text{.} \end{aligned}$$ Thus, we deduce that the contributions in Eq. are given by $$\begin{aligned} \Delta_{\mathrm g}(\mathbf{x}_A,\mathbf x_B)&=\Delta^{(1)}_{\mathrm g}(\mathbf{x}_A,\mathbf x_B)\text{,}\\ \Delta_{\mathrm r}(\mathbf{x}_A,\mathbf x_B)&=\Delta^{(1)}_{\mathrm r}(\mathbf{x}_A,\mathbf x_B)+\Delta^{(2)}_{\mathrm r}(\mathbf{x}_A,\mathbf x_B)\text{.}\end{aligned}$$ Then, theorems \[th:theo6\] and \[th:theo8\] together with Eqs. and allow us to determine the refractive and the gravitational contributions up to the appropriate order. Time transfer function and Doppler ---------------------------------- Using the fact that spacetime is stationary, we first focus on the gravitational time delay. By making use of theorem \[th:theo8\], we soon arrive to the well-known formula $$\Delta_{\mathrm g}(\mathbf{x}_A,\mathbf x_B)=\frac{2R_{AB}}{c^2}\int_0^1U(\mathbf z_-(\lambda))\,{\mathrm{d}}\lambda\text{,}$$ which leads after integration to the Shapiro delay [@1964PhRvL..13..789S] $$\Delta_{\mathrm g}(\mathbf{x}_A,\mathbf x_B)=\frac{2Gm_\oplus}{c^2}\mathrm{ln}\left(\frac{r_A+r_B+R_{AB}}{r_A+r_B-R_{AB}}\right)\text{.}\label{eq:delShap}$$ We introduced the notations $r_{A/B}=|\mathbf{x}_{A/B}|$. The first-order refractive contribution is derived from theorem \[th:theo6\] and is given by $$\Delta^{(1)}_{\mathrm{r}}(\mathbf{x}_A,\mathbf x_B)=R_{AB}\int_0^1N(\mathbf z_-(\lambda))\,{\mathrm{d}}\lambda\text{.} \label{eq:delr1GCRS}$$ We find almost similar expression for the atmospheric delay in [@inproceedingsMendes; @2002GeoRL..29.1414M; @2004GeoRL..3114602M; @2010ITN....36....1P]. When applied to the Earth’s neutral atmosphere, it is common to define the refractivity within a factor $10^6$ and to split it into a hydrostatic and a non-hydrostatic components [@2010ITN....36....1P]. The main differences stand, firstly, in the integration path which is performed along the Euclidean path between the emitter and the receiver in Eq. . This holds true even for non-zenithal observations. Instead, in the literature the atmospheric delay is usually computed at zenith, and then, mapping functions are used to convert the zenithal delay into a delay in the line-of-sight direction as discussed in [@2002GeoRL..29.1414M]. In our case, the first-order refractive delay is the well-known excess path delay due to the change of the phase velocity experienced by the signal during the crossing of the dielectric medium. The geometric delay due to the refractive bending of the ray arises at the post-linear order as we shall see in the next. The other difference stands in the upper limit of integration path in Eq. . However, considering that the refractive region is bounded to a certain domain $\mathcal{D}$ of spacetime, the integration out of $\mathcal{D}$ results in integrating null refractivity which thus does not contribute to the final results. In that respect, the difference in the upper integration limit is only superficial. According to theorem \[th:theo6\], the second-order is given by $$\begin{aligned} \Delta^{(2)}_{\mathrm r}(\mathbf{x}_A,\mathbf x_B)&=\frac{R_{AB}}{2}\int_0^1\Big\{\big(N^2-4N\xi^iN_{AB}^i\big)_{\mathbf z_-(\lambda)}\nonumber\\ &-\big[\partial_i\Delta^{(1)}_{\mathrm r}\partial_i\Delta^{(1)}_{\mathrm r}\big]_{(\mathbf z_-(\lambda),\mathbf x_B)}\Big\}{\mathrm{d}}\lambda\label{eq:delr2GCRS}\end{aligned}$$ where $\partial_i\Delta^{(1)}_{\mathrm r}$ is computed by differentiating Eq. $$\begin{aligned} \big[\partial_i\Delta^{(1)}_{\mathrm r}\big]_{(\mathbf x,\mathbf x_B)}&=-\frac{(\mathbf x_B-\mathbf x)^i}{|\mathbf x_B-\mathbf x|}\int_0^1N(\mathbf y_-(\mu,\mathbf x))\,{\mathrm{d}}\mu\nonumber\\ &+|\mathbf x_B-\mathbf x|\int_0^1\mu\big[\partial_i N\big]_{\mathbf y_-(\mu,\mathbf x)}{\mathrm{d}}\mu\text{.}\label{eq:DPdel}\end{aligned}$$ We have introduced $$\mathbf y_-(\mu,\mathbf x)=(1-\mu)\,\mathbf x_B+\mu\,\mathbf x\text{,}$$ which, in the case where $\mathbf x=\mathbf z_-(\lambda)$, reduces to $$\mathbf y_-(\mu,\mathbf z_-(\lambda))=\mathbf z_-(\mu\lambda)\text{.} \label{eq:ym}$$ We can rearrange Eq. by first noticing that the light-dragging contribution can be further simplified. Indeed, after making use of Eq. , it may be seen that $$\big(\xi^iN_{AB}^i\big)_{\mathbf{z}_-(\lambda)}\equiv\xi^i(\mathbf x_B)N_{AB}^i$$ which is obviously independent of $\lambda$. Then, substituting for $\partial_i\Delta^{(1)}_{\mathrm r}$ from Eq. into while accounting for and , one can apply the following change of variables $\mu'=\mu\lambda$. Finally, by integrating by parts the double integrals, we infer the post-linear refractive order $$\begin{aligned} \Delta^{(2)}_{\mathrm{r}}(\mathbf{x}_A,\mathbf x_B)&=\Delta^{(2)}_{\mathrm{r,exc}}(\mathbf{x}_A,\mathbf x_B)+\Delta^{(2)}_{\mathrm{r,geo}}(\mathbf{x}_A,\mathbf x_B)\nonumber\\ &+\Delta^{(2)}_{\mathrm{r,drag}}(\mathbf{x}_A,\mathbf x_B)\label{eq:delr2}\end{aligned}$$ where we have introduced \[eq:delr2GCRSbis\] $$\begin{aligned} \Delta^{(2)}_{\mathrm{r,exc}}&(\mathbf{x}_A,\mathbf x_B)=\frac{R_{AB}}{2}\int_0^1N^2(\mathbf z_-(\lambda))(1+\mathrm{ln}\lambda)\,{\mathrm{d}}\lambda\text{,}\label{eq:delr2GCRSexc}\\ \Delta^{(2)}_{\mathrm{r,geo}}&(\mathbf{x}_A,\mathbf x_B)=\frac{R_{AB}}{2}\int_0^1\Big\{\lambda R_{AB}^2\big[\partial_i N\partial_i N\big]_{\mathbf z_-(\lambda)}\nonumber\\ &-2R_{AB}N_{AB}^i \big[N\partial_i N\big]_{\mathbf z_-(\lambda)}\Big\}\lambda\mathrm{ln}\lambda\,{\mathrm{d}}\lambda\text{,}\label{eq:delr2GCRSgeo}\end{aligned}$$ and $$\Delta^{(2)}_{\mathrm{r,drag}}(\mathbf{x}_A,\mathbf x_B)=-2\xi^i(\mathbf x_B)N_{AB}^i\Delta^{(1)}_{\mathrm r}(\mathbf{x}_A,\mathbf x_B)\text{.} \label{eq:delr2GCRSdrag}$$ We see that the post-linear approximation of the refractive time delay function can be split into three components. The first one, namely Eq. , is the second-order correction to the excess path delay . The second component, that is Eq. , is the geometric delay which accounts for the bending of the ray. These two components together with Eq. , constitute the static part of the refractive time delay $$\begin{aligned} \Delta_{\mathrm{r,stat}}(\mathbf x_A,\mathbf x_B)&=\Delta_{\mathrm r}^{(1)}(\mathbf x_A,\mathbf x_B)+\Delta_{\mathrm{r,exc}}^{(2)}(\mathbf x_A,\mathbf x_B)\nonumber\\ &+\Delta_{\mathrm{r,geo}}^{(2)}(\mathbf x_A,\mathbf x_B)\text{,}\label{eq:delrstat}\end{aligned}$$ namely the refractive part of delay that would be measured or modeled in a frame comoving with the media. Instead, the last term in Eq. , namely Eq. , is the delay due to the dragging of light caused by the motion of the dielectric medium. Interestingly, when one considers that the main component of the motion is due to the steady rotation of the atmosphere in GCRS coordinates, we may emphasis that the light-dragging component shows up about the same order than the geometric delay. Even more interesting is the fact that the light-dragging contribution can be expressed as a geometric factor scaling the static refractive part of the previous order (see Eq. ). This fact is not a specificity of the post-linear approximation but must hold true for higher order terms too. Indeed, it results from the really specific form of the refractive components $\kappa^{\mu\nu}$. As a matter of fact, the components $\kappa^{0i}$ can always be written such as $$\kappa^{0i}=\kappa^{00}\xi^i\text{.}$$ Therefore, because the scalar product $\xi^iN_{AB}^i$ is independent of the path of integration for a steady rotating atmosphere, the integration of $\kappa^{0i}N_{AB}^i$ reduces to $$\xi^i(\mathbf x_B)N_{AB}^i\int_0^1(\kappa^{00})_{\mathbf z_-(\lambda)}{\mathrm{d}}\lambda$$ where the integrated term corresponds to the static part of the refraction. Solving the line integrals in Eqs. , , and for a realistic index of refraction is not an easy task. Moreover it is beyond the scope of this paper which aims at introducing a recursive method allowing one to determine the integral form of the time transfer functions up to any order in optical spacetime. For this reason, we address the effective resolution of the line integrals to future work. Hereafter, we derive the range transfer function at the post-linear approximation from Eqs.  by making use of Eqs. , , and $$\begin{aligned} \mathcal{R}(\mathbf x_A,\mathbf x_B)&=R_{AB}+\Delta_{\mathrm{g}}(\mathbf{x}_A,\mathbf x_B)\nonumber\\ &+D(\mathbf x_A,\mathbf x_B)\,\Delta^{(1)}_{\mathrm{r}}(\mathbf{x}_A,\mathbf x_B)\nonumber\\ &+\Delta^{(2)}_{\mathrm{r,exc}}(\mathbf{x}_A,\mathbf x_B)+\Delta^{(2)}_{\mathrm{r,geo}}(\mathbf{x}_A,\mathbf x_B)\label{eq:RTFpl}\end{aligned}$$ where we have introduced the light-dragging coefficient $D(\mathbf x_A,\mathbf x_B)$ being defined by $$D(\mathbf x_A,\mathbf x_B)=1-2\xi^i(\mathbf x_B)N_{AB}^i\text{.} \label{eq:Kdrag}$$ According to the previous discussion, we can rewrite Eq. , within the same accuracy, such as $$\begin{aligned} \mathcal{R}(\mathbf x_A,\mathbf x_B)&=R_{AB}+\Delta_{\mathrm{g}}(\mathbf{x}_A,\mathbf x_B)\nonumber\\ &+D(\mathbf x_A,\mathbf x_B)\,\Delta_{\mathrm{r,stat}}(\mathbf{x}_A,\mathbf x_B)\text{.}\label{eq:RTFplD}\end{aligned}$$ The time transfer function can be directly obtained by making use of Eq. $$\begin{aligned} \mathcal{T}(\mathbf x_A,\mathbf x_B)&=\frac{1}{c}\Big[R_{AB}+\Delta_{\mathrm{g}}(\mathbf{x}_A,\mathbf x_B)\nonumber\\ &+D(\mathbf x_A,\mathbf x_B)\,\Delta_{\mathrm{r,stat}}(\mathbf{x}_A,\mathbf x_B)\Big]\label{eq:TTFpl}\end{aligned}$$ where we recall that $\Delta_{\mathrm{r,stat}}$ is given in Eq. . Let us emphasis how remarkably simple result is. As a matter of fact, the dragging coefficient is just a geometrical factor scaling the static part of the refractive delay. Indeed, in a covariant theory such like general relativity, let us recall that the light-dragging contribution is naturally handled through the cross components of the Gordon’s metric. Moreover, let us emphasis that the demonstration which ends up with Eq.  is perfectly independent of the refractive profile inside the steady rotating optical medium. In comparison, a derivation of the light-dragging effect using perturbation equations applied to geometrical optics [@2019AeA...624A..41B] requires heavier calculations (the integration must be performed along hyperbolic path) highlighting the advantage of using the covariant formalism developed so far. From the range or the time transfer functions, we can derive the expression of the frequency transfer within the post-linear approximation as well. After inserting Eq.  within , we deduce \[eq:qAqB\] $$\begin{aligned} q_A&=1-\beta^i_AN_{AB}^i+\beta^i_A\frac{\partial\Delta_{\mathrm{g}}}{\partial x_A^i}+\widehat\beta^i_A\frac{\partial\Delta_{\mathrm{r,stat}}}{\partial x_A^i}\nonumber\\ &+\beta^i_A\frac{\partial D}{\partial x_A^i}\Delta_{\mathrm{r,stat}}\text{,}\end{aligned}$$ and $$\begin{aligned} q_B&=1-\beta^i_BN_{AB}^i-\beta^i_B\frac{\partial\Delta_{\mathrm{g}}}{\partial x_B^i}-\widehat\beta^i_B\frac{\partial\Delta_{\mathrm{r,stat}}}{\partial x_B^i}\nonumber\\ &-\beta^i_B\frac{\partial D}{\partial x_B^i}\Delta_{\mathrm{r,stat}}\end{aligned}$$ where we have introduced two artificial dragging coordinate velocities defined by $$\widehat\beta^i_{A/B}=D(\mathbf x_A,\mathbf x_B)\,\beta^i_{A/B}\text{.} \label{eq:velD}$$ Most of the time, while modeling range and Doppler observables in GCRS coordinates, the dragging coefficient is arbitrarily fixed to $D=1$. In order to investigate the consequences, we close the section by discussing orders of magnitude and variabilities due to the light-dragging contribution in the expressions of the time and the frequency transfers. Light-dragging magnitude and variability ---------------------------------------- In GCRS coordinates, the velocity of the fluid medium at $\mathbf x_B$ is given by Eq. , that is $$\xi^i(\mathbf x_B)=\frac{\omega_\oplus r_B}{c}\,e^{ijk}S_\oplus^jn_B^k \label{eq:3velmedN}$$ where $\mathbf n_B=\mathbf x_B/r_B$. For a ground-based receiver, we have $r_B=R_\oplus$ and the light-dragging coefficient becomes $$D(\mathbf x_A,\mathbf x_B)=1-\frac{2\omega_\oplus R_\oplus}{c}\,(\mathbf S_\oplus\times\mathbf n_B)\cdot\mathbf N_{AB}\text{.} \label{eq:KdragGCRS}$$ Thus, the maximum value of $D-1$ is about $$\frac{2\omega_\oplus R_\oplus}{c}\simeq 3.099\times 10^{-6}\text{.} \label{eq:Knum}$$ A typical value of the static refractive delay in the zenith direction is approximately $2.5\,\mathrm m$ and can reach $15\,\mathrm m$ for elevation angle of $10\degres$ [@1997JGR...10220489C; @doi1010292006JB004834]. Therefore, the light-dragging contribution to the time transfer is expected to remain lower that $0.05\,\mathrm{mm}$ in GCRS coordinates. However, for experiments whose data are mainly analyzed in the Barycentric Celestial Reference System (BCRS), the velocity of the media possesses an orbital components which is of the order of $30\,\mathrm{km}\cdot\mathrm{s}^{-1}$. Thus, the maximum value of $D-1$ becomes of the order of $2\times10^{-4}$, and the dragging contribution can reach $3\,\mathrm{mm}$ in BCRS coordinates. Experiments such like Satellite or Lunar Laser Ranging (SLR or LLR) are currently operating at the millimeter and the centimeter level of precision on range measurements . Therefore, the light-dragging effect is just below the threshold of visibility on both experiments. However, as it may be inferred from Eq. , the effect is mainly suppressed in the case of a round-trip light path. In other words, it might play a role only for one-way and three-way observations. From Eq. , considering a slowly varying refractivity, we can infer that $$\beta_A^i\frac{\partial\Delta_{\mathrm{r,stat}}}{\partial x_A^i}\sim\frac{\ell}{R_{AB}}N_0(\beta_A^iN_{AB}^i)\sim10^{-5}\,(\beta_A^iN_{AB}^i)\text{.}$$ Therefore, for a one-way frequency transfer experiment, the static atmospheric contribution relative to the classical effect $(\beta_A^iN_{AB}^i)$, represents one part in $10^{5}$. Then, the contribution due to the dragging velocity in Eqs. is approximately given by $$\widehat{\beta}_A^i\frac{\partial\Delta_{\mathrm{r,stat}}}{\partial x_A^i}\sim10^{-5}\,(\widehat{\beta}_A^iN_{AB}^i)\text{.}$$ Making use of Eqs. and , one infers that, in GCRS coordinates, the light-dragging contribution relative to the static atmospheric effect, represents one part in $10^6$ and one part in $10^{11}$ relative to the classical effect. On the opposite, if we take a look at orders of magnitude in BCRS coordinates, the light-dragging contribution relative to the the static atmospheric effect, reaches one part in $10^4$ and one part in $10^9$ relative to the classical effect. Therefore, for typical spacecraft’s velocities of $10^{-5}$ and $10^{-4}$ in GCRS and BCRS, respectively, one infers that the effect of the light-dragging contribution produces a fractional frequency change of the order of one part in $10^{16}$ in GCRS coordinates and one part in $10^{13}$ in BCRS coordinates. For one-way radio links, these fractional frequency changes translate into radio signal frequencies at the level of $1\,\mu\mathrm{Hz}$ for $\mathrm{X}/\mathrm{Ka}$-bands and $0.1\,\mu\mathrm{Hz}$ for $\mathrm{S}$-band in GCRS coordinates. In BCRS coordinates, the frequencies of the radio signal due to the dragging of light should arise at $1\mathrm{mHz}$ for $\mathrm{X}/\mathrm{Ka}$-bands and $0.1\,\mathrm{mHz}$ for $\mathrm{S}$-band. The correspondence in term of velocity precision in the Doppler is at the level of $0.01\,\mu\mathrm{m}\cdot\mathrm{s}^{-1}$ and $10\,\mu\mathrm{m}\cdot\mathrm{s}^{-1}$ in GCRS and BCRS coordinates, respectively. Past and future space missions like Cassini [@2003Natur.425..374B; @Kliore2004; @2007IJMPD..16.2117I], BepiColombo [@2002PhRvD..66h2001M; @2009AcAau..65..666I], or JUICE [@Grasset20131] have reached or will reach the level of the $\mu\mathrm{m}\cdot\mathrm{s}^{-1}$ for the Doppler. Therefore, the light-dragging effect is clearly at the threshold of visibility in Doppler observables and should be modeled in data reduction softwares in a close future. In order to understand what could be the signature of an unaccounted light-dragging effect, let us now focus on the computation of the time variability of $D(\mathbf x_A,\mathbf x_B)$. For a ground-based instrument, the spatial coordinates expressed in an Earth centered frame are given by $\mathbf x_B=(R_\oplus,\phi_B,\lambda_B)$, where $\phi_B$ is the latitude and $\lambda_B$ the longitude of the instrument on the surface of the Earth. The variable part in Eq. is better understood if we introduce $(a,e,\iota,\varOmega,\omega,f)$ denoting the set of Keplerian elements of the emitter. In GCRS coordinates the direction $\mathbf n_A$ of the emitter is given for instance in Eq. (3.42) of [@2014gravbookP]. Then, the expression of the light-dragging coefficient reads as follows $$\begin{aligned} &D-1=\frac{\omega_\oplus R_\oplus}{c}\frac{a(1-e^2)}{R_{AB}}\frac{\cos\phi_B}{(1+e\cos f)}\nonumber\\* &\times\Big\{\sin\varOmega\big[I_-\cos(F_++P_+)+I_+\cos(F_-+P_-)\big]\nonumber\\* &-\cos\varOmega\big[I_-\sin(F_++P_+)+I_+\sin(F_-+P_-)\big]\Big\}\end{aligned}$$ where we have set $$I_\pm=1\pm\cos\iota\text{,}$$ and $$F_\pm=f\pm \omega_\oplus t\text{,} \qquad P_\pm=\omega\pm\lambda_B\text{.}$$ Considering a quasi-circular orbit ($e\ll 1$), we have $r_A=a+\mathcal{O}(e)$ and $$f=n (t-t_0)+\mathcal{O}(e)$$ where $t_0$ is the time of perigee passage and where $n$ is the mean motion being given by Kepler’s third law $$n=\sqrt{\frac{Gm_\oplus}{a^3}}\,\text{.}$$ Therefore, the magnitude of $D-1$ oscillates with frequencies $n\pm\omega_\oplus$ around zero and $10^{-4}$ (maximum amplitude of the orbital barycentric velocity) in GCRS and BCRS coordinates, respectively. The peak to peak amplitude is of the order of $10^{-6}$ in both reference systems. In the limit case where $\lim_{a\to\infty}n=0$, the same magnitudes oscillate at diurnal frequency. Consequently, while modeling the time and frequency transfers using Eqs. and in GCRS or BCRS coordinates, the fact of imposing $D=1$ leads to an unaccounted contribution which may lead to systematic errors for instance in the estimations of the spacecraft velocity (considering Eq. ) or in the receiver coordinates (considering that diurnal signatures mainly concern ground-based stations). This last example could be particularly relevant for ground-based techniques operating within the International Earth Rotation and Reference System Service (IERS) for which an error in the estimation of the station coordinates can result in a bias in the determination of the ITRF. Conclusion ========== This paper generalizes the algorithmic approach introduced in [@2008CQGra..25n5020T] by making the time transfer functions formalism applicable in optical spacetime. The main results stand in the theorems \[th:theo4\]–\[th:theo11\] which allow one to determine the integral form of the time transfer functions up to any order. The great benefit of using the time transfer functions formalism rely on the fact that all integrals in theorems \[th:theo6\]–\[th:theo11\] are line integrals taken along the zeroth-order null geodesic path between the emitter and the receiver, independently of the order which is considered. In optical spacetime, the method requires us to know the order of magnitudes of both the gravitational and the refractive perturbations. Then, one can deduce the integer parameter $s$ from Eq. and use theorems \[th:theo4\]–\[th:theo5\] in order to determine the general expansion of the total time delay functions. The different components are the gravitational, the refractive, and the coupling contributions. Each of them is determined recursively making use of theorems \[th:theo7\]–\[th:theo11\]. We emphasis that these theorems have been derived assuming i) a post-Minkowskian expansion and ii) a general expansion in term of an arbitrary refractivity $N_0$. Both choices are motivated by the quasi-Minkowskian path regime which is assumed throughout the paper. We have illustrated the method by determining the integral form of the time transfer function up to the post-linear approximation. We have considered the case of a one-way transfer between a low orbit emitter and a receiving station on the Earth’s surface. We have shown that the time and frequency transfers are both impacted by the light-dragging effect due to the motion of the atmosphere, as seen from a frame which is not comoving with the flowing optical media. With respect to other methods [@2019AeA...624A..41B], we have highlighted the great advantage of the covariant formalism developed in this paper which naturally takes into account the effect of the dragging of light. In addition, we have shown that the light-dragging contribution is independent of the refractive profile which is considered. At the end of the day, the dragging component reduces to a geometrical factor which scales the static part of the atmospheric time delay (where the term static refers to the delay which would be measured in a frame comoving with the refractive medium). Concerning the frequency transfer, we have shown that the light-dragging contribution scales the coordinate velocities of both the emitter and the receiver resulting in the introduction of artificial dragging coordinate velocities. Finally, we have discussed the necessity, in a close future, for taking into account the dragging of light in data reduction softwares modeling the time and frequency transfers within inertial frames (e.g. GCRS and BCRS). Acknowledgments {#acknowledgments .unnumbered} =============== The author is grateful to the university of Bologna and to Italian Space Agency (ASI) for financial support through the Agreement 2013-056-RO in the context of EAS’s JUICE mission. The author is also thankful to A. Hees, P. Teyssandier, and C. Le Poncin-Lafitte from SYRTE in Observatoire de Paris for interesting discussions and valuable comments about a preliminary version of the manuscript. General expansion of $\gamma_{\mu\nu}$ {#sec:genexp} ====================================== The covariant components of $\gamma_{\mu\nu}$ are determined from the inverse conditions which lead to the following implicit expression $$\gamma_{\mu\nu}=-g_{\mu\alpha}g_{\beta\nu}\kappa^{\alpha\beta}-g_{\mu\alpha}\kappa^{\alpha\beta}\gamma_{\beta\nu}\text{.} \label{eq:gmrec}$$ Usually, assuming that $\gamma_{\mu\nu}=f(n)w^{\mu}w^{\nu}$ with $f(n)$ being a sought function of the index of refraction and using Eq. , we infer Eq. . However, the situation slightly changes if we expand the contravariant components $\kappa^{\mu\nu}$ as it is done in Eq. . At the same time, we have assumed that the physical spacetime metric, which is given in Eq. , satisfies a post-Minkowskian expansion (see Eq. ). Thus, considering that the refractive components are the dominant order according to Eq. for $s\in\mathbb{N}_{>0}$, we deduce that the covariant components $\gamma_{\mu\nu}$ satisfy the following expansion $$\gamma_{\mu\nu}(x,N_0,G)=\sum_{l=1}^{\infty}\gamma_{\mu\nu}^{(l)}(x) \label{eq:gamPM}$$ where the quantities $\gamma_{\mu\nu}^{(l)}$ can be recursively determined from Eq. , that is \[eq:kapPMord\] $$\begin{aligned} \gamma_{\mu\nu}^{(1)}&=-\eta_{\mu\alpha}\eta_{\beta\nu}\kappa^{\alpha\beta}_{(1)}\text{,}\label{eq:kapPMord1}\\ \gamma_{\mu\nu}^{(q)}&=-\eta_{\mu\alpha}\eta_{\beta\nu}\kappa^{\alpha\beta}_{(q)}-\eta_{\mu\alpha}\sum_{m=1}^{q-1}\kappa^{\alpha\beta}_{(m)}\gamma_{\beta\nu}^{(q-m)}\label{eq:kapPMord2}\end{aligned}$$ for $2\leqslant q\leqslant s$, and $$\begin{aligned} \gamma_{\mu\nu}^{(s+1)}&=-\eta_{\mu\alpha}\eta_{\beta\nu}\kappa^{\alpha\beta}_{(s+1)}-2\eta_{\mu\alpha}\kappa^{\alpha\beta}_{(1)}h_{\beta\nu}^{(1)}\nonumber\\ &-\eta_{\mu\alpha}\sum_{m=1}^{s}\kappa^{\alpha\beta}_{(m)}\gamma_{\beta\nu}^{(s-m+1)}\text{,}\label{eq:kapPMord3}\\ \gamma_{\mu\nu}^{(s+q)}&=-\eta_{\mu\alpha}\eta_{\beta\nu}\kappa^{\alpha\beta}_{(s+q)}-2\eta_{\mu\alpha}\kappa^{\alpha\beta}_{(q)}h_{\beta\nu}^{(1)}\nonumber\\ &-\eta_{\mu\alpha}\sum_{m=1}^{s+q-1}\kappa^{\alpha\beta}_{(m)}\gamma_{\beta\nu}^{(s-m+q)}\nonumber\\ &-h_{\mu\alpha}^{(1)}\sum_{m=1}^{q-1}\kappa^{\alpha\beta}_{(m)}\gamma_{\beta\nu}^{(q-m)}\label{eq:kapPMord4}\end{aligned}$$ for $2\leqslant q\leqslant s$, and $$\begin{aligned} \gamma_{\mu\nu}^{(2s+1)}&=-\eta_{\mu\alpha}\eta_{\beta\nu}\kappa^{\alpha\beta}_{(2s+1)}-2\eta_{\mu\alpha}\kappa^{\alpha\beta}_{(s+1)}h_{\beta\nu}^{(1)}\nonumber\\ &-2\eta_{\mu\alpha}\kappa^{\alpha\beta}_{(1)}h_{\beta\nu}^{(2)}-h_{\mu\alpha}^{(1)}\kappa^{\alpha\beta}_{(1)}h^{(1)}_{\beta\nu}\nonumber\\ &-\eta_{\mu\alpha}\sum_{m=1}^{2s}\kappa^{\alpha\beta}_{(m)}\gamma_{\beta\nu}^{(2s-m+1)}\nonumber\\ &-h_{\mu\alpha}^{(1)}\sum_{m=1}^{s}\kappa^{\alpha\beta}_{(m)}\gamma_{\beta\nu}^{(s-m+1)}\text{,}\label{eq:kapPMord7}\end{aligned}$$ and $$\begin{aligned} \gamma_{\mu\nu}^{(2s+q)}&=-\eta_{\mu\alpha}\eta_{\beta\nu}\kappa^{\alpha\beta}_{(2s+q)}-2\eta_{\mu\alpha}\kappa^{\alpha\beta}_{(q)}h_{\beta\nu}^{(2)}\nonumber\\ &-2\eta_{\mu\alpha}\kappa^{\alpha\beta}_{(s+q)}h_{\beta\nu}^{(1)}-h_{\mu\alpha}^{(1)}\kappa^{\alpha\beta}_{(q)}h^{(1)}_{\beta\nu}\nonumber\\ &-\eta_{\mu\alpha}\sum_{m=1}^{2s+q-1}\kappa^{\alpha\beta}_{(m)}\gamma_{\beta\nu}^{(2s+q-m)}\nonumber\\ &-h_{\mu\alpha}^{(1)}\sum_{m=1}^{s+q-1}\kappa^{\alpha\beta}_{(m)}\gamma_{\beta\nu}^{(s+q-m)}\nonumber\\ &-h_{\mu\alpha}^{(2)}\sum_{m=1}^{q-1}\kappa^{\alpha\beta}_{(m)}\gamma_{\beta\nu}^{(q-m)}\label{eq:kapPMord8}\end{aligned}$$ for $2\leqslant q\leqslant s$, and $$\begin{aligned} \gamma_{\mu\nu}^{(ps+1)}&=-\eta_{\mu\alpha}\eta_{\beta\nu}\kappa^{\alpha\beta}_{(ps+1)}-2\eta_{\mu\alpha}\sum_{m=0}^{p-1}\kappa^{\alpha\beta}_{(ms+1)}h_{\beta\nu}^{(p-m)}\nonumber\\ &-\sum_{m=1}^{p-1}h_{\mu\alpha}^{(p-m)}\sum_{n=0}^{m-1}\kappa^{\alpha\beta}_{(ns+1)}h^{(m-n)}_{\beta\nu}\nonumber\\ &-\eta_{\mu\alpha}\sum_{m=1}^{ps}\kappa^{\alpha\beta}_{(m)}\gamma_{\beta\nu}^{(ps+1-m)}\nonumber\\ &-\sum_{m=1}^{p-1}h_{\mu\alpha}^{(p-m)}\sum_{n=1}^{ms}\kappa^{\alpha\beta}_{(n)}\gamma_{\beta\nu}^{(ms+1-n)}\label{eq:kapPMord9}\end{aligned}$$ for $p\geqslant 3$, and finally $$\begin{aligned} \gamma_{\mu\nu}^{(ps+q)}&=-\eta_{\mu\alpha}\eta_{\beta\nu}\kappa^{\alpha\beta}_{(ps+q)}-2\eta_{\mu\alpha}\sum_{m=0}^{p-1}\kappa^{\alpha\beta}_{(ms+q)}h_{\beta\nu}^{(p-m)}\nonumber\\ &-\sum_{m=1}^{p-1}h_{\mu\alpha}^{(p-m)}\sum_{n=0}^{m-1}\kappa^{\alpha\beta}_{(ns+q)}h^{(m-n)}_{\beta\nu}\nonumber\\ &-\eta_{\mu\alpha}\sum_{m=1}^{ps+q-1}\kappa^{\alpha\beta}_{(m)}\gamma_{\beta\nu}^{(ps+q-m)}\nonumber\\ &-\sum_{m=0}^{p-1}h_{\mu\alpha}^{(p-m)}\sum_{n=1}^{ms+q-1}\kappa^{\alpha\beta}_{(n)}\gamma_{\beta\nu}^{(ms+q-n)}\label{eq:kapPMord10}\end{aligned}$$ for $p\geqslant 3$ and $2\leqslant q\leqslant s$, where $p$ and $q$ are determined from $l$ using Eqs. . [^1]: This can be easily shown from Eq. using the inverse conditions $\bar g_{\mu\sigma}\bar g^{\sigma\nu}=\delta_\mu^\nu$. [^2]: The reception and the emission time delay functions in [@2008CQGra..25n5020T] are denoted $\Delta_r/c$ and $\Delta_e/c$, respectively. Instead, in this work, we use $\Delta/c$ and $\Xi/c$ in order to keep incoming indices notations as clear as possible. [^3]: We recall that the time component of $x_A$ is constrained by the fact that the past null cone at $x_B$ intersects the world line $\mathbf x=\mathbf x_A$ at only one point of coordinates $\bm x_A=(ct_A,\mathbf{x}_A)$. This is true as long as the null geodesic is quasi-Minkowskian. [^4]: Hereafter, $\mu$ is an affine parameter and does not represent the permeability of the dielectric medium anymore. [^5]: Results assuming that the gravitational perturbation is the leading term can easily be derived by applying the exact same following approach. However, in that case, it must be noted that $s$ should be introduced switching the numerator and the denominator in the right-hand side of Eq. .

--- author: - 'S. Aalto' - 'F. Costagliola' - 'S. Muller' - 'K. Sakamoto' - 'J. S. Gallagher' - 'K. Dasyra' - 'K. Wada' - 'F. Combes' - 'S. García-Burillo' - 'L. E. Kristensen' - 'S. Martín' - 'P. van der Werf' - 'A. S. Evans' - 'J. Kotilainen' bibliography: - 'n1377\_ALMA\_aalto.bib' date: 'Received xx; accepted xx' title: 'A precessing molecular jet signaling an obscured, growing supermassive black hole in NGC1377?[^1]' --- Introduction {#s:intro} ============ The growth of central baryonic mass concentrations and their associated supermassive black holes (SMBHs) are key components of galaxy evolution [e.g. @kormendy13]. The underlying processes behind the evolution of the SMBH and how it is linked to its host galaxy and its interstellar gas are, however, not well understood. In addition, it is not clear how SMBHs can grow despite the energy/luminosity of accretion that leads to gas expulsion from the region. Massive molecular outflows powered by AGNs and bursts of star formation are suggested as being capable of driving out a large fraction of the galaxy’s cold gas reservoir in only a few tens of Myr [e.g. @nakai87; @walter02; @feruglio10; @sturm11; @aalto12a; @combes13; @bolatto13; @cicone14; @sakamoto14; @garcia14; @aalto15a; @alatalo15; @feruglio15]. To maintain nuclear activity and growth, an inflow of gas from larger radii is therefore required. Cold molecular gas has been proposed as an important source of fuel for SMBH growth since the accretion of hot gas is meant to be inefficient and slow [@blandford99; @nayakshin14]. However, it is not known how the cold gas is deposited into the inner nucleus of the galaxy. This angular momentum problem is similar for the growth of SMBHs and the formation of stars [@larson09] and is even more severe for SMBHs because they are smaller than stars in relation to the size of the system in which they form. Thus, the mass that SMBHs may achieve is likely to be strongly regulated by the efficiency of angular momentum transfer during the fuel process. In protostars there is strong evidence of a physcal link between infall and outflow [e.g. @arce] and angular momentum can be transferred by molecular jets and outflows. A link betwen infall and outflow seems to also exist for galaxy nuclei and AGNs [e.g. @davies14; @garcia14] Chaotic inflows of cold gas clumps with randomly oriented angular momenta have been suggested as alternatives to large scale disks in feeding the growth of the SMBH [@king07; @gaspari13; @nayakshin12]. In this scenario, SMBH growth may occur primarily through multiple small-scale accretion events, rather than continous accretion [e.g. @king07] leading to AGN luminosity variations on time scales of $10^3 - 10^6$ yr [@hickox14]. A somewhat contrasting picture is that angular momentum may be effectively transported by, for example, bars and spiral density waves on large and small scales (see e.g. discussion in @garcia14). AGN luminosity and nuclear growth is therefore expected to vary depending on the interplay between mode of accretion, outflow, and winds. To test how gas inflow and the feedback of central activity influences the growth of SMBHs it is important to study galaxies in early or transient phases of their nuclear evolution. NGC1377 is a likely example of such a system. It belongs to a small subset of galaxies that has a pronounced deviation from the well-known radio-to-FIR correlation, having excess FIR emission compared to the radio ($q>3$; $q$=log[\[FIR/3.75$\times 10^{12}$ Hz\]/$S_{\nu}$(1.4GHz)]{} [@helou85]). These FIR-excess and radio-quiet galaxies are rare. @roussel03 find that they represent a small fraction (1%) of an infrared flux-limited sample in the local universe, such as the IRAS Faint Galaxy Sample. Their scarcity is likely an effect of the short time spent in the FIR-excess phase, making them ideal targets for studies of transient stages of AGN, starburst, and feedback. The extremely radio-quiet FIR-excess galaxy NGC 1377 ---------------------------------------------------- NGC 1377 is a member of the Eridanus galaxy group at an estimated distance of 21 Mpc (1=102 pc) and has a far-infrared luminosity of $L_{\rm FIR}=4.3 \times 10^9$ L$_{\sun}$ [@roussel03]. In stellar light, NGC 1377 has the appearance of a regular lenticular galaxy [@rc3] although @heisler94 and @roussel06 find a faint dust lane that extends along the southern part of the minor axis. NGC 1377 is the most radio-quiet, FIR-excess galaxy known to date with radio synchrotron emission being deficient by at least a factor of 37 with respect to normal galaxies [@roussel03; @roussel06]. Interestingly, H II regions are not detected through near-infrared hydrogen recombination lines or thermal radio continuum even though faint optical emission lines are present [@roussel03; @roussel06]. Deep mid-infrared silicate absorption features suggest that the nucleus is enshrouded by large masses of dust [e.g. @spoon07]. This supports the notion that NGC1377 may be in a transient phase of its evolution since a more advanced nuclear activity is expected to have cleared out the enshrouding material. It has been suggested that the compact IR nucleus may be the site of a nascent ($t<$1 Myr) opaque starburst [@roussel03; @roussel06] or of a buried AGN [@imanishi06; @imanishi09]. High resolution SMA CO 2–1 observations revealed a large central concentration of molecular gas and a massive molecular outflow [@aalto12b] that appeared to be young ($\sim1.4$ Myr). The extremely high nuclear dust and gas obscuration of NGC1377 aggravates the determination of the nature of the nuclear activity and the driving force of the molecular outflow, but the extraordinary radio deficiency implies transient nuclear activity . We used the Atacama Large Millimeter/submillimeter Array (ALMA) to observe CO 3–2 at high resolution in NGC1377 aiming to determine the nature of the buried source and the structure and evolutionary status of the outflow. Here we present the discovery of a high-velocity, extremely collimated and precessing molecular jet in NGC1377. Our results show that the nuclear source is likely an AGN and that we are either witnessing a faint radio jet driving a molecular collimated outflow, or a jet powered by cold accretion. The nuclear activity of NGC1377 may be fading, or the large nuclear concentration of gas and dust signify that the major AGN event has not yet occured. We also discuss how the gas transfer in the moleular jet may instead foster gas recycling and how this process may promote SMBH growth. Observations {#s:obs} ============ ![\[f:mom\] CO 3-2 moment maps. Left: Integrated intensity (mom0) where contours are 1.7$\times$ (1, 2, 4, 8, 16, 32, 64) Jy  beam$^{-1}$. Colours range from 0 to 172 Jy  beam$^{-1}$. Centre: velocity field (mom1) where contours range from 1690  to 1820  in steps of 10 . Right: Dispersion map (mom2) where contours are 4.4$\times$(1, 3, 5, 7, 9, 11, 13) . Colours range from 0 to 66 . The cross indicates the position of the 345 GHz continuum peak (see Table \[t:flux\]). ](n1377_co32_mom.pdf) ![\[f:jet\] CO 3–2 integrated intensity image where emission close to systemic velocity (1700 - 1760 ) is shown in greyscale (ranging from 0 to 70 Jy ). The high velocity ($\pm$80 to $\pm$150 ) emission from the molecular jet is shown in contours (with the red and blue showing the velocity reversals). The contour levels are 1.0$\times$(1, 2, 3, 4, 5, 6, 7, 8, 9) Jy  beam$^{-1}$. The dashed lines indicate the jet axis and the inferred orientation of the nuclear disk. The CO 3–2 beam is shown as a grey ellipse in the bottom left corner. The vertical bar indicates a scale of 100 pc. ](n1377_jet_PA.pdf) ![\[f:pv\] Position-velocity (PV) diagrams showing gas velocities in five different slits: (A) along the jet axis; (B) Perpendicular to the jet axis through the nucleus; (C) Perpendicular to the jet axis at 1.2 to the north; (D) Perpendicular to the jet axis at 0.25 to the north; (E) Perpendicular to the jet axis at 0.25 to the south; (F) Perpendicular to the jet axis at 1.2 to the south. Contour levels are 3.1$\times$(1, 3, 5, 9, 18, 36) mJy beam$^{-1}$ thus the first level is at 4$\sigma$. The colour scale range from 2 to 156 mJy beam$^{-1}$. ](N1377_pvfig_2.png) Observations of the CO J=3–2 line were carried out with ALMA (with 35 antennas in the array) on 2014 August 12, for about half an hour on-source and with good atmospheric conditions (precipitable amount of water vapour of $\sim$0.5 mm). The phase centre was set to $\alpha$=03:36:39.074 and $\delta$=$-$20:54:07.055 (J2000). The correlator was set up to cover two bands of 1.875 GHz in spectral mode, one centred at a frequency of $\sim$344.0 GHz to cover the CO J=3–2 line (in the lower side band), and the other centred at 354.3 GHz to cover the HCO$^+$ J=4–3 and HCN $J=4-3$ $v=0$ and $v=1f$ lines (in the upper side band). The velocity resolution for these bands was 1.0 km/s after Hanning smoothing. In addition, two 2 GHz bands were set up in continuum mode, i.e., with a coarser velocity resolution of $\sim$27 , centred at 342.2 and 356 GHz, respectively. The bandpass of the individual antennas was derived from the quasar $J0423-0120$. The quasar $J0340-2119$ ($\sim$0.3 Jy) was observed regularly for complex gain calibration. The absolute flux scale was calibrated using the quasar $J0334-401$. The flux density for $J0334-401$ was extracted from the ALMA flux-calibrator database. After calibration within the CASA reduction package, the visibility set was imported into the AIPS package for further imaging. The synthesized beam is $0.{\mbox{$''$}}25 \times 0.{\mbox{$''$}}18$ (25$\times$18 pc for NGC1377) with Briggs weighting (parameter robust set to 0.5) and the resulting data has a sensitivity of 0.8 mJy per beam in a 10 (12 MHz) channel width. Results ======= [ll]{} &\ \ Position$^b$ (J2000) & $\alpha$: 03:36:39.073 ($\pm$ 0.01)\ & $\delta$: -20:54:07.05 ($\pm$ 0.01)\ Peak flux density$^c$ & 156 $\pm$ 1 (mJybeam$^{-1}$)\ Flux &\ (central beam) & $17.4 \pm 0.05$ (Jy  beam$^{-1}$)\ (molecular jet)$^d$ & $23.2 \pm 0.5$ (Jy )\ (whole map) & $159 \pm 0.5$ (Jy )\ \ Molecular mass$^e$ &\ (central beam) & $1.8 \times 10^7$\ (molecular jet) & $2.3 \times 10^7$\ (whole map) & $16 \times 10^7$\ \ \ [*a)*]{} Listed errors are 1$\sigma$ rms. [*b)*]{} The position of the peak 345 GHz continuum emission and of the CO 3–2 integrated intensity. The peak $T_{\rm B}$ is at $\alpha$:03:36:39.072 $\delta$:-20:54:07.06 at $V_c$=1730 . [*c)*]{} The Jy to K conversion in the $0.{\mbox{$''$}}25 \times 0.{\mbox{$''$}}18$ beam is 1 K=4.6 mJy. The peak $T_{\rm B}$ is 34 K corresponding to 156 mJy. [*d)*]{} The jet flux is integrated from $\pm$(60 to 200) where the blueshifted flux is 5.5 and the redshifted 17.7 Jy . [*e)*]{} The H$_2$ mass $M$(H$_2$)=$1 \times 10^4 \, S({\rm CO} 1-0) \Delta \nu \, D^2$ ($D$ is the distance in Mpc, $S \Delta \nu$ is the integrated CO 1–0 line flux in Jy ) for a conversion factor $N$(H$_2$)/$I$(CO 1–0)=$2.5 \times 10^{20}$ $\cmmt$). Since we have CO 3–2 we have to correct for the frequency dependence of the brightness temperature conversion. If CO 3–2 and 1–0 have the same brightness temperature (thermal excitation, optically thick) the correction factor is 1/9. However, usually the CO emission is subthermally excited and the brightness temperature ratio is expected to be about 0.5 for a giant molecular cloud. Hence the correction factor we apply is 1/4.5 and $M$(H$_2$)=$2.2 \times 10^3 \, S({\rm CO}\, 3-2) \Delta \nu \, D^2$. The inferred H$_2$ column density in the central beam is $N$(H$_2$)=$3 \times 10^{24}$ $\cmmt$. CO 3–2 moment maps ------------------ The CO 3–2 integrated intensity (moment 0) map, velocity field (moment 1) and dispersion map (moment 2) are presented in Fig \[f:mom\]. We smoothed to two channel resolution, then for the moment 0 map we clipped at the 3$\sigma$ level, and for the moment 1 and 2 maps we clipped at 4$\sigma$. The velocity centroids were determined through a flux-weighted first moment of the spectrum of each pixel, therefore assigning one velocity to a spectral structure. The dispersion was determined through a flux-weighted second moment of the spectrum of each pixel. This corresponds to the one dimensional velocity dispersion (i.e. the FWHM line width of the spectrum divided by 2.35 for a Gaussian line profile) The integrated intensity map shows centrally peaked emission with some structure extending radially from the centre up to a radius of $\sim$1.5 (150 pc). An estimated 11% of the emission is emerging from the inner 25$\times$18 pc (see Table \[t:flux\]). The velocity field is complex and implies that the maximum velocity shifts occur outside the nucleus. There is evidence for a shallow east-west velocity gradient around the nucleus. The moment 2 map reveals a striking, narrow 3 long feature of high dispersion. The CO emission clearly delineates two separate structures (Fig. \[f:jet\]): an extremely well collimated [*jet-like*]{} structure, which, essentially, is visible at high velocities and large-scale emission at low velocities, which surrounds the high velocity jet-like feature. We interpret the high velocity feature as a molecular jet (see Sect. \[s:jet\]) and we will refer to it as such in the text below. The high velocity gas – a molecular jet {#s:hivel} --------------------------------------- The high velocity (projected velocities 60-150 ) gas (Fig. \[f:jet\]) is aligned in a $\pm$1.5 ($\pm$150 pc) long, highly collimated, jet. It has an unresolved width ($d<$20 pc - set by the limit of our resolution) and a position angle PA=10$^{\circ}$. In Fig. \[f:jet\] we show that near the nucleus (within 0.5) emission at redshifted velocities is on the southern side and emission at blueshifted velocities are found to the north. Further along the axis (beyond 0.5) this reverses. Systemic and low-velocity gas {#s:lovel} ----------------------------- The systemic and low-velocity gas (projected velocities 0-60 ) consists of a bright central disk-like feature with PA=105$^{\circ} \pm 5^{\circ}$ and larger scale emission extending primarily along the minor axis of NGC1377. Along the PA of 105$^{\circ} \pm 5^{\circ}$ there is an east-west velocity shift of $\sim$50 . The low-velocity emission surrounds the molecular high-velocity jet in a butterfly-like pattern (Fig. \[f:jet\] ). Most of the CO 3–2 flux of NGC1377 emerges from this minor axis structure (Table \[t:flux\]). The minor axis extent of the systemic emission is similar to that of the high-velocity molecular jet, but we note that at zero velocities, negatives in the map indicate that some flux is missing from extended emission. The maximum recoverable scale of our observations is of the order of $\sim5$. Position-velocity (PV) diagrams ------------------------------- In Fig \[f:pv\] we present five PV diagrams to show the distinct structure of the high-velocity emission in relation to that of the systemic and low-velocity gas. The PV diagram along the jet axis (A) shows the velocity reversals. Near the nucleus the highest velocity is blueshifted to the north and redshifted to the south. The maximum velocities occur about 0.25 (25 pc) away from the nucleus. Further away (1.2) the highest velocity is now redshifted on the north side and blueshifted to the south. The PV diagram also shows that the CO emission peaks strongly in the nucleus and that the emission along the jet axis is clumpy. The clumps are unresolved in the CO 3–2 beam and from the Jy to K conversion in Table \[t:flux\] we find that the clumps have brightness temperatures of $T_{\rm B}$(CO 3–2)=1 – 8 K. The CO 3–2 line widths of the gas clumps in the jet are high ranging from 50 to $\sim$150 , which is evident in the PV diagram along the jet axis, as well as in the PV diagrams cut across the jet (C-F in Fig \[f:pv\]). We show four PV diagrams oriented perpendicular to the jet: two at distance $\pm$ 0.25 from the nucleus (D and E) and two 1.2 from the nucleus (C and F). They were selected at the locations of highest velocities in the gas along the jet - and also to show the switch in orientation of the high-velocity gas (the velocity reversals). In PV diagrams D and E the distinction between the narrow, unresolved high-velocity gas from the extended emission (on scales of $\sim 2"$ (200 pc)) of the low-velocity gas is clear. On the north side the high-velocity blueshifted emission in (D) is narrower than the redshifted high-velocity emission further out (C). A similar pattern is seen to the south where the narrow redshifted emission near the nucleus (E) is more confined than the blueshifted emission further from the nucleus (F). Here, there is also some emission at near-systemic velocities as well as an additional blueshifted component. Comparing D and E we find that the low-velocity gas to the north is slightly redshifted with respect to systemic velocity and to the south the emission is somewhat blueshifted. We also present a PV diagram across the nucleus that is perpendicular to the jet component (B). Again, it shows the central concentration of the CO 3–2 emission, broad unresolved emission on the nucleus, and narrower emission extending to the east and west of the nucleus. There is a low velocity shift from east to west of $\pm$25 . Nuclear gas {#s:nuclear} ----------- Velocities in the nucleus span a total of 300 . It is not clear what amount of this constitutes rotation of a circumnuclear disk and what amount stems from the outflowing gas in the jet. The velocities in the moment 1 map (Fig \[f:mom\]) do not show much rotation around the nucleus. The nuclear emission is broad but unresolved in space, and the velocity outside of the nucleus drops quickly with radius along the major axis of the galaxy, as is evident in PV diagram (B) in Fig \[f:pv\]. From the CO luminosity, we infer an H$_2$ column density of $N$(H$_2$)=$3 \times 10^{24}$ $\cmmt$ (Table \[t:flux\]) towards the nucleus. This would imply that the nucleus of NGC1377 is Compton thick and similar to the nuclei of other extremely obscured early type disk galaxies, such as NGC4418 [@sakamoto13; @costagliola13], IC860 and Zw049.057 [@falstad15; @aalto15b], but more studies are required to confirm this high $N$(H$_2$) for NGC1377. Apart from CO 3–2 we also detected HCO$^+$ and H$^{13}$CN $J=4-3$ and vibrationally excited HCN $J=4-3$ $\nu_2=1f$ ($T = E_{\rm l} / k$=1050 K) The vibrationally excited lines is a factor of 20-30 times fainter than CO 3–2 in the nucleus, but its detection is consistent with a large $N$(H$_2$) and the presence of very hot gas and dust [@aalto15b]. We also detect lines at redshifted frequency $\nu$=342.26 and 344.5 GHz. The identification of these lines is not clear but we tentatively identify the first as HC$^{15}$N $J=4-3$ and the second either as vibrationally excited HC$_3$N $J=38-37$ $\nu_4=1$, $\nu_7=1$, or as SO$_2$. We present spectra and a brief discussion of the line identification in Appendix \[s:A1\]. Continuum --------- We merged all line-free channels in our observations into a 0.8mm continuum image (Fig. \[f:cont\]). It consists of a compact component and some extended emission. In the $0.{\mbox{$''$}}25 \times 0.{\mbox{$''$}}18$ beam, the deconvolved FWHM size is 0.25 $\times$ 0.09 and a position angle PA=104 $\pm$ 5$^{\circ}$. The continuum is faint (1.3$\pm$0.1 mJy beam$^{-1}$ peak and 2.2$\pm$0.3 mJy integrated). The rms is 0.045 mJy. The continuum and CO 3–2 peak in the same position, which we assume is the nucleus of the galaxy. ![\[f:cont\] 0.8mm continuum (merged 342, 349, 356 GHz line-free channels). Contour levels are 0.14 $\times$(1,2,3,4,5,6,7) mJy beam$^{-1}$. The lowest contour is at 3$\sigma$. The cross indicates the continuum peak position (see Table \[t:flux\]). ](n1377_cont.pdf) Discussion ========== The high velocity gas: a precessing molecular jet? {#s:jet} -------------------------------------------------- We interpret the high velocity CO 3–2 emission as emerging from a highly collimated and ordered molecular jet. The striking velocity reversals along its symmetry axis are consistent with those of jet precession [e.g. @rosen04]. The maximum velocity swings from 1590 to 1910 north of the nucleus (Fig \[f:pv\] figure (A)) and from 1920 to 1650 to the south. Thus on average the shift is 300 . The velocity shifts to the north and south appear fairly symmetric, which suggests that the symmetry axis of the jet should be relatively close to the plane of the sky and thus launched from a highly inclined disk. The 0.8mm continuum image (Fig. \[f:cont\]) implies a nuclear disk of inclination 70$^{\circ} \pm$ 10$^{\circ}$ and a FWHM radius of 13 pc (although we caution that the continuum emission is faint and only marginally resolved). In addition, the nuclear CO emission lines are broad with an unresolved dynamics, which is also consistent with the notion of a compact, highly inclined nuclear disk. ### Simple models ![\[f:pvmodel\] Simple schematic jet model where we have rotated the jet symmetry axis from PA=10$^{\circ}$ to 0$^{\circ}$. Top: To the right, the northern part of the jet viewed face on. The curve indicates the pattern of the jet path on the sky and the blue and red colours indicate blue- and redshifted emission in the sight-line. The precession angle here is $\theta$=15$^{\circ}$ and the arrow indicates where the CO 3–2 emission in the jet ends. The right panel shows the jet viewed from an angle of 45$^{\circ}$ to illustrate its 3D nature. Bottom: The observed PV diagram along the jet axis (Panel (A) in Fig. \[f:pv\]) with the superposed tracks of a precessing jet of $\theta$=15$^{\circ}$ and outflowing velocity $v_{\rm out}$=390  and $v_{\rm out}$=520  indicated with dashed curves. (These values are within the range for $\theta$ and $v_{\rm out}$ discussed in Sect. \[s:parameters\]). We assume $v_{\rm out}$ to be constant and a jet without width. ](jet_new_2.png "fig:") ![\[f:pvmodel\] Simple schematic jet model where we have rotated the jet symmetry axis from PA=10$^{\circ}$ to 0$^{\circ}$. Top: To the right, the northern part of the jet viewed face on. The curve indicates the pattern of the jet path on the sky and the blue and red colours indicate blue- and redshifted emission in the sight-line. The precession angle here is $\theta$=15$^{\circ}$ and the arrow indicates where the CO 3–2 emission in the jet ends. The right panel shows the jet viewed from an angle of 45$^{\circ}$ to illustrate its 3D nature. Bottom: The observed PV diagram along the jet axis (Panel (A) in Fig. \[f:pv\]) with the superposed tracks of a precessing jet of $\theta$=15$^{\circ}$ and outflowing velocity $v_{\rm out}$=390  and $v_{\rm out}$=520  indicated with dashed curves. (These values are within the range for $\theta$ and $v_{\rm out}$ discussed in Sect. \[s:parameters\]). We assume $v_{\rm out}$ to be constant and a jet without width. ](pvmodel.png "fig:") ![\[f:model\] Contour plots of integrated red- and blueshifted emission ($>\pm$60 ) for ALMA data (right) and model of precessing jet (left): The precession angle is 10$^{\circ}$ (centre) and 25$^{\circ}$ (right) the inclination of the model precession axis along the line of sight is close to zero. ](chan_model.png) The PV diagram along the symmetry axis of a precessing jet of constant outflow velocity shows the projected velocity oscillate[^2] as the jet alternates its direction towards and away from the observer. This is demonstrated schematically in Fig. \[f:pvmodel\] where we show the resulting PV diagram of a simple model with a precession angle $\theta$=15$^{\circ}$, the inclination of the precession axis is zero (i.e. in the plane of the sky), and the outflow velocity $v_{\rm out}$ is constant. The precession has gone through slightly more than half a period. The first maximum velocity occurs 0.25 above the nucleus when the jet is most pointed towards us, implying that the jet close to the nucleus is seen at an angle. We require higher spatial resolution to carry out proper model fits to the jet properties. However, for illustrative purposes we present two model maps in Fig. \[f:model\], showing what the model above would look like if we presented it in contour plot form i.e. similar to the high velocity contours in Fig. \[f:jet\]. We show two scenarios: one with precession angle $\theta$=25$^{\circ}$ and $v_{\rm out}$=260 , the other with $\theta$=10$^{\circ}$ and $v_{\rm out}$=600 , to demonstrate the effect of the precession angle on the high-velocity contour plots. Here the inclination of the precession axis to the declination axis is 10$^{\circ}$ and, to the plane of the sky, it is zero. We assume a jet width of 0.4. ### Jet parameters {#s:parameters} The PV diagrams across the jet will show features that are broad in velocity if the cut includes the maximum projected velocity. For these velocities, the spatial extent will be the lowest and the emission will be narrowest near the base of the jet, while the emission at maximum velocity will be broader further away from the nucleus owing to the precession of the jet. This effect can be seen in Fig. \[f:pv\] (panels (C) and (D)) where the blueshifted jet component near the nucleus in (D) is narrower than the redshifted jet component further away (to the north) from the centre. In addition, the position of the redshifted jet component is shifted to the east, compared to the blueshifted component. This also gives the jet precession direction implied in Fig. \[f:pv\]. The offset of the redshifted jet component to the north can be used to estimate the precession angle. The maximum velocity is expected to be completely aligned with the jet axis, but emission at lower redshifted velocities are coming in from the east (also showing the direction of the precession). The east-offset implies a precession angle $\theta$=10$^{\circ}$-25$^{\circ}$, but this is, of course, very uncertain since we only have slightly more than half a turn of the jet. The unprojected outflow velocity $v_{\rm out}$ depends on $\theta$ and the observed maximum projected velocity $v_{\rm proj}$. This may either be done by selecting the velocity at the 3$\sigma$ contour or the velocity of the brighter clumps. This gives a rough span to $v_{\rm proj}$ of 100-150 . The outflow velocity should, therefore, lie in the range $v_{\rm out}$=240-850  with a precession period $P$=0.3-1.1 Myr. The dynamical age of the full length of the molecular jet appears to be short. The jet can be traced out to $\sim$150 pc and for $v_{\rm out}$=240-850  the time scale ranges between $t$=0.2 and 0.7 Myr. ### Launch region The molecular jet emerges from the nucleus and its width is unresolved, which results in a launch region of the jet inside $r$=10 pc. The nuclear rotation is also unresolved, but from the PV diagram we estimate a rotational velocity of $\sim$110  and, if this occurs ar $r$=10 pc, the rotational timescale is $\sim$1 Myr. The precession period must be longer than the rotational timescale of the jet-launching region and hence the jet is very likely launched close to the nucleus, within the inner few pc. ### Origin of precession Jet precession may occur in a variety of astrophysical objects, including low-mass star formation in the Galaxy (L1157 @gueth96 [@kwon15]), (NGC 1333-IRAS4A @Santangelo15, (L 1551 IRS 5 @fridlund94), ( IRAS 16293-2422 @kristensen13); Galactic micro-quasars (SS433 @blundell05 and 1E 1740.7-2942 [@luqueescamilla15]), and AGN radio jets [e.g. @veilleux93; @steffen97; @martividal11; @pyrzas15]. Jet precession may be caused by a warped accretion disk [e.g. @greenhill03] i.e. by the misalignment between the spin orientation of the black hole and the surrounding accretion disk [e.g. @bardeen75; @lu05] and an accretion flow that is transporting in gas of misaligned angular momentum [@krolik15], but see also the discussion in @nixon13. Alternatively, in a SMBH binary system, jet precession may be caused by geodetic precession of the spin axis of the primary rotating SMBH being misaligned with the binary total angular momentum, or by inner disk precession (owing to the tidal interaction of an inclined secondary SMBH). Interestingly, the presence of a nuclear gas and dust concentration and a precessing molecular jet can aid the coalescence of the SMBHs into resolving the “final-parsec problem” [@milo03; @aly15]. The post-starburst spectrum of NGC1377 (Gallagher et al in prep.) could perhaps be linked to a past merger event that left left an SMBH binary in the heart of NGC1377. ### Other explanations In Appendix \[s:A2\] we discuss potential alternative explanations for the high-velocity gas emission structure and why we find them less likely (with current information) than the precessing jet model presented here. Low-velocity gas {#s:low} ---------------- The extremely simple jet model cannot explain all the features we see in the PV diagrams (Fig. \[f:pv\]). Perpendicular to the jet axis (panels C-F in Fig. \[f:pv\]), we see the jet emission as a broad velocity feature and narrow in space. However, there is also more spatially extended emission at low velocities (panels (D) and (E)). In PV diagram (A), the lower velocity emission occurs as straight lines to the north (in particular) but also in the south. There is also an extra component at 1800  to the south next to the blueshifted part of the jet. This emission cannot be directly explained by a simple model of a precessing jet and may emerge from a background disk, a molecular wide-angle wind, or it is caused by interaction and entrainment by the jet. For example a bow shock can arise by the formation of an internal working surface within the jet at positions of strong velocity discontinuity, and as the high velocity jet interacts with the surrounding medium [@raga93; @gueth99; @cliffe96; @santiago09]. The structure and velocity of the ambient gas may become complex owing to, for example, the action of the global bow shock and gas sweeping into the wakes of the jet turns. Dynamical simulations of precessing gaseous jets have been carried out by @raga01. They present PV diagrams perpendicular to their simulated jet (their Fig. 5) and find that the transverse spatial extent of the emitting region is larger at lower radial velocities. In their simulations, this is due to the presence of bow shock wings trailing behind each internal working surface. These bow shocks result in transverse extended emission of low radial velocities which forms a ’halo’ component. There is a striking similarity between the Fig. 5 of @raga01 and our PV diagrams that are perpendicular to the jet. The low-velocity gas has redshifted velocities north-east of the jet and blueshifted velocities to the south-west. The angle between the most red- and blueshifted gas is PA=40-45$^{\circ}$ and the velocity shift is $60 \pm 20$ . Apart from this gradient, there is no significant net shift in velocity between the north and the south (with deviations at the ends of the jet and at the edges of the map). There is a small (10-20 ) east-west gradient which is somewhat larger (50 ) at the disk major axis. The PA=40-45$^{\circ}$ velocity structure can be caused by the jet entraining and accelerating a very slow, wide-angle minor axis molecular outflow and/or that it is interacting with gas already entrained before. Another possibility is that there is a wind, which is unrelated to the jet and which originates in a disk warped about 20$^{\circ}$, compared to the nuclear disk. This orientation is however not consistent with that of the optical dust absorption features south of the nucleus of NGC1377 ( Fig. 1 in @roussel06, Fig.4c in @heisler94). The dust structures have a v-shaped morphology (opening angle of $\sim$90$^{\circ}$) and are oriented almost perpendicular to the stellar disk. They may be caused, forexample, by the precessing jet bow shocks. Comparing previous results for NGC1377 -------------------------------------- In our previous paper on NGC1377[@aalto12b], we suggested that the molecular outflow seen in CO 2–1 is biconic with an opening angle of 60$^{\circ}$- 70$^{\circ}$, an outflow mass $>1 \times 10^7$ , and an outflow velocity of 140 . These observations were carried out with three times poorer spatial resolution and about ten times lower flux sensitivity than the ALMA CO 3–2 data presented here. The CO 2–1 dispersion map has a cross-like structure that we used as a basis to suggest the biconic outflow. In the ALMA data, high dispersion is found only along a structure that we now identify as a molecular jet. It is interesting to note that the position angle of the outflow in the lower resolution CO 2–1 map is around PA=40$^{\circ}$, while the CO 3–2 jet has a PA of 10$^{\circ}$. The lower resolution SMA data has likely picked up the velocity shift in the low-velocity gas discussed above (Sect. \[s:low\]) which we propose is caused by jet entrainment (or, less likely, an inclined wide-angle flow). Further studies will reveal more on the origin of this gas component. Mass and outflow rate --------------------- [*The molecular jet:*]{} The molecular mass in the high velocity gas is estimated as $M_{\rm j}({\rm H}_2)=2.3 \times 10^7$ , assuming a standard CO to H$_2$ conversion factor (see Table \[t:flux\]). For $v_{\rm out}$ between 240 and 850 , we estimate the mass outflow rate in the jet at 9 – 40  yr$^{-1}$. This results in a momentum flux of (14 - 200)$L/c$, which is very high and exceeds values typically seen in cases of AGN feedback [@cicone14; @garcia14]. However, since we use a standard conversion factor, the H$_2$ mass may have been overestimated. If the gas is turbulent, and the individual gas clouds unbound, the conversion factor may have to be adjusted down by a factor of 10 [e.g. @dahmen98]. [*The low-velocity outflow:*]{} The mass and velocity in the low-velocity outflow (Sect. \[s:low\]) is difficult to estimate since there is the possibility of contamination by a background disk and the morphology and velocity structure are complicated. But if we assume that all the CO 3–2 flux above and below the stellar disk belong to the slow outflow, it would constitute 40% of the total CO 3–2 flux detected in NGC1377. For a standard conversion factor (Table \[t:flux\]) this implies $M_{\rm slow}({\rm H}_2)=6 \times 10^7$ . About a third of this is associated with the entrained (alt inclined wind) part of the flow, with projected velocities $\pm 30$ . A generous estimate of $v_{\rm out} \sim$50  over 100 pc implies that 10  yr$^{-1}$ may be lifted off the midplane of NGC1377. This number is highly uncertain. What is powering the molecular jet? ----------------------------------- ### Accretion {#s:accretion} Jets are generally identified with accretion [@blandford98; @konigl00; @hujeirat03; @sbarrato14] and are likely launched by magnetohydrodynamic processes from the accretion disk and/or the central object. The molecular mass is a crucial ingredient in determining the energetics, nature, and evolutionary stage of the molecular jet. We have to resort to a CO to $M$(H$_2$) conversion factor to determine the molecular mass and we have two limiting cases: A [*massive jet*]{} where $M_{\rm j}$(H$_2$) $\sim 10^7$  or a [*light jet*]{} with $M_{\rm j}$(H$_2$) $\sim 10^6$ . Below we discuss possible driving scenarios in relation to a massive or a light jet. [*Entrainment by a radio jet* ]{} Powerful radio jets are launched when an SMBH is growing through hot accretion which is an inefficient accretion at low rates ($<$1% Eddington) [@mcalpine15]. This is also referred to as radio mode AGN feedback. Radio jet production has been found for high Eddington rates where the jet powers do not exceed the bolometric luminosity of their AGNs [@sikora13]. The jet may entrain molecular gas from the disk of the host galaxy (NGC1266 [@alatalo11], IC5063 [@morganti15], M51 [@matsushita07] and NGC1068 [@garcia14]) or the molecular gas may form in the jet itself through rapid post-shock cooling [@morganti15]. Observed molecular gas distributions associated with these jets tend to be patchier than the more coherent molecular structure of NGC1377. A relativistic radio jet ploughing through a thick disk of gas, is likely to heat and ionise it, and thus form a wide cocoon of multi-phase and turbulent gas mixture, as simulated by @wagner11). As shown in @dasyra15, this kind of cocoon is both pushing on the surrounding gas and has forward and scattered flows that may lead to complicated velocity patterns. However, NGC1377 is the most radio-quiet (with respect to the IR luminosity) galaxy found so far and its radio power is very low. (A similar case with faint radio emission associated with molecular jets may be the double, collimated bipolar outflows of the luminous merger NGC3256 [@sakamoto14]). We can use the limit to the 1.4 GHz radio luminosity [@roussel06] and the relation between jet power and 1.4 GHz luminosity [@birzan08] to estimate the energy in a potential radio jet in NGC1377. We find that it amounts to $<$10% of the mechanical energy in the massive molecular jet. A short burst of hot accretion in the nucleus may have led to the formation of a radio jet that then faded very rapidly without re-acceleration of electrons in the jet itself. If the synchrotron life time is $t_s=8 \times 10^8\, B^{-2} \gamma^{-1}$ (where $B$=$B$-field, $\gamma$=Lorentz factor, [@xu00]) a reasonable combination of $B$ and $\gamma$ can result in a jet lifetime of 0.5-1 Myr. Also, it is conceivable that heavily mass-loading a radio jet with dense molecular gas may lead to the quenching of the non-thermal radio emission. In addition, @godfrey16 recently suggested that jet power and radio luminosity may only be weakly correlated for cases where the jet energy is being used to, for example, drive shocks. In the case of the light jet it is feasible that there would be enough radio power to carry the gas out without invoking a fading or underluminous radio jet. [*Cold gas accretion*]{} The jet may be a hydromagnetic disk-wind (or an accretion X-wind) similar to the extremely collimated molecular outflows found in accreting low-mass protostars [e.g. @konigl00; @codella14; @kristensen15]. Its torque could efficiently extract disk angular momentum and gravitational potential energy from the molecular gas. The jet may be powered by accretion onto the central object and/or infalling gas onto the nuclear disk. Assuming that the $5 \times 10^9$  of NGC1377 emerges from a growing SMBH, the accretion rate would be $\sim 10^{-3} -10^{-2}$  yr$^{-1}$ ($L$=${\epsilon \over c^2} {dM \over dt}$ where $\epsilon$=0.1 onto a $10^6$  SMBH). This is 10% of the inferred Eddington luminosity of the SMBH [@aalto12b] and is a relatively high rate, placing it in the quasar mode of accretion [@mcalpine15]. But it may require an Eddington or super-Eddington accretion rate to produce the mass-outflow rate we see (even in the case of the light jet), implying that the level of SMBH accretion has dropped recently. A jet may also be powered by accretion onto a nuclear disk. The wind energy is derived from the gravitational energy released from the disk through gas rotation and a coupled magnetic field. The extracted angular momentum allows cold molecular gas to sink further towards the nucleus. In the case of the massive jet, it is not clear how the current rotational energy of the disk could continue to sustain the outflow since $M_{\rm j}$ would be equal to that in the disk inside its launching region. The binding energy of the jet is similar to the binding energy of the disk and the outflow speed is at least twice that of the rotational velocity (unless the jet is actually launched very close to the nucleus from a Keplerian disk). In the case of the light jet, however, the jet-binding energy would be much less than that of the disk, and there would be enough rotational energy to sustain the outflow. We note that the molecular jet is observed as being lumpy which may be due to internal and external shocks, or the condensations are gas clumps that originate in separate accretion/outflow events. If so, the energetics of the outflow may be different to that of the steady flow scenario we assume above. ### Other scenarios [*Radiation pressure from dust?*]{} Recent work by @ishibashi15 suggests that large momentum flux outflows ($>10 L/c$) can be obtained in radiation pressure driven outflows if radiation trapping is taken into account. However, it is not clear how radiation pressure would result in a jet-like feature since it should give rise to a more wide-angle wind. @wada15 finds that dusty, biconical outflows (opening angles 45$^{\circ}$ – 60$^{\circ}$) can be formed as a result of the radiation feedback from AGNs. It is conceivable that this may be happening in NGC1377, in addition to the jet. [*Starburst winds?*]{} In @aalto12b we discuss the faintness of the star formation tracers (such as optical, NIR and radio emission) of NGC1377. We find that the upper limits on, for example, the 1.4 GHz continuum imply that star formation falls short by at least one order of magnitude in explaining the momentum flux in the molecular outflow detected with the SMA. Is the molecular jet signaling nuclear growth or quenching? {#s:growth} ----------------------------------------------------------- There is large molecular mass in the nucleus of NGC1377, which appears to be linked to a current SMBH accretion at a respectable rate of $\sim$10% Eddington. So the question is: has the molecular jet action quenched the nuclear activity, or did it promote it? [*Light jet:*]{} Both scenarios discussed in Sect. \[s:accretion\] could power the jet and enable SMBH accretion. A light jet has removed only 10% of the disk mass while it may have transported a substantial amount of angular momentum away from the gas in the disk, allowing it to sink closer to the SMBH. The molecular jet offers a way for the cold gas to shed itself of excess angular momentum, which could promote nuclear accretion from a disk. In this scenario, the inflowing gas clouds do not have to have randomly oriented angular momenta to facilitate accretion. There is no evidence that star formation has hindered the gas flow toward the nucleus of NGC1377. Instead there appears to be a mechanism that prevents stars from forming in the high gas surface density nuclear region. Higher resolution studies will hopefully find and resolve the inflowing gas component in NGC1377. [*Massive jet:*]{} Current rates of accretion would be difficult to reconcile with a large mass outflow rate. Nuclear activity in the form of radio luminosity, or other forms of accretion luminosities, are low and are, perhaps, a signature of quenching. The turning off of the nuclear activity would have to have been abrupt since the molecular jet can be traced almost all the way down to the centre. Furthermore, the large masses of molecular gas are surprising since it is not clear why the activity would turn off with 30% of the nuclear fuel still in place. A possible explanation could be that there has been a recent substantial inflow of molecular gas. The discussion above rests on the assumption that most of the FIR emission originates near the SMBH and is the result of the accretion. However, if the FIR emission is, instead, related to the jet-ISM interaction in an extremely dense medium, then the SMBH would be in the hot accretion mode instead, but with its synchrotron quenched by the interaction. If so, we are witnessing the early stages of jet feedback before it has cleared its environment. ### What is the fate of the molecular gas? A precessing jet has the potential to impact and stir up a large volume of ambient gas. In NGC1377 the jet appears to entrain gas in a slow moving outflow, possibly in combination with a wide-angle wind. It is, however, unlikely that the gas in the low-velocity outflow can leave NGC1377 since even an optimistic estimate of its outflow speed is below the bulge escape velocity $v_{\rm esc}$ for NGC1377 [@aalto12b]. Instead, gas may circulate back to the midplane of NGC1377 where it could eventually participate in star formation or another cycle of nuclear growth. The molecular jet appears to be a young structure with a dynamical age $<$1 Myr (Sect. \[s:parameters\]). The estimated $v_{\rm out}$=240-850   is higher than $v_{\rm esc}$ for NGC1377. We find no high-velocity molecular gas outside 200 pc and this would be consistent with the notion that the jet has been caused by a recent accretion event in the nucleus. This would also be consistent with the high nuclear concentration of molecular gas. However, if the molecular gas becomes dissociated at 200 pc, we may simply be observing the inner denser part of an older outflow event and, if the gas is not slowing down, it may escape the galaxy. Yet another alternative is that the jet is rapidly decelerating and its gas is grinding to a halt at its end. The v-shaped optical dust lane is roughly 2-3 times longer than the molecular jet/outflow structure, which implies that the molecular jet is part of a somewhat older structure. Our results demonstrate that outflows/jets even from low-power AGNs can have substantial impact on the evolution of the galaxy, also beyond the innermost pc. We require the high resolution, dynamic range and sensitivity of ALMA to reveal the presence of the molecular jet and to separate it from surrounding emission. Determining the molecular mass in the jet will provide an important clue as to whether the jet is a signature of growth or quenching of the nuclear activity. More detailed studies will also reveal how the jet impacts its environment and entrains gas and dust. Conclusions =========== With high resolution (0.2 $\times$ 0.18) ALMA CO 3–2 observations of the nearby extremely radio-quiet galaxy NGC1377, we have discovered a high velocity, collimated molecular jet with a projected length of $\pm$150 pc. Along the jet axis we find strong velocity reversals where the projected velocity swings from -150  to +150 . A simple model of a molecular jet precessing around an axis close to the plane of the sky can reproduce the observations. The velocity of the outflowing gas is difficult to constrain due to the velocity reversals but we estimate it to be between 240 and 850  and the jet to precess with a period $P$=0.3-1.1 Myr. The jet is launched close to the nucleus inside a radius $r<10$ pc and its molecular mass lies between $2 \times 10^6$ ([*light jet*]{}) and $2 \times 10^7$  ([*massive jet*]{}) depending on which CO to $M$(H$_2$) conversion factor is adopted. There is also a wide-angle structure of CO emission along the minor axis which may be a slower molecular outflow. A substantial fraction of the CO flux is located here and the estimated mass of the minor axis outflow is $6 \times 10^7$ . Its velocity structure is consistent with parts of the wind being entrained by the jet, or that there is a molecular wind inclined by 30$^{\circ}$ with respect to the jet. We discuss potential powering mechanisms for the molecular jet. It may be gas entrained by a very faint radio jet, or it is driven by an accretion disk-wind similar to those found in protostars. It is important to better constrain the jet molecular mass. Given the possibility of either a light or a heavy jet, it is difficult to draw conclusions on whether the jet is quenching the nuclear activity or, instead, is enabling it. The nucleus of NGC1377 harbours intense embedded activity and, if the current IR luminosity is powered by a growing SMBH, it would have an accretion rate of $\sim$10% Eddington. But the origin of the FIR luminosity still needs to be determined. The light jet would only have driven out 10% of the nuclear gas which should not (yet) significantly impact the fueling of the activity. It seems, however, unlikely that a massive jet could have been powered by the current activity and this may be a sign of rapid quenching. In this case, the large mass of H$_2$ in the nucleus is surprising and may be caused by a recent massive influx of gas. A fraction of the outflowing gas may return to the inner region of NGC1377 to fuel further nuclear growth. NGC1377 is the first galaxy with evidence for a precessing, highly collimated molecular jet. The extreme $q$-value for NGC1377, the short apparent time-scale of the molecular jet ($<$1 Myr), and the gas-rich nucleus are all signs consistent with the notion that we are seeing NGC1377 in a transient phase of its evolution. NGC1377 offers a unique opportunity for detailed studies of the processes that feed, promote and quench nuclear activity in galaxies. Further studies are required to determine the age of the molecular jet, driving mechanism, its mass and the role it plays in the growth of the nucleus of NGC1377. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2012.1.00900.S. ALMA is a partnership of ESO (representing its Member States), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. We thank the Nordic ALMA ARC node for excellent support. SA acknowledges support from the Swedish National Science Council grant 621-2011-4143. F.C. acknowledges support from Swedish National Science Council grant 637-2013-7261 KS was supported by grant MOST 102-2119-M-001-011-MY3 SGB thanks support from Spanish grant AYA2012-32295. JSG thanks the Chalmers University for the appointment of [*Jubileumsprofessor*]{} for 2015. SA thanks S. König, [Å]{}. Hjalmarson, R. Haas and S. Bourke for discussions of the manuscript. Spectra {#s:A1} ======= ![\[f:spec\] Spectra of the nuclear emission in NGC1377. Dashed vertical line indicates $v$=1740 . Top panels: CO 3-2 in high- (left) and low- (right) resolution spectral mode. The dashed red box of the right panel indicates the zoomed-in region in the next panel. Centre: Zoomed-in spectrum showing detections of H$^{13}$CN $J=4-3$ and either HC$_3$N $J=38-37$ $\nu_4=1$, $\nu_7=1$ or SO$_2$ 16(4,12)-16(3,13). Bottom: Panels showing detections of HCO$^+$ $J=4-3$, HCN $J=4-3$ $\nu_2=1f$ (left) and a line that we tentatively identify as HC$^{15}$N $J=4-3$. All spectra apart from in panels 2 and 3 have Gaussian smoothed with FWHM of two channels. In panels 2 and 3 there has been no smoothing but frequency resolution is reduced by a factor of 3 (these data stem from another spectral window than that presented in the first panel). ](n1377_spec.pdf) In Fig. \[f:spec\] we present spectra towards the nucleus of NGC1377. Apart from CO 3–2 we detect HCO$^+$, H$^{13}$CN $J=4-3$, vibrationally excited HCN $J=4-3$ $\nu_2=1f$ ($T = E_{\rm l} / k$=1050 K). We detect a line at $\nu$=345.5 GHz which is either vibrationally excited HC$_3$N $J=38-37$ $\nu_4=1$, $\nu_7=1f$ ($T = E_{\rm l} / k$=1891 K) or it is SO$_2$ 6(4,12)-16(3,13) ($T = E_{\rm l} / k$=148 K). In addition we detect a line at redshifted frequency $\nu$=342.26 GHz which we tentatively identify as HC$^{15}$N $J=4-3$. In this case, the line would peak at $v$=1670  and thus be blueshifted with respect to the other lines by 60 . This type of shift could be caused by excitation, optical depth and/or abundance gradients and should be investigated in further studies since it may hold another clue to the nature of the nuclear emission of NGC1377. Other potential explanations to the high velocity CO 3–2 emission {#s:A2} ================================================================= [*An orbiting object and/or two jets?*]{} Velocity variations in a PV diagram may also be caused by a jet launched from an orbiting object. In this case the velocity reversals can be dominated by the orbital motion in a near edge-on plane of rotation. A possibility would be a jet launched from one of two orbiting SMBHs. @masciadri02 have discussed the similarities and differences between orbiting and precessing jets. However, without jet precession the velocity pattern will not fit the structure we see in the observed PV diagram - unless the jet symmetry axis is misaligned with respect to the axis of the plane of rotation. Both SMBHs could have jets and a combination of orientation and length of the jets could be put together to reproduce the observed PV diagram. However, this seems unlikely compared to the relatively simple scenario of one single precessing jet. [*Jet shocks?*]{} A pulsed jet will have a sawtooth like pattern in its PV diagram along the jet major axis [e.g. @santiago09]. This pattern is caused by axial compression and lateral ejection of material inside the internal working surface. @santiago09 point out that the effects are localized within the jet so it is not obvious how it would give rise to the large scale shifts and gradients seen here. However, internal and external shocks would be important for the jet of NGC1377 and thus influence its velocity structure. [*A bicone projection?*]{} Is it possible that the velocity reversals we observe in the PV diagram (A) (Fig \[f:pv\]) is a projection effect, instead of the emission from a collimated jet? A tilted wide-angle biconical outflow may result in projected, foreshortened blueshifted emission from the lower end of the cone, and redshifted (more elongated) emission from the back side of the cone (and vice versa on the underside of the cone). PV diagrams of these scenarios are, for example, presented by (e.g.) @cabrit86, @das05 and @storchi10. In Fig. \[f:cone\] we show a sketch of a cone (displayed from two angles) with the northern part tilted towards us. Schematic PV diagrams along and transverse to the projected cone symmetry axis are shown in Fig. \[f:cone\_pv\]. (Note that it is a very simple cone model with uniform density. The PV diagram would be much more complicated for a non-uniform cone, or multiple cones.) The resulting PV diagram along the main axis has two scenarios: one with constant outflow velocity, and one where the gas is first accelerating and then decelerating. It is likely possible to find an outflow velocity scenario that can at least produce a reasonable fit to the PV along the jet axis ((A) in Fig \[f:pv\]), if the northern cone is tilted towards us and the opening angle is large, $>$$45^{\circ}$. This orientation of a wide angle cone is, however, inconsistent with the optical dust structure found by @roussel06 (their Fig. 1) and @heisler94 (their Fig.4c), which would require the northern cone to be tilted away from us. Another important argument against the cone-projection model is the shape of the observed PV diagrams transverse to the jet symmetry axis (panels C- F in Fig. \[f:pv\]). For a cone, the PV diagrams perpendicular to the major axis will always be ellipses (see schematic PV diagrams in Fig. \[f:cone\_pv\]). And, when the top cone is tilted towards us, there should be a broad (in space) blueshifted emission component to the north. The observed PV diagrams transverse to the jet axis (Fig. \[f:pv\]) do, however, not show this structure. In Fig. \[f:transverse\] we show the PV diagrams D and E from Fig. \[f:pv\] with the expected PV diagram of a tilted cone indicated by dashed lines. Instead of tracing out the curved front ellipse, the maximum velocity is structured in a spatially unresolved tounge-like shape of broad emission. ![\[f:cone\] Sketch of a hollow cone with opening angle 60$^{\circ}$ and tilt angle 40$^{\circ}$ towards the observer. ](cone_cartoon_view.pdf) We note that this exercise is not an attempt to model the minor-axis structure of the low-velocity gas as discussed in Sect. \[s:low\]. The low-velocity gas may (at least partially) originate in a cone-like slow outflow, which we suggest is interacting with the molecular jet. An attempt to link it to the optical dust structure mentioned above would require its southern part to be at least slightly directed towards us. ![\[f:cone\_pv\] Schematic PV diagrams of a cone with its top (northern) part tilted towards us. Top panel: Cut perpendicular to the cone axis showing the elliptical PV diagram through the cone. Bottom panel: Cut along the cone major axis. We show two simplified cases: The straight solid lines show the PV diagram of outflowing gas along the cone walls of constant velocity. The dashed lines show the generic PV diagram along the axis of a cone where the gas is first linearly accelerating and then decelerating to zero velocity. ](cone_cartoon_pv.pdf) ![\[f:transverse\] PV diagram, showing gas velocities in a slit across the jet axis at $\pm$0.3. Contour levels are 3.1$\times$(1,2,4,8,16,32) mJy beam$^{-1}$. The colour scale ranges from -11 to 156 mJy beam$^{-1}$. The dashed semi-ellipticals show the PV diagram expected from a projected wide-angle cone It is clear that this model does not fit the data. ](n1377_transverse.pdf) [^1]: Based on observations carried out with the ALMA Interferometer. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. [^2]: We use the formalism by @wu09 to describe the line-of-sight velocity change along the symmetry axis of the jet: $v_{\rm LOS}$=$v_{\rm LSR} \pm v_{\rm out}[\cos \theta \sin i + \sin \theta \cos i \cos (2\pi l/\lambda +\phi_0)]$, where $v_{\rm LOS}$ is the observed line-of-sight velocity, $v_{\rm out}$ is the outflow velocity in the jet, $i$ is the inclination of the jet symmetry axis to the plane of the sky, $\theta$=precession angle, $l$=distance from the nucleus, $\lambda$=precession length scale, $\phi_0$=initial phase at the nucleus.

--- abstract: 'TIMMI2 diffraction–limited mid-infrared images of a multipolar proto-planetary nebula IRAS 16594$-$4656 and a young \[WC\] elliptical planetary nebula IRAS 07027$-$7934 are presented. Their dust shells are for the first time resolved (only marginally in the case of IRAS 07027$-$7934) by applying the Lucy-Richardson deconvolution algorithm to the data, taken under exceptionally good seeing conditions ($\leq$0.5). IRAS 16594$-$4656 exhibits a two-peaked morphology at 8.6, 11.5 and 11.7 $\mu$m which is mainly attributed to emission from PAHs. Our observations suggest that the central star is surrounded by a toroidal structure observed edge-on with a radius of 0.4$\arcsec$ ($\sim$640 AU at an assumed distance of 1.6 kpc) with its polar axis at P.A.$\sim$80, coincident with the orientation defined by only one of the bipolar outflows identified in the HST optical images. We suggest that the material expelled from the central source is currently being collimated in this direction and that the multiple outflow formation has not been coeval. IRAS 07027$-$7934 shows a bright, marginally extended emission (FWHM=0.3$\arcsec$) in the mid-infrared with a slightly elongated shape along the N-S direction, consistent with the morphology detected by HST in the near-infrared. The mid-infrared emission is interpreted as the result of the combined contribution of small, highly ionized PAHs and relatively hot dust continuum. We propose that IRAS 07027$-$7934 may have recently experienced a thermal pulse (likely at the end of the AGB) which has produced a radical change in the chemistry of its central star.' author: - 'D. A. García-Hernández, A. Manchado,P. García-Lario, A. Benítez Cañete, J. A. Acosta-Pulido, and A. M. Pérez García' title: | Revealing the mid-infrared emission structure of\ IRAS 16594$-$4656 and IRAS 07027$-$7934[^1] --- Introduction ============ Planetary nebulae (PNe) are the result of the evolution of low- to intermediate-mass stars (0.8–8 M$_{\odot}$). These stars experience a phase of extreme mass loss during the previous asymptotic giant branch (AGB) that causes the ejection of the stellar envelope. When this mass loss ceases the AGB phase ends and the star evolves into a short-lived evolutionary stage called the ‘post-AGB’ or ‘proto-PN’ (PPN) phase just before the star becomes a PN. At present, the formation of axisymmetric structures in PNe (ranging from elliptical to bipolar) is believed to be completed by the end of the AGB phase (Balick & Frank 2002; Van Winckel 2003). But unlike for PNe, the study of PPNe is more difficult since their central stars (CSs) are usually too cool to photoionize the gas. Therefore, we cannot study the formation of axisymmetric morphologies in PPNe by mapping the ionized gas. We must use alternative techniques based on the analysis of: (i) the light scattered by the surrounding dust at optical wavelenghts; (ii) the neutral molecular gas in the envelope in the near-infrared (H$_2$), submilimeter (e.g. CO) or radio domain (e.g OH, SiO, H$_{2}$O, CO); and (iii) the dust emission emerging at mid- to far-infrared wavelengths. PPNe generally show a double-peaked spectral energy distribution (SED) (Kwok 1993; Volk & Kwok 1989; van der Veen, Habing & Geballe 1989) with the photospheric emission coming from the central star dominating in the optical range and a strong infrared excess indicating the presence of a cool detached envelope (T$_{d}$$\sim$150–300 K). This strong infrared excess is produced by the thermal emission of the dust present in their circumstellar shells previously expelled during the AGB phase. The processes that lead to a wide variety of different morphologies observed in PNe (e.g. Manchado et al. 2000) are, however, still unknown. Several mechanisms have been proposed: the interaction of stellar winds (e.g. Mellema 1993), binary systems as central stars (e.g. Bond & Livio 1990; Morris 1987), non-radial pulsations (e.g. Soker & Harpar 1992) or the influence of magnetic fields (e.g. Pascoli 1992; Soker & Harpar 1992; García-Segura et al. 1999). To establish which one(s) of the above is the dominant process, it is essential to study these morphologies as early as possible after the departure from the spherical symmetry takes place, that is, in the PPN phase. Only recently, with the help of high spatial resolution observations it has been possible to study the intrinsic axisymmetric nature of the dust shells around a few compact PPNe at subarcsec level (Meixner et al. 1997; Meixner et al. 1999; Ueta et al. 2001). In this paper, we present for the first time mid-infrared images (8–13 $\mu$m) at subarcsec level of a PPN IRAS 16594$-$4656 (hereafter I16594) and of a very young \[WC\] PN IRAS 07027$-$7934 (hereafter I07027) with the aim of mapping the dust emission originated in the innermost regions of their circumstellar dust shells. The observations made in the mid-infrared are presented in Sect. 2 while the data reduction process is described in Sect. 3. We show the results obtained in Sect. 4, which are later discussed in Sect. 5. The main conclusions derived from our analysis are given in Sect. 6. Mid-infrared Observations ========================= The observations were carried out on 2001 October 9 and 10, using the imaging mode of TIMMI2 (Reimann et al. 2000; Käufl et al. 2003) attached to the ESO 3.6m telescope (La Silla, Chile). TIMMI2 has an array of 320 $\times$ 240 pixels with a pixel scale of 0.2$\arcsec$ $\times$ 0.2$\arcsec$ resulting in a field of view of 64$\arcsec$ $\times$ 48$\arcsec$. The observational conditions were very good (photometric and with a stable seeing of around 0.5$\arcsec$) and, thus, we could obtain mid-infrared images (at 8.6 $\mu$m \[N1-filter\], 11.5 $\mu$m \[N11.9-filter\] and 11.7 $\mu$m \[SiC-filter\]) of I16594 and I07027 at the diffraction limit of the telescope. The standard nodding/chopping observational technique was used in order to cancel the thermal emission from the atmosphere and from the telescope. An on-chip nodding/chopping throw of 15" along the north-south direction was selected. Due to the short integration times required to avoid saturation in the mid-infrared, each image is a combination of a large number of individual sub-images $\sim$50–100, each one with an integration time between 18 and 40 ms and a chopping frequency around 6 Hz. Total on-source integration times were typically of $\sim$2 minutes. The mid-infrared photometric standard stars HD 29291, HD 156277, HD 196171 and HD 6805 (Doublier et al. 2004) were also observed at different air masses every night to determine the photometric flux conversion from ADUs (Analog to Digital Units) to Jy and to measure the telescope point spread function (PSF). Data Reduction ============== The data reduction process includes bad pixel correction and the combination of all the images into one single image per filter using standard tasks in IRAF[^2]. The flux calibration was made using the conversion factors derived from the observation of standard stars at different air masses. The variation of these conversion factors with air mass was slightly different during the two nights of observation due to the different atmospheric conditions. On October 9 this variation was very small ($<$10%), so a single averaged conversion factor was used for all the observations performed during the night. However, on October 10 we found larger variations ($\sim$30%). Thus, for I07027 we took the conversion factors derived from the observations of the standard star HD 156277, observed closer in time and at a similar air mass. For each filter the size and morphology of the target stars in our programme was compared with a mean PSF derived from the observation of the standard stars used for the flux calibration. The object name, observing date, filters, central wavelength and width of the different filters used, total on-source integration time, object size, PSF size, integrated and peak fluxes, are listed in Table 1. From the internal consistency of the measurements made on the standard stars we estimate that the photometric uncertainty of our observations is of the order of $\sim$10%. The observed PSFs are dominated by diffraction effects. Thus, the Lucy-Richardson deconvolution algorithm as implemented in IRAF (task [lucy]{}) was used in order to recover the actual emission structure of the targets in each filter and remove the effects induced by the telescope PSF, which was found to be very stable. In order to study the goodness of the deconvolution process the standard stars observed for flux calibration purposes were also deconvolved with the same PSF used for the target stars. This exercise is useful to confirm that the deconvolution process does not introduce undesired artifacts. We found that in all cases the deconvolved images of the standard stars spread over only one or two pixels, showing a quasi-point-like brightness distribution. As an example, one of the deconvolved standard stars is shown together with its corresponding raw image in Figure 1. Results ======= Mid-infrared Morphology of IRAS 16594$-$4656 -------------------------------------------- Figure 2 shows the raw images of I16594 taken with TIMMI2 (left panel) together with the images obtained after applying the Lucy-Richardson deconvolution algorithm (right panel) above mentioned. The PSFs used for the deconvolution are also shown for comparison. I16594 shows already in the raw images an extended elongated morphology surrounding a complex inner core emission (with a FWHM=1.5" or 3 times the PSF FWHM) which is resolved in more detail after deconvolution. The source center has been determined by averaging the central coordinates of the elliptical isophotes within 20–40% of the peak intensity (this way we avoid any contamination from the core structure). The images displayed in Figure 2 have been centered at this position. The deconvolved 8.6, 11.5 and 11.7 $\mu$m images are shown in the right panel of Figure 2. The overall elliptical shape of the nebula is clear in all three filters with its major axis oriented along the east-west direction (P.A.$\sim$80) and extends out to at least 3.5$\arcsec$ $\times$ 2.1$\arcsec$ at 5% of the peak intensity (10-$\sigma$ level above the sky background) at 8.6 $\mu$m. In addition, a conspicuous double-peaked morphology in the innermost region of the nebulosity is also recovered, suggesting the presence of an equatorial density enhancement (e.g. a dust torus). The two detected peaks are oriented approximately along the north-south direction (P.A.$\sim$$-$10) perpendicular to the axis of symmetry defined by the outer elliptical emission. The measured separation between the two peaks is always 0.8$\arcsec$, independent of the filter considered (see Figure 3). In addition, the north-peak (P.A.$\sim$$-$10) is about a factor 2 brighter than the south peak (see Figure 3). This finding is not unique to I16594. Indeed, Ueta et al.(2001) found a similar asymmetric profile in IRAS 22272$+$5435. The origin of this asymmetric appearance of the dust torus is still unclear. Ueta et al. (2001) argued that this can be attributed to asymmetric mass loss and/or an inhomogeneity in the dust distribution. We are confident that the deconvolved structure is real because a very similar emission structure is observed in all three filters. Note that a negligible contribution to the observed flux at 10 $\mu$m is expected from the central star of I16594 if this has a B7 spectral type as suggested by Van de Steene, Wood & van Hoof (2000). Hrivnak, Kwok & Su (1999) found that the central star contributes only 3% to the total flux detected in the mid-infrared. We can therefore safely assume that we are just observing the emission structure of the dust in the shell alone. The mid-infrared morphology seen in the deconvolved images is thus interpreted as the evidence of the presence of a dusty toroidal structure with a 0.4$\arcsec$ radius size seen nearly edge-on. Recently, Ueta et al. (2005) based on polarization data also suggest an orientation of the dust torus close to edge-on and they indicate that an inclination angle of roughly 75 ${\hbox{$^{\circ}$}}$ with respect to the line of sight is derived from a 2-D dust emission model. This adds I16594 to the short list of PPN where a similar dust torus has been resolved at subarcsec scale. Mid-infrared Morphology of IRAS 07027$-$7934 -------------------------------------------- The mid-infrared morphology of I07027 is clearly less complex than the one observed in I16594 as deduced already from the raw images shown in Figure 4 (left panel). In this case, only a slightly extended and asymmetric source is detected in the two images available, which correspond to the filters N1 and N11.9 (centered at 8.6 and 11.5 $\mu$m, respectively). This can just be due to the larger distance to this source (see Section 5.2.2). Note, that the low level extension to the west of the peak emission seen in both filters seems to be a PSF effect, as it is also observed in the standard stars, although not so prominent. The deconvolved images of I07027 in the 8.6 and 11.5 $\mu$m filters are shown in Figure 4 (right panel). After the deconvolution, a very bright and slightly elongated, marginally extended (FWHM=0.3$\arcsec$) emission core is recovered in both filters, oriented along the north-south direction. A similar orientation is observed in recent HST-NICMOS images of I07027 taken in the near-infrared (see Section 5.2). Unfortunately, our TIMMI2 data cannot confirm precisely whether the mid-IR emission peak is exactly coincident with the near-IR emission peak. We propose that the emission core which is detected in the mid-infrared must be coincident with the location of the central star, as the peak emission is observed exactly at the centre of the slight extended emission. Discussion ========== IRAS 16594$-$4656 ----------------- ### IRAS 16594$-$4656 in the literature I16594 (=GLMP 507) was first identified as a PPN candidate on the basis of its IRAS colors by Volk & Kwok (1989) and van der Veen, Habing & Geballe (1989). It shows a double-peaked spectral energy distribution dominated by a strong mid- to far- infrared dust emission component which is much brighter than the peak in the near-infrared (Van de Steene, van Hoof & Wood 2000). The first indication of the C-rich chemistry of I16594 was the detection of CO molecular emission in its envelope (with V$_{exp}$$\sim$16 km s$^{-1}$) by Loup et al. (1990), and the non-detection of OH maser emission (te Lintel Hekkert et al. 1991). More recently, García-Lario et al. (1999) studied the ISO spectrum of this source and confirmed this classification based on the detection of the characteristic IR emission features generally attributed to PAHs (at 3.3, 6.2, 7.7, 8.6 and 11.3 $\mu$m) together with relatively strong features at 12.6 and 13.4 $\mu$m which indicates a high degree of hydrogenation in these PAHs. The ISO spectrum also reveals the presence of strong 21, 26 and 30 $\mu$m dust emission features (see Figure 5), adding I16594 to a short list of known PPNe displaying this set of still unidentified features. The optical spectrum of I16594 shows only the hydrogen Balmer emission lines over an extremely reddened stellar continuum (E$_{B-V}$=1.8, Van de Steene & van Hoof 2003) consistent with a B7 spectral type if dereddened. HST optical images show the presence of a bright central star surrounded by a multiple-axis bipolar nebulosity (seen in scattered light) with a complex morphology at some intermediate viewing angle (see Figure 6). The size of this optical nebulosity is 6.3$\arcsec$ $\times$ 3.3$\arcsec$ at 3$\sigma_{sky}$ level (Hrivnak, Kwok & Su 1999). In the literature there are several indications of the presence of a circumstellar disc or a torus (an equatorial density enhancement) around I16594. The highly collimated structure seen in the HST optical images and the non-detected radio-continuum emission ($<$10 $mJy$) by Van de Steene & Pottasch (1993) suggest that the emission lines observed in the optical spectrum are the result of shock excitation produced by a fast bipolar wind from the central source in interaction with the slow AGB wind. In agreement with this hypothesis García-Hernández et al. (2002) reported the detection of H$_2$ shock-excited emission in I16594, later confirmed by Van de Steene & van Hoof (2003) through a more detailed analysis of the H$_2$ spectrum. They postulate that the H$_2$ emission originates mainly where the stellar wind is funnelled through a circumstellar disc or torus. More recently, Hrivnak, Kelly & Su (2004) presented HST-NICMOS near-infrared images of I16594 which show that this emission is originated in regions where shocks must be taking place. Polarization measurements originally taken by Su et al. (2003) and later analyzed by Ueta et al. (2005), who presented a PSF subtracted map of the polarized light, suggest the presence of an equatorial enhancement in I16594 as well. However, Van de Steene, van Hoof & Wood (2000) failed to detect any extended emission in their N-band TIMMI images of I16594 in a previous attempt to search for mid-infrared emission coming from this torus, but they observed the source with a lower spatial resolution (pixel scale of 0.66$\arcsec$), and under poor weather conditions. ### A Dusty Toroidal Structure around IRAS 16594$-$4656 There exists more than a dozen PPN shells that have been resolved in the mid-infrared so far. However, only a few of them show some structure at mid-infrared wavelengths. Meixner et al. (1999) found two different classes of mid-infrared morphologies. They distinguish those sources with a mid-infrared core/elliptical structure from those with a toroidal one and they argue that this morphological dichotomy is due to a difference in optical depth. In their sample there are only 4 out of 6 toroidal PPN/PNe in which the central dust torus is well resolved in two emission peaks. This work adds I16594 to this short list. In Table 2 we list the few known toroidal PPN/PNe sample together with some of their main observational characteristics, such as the spectral type of the central star, C/O ratio, evolutionary classification, optical morphology, and the list of mid-infrared dust emission features detected. An inspection of Table 2 clearly indicates that I16594 is now a toroidal-PPN with the earliest spectral type known. The other PPNe with mid-infrared toroidal structures have all F-G spectral types, while IRAS 21282$+$5050 is already a young PN with an O9-type central star. It seems that many of these sources show a C-rich chemistry (indicated by the presence of PAH emission features) but the number of objects considered is still small and the statistics are very poor. It is interesting to remark the fact that all mid-infrared toroidal-PPN/PNe have bipolar/multipolar optical morphologies where the central star is clearly seen. In contrast, the central star is rarely seen in the mid-infrared core/elliptical class sources described by Meixner et al. (1999) and almost all of them display bipolar morphologies in the optical. In addition, the mid-infrared core/elliptical sources are typically O-rich and show deep silicate absorption features at 9.8 $\mu$m in their mid-infrared spectra, indicating that they may be optically thick at mid-infrared wavelengths (Meixner et al. 1999). The different optical morphology (with or without a visible central star) and the apparent differences in dust properties (optical thickness in the mid-infrared) suggest that mid-infrared toroidal PPNe might be surrounded by a dust torus which is optically thin at mid-infrared wavelengths and, thus, not able to obscure the central star in the optical domain, while, in contrast, mid-infrared core/elliptical PPNe would be surrounded by an optically thick dust torus/disk which would completely obscure the central star in the optical (Meixner et al. 1999, 2002; Ueta et al. 2000, 2003). Our deconvolved mid-infrared images of I16594, of much better quality than those previously reported by Van de Steene, van Hoof & Wood (2000), reveal directly for the first time the presence of an optically thin dusty toroidal structure with a radius of 0.4$\arcsec$. Unfortunately, the distance determinations to I16594 are quite uncertain and, thus, a direct transformation of this observed size into an absolute physical value is not straightforward. Estimations based on the observed reddening are hampered by the fact that the overall extinction is always a combination of interstellar and circumstellar reddening. And in the case of I16594 there seems to be a considerable contribution from the circumstellar component. Calculations made by Van de Steene & van Hoof (2003) based on the intrinsic colors expected for a B7 central star in the optical and in the near-infrared suggest a total extinction of A$_{V}$=7.5 mag with R$_{V}$=4.2. With this value for the extinction and the flux calibration from the Kurucz model a distance of (2.2$\pm$0.4) L$_4$$^{1/2}$ kpc is obtained, where L$_4$ is in units of 10$^4$ L$_{\odot}$. We have tried to derive our own distance estimate to I16594 based on the analysis of the overall SED, from the optical to the far-infrared. For this we put together the IRAS fluxes at 12, 25, 60 and 100 $\mu$m, the near-infrared JHKL magnitudes from @gl97 and the BVRI magnitudes from @hr99. The observed BVRI and JHKL fluxes were corrected for extinction using the total extinction of A$_{V}$=7.5 mag determined by @vv03 and the extinction law from @ca89. Then, a distance-dependent luminosity was obtained by integrating the observed flux at all wavelengths and extrapolating the IRAS fluxes to the infinite following Myers et al. (1987). This way, a distance of 2.1 L$_4$$^{1/2}$ kpc is obtained, in very good agreement with the previous determination by Van de Steene & van Hoof (2003). Assuming a luminosity of 6,000 L$_{\odot}$, which is the theoretical luminosity expected for a post-AGB star with a core mass of 0.60 M$_{\odot}$ (Schönberner 1987), a distance of 1.6 kpc to I16594 is derived, value that will be adopted in the following discussion. The value of 0.60 M$_{\odot}$ is chosen for the mass of the core because the mass distribution of planetary nebulae central stars is strongly peaked at this value (Stasynska, Gorny & Tylenda 1997). At a distance of 1.6 kpc, the extended emission detected in our deconvolved mid-infrared images of I16594 would correspond to a dusty toroidal structure with a radius of $\sim$640 AU. Assuming that the CO emission detected towards I16594 is a good tracer of the dusty torus structure and considering the CO expansion velocity of 16 km s$^{-1}$ measured by Loup et al. (1990), a dynamical age of the dusty torus structure of $\sim$190 yr can be estimated. This dynamical age is quite consistent with a source which has left the AGB very recently. ### Dust Temperature The radiation transfer equation in the interior of a dust cloud adopts a simple form when the energy source is a single exciting star under the optically thin approximation and assuming thermal equilibrium. Under these conditions, the mean color temperature of the dust can be obtained from a simple equation (see e.g. Evans 1980) which relates the measured fluxes S$_{\nu1,2}$ at two different wavelenghts $\lambda_{1,2}$ and the dust emissivity index (which depends on the assumed dust model), assuming a homogeneous dust distribution throughout the cloud. It should be noted that the assumption of the central star as the only source of energy for dust heating is appropiate for I16594 because direct stellar ratiation is the dominant heating source for the circumstellar dust grains in the shell. In particular, we have investigated whether dust heating due to line emission could also contribute to the observed emission and found that this effect is negligible, as the value of the Infrared Excess(IRE) for I16594, defined as the ratio between the observed total far infrared flux and the expected far infrared flux due to absorption by dust of Ly $\alpha$ photons (see e.g. Zijlstra et al.1989), is $\sim$ 400. In principle, one could construct color temperature maps for the dust from the analysis of the 8.6 and 11.5 $\mu$m TIMMI2 images of any given source, as long as these bands are representative of the dust continuum emission. Unfortunately, in the case of I16594, the 8.6 and 11.5 $\mu$m emission is strongly affected by the PAH emission features which are clearly visible in the ISO spectrum (see Figure 5). Thus, the temperature values derived this way are not expected to represent realistic estimations of any physical temperature in the shell. The same problem is found if we try to derive the dust temperature from the IRAS photometry at 12 and 25 $\mu$m, since both filters are also strongly affected by the presence of dust features, as ISO spectroscopy reveals. This is confirmed by the strongly different mean dust temperatures T$_{8.6/11.5}$ of 227 K and T$_{12/25}$ of 129 K, derived (assuming a dust emissivity index of 1) from our mid-infrared data and from the IRAS photometry at 12 and 25 $\mu$m, respectively. A more reliable dust temperature can be directly estimated from the observed size of 0.4$\arcsec$ for the inner radius of the dusty torus assuming that this is the equilibrium radius for the bulk of the dust emitting at mid-infrared wavelengths. Based on the formula worked out by Scoville & Kwan (1976), this can be calculated using the equation: $$T_{d} = 1.64f^{-1/5}r_{eq}^{-2/5}L_{*}^{1/5}$$ where *T$_{d}$* is the dust temperature in K, *f* is the emissivity of the dust, r$_{eq}$ is the equilibrium radius in $pc$, and *L$_{*}$* is the source luminosity in L$_{\odot}$. We decided to use for our calculations a basic dust model composed by hydrogenated amorphous carbon grains (HACs; type BE of Colangeli et al. 1995), whose emissivity index is $\sim$1. The selection of this dust model to reproduce the dust continuum emission observed in I16594 is justified by the presence of highly hydrogenated PAHs in the ISO spectrum (García-Lario et al. 1999). In order to calculate the dust emissivity, the mass extinction coefficient value for hydrogenated amorphous carbon was taken from appendix A of Colangeli et al. (1995) at the central wavelength between the two filters. Then, a typical grain density of 1.81 g cm$^{-3}$ (Koike, Hasegawa & Manabe 1980) was assumed. Finally, this quantity was multiplied by a dust grain size in the range 0.001-0.1 $\mu$m obtaining a dust emissivity *f*. Note that a dust grain size of 0.01 $\mu$m is a reasonable mid-range size for circumstellar carbon dust (e.g. Jura, Balm & Kahane 1995). The dust temperature in thermal equilibrium at 0.4$\arcsec$ (or $\sim$640 AU at the assumed distance of 1.6 kpc) can then be derived using the dust emissivity *f* and the assumed luminosity of the source (6,000 L$_{\odot}$). This way, a dust temperature T$_{d}$=237 K is found for a mid-range dust grain size of 0.01 $\mu$m. A smaller or a larger dust grain size of 0.001 and 0.1 $\mu$m would imply dust temperatures of 376 and 150 K, respectively. We are conscious that the assumption of spherical geometry may not be valid for the circumstellar envelope of I16594 where the emission is clearly asymmetric and the geometry assumes a toroidal shape, according to our mid-IR images. Note that adopting a more complex, axysimmetric geometry would esentially translate into grains being more effectively heated in the biconical opening angle defined by the dust torus because of the different local optical depth. In spite of this, our simple model can be used irrespective of the shell geometry when applied to dust grains at the inner radius of the shell. The use of more detailed axysimmetric, multiple grain size models is beyond the scope of this paper. In addition, the presence of a dust torus close to the star with respect the spherical case mainly influences the optical and near-infrared radiation. A large effect on the mid- to far-IR emission is not expected (see e.g. Ueta & Meixner 2003). In this sense, spatially unresolved SEDs do not provide any spatial information necessary to constrain the geometry and inclination angle of the PPN dusty shells. ### Comparison with ISO data Another dust temperature estimate can be derived by fitting one (or more) blackbodies to the available ISO data by considering fluxes representative of the underlying continuum at carefully selected wavelengths not affected by any dust feature. We did this by selecting the ISO fluxes at 6.0, 9.4, 14.3, 18.0, and 45.0 $\mu$m plus the IRAS fluxes at 60 and 100 $\mu$m. The best fit to the overall SED is obtained with a combination of two blackbodies (with an emissivity index of 1) with temperatures of 273 K and 130 K, respectively, as we can see in Figure 5, where we display the SED of I16594 from 1 to 100 microns together with the two blackbodies. We find that actually the warm component (at 273 K) dominates in the wavelength range of the N1-filter (at 8.6 $\mu$m) while the cool component (at 130 K) dominates in the N11.9-filter range (at 11.5 $\mu$m). Overimposed on the continuum emission, strong PAH features are also clearly contributing to the observed emission. Note that the PAH emission features observed at the ISO short wavelengths as well as the dust features at 21, 26 and 30 $\mu$m, the latter extending from 20 to 40 $\mu$m, are intentionally excluded from the fitting because they are not representative of the dust continuum emission. In particular, the 30 $\mu$m feature overlaps with the 26 $\mu$m feature, and even with the 21 $\mu$m feature and the continuum level is well below the flux detected by ISO at 22–24 microns (see e.g. the analysis of the similar sources IRAS 20000$+$3239 and HD 56126 shown in Fig. 10 of Hony, Waters & Tielens 2002). At present, most of these features remain still unidentified, although several possible carriers have been proposed in the literature, e.g. fullerenes, TiC, SiC for the 21 $\mu$m feature (García-Lario et al. 1999; von Helden et al. 2000; Speck & Hofmeister 2003); MgS for the broad 30 $\mu$m feature (Hony, Waters, & Tielens 2002 and references therein). Using the above two dust temperatures we can estimate the size of the dust grains which are expected to emit in equilibrium at the distance of 0.4$\arcsec$ from the central star which is derived from our mid-IR images. This is found to correspond to small dust grains with a size of 0.005 $\mu$m in the case of the warm dust component emitting at 273 K which dominates at 8.6 $\mu$m (comparable to the typical size of small PAH clusters). Dust grains with the same size emitting at 130 K (note that this cold dust emission dominates at 11.5 and 11.7 $\mu$m) would need to be located at $\sim$4075 AU from the central star, which corresponds to a projected $\sim$2.5$\arcsec$ on the sky at the assumed distance. This is considerably beyond the observed extension of the inner shell in the mid-infrared. A surface brightness of $\sim$350 mJy/pixel can be roughly estimated for the continuum emission expected under these conditions, well above ($\sim$50-$\sigma$) our detection limit. The fact that we do not detect this extended emission in our images suggests that the angular size of the region giving rise to the bulk of the hot dust emission is much smaller than that of the region emitting at 130 K. Note that, assuming e.g. that the cold dust emission extends homogeneously over the larger aperture used by ISO, we find that the surface brightness would be just below the 3-$\sigma$ level of the sky background and, as such, undetectable in our TIMMI2 images. The similar extension and morphology of the mid-infrared emission observed at 8.6, 11.5 and 11.7 $\mu$m suggests that the contribution from PAHs observed in the ISO spectrum must be dominant in our TIMMI2 images, and that these PAHs may be well mixed with the small, hot dust grains responsible for the underlying continuum, being mainly distributed along the torus. Considering the information available and the limited spectral coverage, an alternative scenario which cannot be ruled out completely might be that both small, hot dust grains and large, cold dust grains could be co-located in the dust torus. This would be possible if a larger grain size ($\geq$0.1 $\mu$m) is assumed for the cold dust. Note that, in a non-spherical (torus) distribution of the dust, the shielding can become very efficient and the density very high in the outer equatorial regions, where the dust can grow and get colder, protected both from the radiation from central star and from the ISM UV radiation field. This would explain the larger size of the cold dust grains in the torus. In contrast, small, hot dust grains are expected to dominate in the inner boundary of the torus. Unfortunately, the spatial resolution of our images is not enough to resolve the grain size distribution within the torus. ### Collimated outflows in IRAS 16594$-$4656 I16594 has also been observed by the HST in the optical, through the broad F606W continuum filter with the Wide Field Planetary Camera (WFPC2) under proposal 6565 (P.I.: Sun Kwok), and in the near-infrared, through the narrow F212N (H$_2$) and F215N (H$_2$-continuum) filters with the Near Infrared Camera and Multi Object Spectrometer (NICMOS) under proposal 9366 (P.I.: Bruce Hrivnak). In the optical, I16594 shows a flower-shaped morphology where several petals (or bipolar lobes) can be identified at the opposite sides of the central star with different orientations, which has been suggested to be a result of episodic mass ejection (Hrivnak, Kwok, & Su 1999). Similar structures have also been detected in other PPNe (e.g. Hen 3-1475; Riera et al. 2003) and in more evolved PNe (e.g. NGC 6881; Guerrero & Manchado 1998) and they have been interpreted as the result of episodic mass loss from a precessing central source (e.g. García-Segura & López 2000). From the HST optical images (taken from the HST Data Archive) we identify pairs of elongated structures with at least four different bipolar axes at P.A.$\sim$34${\hbox{$^{\circ}$}}$, $\sim$54${\hbox{$^{\circ}$}}$, $\sim$84${\hbox{$^{\circ}$}}$ and $\sim$124${\hbox{$^{\circ}$}}$. In Figure 6 we have displayed the contour map of the deconvolved mid-infrared images of I16594 obtained with TIMMI2 in the N1 and N11.9 filters overlaid on the optical HST-WFPC2 image taken in the F606W filter. Remarkably, we can see that the axis of symmetry defined by the mid-infrared emission nicely coincides with only one of the bipolar axes that can be identified in the optical images, in particular with that oriented at P.A.$\sim$84${\hbox{$^{\circ}$}}$. If this emission is a good tracer of the hot dust in the envelope and we accept that this hot dust must have been recently ejected from the central star we can interpret the observed spatial distribution in the mid-infrared as the result of the preferential collimation of the outflow material along this direction in the most recent past. Remarkably, the H$_2$ shocked emission detected with HST-NICMOS in the near-infrared is also found mainly distributed following the same bipolar axis (Hrivnak, Kelly & Su 2004) and nicely coincides with the mid-IR emission seen in our TIMMI2 images. This is shown in Figure 7, where the H$_2$ continuum-subtracted HST-NICMOS image is shown together with a contour map of the deconvolved mid-infrared image taken in the N11.9 filter. Note that the H$_2$ image (at 2.122 $\mu$m) showed in Figure 7 was continuum-subtracted using the HST-NICMOS image taken in the adjacent continuum at 2.15 $\mu$m (both images were also taken from the HST Data Archive). Interestingly, we found that the H$_2$ emission is mainly coming from the walls of the bipolar lobe oriented at P.A.$\sim$84${\hbox{$^{\circ}$}}$ identified in the HST optical images. In addition, four additional clumps of much weaker H$_2$ emission are detected at the end of each of the other two point-symmetric outflows associated to I16594 (Hrivnak, Kelly & Su 2004). The stronger emission detected along the walls of this bipolar lobe suggests that the interaction of the fast wind from the central star with the slowly moving AGB wind is currently taking place preferentially also along this axis of symmetry. This suggests that the formation of the multiple outflows observed in I16594 has not been simultaneous. The rest of bipolar outflows observed at other orientations in the optical images taken with HST must then be interpreted as the result of past episodic mass loss ejections. As such, they must contain much cooler dust grains which are then only detectable in the optical because of their scattering properties. IRAS 07027$-$7934 ----------------- ### IRAS 07027$-$7934 in the literature I07027 (=GLMP 170) is a very peculiar young PN. It has a central star that was classified by Menzies & Wolstencroft (1990) as of \[WC11\]-type. At present, there are only about half a dozen PNe with a central star classified as \[WC11\]. They all have stellar temperatures between $\sim$28,000 and 35,000 K (Leuenhagen & Hamann 1998) and are supposed to be in the earliest observable phase of its PN evolution, soon after the onset of the ionization in their circumstellar envelopes. I07027 is also among the brightest IRAS PNe and it has IRAS colors similar to other young PNe (Zijlstra 2001). The youth of I07027 as a PN is also evidenced by the detection of OH maser emission at 1612 MHz (Zijlstra et al. 1991), which is usually observed in their precursors, the OH/IR stars, but very rarely in PNe. The OH emission is single-peaked, which is interpreted as being detected only coming from the blue side of the shell, as the consequence of the ionized inner region being optically thick at 1612 MHz. This is supported by the shift in velocity with respect to the CO emission, which has also been detected toward this source, and from which an expansion velocity of 14.5 km s$^{-1}$ is derived (Zijlstra et al. 1991). The detection of strong PAH features and crystalline silicates in the ISO spectrum (Cohen et al. 2002; Peeters et al. 2002) indicates the simultaneous presence of oxygen and carbon-rich dust in the envelope. Remarkably, all other \[WC\] CSPNe observed with ISO show a mixed chemistry as well (Cohen et al. 2002) but I07027 is the only known \[WC\] star belonging to the rare group of PNe with OH maser emission, and therefore it links OH/IR stars with carbon-rich PNe. Zijlstra et al. (1991) published an H$\alpha$ image of I07027 taken with the ESO 3.5m NTT telescope. This image shows a stellar core with non-gaussian wings extending to a maximum diameter of about 15$\arcsec$, which may be mostly due to light scattered by neutral material and dust grains in the envelope. García-Hernández et al. (2002) detected H$_2$ fluorescence-excited emission from this source, in agreement with the round/elliptical H$\alpha$ morphology of the nebula and the temperature of the central star. I07027 had never been imaged in the mid-infrared before. Thus, our observations are the first attempt to reveal the spatial distribution of the warm dust in this peculiar object. ### The Marginally Extended Mid-infrared Core of IRAS 07027$-$7934 The deconvolved mid-infrared images of I07027 displayed in Figure 4 show a slightly extended emission at 8.6 and 11.5 $\mu$m. This mid-infrared emission is only marginally resolved (with a FWHM=0.3$\arcsec$ as compared to the typical PSF size of FWHM$\leq$0.2$\arcsec$ measured in the deconvolved standard stars) and is elongated along the north-south direction. Zijlstra et al. (1991) predicted for this source a radio flux density of 10 $mJy$ assuming E$_{B-V}$=1.1 and T$_{e}$=10$^4$ K. In addition, by using a plausible radio brightness temperature of 10$^3$ K they predicted an angular diameter of $\sim$0.3$\arcsec$ for the ionized region. This size for the ionized region is consistent with the measured size of the bright mid-infrared core seen in our deconvolved images of I07027. Unfortunately, there are no HST images of I07027 available in the optical but it has very recently been observed in the near-infrared through the broad F110W (J-band) and F160W (H-band) continuum filters with NICMOS under proposal 9861 (P.I.: Raghvendra Sahai). In Figure 8 we have displayed the still unpublished near-infrared HST images of I07027 (taken from the HST Data Archive) together with the contour levels of the deconvolved mid-infrared image taken by us in the N11.9 filter. In the F160W filter, I07027 shows a bright extended core (with FWHM=0.25$\arcsec$), which is slightly elongated along the north-south direction, in agreement with the mid-infrared structure seen in our deconvolved TIMMI2 images. This core is surrounded by a fainter elliptical nebulosity extended along the NW-SE direction with a total size of $\sim$1.6$\arcsec$ $\times$ 2.1$\arcsec$ at 1% of the peak intensity. The HST image in the F110W filter shows a slightly less extended emission of $\sim$1.1$\arcsec$ $\times$ 1.5$\arcsec$ (at 1% of the peak intensity) and shows a very similar morphology. In this case, a central point source is clearly detected which corresponds very probably to the central star, which is barely detected in the F160W image. Similarly to what we did for I16594 we have also estimated the distance to I07027. In this case we constructed the SED of I07027 by combining the available IRAS fluxes at 12, 25, 60 and 100 $\mu$m with the JHKL and BVRI photometry taken from García-Lario et al. (1997) and Zijlstra et al. (1991), respectively. The observed fluxes were also corrected for reddening adopting the extinction law from Cardelli, Clayton & Mathis (1989) and the value of E$_{B-V}$=1.1 derived by Zijlstra et al. (1991) through the measurement of nearby stars, with R$_{V}$=3.1. Then, a distance of 4.1 L$_4$$^{1/2}$ kpc is obtained. Note that I07027 is located at a much higher galactic latitude (b=$-$26) than I16594 (b=$-$3) and at such high galactic latitudes so much interestellar reddening is unexpected. Thus, we interpret that the observed reddening E$_{B-V}$=1.1 is mainly circumstellar in origin. On the other hand, most of the flux is emitted in the infrared where the effect of the interstellar/circumstellar extinction is mild. This is probably the reason why a very similar luminosity of 4.2 L$_4$$^{1/2}$ kpc was obtained by Surendiranath (2002), who derived this value by integrating the photometric fluxes from 0.36 $\mu$m to 100 $\mu$m, but without introducing any correction for extinction. Assuming a standard luminosity of 6,000 L$_{\odot}$ for I07027, a distance of 3.2 kpc is derived, in agreement with the distance of 3–5 kpc suggested by Zilstra et al. (1991). At this distance, the core size would correspond to $\sim$960 AU. ### Dust Temperature As for I16594, stellar light must be the dominant heating source for the circumstellar dust grains in I07027. This is confirmed by Zijlstra et al. (1991), who derived an Infrared Excess (IRE) of 93 for this source, indicating that dust heating by line emission can also be neglected in I07027. Again, we cannot interpret our mid-infrared observations of I07027 in terms of dust temperatures in the shell because the ISO spectrum of I07027 (Cohen et al. 2002) shows that the 8.6 and 11.5 $\mu$m filters are also heavily affected by strong PAH emission features. In particular, the PAH emission features around 8 $\mu$m (at $\sim$7.7 and 8.6 $\mu$m) are much stronger in this case than the feature located at 11.3 $\mu$m. Thus, the dust temperature values derived would be unrealistically high. Using IRAS data and assuming a dust emissivity index of 1, a mean dust temperature T$_{12/25}$ of 148 K is derived, while the TIMMI2 data gives a T$_{8.6/11.5}$ of $\sim$363 K. The strong differences in the derived temperatures confirm that the PAH emission is dominating the emission observed in the mid-infrared. In contrast to I16594, the ISO spectrum of I07027 shows much weaker emission features at 12.6 and 13.4 $\mu$m, which are the signatures of the CH out-of-plane bending vibrations for hydrogens in positions duo and trio, respectively (Pauzat, Talbit & Ellinger 1997) and indicate that the PAH population in I07027 is largely dehydrogenated. Then, for the modelling of the dust emitting at mid-IR wavelengths we made the same assumptions as in the case of I16594 (see Section 5.1.3) but this time we adopted a composition dominated by dehydrogenated amorphous carbon grains (type ACAR of Colangeli et al. 1995). Under these assumptions and taking into account the dust equilibrium radius to be consistent with the 0.3$\arcsec$ (or $\sim$960 AU at the assumed distance of 3.2 kpc) of the shell (which is the radius at $\sim$90% of the peak intensity) seen in our mid-infrared deconvolved images we obtain a dust temperature of 219 K for a mid-range dust grain size of 0.01 $\mu$m. For a larger dust grain size of 0.1 $\mu$m a smaller dust temperature of 138 K is derived while a smaller dust grain size of 0.001 $\mu$m yields a dust temperature of 347 K. Note that if the dust grains were located closer to the central star than the 0.3$\arcsec$ derived from our mid-infrared images, the dust temperatures above derived should then be considered as lower limits. ### Comparison with ISO data The validity of the range of possible dust temperatures derived from our TIMMI2 observations can be further explored by looking at the ISO spectrum originally published by Cohen et al. (2002). In a similar way as we did for I16594, the SED can be fitted by a two-component dust continuum with temperatures of T$_{BB1}$=430 K and T$_{BB2}$=110 K, respectively. The warm component in this case completely dominates in the wavelength range of our TIMMI2 observations (where also strong PAH features are found) while the cool component dominates at longer wavelengths, where crystalline silicate dust features are also detected on top of the continuum emission. By forcing the dust equilibrium radius to be consistent with the 0.3$\arcsec$ seen in our mid-IR deconvolved images, we need to assume in this case a very small grain size of $<$0.001 $\mu$m in order to reproduce the dust temperature of 430 K derived from the ISO spectrum. This suggests that the mid-infrared emission at $\sim$430 K must be the result of the combined contribution of small PAH molecules, located very close to the central star, and relatively hot dust continuum. In this case, the PAH population must be subject to a relatively strong UV field, consistent with the narrow features detected by ISO (in contrast to the broader features observed in I16594). Actually, the ISO spectrum of I07027 shows that the PAH emission features at 3.3 and 11.3 $\mu$m are weak compared with the emission features located at 6.2, 7.7, and 8.6 $\mu$m, which is also indicating a high degree of ionization in the population of PAHs (see Figure 2 in Allamandola, Hudgins & Sandford 1999). Note, however, that if the UV radiation field becomes too strong the PAH molecules can be destroyed, especially the small ones with a size $\sim$20$-$30 carbon atoms (see e.g. Allain, Leach & Sedlmayr 1996). This means that the C-rich dust seen in the ISO spectrum subject to the UV irradiation coming from the central star must be shielded from the stronger ISM UV radiation field by the outer layers of the circumstellar shell, where the OH maser emission is originated. For the cool dust emitting at 110 K, a different dust model was assumed, composed mainly of astronomical silicates. This choice takes into account the O-rich nature of the crystalline silicates detected in the ISO spectrum at wavelengths longer than 25 $\mu$m. The crystalline silicates are expected to be formed in the circumstellar dust shells of evolved stars at temperatures in the range 60$-$160 K (Molster et al. 2002b). The mean emissivity value adopted between 25 and 60 $\mu$m was taken from Figure 5 in Draine & Lee (1984). If we try to confine this cool O-rich dust to the observed extension of 0.3 $\arcsec$ we would need to adopt a very large dust grain size of $>$0.1 $\mu$m. Note that for this O-rich cool component we derive dust equilibrium radii of $\sim$0.02, $\sim$0.05 and 0.17 pc (or 1.1$\arcsec$, 3.4$\arcsec$ and 10.4$\arcsec$ at 3.2 kpc) for dust grain sizes of 0.1, 0.01 and 0.001 $\mu$m, respectively, which in all cases are inconsistent with our mid-infrared observations. These calculations indicate that independent of the dust grain size considered, the O-rich cool dust must be located much farther away from the central source than the C-rich warm dust emission (at 430 K) detected in the mid-infrared This different relative distribution of O-rich and C-rich dust would also be consistent with the detection of OH maser emission from the outer shell and suggests that the material expelled by the central star during the previous AGB phase was predominantly O-rich. ### Evolutionary Status of IRAS 07027$-$7934 At present, the evolutionary status of I07027 is not well understood. The hydrogen-deficiency of the central star together with the mixed dust chemistry (C-rich and O-rich) is a common finding among the limited sample of known \[WC\] PNe (De Marco & Soker 2002; Cohen et al. 2002). The most promising scenarios to explain the current observational properties of this rare class of PNe are: (i) the so-called ‘disk-storage’ scenario (Jura, Chen & Plavchan 2002; Yamamura et al. 2000); (ii) a final thermal pulse while the star was still in the AGB; or (iii) a late thermal pulse during the post-AGB evolution (Herwig et al. 1997, 1999; Herwig 2000, 2001; Blöcker 2001). The disk-storage scenario invokes the presence of a binary system in which the O-rich silicates are trapped in a disk formed by a past mass transfer event, with the C-rich particles being more widely distributed in the nebula as a result of recent ejections of C-rich material. This type of dusty disk structures have been detected in some PPN/PNe with binary \[WC\] central stars like CPD$-$56${\hbox{$^{\circ}$}}$8032 (De Marco, Barlow & Cohen 2002) or in the Red Rectangle (HD 44179) (Waters et al. 1998), but no firm evidence of the presence of any disk-like structure nor of the binarity of I07027 exists yet. Both a final thermal pulse in the AGB and a late thermal pulse during the post-AGB phase can eventually produce a sudden switch to a C-rich chemistry and a strong stellar wind, which is also characteristic of these \[WC\] CSPNe. However, because of the short lifetime of stars in the post-AGB phase, the latter is expected to be a rare phenomenon. Models predict that post-AGB stars which experience a late thermal pulse evolve back into the AGB (the so-called “*born-again*” scenario, e.g., Herwig 2001; Blöcker 2001). As a result of this, they show a fast spectroscopic evolution in the H-R diagram as well as peculiar spectroscopic features (e.g., Asplund et al. 1999; Lechner & Kimeswenger 2004; Hajduk et al. 2005) which are not observed in I07027, nor in any other known \[WC\] CSPNe. A final thermal pulse in the AGB phase seems to be a more plausible explanation since it does not require the assumption of exotic scenarios. As we have discussed in Section 5.2.4, the emission detected in our mid-infrared images can be mainly attributed to ionized PAHs plus thermal emission from relatively warm dust ($\sim$430 K) located very close to the central source. The OH maser emission detected by Zijlstra et al. (1991) supports the idea that the envelope of I07027 was until very recently O-rich. It is very difficult to explain how a low-mass disk around a binary system, which could act as an oxygen-rich reservoir, may be able to sustain such a luminous maser emission. Attending to geometry considerations, Zijlstra et al. (1991) suggests that the star must have changed its chemistry within the last 500 yrs. I07027 may have experienced a final thermal pulse in the AGB which has produced the recent switch to a C-rich chemistry. All C-rich material would then be warm as a consequence of its very recent formation and, thus, located very close to the central source (as it is actually observed) while the cooler O-rich material ejected during the previous AGB phase is then found now only farther away from the central source. In contrast, the typical disk sources with dual chemistry which are known to be binary systems show a completely different relative distribution of O-rich and C-rich dust. Waters et al. (1998) found that the PAH emission at 11.3 $\mu$m has a clumpy nature and comes from the extended nebula around HD 44179, while the O-rich material is located in a circumbinary disk. More recently, the bipolar post-AGB star IRAS 16279$-$4757 has been studied in the mid-infrared by Matsuura et al. (2004). They found that the PAH emission is enhanced at the outflow, while the continuum emission is located towards the center. Thus, they suggest the presence of a dense O-rich torus around an inner, low density C-rich region and a C-rich bipolar outflow resembling the morphology attributed to HD 44179. The observational characteristics of I07027 indicate a totally different formation mechanism, which are only consistent with a very recent change of chemistry from O-rich to C-rich. Conclusions =========== We have presented diffraction limited mid-infrared images of the PPN I16594 and the \[WC\] PN I07027 at 8.6, 11.5 and 11.7 $\mu$m taken under exceptionally good seeing conditions ($\leq$0.5). By applying the Lucy-Richardson deconvolution algorithm, we have resolved, for the first time, the subarcsecond dust shell structures around both objects. I16594 displays two emission peaks in the innermost region of the circumstellar dust shell at the three wavelengths observed. This two-peaked mid-infrared morphology is interpreted as an equatorial density enhancement revealing the presence of a dusty toroidal structure with a 0.4$\arcsec$ radius size (or $\sim$640 AU corresponding to a dynamical age of $\sim$190 yr at the assumed distance of 1.6 kpc). The observed size is used to derive the dust temperature at the inner radius of the shell. This result has been combined with the information derived from the ISO observations of I16594 to conclude that the mid-infrared emission detected in our TIMMI2 images must be dominated by PAH molecules or clusters which must be mainly distributed along the torus, as suggested by the similar size and morphology observed in all filters. We have also found that the axis of symmetry observed in the mid-infrared is well aligned with only one of the bipolar outflows (at P.A.$\sim$84${\hbox{$^{\circ}$}}$) seen as optical reflection nebulae in the optical HST images. We suggest that the multiple outflow formation has not been coeval and that, at present, the outflow material is being ejected in this direction. Consistently, the H$_2$ shocked-emission seen in the HST NICMOS image is mainly distributed along the same bipolar axis where the fast post-AGB wind is interacting with the slow moving material ejected during the previous AGB phase. The presence of several other bipolar outflows at a variety of position angles may be the result of past episodic mass loss events. I07027 exhibits a slightly asymmetric mid-IR emission core which is only marginally extended along the north-south direction with FWHM=0.3$\arcsec$ at 8.6 and 11.5 $\mu$m. This is the same orientation observed in recent HST images of the source taken in the near-infrared. The mid-infrared emission is attributed to a combination of emission from highly ionized, small PAH molecules plus relatively warm dust continuum located very close to the central star. The characteristics of the PAH emission observed in the ISO spectrum are also consistent with this interpretation. Taking into account the spatial distribution of the C-rich material deduced from our observations and because the OH maser emission from I07027 is expected to be located in the external and cooler regions, we propose that the dual chemistry observed in I07027 must be interpreted as the consequence of a recent thermal pulse (probably at the end of the previous AGB phase) which has switched the chemistry of the central star from the original O-rich composition to a C-rich one within the last 500 yrs. 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A. 2001, Ap&SS, 275, 79 [ccccccccc]{} IRAS 16594$-$4656&2001 Oct 9&N1&8.6(1.67)&120.4&$\sim$1.7 x 1.6&0.57&14&5\ $\dots$&$\dots$&N11.9&11.5(1.89)&107.5&$\sim$1.8 $\times$ 1.6&0.73&40&13\ $\dots$&$\dots$&SiC&11.7(3.21)&161.3&$\sim$1.8 $\times$ 1.6&0.75&35&11\ IRAS 07027$-$7934&2001 Oct 10 &N1&8.6(1.67)&120.4&0.65&0.57&19&23\ $\dots$&$\dots$&N11.9&11.5(1.89)&107.5&0.83&0.73&19&17\ [cccccccccc]{} 07134$+$1005 & R & 1& F5 Iab & C & PPN & S+B&y&y&1\ 16594$-$4656 & R & 2& B7 & C & PPN & S+M&y&y&2\ 17436$+$5003 & R & 3& F3 Ib & O & PPN & S+B&n&n&3\ 19114$+$0002 & R$^{*}$ & 4& G5 Ia & O & PPN/SG & S+M&n&n&3\ 21282$+$5050 & R & 1& O9& C & Young PN & S+B&y&n&4\ 22223$+$4327 & U & 4& G0 Ia & C & PPN & S+M &y&y&5\ 22272$+$5435 & R & 5& G5 & C & PPN & S+M&y&y&6\ [^1]: Based on observations collected at the European Southern Observatory (La Silla, Chile), on observations made with ISO, an ESA project with instruments funded by ESA Member States (especially the PI countries: France, Germany, the Netherlands and the United Kingdom) with the participation of ISAS and NASA, and on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555 [^2]: The Image Reduction and Analysis Facility software package (IRAF) is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation